Numerical Modelling in Damage Mechanics_edited by Kh. Saanouni _ All937572640Limit

Numerical Modelling in

DamageMechanics

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I N N O V A T I V E T E C H N O L O G Y S E R I E S

Numerical Modelling in

DamageMechanics

edited byKhemais Saanouni

London and Sterling, VA

First published in 2001 by Hermes Science Publications. ParisFirst published in Great Britain and the United States in 2003 by Kogan Page Science, animprint of Kogan Page LimitedDerived from Revue europeenne des elements finis, Numerical Modelling in DamageMechanics, NUMEDAMW. Vol. 10, no. 2-3-4.

Apart from any fair dealing for the purposes of research or private study, or criticism orreview, as permitted under the Copyright, Designs and Patents Act 1988, this publication mayonly be reproduced, stored or transmitted, in any form or by any means, with the priorpermission in writing of the publishers, or in the case of reprographic reproduction inaccordance with the terms and licences issued by the CLA. Enquiries concerning reproductionoutside these terms should be sent to the publishers at the undermentioned addresses:

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The right of Khemais Saanouni to be identified as the editor of this work has been asserted byhim in accordance with the Copyright, Designs and Patents Act 1988.

ISBN 1 9039 9619 8

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Contents

ForewordKhemais Saanouni vii

1. Some Aspects of a Gradient Damage FormulationTina Liebe, Paul Steinmann and Ahmed Benallal 1

2. On the Numerical Modelling of Ductile Damage with an ImplicitGradient-enhanced FormulationMarc G.D. Geers, Roy A.B. Engelen and Rene J.M. Ubachs 19

3. Adaptive Analysis based on Error Estimation for NonlocalDamage ModelsAntonio Rodriguez-Ferran, Ivan Arbos and Antonio Huerta 39

4. Mathematical and Numerical Aspects of an Elasticity-basedLocal Approach to FractureR.H.J. Peerlings, W.A.M. Brekelmans, M.G.D. Geers and R. de Borst 55

5. Numerical Aspects of Nonlocal Damage AnalysesClaudia Comi and Umberto Perego 75

6. Computational Issues and Applications for 3D Anisotropic DamageModelling: Coupling Effects of Damage and Frictional SlidingDamien Halm, Andre Dragon and Pierre Badel 93

7. Energy Dissipation Regarding Transient Response ofConcrete Structures: Constitutive Equations Coupling Damageand FrictionFrederic Ragueneau, Jacky Mazars and Christian La Borderie 111

8. Numerical Analysis of Failure in Sheet Metal Forming withExperimental ValidationMichel Brunet, Fabrice Morestin and Helene Walter 127

vi Numerical Modelling in Damage Mechanics

9. Damage in Sheet Metal Forming: Prediction of NeckingPhenomenonNathalie Boudeau, Arnaud Lejeune and Jean-Claude Gelin 147

10. Anisotropic Damage Applied to Numerical Ductile RupturePatrick Croix, Franck Lauro, Jerome Oudin and Jens Christlein 165

11. Numerical Aspects of Finite Elastoplasticity with IsotropicDuctile Damage for Metal FormingKhemais Saanouni, Abdelhakim Cherouat and Youssef Hammi 183

12. 3D Nonlocal Simulation of Ductile Crack Growth: A NumericalRealizationHerbert Baaser and Dietmar Gross 209

13. On the Theory and Computation of Anisotropic Damage atLarge StrainsAndreas Menzel and Paul Steinmann 225

14. On the Numerical Implementation of a Finite Strain AnisotropicDamage Model based upon the Logarithmic RateOtto Timme Bruhns and Christian Ndzi Bongmba 241

15. Ductile Rupture of Aluminium Sheet MaterialsJacques Besson, Wolfgang Brocks, Olivier Chabanet and Dirk Steglich 259

16. On Identification of Small Defects by Vibration TestsYitshak M. Ram and George Z. Voyiadjis 275

17. Multi-scale Non-linear FE2 Analysis of Composite Structures:Damage and Fiber Size EffectsFrederic Feyel and Jean-Louis Chaboche 291

Index 317

Foreword

This publication contains seventeen selected papers derived from the thirty four paperspresented during the Euromech Colloquium "Numerical Modelling in DamageMechanics" held in Troyes at the University of Technology of Troyes, October 2000,with Professors JL Chaboche, K Saanoumi and P Steinmann as co-chairmen.

Damage mechanics has now reached a high degree of maturity and is currently used formany different applications connected with numerical simulation techniques. Manyattempts have been made to build efficient numerical tools for damage initiation andgrowth simulation in mechanical structures under both small and large deformationhypotheses. The objectives of the colloquium published here were to gather recentadvances in numerical and computational aspects of damage mechanics, and also tostimulate current research and future challenges in this field.

We invite the reader to make his own explorations. Simply, we hope that the reader willfind this publication of much intertest and a stimulus to further research.

Finally, we should like to thank all the contributors to the Euromech 417 colloquium, onwhose proceedings this publication is based.

Khemais SaanouniGSM/LASMIS

Universite de Technologie de Troyes


Chapter 1

Some Aspects of a Gradient DamageFormulation

Tina Liebe and Paul SteinmannChair for Applied Mechanics, Department of Mechanical Engineering, University ofKaiserslautern, Germany

Ahmed BenallalLaboratoire de Mecanique et Technologie, LMT-Cachan, Cachan, France


Aspects of a Gradient Damage Formulation 3

1. Introduction

Softening at the continuum level due to damage accumulation mimics deteriora-tion processes within the material at the micro scale. As a consequence of softening,damage tends to accumulate within narrow bands, so called localized zones. In ex-periments these localization zones display a finite width which is related to the microstructure of the material. Upon further loading localized zones then most often form aprecursor to the final rupture of the material. In a standard continuum description andin particular in the corresponding numerical solution schemes no finite width is ob-tained, instead pathologically mesh dependent solutions are observed upon refinementof the discretization.

Among the most effective remedies against the unphysical behavior displayed bya softening standard continuum and its numerical computation nonstandard contin-uum theories have been proposed which incorporate higher gradients of those quan-tities which are responsible for softening. Physically motivated gradient models incrystal plasticity were proposed, e.g. by Steinmann [STE 96] and Menzel & Stein-mann [MEN 00]. Gradient dependent models, whereby the gradient dependence isessentially incorporated in the loading surface by the Laplacian of an internal variable,were treated by e.g. Comi [COM 96], de Borst, Benallal & Heeres [BOR 96a], Benal-lal & Tvergaard [BEN 95]. The well-posed initial boundary value problem for acontinuum model was studied by Benallal, Billardon & Geymonat [BEN 93].

A variety of numerical strategies, different from the one proposed in this work,were investigated e.g. by Sluys, de Borst & Muhlhaus [SLU 93], Parnin [PAM 94],de Borst & Pamin [BOR 96b], Peerlings et. al [PEE 96], Steinmann [STE 99], Comi[COM 99].

In this contribution the essential ingredient of gradient damage is an additionalequation represented by the damage condition containing the quasi-nonlocal energyrelease rate. A noteworthy feature from the numerical point of view is thus the treat-ment of the damage field as an independent variable.

2. A gradient damage formulation

As a simple phenomenological measure of micro defect interactions we mightconsider the gradient of the damage field d = Vzd, which we include in the freeHelmholtz energy \I> = \&(d, c,d) of the standard local damage model. Moreover,the model is based on a dissipation potential and the postulate of maximum dis-sipation. Therefore healing processes are excluded and a thermodynamically con-sistent approach is envisioned. Thereby, due to the extension of the classical lo-cal theory with the damage gradient contribution, the local dissipation inequalityT> = Yd + Y • d + P > 0 for the whole body B incorporates the nonlocality residualP, which, according to the arguments by Polizzotto & Borino [POL 98] satisfies theinsulation condition fBd Pd V = 0 for the actively damaged part of the whole body

4 Numerical Modelling in Damage Mechanics

Table 1. Key Ingredients of Gradient Damage

Bd C B. Thereby, the assumption of a bilinear form for the dissipation power T> = Yddetermines the quasi-nonlocal energy release rate Y = Y (e, d, d) as conjugated to theevolution of the independent arbitrary damage variable field in Bd C B. Moreover,applying the insulation condition, integration by parts and invoking Gauss theoremon the nonlocality residual yields a constitutive boundary condition (homogeneousNeumann b.c.) on dBd

xt C dB for the vector field Y — Y (d} which is thermody-namically conjugated to the gradient of the damage variable d and which we tend todenote as the damage flux. In addition to that, it results also in the so-called continuityboundary condition d = 0, which is imposed on dBfnt with dBd = dBd

xt U dBfnt.Thus, compatibility between the evolution of the damage variable and its gradient isautomatically assured. The quasi-nonlocal energy release rate essentially contains thedivergence of the damage flux divY in addition to the local energy release rate Y. Fi-nally it can be stated that the damage condition and the Kuhn-Tucker conditions retainthe same structure as for the local case. Therefore, we end up with a coupled problemfor the two primary unknown fields x and d which have to satisfy a partial differentialequation and an inequality constraint simultaneously, as will be shown in the sequel.The key ingredients of our gradient damage formulation are summarized in Table 1.

(1) Free Energy, (2) Dissipation Inequality, (3) Insulation Condition of Nonlocal-ity Residual, (4) Macroscopic Stress, (5) Energy Release Rate, (6) Damage Flux,(7) Quasi-Nonlocal Energy Release Rate, (8) Constitutive Boundary Conditions,(9) Continuity Boundary Conditions, (10) Damage Condition, (11) Kuhn-TuckerConditions


3. Strong form of the coupled problem

To set the stage for the following developments we first summarize the pertinentset of equations for the solution of the coupled boundary value problem in strong form.

Let B denote the configuration occupied by a solid body. Then the displacementfield u = u(x) and the damage field d = d(x) are parameterized in terms of theplacements x € B. These two primary fields are determined by the simultaneoussolution of a partial differential equation and a set of Kuhn-Tucker-complementaryconditions. The boundary dB to B with outward normal n is subdivided into disjointparts whereby either Neumann or Dirichlet boundary conditions for the two solutionfields u(x) and d(x) are prescribed. The residua of the resulting coupled problem instrong form are displayed in Table 2.

Table 2. Strong Form of the Coupled Problem

4. Weak form of the coupled problem

As a prerequisite for a finite element discretization the coupled nonlinear bound-ary value problem has to be reformulated in weak form. Therefore, the equationsin strong form are tested by the corresponding virtual quantities to render the virtualwork expression, see Table 3.

Note that the decomposition of the solution domain B into an active damagedand an inactive elastic domain B = Bd U Be and 0 = Be n Bd is indeed a quiteimplicit definition at this stage since one has to test for all possible combinations of

(1) Balance of Linear Momentum, (2) Kuhn-Tucker Conditions, (3) AdditionalCompleteness and Non-Overlapping Requirements, (4) Elastic Solution Domain,(5) Damaged Solution Domain


Table 3. Weak Form of the Coupled Problem

supports with all admissible test functions. Furthermore, it is remarkable that theabove decomposition corresponds to the pointwise complementary condition d(p = 0.

5. Discretized form of the coupled problem

The above set of equations has to be discretized in time and space, whereby weapply without loss of generality the implicit Euler backward method and resort to thestandard Bubnov-Galerkin finite element method. Then the temporal integration of theprimary variables u and d renders a discretized temporal update for the values un+1and dn+1. Furthermore, based on the iso-parametric concept, the displacement fielduh\Bf - Y^kNxuk € HI(&) together with its variation (W11Be = £fc A^foi* eHi(B) is elementwise expanded in terms of the nodal values Uk and 6uk by the sameshape functions as the geometry xh\sf = ^k N%Xk. Moreover, the damage fielddh\Be = ^kNddk e HI(&) together with its variation 6dh\Be = ^kN^6dk €Hi (B) is elementwise expanded by independent shape functions in terms of the nodalvalues dk and Sdk- Likewise, the test function 5<p\Be = Xlfc^d^fc € I/2(#) isdiscretized by the same shape functions as for the damage field in terms of nodal

(1) Weak Form of Balance of Linear Momentum, (2) Weak Form of Kuhn-TuckConditions, (3) Additional Completeness and Non-Overlapping Requirements, (Elastic Solution Domain, (5) Damaged Solution Domain


values 6ipk. The corresponding discrete algorithmic equations of the coupled problemare given in Table 4.

Table 4. Discretized Form of the Coupled Problem

Note that now the discrete algorithmic decomposition of the node point set is in-deed a complete explicit definition since one only has to check separately all nodepoints K G B. Furthermore, it is remarkable that the above discrete algorithmic de-composition corresponds to the discrete algorithmic complementary condition&Rd

KR% = 0 Vtf in B.

The initially unknown decomposition of the discretization node point set into ac-tive and inactive subsets B = B£+I U B^+1 at time step tn+i is determined iterativelyby an active set search. Thereby, the strategy is borrowed from convex nonlinear pro-gramming as is frequently used e.g. in multi-surface and crystal plasticity.

(1) Discrete Algorithmic Balance of Linear Momentum, (2) Discrete Algorith-mic Kuhn-Tucker Conditions, (3) Additional Completeness and Non-OverlappingConditions, (4) Discrete Algorithmic Elastic Solution Domain, (5) Discrete Algo-rithmic Damaged Solution Domain


An efficient algorithm for the solution of the coupled problem stated in the abovesections is offered by a monolithic iterative strategy. Here, the discrete algorithmicbalance of linear momentum together with the discrete algorithmic Kuhn-Tucker con-ditions are solved simultaneously within a typical Newton-Raphson scheme.

6. Constitutive update

Typically, a strain-driven constitutive update algorithm has to provide the updateddependent variables, like stress, damage flux, etc at time tw+1. Moreover, its consistentlinearization is essential in order to set up the appropriate global iteration matrix forthe quadratically converging global Newton-Raphson strategy.

The constitutive update of the simplest geometrically linear damage prototypemodel for given en+i, dn+\ is summarized in Table 5. Note that despite its implicitcharacter the constitutive update does not rely on local iterations usually employed instandard update algorithms.

Table 5. Update Algorithm for Gradient Damage

Note that the damage variable d is a given input for the update of the internalvariable K. Thereby, for convenience of exposition we use here a simple exponential-type evolution law for the damage evolution, which allows a closed form update forthe internal variable K. Otherwise, an additional local iteration for K = 0"1 (d) wouldbecome necessary but does not limit the generality of the formulation proposed here. Itis remarkable that the linearization of the constitutive update, i.e. the tangent operatorresults in a symmetric global iteration matrix.

Note that the update algorithm in the local case varies significantly. Here, only thestrains en+1 are given and in a first step the local energy release rate Yn+1 is computed.Based on this the history variable kn+1 is determined from the maximum of the new

(1) Nominal Stress, (2) Effective Stress, (3) Damage Flux, (4) Local Energy Re-lease Rate, (5) Internal Variable Update


local energy release rate Yn+1, the old value Kn or the initial damage threshold KQ,respectively. Finally, the updated damage variable dn+1 is computed from the newhistory variable KN+1. Thus in contrast to the gradient update algorithm the damagevariable d is a dependent variable in the local case.

7. Examples

In the above sections the theory as well as the numerics were outlined for a gradi-ent damage formulation. This is now applied to computational examples showing theperformance of the elaborated model by modifying the gradient parameter as well asdiscretization density in deterioration processes.

7.1. One-dimensional bar under uniaxial tension

Figure 1. 1-D-Model Problem: Bar under Uniaxial Tension

As a model problem we will examine in the sequel the bar in Fig. 1 under uniax-ial tension for the sake of demonstration. The problem statement, which includes aslight material imperfection in the middle of the bar, is taken from Feedings, de Borst,Brekelmans & de Vree [PEE 96], whereby homogeneous Neumann boundary condi-tions for the damage flux were prescribed at the boundary. The material is modeledbased on a linear elastic gradient damage formulation with a simple exponential-typeevolution law for the damage evolution. The material parameters for the followingexamples are summarized in Table 6.

The total bar is discretized with 80, 160, 320, 640, 1280 and 2560 elements.Thereby, due to the symmetry in the problem statement only one half of the bar is con-sidered. The load is applied using arclength control enabling one to trace the post-peakbranch of the load-deflection curves. The main objective is to show the performanceof the gradient model. Therefore as a comparison, the local model is also addressed.For different possibilities of discretization techniques for the local and quasi-localcase we refer to Liebe and Steinmann [LIE 01]. Likewise, a two-field finite elementformulation for elasticity coupled to local damage was proposed by Florez-Lopez et.al [FLO 94]. In this work, we focus on the classical approach in local damage with


Table 6. Material Parameters

linear element expansions for the displacement. Hereby, the local damage variablefield is not separately discretized. The element type for gradient damage reflects acontinuous expansion in both the displacement as well as the damage variable field.Hereby, it appears that the choice of linear expansions in both discretized fields yieldsthe most effective and efficient results. This can be explained by considering the dis-cretized Kuhn-Tucker conditions, which seem to be mainly affected by the choice ofdiscretization order. Using quadratic expansions for the displacements renders piece-wise linear strains and would result in a quadratic expansion of the elastically storedenergy Y. This quantity would then be coupled with a highly nonlinear history vari-able expression AC and a piecewise constant damage gradient, which causes oscilla-tions in both the damage variable distribution as well as in the load-deflection curves.Therefore, we use linear-linear approximations (P°1P°1) for the following examplesin gradient damage, which give stable results. The different element formulations aredescribed in Table 7.

Damage For-mulation

localgradient

DiscretizationVariableuu, d

Continuity of Ap-proximation

C°CQ/C°

Element Type

P° 1 ExpansionP°1P°1 Expansion

Table 7. Classification of Element Formulations

Firstly, as a reference for the gradient model we investigate the local damage case.Here, in order to trigger localization we additionally introduced a graded imperfectionin the middle of the bar. Hereby, the first element has the lowest elastic modulus andthe neighboring elements a slightly increased elastic modulus Eg = 9500.0-/V/mm2

compared to the rest of the bar elements with the highest elastic modulus. The re-

(1) Elastic Modulus, (2) Reduced Elastic Modulus, (3) Initial Damage Threshold,(4) Exponential Hardening Modulus


suiting load-deflection curves for the classical local P°l element type are displayedin Fig. 2. The typical deficiency in terms of a quasi-lack of convergence in the post-peak branch of the curves can be observed upon mesh refinement. This is even moreemphasized in Fig. 3 depicting the corresponding distribution of the damage variable,whereby a concentration of damage evolution is accumulated in only one element.

Figure 2. Load versus Deflection (P°l)

Secondly, to overcome the lack of discretization invariance the following examplesare based on the incorporation of the gradient regularization in the constitutive modelas described in the previous sections. First we show the quasi-mesh independence for aconstant gradient parameter c = 100.0 upon mesh densification, see Fig. 4 and Fig. 5.Clearly, also for different gradient parameters the solution converges upon mesh den-sification. Thereby, higher values of the gradient parameter render a somewhat moreductile post-peak behavior, see Fig. 6 and Fig. 7. In any case, the corresponding dis-tribution of the localized zone is convergent.

Note that the influence of modifying the gradient parameter results in a variationof ductility in the load-deflection curves, see Fig. 6 as well as in the damage variabledistribution, see Fig. 7. Hereby, the regularizing effect of the incorporation of gradi-ents into the damage model is obvious as the jumps in the damage variable distributionin the local model are smoothed out in the gradient one. Nevertheless, the overall so-lution shows a shrinkage of the localized band width upon further loading into a crackline mode, i.e. a gradual transition from a damaged zone into a line crack.


Figure 3. Damage Variable Distribution (P°1)

Figure 4. Load versus Deflection for c = 100.0 (P°1P°1)


Figure 5. Damage Variable Distribution for c = 100.0 (P°1P°1)

Figure 6. Load versus Deflection for constant mesh discretization (640 elements)and varied c = 0.0,0.1,1.0,10.0,100.0


Figure 7. Damage Variable Distribution for constant mesh discretization (640elements) and varied c = 0.0,0.1,1.0,10.0,100.0

7.2. Two-dimensional panel under uniaxial tension

Finally, in order to show the performance of the damage gradient formulation in 2dwe investigate the panel in Fig. 8 under tension. Again we have included a slight ma-terial imperfection in the center elements. The material is modeled by analogy to theId example, see Table 6. The bar is discretized with 20x10 and 40x20 Q1Q1-elements(continuous approximation of both displacement field and damage field). Again, wefocus here on the damage variable distribution which emphasizes the convergent per-formance of the gradient damage formulation as displayed in Fig. 9 and Fig. 10.

Figure 8. 2-D-Model Problem: Panel with Center-Imperfection


Figure 9. Damage Distribution shortly before reaching d=I, c=100, coarse mesh

Figure 10. Damage Distribution shortly before reaching d-l, c=100, fine mesh

8. Conclusion

We have derived a theoretical formulation and the corresponding discretized algo-rithmic format of a gradient damage model. Based on a positive domain dissipationand the postulate of maximum dissipation we end up with algorithmic Kuhn-Tuckerconditions in dependence on the quasi-nonlocal energy release rate, which is conju-gated to the damage evolution. On the numerical side, due to this special structure, anactive set search becomes necessary for the monolithic iterative solution of the cou-pled problem within a typical Newton-Raphson strategy. Nevertheless only standardFE-data structures and corresponding FE-modules are involved. Moreover, we end upwith a symmetric iteration matrix avoiding the use of a secant stiffness matrix as typ-


ically adopted in nonlocal models. In addition to that, other gradient damage modelsusually result in a non-symmetric tangent operator, see e.g. Peerlings et al. [PEE 96].

Considering a model problem of an one-dimensional bar under uniaxial tensionwe firstly investigated the classical local element formulation with only continuouselement expansions for the displacement. Here, the local theory resulted in spuriousmesh dependence in particular for the damage variable distribution. This could onlybe remedied by using the gradient formulation with gradient parameters c > 0. Forverification we investigated the behavior for c — 1.0,10.0,100.0. Thereby, it could benoted that with increasing gradient parameter the solution becomes somewhat moreductile. In any case, mesh densification renders mesh objective results and convergentdistributions of the damage variable field in both Id as well as 2d computations. Itis remarkable that a gradual transition from a damaged zone into a line crack can beobserved in the load-deflection curves in contrast to standard gradient models.

Therefore, it was emphasized that based on the theory and numerics underlyingthe gradient model advocated here, the regularization effect in damage is consider-able. Moreover, the simultaneous solution of the discrete algorithmic Kuhn-Tuckerconditions in addition to the discretized algorithmic balance of linear momentum of-fers an elegant solution strategy in the numerical treatment of gradient damage. Inparticular it is notable that the additional discrete algorithmic loading and unloadingconditions complemented by an active set search are implemented on a nodal basis,which is in contrast to alternative gradient-enhanced formulations.

9. References

[BEN 93] BENALLAL, A., BILLARDON, R. AND GEYMONAT, G. «Bifurcation andlocalization in rate-independent materials: some general considereations». InCISM Lecture Notes N° 327, pages 1-44. Springer-Verlag, 1993.

[BEN 95] BENALLAL, A. AND TVERGAARD, V. «Nonlocal continuum effects onbifurcation in the plane strain tension-compression test». J. Mech. Phys. Solids,43:741-770,1995.

[BOR 96a] DE BORST, R., BENALLAL, A. AND HEERES, O. «A gradient-enhanceddamage approach to fracture». J. Phys. IV, 6:491-502,1996.

[BOR 96b] DE BORST, R. AND PAMIN, J. «Some novel developments in finite el-ement procedures for gradient-dependent plasticitiy and finite elements». Int. J.Num. Meth. Eng., 39:2477-2505,1996.

[COM 96] COMI, C. «A gradient damage model for dynamic localization problems».Rend. Sc. Istituto Lombardo, A 130:119-141,1996.

[COM 99] COMI, C. Computational modelling of gradient-enhanced damage inquasi-brittle materials». Mech. Cohes.-Frict. Mater., 4:17-36, 1999.


[FLO 94] FLOREZ-LOPEZ, J., BENALLAL, A., GEYMONAT, G. AND BILLARDON,R. «A two-field finite element formulation for elasticity coupled to damage».Com/?. Meth. Appl. Mech. Eng., 114:193-212,1994.

[LIE 01] LIEBE, T. AND STEINMANN, P. «Theory and numerics of a thermodynam-ically consistent framework for geometrically linear gradient plasticity». Int. J.Num. Meth. Eng., in press 2001.

[MEN 00] MENZEL, A. AND STEINMANN, P. «On the continuum formulation ofhigher gradient plasticity for single and polycrystals». J. Mech. Phys. Solids,48:1777-1796,2000.

[PAM 94] PAMIN, J. «Gradient-dependent plasticity in numerical simulation of lo-calization phenomena». Diss., Delft University of Technology, 1994.

[PEE 96] PEERLINGS, R., DE BORST, R., BREKELMANS, W. AND DE VREE, J.«Gradient enhanced damage for quasi-brittle materials». Int. J. Num. Meth. Eng.,39:3391-3403,1996.

[POL 98] POLIZZOTTO, C. AND BORING, G. «A thermodynamics-based formula-tion of gradient-dependent plasticity». Eur. J. Mech. A/Solids, 17:741-761,1998.

[SLU 93] SLUYS, L., DE BORST, R. AND MUEHLHAUS, H. «Wave propagation, lo-calization and dispersion in a gradient-dependent medium». Int. J. Solids Struct.,30:1153-1171,1993.

[STE 96] STEINMANN, P. «Views on multiplicative elastoplasticity and the contin-uum theory of dislocations». Int. J. Eng. Sci., 34:1717-1735,1996.

[STE 99] STEINMANN, P. «Formulation and computation of geometrically non-linear gradient damage». Int. J. Num. Meth. Eng., 46:757-779,1999.


Chapter 2

On the Numerical Modelling of DuctileDamage with an ImplicitGradient-enhanced Formulation

Marc G.D. Geers, Roy A.B. Engelen and Rene J.M. UbachsDepartment of Mechanical Engineering, Eindhoven University of Technology, TheNetherlands


Gradient-enhanced Ductile Damage 21

1. Introduction

The numerical modelling of ductile damage and fracture in engineering materialsand particularly in metals has gained considerable interest in recent years. Manyof the available solution strategies focus on the initiation process, which is of greatimportance in manufacturing processes where any damage is to be avoided. However,there is also an important class of practical problems, in which damage, failure andcrack propagation are essential steps in the production process, e.g., cutting, blanking,drilling, etc. Analysis of these processes is mainly based on phenomenological know-ledge. Lengthy trial and error procedures along with a variety of empirical guidelinesare used to develop and optimize the process. Nowadays, it is expected that numer-ical techniques may be a versatile alternative, provided that they are capable to givea reliable and adequate description of the ductile fracture process, i.e., initiation andpropagation of a crack in a ductile material.

It is known from the analysis of damage and fracture in the infinitesimal de-formation theory, that the adequate modelling of the evolution of damage requiresa higher-order continuum in which either a gradient or a nonlocal approach is used. Inthe context of ductile damage, explicit gradient-enhanced small deformation theorieshave been developed in the past ten years, see for instance [DB 92, PAM 94, SVE 97,RAM 98]. So far, only few extensions to a large deformation framework are avail-able, e.g., [MIK 99] where explicit gradients have been incorporated in the constitutivemodel. Integral nonlocal approaches applied to softening plasticity have been studiedin [NIL 98] on the basis of thermodynamical considerations. Steinmann [STE 99] in-vestigated a geometrically nonlinear gradient damage formulation, applicable to rub-berlike materials. In contrast to the explicit gradient enhancements used in the citedgradient plasticity models, this paper is based on the use of an implicit (and hence non-local) gradient formulation. The nonlocal character of this gradient formulation hasbeen proven recently by Peerlings et al. [PEE 99], whereas nonlocal damage modelshave been used with considerable success by several authors [BAZ 88, TVE 95]. Thesmall deformation solution of the implicit gradient version has now been elaboratedand tested by Engelen et al. [ENG 01]. Only basic features of this solution strategywill be reviewed here.

Several finite plasticity formulations nowadays exist [NAG 90, MIE 98b, MIE 98a,ALF 98]. Many models are based on a hypoelastic stress response, which is com-monly obtained by generalization of the corresponding infinitesimal framework. It isknown that elastic deformations need to be small in order to apply such models, sinceno stored energy function exists that ensures true elastic behaviour (i.e., reversibledeformations without energy dissipation). This restriction does not hold for a hypere-lastic stress response, where a stored energy function does exist, that depends on theinvariants of the right Cauchy-Green deformation tensor for isotropic materials. Onthe basis of this approach, hyperelasto-plasticity frameworks have been proposed andimplemented by Simo et al. [SIM 85, SIM 88a] and Simo [SIM 88b]. The volumetricand deviatoric response is fully decoupled on the level of the stored energy potential.This hyperelasto-J2-plasticity model is taken as the point of departure for the incor-


poration of a ductile damage evolution. A major difficulty in numerical descriptionsof ductile damage is the adequate incorporation of physical mechanisms. In metals,ductile damage starts if the number of dislocation barriers prevents further plastific-ation and leads to void initiation, growth and coalescence, the underlying damagemechanisms of the frequently applied Gurson model [GUR 77] . A lot of empiricaland micromechanical research has been performed in this area, which has to embed-ded in the damage initiation and evolution in a later stage. This paper subsequentlypresents the small and large deformation framework, the incorporation of a nonlocallydriven ductile damage variable in the field function, some computational aspects forthe large deformation model, as well as several numerical examples for different cases.A discussion on the use of nonlocal models in the presence of large deformations isalso made, where a material or a spatial framework do not lead to the same interpret-ation of the underlying 'material' length scale that is mostly used as a constant in thesmall deformation context. Note that the ductile damage parameter appears as an ad-ditional internal state variable, a concept that is widely used in internal state variableplasticity and damage [LEM 90, KRA 96].

2. Underlying elasto-plasticity formulations

2.1. Small deformation elasto-plasticity model

The elasto-plastic framework used within the infinitesimal deformation assump-tion is a standard isotropic von Mises elasto-plastic model. The constitutive relationfor the stress rate tensor versus the elastic strain rate tensor is typically given by

The plastic state is characterized with a yield function /

where aeq is the von Mises equivalent stress

and cry the yield stress. A linear or a nonlinear hardening rule may be considered, e.g.,

In here, sp is the effective plastic strain, h is the linear hardening modulus and /IQQ,ho, 8 > 0 are nonlinear hardening parameters. The effective plastic strain is definedby


The plastic strain rate tensor ep is extracted from an associate flow rule, given by

Such a multiplicative split implies the existence of an intermediate state whichis obtained if the current state £2 is relaxed to a (local) stress-free configuration Qp,

It can be easily shown that the equations [2], [3], [6] and [7] lead to the followingidentity

The set of equations is completed with the standard Kuhn-Tucker loading/unloadingrelations

and the consistency condition

The solution of this elasto-plastic problem follows standard rules, which can befound in many textbooks.

2.2. Large deformation hyperelasto-plasticity model

The large deformation formulation which is used in this paper is based on a rate-independent hyperelasto-plastic model presented by [SIM 88a, SIM 88b, SIM 98], inwhich a correction has to be made in order to comply with the assumed isochoricplastic flow. The model is presented in a format that tends towards the infinitesimalsolution of the previous section for small deformations. This large deformation modelpresents many features which are well-known in the classical infinitesimal theory ofplasticity, although it has not been obtained through ad hoc extensions of the smallstrain theory. It is based on the adequate implementation of finite deformation kin-ematics in the elastic part and through the application of general principles of asso-ciative plasticity in the plastic part. A hyperelastic stress-strain relation is used forthe elastic predictor. In this section, the essential algorithmic steps are highlighted, aswell as the modifications made with respect to Simo's original model. The numericalimplementation then follows identical lines as given in [SIM 88b].

Based on micromechanical considerations of crystallographic slip, a multiplicativedecomposition of the deformation gradient tensor F is performed according to


where only plastic deformations exist. Note that Fp and Fe are only defined up toan arbitrary rigid body motion of the intermediate state. The different configurationsand the well-known associated kinematic tensors (total, elastic, plastic right or leftCauchy-Green deformation tensors, stretch tensors and strain tensors) are shown inFigure 1. The following pull-back push-forward relations are then valid

Figure 1. The multiplicative split and associated deformation tensors

The elastic left Cauchy-Green deformation tensor Be is the push-forward (with F)of the inverse plastic right Cauchy-Green deformation tensor C~l. The (objective andcovariant) Lie-derivative of Be is the push-forward of the material time derivative ofC-1p, which also holds for their corresponding spatial and material deviatoric parts,respectively given by


Assuming that the plastic flow is isochoric means that the volume change ratio Jdepends on elastic deformations only

The yield function is the classical von Mises-Huber function, formulated in termsof the Kirchhoff stress tensor

The effective plastic strain sp is now defined with respect to the elastic left Cauchy-Green deformation tensor

The kinematic constraint which follows from this assumption can be rephrased as

which is different from the facilitating assumption made by [SIM 88a], where

was forwarded. Unfortunately, satisfaction of [19] does not lead to an isochoric plasticflow. Equation [18] should have been used instead.

The Kirchhoff stress tensor i is computed from the elastic deformations, using anisotropic hyperelastic stress-strain response.

which closely resembles equation [2]. The von Mises equivalent stress teq is definedas in the infinitesimal case by

where K and G equal the bulk and shear modulus respectively. The stored energyfunction W that corresponds to this hyperelastic relation is given by

The earlier proposed linear and a nonlinear hardening rules now read


where a is a coefficient that will be determined later in order to equate the large andsmall deformation models in the case of infinitesimal displacements. Simo [SIM 88a]derived an associative flow rule for this hyperelasto-plastic formulation, through theapplication of the principle of maximum plastic dissipation. The following deviatoricflow rule was obtained

Consequently, the flow rule [27] in this infinitesimal limit case reads

by means of which the following simplifications can be made if the limit towards thinfinitesimal framework is taken

The plastic incompressibility condition for infinitesimal displacements becomes

The relation between the plastic multiplier y and the effective plastic strain ep maybe extracted by combining equations [22], [23], [26] and [27]

The formulation of the model is again completed with the standard Kuhn-Tuckerloading-unloading conditions and the consistency condition

Hence, if the coefficient a equals 3, all equations coincide with the infinitesimalelasto-plasticity framework.

The algorithm is completed with an elastic predictor - plastic corrector scheme,see [SIM 88b, SIM 98] for a similar example. The algorithm can be implemented in afinite element framework without difficulty, including the desired consistent tangentoperator.


3. Incorporating ductile damage

The addition of ductile damage during plastic flow is based on a progressive re-duction of the yield stress once the failure process initiates. Within an isotropicframework, this can be achieved with one single damage variable, in a similar wayto the stiffness reduction in damage mechanics. Similar arguments as the one usedby Kachanov can be used to motivate such an approach, since the initiation of mi-crocracks (intergranular) and voids can be observed in this stage of the deformation.Based on the knowledge which has been acquired in the field of the computationalmodelling of material instabilities, it is now known that a continuum solution can onlybe obtained if the principle of local action is abandoned. A higher-order or a nonlocalconstitutive theory [DB 92, PAM 94, SVE 97, RAM 98, GEE 00] has to be used, inorder to obtain a set of well-posed partial differential equations. In this paper, an im-plicit gradient enrichment is used, which combines the computational efficiency ofgradient type theories with the integral nonlocal concept. It has been shown that suchan implicit approach presents a true nonlocal character [PEE 99], in the sense thatan equivalent integral format exists, in which the nonlocal variable is a long-rangeweighted average of field of local variables. The solution strategy for the infinitesimalcase will be highlighted briefly, after which the extension towards the hyperelasto-plastic formulation is scrutinized. Computational details and algorithmic aspects forthe infinitesimal elasto-plasticity framework can be found in [ENG 01]. Note that theinherent relation between this ductile damage variable and its associated thermody-namic kinematic quantity is not further considered here, where the nonlocal characterof the damage requires special attention, see [NIL 98, GB 99, GAN 99].

3.1. Small deformations

Assuming a fully isotropic material behaviour, a ductile damage parameter 0 ô)p < 1 is introduced, which leads to a gradual reduction of the yield stress in thesoftening stage. Void nucleation, void growth and coalescence, growing out to cracksare the underlying physical mechanisms. The yield function given in [2] is now trans-formed to the following damage-sensitive yield function

Evidently, damage will affect the stress tensor, but unlike damage mechanics theductile damage does not enter the hyperelastic constitutive relation and deformationsare thus not reversible. The ductile damage w>p is computed from a history variable K,which is the ultimate value of the nonlocal variable ifs in the deformation history ofthe considered material point. The 'implicit gradient' approach refers to the equationthat is used to extract the nonlocal field variables ijr from the field of their local coun-terparts ty. This is done through the solution of a partial differential equation of theHelmholtz type


along with the Neumann boundary condition

V\jr-n = 0 on the boundary F with outward normal n [35]

The parameter l is a length parameter (often called the intrinsic length) that isrelated to the size of the influence zone of the nonlocal averaging function. De-termining ^ from such a gradient formulation is equivalent to an integral nonlocalformat, see [BAZ 88], where the nonlocal variable is computed as a weighted averageof the local field. Note that the Laplacian used in the nonlocality equation [34] canbe defined with respect to the undeformed or deformed state. In the case of infinites-imal displacements however, this difference is not relevant. The relation between theductile damage and the nonlocal field variable is governed by a damage evolution lawU)P(K) which quantifies the damage growth in terms of the field of kinematic variables.Phenomenological examples of such laws are

In the case of infinitesimal displacements, the constitutive equations for elasto-plasticity enriched with this ductile damage approach have been implemented in acomputational strategy and solved within a finite element framework [ENG 01]. It isshown that such a formulation is well suited to solve softening and failure in ductilematerials up to complete failure.

3.2. Large deformations

3.2.1. Material and spatial nonlocality

Extending the damage evolution proposed in the infinitesimal case to large deform-ations raises a fundamental problem with respect to the nature of the nonlocality. Thisis easily understood from the Helmholtz equation [34], which may take two possible

where the damage grows respectively linearly or exponentially (initially fast increase)or towards its ultimate value 1 at failure. The parameters /J, /c, and KC are materialparameters. These equations only influence the evolution of damage once it has initi-ated. The initiation of damage is controlled by the proper scalar function for the localfield variable \js. Its choice should be founded on micromechanical considerations andknown mechanisms in ductile damage initiation and evolution, which is still subject offuture research. In this contribution, the damage controlling field variable \fr is takenequal to the effective plastic strain measure ep, i.e.,


formats in the geometrically nonlinear case, i.e.,

where the material length parameter l0 is constant. Nonlocal spatial averaging (oneaverages over distances that are fixed in space and independent of the deformation ofthe material) is thus performed over a volume that is constant in space but variablein the material, while nonlocal material averaging (one averages over the initial un-deformed distances between the material particles or voids) is applied over a constantmaterial volume. In the case of tension (A. > 1, typically the case with tensile cracks)the material volume over which the spatially nonlocal kernel acts vanishes, while itbecomes extremely large in the case of compression (0 < X < 1).

The first equation corresponds to the Lagrangian (or material) averaging of thelocal field, while the second equation reflects the Eulerian (or spatial) averaging case.Steinmann [STE 99] already addressed this difference in his analysis for large de-formation quasi-brittle damage models. He showed that only the Lagrangian averagingsolution seemed to inherit the properties of the infinitesimal model.

Equation [40] can be pulled back towards the Lagrangian configuration, whichyields

Besides a symmetric term that depends inversely on the deformation, a secondnon-symmetric term appears. Clearly, the relative influence of both terms graduallychanges with the deformation, which may be particularly important in localizationzones. This is well illustrated with the one-dimensional counterpart of [41].

where X is the material coordinate, jc the spatial coordinate and A the stretch ratio. Inlocalization zones |^ becomes large, while the influence of A. varies from traction tocompression. The coefficients are small in tension and large in compression. In thecase of locally uniform stretch, the non-symmetric term locally equals zero and thenonlocal averaging equation can be written as

The material length parameter £Q = tfk now depends on the deformation, whichis in contrast to the one-dimensional Lagrangian equation derived from [39]


A spatial nonlocal formulation [40] with a constant spatial averaging volume leadsto a nonlocal variable that tends to the local variable with increasing tensile deforma-tions. It is also clear that spatial and material nonlocal formulations will behave quitedifferently in tension or compression. Note that equation [34] has already been usedwith a non-constant length scale within an infinitesimal damage-mechanics frame-work, see [GEE 98, GEE 00]. It was shown that a length scale that decreases withdeformation leads to a solution that tends towards the local ill-posed solution uponcomplete damage. In fact, local deformations in the neighbourhood of cracks willalways tend to large values, while they tend to zero in the surrounding unloading ma-terial. Although material nonlocality seems more relevant than spatial nonlocality,there is no obvious reason why the nonlocal kernel should be independent of the de-formation. The non-trivial answer to this question should ensue from micromechanicsor physics and certainly constitutes a challenge for future research.

3.2.2. Computational predictor-corrector algorithm

The solution of the damage-enhanced material behaviour is performed with anelastic predictor - plastic corrector scheme during a time increment from t to t +At. The predictor corresponds to a fully elastic increment Af, i.e., the plastic flowincrement is zero. The intermediate stress free configuration is thus preserved in thatstate.

and hence the trial Kirchhoff stress tensor as

Note that the ductile damage does not evolve in the predictor phase, since thehyperelastic relation is independent from wp.

The next step consists in the integration of the flow rule, which is performed witha pull-back and push-forward procedure in order to ensure incremental objectivity.The objective Eulerian flow rule is pulled-back to the invariant Lagrangian configura-tion, after which a time discretization is carried out using an implicit Euler backwardscheme. The discretized Lagrangian flow rule is then pushed forward to the spatial

The small star symbol is used to indicate that the considered quantity correspondswith the predictor state. The incremental deformation tensor FA — [Ft+^']-[F']~l,quantifies the deformation at time t + Af with respect to the configuration at time t.The predictor of the isochoric left Cauchy-Green deformation tensor is then obtainedthrough

Repeating the pull-back/integration/push-forward scheme to rate equation [28]gives for a = 3

The increment of the plastic flow Ay may now be determined from equation [49]by taking the square root of the double inner product of this equation with itself, andmaking use of the definition of the equivalent Kirchhoff stress req and the deviatoricpart of r in equation [20], which yields


description again. Following this procedure with respect to [27], followed by a mul-tiplication with [jt+At]~2/3 and substitution of the elastic predictor [46] permits oneto rephrase the discretized flow rule in the current state as

which may also be written in the following format

Using a linear hardening law[24] enriched with ductile damage, i.e., the form ry =(TyO + /iep)(l — o)p), with a constant hardening modulus h, and making use of the

trial value / = req — (Tyo + he'p)(l — a)'p) of the yield function permits to rewriteequation [51] and extract the plastic multiplier by means of [50]

For the nonlinear hardening rule [25], equation [51] has to be solved iteratively forAy with a local Newton scheme. Once Ay is determined, the equivalent von Misesstress can be extracted directly from [51]. Alternatively, it can be noticed from the

equations [49] and [51] that Nd = *Nd, which permits one to compute [-Bg]t+At andthus Tt+At directly from equation [48].

The spherical part of the flow rule is fully determined through the isochoric plasticflow assumption. Plastic incompressibility is enforced by computing the spherical partsuch that det(fle) = J2 holds. Using the principal invariants jf = 0, J%, J$ of Bd

e,which can be computed after the return mapping of the deviatoric part, it is easy toshow that the trace of Be is the solution of the following cubic polynomial equation


The solution tr (Be) of this equation, and the result of the return mapping schemein terms of Bd

e, permits the determination of the complete elastic left Cauchy-Greendeformation tensor. Note that the elastic deformation tensor Fe is not computedexplicitly in this algorithm, which means that the unknown rotation tensor, up towhich the intermediate configuration is defined, does not have to be quantified.

The solution strategy presented above can be implemented in a finite elementframework without great difficulty, since it is a natural combination of the small de-formation framework [ENG 01] and Simo's work [SIM 88a, SIM 88b]. In any case,i.e., material or spatial nonlocality, a consistent tangent operator can be determined.

4. Examples and comparisons

4.1. Small deformation analysis

Several examples for small deformation gradient-enhanced ductile damage werealready given in [ENG 01]. Only one example is therefore presented here. A two-dimensional plate is loaded in compression as indicated in Figure 2, where the materialparameters are also given. The problem has been investigated with several discretiz-

Figure 2. Compression of a plate with an initial imperfection

ations, where the expected mesh-objectivity of the result has been confirmed. Thedevelopment of the shear bands and the localization process in the softening branch isillustrated in Figure 3, where the effective plastic strain is depicted. It can be noticedthat the softening branch is modelled up to complete failure, where the deformation


Figure 3. Shear band development and localization in a softening plate

localizes in a zone that expands to certain width determined by l, after which thefurther localization takes place in a smaller band that narrows progressively.

4.2. Large deformation analysis

A first trivial case, that validates the assumptions made in the infinitesimal frame-work and its extensions to the geometrically nonlinear framework, is found by com-paring the plate compression example for both frameworks. The associated force dis-placement curves are depicted in Figure 4. It can be noticed that results almost overlap,which is essentially due to the fact that deformations remain small in this example. Ifapplications towards metals are envisaged, deformations inevitably become much lar-ger. To illustrate this, an axisymmetric tensile bar has been modelled, for which thecharacteristics are listed in Table 1 (geometry was taken from Simo [SIM 88b]). Noimperfection was used, since physical softening will be triggered automatically by thecross-section reduction in the necking area. The tensile bar fails due to geometricaland physical softening in the necking zone. The analysis was again made with boththe infinitesimal and the large deformation framework. Results are shown in Figure 5.Large differences now appear, soon after the initial yield point. Note that this stronglyinfluences the localization of deformation and the softening behaviour of the materialin the failure stage.


Figure 4. Plate compression solved with the infinitesimal and geometricallynonlinear approach

LengthRadiusBulk modulusShear modulusInitial yield stressResidual flow stressHardening modulusInitial Ki (wp = 0)Critical KC (wp =.;1)Softening slope parameterLength scale

LRKG

TyO

fyoohKi

KC

ftt

53.3 mm6.4mm164 GPaSOGPa450 MPa715 MPa129 MPa5%150%110mm

Table 1. Characteristics of the axisymmetric tensile bar


Figure 5. Axisymmetric tensile bar under failure

The same example was used to make a comparison between material and spatialaveraging. The force-displacement curve is given in Figure 6. In contrast to the obser-vations made by Steinmann [STE 99], differences are rather small in elasto-plasticity.This is mainly due to the fact that deformations are irreversible here. Furthermore, thepull-back analysis of the spatial averaging solution points out that differences can beexpected if deformations are very large, i.e., at the end of the crack initiation processwhich is at the very end of the failure stage of a tensile bar. No singularities are presentin this example and no crack propagation occurs, which means that these results maynot be generalized ad hoc. Future work will undoubtedly clarify this point.

5. Conclusions

A small and a large deformation elasto-plasticity framework, enhanced with anisotropic ductile damage variable, has been presented. The solution strategy has beenemphasized, where it has been shown that this formulation is particularly well suitedto model damage initiation and evolution in real engineering problems. In spite of thephenomenological character of the framework, it overcomes the well-known problemsin continuum modelling of damage which were not solved at this level yet for largedeformation elasto-plastic behaviour. Future research must address issues like damageinitiation and evolution in terms of the complex deformation history (including the


Figure 6. Axisymmetric tensile bar with material/spatial nonlocality

influence of hydrostatic pressure, micromechanical theories, e.g., Gurson, etc.), aswell the use of advanced remeshing techniques that allow the transition of smoothdamage zones into discrete cracks.

6. References

[ALF 98] ALFANO G., ROSATI L., VALOROSO N., "A displacement-like finite element modelfor J2 elastoplasticity: variational formulation and finite-step solution", Computer Methodsin Applied Mechanics and Engineering, vol. 155, 1998, pp. 325-358.

[BAZ 88] BA£ANT Z. P., PIJAUDIER-CABOT G., "Nonlocal continuum damage, localizationinstability and convergence", Journal of Applied Mechanics, vol. 55, 1988, pp. 287-293.

[DB 92] DE BORST R., MUHLHAUS H. B., "Gradient-dependent plasticity: Formulation andalgorithmic aspects", International Journal for Numerical Methods in Engineering, vol. 35,1992, pp. 521-539.

[ENG01] ENGELEN R. A. B., GEERS M. G. D., BAAIJENS F.P.T., "Nonlocal implicitgradient-enhanced softening plasticity", International Journal of Plasticity. Accepted.

[GAN 99] GANGHOFFER J.F., SLUYS L.J., DE BORST R., "A reappraisal of nonlocal mech-anics", European Journal of Mechanics - A/Solids, vol. 18 1-2, 1999, pp. 17—46.


[GB 99] G. BORING P. FUSCHI, POLIZZOTTO C., 'Thermodynamic approach to nonlocalplasticity related variational principles", Journal of Applied Mechanics, vol. 66 4, 1999, pp.952-963.

[GEE 98] GEERS M. G. D., DE BORST R., BREKELMANS W. A. M., PEERLINGS R. H. J.,"Strain-based transient-gradient damage model for failure analyses", Computer Methods inApplied Mechanics and Engineering, vol. 160 1-2, 1998, pp. 133-154.

[GEE 00] GEERS M. G. D., PEERLINGS R. H. J., BREKELMANS W. A. M., DE BORSTR., "A comparison of phenomenological nonlocal approaches based on implicit gradient-enhanced damage", Acta Mechanica. Accepted.

[GUR77] GURSON A.L., "Continuum theory of ductile rupture by void nucleation andgrowth: Part 1 - yield criterion and flow rules for porous ductile media", Transactionsof ASME Journal of Engineering Materials Technology, vol. 99 1, 1977, pp. 2-15.

[KRA 96] KRAUSZ A.S., KRAUSZ K., eds., Unified Constitutive laws of plastic deformation,San Diego, Academic Press, 1996.

[LEM 90] LEMAITRE J., CHABOCHE J.L., Mechanics of Solid Materials, Cambridge Univer-sity Press, 1990.

[MIE 98a] MlEHE C., "A constitutive frame for elastoplasticity at large strains based on thenotion of a plastic metric", International Journal of Solids and Structures, vol. 30, 1998,pp. 3859-3897.

[MIE 98b] MlEHE C., "A formulation of finite elastoplasticity based on dual co- and contra-variant eigenvector triads normalized with respect to a plastic metric", Computer Methodsin Applied Mechanics and Engineering, vol. 159, 1998, pp. 223-260.

[MIK 99] MiKKELSEN L. P., "Necking in rectangular tensile bars approximated by a 2-dgradient dependent plasticity model", European Journal of Mechanics A/Solids, vol. 18,1999, pp. 805-818.

[NAG 90] NAGHDI P. M., "A critical review of the state of finite plasticity", Journal of AppliedMathematics and Physics, vol. 41, 1990, pp. 315-387.

[NIL 98] NlLSSON C., "On nonlocal rate-independent plasticity", International Journal ofPlasticity, vol. 14 6, 1998, pp. 551-575.

[PAM 94] PAMIN J., Gradient-dependent plasticity in numerical simulation of localizationphenomena, Ph.D. thesis, Delft University of Technology, Delft, 1994.

[PEE 99] PEERLINGS R. H. J., GEERS M. G. D., DE BORST R., BREKELMANS W. A. M.,"A critical comparison of nonlocal and gradient-enhanced continua", International Journalof Solids and Structures. Submitted.

[RAM 98] RAMASWAMY S., AVARAS N., "Finite element implementation of gradient plas-ticity models. part ii: Gradient-dependent evolution equations", Computer Methods in Ap-plied Mechanics and Engineering, vol. 163, 1998, pp. 33-53.

[SIM 85] SlMO J. C., ORTIZ M., "A unified approach to finite deformation plasticity based onthe use of hyperelastic constitutive equations", Computer Methods in Applied Mechanicsand Engineering, vol. 49, 1985, pp. 221-245.


[SIM88a] SlMO J. C., "A framework for finite strain elastoplasticity based on maximumplastic dissipation and the multiplicative decomposition: Part i. continuum formulation",Computer Methods in Applied Mechanics and Engineering, vol. 66, 1988, pp. 199-219.

[SIM885] SlMO J. C., "A framework for finite strain elastoplasticity based on maximumplastic dissipation and the multiplicative decomposition: Part ii. computational aspects",Computer Methods in Applied Mechanics and Engineering, vol. 68, 1988, pp. 1-31.

[SIM 98] SlMO J. C., HUGHES T. J. R., Computational Inelasticity, Interdisciplinary AppliedMathematics, Springer-Verlag, 1998.

[STE 99] STEINMANN P., "Formulation and computation of geometrically non-linear gradientdamage", International Journal for Numerical Methods in Engineering, vol. 46, 1999, pp.757-779.

[SVE 97] SVEDBERG T., RUNESSON K., "A thermodynamically consistent theory of gradient-regularized plasticity coupled to damage", International Journal of Plasticity, vol. 13 6-7,1997, pp. 669-696.

[TVE 95] TVERGAARD V., NEEDLEMAN A., "Effects of nonlocal damage in porous plasticsolids", International Journal of Solids and Structures, vol. 32 8-9, 1995, pp. 1063-1077.

Chapter 3

Adaptive Analysis based on ErrorEstimation for Nonlocal DamageModels

Antonio Rodriguez-Ferran, Ivan Arbos and Antonio HuertaDepartament de Matematica Aplicada III, Universitat Politecnica de Catalunya,Barcelona, Spain


Error Estimation for Nonlocal Damage Models 41

1. Introduction

Damage models are nowadays a standard approach to model the failure of concreteand other quasi-brittle materials [LEM 90]. To avoid the pathological mesh depen-dence of numerical simulations carried out with local models, various regularisationtechniques may be used [BOR 93]. One possibility, considered in this work, is the useof nonlocal damage models [PIJ 87, BA2 88, MAZ 89]. The basic idea of nonlocalmodels is that the damage parameter that describes the loss of stiffness depends onthe strain state in a neighbourhood (associated to a characteristic length) of the pointunder consideration.

To ensure the quality of the finite element solution, an adaptive strategy based onerror estimation was recently proposed by the authors [ROD 00]. A salient featureof the approach proposed in that reference is the extension of an existing residual-type nonlinear error estimator [DIE 98, HUE 00] to the context of nonlocal damagemodels, where tangent stiffness matrices are not readily available.

Attention is focused here on the fact that the error estimator is based on localcomputations over elements and so-called patches. It will be shown that it is importantto account for the nonlocality of the damage model when solving these local problems.These leads to a slight modification of the nonlocal damage model to be used duringerror estimation.

An outline of paper follows. Nonlocal damage models are briefly reviewed in sec-tion 2. Section 3 is devoted to the error estimator. After reviewing its main ingredientsin section 3.1, section 3.2 discusses the required modification of the damage model sothat its nonlocality is taken into account during error estimation. The resulting adap-tive strategy is illustrated in section 4 by means of two numerical examples involvingthe single-edge notched beam test. The concluding remarks of section 5 close thepaper.

2. Nonlocal damage models

The basic features of nonlocal damage models are briefly reviewed in this section.For the sake of clarity, only isotropic elastic-damage models are considered. Thesesimple models are sufficient to illustrate how the error estimator based on local com-putations takes into account the nonlocality of the model. However, the approach canbe extended to more complex nonlocal damage models, incorporating, for instance,anisotropy and/or coupling with plasticity [MAZ 89].

The loss of stiffness associated with mechanical degradation of the material isrepresented by a parameter D, according to

where a and e are respectively the Cauchy stress tensor and the small strain tensor,and C is the tensor of elastic moduli (E: Young's modulus; v: Poisson's coefficient).


Parameter D ranges between 0 (virgin material, with elastic stiffness) and 1 (com-pletely damaged material, with no stiffness). For computational purposes, an upperbound of .Dmax = 0.99999 is set. In this manner, zero stiffness is avoided and it is notnecessary to remove the fully damaged elements from the mesh.

It is assumed that D depends on a state variable Y, which in turn depends on thestrains:

Damage starts above a threshold Y0 (that is, D = 0 for Y < YQ) and it cannotdecrease (that is, D > 0).

To define a particular model, it is necessary to specify the definition of the statevariable, equation [2], and the evolution law for damage, equation [5].

In the modified von Mises model [VRE 95] Y depends on the first strain invariant/i, the second deviatoric strain invariant J-2 and the ratio k of compressive strengthto tensile strength. Regarding the damage evolution for Y > YQ, an exponential law[PIJ 91, ASK 00] is used. The modified von Mises model is summarized in Table 1.More details about this model can be found in [VRE 95, PEE 98].

3. The error estimator

In order to control the finite element discretization errors, an adaptive strategy isemployed [ROD 00]. It is based on the combination of a residual-type error estimator[DIE 98, HUE 00] and /i-remeshing. The error distribution is computed with the errorestimator and translated into desired element sizes with a so-called optimality criterion[DIE 99]. An unstructured quadrilateral mesh generator [SAROO] is then used tobuild a mesh with the desired sizes. This iterative process stops (typically after 2 to 4iterations) when the relative error of the solution (i.e. energy norm of the error dividedby energy norm of the solution) is below a prescribed threshold set a priori.

The basic idea of nonlocal damage models is averaging the state variable Y in theneighbourhood of each point. In this manner, the nonlocal state variable Y is obtained:

The weight function a, which depends on the distance d to the point under consid-eration, is typically the Gaussian

where the characteristic length lc is a material parameter of the nonlocal damagemodel, which acts as a localization limiter and can be associated to the grain size[PIJ 91]. The nonlocal state variable drives the evolution of damage,


Table 1. Modified von Mises model

3.1. A residual-type error estimator based on local computations

The error estimator used in this work was first developed for linear problems[DIE 98] and later extended to nonlinear problems [HUE 00, DIE 00]. A detailedpresentation and analysis can be found in these references. Here, only a brief reviewis given.

Using a mesh of characteristic size H, the finite element method provides thediscrete nonlinear equilibrium equation

State variable:

Damage evolution:

The error in displacements is defined as the difference between the two solutions:

Note, however, that computing u^ is computationally much more expensive thancomputing uH, because it involves solving the nonlinear problem over the fine mesh,see equation [7]. For this reason, the basic idea of the error estimator is to approximateeu by low-cost local computations over subdomains. This is a standard strategy inresidual-type error estimators.

The proposed approach consists of two phases. First, a simple residual problemis solved inside each element of the coarse mesh (interior estimate). Note that ele-ments are the natural subdomains for the local computations. To avoid the expen-sive flux-splitting procedures of other residual-type estimators, homogeneous Dirich-let boundary conditions are prescribed for each element (that is, eu = 0 in the elementboundary).

where the unknown is the nodal displacement vector uH, fintH(uH) is the vector of

nodal internal forces associated with uH and fextH is the discretized external force

term.

To estimate the error in uH, a finer mesh of size h (h <g. H) is used as reference.On this finer mesh, the problem reads


Figure 1. Patch associated to a node of the coarse mesh subdivided into 4 x 4 elementsof the fine mesh

Of course, the error is not really zero in the element boundary. For this reason, asecond set of simple problems is solved. The idea is to use a different partition of thecomputational domain into subdomains. A natural choice is to associate these subdo-mains, called patches, to the nodes of the coarse mesh. Since four-noded quadrilateralelements are used, a patch consists of one-fourth of each element sharing the node,see Figure 1. By combining equations [7] and [8], the nonlinear problem to be solvedon every element and every patch can be recast as

These local problems are solved over the fine mesh of size h. Every element of thecoarse mesh H is subdivided into 4 x 4 elements of size h (i.e. the fine mesh is nestedinto the coarse mesh, with h = H/4). Due to the patch definition, every patch alsoconsists of 4 x 4 elements of size h, see Figure 1. For the iterative solution of equation[9], UH is taken as an initial approximation to uh (that is, the initial approximationfor the error is eu = 0). This means that the final state obtained with the coarsemesh of size H is taken as the initial state for solving the local problem over eachelement/patch with the fine mesh of size h.

As a consequence, information must be projected from the coarse mesh H to thefine mesh h. To ensure the consistency between the various projected fields, the fol-lowing projection strategy is employed: (1) Displacements and damage are projectedover the fine mesh. To project the nodal field of displacements, the finite element ap-proximation based on mesh H is used. The damage field must be projected from theGauss points of the coarse mesh to the Gauss points of the fine mesh. A very simpleand efficient strategy is used: the value of damage at each Gauss point of the coarsemesh is assigned to all the Gauss points of the four corresponding elements of thefine mesh, see Figure 2. With this projection strategy, the risk of unrealistic values ofdamage (i.e. D < 0 or D > 1) is precluded; (2) strains and stresses are not projected,but computed from the projected displacements and damage. In this manner, the con-sistency between all the "projected" quantities (displacements, damage, strains andstresses) is guaranteed. To keep the notation simple, these fields are still denoted withan H subscript denoting they are associated with the solution of the global problem overthe mesh of size H, even though they are now supported by the mesh of size h.


Figure 2. Projection strategy for the damage field. The value at each Gauss point ofthe element of size H is assigned to the four associated elements of size h

3.2. Accounting for the nonlocality of the model

At this point, it is important to remark that the proposed error estimator for nonlo-cal damage models is based on local computations over subdomains (i.e. elements andpatches). The nonlocality of the damage model must be accounted for when solvingthe local problems. Note that, upon mesh refinement, the element size may becomesmaller that the characteristic length lc. With the proposed approach, the interactionbetween adjacent elements is considered (thanks to the loop over patches, that overlapelements), but not the interaction between more distant elements. However, this is notregarded as a significant drawback of the suggested approach; due to the weightingfunction a of the nonlocal average, see equation [4], the error in one element has alimited influence on the error in distant elements. Moreover, accounting for the inter-action between distant elements during the error estimation would destroy the mostattractive feature of the suggested approach: it consists of solving independent localproblems. Note, for instance, that the error estimation algorithm has a computationalcost of O(N) (with N the number of elements) and can be parallelized.

On the other hand, it is essential that these local problems are solved taking intoaccount the current mechanical properties (i.e. the damaged stiffness) of each el-ement/patch. As discussed in the following, this implies that the nonlocal damagemodel must be slightly modified.

The standard and the modified nonlocal damage models are summarized in Table2. In the standard model —that is, the one used for solving the global problem, seeequation [6]—, the error in strains e£ is computed as the symmetrized gradient of theerror in displacements eu and added to the strains £H to produce the strains £h overthe element/patch. After that, the local state variable Yh is computed and averagedinto the nonlocal state variable Yh. Finally, damage Dh is obtained. Note that thenonlocal average that transforms Yh into Yh is over a local support (the element/patchunder consideration). This fact leads to non-physical responses, especially in zones


Table 2. Standard and modified nonlocal damage models

of large damage gradients. Assume, for instance, that the error in strains is small ande/i « £H- A small variation in Y is also expected. However, it may happen thatYh ^ YH=, because Yh. contains no information about nearby zones.

This point is illustrated in Figure 3, which depicts the local state variable, the non-local state variable and the damage parameter for a given time increment in a zone ofthe coarse mesh with large gradients. The circled element has a very small local statevariable YH, see Figure 3(a), below the threshold Y0. However, since the elements tothe right have large values of YH , it has a relatively large (above Y0) nonlocal statevariable YH, see Figure 3(b), which leads to damage, see Figure 3(c). If the standardmodel is used to solve the local problem on the circled element during error estima-tion, a small error in strains leads to a small variation in the local state variable which,after nonlocal averagingôver the element, results in a low value of the nonlocal statevariable (that is, Yh <& YH). As a consequence, damage cannot increase in the circledelement during error estimation. When estimating the error for the circled element,the nonlocal state variable YH, rather than the local state variable YH, is representativeof its mechanical properties.

For this reason, a modification of the nonlocal damage model is proposed here, seeTable 2. The difference resides in the way the nonlocal state variable Yh is computed.By means of a first-order Taylor expansion, the local state variable Yh is expressed as

Standard model

ee-V s(eu)

eh = £H + e£

Yh = Y(eh)

Yh-^Yh

(-

Dh = D(Yh)

Modified model

Error in strains ee

Strain £h

Local variable Yh

= Vs(eu)

= £H + ee

*YH+ ^(£H)e.

ey: Error in Y

Nonlocal variable ey — > e~ ; Yh = YH + e~

— >: nonlocal average over local support)

Damage Dh = D(Yh)


YH plus an error term ey. The derivative ^ is computed analytically by means ofthe chain rule

4. Numerical examples: the single-edge notched beam

The proposed adaptive strategy is illustrated here by means of the single-edgednotched beam (SENB) test [CAR 93]. The geometry, loads and supports are shown inFigure 4. A plane stress analysis is performed. The concrete beam is modelled withthe modified von Mises model with exponential damage evolution, see Table 1. The steelloading platens are assumed to be elastic. Two sets of material parameters are used,see Table 3. For material 1, there is a significant post-peak softening in the stress-strainlaw for concrete. For material 2, on the contrary, the softening is very slight, so theresidual strength almost coincides with the peak strength [PEE 98].

Table 3. The two sets of material parameters: (a) large softening; (b) very slightsoftening

In equation [10], EI denotes a component of the strain vector. All the derivativesin the RHS are very simple to compute from the definition of the local state variableY, see Table 1 and of the strain invariants I1 and J2.

The error term ey is averaged over the element/patch into e~. As a consequence,

Yh is computed as the addition of a reference value YH, which describes the realdamaged stiffness, and an error term e~. For doing so, it is necessary to project —bymeans of the same projection strategy used for the damage field, see Figure 2—YH

and ^(CH) into the fine mesh.

With this modified model, a small variation in strains does result in a small vari-ation in the nonlocal state variable (that is, Yh ~ YH). Going back to Figure 3, thismeans that the damage level of the circled element may either remain constant (forYh < YH) or increase (for Yh > YH) during error estimation.

To sum up: the standard model is not capable of capturing the spread of the dam-aged zone associated to error estimation.

ParameterEV

Y0

ABlc

Material 1Concrete Steel28000MPa 280000MPa0.1 0.21.5 x 10-4

0.8900010mm

Material 2Concrete Steel35000MPa 350000MPa0.2 0.26.0 x 10-5

0.08820010mm


Figure 3. Fields in a zone of large gradients: (a) local state variable Y; (b) nonlocalstate variable Y; (c) damage. The damage threshold is Y0 — 1.5 x 10-4

Figure 4. Single-edge notched beam: problem statement. All distances in mm


Figure 5. SENB test with material 1, initial approximation in the adaptive process. (a)Mesh 0: 659 elements and 719 nodes; (b) final damage distribution; (c) final deformedmesh (x 300); (d) error distribution. The global relative error is 3.96%

4.1. Test with material 1

The results with material 1 are shown in Figures 5 to 7. The initial mesh is shownin Figure 5(a). Note that this mesh is relatively coarse, with only one element in thenotch width. The final damage distribution and deformed mesh (amplified 300 times),corresponding to a CMSD (crack-mouth sliding displacement) of 0.08 mm, is depictedin Figure 5(b). The curved crack pattern observed in experiments [CAR 93] is clearlycaptured. The error estimation procedure discussed in section 3.2 is employed tocompute the error field of Figure 5(d). The error is larger in the damaged zone andnear the loading platens. The global relative error (i.e. energy norm of the error indisplacements over the energy norm of displacements) is 3.96%, above a threshold seta priori of 2%, so adaptivity is required.

The error field of Figure 5(d) is translated into the mesh of Figure 6(a). Note theelement concentration in the crack and the central supports. This finer mesh leadsto a better definition of the damaged zone, see Figure 6(b). The error estimator nowdetects that the largest errors are associated to the edges of the cracked zone, seeFigure 6(d). The global relative error of 2.11% is still slightly above the error goal, soanother adaptive iteration is performed. The outcome of this second iteration is shownin Figure 7. The qualitative results of iteration 1 are confirmed: (1) small elements areneeded to control the error in the damaged zones and close to the loading platens and(2) error is larger in the edges than in the centre of the crack. The global relative errorof 1.77% is below the threshold of 2%, so the adaptive iterative process stops.

The relation between damage and error is illustrated by Figure 8, which depictsprofiles of these two fields along the crack. Note that the two error peaks are associatedto the edges of the damaged zone (i.e. large damage gradients). This indicates that thedamage gradient is a good error indicator [HUE 99] for these models.


Figure 6. SENB test with material 1, after one iteration in the adaptive process. (a)Mesh 1: 1155 elements and 1228 nodes; (b) final damage distribution; (c) final de-formed mesh (x300); (d) error distribution. The global relative error is 2.11%

Figure 7. SENB test with material 1, after two iterations in the adaptive process.(a) Mesh 2: 1389 elements and 1469 nodes; (b) final damage distribution; (c) finaldeformed mesh (x300); (d) error distribution. The global relative error is 1.77%

4.2. Test with material 2

The SENB test is now reproduced with material 2, see Table 3. The small value ofparameter A leads to a stress-strain law with almost no softening. A very similar lawhas been employed to simulate the SENB test with gradient-enhanced damage models[PEE 98].

The results are shown in Figures 9 to 11. The initial mesh is the same as before,see Figure 9(a). The change in the material parameters lead to a completely differentfailure pattern, dominated by bending of opposite sign in the two halves of the beam,see Figures 9(b) and 9(c). A crack at the notch tip is also initiated, but it is only a


Figure 8. SENB test with material 1. Profiles of damage (dashed line) and error (solidline) across the crack. The two error peaks are associated to large damage gradients

Figure 9. SENB test with material 2, initial approximation in the adaptive process. (a)Mesh 0: 659 elements and 719 nodes; (b) final damage distribution; (c) final deformedmesh (x300); (d) error distribution. The global relative error is 3.66%

secondary mechanism. The error estimation procedure has no difficulties in reflectingthe change in the failure mode, see Figure 9(d). The global relative error is 3.66%, soadaptivity is required.

Figures 10 and 11 illustrate the adaptive process. Note that meshes 1 and 2 arequite different from the ones obtained with material 1. The global relative errors are2.46% and 2.13%. This value is still slightly above the threshold of 2%. However,an additional iteration is considered not necessary for the illustrative purposes of thistest.

A final comparison between the two sets of material parameters is offered by Figure12, where the total load is plotted versus the CMSD for meshes 0 and 2. The results


Figure 10. SENB test with material 2, after one iteration in the adaptive process, (a)Mesh 1: 776 elements and 848 nodes; (b) final damage distribution; (c) final deformedmesh (x300); (d) error distribution. The global relative error is 2.46%

Figure 11. SENB test with material 2, after two iterations in the adaptive process, (a)Mesh 2: 870 elements and 954 nodes; (b) final damage distribution; (c) final deformedmesh (x300); (d) error distribution. The global relative error is 2.13%

obtained with material 1 —a peak load of around 60 kN and post-peak structuralsoftening, see Figure 12(a)— are in good agreement with the experiments [CAR 93].With material 2, on the other hand, the peak load is quite higher and no softening isobserved, see Figure 12(b).

5. Concluding remarks

An adaptive strategy based on error estimation for nonlocal damage models hasbeen presented. The constitutive model has been slightly modified in order to accountfor its nonlocality during the error estimation procedure, see Table 2. The basic idea


Figure 12. Total load versus crack-mouth sliding displacement (CMSD) for meshes0 (solid line) and 2 (dashed line): (a) with material 1; (b) with material 2 (see Table3)

of the modification is that the error in the local state variable, rather than the variableitself, is averaged. By doing so, the error estimation takes into account the real me-chanical properties of the damaged material, while retaining its most attractive feature:it consists in solving simple, independent problems over elements and patches.

The resulting adaptive strategy has been illustrated by means of the single-edgenotched beam test. With two sets of material parameters leading to very different fail-ure modes, h-remeshing concentrates elements where needed according to the errorestimator, until the global relative error falls below an error threshold. By keeping thediscretization error under control, it is possible to ensure the quality of the FE solutionand assess the influence of the material parameters in an objective way.

Acknowledgements

The partial financial support of the Ministerio de Ciencia y Tecnologia (grant num-bers: TAP98-0421, 2FD97-1206) is gratefully acknowledged.

6. References

[ASK 00] ASKES H., SLUYS L., "Remeshing strategies for adaptive ALE analysis of strainlocalisation", European Journal of Mechanics A/Solids, vol. 19, 2000, p. 447-467.

[BAZ 88] BA2ANT Z., PIJAUDIER-CABOT G., "Nonlocal continuum damage localization in-stability and convergence", Journal of Applied Mechanics, vol. 55, 1988, p. 287-293.


[BOR93] DE BORST R., SLUYS L., MCLHAUS H.-B., PAMIN J., "Fundamental issues infinite element analysis of localization of deformation", Engineering Computations, vol. 10,1993, p. 99-121.

[CAR 93] CARPINTERI A., VALENTE S., FERRARA G., MELCHIORRI G., "Is mode II frac-ture energy a real material property?", Computers and Structures, vol. 48, 1993, p. 397-413.

[DIE 98] DfEZ P., EGOZCUE J., HUERTA A., "A posteriori error estimation for standard finiteelement analysis", Computer Methods in Applied Mechanics and Engineering, vol. 163,1998, p. 141-157.

[DIE 99] DfEZ P., HUERTA A., "A unified approach to remeshing strategies for finite element/i-adaptivity", Computer Methods in Applied Mechanics and Engineering, vol. 176, 1999,p. 215-229.

[DIE 00] DfEZ P., ARROYO M., HUERTA A., "Adaptivity based on error estimation for vis-coplastic softening materials", Mechanics of Cohesive-Frictional Materials, vol. 5, 2000,p. 87-112.

[HUE 99] HUERTA A., RODRIGUEZ-FERRAN A., DfEZ P., SARRATE J., "Adaptive finite el-ement strategies based on error assessment", International Journal for Numerical Methodsin Engineering, vol. 46, 1999, p. 1803-1818.

[HUE 00] HUERTA A., DfEZ P., "Error estimation including pollution assessment for nonlin-ear finite element analysis", Computer Methods and Applied Mechanics in Engineering,vol. 181, 2000, p. 21-41.

[LEM 90] LEMAITRE J., CHABOCHE J.-L., Mechanics of solid materials, Cambridge Uni-versity Press, Cambridge, 1990.

[MAZ89] MAZARS J., PIJAUDIER-CABOT G., "Continuum damage theory: application toconcrete", Journal of Engineering Mechanics, vol. 115, 1989, p. 345-365.

[PEE 98] PEERLINGS R., DE BORST R., BREKELMANS W., GEERS M., "Gradient-enhanceddamage modelling of concrete fracture", Mechanics of Cohesive-Frictional Materials,vol. 3, 1998, p. 323-342.

[PIJ 87] PIJAUDIER-CABOT G., ZANT Z. B., "Nonlocal damage theory", Journal of Engi-neering Mechanics, vol. 118, 1987, p. 1512-1533.

[PIJ 91] PIJAUDIER-CABOT G., MAZARS J., "Steel-concrete bond analysis with nonlocalcontinuous damage", Journal of Structural Engineering, vol. 117, 1991, p. 862-882.

[ROD 00] RODRfGUEZ-FERRAN A., HUERTA A., "Error estimation and adaptivity for non-local damage models", International Journal of Solids and Structures, vol. 37, 2000,p. 7501-7528.

[SAR 00] SARRATE J., HUERTA A., "Efficient unstructured quadrilateral mesh generation",International Journal for Numerical Methods in Engineering, vol. 49, 2000, p. 1327-1350.

[VRE95] DE VREE J., BREKELMANS W., VAN GILS M., "Comparison of nonlocal ap-proaches in continuum damage mechanics", Computers and Structures, vol. 55, 1995,p. 581-588.

Chapter 4

Mathematical and Numerical Aspects ofan Elasticity-based Local Approach toFracture

R.H.J. Peerlings, W.A.M. Brekelmans and M.G.D. GeersDepartment of Mechanical Engineering, Eindhoven University of Technology, TheNetherlands

R. de BorstDepartment of Aerospace Engineering, Delft University of Technology, TheNetherlands


An Elasticity-based Local Approach to Fracture 57

1. Introduction

Component failure due to the formation and growth of cracks is traditionally mod-elled using fracture mechanics. Fracture mechanics theory uses global criteria to deter-mine under which conditions a pre-existing crack will grow and thus lead to completefracture of the component. Additional criteria predict the rate and direction of crackgrowth. In situations where the surrounding material is relatively unaffected by thepresence of a crack, this type of modelling is highly successful. In many engineer-ing materials, however, the concentration of deformation and stress near the cracktip produces irreversible changes in the microstructure of the material and in the mi-crostructural processes which govern its behaviour. Examples are plastic flow and/orvoid formation in ductile materials, microcracking in concrete and fibre pull-out ordelamination in fibre-reinforced polymers. In return, these changes may have a con-siderable influence on - or indeed govern - the crack growth process.

In situations where interactions between crack and microstructural damage playan important role, fracture mechanics may not be the most suitable modelling tool.A more natural treatment of these problems is provided by the so-called local ap-proach to fracture, in which the change of material behaviour is modelled explicitly[LEM 86, CHA 88]. The development and growth of a crack is regarded as the ulti-mate consequence of the local degradation process. A crack is represented by a regionin which the material integrity has been completely lost and which therefore cannotsustain any stress. The internal boundary which describes the crack contour expandswhen material in front of the crack tip fails. As a result, no separate fracture criteria areneeded: the rate and direction of crack growth follow from the constitutive behaviour.As an additional advantage, crack initiation and crack growth can be described withinthe same framework, so that it is not necessary to define an - often arbitrary - initialcrack.

In local approaches to fracture, the degradation of material properties is often mod-elled using continuum damage mechanics [KAC 58, RAB 69, CHA 88, LEM 96]. Intheir standard form, this and other types of degradation modelling (e.g., softeningplasticity) are usually not suited to describe crack initiation and crack growth, be-cause they cannot properly describe the accompanying localised deformations. As aconsequence, the deformation and damage growth are often observed to localise in asurface (i.e., in a vanishing volume) right at - or even before - the onset of fracture.In numerical analyses, this localisation results in an extreme sensitivity to the spatialdiscretisation of the problem. Upon refinement of the discretisation, the solution con-verges to one in which the fracture process is instantaneous and does not dissipate anyenergy. It is emphasised that this pathological behaviour is not due to the numericaltreatment, but to a shortcoming of the underlying continuum modelling. It can beremoved by introducing nonlocality in the constitutive modelling, either by integralterms or by higher-order gradients [BAZ 84, PC 87, FRE 96, PEE 96]. The enrichedcontinuum formulations which are thus obtained preclude the localisation of deforma-tion in a vanishing volume and the resulting instability. As a result, crack growth ratepredictions remain finite and a positive amount of energy is dissipated.


Finite element analyses using nonlocal continuum models are mesh-insensitiveonce the discretisation is sufficiently fine to accurately capture the solution. However,these methods require some modifications of the standard algorithms. The nonlocalityintroduces either an additional integration step or an extra set of equations. Specialattention must be paid to the treatment of the additional terms near boundaries. This isparticularly true in fracture problems, where boundary conditions must also be appliedat the internal boundary between the crack and the remaining material. Furthermore,frequent remeshing and an adaptive step size selection may be needed to accuratelydescribe the crack path and special solution control techniques may be required to dealwith instabilities and bifurcations [PC 91, GEE 99, PEE 99, PEE 00].

This paper summarises the algorithmic ingredients which are essential for reliableand accurate fracture analyses using the continuum approach. An elasticity-baseddamage framework is used here (Section 2), but most issues are equally relevant inother degradation models (e.g., softening plasticity). Emphasis is on preventing patho-logical localisation and mesh sensitivity. Since these phenomena find their origin inthe mathematical (continuum) modelling, it is useful to first study the difficulties aris-ing at this level; this is done in Section 3. Section 4 shows how these difficultiescan be avoided by using an enriched formulation. Some aspects of the finite elementimplementation are discussed in Section 5 and results are given in Section 6.

2. Constitutive modelling and the local approach to fracture

Continuum damage mechanics uses a set of continuous damage variables to repre-sent microstructural defects (microcracks, microvoids) in a material. If it is assumedthat the development of damage does not introduce anisotropy, a single, scalar damagevariable can be used to describe the local damage state. This damage variable D isdefined such that 0 < D < 1, where D = 0 represents the initial, undamaged materialand D = 1 represents a state of complete loss of material strength. If the influenceof damage is added to standard linear elasticity, the classical stress-strain relation ofelasticity-based damage mechanics is obtained [LEM 90]:

where cr,7 denotes the Cauchy stress components, Cijkl the standard elasticity ten-sor and Skl the linear strains. An important application of relation [1] is to quasi-brittle fracture (e.g., concrete, fibre-reinforced polymers). Here, however, we will useit to model high-cycle fatigue, in which plastic deformations also remain negligible[PAA 93, PEE 99, PEE 00].

In the fatigue model, the growth of damage is related to the local deformation ofthe material. For this purpose a damage loading function is introduced in terms of thestrain components:


with s a positive, scalar equivalent measure of the actual strain state and K a thresholdvariable, which is taken constant here: K = K0. For the equivalent strain the von Misesstrain, scaled such that it equals the axial strain in uniaxial tension, will be used:

The damage variable remains constant when / < 0; the behaviour is then linearelastic. Notice that positive values of K0 therefore imply the existence of a fatiguethreshold. Strain states for which / > 0 lead to damage growth only for continueddeformation and as long as the critical value D = 1 has not been reached, i.e., if f > 0and D < 1. When these three conditions are satisfied, the damage rate is governed byan evolution law which reads in its most general form

where

where the dependence of the damage growth rate on the equivalent strain rate has beentaken to be linear in order to avoid rate effects.

It is immediately clear from equation [1] that no stresses can be transferred forD = 1. In the local approach to fracture this critical state is used to represent acrack by a region of completely damaged material (fic in Figure 1). In the remainingpart of the domain, and particularly next to the crack, some (noncritical) damage mayhave been developed (fid in the figure), while other areas may still be unaffected bydamage (fi0). In the latter region the material has retained its virgin stiffness. Underthe influence of further straining the damage variable will increase in those parts ofthe body where the conditions for damage growth are met. This will often be thecase particularly near the strain concentration in front of the crack tip. When thecritical value D = 1 is reached in this region, the completely damaged zone fic willstart to expand, thus simulating crack growth. The direction and rate of crack growthare governed by the damage growth locally near the crack tip, hence the term localapproach to fracture.

For reasons of computational efficiency, the stress-strain behaviour is sometimespartly uncoupled from the growth of damage. The material in the damaged zonefid then retains its virgin stiffness until the damage variable becomes critical. Uponreaching D = 1, the elastic stiffness is then suddenly decreased from its virgin valueto zero. This method is sometimes referred to as uncoupled or semi-coupled approach,while the full model, in which the stresses are governed by relation [1], may be calledcoupled or fully coupled [PAA 93, LEM 96]. It is obvious that the uncoupled approachcan only be followed if the influence of damage prior to failure is relatively small,which may be true in the high-cycle fatigue case considered here. However, this alsocancels part of the advantage of the local approach, namely that it can account for theinfluence of material damage on crack growth.

Figure 1. Damage distribution in a continuum

Both in the coupled and uncoupled approach it is important to realise that the local,complete loss of strength in £2C implies that stresses are zero for any arbitrary deforma-tion (see equation [1]). As a consequence, the equilibrium equations are meaninglessin this region. This can be seen for the elasticity-based damage model by substitutingrelations [1] in the standard equilibrium equations


Making use of the right minor symmetry of the elasticity tensor (i.e., Cijkl = Cijlk)the equilibrium equations can then be written as the system of second-order partialdifferential equations

For a given damage field D(x) < 1, the displacement components Uk can be de-termined from this differential system and the corresponding kinematic and dynamicboundary conditions. In a crack however, where D = 1, both terms in [7] vanish.Consequently, the partial differential equations degenerate and the boundary valueproblem becomes ill-posed. This indefiniteness must be avoided by limiting the equi-librium problem to the subdomain £2 = £0 Ud where D < 1. At the boundarybetween crack and remaining material the natural boundary condition nicij = 0 mustbe applied, with the vector n normal to the boundary. A free boundary problem is thusobtained, in which the position of the internal boundary (the crack front and crackfaces) follows from the growth of damage, see also [BUI 80].

As the damage variable grows, the second term in this expression may becomeof the same order as the first term, so that condition [8] may indeed be met at somepoint. The vector n is then the normal to a characteristic surface segment of the set ofrate equilibrium equations. Since solutions of linear partial differential equations withsmooth coefficients can have discontinuities or discontinuous derivatives only acrosscharacteristic surfaces, this opens the possibility of jumps in the velocity solution. Adiscontinuity of the velocity across a characteristic surface results in a strain rate sin-gularity on this surface, which in turn renders the damage rate singular (see equation[5]). This means that for continued loading the damage variable immediately becomescritical on the characteristic segment and thus that instantaneous failure of this surfaceis predicted. In order to follow the stress drop resulting from this instantaneous lossof stiffness, material adjacent to the characteristic segment must unload elastically, sothat the growth of damage indeed localises in the surface segment.

Loss of ellipticity plays an important role in localisation of deformation and dam-age in static fracture, where damage growth is fast right at the onset [PEE 96]. Infatigue, however, the initial growth of damage is usually slow and the ellipticity of therate equilibrium problem is therefore preserved until near the end of the fatigue life.

3. Localisation and mesh sensitivity

Analyses using the local approach as described in the previous section will usuallyquickly result in a situation where all further growth of damage is concentrated in asurface. Since this means that the volume of material that participates in the damagedevelopment vanishes, no work is needed for the crack to propagate, even if the spe-cific work needed by the damage process is positive. Furthermore, the crack traversesthe remaining cross-section instantaneously, instead of by a small increment in everyloading cycle. From a mathematical point of view, two phenomena play a role in thispathological localisation of damage growth: loss of ellipticity of the rate equilibriumequations and singularity of the damage rate. The latter cause is often not recognisedin the literature, but is actually the most important in crack growth analyses, since it isresponsible for the instantaneous and perfectly brittle crack growth. Loss of elliptic-ity, on the other hand, may result in premature initiation of cracks, before the damagevariable has reached the critical level D = 1 in a stable way.

The rate equilibrium equations lose ellipticity when the acoustic tensor n iC i jk ln l,where Cijkl denotes the tangent stiffness, becomes singular for some unit vector n,i.e., when

For the fully coupled fatigue damage model the tangential stiffness Cijkl is givenby

A nElas t i c i t y -based Loca l Approach t o Frac tu re 61

where X, u denote Lame's constants. Expression [10] is always positive and the prob-lem therefore remains elliptic until D = 1. As a result, the damage growth remainsstable and affects a finite volume throughout the crack initiation phase in fatigue.

Once a crack has been initiated, however, the continued growth of damage nev-ertheless localises in a surface and the predicted crack growth becomes nonphysicalagain. This localisation during crack growth is not due to loss of ellipticity, but is re-lated to the strain singularity which is inevitably present at the tip of the crack. Oncethe damage variable becomes critical in a certain point and a crack is thus initiated, thedisplacement and velocity must become discontinuous across this crack. This impliesthat the strain (rate) field at the crack tip becomes singular. Since the damage growthrate is directly related to the equivalent strain, the strain singularity at the tip resultsin an infinite damage rate. For continued deformation all stiffness is therefore lost in-stantaneously at the most critical point in front of the crack tip and the crack thus startsto propagate. Since the material adjacent to the crack must unload elastically in orderto follow the resulting stress drop, the width of the crack remains zero. This impliesthat the strain and damage growth rate at the crack tip remain singular as the crackgrows and consequently that the crack grows at an infinite rate. No work is needed inthis fracture process, since it involves damage growth in a vanishing volume.

It is emphasised once more that this mechanism of damage localisation and instan-taneous crack growth is activated even if the rate equilibrium equations remain ellipticuntil D = 1. Loss of ellipticity may cause premature initiation of a crack and thusresult in perfectly brittle crack growth, but the pathological propagation behaviour isnevertheless due to the singularity of the damage rate at the crack tip. This is true evenin models of static fracture, but the problem is aggravated in this case by the fact thata crack is initiated shortly after the onset of damage as a result of loss of ellipticity.

Finite element solutions try to follow the nonphysical behaviour of the continuummodel as described above, but are limited in doing so by their finite spatial resolution.In standard finite element methods the displacement field must be continuous. Thedisplacement jumps and singular strains of the actual solution can therefore only beapproximated by high, but finite displacement gradients in the finite element solution.As a consequence, a finite volume is involved in the damage process, and a positiveamount of energy is dissipated. Also, because the damage growth rate at the tip of thedamage band remains finite, the crack propagates at a finite velocity. When the spatialdiscretisation grid is refined, however, the finite element approximation becomes moreaccurate in the sense that the displacement gradients which describe the discontinuitiesbecome stronger. Consequently, the predicted fracture energy becomes smaller and thecrack propagates faster. In the limit of vanishingly small elements, the actual solution


Indeed, it can be easily seen that for the uncoupled approach loss of ellipticity cannotoccur before the damage variable becomes critical. Since the tangential stiffness ten-sor Cijkl equals the elasticity tensor Cijkl in this case, the characteristic determinantequals


is retrieved, i.e., a vanishing fracture energy and an infinite crack growth rate. Thisconvergence of the finite element approximation to the actual, nonphysical solution ofthe problem is the origin of the apparent mesh sensitivity of damage models and othercontinuous descriptions of fracture.

An example of the apparent mesh sensitivity is given in Figure 2. The diagramshows the steady-state fatigue crack growth rate predicted by a finite element analysisversus the size of the elements which were used in the analysis. The problem geome-try, loading conditions and modelling for which these results have been obtained willbe detailed in Section 6. Fully coupled as well as uncoupled analyses have been done.The dependence of the crack growth per cycle da/dN on the element size h is quitestrong in both approaches: a decrease of the element size by roughly one decade leadsto an increase of the crack growth rate by almost three decades. In the limit h —> 0the crack growth rate clearly goes to infinity, as predicted by the discussion above.

Figure 2. Predicted fatigue crack growth rate versus element size

4. Nonlocal modelling

An effective method to avoid pathological localisation of damage is to add non-local terms to the constitutive model. This approach has been successfully appliedto damage models of a number of failure mechanisms [BAZ 84, PC 87, SAA 89,TVE95]. The spatial interactions resulting from the nonlocality prevent the dam-age growth from localising in a surface. Instead, the damage growth occupies a finiteband, the width of which is related to the internal length scale provided by the non-locality. In its traditional, integral form, nonlocality can be introduced in the fatiguemodel of Section 2 by rewriting the loading function [2] and the damage evolution law


[5] in terms of a new field variable, the nonlocal equivalent strain e :

to define the nonlocal equivalent strain s [PEE 96]. The solution of this problem canformally be written as:

where G(y; x) denotes the Green's function associated with it [PEE 99, PEE 01]. Ex-pression [16] takes exactly the same form as equation [13] for the nonlocal model.This means that the enhanced damage model based on the differential equation [14] isa member of the class of nonlocal models defined by [13]. The parameter c in equa-tion [14], which is of the dimension length squared, sets the internal length scale ofthe model and thus determines the degree to which damage growth localises.

It should be noted that, in the presence of cracks, equation [14] is defined only inthe domain Q. where the damage variable has not yet become critical. This is notonly natural, since the equilibrium problem is defined only on £2, but also necessarybecause the right-hand side s is not uniquely defined in the cracked region as a resultof the indefiniteness of the displacement field (Section 2). The boundary condition[15] associated with equation [14], as well as the standard boundary conditions for theequilibrium problem, must therefore be defined on the boundary f of £2. This meansthat they are imposed not only at the boundary of the problem domain, but also atthe internal boundary which represents the crack contour and that they move with thecrack contour as the crack grows. In terms of the original nonlocal formulation, theintegration in equation [13] must be limited to £2 and must therefore be re-evaluated

The nonlocal equivalent strain is defined by (cf. [PC 87])

with VK(y; x) a weight function which usually decays rapidly with the distance |y — x|.The nonlocality is apparent from equations [12] and [13]: the damage growth and thusthe stresses in x are influenced by the value of the strain in other points y = x.

Instead of the integral definition [13] of the nonlocal strain, we will use the bound-ary value problem given by


as the crack grows. The necessity of this identical treatment of internal and externalboundaries does not seem to have been recognised in the literature. If the - nonphys-ical - strains in the crack are included in the integral in [13] the computed nonlocalequivalent strain and damage rate at the crack tip are too high, resulting in an overpre-diction of the rate of crack growth. Furthermore, since the nonlocal strain at the crackfaces continues to increase as the crack opens, the damage variable continues to growat the crack faces. As a consequence, the width of the crack region continues to in-crease along the entire crack surface, until it finally occupies the entire domain. Botheffects can also be observed in numerical analyses if the cracked zone is not properlyseparated from the remaining material.

It has been argued in Section 3 that the standard, local damage model predicts theimmediate initiation of a crack when the displacement field becomes discontinuousupon loss of ellipticity of the rate equilibrium problem. It can be easily shown forthe nonlocal model that this loss of ellipticity no longer occurs. The characteristicdeterminant associated to the set of rate equilibrium equations and equation [14] isgiven by [PEE 99]

where C*ijkl (i, l = 1,2, 3, j, k = 1, 2, 3, 4) contains the coefficients of the second-order derivatives in the combined set of equations. Expression [17] is positive for all nas long as D < 1. This means that the partial differential system is elliptic throughoutthe initiation phase. However, when D = 1 somewhere in the domain, and a crack isinitiated, a strain singularity may be unavoidable at the crack tip. It is important thatthe damage growth rate remains finite, because the crack growth would otherwise beinstantaneous (see Section 3). Since the damage growth rate depends on the nonlocalequivalent strain e in the nonlocal formulation, this implies that e must remain finiteat the crack tip in order to have a finite crack growth rate.

An analytical expression for the nonlocal strain can be obtained for a crack in aninfinite, linear elastic medium. It is emphasised that this situation is not entirely rep-resentative of a crack in a damaged medium, because the influence of damage on thedeformation near the crack tip is not accounted for. However, the analysis is illustra-tive of the way in which the nonlocality removes the damage rate singularity and thuslocalisation of damage in a surface. A plane crack in an infinite medium is considered,which is loaded in mode I. Furthermore, a plane stress state is assumed throughout themedium. The asymptotic local equivalent strain field can then be determined fromlinear fracture mechanics and shows the usual r-1/2-singularity:

The nonlocal equivalent strain is now obtained by solving the boundary value prob-lem [14]-[15] for this source term. This results for the nonlocal strain at the crack tip


in [PEE 99]:

with r(a) the gamma function. This expression is indeed finite for c > 0, so thatthe damage growth rate at the crack tip remains finite in the enhanced model. Thisin turn means that in this simplified situation a finite crack growth rate is obtainedinstead of the instantaneous fracture predicted by the standard, local damage model.As was mentioned earlier, this does not necessarily imply that singularities are avoidedalso in the full, coupled model, where the singularity of e may be stronger. However,numerical simulations (see Section 6) seem to indicate that this is indeed the case.

5. Aspects of the finite element implementation

In mathematical terms the essential difference between the nonlocal damage for-mulation introduced in Section 4 and the classical, local damage models consists ofthe additional linear partial differential equation [14]. This equation must be solvedsimultaneously with the standard equilibrium equations. For finite element implemen-tations this means that e must be discretised in addition to the displacement compo-nents. The discrete form of [14] follows from the standard transition to a weak formand Galerkin discretisation of the nonlocal equivalent strain [PEE 96]:

where the matrices N and B contain the interpolation functions of the nonlocal equiv-alent strain and their derivatives and the column matrix e contains the nodal values ofs. The discrete form of the equilibrium equations follows in the standard way:

with the matrices N and B containing the displacement interpolation functions andtheir derivatives, respectively, and the column matrices a and t the Cauchy stressesand boundary tractions. The finite element interpolations of the displacements andthe nonlocal strain need to satisfy only the standard, Co-continuity requirements. Theorder of each of the interpolations can be selected independently, although some com-binations may result in stress oscillations [PEE 99].

Since damage growth is defined by relation [12] in a rate format, it must be in-tegrated over each time increment of the numerical analysis in order to obtain thedamage at the end of the increment. Standard integration rules may be used for this


purpose, but in high-cycle fatigue analyses it may be advantageous to use a more so-phisticated integration, which takes into account the cyclic character of the loading[PEE 99, PEE 00]. After discretisation in time by either method, [20] and [21] be-come a set of nonlinear algebraic equations, which can be solved for instance usinga Newton-Raphson scheme, see references [PEE 99, PEE 00] for details. It is at thispoint that the gain in efficiency of an uncoupled approach becomes apparent: if theeffect of damage growth on the stiffness is neglected while the damage variable isnoncritical, the problem will often remain linear as long as no additional elements fail(equation [20] may be nonlinear for nonproportional loading and for some equivalentstrain definitions). As a result, the tangent stiffness matrix remains constant and oneiteration suffices in each increment to reach equilibrium.

It has already been argued that the cracked region, £2C, should not be part of theequilibrium problem domain because the equilibrium equations are not meaningfulin it. Accordingly, the equilibrium equations and the additional equation [14] aredefined only on the domain & = Q \ £2C and boundary conditions are provided at theboundary f of £2. For the finite element formulation this means that the discretisationof the equilibrium problem must also be limited to the noncritical domain £2. Thedifficulty is, however, that this effective domain will gradually shrink as the predictedcrack growth progresses. Consequently, the problem domain must be redefined inthe numerical analysis for each increment of crack growth and a new finite elementdiscretisation must be defined. This remeshing is often avoided by using the originaldomain Q even if this domain contains a crack. The material in the crack is thengiven a small residual stiffness in order to avoid singularity of the discrete equilibriumequations. It is then argued that the stresses which are still transferred by the crackinfluence equilibrium only marginally if the residual stiffness is sufficiently small.This may indeed be true in local damage models, in which the nonphysical strains inthe crack do not influence the surrounding material. But if this approach is followedfor nonlocal damage models, the nonlocal equivalent strain maps the large strainswhich may be computed in the cracked region onto the surrounding material in whichthe damage variable is not (yet) critical. This does not only result in faster growthof damage in front of the crack and consequently in higher predicted crack growthrates, but also in damage growth at the faces of the crack, thus causing the thicknessof the crack region to increase unboundedly. The numerical implementation shouldtherefore reflect the mathematical separation of the crack and the remaining domainby adapting the finite element mesh to the growth of the crack. This can be achievedwithout full remeshing of the problem if completely damaged elements are removedfrom an otherwise fixed finite element mesh. However, the crack contour alwaysfollows the (initial) grid lines in this approach, which means that a fine discretisationis needed in a relatively large region.


6. Application

The nonlocal damage formulation has been used to model crack initiation andgrowth due to fatigue. Reference is made to [PEE 99, PEE 00] for details of the dam-age modelling. The problem geometry of Figure 3 has been considered. The thicknessof the specimen is 0.5 mm. The lower edge of the specimen is fixed in all directions,while fully reversed vertical displacement cycles with an amplitude of 0.0048 mmare forced upon its top edge. Because of symmetry, only half of the specimen hasbeen modelled in the finite element analyses. The reference mesh contains a regu-lar grid of elements with an edge length h = 0.04 mm in an area of approximately0.65 x 0.12 mm2 at the notch tip (indicated in Figure 3). The discretisation has beensuccessively refined in this area to h = 0.02, 0.01 and 0.005 mm. Quadrilateral plane-stress elements with bilinear displacement and nonlocal strain interpolations and aconstant damage variable have been used. Both fully coupled and uncoupled analyseshave been done. In these analyses, elements were removed when the damage vari-able exceeded 0.999999, after which led to this critical damage value was recomputedstarting from the converged state in the previous increment [PEE 99].

Figure 4 shows the crack initiation and growth process as simulated using thefinest of the four meshes and the coupled approach. The area which is shown in thisfigure is the refined area indicated in Figure 3. The stress concentration at the notchtip initially leads to a concentration of damage at the tip. After 4210 cycles a crackis initiated, i.e., the damage variable becomes critical in an element which is thenremoved from the mesh. For continued cycling the crack grows along the symmetryaxis. The crack width decreases as the damage zone which was formed before crackinitiation is traversed. Beyond this damage zone the crack width becomes stationaryat 0.04 mm, which is of the same order as the internal length ^fc = 0.1 mm.

Figure 3. Problem geometry and loading conditions of the fatigue problem (dimen-sions in mm)


Figure 4. Damage and crack growth at the notch tip in the h = 0.005 mm mesh(coupled approach)

The influence of the finite element discretisation on the crack shape is shown inFigure 5, in which the final crack pattern has been plotted for the four discretisations.The coarsest mesh (Figure 5(a)) gives a rather crude approximation of the crack shapeand necessarily overestimates the width of the steady-state part of the crack becausethis width is smaller than the element size. But the h = 0.02 and 0.01 mm meshes givea good approximation of the crack shape in the finest discretisation. The steady-statewidth of the crack does not vary between the three finest discretisations. The finaldamage and crack patterns obtained with the uncoupled approach are almost identicalto the ones shown in Figure 5. However, there is a slight difference in the numberof loading cycles needed to reach these states. This is illustrated in Figure 6, whichshows the length of the crack, a, versus the number of loading cycles, N, for thefour meshes in the coupled as well as the uncoupled approach. For an increasinglyrefined discretisation the growth curves converge to a response with a finite number of


Figure 5. Final crack pattern in the (a) h = 0.04 mm, (b) h = 0.02 mm, (c) h =0.01 mm and (d) h = 0.005 mm meshes (coupled approach)

cycles to crack initiation and a finite growth rate, instead of the instantaneous growthpredicted by the local model. In the uncoupled model, the crack is initiated slightlylater and grows slightly slower than in the fully coupled model. This is due to thefact that the damage in front of the crack tip has no influence on the deformation,which is therefore smaller. This results in a smaller damage rate and thus in slowercrack growth. The steady-state crack growth rate obtained in both approaches hasbeen plotted versus the element size h in Figure 7. In contrast with the local damagemodel (Figure 2) the growth rate in the nonlocal models becomes practically constantas the element size is reduced.


Figure 6. Influence of the element size on the predicted crack growth in the nonlocaldamage model

Figure 7. Influence of the element size on the steady-state crack growth rate

7. Discussion and concluding remarks

A key issue in the development of fracture models based on a continuum damageapproach is their ability to correctly describe the localised deformations which are typ-ical of fracture problems. If this issue is not properly addressed, the damage processwhich represents the initiation and growth of cracks tends to localise in a vanishingvolume. A perfectly brittle response is then obtained, even if the constitutive relations


have been designed to show a gradual loss of strength. This pathological localisationof damage is not so much caused by loss of ellipticity of the rate equilibrium equa-tions, but rather due to singularities at the crack tip. It therefore occurs not only infully coupled analyses, but also in uncoupled analyses, in which the damage variabledoes not immediately affect the constitutive behaviour. As a result of the singularities,the material in front of the crack fails immediately and in a vanishing volume, even ifthe rate equilibrium equations do not first lose ellipticity. The crack traverses the re-maining cross section at an infinite growth rate and the thickness of the correspondingdamage band is zero.

The nonphysical behaviour of the standard models can be effectively removed bythe introduction of nonlocality in the constitutive relations. This can be achieved byincluding an additional partial differential equation in the equilibrium problem. As aresult, the localisation of damage is limited to the scale of the intrinsic length which isintroduced by the nonlocality. Crack growth is no longer instantaneous and a positivevolume takes part in the damage process which describes the crack growth. This alsomeans that a positive amount of work is needed for the crack growth and that thefracture process is thus no longer perfectly brittle.

Additional boundary conditions must be provided in the nonlocal model, not onlyat the boundary of the problem domain, but also at the internal boundary which de-scribes the crack contour. The latter ensures that the crack is well separated from theremaining part of the continuum and that nonphysical deformations which may becomputed in the cracked region do not affect the growth of damage at the crack faces.The numerical implementation of the nonlocal model must reflect this separation. Thismeans that the spatial discretisation of the equilibrium problem must be adapted foreach increment of crack growth. If this separation is not made rigorously, the damagegrowth rate may be overestimated and nonphysical damage growth may be predictedat the faces of the crack. In this contribution, a rigorous but crude approach has beenfollowed: completely damaged elements are removed from the finite element mesh.Meaningful, mesh-objective numerical solutions have been obtained with this tech-nique for the nonlocal formulation of the coupled as well as the uncoupled problem.Although reliable and useful for development purposes, the approach is not very suit-able for practical problems. The location of crack initiation and the direction of crackgrowth are usually not known in advance. In this case, adaptive spatial discretisationtechniques are needed to follow the free boundary which represents the crack contourand to accurately describe the high deformation gradients at its tip.

8. References

[BAZ84] BA£ANT Z.P., BELYTSCHKO T., CHANG T.P., "Continuum theory for strain-softening", J. Eng. Mech., vol. 110, 1984, p. 1666-1692.

[BUI 80] BuI H.D., EHRLACHER A., "Propagation of damage in elastic and plastic solids",D. Francois, et al., eds., Advances in Fracture Research, 1980, p. 533-551.


[CHA 88] CHABOCHE J.L., "Continuum damage mechanics. Part I - General concepts; PartII - Damage growth, crack initiation, and crack growth", J. Appl. Mech., vol. 55, 1988, p.59-64, 65-72.

[FRE 96] FREMOND M., NED JAR B., "Damage, gradient of damage and principle of virtualpower", Int. J. Solids Struct., vol. 33, 1996, p. 1083-1103.

[GEE 99] GEERS M.G.D., "Enhanced solution control for physically and geometrically non-linear problems, parts I and II", Int. J. Num. Meth. Eng., vol. 46,1999, p. 177-204, 205-230.

[KAC 58] KACHANOV L.M., "On the time to failure under creep conditions", Izv. Akad. Nauk.SSSR, Old. Tekhn. Nauk., vol. 8, 1958, p. 26-31. In Russian.

[LEM 86] LEMAITRE J., "Local approach to fracture", Eng. Fract. Mech., vol. 25, 1986, p.523-537.

[LEM 90] LEMAITRE J., CHABOCHE J.-L., Mechanics of Solid Materials, Cambridge, Cam-bridge University Press, 1990.

[LEM 96] LEMAITRE J., A Course on Damage Mechanics, Berlin, Springer, 2nd edn., 1996.

[PAA93] PAAS M.H.J.W., SCHREURS P.J.G., BREKELMANS W.A.M., "A continuum ap-proach to brittle and fatigue damage: theory and numerical procedures", Int. J. SolidsStruct., vol. 30, 1993, p. 579-599.

[PC 87] PIJAUDIER-CABOT G., BAZANT Z.P., "Nonlocal damage theory", /. Eng. Mech.,vol. 113, 1987, p. 1512-1533.

[PC 91] PIJAUDIER-CABOT G., HUERTA A., "Finite element analysis of bifurcation in non-local strain softening solids", Comp. Meth. Appl. Mech. Eng., vol. 90, 1991, p. 905-919.

[PEE 96] PEERLINGS R.H.J., DE BORST R., BREKELMANS W.A.M., DE VREE J.H.P.,"Gradient-enhanced damage for quasi-brittle materials", Int. J. Num. Meth. Eng., vol. 39,1996, p. 3391-3403.

[PEE 99] PEERLINGS R.H.J., Enhanced damage modelling for fracture and fatigue, Ph.D.thesis, Eindhoven University of Technology, Eindhoven, The Netherlands, 1999.

[PEE 00] PEERLINGS R.H.J., BREKELMANS W.A.M., DE BORST R., GEERS M.G.D.,"Gradient-enhanced damage modelling of fatigue", Int. J. Num. Meth. Eng., vol. 49, 2000,p.1547-1569.

[PEE 01] PEERLINGS R.H.J., GEERS M.G.D., DE BORST R., BREKELMANS W.A.M., "Acritical comparison of nonlocal and gradient-enhanced softening continua", Accepted forpublication.

[RAB 69] RABOTNOV Y.N., Creep Problems in Structural Members, Amsterdam, North-Holland, 1969.

[SAA 89] SAANOUNI K., CHABOCHE J.-L., LESNE P.M., "On the creep crack-growth pre-diction by a non local damage formulation", Eur. J. Mech. A/Solids, vol. 8, 1989, p. 437-459.

[TVE 95] TVERGAARD V., NEEDLEMAN A., "Effects of non-local damage in porous plasticsolids", Int. J. Solids Struct., vol. 32, 1995, p. 1063-1077.


Chapter 5

Numerical Aspects of Nonlocal DamageAnalyses

Claudia Comi and Umberto PeregoDepartment of Structural Engineering, Politecnico of Milan, Italy


Aspects of Nonlocal Damage Analyses 77

1. Introduction

In many instances of practical interest, the initiation of fracture is preceded by asignificant strain localization phase in which the material is macroscopically integerand inelastic phenomena tend to be confined in a narrow region. In this phase, theuse of continuum models with softening, like e.g. damage models, is justified.However, the strain softening behavior due to the development of material damageis well known to produce unrealistic mesh sensitivity in standard finite elementapplication. Zero energy dissipation is expected in the limit since strains tend tolocalize on a zero volume region as the mesh is refined. In statics, the failure ofclassical discretization methods can be explained, from the mathematical point ofview, with the boundary value problem losing ellipticity as a consequence of thesoftening material behavior. The ill-posedness of the boundary value problemsreflects the fact that standard continuum mechanics theories are not appropriatewhen the microscopic material heterogeneity is characterized by an internal lengthwhich is not negligible if compared to the typical macroscopic length of thestructure, so that the range of the microscopic interaction forces has to be consideredlarge with respect to the macroscopic scale (see e.g. [GAN 00] for a recentdiscussion).

Among the several regularization techniques proposed in the literature, one ofthe most computationally convenient seems the one based on the formulation of anonlocal continuum (see [ERI 81] for nonlocal plasticity). The idea is that the longrange nature of the microscopic interaction forces is taken into account on themacroscale by expressing the material constitutive law in terms of one or morenonlocal variables defined as suitable weighted averages of their local values overthe interaction domain. In the formulation of a nonlocal model, several choices haveto be made such as the definition of the nonlocal variable (variables), the definitionof the weight function and the definition of the interaction domain.

The adopted choices have important numerical consequences in finite elementimplementations (see [JIR 98] for a discussion of other aspects): the corrector phaseof the iterative procedure, typically carried out at each Gauss point separately, maycease to be local [STR 96]; the consistent tangent matrix becomes non-symmetric[BAZ 88], [PIJ 95], [JIR 99]. The lack of symmetry has important consequencesboth from the theoretical and computational point of view. In particular a nonsymmetric model is not suitable for variational approaches and non symmetricsolvers have to be used in numerical applications, with a consequent increase incomputing costs.

In the present paper the discussion is confined to isotropic damage models.Within this context, it is shown that it is possible to formulate a very generalisotropic local model endowed with a symmetric consistent tangent matrix. Themodel considered is based on the definition of two damage variables affecting theshear and bulk moduli separately. The consistent tensor of tangent elastic moduli is


derived and it is shown that it is symmetric provided that associative evolutionequations are assumed for both damage and kinematic internal variables.

A nonlocal formulation of the model is then proposed, based on thethermodynamic formulation of Borino et al. [BOR 99], [BEN 00]. While in[BEN 00] a kinematic internal variable was assumed as the primal nonlocal variable,in the model here proposed the primal nonlocal variables are the damage variables.Following [BOR 99], the nonlocality is transferred onto the conjugate variableswhich in the present case are the energy release rates, by means of an energyequivalence which allows one to eliminate the so called nonlocality residual [ERI81]. Itis shown that, unlike in [BEN 00], the proposed model maintains the attractivefeature that all constitutive computations can be performed locally, at the Gausspoint level [PIJ 87], [COM 00], and that it gives rise to a symmetric finite elementtangent stiffness matrix.

A one-dimensional problem is studied for a simplified version of the model, witha single damage variable. The results obtained with the proposed dual nonlocalformulation and with the standard nonlocal formulation of [COM 00] are compared.

2. A "symmetric" isotropic local damage model

Let e=e-l/3Iev be the deviatoric part of the strain tensor e, ev being itsvolumetric part and I the second order identity tensor. The free energy densitypotential under isothermal conditions for the proposed damage model is defined as

where G0 and K0 are the initial elastic and shear moduli, respectively, dc and dK

are shear and volumetric damage variables and ^ is a scalar variable of kinematicnature. The state equations defining the conjugate static variables are given by

where s = a-lp is the stress deviator and p = l/3crkk is the mean stress; % is astatic internal variable and YG , YK represent the elastic energy release rates.

The activation of damage is governed by the following activation functions andloading-unloading conditions


7 being a scalar dissipation multiplier. The associative evolution equations aregiven by

Finally, the rate of dissipation density is given by

NOTE 1. - The presence of separate damage variables dG, dK adds flexibility tothe model. The activation function may be defined in a form more suited formaterials with non-symmetric tension-compression behavior like concrete and theseparate evolution equations for deviatoric and volumetric damages allow for avarying Poisson's coefficient while preserving the isotropic nature of the model.

NOTE 2. - The scalar internal variable £ accounts for material rearrangements atthe microscale due to damage development. Damage is the only dissipationmechanism considered in this model.

In finite element applications, the constitutive law is integrated within a time-step in the corrector phase of the iterative procedure, according to a Eulerbackward-difference scheme. This implies computing all derivatives in [4] at the endof the step. At the end of the corrector phase, a relation between stress and strainincrements is implicitly obtained: Aa = Aa(As,). In the subsequent predictor phase,the consistent tangent elastic tensor is computed by differentiating this relationunder the assumption of continuous loading in the increment, i.e.

with all quantities evaluated at the end of the step. Taking into account eqs. [43 and23], one has

lc being a material internal length related to the width of the localization zone. Theparticular definition of the weighting function W accounts for the effect of theboundary on the nonlocal interaction at the microscale and allows the reproduction ina simple way of a uniform field. In other words, if YG is constant over the body, it seems

with:

3. Nonlocal version of the "symmetric" isotropic damage model

The nonlocal version of the model is obtained substituting one of the constitutivevariables by its weighted average over the whole domain Q. of the structure. Theaveraged quantity reflects the effect of the interaction at the microscale between thematerial point considered and the neighboring points. The decaying effect of theinteraction with the distance is taken into account by the weighting function. In theliterature, there exist several proposals concerning the choice of the nonlocalvariable (see [JIR 98] and [GAN 99] for a recent discussion on the subject). Fromthe computational standpoint, the most convenient choices are those which allow one tocarry out the constitutive calculations locally at each Gauss point, withoutintroducing any coupling at constitutive level as, e.g., in [PIJ 87] and [COM 00],where the strain invariants have been selected as nonlocal variables. In the presentcontext, this would imply defining two nonlocal variables as follows

where I®I denotes the fourth order symmetric identity tensor of components


The explicit expression of the consistent tangent elastic tensor is then obtained as:


logic and desirable that also YG be uniform. However, the adopted definition of Win [10]| is such that Vy(x,s)= W(s,x). This lack of symmetry of the weightingfunction entails that also the consistent tangent operator is not symmetric for thenonlocal model [BAZ 88], [PIJ 95], [JIR 99] even if the consistent tangent operatorof the underlying local model is symmetric.

A more rigorous treatment of the boundary effect could be used to employhomogeneization techniques for periodic structures in the proximity of geometricboundaries (see e.g. [LEG 97]).

The non-symmetric nonlocal version of model [l]-[5] is governed by eqs. [1], [2]and by the following activation conditions and evolution equations

where:

The dissipation rate density takes the expression:

P being the so called nonlocality residual representing the energy exchangedbetween the considered material point and other points belonging to its interaction

A symmetric nonlocal formulation of the same local damage model can beachieved following the thermodynamic nonlocal approach of Borino et al.[BOR 99], [BEN 00]. An application of that theory to the present model whichpreserves the computational advantages of the above non-symmetric nonlocal modelis obtained by assuming that the damage variables are the variables reflecting at themacroscale the microscopic interaction due to the heterogeneity of the material and,therefore, have to be considered nonlocal [BAZ 88]. Hence, one can set:

Figure 1. a) One-dimensional weight functions W(x,s) and W*(x,s) for varying

position x over a bar for l/L=0.2; (b) function \ W*(x,s) ds for varying

characteristic length lc


domain due to the intrinsic nonlocality of the developing damage mechanism. Thefact that the system is thermodynamically isolated implies the following insulationcondition [ERI 81], [BOR 99]

The insulation condition allows elimination of the nonlocality residual and totransfer the nonlocality onto the dual variables of the nonlocal damage variablesdefined in the model, i.e. the energy release rates YG and YK . One can write

From the insulation condition [15] it follows that-

Having in mind the definitions [13], from eq. [17] one obtains

where W * is the adjoint function of W, i.e.


A plot of the weight functions W and W* centered at various positions over a barof length L is shown in Figure la where the influence of the boundary on the shapeof the weight function is also evidenced. The activation function and the evolutionequations [3] and [4] are now written in terms of the dual nonlocal variables as inthe non-symmetric non-local model [11]

be the vector of internal equivalent nodal forces and let u be the vector of nodal

displacements in a finite element discretization. Let Ng be the total number of Gauss

points used to carry out the numerical integration over all the elements in the mesh.

However, in the absence of damage, it appears to be an obvious requirement thata uniform strain field generates a uniform field of strain energy release rates YG and

YK. Therefore, in the applications, the weight function W in eq. [10]1 will be used

for YG and YK, while the weight function W* in [19] will be used for dG and dK ,

satisfying in this way condition [17].

The algorithmic tangent matrix can be computed for the dual nonlocal modelfollowing the procedure proposed by Jirasek [JIR 99]. Let

NOTE 3. - A dual nonlocal damage model based on the thermodynamic approachhas been presented by Benvenuti et al. [BEN 00]. In their model, however, only thekinematic internal variable £ has a nonlocal nature, while the damage variable islocal. While the issue of the most appropriate choice seems to be still open from themechanical point of view, from the computational standpoint the definition of anonlocal kinematic internal variable leads to a nonlocal constitutive problem in thecorrector phase of a standard finite element implementation. On the contrary, theintegration of eqs. [20] leads to the same local, and therefore computationallyconvenient, problem as in the non-symmetric model [11] [COM 00].

NOTE 4. - The weight function used for the definition of the dual nonlocal variables in[18] does not allow for the reproduction of a uniform field as, in general (see Figurelb),


and let Nactg be the number of Gauss points where f(YG,YK, X) = 0 and y > 0 at

the end of the correction phase. Let us also define the following quantities at Gauss

points q and p

where A is defined in [7]2. One then obtains the symmetric elasto-damage tangentmatrix Ked

where wp denotes the Gauss weight at Gauss point p and

and PT = {1/3 1/3 1/3 0 0 0}. It should be noted that in [24] the index p runs

over the whole set of Gauss points. This is because the global damage variable at a

point varies as a consequence of the variation of the local damage at any point in the

body. Thus, even though at a point one has y = 0 and the material point unloads

elastically, at the same point one has d = 0 if there is at least one active point in the

structure. By contrast, the index q runs only over the active Gauss points since it

concerns the dependence of the nonlocal damage variables at point p on their

corresponding local variables which are zero at inactive Gauss points. On the basis

of these considerations and noting that, while Wpq = Wqp and W*pq = W*qp, one has

WpqW*i = W^W^ , the symmetry of Ked can be easily assessed.

represent the deviatoric and volumetric contributions of the same Gauss point to theinitial undamaged stiffness matrix. In [25], D0 is the matrix of initial elastic moduli,BG and BK are compatibility matrices such that:


4. A simple nonlocal damage model

To study the effects of the dual nonlocal regularization described in the previousSection a simple model, with only one damage variable d, is considered.Applications of the two damage variables model to concrete problems will bepresented in a forthcoming paper. The simplified model is based on the followingfree energy density:

where DO is the undamaged elastic tensor and k, c and n are material parameters. Thestate equations are given by:

Figure 2. Stress-strain behavior for the simple damage model for varying n

The activation function, loading-unloading conditions and evolution equationsare defined as:

and therefore the kinematic internal variable £ coincides with the damage variable d.

5. One-dimensional numerical application

The simplified damage model is used for the simulation of a tensile test on aprismatic bar. The problem data and geometry, together with the adopted meshes areshown in Figure 1. To trigger the damage localization, the elements at the left

The symmetric consistent tangent matrix can be computed for the simplifiedmodel following the same procedure as in the previous Section


The local model is such that, in one dimension, the stress vanishes onlyasymptotically, for e —> °°, but with a bounded fracture energy density. This can be

seen by confining the model to one dimension. For e > E0, £0 being the strain at thelinear elastic limit, from the condition/= 0, one has (Figure 2)

E denoting Young's modulus. The fracture energy density is defined as

If / denotes the integrand in [31]2, the boundedness of gf can be establishednoting that


boundary have been slightly weakened. The problem has been solved adopting twononlocal approaches: (model A) the non-symmetric approach of Comi [COM 00]based on the definition of nonlocal strain invariants (in this simple case coincidingwith the energy release rate); (model B) the symmetric dual nonlocal approach ofBorino et al. [BOR 99] in the form discussed in Section 3. Note that different valuesof lc have been adopted for the two models to obtain comparable damageaccumulation in the part of the bar where unloading occurs after localization.

The parameter lc can be identified using a back-analysis technique based on onedimensional tests where the width of the process zone is measured. Alternatively, ananalytical approach can be pursued where lc is related to the length of the stationaryharmonic localization wave (see e.g. [SLU 93]). This type of study has still to becarried out for the symmetric nonlocal model considered here.

Figure 3. One-dimensional test problem: geometry, adopted meshes and materialdata

As shown in Figure 4, both approaches provide an effective regularization of theproblem as the results in terms of reaction force versus imposed displacementrapidly converge towards a mesh independent solution. From Figure 4, it appears thatthe dual regularization technique produces an initially more ductile response with asubsequent very steep drop of the reaction force. The displacement controlledanalysis cannot proceed further due to a global snap-back behavior which is notobserved in the analysis with model A regularization. Both behaviors can beobserved in uniaxial tension tests depending on the material properties and testingconditions. Since the present numerical test does not simulate a physical experiment,it is not possible to assess which one of the two results is more realistic.

The longitudinal strain evolution obtained by means of the two regularizations isshown in Figure 5. While model A regularization gives rise to a sharp strainlocalization, the model B technique produces a smoother profile with a much lowerpeak value developing at a significant distance from the boundary, where someelements have been weakened. This is a consequence of the effect of the boundary


due to the particular shape of the weight function as already mentioned in Section 3(see Figure 1).

Figure 4. Reaction per unit cross-section area versus imposed displacement withnonlocal models A and B: convergence with mesh refinement

Figure 5. Strain evolution for imposed displacement u: (a) model A; (b) model B

The difference is less pronounced in terms of local damage profiles, as illustratedin Figure 6a for a displacement u = 0.0168 mm. Again, with model B regularizationthe damage peak is offset with respect to the boundary. It should also be noted thatfor equal imposed displacement u, the model A regularization leads to a higherdamage peak. The comparison between the local and nonlocal damages in model Banalysis is shown in Figure 6b. It can be noted that the nonlocal damage presents asharper peak though at almost the same value of the local one.


Finally, the stress profiles are shown in Figure 7. It turns out that the dual regular-ization has the beneficial effect of reducing the stress oscillation caused by theweighting process [JIR 99]. Furthermore, in both cases stress oscillation tends todecrease as the mesh is refined.

Figure 6. Imposed displacement u=0.0168 mm: (a) damage profiles d (x) formodels A and B; (b) local d (x) and non local d (x) damage profiles for model B

Figure 7. Stress distributions along the bar for model A and model B upon meshrefinement


6. Conclusions

The finite element implementation of a family of isotropic nonlocal damagemodels has been discussed. Attention has been focussed on the issue of thesymmetry of the consistent tangent operator.

A rather general isotropic local damage model based on two damage variablesaffecting separately the shear and bulk moduli has been presented. The explicitexpression of the consistent tangent matrix has been derived and it has been shownthat symmetry is obtained provided that associative evolutions are postulated for thedamage and the internal variables. Then the model has been re-formulated as anonlocal model following the approach proposed in [PIJ 87] and [COM 00] whichconsists of assuming as nonlocal variable the elastic energy release rate. This has theadvantage that all constitutive calculations can be carried out separately at eachGauss point during the corrector phase of the standard finite element iterativeprocedure. The consistent tangent matrix for the considered nonlocal model is wellknown to be non-symmetric [BAZ 88], [JIR 99].

A nonlocal version of the same model, based on the thermodynamically foundednonlocal theory recently put forward by Borino et al. [BOR 99] and preserving thesymmetry of the underlying local model has also been formulated. In this newversion of the model, the nonlocal nature, originally conferred to the damagevariables, is transferred to their conjugate variables, the energy release rates, on thebasis of an energy equivalence which allows one to eliminate the so called nonlocalityresidual. The explicit expression of the finite element tangent stiffness matrix of thenew nonlocal model has been derived and it has been shown to be symmetric.

A one-dimensional test has been carried out for a simpler nonlocal model basedon a single damage variable. The regularization property of the dual nonlocalformulation has been assessed even though the issue of the influence of theboundary conditions with the development of a significant boundary layer seems todeserve further consideration.

Acknowledgements

This work has been carried out within the framework of the joint co-financingMURST and LSC-Politecnico of Milan program.

7. References

[BAZ 88] BA£ANT Z.P., PUAUDIER-CABOT G., "Nonlocal continuum damage, localizationinstability and convergence", Journal of Applied Mechanics, vol. 55, 1988, p. 287-293.


[BEN 00] BENVENUTI E., BORINo G., TRALLI A., "A thermodynamically consistent non-localformulation for elasto-damaging materials: theory and computations", Proceedings ofECCOMAS 2000, Barcelona, Spain, 11-14 September 2000.

[BOR 99] BORINO G., FUSCHI P., POLIZZOTTO C., "A thermodynamic approach to nonlocalplasticity and related variational principles", Journal of Applied MechanicsI, vol. 66,1999, p. 952-963.

[COM 00] COMI C., "A nonlocal model with tension and compression damage mechanisms",to appear in European Journal of Mechanics A/Solids, 2000.

[ERI 81] ERINGEN A.C., "On nonlocal plasticity", International Journal of EngineeringScience, vol. 19, 1981, p. 1461-1474.

[GAN 99] GANGHOFFER J.F., SLUYS L.J., DE BORST R., "A reappraisal of nonlocalmechanics", European Journal of Mechanics A/Solids, vol. 18, 1999, p. 17-46.

[GAN 00] GANGHOFFER J.F., DE BORST R., "A new framework in nonlocal mechanics",International Journal of Engineering Science, vol. 38, 2000, p. 453-486.

[JIR 98] JIRASEKM., "Nonlocal models for damage and fracture: comparison of approaches",International Journal of Solids and Structures, vol. 35, 1998, p. 4133-4145.

[JIR 99] JIRASEK M., "Computational aspects of nonlocal models", Proceedings of ECCM 99,Munchen, Germany, August 31-September 3, 1999.

[LEG 97] LEGUILLON D., "Comparison of mached asymptotics, multiple scalings andaverages in homogenization of periodic structures", Math. Models Meth. Appl. Sci., vol. 7,1997, p. 663-680.

[PIJ 87] PUAUDIER-CABOT G., BAZANT Z.P., "Non local damage theory", Journal ofEngineering Mechanics, vol. 113, 1987, p. 1512-1533.

[PIJ 95] PUAUDIER-CABOT G., "Non local damage", in Continuum Models for Materials withMicrostructure, H.-B. Muhlhaus (ed.), New York, Wiley, 1995, p. 105-143.

[SLU 93] SLUYS L.J., DE BORST R., MUHLHAUS H.-B., "Wave propagation, localization anddispersion in a gradient-dependent medium", Int. J. Solids Struct., vol. 30, 1993, p. 1153-1171.

[STR 96] STROMBERG L. RISTINMAA M., "FE-formulation of a nonlocal plasticity theory",Computer Methods in Applied Mechanics and Engineering, vol. 136, 1996, p. 127-144.


Chapter 6

Computational Issues and Applicationsfor 3D Anisotropic Damage Modelling:Coupling Effects of Damage andFrictional Sliding

Damien Halm and Andre DragonLaboratoire de Mecanique et de Physique des Materiaux, Ecole NationaleSuperieure de Mecanique et d'Aerotechnique, France

Pierre BadelElectricite de France, Division Recherche et Developpement, France


This paper addresses some issues concerning the modelling of the behaviour ofquasi-brittle materials, comprising some rocks, concrete, ceramics, etc. These mate-rials share the same damage process, namely the generation and growth of decohesionmesosurfaces (mesocracks). This phenomenon induces a degradation of the effectiveproperties of the material. Besides, the generally oriented nature of flaws gives riseto a number of characteristic events such as induced anisotropy, volumetricdilatancy, irreversible stress/strain effects, dissymmetry between tension andcompression, unilateral behaviour due to crack opening/closure, dissipative frictionalsliding on closed mesocrack lips, etc. The purpose of this paper is to summarize mostsalient features of a model capable of taking into account most of the abovephenomena concentrating specially on its numerical implementation and applicationsfor a set of engineering problems concerning concrete structures.

The postulate of combining both physical pertinence and numerical simplicity ledthe authors to search a third way between micromechanical and phenomenologicalapproaches: the former propose an accurate picture of the real mechanisms but their useis frequently limited to particular loading paths due to inherent complexitiesencountered; the latter are generally designed to be easily implanted in finite elementcodes but suffer from a lack of physical motivation. Section 2 of this paper describes a3D damage model by mesocrack growth, originally proposed by Dragon [DRA 94],and recently developed by Halm and Dragon [HAL 96], [HAL 98]. Its particularitylies in its modular nature, with two main parts:

- A first step deals with the modelling of the mesocrack growth as well as withthe moduli recovery phenomenon due to crack closure (unilateral effect). Theemphasis is put on the stress continuity requirement when passing from open toclosed cracks (and vice versa). Thus, f. ex., tension-compression cycles can bemodelled.

-The second level couples damage with a second dissipative phenomenon,namely frictional sliding on closed mesocracks and allows to simulate more complexloading paths (torsion, f. ex.).

The purpose of the model depicted in Section 2 is to provide an efficient tool forresolving boundary-value problems involving non linear behaviour of quasi-brittlesolids. Thus, great care is taken with the accuracy and simplicity of the numericalintegration scheme related to both independent mechanisms as well as to the coupledmodel. It is worth noting that the use of an implicit integration scheme for damageleads to the resolution of a linear equation, while classical elastoplastic modelsrequire more complex numerical treatment. Moreover the low degree of couplingbetween the two equations governing respectively damage and sliding evolutionsavoids having to solve an intricate non linear system. Details are given in Section 3.

In order to illustrate the pertinence of the coupled model and the efficiency of theintegration algorithm, the constitutive equations have been introduced in

3D Anisotropic Damage Modelling 95

1. Introduction


Code_Aster, the Finite Element code developed by Electricite de France. Section 4provides comments on some boundary-value problems underscoring the applicabilityof the model for efficient structural analyses of concrete structures.

2. Anisotropic damage and sliding model

This section outlines the salient features of the anisotropic damage model byDragon et al. [DRA 94], [HAL 96], [HAL 98]. The particularity of this model lies inits modular structure, each part dealing with a given dissipative mechanism: damageby mesocrack growth (with unilateral behaviour) and frictional sliding on closedmesocrack lips. The behaviour of the mesocracked material is assumed to be rate-independent, isothermal and restrained to small strain.

2.1. Damage by mesocrack growth and unilateral behaviour

The model at stake here aims at describing the progressive mesocrack-inducedanisotropic degradation and related behaviour of elastic quasi-brittle solids. It isbased on a series of assumptions combining micromechanical considerations andmacroscopic formulation:

(i) Damage is described by a single internal variable, a second-order tensor Dconveying information on crack orientation:

where ni stands for the normal of the i-th set of parallel cracks and d(i)(S) is adimensionless scalar function proportional to the extent S of decohesion. The form[1] derives from micromechanical considerations [KAC 92]. From a macroscopicpoint of view, Onat and Leckie [ONA 88] prove that D must be an even function ofni, and then at least quadratic. The spectral decomposition of D leads to:

Expression [2] can be macroscopically interpreted as follows: any system ofmicrocracks can be reduced to three equivalent orthogonal sets of crackscharacterized by densities Dk and normal vectors vk.

NOTE - Unlike the case of « 1-d » models (the value of d is then bounded by 0and 1), values of the Dy-components within the relative tensorial representationcannot be straightforwardly interpreted in the same simplistic manner. In fact, whenconsidering the scalar dimensionless density function d'(S) as a part of the


micromechanical interpretation of the damage tensor D, one can - for a particularnature of defects considered (e.g. penny-shaped microcracks) - interpret d'(S) in

terms of the conventional crack density

theoretically vary within the interval [0,1]. So, one can state that Dij-componentsvalues take their micromechanically licit values in the interval [0,1] while theeffective control of the evolution equations (including their algorithmicmanagement) and local instability phenomena generated by the CDM model puteffective limits well below this conceptual absolute bound of unity. That is why sucha damage model has to be associated with tools of detection of relevant localinstabilities (i.e. localisation bifurcation) in the context of computational algorithmsfor efficient structural analysis. This association has been achieved for the first levelof the model (frictionless damage model without unilateral behaviour), see[DRA 94]: it allows one to correctly predict the incipience of localisation phenomenawithin 3D framework. The localisation detection is not treated in this paper.

(ii) Micromechanical studies [KAC 92] show that 3D damage configurationsshould be rigorously described not by the single variable D_[l], but by two damageparameters, namely D and its extension to the fourth-order D :

However, when cracks are open, the influence of D can be neglected and thesingle variable D appears sufficient to model the degradation of solids containingcracks. Under compressive loading, favourably oriented cracks may close,Jeading toan elastic moduli recovery phenomenon. In this case, the contribution of D into theoverall elastic properties can no longer be neglected. In order to maintain themacroscopic interpretation [2], the complementary fourth-order entity (named D)necessary to account for the unilateral effectjs directly built with the eigenvalues andeigenvectors of D and slightly differs from D :

Note that there is no new information in D with respect to D, so D is notconsidered as a new damage variable.

(iii) One assumes the existence of a thermodynamic potential (free energy perunit volume w), function of strain e, damage D and the fourth-order damageparameter D, and generating a form of elastic orthotropy for D * 0, in connectionwith the three eigensystems [2]. Assuming linear elasticity and non interactionbetween cracks, the tensorial functions representation theory [BOE 78] gives thegeneral form of the terms entering w(e,D, D (D)):

In such case, d(S) can


H stands for the classical Heaviside function and activates or deactivates the D -term depending on whether the k-th equivalent set of mesocracks is open (vk.e. v'SO)or closed (vk.e. vk<0). The proof for the form of the opening/closure criterionvk.e. vk=0 can be found in [HAL 96]. A, and )H are the classical Lame constants; aand 3 are material constants related to modified elastic moduli for a given damagestate. The factor (oc+2|3) in front of the D-term is obtained by assuming a totalstiffness recovery in the direction normal to the closed crack. The linear term,reading g tr(e.D), generates residual phenomena for D*0. The elastic stress a and thedamage thermodynamic force FD are determined by partial derivation:

The forms of w, CT and F respect the continuity conditions for multilinear elasticity[CUR 95], so that these functions remain continuous despite the presence of H.

(iv) The evolution of D, corresponding to the brittle, splitting-like crack kinetics,has been found to follow the normality rule with respect to a criterion in the space ofcomponents of the proper thermodynamic force FD. The damage evolution is thusapparently following the principle of maximum dissipation and is related here totensile (positive) straining e* and to actual damage pattern. It should be stressedhowever that the particular damage criterion proposed in [DRA 94] f(FD,D)<0 isexplicitly dependent on the part FD1+ = -ge+ = FD-FD2-FD1" of the driving force FD.FD1 is the strain energy release rate term related to residual effects: FD1 = -ge, FD2

represents the remaining recoverable energy release rate. The former term isdecomposed into the splitting part FD1+ = -g£+, 6+ = P+:e, with P+ a positive fourth-order projection operator selecting positive eigenvalues from strain, and the non-splitting part FDI" = -g(e-e+). The damage criterion and rate-independent damageevolution law are thus as follows:


Note that the damage model including the unilateral effect necessitates theidentification of eight material constants only, which can be relatively easilydetermined, see f. ex. [HAL 01].

2.2. Fractional sliding on closed mesocrack lips

Even if it takes into account the unilateral effect the previous model does notrestore the shear moduli when cracks close, assuming thus that cracks are perfectlylubricated. Because of the roughness of the crack lips and the consecutive friction,this assumption appears too strong: experimental data involving loading-unloadingcycles for specimens undergoing frictional sliding on the lips (torsional tests forexample) exhibit a shear moduli recovery in the direction parallel to the crack plane,due to blocking of crack lips displacement. The work by Gambarotta andLagomarsino [GAM 93] proposes a 3D micromechanical model for this phenomenonwhich constitutes progress with respect to some earlier 2D attempts. This sectionprovides a macroscopic formulation suitable for boundary-value problems involvingfrictional sliding. It is built within the same thermodynamic framework as fordamage and is based on following hypotheses:

(i) Sliding occurs within the crack plane. A micromechanical study ([KAC 92],considering this time that crack displacement has no opening component) leads tothe following possible expression for the sliding variable:

S1 stands for the cracked surface of the i-th set of parallel mesocracks of normaln1, £' the sliding following the direction g1, V the representative volume element. Asthe influence of D reduces to that of three equivalent sets according to [2], y can bewritten in the analogous manner:


where vk, k = 1,2,3 are the D-eigenvectors.

(ii) Frictional blocking induces a macroscopic recovery of the shear moduli. Inexpression [3], the degradation of the shear moduli is related to the p-term in thefirst line. Invariants involving Y will thus replace the previous (3-term in the freeenergy w(e,D,y) taking into account this additional dissipative phenomenon. Due tothe particular structure of D and y and the fact that only simultaneous (Y,D)-invariants enter w, two additional invariants convey useful information: tr(e.y.D) andtr(y.Y.D).

(iii) According to the points (i) and (ii), the following expression is proposed forthe free energy of the solid containing sliding cracks:

with Lk =vk ®vk ®vk ®vk and Dk = D k v k ® v k . The coefficients 4(3 and -2(3in the last line have been calculated by assuming: (1) the continuity between theexpressions of w corresponding respectively to open and closed cracks, (2) sliding Yis equal to the strain e in the crack plane at the very closure moment. The elasticstress as well as the thermodynamic force related to D contains the contribution ofeach equivalent set (open or closed, sliding or blocked):

The thermodynamic force related to sliding concerning a particular equivalent setis:

(iv) The model considers frictional non-sliding/sliding phenomena on mesocracklips on a macroscopic scale by an approach similar to that to damage. Although it iswidely employed in many models [HOR 83], [GAM 93], etc., the Coulomb's criterion


is not suitable in this context because of its micromechanical formulation. Thepertinent quantity governing sliding on an equivalent system k is the thermodynamicforce FYk (which can be physically interpreted as the sliding energy release rate). Oneassumes that the sliding criterion explicitly depends on the norm of the tangentialpart F^1* of the force F^k and on the normal strain vk.e.vk. Unlike Coulomb's law,the normality rule with respect to the function defining the reversibility domain hasbeen found to keep a strong physical sense: it indicates a connection between y andF^ indicating that sliding occurs in the crack plane (as long as damage axes do notrotate). The sliding convex reversibility domain hk can be written as:

where p is a friction coefficient in the sense meant by the above thermodynamicforce (tangential component) - normal strain relationship, and:

The normality rule gives:

2.3. Damage and sliding coupling

The both dissipative phenomena (damage and frictional sliding) describedindependently in the previous paragraphs may occur simultaneously under particularloading paths. One assumes that the splitting-like kinetics considered inParagraph 2.1. is still valid for closed sliding cracks even when they branch: after ashort transitional distance, cracks tend to grow perpendicularly to positive principalstrain direction (see, f. ex., [BAR 97]). The sliding evolution law needs a rewriting,especially when the principal axes of Dk rotate (for Dk-non-proportional loadingpaths): in this case, sliding tends to depart from the crack plane and thus the drivingforce for sliding has to incorporate not only the tangential part F7™ of Fyk but also afraction of the normal part FyNk. Let be the following partition of FYk:

Fk is the appropriate part of FYk to enter the expression of the criterion hk takinginto account Dk-axes rotation and avoiding discontinuities when cracks open[HAL 98]. Note that in the case of proportional loading paths (i.e. yk:Dk=0), the termFk reduces to F^.


The normality assumption leads to:

NOTE - It was obvious to the present authors that controlling the followingeffects: (i) damage-induced anisotropy, (ii) damage related volumetric dilatancy, (iii)damage related residual effects, (iv) rigorous 3D treatment of the unilateral problem,(v) idem for the frictional resistance and sliding effect for closed microcracksinvolving the dissipative coupling with damage, should first have been embracedwithin the framework of classical local approach. The non locality of constitutiveequations, which can be now postulated for particular purposeful aspects of themodel, would allow enlarging its domain of pertinence by e.g. casting the underlyinghypothesis of non interacting microcrack in the actual one and treat the problems ofclustering and related enhancement vs. shielding microcracks interactions. This isplanned as further work.

3. Numerical treatment

In order to treat complex boundary-value problems, an accurate numerical toolhas to be associated with the previous model. The strong non linearity of the damageand frictional sliding mechanisms requires a time integration algorithm for theevolution of the damage variable D and of the sliding variable Y- This sectionsummarises the local (i.e. for each integration point of a finite elementdiscretization) integration scheme for both evolution laws [4] and [7], After dealing

separately with damage and sliding mechanisms respectively, the coupling of both isconsidered.

3.1. Local integration for the damage model

Let I be the time interval [0,T], with the partition I = U= [t r , t r + 1]. Given the

mechanical state q, = (e^D^rA) at time tr and the prescribed strain increment Ae (suchas eT+1 = 8T + Ae, the integration problem amounts to calculating the statqr+i = (er+i,Dr+1,Y

kr+i,ar+1). Since only damage evolution is concerned in this paragraph,

Y* is considered constant (ykr = Y'H-I)- The tensors eî, Dr+1 and or+1 are determined by:

with G0 standing for relation [5]. This calculation comprises two steps,

(i) Elastic prediction: First, the increment is assumed elastic, i.e. AD = 0. Onechecks whether the mechanical state (e^Dr) meets the condition:

If [9] is satisfied, the elastic prediction coincides with the solution of theproblem. Then,

and CH-I is calculated by [8]. Otherwise, if f(er+i,Dr)>0, the mechanical state has to becorrected in order to determine the increment AD.

(ii) Non linear correction: The incremental formulation leads to the followingformulation for the damage evolution:

The increment AD depends on e+r+i and Dr+1, i.e. the value of e+ and D at the end

of the integration interval [tr,tr+i]. This assumption corresponds to a fully implicitintegration scheme, which is known for its unconditional stability [ORT 85]whatever Ae is. It is worth noting that unlike for most of elastoplastic models, theimplicit scheme is well adapted to the damage model presented in Section 2.1: infact, the damage multiplier increment AAD is obtained by solving the criterionf(e+

r+i.Dr+i) = 0 which reduces to a linear equation whose solution is:



3.2. Local integration for the sliding model

In this paragraph, damage is assumed constant (Dr+i = Dr). The sliding integrationfollows the same scheme as for damage.

(i) Elastic prediction: The step is assumed elastic (ykr+i = Yk

r)- The damageeigenvalues Dk

r+1 and eigenvectors vkr+i are known, so the value of Fk

r+i entering his given by:

Then the value of hk is checked:

If hkr+1<0, y'V+i = Ykr and the mechanical state is fully determined. Otherwise (if

hkr+|>0), the frictional sliding evolution undergoes the following correction.

(ii) Non linear correction: The sliding increment Ayk is obtained by solving thefollowing system:

Equations [10] and [11] stand respectively for equations [6] and [7]. Again, thissystem corresponds to an implicit integration scheme. But while the damageintegration reduced to a linear equation, the above system remains non-linear and itssolving necessitates a Newton-Raphson algorithm.

3.3. Fully coupled model

Both above dissipative mechanisms (damage and frictional sliding) mayoccur simultaneously along particular loading paths. The problem is then todetermine the coupled increments AD and Ayk simultaneously. The integration ishere facilitated by the low degree of effective connection between f on one hand andhk on the other: whereas hk is a function of D and y\ f only depends on D and theequation f = 0 can be solved without explicit reference to sliding. The generalalgorithm is as follows:

(1) The value of vk.e.vk of the normal strain for each equivalent microcracksystem is checked.


(2) If vk.e.vk>0, the corresponding system is open; sliding does not occur and ADis calculated as described in Paragraph 2.1.

(3) If vk.e.vk<0, the corresponding system is closed and may slide. Both criteria f<0, hk<0 are checked; AD and Ayk are calculated, if necessary, by solvingsuccessively f = 0 for the damage evolution and later hk= 0 for the sliding one.

NOTE- The numerical algorithm is apparently standard and this constitutesparadoxically a non negligible contribution: the model deals with two stronglycoupled dissipative phenomena (damage by mesocrack growth, frictional sliding)with only nine material constants. However, the particular structure of the equationsgoverning the evolution of the internal variables, optimised somewhat by themodelling procedure, allows a classical backward difference time integration schemeto be efficient enough in spite of the complexity of the mechanisms at stake.

4. Numerical example

This section presents an application of the model for structural analysis. Thethree major phenomena, i.e. degradation, unilateral effect and frictional sliding areillustrated. A numerical simulation of boundary-value problem requires an efficienttool including a reliable model (in the case of damage model with unilateral effect,great care must be taken of the continuity of the response) and of an efficientintegration scheme. With the implicit scheme used, the tangent operator has a greatinfluence on the time needed by the simulation.

4.1. Geometry and loads

The numerical test is carried out on a slab with a symmetrical double edge notch.The material constants are given in Table 1. This set has been identified for aFontainebleau sandstone. The geometry of the specimen is described in Figure 1.The structure is constrained against x- and y-displacements along the lower edge Ar

A2 and the lower half A2-B2 of the right-hand side. Stress is applied on the top faceAs-A4 and the upper half of the left-hand side A4-B4 via sheets considered to beinfinitely rigid that are stuck to the test specimen. The upper face A3-A4 ismaintained in the horizontal position and the left face A4-B4 in the vertical position.The test specimen is first subjected to a positive displacement of A3-A4, then acompressive force Pn is applied; finally, in addition to the compressive force, ashearing force Ps is applied. The slab is meshed by QUA4 elements under thehypothesis of plane stress. The mesh concerned appears in Figure 2: while relativelyrough, it has been refined in the critical areas, i.e. in the central band and around thenotches.


Figure 1. Geometry of the test specimen

X (MPa)

26250

H (MPa)

17500

a (MPa)

1 900

P (MPa)

-20 400

g (MPa)

-110

Co (MPa)

0.001

Ci (MPa)

0.55

B ( l )

0

p (MPa)

2500

Table 1. Constitutive parameters

4.2. Mesocrack growth

Figure 2 shows a damage map during the first stage of the loading history, i.e.tension (by displacement imposed) on A3-A4. More precisely, the damage presentedhere is the component Dyy of the damage tensor. Due to the strongly brittle behaviour ofsandstone, damage rapidly localizes around the notches for a low level of Dyy

(maximum about 0.12) and becomes quasi-negligible in the central slab section.

Figure 2. Dyy map


4.3. Unilateral effect

The unilateral effect is observed in Figure 3, (force Pn vs. difference of thevertical displacement of the two edge points in the right notch). After degradationin the first stage of the loading history, the unloading stage exhibits two majoreffects: first the appearence of residual strain-like quantity 5 for Pn = 0 and secondthe stiffening of the material caused by the mesocrack closure.

Figure 3. Pn vs. 8 (difference of the vertical displacement of the two edge points inthe right notch)


4.4. Frictional sliding

The first stage of the loading generates microcracks, principally concentratedaround the notches. After crack closure (stage 2), a shear loading is applied. Figure4a shows the intensity of the frictional sliding: even if numerous microcracks arelocated close to the notches, one observes a high density of sliding cracks within aband crossing the sample. Even the central zone, damaged to slighter degree thannear-notch zones, is affected by this effect. However, due to the very low level ofdamage, the incipience of frictional sliding does not notably influence thedistribution and the level of the stress (f.ex., Von Mises stress, Figure 4b): thedifference between the values of Von Mises stress with or without frictional slidingdoes not exceed a few percent. The damage localization phenomenon acts here as aninhibitor for the sliding mechanism. This precocious influence of localization iscorroborated by recent works [GIR 00]. Frictional sliding may influence moredrastically the stress distribution for more ductile materials such as some concretes.Further work (some of which being under way) deals with this subject.

Figure 4a. Sliding Yxy map Figure 4b. Von Mises stress map

5. Conclusion

The purpose of the model depicted in this paper is to provide the engineer with anefficient while physically motivated tool for structural analysis: it seems that areasonable compromise has been found between the pertinence of the 3D theoreticalformalism and its applicability for industrial boundary-value problems. The constitutiveequations and the required continuity of the stress-strain response stem from the tensorfunctions representation theory and the multilinear functions theory, the latter applied tomanaging unilateral effects linked to damage deactivation. Although two coupleddissipative phenomena - mesocrack growth and frictional sliding on closed mesocracklips - are considered, the low degree of numerical coupling between the respectiveequations governing these two mechanisms allows convenient algorithmic treatmentand FE implementation; an example of application of the model shows its capacity to


illustrate the mesocrack growth and the recovery of effective properties. Further worksattempt to clarify whether the localization effects are premature compared to intrinsicmaterial and structural response, i.e. whether they represent a specific excessive modeltendency to be amended. It could be done, f. ex., by introducing some rate-dependanceinto the model which would in this manner account for genuine viscosity of engineeringmaterials like concrete and would contribute as a regularizing factor for numericalcalculations. Another aim of prospective short term research is to quantify on abroader basis the effects induced by frictional sliding.

6. References

[BAR 97] BARQUINS M., CHAKER C., PETIT J.P., « Influence du frottement sur le branchementde fissures a partir de defauts obliques soumis a une compression uniaxiale », C.R. Acad.Sci. Paris, L 324, Sene lib, 1997, p. 29-36.

[BOE 78] BOEHLER J.P., « Lois de comportement anisotrope des milieux continus », J. Meca.,Vol. 17, 1978, p. 153-190.

[CUR 95] CURNIER A., HE Q., ZYSSET P., « Conewise linear elastic materials », J. Elasticity,Vol. 37, 1995, p. 1-38.

[DRA 94] DRAGON A., CORMERY F., DESOYER T., HALM D., « Localized failure analysis usingdamage models », [dans] Localization and bifurcation theory for soils and rocks, EdChambon R. et al., Balkema, Rotterdam, 1994, p. 127-140.

[GAM 93] GAMBAROTTA L., LAGOMARSINO S., «A microcrack damage model for brittlematerials », Int. J. Solids Structures, Vol. 30, 1993, p. 177-198.

[GIR 00] GIRAUD F., Etude de validation de la loi de comportement elasto-endommageabledu Pr Dragon, IFP-Report 53 185, 2000.

[HAL 96] HALM D., DRAGON A., « A model of anisotropic damage by mesocrack growth;unilateral effect », Int. J. Damage Mech., Vol. 5, 1996, p. 384-402.

[HAL 98] HALM D., DRAGON A., « An anisotropic model of damage and frictional sliding forbrittle materials », Ear. J. Mech. A/Solids, Vol. 17, 1998, p. 439-460.

[HAL 01] HALM D., DRAGON A., « Modelisation de I'endommagement par mesofissuration dugranite », Revue Francaise de Genie Civil, soumis.

[HOR 83] HORII H., NEMAT-NASSER S., « Overall moduli of solids with microcracks: load-induced anisotropy », J. Mech. Phys. Solids, Vol. 31, 1983, p. 151-171.

[KAC 92] KACHANOV M., « Effective elastic properties of cracked solids.-critical review ofsome basic concepts », ASME Appl. Mech. Rev., Vol. 45, 1992, p. 304-335.

[ONA 88] ONAT E.T., LECKIE F.A., « Representation of mechanical behavior in the presenceof changing internal structures J. Appl. Mech., Vol. 55, 1988, p. 1-10.

[ORT 85] ORTTZ M., POPOV E.P., «Accuracy and stability of integration algorithms forelastoplastic constitutive relations », Int. J. Num. Meth. Eng., Vol. 21, 1985, p. 1561-1576.


Chapter 7

Energy Dissipation RegardingTransient Response of ConcreteStructures: Constitutive EquationsCoupling Damage and Friction

Frederic Ragueneau and Jacky MazarsLaboratoire de Mecanique et Technologic, ENS de Cachan, Universite Pierre etMarie Curie, Cachan, France

Christian La BorderieLaSAGeC, ISA du BTP, Universite de Pan et des Pays de I'A dour, Anglet, France


1. Introduction

Transient non-linear computations of structures need the use of energydissipation tools for both physical and numerical reasons. One of the majordrawbacks in such analysis lies in the expression of the damping matrix. Severalkinds of methods can be used to achieve that purpose, such as viscous or hysteretic[BAT 82]. A more realistic approach consists of better modelling of the internaldissipation. Recent experiments on reinforced concrete mock-ups subjected toseismic loading permitted appreciation of the strong interaction between state offailure and resulting global damping [QUE 98]. Most of the constitutive models areable to reproduce realistically the behaviour of concrete in the non-linear range,based on damage mechanics, on plasticity theory or using the microplane concept[JU 89] [KRA 81]. They often ensure predictive computations in the static case butdo not easily take into account a main cyclic characteristic: the influence ofheterogeneities and roughness of the crack surfaces. At a fixed level of damage,concrete still exhibits dissipation due to the frictional sliding between cracksurfaces. This property can be experimentally observed for a specimen during cyclicsolicitations through the hysteresis loops.

A new constitutive relation for concrete material including residual hystereticloops at a fixed level of damage is proposed. Derived from thermodynamics, it allowscoupling of the state of cracking with the hysteretic dissipation induced by the cracksurfaces sliding. Based on damage mechanics, a particular Helmholtz free energyallows introduction of a coupling of the level of damage in one direction to a fric-tional stress. The main assumption postulated to describe the hysteretic behaviour isthat cracks surfaces created after fracture will not open anymore following a perfectsurface but will slide depending on their roughness. This phenomenon induces theoccurrence of a sliding stress which prevents the crack from going on opening easily.In another sense, the consumed energy during cracking is not entirely dissipated butpart of it is stored in the sliding potential. This frictional stress (os) is assumed tohave a plasticity-like behaviour associated with a non-linear kinematic hardening. Aparticular dissipative potential allows the description of dilatancy, fundamentalfeature of geo-materials like concrete, sand or rocks.

This model has been implemented in the finite element code EFICOS based on amultilayered beam kinematics approach. Concerning the constitutive equations, wechose the implicit return mapping algorithm. This approach of structural compu-tation allows one to take into account at the local level refined material models,avoiding at the same time heavy analysis. In order to validate our approach ofstructural dissipation, comparisons between computations and experiments aremade. This communication focuses on the 3D expression of the local model and onnumerical comparisons of energy dissipation at the structural level pointing out theinfluence of frictional sliding regarding global damping.

Energy Dissipation Regarding Concrete Structures 113

Initially introduced by Kachanov [KAC 58] for creep failure problems,damage mechanics formulation requires the addition of a new internal variable inorder to represent the macroscopic loss of stiffness [LEM 90]. This can be achievedin many ways. The classical one is to relate the damaged material's and the intactmaterial's elastic properties. Defining a damage variable requires reducing the rankof damage operator while maintaining as well as can be the experimental propertiesof the material. In order to reach this objective, the effective stress concept isintroduced. It induces a relation between the stress tensor and the effective stresstensor through a strain equivalence (the effective stress which, when applied to theundamaged material, produces the same strain).

Therefore, in the basic form, a scalar variable can be used for the sake ofsimplicity. Numerous authors have proposed several expressions of different ranksfor the damage variable. The fourth-order tensor was suggested by Chaboche.Second-order tensors are more frequently introduced [DRA 94]. In this case, aproblem exists in that the symmetry of the elasticity operator is no longer ensured, asit depends on the way one defines the previous relation [MUR 78]. Cordebois[COR 79] postulated a state potential which depends only on the effective stress.Therefore, the symmetry of the elasticity operator is obtained through this energyequivalence.

The pursuit of a physical and realistic description of the oriented crack growth inconcrete without neglecting the simplicity requirement led to a second-orderdamage tensor formulation. In order to make the subsequent numericalimplementation of the model in a finite element code easier, Helmholtz's strain-based free energy has been chosen. Considering a particular definition for thedamage variable d, an effective strain tensor is defined on the principal axis of thedamage tensor as follows,

Where e is the effective strain tensor, e the second order strain tensor (symmetricpart of the displacement gradient field) and d the symmetric second order tensorrelated to damage phenomena. Because of symmetry conditions on the resulting stresstensor, the expression in the previous relation denotes a symmetrization. This effectivestrain, introduced directly into the state potential, allows description of an elasto-damage material exhibiting orthotropic cracks:

2.1. State potential

2. Crack growth and damage



with (j, and A,, the Lame coefficients defined for the undamaged material, p isthe material density and \f/d is the Helmoltz free energy.

2.2. Damage criteria and evolution laws

Based on experimental investigations, damage for brittle materials such asconcrete is governed principally by their tensile behavior. To take into account thisasymmetry, two damage tensors must be introduced. The splitting between thetensile and the compressive damage tensors is achieved through the sign of thesliding strains (defined in the next section) expressed in their respective principaldirections:

(tensorial functions), A is the tensor of strain eigenvalues and P the transformationmatrix. H is the Heaviside function.

The level of damage is governed by the value of positive strains. The evolutionequation assumed to be expressed in its incremental form as follows, is written inthe principal axes of the incremental strains:

Bt is a material parameter driving the slope of the softening branch. Theassociated damage criterion is also expressed in the principal axis of the straintensor:

edo is the initial tensile yield strain, usually equal to 1.10"04. K"(£,) is thehardening variable. Such a criterion is similar to the so-called St-Venant's criterionin the principal stress space.

The use of a damage evolution law based on the state of total strain is consistentwith the meaning of the damage variable expressed in the elasticity law.

Compressive damage in a particular direction is considered only as aconsequence of the tensile behavior of the material and, therefore, is taken equal to afunction of the state of tensile cracking along the orthogonal directions.


ft is a material parameter connecting the damaged Young's moduli for twoorthogonal directions. The comparison of apparent Young's moduli in thelongitudinal and radial directions allows the measurement of p.

3. Inelasticity and friction coupled to damage

3.1. State potential

The main assumption used to describe the hysteretic behavior is that crack lipsarising from fracture do not open further along a perfect surface but slide dependingon their rugosity. This phenomenon induces a sliding stress which prevents thecrack from continuing to open easily. In other words, the energy consumed duringcracking is not entirely dissipated but part of it is stored in the sliding potential.Thus, it is through the damage variable that this energy shift can be obtained.Each dissipative nonlinear phenomena needing its own internal variable, a measureof the sliding with friction will be defined thanks to a particular second order newvariable: the sliding strain es. Following the same methodology as in section 2,sliding is integrated into the behavior through an equivalent strain which couplesdamage and elasticity of the sliding surface.

In that way, the total state potential is written as follows,

a is the internal variable associated with the kinematic hardening phenomenonand b is a material parameter.

One can easily recognize a classical elasto-damage coupling and a new termallowing the energy to shift from the elasto-damageable part to the frictional slidingpart. The coupling between sliding and cracking is made possible thanks to thepresence of the damage variable as a multiplier in the second element of the right-hand side of [8].


3.2. State laws

In order to define the state laws, the model has to be thermodynamicallyadmissible: it must comply with the positiveness of the dissipated energy. Startingfrom the Clausius-Duhem inequality, and assuming that the state potential islinearized around the current value of every state variable, the state laws areexpressed as follows:

The stress tensors can be derived as

And the sliding stress tensor, associated to the sliding strain:

The back stress is defined as:

We can observe that the total stress is divided into two parts: a classical elasto-damage component and a sliding component. Damage is classically controlled bythe elasto-damage stress and the sliding strain is linked only to the sliding part of thestress. This kind of partitioning, in conjunction with the two failure surfaces, allowsthe description of a hysteretic behavior at a fixed level of damage. Details of thecomplementary and evolution laws will be developed in the next section. At thisdescription level, such an approach could be compared to multi-surface modeling[MRO 67], except for the fact that the surfaces are not expressed in the same space(strain space for damage and stress space for sliding). The sliding and plastic strainsbeing different, the thermodynamic forces associated with the total strain and thesliding strain are different. Such a formulation differs greatly from the classicalplasticity-damage coupling. This choice of introducing damage into the slidingstress is guided by the idea that all inelastic phenomena in concrete result from thecracks' growth. The thermal aspects are presently ignored.

3.3. Sliding criteria and plastic potential

The sliding part of the constitutive relation is assumed to represent a plasticity-like behavior. In order to reproduce the hysteresis loops, nonlinear kinematic


hardening is considered. Initially introduced by Armstrong & Frederick [ARM 66]and recently developed further [CHA 93], it allows the formulation to overcome themajor drawback of Prager's kinematic hardening law, i.e. the linearity of the statelaw defining the forces associated with kinematic hardening. The nonlinear termsare added in the dissipative potential. The sliding criterion takes the classical form:

J2\0S—X) i-e. Von Mises' equivalent stress was chosen to a firstapproximation in order to keep simplicity and adequacy to the classical numericalalgorithm for constitutive laws implementation. The specific aspects of geo-materials non-linear behaviour have been introduced in the plastic potentials

Classical plasticity, in order to govern the evolution of the internal variables,requires the definition of a dissipative potential. The expectation of nonlinearkinematic hardening imposes the use of a non-associated flow rule:

with /! = o Tr[as], the first invariant of the sliding stress tensor. It enables one

to take into account the dilatancy phenomenon observed experimentally ongeomaterials such as concrete or rocks, a and c are material parameters. As a resultof the normality rules, the evolution laws of the internal variables are expressed asfollows, thus ensuring that the dissipation is positive:

A is the plastic multiplier, determined by the consistency condition.

3.4. Unilateral effects

In the case of cyclic loading, a model has to take into account the crack closurephenomenon which generates the concurrent stiffness recovery. Some damagemodels have been extended to take into account the crack closure conditions byinducing different behaviors in tension and in compression. A general and rigorousframework concerning the unilateral condition, based on a particular decompositionof the elastic energy introducing a deviatoric behavior and a spherical one, is givenby Ladeveze [LAD 83]. Other kinds of anisotropic damage models based on amesocrack description can also deal with the crack closure condition [HAL 96].

In our case, the need to couple damage with sliding of the crack surfaces led usto the introduction of a second-order damage tensor. This was achieved through a


particular effective strain [1] [7] introduced into the elastic free energy for thedamage part as well as for the sliding part. This enables a physical description of thematerial behavior (oriented cracking, inelasticity, nonlinear unloading) but is not ingood agreement with the unilateral condition. This is due to the presence of tensorialdamage in the volumetric part of the stress [9]. Concerning the radial loading casesin our analysis, a simplification of the previous model has to be achieved. The sameformulation is retained, but an isotropic evolution of the damage tensors is nowassumed. For the damage part, the simplified model boils down to two scalardamage variables allowing one to account for stiffness restoring after tensile degrada-tion keeping at the same time the continuity of the stress-strain law. Figure 1 showsthe response of the model subject to crack closure following a tensile loading path:

Figure 1. Compression/tension response. Unilateral effect and stiffness recovery.The thickness of the hysteresis loops is proportional to the state of cracking

4. Numerical implementation

4.1. Finite element code: EFiCoS

The choice of using a multilayered f.e. configuration combines the advantage ofusing beam type finite elements with the simplicity of uniaxial behavior. Each finiteelement is a beam which is discretized into several layers. The basic assumption is thatplane sections remain plane (Bernoulli's kinematic) allowing one to consider auniaxial behavior of each layer. The local constitutive equations are integrated foreach layer of a cross section. Concerning reinforced concrete structures, reinforce-ment steel bars are introduced with special layers, the behavior of which is a combi-nation of those of concrete and steel. A mixing homogenized law is considered.


For each element a secant matrix K is assembled from the relation between Oand £ . The non-linear behaviors appear in the second member of the equilibriumequation and are subtracted from the vector of generalized forces:

in which U is the vector of nodal displacements, Fext is the vector of externalforces and Fin is the vector of inelastic forces. Such a formulation, based on theinitial secant stiffness matrix algorithm, is well suited to our modeling. Afterreaching the peak-load, it prevents the iteration matrix from becoming singular.Concerning dynamic analysis, the same framework is used. The seismic loading isapplied by the mean of an accelerogram at the basis of the structure. A doubleintegration of this signal allows one to determine the ground displacement. In such away, the equations of motions are solved in the global coordinates and allows one todeal with computations needing different accelerograms exciting different parts of asame structure at the same time. For stability and precision reasons, a classicalNewmark algorithm has been implemented to solve the equation of motion [NEW59]. The expression of the discretized displacements and celerities are derived as:

Choosing 7 = 1/2 and )3 = 1/4 prevent us from any numerical dissipation andensures an unconditional scheme stability. This expressions are introduced in theequations of motions at step t + A? imposing an implicit integration scheme.

4.2. Constitutive law implementation

By defining two different surfaces, one can integrate the damage tensor undertraction explicitly. For uniaxial compression, the algorithm becomes implicit due to thedependence of the compressive damage on the radial extension, in which case a kindof plane stress procedure has to be introduced. Concerning the sliding stress, a classicalimplicit analysis has to be carried out. Among the different methods available for thispurpose (Euler's backward or mid-point rules algorithm solved by an iterative Newtonmethod), we chose the classical form of the so-called "return mapping" algorithm[ORT 86]. Indeed, it ensures convergence in the most efficient way. More detailsconcerning the numerical implementation can be found in [RAG 00].


5. Applications

5.1. Response to uniaxial stress loading

The analysis is performed at the material level. Figure 3 represents the simulationof the uniaxial compression test. One can observe that the model is able to describe thevolumetric response of the material satisfactorily. The material parameters used for theanalysis are: E = 36 000 MPa, v = 0.24, e^ = 1.10"*, Btdbttt = 9 000, Btnduced = 300,P = 12, adirea = 5.0 10" MPa1, bdiim = 1.0 10+1° MPa, ainduce= 2.0 10*" MPa1,binduced = 1.0 10+1° MPa, c = 0.18. The direct and induced qualification are related to thecracking mode: 'direct' means that frictional stress and damage are aligned with thedirection of loading.

The two curves in Figure 1 show the ability of the model to describe thehysteresis loops under traction and compression loading paths. The hystereticdissipation capability of the model can be illustrated by plotting the absorbed energyof an unloading tensile loop against the value of tensile damage in Figure 3. One caneasily appreciate the effect of the coupling between the state of damage and thesliding stress. Despite the lack of physical shape of the yield surface for the 2Dcompression state due to the use of a St Venant's-type criterion, the use of aninduced and direct damage evolution law allows a good modeling of a 3D state ofconfinement. Figure 3 shows such an effect induced by a lateral pressure appliedproportionally to the longitudinal one (radial path).

Figure 2. Dissipated energy versus damage-radial confinement effect


Figure 3. Compression test simulation, longitudinal, orthogonal and volumetricstrain

5.2. Structural applications: seismic case study

This section is dedicated to the contribution of this local material model to theglobal dissipation at the structural level. A dynamic example of a reinforcedconcrete mock-up subjected to an earthquake loading confirms the relevance of ourlocal approach in modeling damping in the nonlinear range. The main purpose ofthe CAMUS experimental program [QUE 98J is to demonstrate the ability ofreinforced concrete bearing walls to sustain seismic loading. To reach this goal, aone-third scale model was tested on the shaking table of CEA. This mock-up wascomposed of two parallel walls connected by 6 square slabs. A heavily reinforcedfooting allowed the mock-up to be anchored to the shaking table. The mock-upplans are shown below:

The mock-up is loaded through horizontal accelerations parallel to the walls.The presence of steel bracing systems at each level disposed perpendicularly to theloading direction prevents any torsion modes occurrence. The accelerograms are

modified in time with a ratio of 1/V3 to take into account the similarity rules. Twotypes of accelerogram are imposed: Nice SI for the far field type earthquake andSan-Francisco (earthquake happened in 1957) for the near field one. The responsespectra may show the difference of these two kinds of earthquakes: on one handNice is very rich in terms of frequencies and on the other hand, San Francisco has athicker effective band width of high accelerations. The complete experimentalsequence, as shown in Figure 5, is: Nice 0.25 g, San Francisco 1.13 g, Nice 0.4 g andNice 0.7 Ig.


A simple modeling of the boundary conditions at the base was obtained using ahorizontal bending beam instead of assuming a perfect anchorage introducing theanchorage and contacts defects. The knowledge of the stiffness measured from theshaking table helped us adjust the vertical stiffness as well as the rotational stiffnessof the elements along the boundary. This kind of modeling allowed us to take intoaccount 2 eigenmodes, which is the minimum for a structure such as CAMUS wherethe second vertical mode plays a major role.

Figure 4. CAMUS mock-up

Figure 5. Accelerograms: experimental complete sequence

No structural Rayleigh damping was introduced into the analysis. The globalbehavior of the mock-up was well-reproduced. The benefit of such a materialmodeling is made more obvious by focusing our comparisons on energy balance.


Figures 6 and 7 show comparisons between the new model with no viscous dampingand a classical damage model [LAB 91 ] with Rayleigh damping in terms of energyequilibrium.

Figure 6. Computations with a classical damage model: dissipation through a globalviscous matrix

Figure 7. Hysteretical energy dissipation: computations without any viscousRayleigh damping


We can notice that for a "traditional" model of damage, the external Rayleighdamping takes part in height of 60 % to the total dissipation of energy. In the case ofthe second model, all energy is dissipated within the material. From a quantitative pointof view, no adjustment is necessary at the level of the structural analysis and from aqualitative point of view, dissipation is relocated in the zones of strong degradations(in the three lowest level) and no more distributed on the whole of the structure.

6. Conclusions

Analyzing a specific case study (the response of a reinforced concrete modelsubjected to seismic loading) allowed us to point out the major features of local non-linear mechanisms which should be integrated in the analysis. Local constitutiveequations based on damage mechanics for concrete have been developed andimplemented in a finite element code dedicated to civil engineering large scalecomputations. The use of the simplified finite element method, allowing parametricalstudies, helped us to understand and analyse the experimental responses by pointingout the influences of physical material features such as frictional sliding betweencrack faces This approach of damage mechanics coupling cracking and frictionalsliding emphasizes the importance of a refined material modeling with regards to theaccuracy and the predictive ability of structural computation tools. The effect ofmodelling the hysteresis loops was analyzed through its contribution to overalldamping. The use of this kind of local model for dynamic and, in particular, seismicloading may, in the non-linear range, make the explicit expression of an oftenarbitrarily-defined damping matrix unnecessary. The role played by this globalviscous damping matrix in the analysis is considerably lessened. Moreinvestigations in the identification of material parameters at the specimen scalesubjected to complex loading paths would allow us to treat the cases of 3D structuresbearing complex solicitations combining shear and torsion effects.

7. References

[ARM 66] ARMSTRONG P.J. and FREDERICK CO., A Mathematical Representation of theMultiaxial Bauschinger Effect, G.E.G.B., Report RD/B/N, 731, 1966.

[BAT 82] BATHE K.J., Finite element procedures in engineering analysis, Prentice Hall (eds),Inc. Englewood Cl.iffs, New Jersey, 1982.

[CHA 93] CHABOCHE J.L., "Cyclic Viscoplastic Constitutive Equations, Part I: AThermodynamically Consistent formulation", J. Appl. Mech., Vol 60, p. 813-821, Dec.1993.

[COR 79] CORDEBOIS J.P. and SIDOROFFF., "Damage Induced Elastic Anisotropy", ColloqueEuromech, 115, Villars de Lans, p. 761-774, 1979.


[DRA 94] DRAGON A., CORMERY T., DESOYER T. and HALM D., "Localised Failure AnalysisUsing Damage Models", In Chambon R. et al. (eds), Localisation and Bifurcation Theoryfor Soils and Rocks, p. 127-140. Rotterdam:Balkema, 1994.

[HALM 96] HALM D. and DRAGON A., "A model of anisotropic damage by mesocrackgrowth; unilateral effect", In. J. Damage Mechs., Vol. 5, Oct. 1996, p. 384-402, 1996.

[JU 89] Ju J.W., "On energy-based coupled elastoplastic damage theories: constitutivemodelling and computational aspects", Int. J. Solids and Structures, 25(7), p. 803-833,1989.

[KAC 58] KACHANOV L.M., "Time of the rupture process under creep conditions", Izv. Akad.Nauk. S.S.R., Otd. Tekh. Nauk., n° 8, p. 26-31, 1958.

[KRA 81] KRACJINOVIC D. et FONSEKA G.U., "The continuous damage theories of brittlematerials, Part I and II", J. ofAppl. Mech., ASME, Vol. 48, p. 809-824, 1981.

[LAB 91] LA BORDERIE CH., Phenomenes unilateraux dans un materiau endommageable.-modelisation et application a 1'analyse de structures en b&on, Ph. D. thesis, Univ. ParisVI, 1991.

[LAD 83] LADEVEZE P., "On an anisotropic damage theory", Proc. CNRS Int. Coll. 351,Villars-de-Lans, Ed. by J.P. Boehler, p. 355-363, 1983.

[LEM 90] LEMAITRE J. and CHABOCHE J.L., Mechanics of solids material, CambridgeUniversity Press, 1990.

[MRO 67] MROZ Z., "On the description of anisotropic workhardening", J. Mech. Phys.Solids, 15, p. 163, 1967.

[MUR 78] MURAKAMI S. and OHNO N., "A constitutive equation of creep damage inpolycristalline metals", I.U.T.A.M. Colloquium Euromech 111, Marienbad, 1978.

[NEW 59] NEWMARK N.M., "A method of computation for structural dynamics", A.S.C.E.Journal of Engineering Mechanics Division., Vol. 85, p. 67-94, 1959.

[ORT 86] ORTIZ M. and SIMO J.C., "An analysis of a new class of integration algorithms forelastoplastic constitutive relations", Int. J. Numer. Meth., Eng. Vol. 23, p. 353-366, 1986.

[QUE 98] QUEVAL J.C., COMBESCURE D., SOLLOGOUB P., COIN A. et MAZARS J., "CAMUSexperimental program. In-plane tests of 1/3 scaled R/C bearing walls", Proc. Xlth ECEE-98, CD-ROM eds Bisch P., Labbe P. et Pecker A., Paris/CNIT La defense, 1998.

[RAG 00] RAGUENEAU F., LA BORDERIE Ch. and MAZARS J., "Damage model for concretelike materials coupling cracking and friction, contribution towards structural damping:first uniaxial application", Mechanics Cohesive Frictional Materials (to appear), 2000.

Chapter 8

Numerical Analysis of Failure in SheetMetal Forming with ExperimentalValidation

Michel Brunet, Fabrice Morestin and Helene WalterLaboratoire de Mecanique des Solides, Villeurbanne, France


Failure in Sheet Metal Forming 129

1. Introduction

The increasing use of sheet-metals with high elastic-limits and with a limitedformability such as aluminium or titanium alloys leads to new problems in thesimulation of the sheet forming processes of these materials. In the experimentsconducted by the authors to determine their Forming Limit Diagrams (FLD), it iscurrently observed that necking is immediately follows by failure and crack alwaysappears. Moreover, the necking of the sheet is hardly visible and consequently,plastic-instability theories alone fail to predict the failure of these sheet-metals.There are several ways to achieve analysis of failure occurrence in sheet-metalforming. One way consists of carrying out a conventional F.E. simulation and bypost-processing the F.E. results, using an experimental necking-failure curve to detectthe zones where risks of cracks can occur.

On the other hand, a large number of macroscopic fracture criteria for failurewhich occurs after necking have been evaluated by many authors consisting ofproducts, integrals and sums of macroscopic stresses and strains. To determine thevalues of these criteria at the onset of failure, both experiments and F.E. simulationsare needed. When applying these criteria, it was found that the main factor affectingthe accuracy is the mode in which failure takes place, mainly under deep-drawing orunder stretching conditions. The equivalent Mises-stress was judged best for theprediction of both deep-drawing and stretch-drawing cracks but the locus ofmaximum equivalent stress does not necessarily coincide with the locus of failurein the sheet. Moreover, the thickness distribution may also indicate the wrong locusof failure since this parameter is operation dependant and there is no materialdependant critical sheet thickness reduction.

Also there is a need in the simulation process to achieve better localization of theonset of failure. This can be expected by the coupled approach where the damageprocess is incorporated into the constitutive relations and necking criterion. Manyinvestigations have shown that ductile fracture involves four successive damageprocesses which are the nucleation of voids from inclusions, void growth, voidcoalescence and cracking propagation. One constitutive equation to account forthese processes is Gurson's model [GUR 77], which was derived in an attempt tomodel a porous isotropic plastic material containing randomly disposed voids. Assuggested by Doege and co-workers [DOE 93], we have already extended theGurson model to anisotropic matrix behaviour and implemented with our shellfinite-elements suitable for simulating sheet-metal forming processes [BRU 96,97].In these papers, the onset of necking may be found numerically by mathematicalconsiderations due to the fact that the strain state gradually drifts to plane strain afterthe onset of load instability [BRU 97,98].

In this paper where a titanium alloy sheet is tested, a refined approach ispresented due to the fact that failure occurs just after a very small necking. If theGurson damage model demonstrates the softening effect of the material, the modelitself does not constitute a fracture criterion. Therefore, a criterion of void


coalescence which determines a critical porosity has to be used to simulate theinitiation of material failure. Tvergaard and Needleman [TVE 84] have introducedthe so-called critical void volume fraction at which voids coalesce, which in thepresent study is first determined by fitting numerically the load-displacement orengineering strain curve of the tensile test. However, the critical void volume is notunique. It depends on the choice of void nucleation model and the correspondingparameters. Moreover, to the authors' knowledge there is no sound theory or methodat present available in the literature for the choice of void nucleation model. Assuggested by Zhang and Neimi [ZHA 94, 95], a second method to determine thecritical porosity is tested by using the modified Thomason plastic limit-load modelof internal necking [THO 85,90]. Fully compatible with the Gurson damagemodel, the main feature of the Thomason void coalescence model is that thematerial failure initiation is a natural process where the void coalescence is notneeded to be fitted beforehand. The finite element analysis of necking-failure of ourNakazima tests on a titanium sheet-alloy for different strain-paths will show thepotential advantage of this criterion.

2. Damage model

The coupled approach where the damage process is incorporated into theconstitutive relations and necking criteria is expected to achieve better localizationof the onset of necking and failure. For example, it is frequently observed in actualproduction processes that steel and aluminium sheets exhibit different forming limitcurves even if both have the similar n-hardening coefficients.

2.1. Extension of Gurson-Tvergaard damage model

Failure in metal forming is mainly due to the development of ductile damage.Needleman and Triantafyllidis [NEE 78] found that the predictions of forming limitfor voided sheets based on the Gurson damage model are qualitatively in accord withexperimental results. By finite element analysis using a membrane theory, Chu[CHU 80a] examined the effects of void growth on forming limit under punchstretching, also Chu and Needleman [CHU 80b] examined the influence of voidnucleation on the forming curves. A primary extension of the Gurson-Tvergaardmodel has been used in the context of plane-stress and orthotropic materialsimplemented in shell finite elements in order to simulate our Marciniack tests byBrunet, Sabourin and Mguil-Touchal [BRU 96]. The model of Gurson is based on theobservation that the nucleation and growth of voids in a ductile metal maymacroscopically be described by extending classical plasticity to cover effects ofplastic dilatancy and pressure sensitivity of plastic flow. Tvergaard [TVE 81,82] hasproposed a first modified form of the original Gurson's yield criterion by introducingthree coefficients q^q^qj in order to better fit the corresponding three-dimensionalfinite element solutions:


In Eq. (1) for micro-voided material, f*(f) is the damage function of the micro-void volume fraction or porosity f and Tvergaard's constants q j = 1.5, q 2 = 1 andq3 =q^ as coefficients of the void volume fraction and pressure terms, instead of

Qi = Q2 = <b = 1 in the original Gurson model. ay describes the hardening of the

fully dense matrix material by ay =h (e p ) and p is the macroscopic hydrostatic

stress, q is the effective Von-Mises stress of the macroscopic Cauchy stress tensor CJwhich is expected to be replaced here by the quadratic orthotropic such as Hill[HIL48] effective stress or non quadratic as: Hill [HIL 79,90] or Barlat and Lian[BAR 89]. The 3/2 factor in the 'cosh' term stands for isotropic material; it must beslightly modified here in order to be consistent with the original paper of Gurson[GUR 77]. Liao, Pan and Tang [LIA 97] have established the modified yieldcriterion for porous sheet metals containing spherical voids based on Hill's quadraticyield criterion to describe the matrix normal anisotropy and planar isotropy. Theclosed-form yield criterion is a function of the anisotropy parameter r whichrepresents the mean ratio of the transverse plastic strain rate to the through thicknessplastic strain rate under in-plane uniaxial loading conditions. For all possible plane-stress conditions, the anisotropic yield function is expressed as:

As anisotropic yield criterion is approximate in nature, it is possible to maintainthe Tvergaard coefficients in Eq. [2] but in the following it is assumed that qt = 1.45.In this case, it is worth noticing that the modified Gurson model only differs from theoriginal one Eq. [1] by:

In sheet metal forming applications, we are generally concerned with plane stressconditions. Consider x, y to be the 'rolling' and 'cross' directions in the plane of thesheet, z is the thickness direction. Based on the Hill quadratic yield function, theyield function q is defined in the orthotropic axes x,y as:

where:


The parameters f, g, h and n are the dimensionless Hill material coefficientswhich are defined in terms of the Lankford coefficients r 0, r45, r 90 as:

The Lankford parameters are determined by three experiments in the variousdirections as pointed out by their different indices. If f = g = h = 1/2 and n = 3/2, theVon Mises isotropic yield function is recovered. The equivalent stress function qgives the current size of the yield surface but due to the anisotropy, the directEulerian constitutive law based on this criterion is not objective. In order to assurethe objectivity, rotating frame formalism is applied. The axes of orthotropy ofthe Hill criterion can be updated by a rotation which can be chosen as the materialspin rate co (co-rotational stress-rate) or from the polar decomposition F = RU(Green-Nagdi stress-rate). Since the elastic strain are assumed to be small and frompractical sheet forming applications, the differences between these differentrotations are very small.

The flow rule is derived from the yield potential Eq. [1] or [2], the presence ofthe hydrostatic pressure in the yield function results in non-deviatoric plastic strains:

The hardening of the fully dense matrix material is described through a = h(ep) •The evolution of ep is assumed to be governed by the equivalent plastic workrelation:

2.2. Damage evolution

The damage model takes into account the three main phases of damageevolution: nucleation, growth and coalescence:


The micro-void volume fraction increment due to nucleation may be expressedby the normal distribution model of Chu and Needleman [CHU 80b]:

In this strain controlled nucleation model, the normal distribution of the

nucleation strain has a mean value £N , a standard deviation SN and fN is the

volume fraction of voids which could nucleate if sufficiently high strains arereached. With the normal distribution, the major part of voids nucleates between theeffective plastic strain values: e p = e N - S N and e p = e N + S N - However, a

continuous nucleation model with one constant can also be chosen in place orcombined with Eq. [9].

Growth of existing voids is based on the apparent volume change and law ofconservation of mass and is expressed as:

Finally, the modification of the yield condition to account for coalescence andfinal material failure is introduced trough the function f*(f) specified by Tvergaardand Needleman [TVE 84]:

With the accelerator ratio:

f * = i/q{ is the ultimate value of f* at ductile rupture, fc is a critical value of the voidvolume fraction when the coalescence of micro-voids occurs and the stress-carryingcapability of the material sharply drops and finally, ff is the void volume fraction forwhich the stress-capability totally vanishes (final failure).

The analysis of equations [8] to [13] shows that the material damage behaviourdepends at least on the values assumed by four to six damage parameters, dependingon the choice of the nucleation model. Consequently, the predictive capability of thedamage mechanics model depends on the effectiveness of finding theseparameters. A optimisation procedure is needed to match the experimental andnumerical finite element results as regards the loads vs. displacement curve in a


tensile test. In this paper, such a first choice has been carried out by means of aninverse identification approach which will be described in paragraph 4. However,whether the critical porosity is a material constant, or whether the critical porosity isindependent of the stress state, is questionable. Moreover, we have found that if thevalue of fc is dependent on the choice of the nucleation model, then the set of

damage parameters must be considered as a whole in this case and not as a set ofindependent material parameters.

3. Computational aspects

3.1. Explicit solution procedure

The four node quadrilateral shell element with five degrees of freedom per nodeand plane-stress state is adopted for the spatial discretization of the sheet. Thethrough thickness shearing stresses are also taken account and in order to avoid thewell known shear locking of this kind of element, the assumed strain field method ofDvorkin and Bathe is used [DVO 84]. A large number of analyses have shown thatsheet forming processes can be analysed successfully by both the implicit staticmethod and explicit dynamic procedure if the latter is run at a relatively low speed(<10 m/s). With the use of a lumped mass matrix, the advantages of the explicitdynamic algorithm is that the stiffness matrix does not need to be formed and thecontact conditions are modelled accurately in a simple manner because of therequirements of small time steps. Moreover the material behaviour can be complexwhich is the case with internal damage variable leading to softening of the material.

3.2. Integration of constitutive equations

It is known that one of the best algorithm for integrating constitutive equations isthe backward Euler or implicit scheme. However, in the case of the plane-stresscondition, the out of plane component of strain is not defined cinematically and mustbe added as an extra unknown in the local Newton iteration scheme. This fact and thepresence of 'cosh' terms in the yield function and flow rule may lead to numericaldifficulties when the damage variable increases rapidly. The authors have chosen asub-stepping scheme on the modified Euler algorithm which incorporates errorcontrol. This approach is suitable with explicit dynamic analysis since it takesadvantage of the small time step required by the overall stability limit.

Then on each sub-step, the following set of incremental forms of equations areused to compute the plane-stress increments:


where £iae} is the elastic stress increment vector and [D] the elastic (3 x 3) matrix

satisfying the plane-stress assumption. From the normality of the flow rule of plasticstrain increments, the plastic multiplier dA, is eliminated with the following set ofequations:

Eq. [1] and Eq. [3] are used to yield:

Defining the gradient vector {a} so that:

where it is found that for plane-stress state:

The equivalent plastic work Eq. [7] gives the effective strain increment:

Use of h' the hardening modulus of the matrix in Eq. [18] and in Eq. [19] leads to:

The plastic out-of-plane strain increment can be now written as:

Notice that if the void volume fraction f = 0 then kj =0 and the plasticincompressibility is recovered.

For the first order Euler algorithm the stress at the end of a sub-step is given by:


and it is the same for each internal state variable, the effective strain and porosity fwhere all quantifies have been evaluated at the stress state {a}k . A more accurate

estimate of {cT/k+j and state variables may be obtained from the modified Euler-

scheme which gives:

where {da}2 > and all quantities are evaluated at the stress state {c?}k+1 - The global

error in the solution may be controlled by ensuring that the relative error for eachsub-step is less than some specified tolerance:

The size of each sub-step is continually updated during the integration procedure tosatisfy Eq.[24] where TOL is a small positive number in the range l.E-03 to l.E-05.

4. Damage parameters identification

To determine the constitutive and damage material parameters of the proposeddamage model, an identification technique must be used. First, the anisotropiccoefficients are evaluated separately by our digital image correlation method(D.I.C.) for strain measurements. These Lankford coefficients r0, r45, r^ are

determined from uniaxial tension tests in the three directions 0, 45 and 90 degrees to

the rolling direction of the sheet. ra is defined as the ratio of width to thickness

strain at a stabilized state of strain measured by the D.I.C. and taking account of theplastic incompressibility:

In this paper, a titanium alloy has been tested where the sheet thickness used was1.2 mm, the D.I.C. method gives the following anisotropic Lankford's values:

The stress-strain curve can only be described to the value of the homogeneous limitstrain and is expected to give ay = h(e ) as the hardening law of the pure matrix

material. The tensile tests, carried out on a set of tensile specimens prepared according


to initial length 140 mm, initial width 20 mm, have allowed us to obtain the flowstress expressions in terms of Swift's law:

where:

The material parameters (nucleation and coalescence) of the previous damagemodel are very difficult to quantify by direct experimental measurements. Aninverse identification is needed by comparing some numerical and experimentalresults and searching for a suitable matching between them. This technique is basedon the determination of the damage parameters minimising the cost functionrepresentative of the correlation between the load vs. displacement or engineeringaxial-strain during a tensile test and numerical finite element simulation. Thepreviously described tensile tests were performed until the ductile rupture of thespecimens and as an example, the load vs engineering axial-strain curve of thetitanium alloy is displayed in Figure 1. The cost function expressed by the leastsquare approximation is:

where p are the damage parameters, FjS1Itl and Fjexp are the simulated and

experimental load responses and n is the number of points considered. Assumingthat such response function in an assigned region of the input parameters has aregular behaviour, for instance it has a unique minimum and it is locally quadratic, itis possible to use known numerical techniques to search for such a minimum. Then sixor four coefficients remain to be determined depending on the nucleation modelchosen.

In the present paper, the three-dimensional numerical analysis has been carriedout for the simulation of the tensile tests with our specific explicit finite elementcode where the modified Gurson model Eq. [2] has been implemented. Theexplicit formulation permits a significant advantage in terms of CPU times for eachcall by the statistical analysis, but this approach requires a particular attention inorder to avoid the occurrence of unacceptable inertial effects when the velocity isartificially increased. Then we have followed the procedure presented by Fratini,Lombardo and Micari [FRA 96], where the response function has been calculatedfor 26 different sets of input damage parameters all around a given starting point, theinitial porosity being fixed. Two to three steps have been required to obtain the newstarting point to develop the so called central composite design method withsmaller incremental values for the damage parameters.


Figure 1. Experimental and numerical points of load vs. engineering axial strain

Figure 2. F.E. mesh and void volume fraction distribution at coalescence


However, in this fast inverse method it is necessary to select the starting pointnot far from the absolute minimum. In this way, the following sets of damageparameters have been obtained:

Normal distribution model: f0=o.0001 SN =0.012 eN =0.076 fN = 0.098f c = 0.044,

Continuous model: fo =o.0001 A0 = 0.14, fc = 0.0274

In Figure 2 the void volume fraction distribution just at coalescence is reportedon the mesh used, showing a clear shear band where one quarter of the specimen isanalysed making use of the symmetry. Due to the assumed symmetries, Figure 2represents two localised necks crossing each other at the centre of the strip but notobserved experimentally since only one localised neck grows in reality [TVE 93].

5. Void coalescence criterion by plastic limit-load model

In line with pure numerical convenience, a constant critical void volume fractionis almost always used in numerical analysis and practical applications using theGurson model. When void nucleation is taken into account, the critical value dependson the choice of the nucleation models and parameters as it has been observed inthe previous paragraph. Figure 1 shows that the two set of parameters give virtuallyidentical prediction of the load-displacement curve but a different critical voidvolume fraction. It is interesting to find that the simple continuous nucleation modelworks as equally well as the more complicated normal distribution model in thisexample.

Therefore, a criterion of void coalescence which determines a critical voidvolume fraction would be useful. As suggested by Zhang and Niemi [ZHA 95], amodified version of the coalescence model by Thomason [THO 85,90] is tested.Thomason has developed a 3D micro-mechanical model of the internal necking ofthe inter-void matrix called plastic limit-load model. What is interesting in theplastic limit-load criterion is that void coalescence is not only related to void volumefraction but also to void-matrix geometry and stress triaxiality.

Assuming that the material containing voids consists of a rigid-plastic non-hardening and isotropic material and using the Rice-Tracey [RIC 69] void growthequations of spherical shape initial voids, the variation in the geometry of the inter-void matrix is calculated using assumed velocity fields. Then the upper boundtheorem is applied to obtain an overestimate of the ratio between the mean stress andthe uniaxial yield stress of the matrix. This ratio a n / CT y is called the plasticconstraint factor by Thomason. If we note An the net area fraction of the inter-voidmatrix in the maximum principal stress direction such that:


This gives the virtual maximum principal stress to initiate the localised necking ofthe inter-void matrix material, which represents the strong dilational plasticbehaviour. If we note al the macroscopic maximum principal stress calculated byany numerical method, the critical condition to initiate the internal necking in a unitcell with a current ellipsoidal void can be postulated as:

By approximating the ellipsoidal void by the equivalent square-prismatic voidand assuming two velocity fields parallel and triangular in the inter-void matrix ofthe unit cell, from the upper-bound theorem Thomason obtained the following typeof empirical relation:

where F and G are constants, N and M are exponents, R X ,R Z are the radii of the

ellipsoidal void and X denotes half the current length of the cell. For an isotropicnon-hardening material with the following values, F = 0.1, G = 1.2, N = 2, M = 0.5,the empirical results have been found to represent a good approximation to theupper-bound constraint factor.

The stress triaxiality used in the original model ranges from 0.5 to 3, which isgreater than the range 0.33 to 1.0 currently observed in sheet-metal forming. Zhangand Niemi [ZHA 94] have found that the original Thomason criterion gave toolarge predictions at low stress triaxiality and proposed a modification which uses themean void radius R in Eq. [30]. This modification greatly decreases the prediction atlow stress triaxiality, while for the high stress triaxiality the predictions are almostthe same. As mentioned by Thomason [THO 90], it is interesting to note that there isno theoretical basis for the validity of plane-stress models of ductile fracture. This isdue to the fact that only very small void-growth strains would be needed to initiatelocalised plane-stress necking at a row of holes. In this paper where we areconcerned with the location of a necking-failure forming limit, that is hardlypreceded by necking, this location is assumed to depend on the critical void volumefraction given by the 3D modified coalescence model of Thomason. As originallysuggested by Thomason [THO 85] and already tested by Zhang and Niemi[ZHA 94,95], but not in the context of anisotropic sheet-metals forming, themodified Gurson model Eq. [2] is used to characterise the macroscopic behaviourassuming that the void grows spherically and to calculate the void and matrixgeometry changes using the current strain and void volume fraction.

During the F.E. analysis, the maximum principal stress is calculated and thenormal strains in the directions of the principal stress-axes X,Y,Z are evaluated in


order to calculate the current void and matrix dimensions where the current voidvolume fraction f is an output of the modified Gurson model. If the initial andcurrent volumes of a unit void containing cell are 1 and V, it is readily shown that:

Denoting Z the direction of the maximum principal stress a, and the current halfintervoid distance X in the direction perpendicular to Z is calculated by:

The net area fraction of the intervoid matrix is evaluated according to the cellmodel as:

Once the equality [30] is satisfied, the void coalescence starts to occur and the voidvolume fraction at this point is the critical value fc in the modified Gurson model

and then in the combined necking-failure criterion as explained in the next section.

6. Necking-failure criterion for anisotropic sheet-metals and example

The strain ratio: (3 = Ae2/Ae, has an evident influence on the internal damage ofsheet metals. At the same level of deformation, it is generally noted that the damageincrement is the greatest at plane strain such that Ae22 = 0 when the localisednecking occurs. The formulation follows our previous works [BRU 97,98], thecriterion is formulated in terms of the principal stresses and their orientation withrespect to the orthotropic axes leading to an intrinsic formulation. At the onset ofload instability (dF, <o ) , the plane stress assumption and plastic incompressibilitygive the maximum force criterion or diffuse necking:

For a given material, if damage starts just after load instability as it is generallyobserved, we assume the inequality in terms of the effective stresses:

If the major principal stress a, is calculated with internal damage coupled aspreviously explained, the inequality in the major principal stress-axis is:


When a sheet is strained under a biaxial tension stress state, development ofdamage will make the strain state gradually drift to plane strain. Similarly, when asheet is deformed under tension-compression condition, in the centre of diffuseneck, the final strain state at the local necking can also approach the plane strain.These observations have been earlier mentioned by Hecker [HEC 72] and discussedby Graf and Hosford [GRA 93] in the context of a theoretical analysis. Since thestate of strain evolves towards the plane strain state, due to the related stress statechange, there is an additional hardening-softening effect. The major stress is afunction of many variables and different possibilities may be considered inconnection with instability. The induced stress rate may be expressed as:

where £{ is the normal strain on the major stress axis, (3 is the strain increment ratio,t is the effective strain rate and 0 is the temperature. If we consider here only theeffects of the normal strain £j and of the strain ratio P, according to Eq. [36], thelocalised necking condition coupled with damage is given by:

An analytical and intrinsic form of the left-hand side of Eq. [38] can beformulated with [HIL 48] quadratic yield surface and the Gurson damage modeland can be found in [BRU 98]. It is worth noticing that the stress state can beevaluated by any others quadratic or non-quadratic flow rules for the coupledplastic-damage F.E. analysis of the sheet forming process.

As an example, the comparison between the proposed necking-failure criterionand our experiments has been obtained from the titanium alloy. ExperimentalNakazima tests (hemispherical punch) have been carried out on notched sheetswith various radii and depths in order to obtain different strain ratios as it can beseen in Figure 3. By a direct implementation into our F.E.-code where the principalstresses and their orientation with respect to the orthotropic axes are calculated ateach time step, the calculation was stopped once Eq. [38] was satisfied. The F.E.analysis of the experiments have been done for the complete formed part as it can beseen in Figure 4 where the meshes of the two blank-holders and the hemisphericalpunch are not represented.


Figure 3. Specimens tested by hemispherical punch for necking-failure analysis

Figure 4. Example of major strain distribution at the onset of necking-failure

Unlike the yield stress and other material constants, the critical void volumefraction is an indirect material parameter which depends on the mathematical formof the constitutive equations and is not a material constant. It depends strongly onstress triaxiality and strain state as it can be seen in Figure 5.


Figure 5. Evolution of the critical void volume fraction using the modifiedThomason formula

Figure 6. Ductile-fracture forming limit for a titanium alloy

The predictions using the present criterion Eq. [38] with the modified Gursondamage model Eq. [2] for anisotropic sheet-metals and the modified Thomasoncoalescence model Eq. 30 are shown in Figure 6 where reasonable agreement isobserved. However, the agreement between experiment and theory is poorest for


plane straining where in this case the theoretical curves are above the experimentalpoints which vary rapidly in a narrow range across the pure plane strain state.

7. Conclusion

A formability analysis of anisotropic sheet-metals presenting ductile-fractureforming limits has been carried out based on the anisotropically extended form ofthe Gurson damage model. The modified model takes into account the anisotropicproperties of the matrix material and has been implemented in our explicit F.E.-code. For shell elements and plane-stress state, a second order explicit integrationscheme with error control has been found to be accurate and robust. The load-displacement curve of the tensile test has been used to identify the damage para-meters using an inverse method. It is difficult to determine the critical void volumefraction which appears to be not a material constant. A more promising approach isto introduce more micromechanism in the damage analysis. To this end, a modifiedform of Thomason's coalescence model has been tested in conjunction with ournecking-failure criterion. The results emphasise that the extension of Gurson's modelcombining the Thomason coalescence mechanism by internal necking is a goodmethod for failure assessment in the design of metal forming processes of metalsheets with ductile-fracture forming limits.

8. References

[BAR 89] BARLAT F., LIAN J., "Plastic behaviour and stretch ability of sheet metals", Int. J. ofPlasticity, Vol. 5, 1989, p. 51-67.

[BRU 96] BRUNEI M, SABOURIN F., MGUIL-TOUCHAL S., "The prediction of necking andfailure in 3D. sheet forming analysis using damage variable", Journal de Physique,Vol. 6, 1996, p. 473-482.

[BRU 97] BRUNET M., MGUIL-TOUCHAL S., MORESTIN F., "Numerical and experimentalanalysis of necking in 3D. sheet forming processes using damage variable", AdvancedMethod in Materials Processing Defects, Studies in Applied Mechanics, 45, ElsevierScience B.V., 1997, p. 205-214.

[BRU 98] BRUNET M., MGUIL-TOUCHAL S., MORESTIN F., "Analytical and experimentalstudies of necking in sheet metal forming processes", J. of Mat. Proc. Tech., Vol. 80-81,1998, p. 40-46.

[CHU 80a] CHU C.C., "An analysis of localised necking in punch stretching", Int. J. SolidsStructure, Vol. 16, 1980, p. 913-921.

[CHU 80b] CHU C.C., NEEDLEMAN A., "Void Nucleation Effects in Bi-axially StretchedSheets", J, ofEng. Mat. Tech., Vol. 102, 1980, p. 249-256.

[DOE 93] DOEGE E., EL-DSOKI T.t SEIBERT D., "Prediction of necking and wrinkles in sheetmetal forming", NUMISHEET'93, 1-3 Sept. 1993, Tokyo, Japan, p. 187-197.


[DVO 84] DVORKIN E.N., BATHE K.J., "A continuum mechanics based four-node shellelement for general non-linear analysis", Eng. Comput., Vol. 1, 1984, p. 77-88.

[FRA 96] FRATINI L, LOMBARDO A., MICARI F., "Material characterisation for the predictionof ductile fracture: an inverse approach", /. of Mat. Proc. Tech., Vol. 60, 1996, p. 311-316.

[GRA 93] GRAF A.F., HOSFORD W.F., "Calculation of Forming Limit Diagrams for ChangingStrain Paths", Metallurgical Transactions, Vol. 24A, 1993, p. 2497-2501.

[GUR 77] GURSON A.L., "Continuum Theory of Ductile Rupture by Void Nucleation andGrowth", J. of Eng. Mat. Tech., Vol. 99, 1977, p. 2-15.

[HEC 72] HECKER S.S., "Sheet Metal Forming and Formability", Proceedings of the 7th'Congress of IDDRG, Amsterdam, Oct. 1972, Paper 5C.

[HIL 48] HILL R., "A theory of the yielding and plastic flow of anisotropic metals", Proc.Roy. Soc., London, 1948, p. 281-297.

[HIL 79] HILL R., "Theoretical plasticity of textured aggregates", Math. Proc. Cambrige PhilSoc., 85, 1979, p. 179-186.

[HIL 90] HILL R., "Constitutive modelling of orthotropic plasticity in sheet metals", J. ofMech. Phys. Solids, Vol. 38, 1990, p. 405-417.

[LIA 97] LIAO K.L., PAN J., TANG S.C., "Approximate criteria for anisotropic porous ductilesheet metals", Mechanics of Materials, 26, 1997, p. 213-226.

[NEE 78] NEEDLEMAN A., TRIANTAFYLLIDIS N., "Void growth and local necking in bi-axiallystretched sheets", J. of Eng. Mater. Tech., Vol. 100, 1978, p. 164-172.

[RIC 69] RICE J.R., TRACEY D.M., "On the ductile enlargement of voids in triaxial stressfields", J. of Mech. Phys. Solids, 17, 1969, p. 201-207.

[THO 85] THOMASON P.F., "A three dimensional model for ductile fracture by the growth andcoalescence of microvoids", Acta metall, Vol. 33, 1985, p. 1087-1095.

THO 90] THOMASON P.F., Ductile fracture of metals, Oxford, Pergamon Press.

[TVE 81] TVERGAARD V., "Influence of voids on shear band instabilities under plane strainconditions", Int. J. of Fracture, Vol. 17, 1981, p. 389-407.

[TVE 82] TVERGAARD V., "On localisation in ductile materials containing spherical voids",Int. J. of Fracture, Vol. 18, 1982, p. 237-249.

[TVE 84] TVERGAARD V., NEEDLEMAN A., "Analysis of the cup-cone fracture in a roundtensile bar", Acta Metall, Vol. 32, 1984, p. 157-169.

[TVE 93] TVERGAARD V., "Necking in tensile bars with rectangular cross-section", Comp.Meth. App. Mech. Eng., Vol. 103, 1993, p. 273-290.

[ZHA 94] ZHANG Z.L., NIEMI E., "Studies on the ductility predictions by different localfailure criteria", Eng. Fracture Mech., Vol. 48, 1994, p. 529-540.

[ZHA 95] ZHANG Z.L., NIEMI E., "A new failure criterion for the Gurson-Tvergaarddilatational constitutive model", Int. J. of Fracture, Vol. 70, 1995, p. 321-334.

Chapter 9

Damage in Sheet Metal Forming:Prediction of Necking Phenomenon

Nathalie Boudeau, Arnaud Lejeune and Jean-Claude GelinLaboratoire de Mecanique Applicquee R. Chaleat, Universite de Franche-Comte/CNRS, Besanôn, France


Damage in Sheet Metal Forming 149

1. Introduction

Necking in sheet metal forming arises quite naturally due to loading conditions,boundary conditions or material inhomogeneities that develop when plastic strainingincreases. In the field, pioneer works have been carried out on the early stages of thedevelopment of mathematical plasticity [HIL 52]. Lots of experimental work hasbeen carried out to characterize the forming ability of the sheet with the introductionof the Forming Limit Diagrams (FLD) [KEE 65] [GOO 68]. Unfortunately, thesecurves are not intrinsic and depend strongly on the strain path that can be verycomplex in industrial forming processes, especially in cases of multi-stage formingprocesses. Arrieux introduced then Stress Limit Curves (SLC) [ARR 89]. Thesecurves seem not to be dependent on the strain path.

Concurrently, theoretical works were developed and numerous models wereproposed. These models can be divided into two classes: models based onhomogeneous continuum and models based on heterogeneous continuum.

The development of numerical methods to simulate the forming processes createsthe necessity to predict necking phenomena from FE results. The commercial codesthat Pamstamp® and Optris® propose plot the principal strain state calculated duringthe FE simulation on the experimental or calculated FLD without taking into accountthe fact that the necking state depends strongly upon the strain path. The early worksbased on the proposition of a criterion independent of the strain path and able topredict necking from FE results are due to two of the authors in [BOU 94], Since then,several approaches have been proposed ([BRU 95], [BRU 97], [HOR 96], [KNO 00],[FRO 98]).

After this short introduction, the second section is devoted to the presentation ofthe linear perturbation and the numerous possibilities of this approach. Finally,section 3 will present the results obtained in two cases: FLD computations andnecking prediction during FE simulations.

2. The perturbation technique: theory and applications

2.1. Theory

Necking phenomenon is considered as an instability of the local equilibrium. Tostudy this instability, the perturbations technique and its first order Taylor'sdevelopment are employed [MOL 85]. Because necking occurs only in regions whereplastic strains are quite important the local problem is strongly non-linear. Moreover,the hypothesis of a rigid-plastic material is applied.

The local equilibrium is defined by:

- the plastic yield locus,

- the hardening law,


- the constitutive law,

- the local equilibrium relationship,

- the compatibility of the deformation,

- the plastic incompressibility.

The hardening law is Hollomon with strain rate dependence:

but other hardening laws can be used.

£0 is defined as <3Y = (J0 (0, £ ) with (7y the initial yield stress.

The local equilibrium conditions are:

where / corresponds to the wideness of the necking band (Figure 1).The strain rate compatibility conditions and the plastic incompressibility are:

A perturbation vector 8 U is introduced to provide a way to detect plastic

instability:

The local equilibrium can take the following form:

where A is a non linear operator and U a vector.Vector U describes the strain and stress state for the mechanical equilibrium:

where 8U° is the amplitude of the perturbation, £ is the spatial part of the

perturbation and r] the temporal part of the perturbation, x and n are vectors that

correspond respectively to the spatial location where necking could occur and to thenormal vector associated with the necking band which could develop at the point inthe material corresponding to the endpoint of the n vector as shown in Figure 1.


Thus if U ° is the solution of [5] the perturbed vector:

has also to satisfy equation [5], In the small perturbations case, the Taylor develop-ment to the first order gives:

A non trivial solution for 8U ° is needed meaning that:

Generally, condition [11] is too severe and a positive value for the threshold £

is used. So the necking criterion becomes:

The expression of equations [2] and [3] are different for 2D- or 3D-stress states. For3D stress state under consideration, the stress tensor and the strain rate tensors are:

Equation [4] remains the same as in the 2D case.The vector position and normal vector to the localised band are given in Figure 1:

The local equilibrium is governed by equation [2] with:

where / corresponds to the thickness of the necking band as shown in Figure 1.

where P,<p,e,uQ (7?) is a polynomial of 7].

Finally instability or necking occurs when:


The case (p = 0 is ignored because it corresponds to the case where the band is

parallel to the sheet plane.

The perturbation of equation [2] taking into account equation [15] followed by afirst order series expansion is given in Appendix 1.

In case of 3D-stress state, there are six equations of compatibility. Only threeequations are linearly independent. These three independent equations have beendetermined with the formal computation software Mathematica®. The perturbationfollowed by Taylor's development of these equations is also given in Appendix 1.

Figure 1. Orientation of the necking band compared to the position vector in thereference frame. Model for necking in the case of3D-stress state

2.2. Applications

The aim of this section is to demonstrate the high flexibility of the linearizedperturbations technique. The application of the method to several plastic yield loci andto damaged material is presented. The numerical results will be presented in thefollowing paragraphs.

2.2.1. Plastic yield locus

There is a large variety of plastic yield loci that can be used to represent the sheetmetal behaviour during stamping or deep drawing. They have been developed to give abetter representation of the material flow in the case of large plastic deformation. Thefirst anisotropic model is due to Hill [HIL 48] who proposed a quadratic model suitablefor the behaviour of standard steel. It allows one to model isotropic, transverseisotropic and orthotropic material that is well suited to describe the behaviour in sheetmetal forming. The use of aluminium alloys in sheet metal-forming to decrease theweight of automotive parts revealed a lack of the Hill 48 criterion to describe the socalled anomaly of aluminium and several others criteria were developed [(BAS 77],[BAN 99], [BAR 91], [GOT 77], [HIL 79], [HIL 93], [HOS 79], KAR 93]).


Banabic's model is a yield criterion for orthotropic sheet metals under plane stressconditions. It has been developed to take into account the particularity of thealuminium alloys which are BCC materials whereas is FCC one [BAN 00].

The yield stress is expressed as following:

where F and *F are functions of the stress tensor components defined below:

The coefficients M,N,P,Q and R are defined as following:

The above equations show that the shape of the yield locus is defined by eightmaterial parameters: a, b, c, d, e, /, g and k .

The k -parameter can take only two integer values: 3 or 4 depending on thecrystallographic structure of the material. For steel and all other FCC materials, k = 3 .To the contrary, for BCC materials such as aluminium alloys, k = 4. Then only sevenmaterial parameters define the Banabic yield criterion.

The constitutive law is given for the first component:

The perturbation and the linearization of equation [20] is given in Appendix 2.

2.2.2. Damaged material

The damage model chosen in this study is an extension of the model proposed byGelin and Predeleanu [GEL 92]. In that case the plastic flow is governed by thefollowing equation:


with:

3The z - parameter can take only two values: z = — in case of isotropic material and

z = 1 in case of orthotropic material described with the Hill 48 plastic criterion. Thecombination of this damage model with Banabic's yield locus has not been studied yet.P_ is a fourth order tensor and is a deviatoric operator. In case of isotropic material,

P_: G_ = devq_ and in case of orthotropic material described with the Hill's fourth order

tensor, P : cr = H_: <7 .

9d is the volume change associated with damage evolution and cr0 is the strain

rate-dependent hardening law [1]. 9d is related to the void volume fraction / by:

- j

The material is then compressible and the mass conservation condition isexpressed as:

trace

The equations obtained by the perturbation of [21] followed by a first orderTaylor development can be found in [BOU 00].

2.2.3. Post-processing of FE results

The necking criterion established with the linearized perturbation technique can beeasily interfaced to FE programs for necking prediction from FE results. Figure 2summarizes the method.


At each step of a FE calculation the program gives the stress and the strain statecorresponding to the current mechanical equilibrium. The calculation of the r] -roots

can be done very quickly and integration points where necking occurs can bedetermined very easily without complicated software developments. The necking zonescan be revealed by post-processing the FE results.

Figure 2. Implementation of necking criterion based on perturbations technique in aFE program

3. Simulations and prediction of necking

Two kinds of results will be presented in this section.

In the first subsection local simulations are carried out to improve the differentmodels used. The 3D- and the ID-stress state approaches are compared. The classical

FLD or the (e *, p) diagram will be used for the illustrations where £ * is the effective

strain at necking and p = £2 / £j is the strain path.

In the second subsection FE simulations of deep-drawing processes are performedand necking predictions are presented and compared to experimental observations.

3.1. Local simulations

3.1.1. 3D modelling of necking

Figure 3 shows the necking strain calculated from 2D or 3D modelling. For the 3Dmodelling no pressure was imposed meaning that <T33 = 0 along the normal to the

sheet metal. For the 2D case, an optimum threshold (different in the thinning domain


than in the expansion one) has been used and for the 3D case an unique threshold hasbeen used.

That explains the difference in shape between the 2D and the 3D FLD in theexpansion domain first and secondly the too high level for the 3D FLD in the expan-sion domain. The strain level is superior in the case of 3D modelling for negativestrain path. This curve demonstrates that 2D and 3D-modelling are strongly different.The difference is due to the fact that necking is searched not only in the plane but alsoin the thickness of the sheet. The increases of the strain level at necking in the thinningdomain are in agreement with [ITO 00]. The results in the expansion domain aresomewhat different compared to results presented in [ITO 00].

Figure 3. Comparison between the 2D and the 3D modelling of necking

3.1.2. Influence of the yield surface

The Banabic yield criterion allows for the distinguishing of materials with a BCCor FCC structure. Necking predictions have been made with the 2D-stress statemodelling working with Hill 48 criterion or Banabic criterion.

First the relationship between the strain path and the stress path has been builtin Figure 4 for steel and aluminium (Table 1). Figure 4 shows that the (OC, p)relationship obtained for steel (FCC structure) is closer to the linear relation-ship obtained for Hill 48 than the one obtained for aluminium (BCC structure).


Ct = (J2 / Cf{ is the stress ratio. That proves that the use of Banabic's plastic yield

surface will allow necking prediction for a larger range of materials.

Figure 4. Illustration of the effect of the plasticity model on the p = f(CC) curves incomparison with the Hill quadratic model (from [LES 00])

Figure 5. Comparison of necking predictions using the Hill quadratic plastic crite-rion and the Banabic one for aluminium (from [LES 00])


Secondly the diagram (§"*, p) has been built for aluminium with the Hill 48 crite-rion and with the Banabic criterion (Figure 5). The strain level at necking for theextreme strain paths are very different and it is shown that the Hill criterion underes-timates the formability. On the contrary, strain level at necking for strain path close toplane strain is diminished a little.

This study shows clearly that the yield surface modelling is very important fornecking evaluation.

Table 1. Material parameters for Banabic's yield locus

3.2. FE simulations and necking prediction

The simulations presented in this section have been performed with Stampform®,the FE code developed in our laboratory and dedicated to the simulation of deep-drawing processes. Several hardening laws and yield criteria are available. In thepresent cases the material is considered isotropic and a Swift hardening law is chosen.

3.2.1. Stamping of an industrial automotive part

This part is obtained in two operations and necking is known to appear in the firststage. The part is a 200 x 250 mm2 sheet metal with 3 mm of thickness. The steelparameters are summarised in Table 2.

Necking zones predicted with the perturbations technique are shown in Figure 6and are in good agreement with experiments. Necking is detected for 28 mm punchdisplacement as ductile fracture is observed for 29 mm of displacement. Thecomparison of necking zones with critical thinning zones shows that necking criteriabased only on geometrical aspects (critical thinning) are not efficient. For the presentexample, a necking criterion based on the perturbations technique is in good agree-ment with experiment.

A b C M N p q R k

Aluminium 0.6512 0.9521 0.0987 0.4881 0.5659 5.2209 -5.2598 98.672 4

Steel 0.2115 0.9941 0.8390 0.5811 0.5571 0.5923 -0.5803 1.1548 3


Young modulus

Poisson coefficient

Yield stress

Hardening law

Damage

Anisotropy

Automotive part

E = 206 800 MPa

v=0.3

0Y = 259 MPa

K= 562.3 MPa

n = 0.241

e0 = 0.0256

/Q = 0 (no damage)

H = 0.6154

F = 0.3846

P = 1.6153

Cross tool

E = 70000 MPa

v=0.3

0Y = 259 MPa

K= 419 MPa

n = 0.237

£„. = 0.0001

/0 = 0 (no damage)

Isotropic

Table 2. Material parameters for deep-drawing simulations

Figure 6. Necking zones predicted in an industrial automotive part

3.2.2. Deep-drawing with a cross-tool

The geometry of both cross-tool and sheet metal is visible in Figure 7. Thealuminium parameters are given in Table 2.


Figure 7. Geometry of the sheet and of the cross-tool

Figure 8 shows the necking zones predicted for a 50 mm punch displacement arein good agreement with experiment.

Figure 8. Necking zones in the cross-tool deep-drawing

4. Conclusions

This paper reveals several important tendencies concerning necking prediction.First the linear perturbations technique is a very flexible method that covers a verylarge domain of applications. The technique initially developed for ID-stress has beenextended to 3D-stress states. The adaptation to more accurate plasticity criterion suchas Banabic's yield locus can be done without problem. As the linear stability analysisanalyses the current state, its implementation in a FE code does not need specific andcomplex developments. All these possibilities have been demonstrated in the paper.


Moreover, this approach can be integrated into quality functions to optimise the deep-drawing processes.

An apparent inconvenience of the approach is the complexity of the analyticalcalculations as it is shown in the appendices. But nowadays this is no more a majorproblem with all the formal calculation software and the numerous interfaces existingbetween these software and the programming languages.

5. References

[ARR 89] ARRIEUX R., Boivin M., "Theoretical determination of the forming limit stresscurve for isotropic sheet materials", Annals of the CIRP, 38/1, 1989, p. 261-264.

[BAN 99] BANABIC D., BALAN T., POHLANDT K., "Analytical and experimental investigationon anisotropic yield criteria", 6th Int. Conf. ICTP'99, Nuremberg, 1999, p. 1411-1416.

[BAN 00] BANABIC D., COMSA D.S., BALAN T., "A new yield criterion for orthotropic sheetmetals under plane-stress conditions", Proc. of TPR 2000, International section, Cluj-Napoca, 11-12 May, 2000, p. 217-224.

[BAR 91] BARLAT F., LEGE D.J., BREM J.C., "A six-component yield function for anisotropicmaterials", Int. J. of Plasticity, Vol. 7, 1991, p. 693-712.

[BAS 77] BASSANI J.L., "Yield characterisation of metals with transversally isotropic plasticproperties", Int. J. Mech. ScL, Vol. 19, 1977, p. 651-654.

[BOU 94] BOUDEAU N., GEUN J.C., "Prediction of the localized necking in 3D sheet metalforming processes from FE simulations", J. of Mat. Process. Tech., V. 45,1994, p. 229-235.

[BOU 00] BOUDEAU N., GELIN J.C., "Necking in sheet metal forming. Influence ofmacroscopic and microscopic properties of materials", Int. J. Mech. Sci., Vol. 42, 2000, p.2209-2232.

[BRU 95] BRUNEI M., ARRIEUX R., NGUYEN NHAT T., "Necking prediction using forminglimit stress surfaces in 3D sheet metal forming simulation", Proc. ofNumiform'95, 1995.

[BRU 97] BRUNEI M., MGUIL-TOUCHAL S;, MORESTIN F., "Numerical and experimentalanalysis of necking in 3D sheet metal forming processes using damage variable", inAdvanced Methods in Materials Processing Defects, edited by M. Predeleanu and P.Gilormini, 1997, p. 205-215.

[FRO 98] FROMENTDM S., Etablissement d'un critere de striction intrinseque des toles etvalidation numerique par simulations d'emboutissage, Ph.D. thesis, University of Metz,France, 1998.

[GEL 92] GELIN J.C., PREDELEANU M., "Recent advances in damage mechanics: modellingand computational aspects", Fourth International Conference on Numerical Methods inIndustrial Forming Processes, edited by Chenot J.L. et al., 1992, p. 89-98.

[GOO 68] GOODWIN G.M., "Application of the strain analysis to steel metal forming in pressshop", La Metallurgia Italiana, n° 8, 1968, p. 767-772.


[GOT 77] GOTOH M., "A theory of plastic anisotropy based on a yield function of fourthorder", Int. J. Mech. Sci., Vol. 19, 1977, p. 505-520.

[HIL 48] HILL R., "A theory of the yielding and plastic flow of anisotropic metals", Proc.Roy. Soc., London, A 193, 1948, p. 281-297.

[HIL 52] HILL R., "On discontinuous plastic states, with special references to localizednecking in thin sheets", J. Mech. Phys. Solids, Vol. 1, 1952, p. 19-30.

[HIL 79] HILL R., "Theoretical plasticity of textured aggregates", Math. Proc. CambridgePhilosophical Soc., Vol. 85, 1979, p. 179-191.

[HIL 93] HILL R., "A User-friendly theory of orthotropic plasticity in sheet metals", Int. J.Mech. Sci., Vol. 15, 1993, p. 19-25.

[HOR 96] HORA P., TONG L., REISSNER J., "Prediction method for ductile sheet metals failurein FE simulation", Third Int. Conf. In Numerical Simulation of 3D Sheet Metal FormingProcesses, edited by J.K. Lee et al., 1996, p. 252-256.

[HOS 79] HOSFORD W.F., "On Yield loci of anisotropic cubic metals", Proc. 7th NorthAmerican Metalworking Conf., SME, Dearborn, MI, 1979, p. 191-197.

[ITO 00] ITO K., SATOH K., GOYA M., YOSHIDA T., "Prediction of limit strain in sheet metal-forming processes by 3D analysis of localized necking", Int. J. Mech. Sci., Vol 42, n° 11,2000, p. 2233-2248.

[KAR 93] KARAFILLIS A.P., BOYCE M.C., "A general anisotropic yield criterion using boundsand a tranformation weighting tensor", J. Mech. Phys. Solids, Vol. 41, 1993, p. 1859-1886.

[KEE 65] KEELER S.P., "Determination of the forming limits in automotive stamping", Sheetmetal industries, Vol 42, n° 461, 1965, p. 683-691.

[KNO 00] KNOCKAERT R., MASSONI E., CHASTEL Y., "Prediction of strain localization duringsheet forming operations", Third ESAFORM Conference on Material Forming, Stuttgart,Deutschland, 2000, II, p. 7-10.

[LES 00] LESTRIEZ P., Prediction de striction localisee par 1'analyse linearisee de stabilite:application au critere de plasticite de BANABIC, Rapport de DBA, Universite deFranche-Comte - ENSMM, France, Septembre 2000.

[MOL 85] MOLINARI A., "Instabilite thermoviscoplastique en cisaillement simple", Journalde Mecanique Theorique etAppliquee, Vol 4, n° 5, 1985, p. 659-684.

6. Appendix

6.1. Appendix 1

The perturbed equations for the local equilibrium in case of 3D-stress state are:


With:

The perturbation of the compatibility equations in case of SD-stress state gives:

6.2. Appendix 2

The perturbed equation for the Banabic yield criterion is the


Chapter 10

Anisotropic Damage Applied toNumerical Ductile Rupture

Patrick Croix, Franck Lauro and Jerome OudinLaboratory for Automation, Mechanical Engineering, Information Sciences andHuman-machine Systems, University of Valenciennes and Hainaut-Cabresis, France

Jens ChristleinAUDIAG, Germany


Numerical Ductile Rupture 167

1. Introduction

Description of damage is required for numerical simulations of sheet metalforming processes, sheet metal stamping or vehicle crash tests in which internaldeterioration plays a significant role.

Damage of porous material can be defined as a collection of permanentmicrostructural changes based on the description of the growth, nucleation andcoalescence of the micro voids. This microscopic approach is described by severaldamage models which are applied to static loading (eg: extrusion, forging, etc)[BEN 93, 95] and dynamic loading (eg: crash, stamping, etc) [LAU 97, 98]. TheGurson damage model [GUR 77], modified by Tvergaard and Needleman[TVE 81, 82, 84], is based on this damage process for isotropic materials. Tointroduce the anisotropy of the material, this damage model has been modified byintroducing Hill's yield stress instead of the von Mises into Gurson-Tvergaardpotential [DOE 95]. The micro void shape, which is frequently expected to be at theorigin of the anisotropic ductility, is taken into account in order to accurately predictan anisotropic damage. This model, based on the improved version of the Gologanu-Leblond-Devaux model [GOL 93, 94, 97], extends the Gurson-Tvergaard modelto take the void shape effect into account. In order to emphasise the role ofanisotropic void growth on ductile rupture, the microvoid shape is taken intoaccount in its growth evolution. The description of the porous material is based onthree internal variables: the microvoid volume fraction, shape and orientation. Themicrovoid volume fraction is defined as the ratio of the microvoid volume to thematerial volume. The microvoid shape corresponds to the difference between theNapierian logarithms of the minor and major semi axes of the microvoids and theorientation of microvoids changes with the rotation of the material. The sensitivityof the damage evolution is analysed in the case of prolate and spherical cavities withdifferent loading directions. The anisotropy of the material is also introduced in theGologanu-Leblond-Devaux model by means of the Hill 48 norm. Consequentlysome of the equations are modified. The new damage model for anisotropicmicrovoided material has been integrated into the three dimensional explicit finiteelement code for non-linear dynamic analysis of structures, PAM-SOLID™, in thecase of convected coordinates shell elements. This paper will describe theconstitutive damage model and its numerical implementation in the finite elementcode. The failure prediction is shown in the case of a non-axisymmetric double V-notched tensile specimen. Different anisotropic ductile ruptures are obtainedaccording to the initial shapes of the cavities and their evolution.


2. Constitutive damage models

2.1. Gurson's model

The Gurson Tvergaard Needleman GTN model (1984) is a quadratic formulationof plastic potential which can also be used as a yield function as follows:

where:

in which aeq is the macroscopic equivalent stress, QM is the elasto-viscoplastic flowstress, am is the mean stress (am = akk / 3), qi and q2 are the two "material"parameters introduced by Tvergaard [TVE 81] in order to converge the model withfull numerical analyses of periodic arrays of voids, f* is the Tvergaard andNeedleman's coalescence function. The original GTN model used the von Misesnorm: ai, = 1.5a'ija'ij where a'is the macroscopic stress deviator. In this study it is

replaced by the Hill 48 norm in order to take the anisotropic plastic behaviour intoaccount[DOE 95], and in plane stress condition take the following form:

with F, H and N the Hill anisotropic parameters.

2.2. Gologanu 's model

This is based on the studies of a material unit cell formed by two confocalspheroids. The Gologanu, Leblond and Devaux's model (GLD) extends the GTNmodel to take the microvoid shape effect into account, and is therefore interesting asit becomes the GTN potential for spherical microvoids.

The void shape is defined by the parameter S = ln (a t /bi) where ai and b1 are

the major and minor axis of the ellipsoidal void. The GLD model is based on theanalysis of an ellipsoidal cavity embedded in a medium Q which has the shape of aconfocal ellipsoid of minor and major semi-axes a2 and b2, the axis of the void isalways collinear to the unit vector ex (Figure 1.). This cavity follows the rollingdirection and is defined in the local frame of each element.

The plastic potential takes the following form:

Figure 1. The microstructure of a prolate cavity

where:

The parameters K, r|, C and X are given by:


and where the eccentricities ei, e2 and the parameter cc2 are deduced from the voidshape parameter S and the void volume fraction f by the following equations:

, with ex, Cy, ez the axis of the frame of the void


The notation • | is used in the original GLD potential to express the calculation

Then the Hill 48 norm presented for Gurson's model is modified and expressedas follows:

For the particular case of spherical voids (S = 0) the parameter K and a^ arerespectively equal to 3/2 and 1/3 and the GLD model is identical to the GTN model.

This model is completed by the equation of the evolution of the internal shapeparameter S given by:

with HT(T,C) a function, dependent on the triaxialityT = aidc/vcfeq) according to

the sign of £ = akk a'u , given by:

of the von Mises norm applied to the deviator stress cr'on

which TI CTH X is added. In this paper the material anisotropy is introduced with the

Hill 48 norm and then a transformation of the deviatoric expression a' + rj an X is

required.

Moreover, considering plane stress strate , and assuming the microvoid direction ex

follows the rolling direction, the deviatoric stress tensor a' + TJ O~H ̂ is expressed as:


and (Xi and a? are obtained by:

where fc is the critical microvoid volume fraction at coalescence onset, fF is themicrovoid volume fraction when ductile fracture occurs. This specific function f*inside the microvoid material potential describes the rapid loss of the stress carryingcapacity due to the coalescence of the neighbouring microvoids, when f reaches thelimit 1/qj.

The microvoids volume fraction evolution has two main phases: the nucleationof the new voids and the growth of existing voids. The microvoids volume fractionrate is expressed by:

The constitution from void nucleation is controlled by plastic strain [CHU 80],and takes the form:

where fN is the nucleated microvoid volume fraction, SN is the Gaussian standarddeviation, EN is the mean effective plastic strain for nucleation and eM is the effectiveplastic strain.

The evolution of the micro structural damage is represented by the current voidvolume fraction f, defined by f = 1-VM/VA, where VA, VM are respectively theelementary apparent volume of the material and the corresponding volume of thematrix.

f* is a function of the void volume fraction f


The growth of existing voids is given by:

The plastic multiplier A is deduced from the consistency condition <|)evp = 0 and

6 =0 leading to solve:Tevp °

The plastic multiplier is finally expressed by

where Ce is the isotropic material tensor and I is the second order identity tensor.

the equivalence between the plastic work dissipated into the porous material and theductile matrix is expressed as follows:

which leads to the following expression of the effective plastic strain rate

in which o is the Cauchy stress tensor, CTM is the elasto-viscoplastic flow stress andDp is the macroscopic plastic strain rate tensor defined in the case of the associatedplasticity by:

and

with


2.3. Numerical implementation

The previous constitutive damage model is implemented in the finite elementcode PAM-SOLIDTM

This code uses an explicit process in which the solution is advanced along thetime axis, along which the velocities are discretised at half time intervals, n"1/2t,n+1/2t... and the displacements and accelerations are discretised at full time intervals,n"'t, nt, n+1t... where n is the number of the time increment. Considering a given timent, the program calculates from the known quantities, which are the nodal velocity""1/2V the nodal displacement "X and the Cauchy stress n"'a; the updated or unknownquantities "a, n+1/2v, n+1X, "a the nodal acceleration and Fint, Fext, the internal andexternal nodal forces, respectively. The new development consists of the stresscalculation described below (Figure 2.).

Assuming the strain rate tensor n~1 /2Eij computed at the previous time increment

as elastic, the corresponding Cauchy stress tensor is updated by central finitedifference

3. Numerical examples

3.1. Unit cells tensile test

In order to study the anisotropic damage evolution several unit cell tensile testscontaining ellipsoidal void shapes are carried out. Two loading directions L and Tare considered which are respectively along the major and minor axis of the voids.

To take only the anisotropic effect of the void shape into account, the isotropicvon Mises criterion is considered and for this the G, F, N Hill's parameter arerespectively equal to 1, 1 and 3 in this study.

Tests with the GTN model and the OLD model are performed on a unit cellmodel with an initial spherical void. Two tests are performed on a unit cell modelwith an initial prolate void according to the loading directions L and T to exhibit theanisotropic damage evolution.

in which n Q^ depends on , in the case of shell elements.

The corresponding potentials are obtained byall the derivation comes from this equation.

then


Figure 2. The modified stress elastic prediction and plastic correction flowchart


Et (MPa)

aM (MPa)

65000

195

647

206

600

218

450

227

300.5

518

Table 1. Flow stress

This analysis carried out with a standard elastoplastic material; the flow stress isdescribed by successive tangent moduli (Table 1).

The usual material parameters are, q1=1.5, q2=l. for the elasto-viscoplasticpotential, fo=10-4 for the initial void volume fraction, fN=0.04, SN=0.1 and eN=0.2 forthe nucleation and the coalescence is not considered with fc = 1. fp = 1. Theseparameters are obtained from the literature [TVE 82], [NEE 85, 87].

The damage evolution of these tensile tests based on the plastic strain ispresented in Figure 3.

Figure 3. Damage evolution based on the plastic strain for the tensile tests

The GTN model could be considered as the reference but it is generally wellknown that it overestimates the damage in the structure. It is directly compared tothe GLD model with initial spherical voids.

The GTN model has spherical voids which stay spherical during thedeformation. With the GLD model the initial spherical voids become ellipsoidal inthe direction of the deformation and consequently the damage evolution is moredifficult and increases slower than with the GTN model, as seen in Figure 3.


Consequently, the damage value is more realistic. In the case of prolate voids, thedamage value is greater for the loading direction T than for the loading direction L

(Figure 3.).

This is really due to the combination of the shape of the voids and the loadingdirection. For the direction L, the prolate voids tend to become more and moreprolate with an increase of the parameter S which corresponds to the ratio betweenthe major and minor axis of the ellipse. For the direction T, on the contrary, theprolate voids tend to become oblate voids with an inversion of the ratio between themajor and minor axis of the ellipse. The parameter S then evolves to S positive to Snegative. Consequently, the area of the voids for this case is greater than with theloading direction L and the evolution of the parameter S is quicker and leads to moredamage and of course to a quicker rupture (Figures 3 and 4).

Figure 4. Shape evolution according longitudinal and transverse loading directions

3.2. Non axisymmetrical double V-notched specimen

The aim of the numerical simulation is to confirm that the OLD model is moreaccurate in predicting crack propagation than the GTN model. Thus, the numericaldescription of crack propagation for a non-axisymmetric double V-notched tensiletest specimen is performed using both the GTN and OLD models. This crackpropagation is obtained by elimination of successive elements which occurs at thecomplete loss of the stress carrying capacity. To avoid numerical divergence thestress in the element is put to zero when it reaches the damage value f = 0.9 fF. Figure6a. presents the initial mesh size.


The porous elasto-viscoplastic material is described as was the previous materialfor the unit cell tensile test. The values of the material parameters are the result of anindustrial requirement and are obtained using an inverse method [LAU 99].This method consists in the identification of the damage parameters by correlatingan experimental and numerical macroscopic measurement strongly dependent on theparameters. The macroscopic measurement is given by means of a tensile test on anotched specimen. It corresponds to the variation of the inner radius of the specimenas a function of the elongation. In order to take the anisotropic aspect of the microvoidshape parameter S into account, three experimental measurements are considered.They correspond to three tensile tests specimen at 0°, 45° and 90° with the rollingdirection. An optimiser is used to find the damage parameters minimising the gapbetween numerical and experimental measurements in all the directions at the sametime. Due to the symmetry of the specimen, one quarter of its finite element modelis used and the result for only one direction (0°) is presented in Figure 5. Thisidentification process is applied for Gurson's parameters as well as Gologanu'sparameters.

Figure 5. a) Finite element modelling, boundary conditions for one quarter of thenotched tensile specimen, b) Comparison of experimental-numerical width evolutionat the bottom of the notch according to the tensile specimen elongation, for aspecimen sample along the rolling direction

The material has an anisotropic behaviour which is introduced by Hill'sparameters G, F and N. The initial void shape used with the GLD model is prolatewith S = ln(4) = 1.3863.


Figure 6. Tensile test of a non-axisytnmetrical double V-notched specimen, a) initialfinite element modelling; b) and c) are the damage distribution at the end of theprocess using the GTN and GLD models respectively

Figure 7. Tensile test of a non-axisymmetrical double V-notched specimen,orientation and shape evolution S at the end of the process for an initial prolate voidshape using the GLD model


Figures 6b. and 6c. show the results from the computations after rupture with theGTN model and the GLD model, respectively. The GTN model shows that cracksare initiated at the bottom of the notches and propagated in opposite directionswithout ever coming into contact. The result with the GLD model differs from thatobtained with the GTN model. The rupture initiation is identical for both models butwith the GLD model the cracks tend to meet each other and the final rupture occursin the middle of the test specimen which is the experimental result. This differenceis due to the shape of the voids and their orientations. Due to the shape of thespecimen, the main plastic strain direction is different in the middle of the V-notches. Consequently, the voids rotate to follow the main axis of deformation andthis leads to a better estimation of the crack propagation. The orientation of voids isdescribed by vectors and presented in Figure 7. The length of these vectors is propor-tional to the shape parameter value.

4. Conclusion

The GTN and GLD models are used to predict the damage evolution occurringwith plastic strain. Both models are based for the description of the porosity of thematerial by the microvoid volume fraction and the prediction of the porous materialflow by a modified yield surface. The evolution of the microvoid volume fractiondue to the growth of existing microvoids and the nucleation of new microvoids istaken into account. The ductile rupture is finally predicted. These models have beenmodified by introducing Hill's potential into the elasto-plastic potential to takethe anisotropy of the material into account. Moreover, the GLD model definesdifferent initial shape of voids (prolate, spherical, and oblate) which change ofshape, and orientation during the deformation.

First of all the two damage models are presented in this paper. Theimplementation of the GLD model in the explicit finite element code PAM-SOLID™ is then explained in more detail. Two different elementary tests arepresented. First of all, a unit cell computation highlights the effect of the shape ofthe voids on the damage evolution and moment of rupture. These tests illustrate that:

- the damage evolution is better predicted with the GLD model than the GTNmodel which generally overestimates the damage;

-in the case of prolate voids, the damage evolution differs if the loadingdirection corresponds to the direction of the major axis or the minor axis of theellipse.

Secondly, tensile tests of non-axisymmetrical V-notched specimens areperformed with the GTN and GLD models. These tests show that the direction of thepropagation of the crack is better predicted with the GLD model due to the changeof orientation of the voids according to the direction of the main plastic strain duringdeformation. Finally, two different ruptures are obtained with the GTN and GLDmodels and the GLD model gives closer results to the experimental ones.


5. References

[BEN 93] BENNANI B., PICART P., OUDIN J., «Some basic finite element analysis ofmicrovoid nucleation, growth and coalescence », Engineering Computations, vol. 10,1993, p. 409-421.

[BEN 95] BENNANI B., OUDIN J., « Backward can extrusion of steels effects of punch designon flow mode and void volume fraction », International Journal of Machine Tools andManufacture, vol. 35, 1995, p. 903-911.

[CHU 80] CHU C., NEEDLEMAN A., « Void nucleation effects in biaxially stretched sheets »,Journal of Engineering Materials and Technology, vol. 102, 1980, p. 249-256.

[DOE 95] DOEGE E., EL-DSOKI T., SEIBERT D., «Prediction in necking and wrinkling insheet-metal forming», Journal of Materials Processing Technology, vol. 50, 1995,p. 197-206.

[GOL 93] GOLOGANU M., LEBLOND J.B., DEVAUX J., « Approximate models for ductilemetals containing non-spherical voids—Case of Axisymmetric prolate ellipsoidalcavities », Journal of the mechanics and physics of solids, vol. 41,1993, p. 1723-1754.

[GOL 94] GOLOGANU M., LEBLOND J.B., DEVAUX J., « Numerical and theoretical Study ofcoalescence of cavities in periodically Voided solids», Computational MaterialModelling, vol. 42, 1994, p. 223-244.

[GOL 97] GOLOGANU M., LEBLOND J.B., PERRIN G., DEVAUX J., «Recent's extensions ofGurson model for porous ductile metals », Continuum micromechanics, 1997, P. Suqueted., New York.

[GUR 77] GURSON A. L., « Continuum theory of ductile rupture by void nucleation andgrowth: Part I —Yield criteria and flow rules for porous ductile media », EngineeringMaterial Technology, vol. 99,1977, p. 2-15.

[LAU 97] LAURO F., BENNANI B., DRAZETIC P., OUDDM J., Ni X., « Damage occurrence underdynamic loading for strain rate sensitive materials », Communications in NumericalMethods in Engineering, vol. 13, 1997, p. 113-126.

[LAU 98] LAURO F., BENNANI B., OUDIN J., Ni X., «Damage occurrence under dynamicloading for anisotropic strain rate sensitive materials », Shock and Vibration, vol. 5, 1998,p. 43-51.

[LAU 99] LAURO F., BENNANI B., CROK P., OUDIN J., «Identification of the damageparameters for anisotropic materials by inverse technique: application to an aluminium »,Proceedings of the International Conference on Advances in Materials and ProcessingTechnologies, AMPT'99, Dublin, 3-6 August 1999, Editors Profs. M.S.J. Hashmi, and DrL. Looney.

[NEE 85] NEEDLEMAN A., TVERGAARD V., « Material strain rate sensitivity in round tensilebar », Proc. Considere Mem. Symp., Salengon J. Ed., Presse de I'ecole Nationale desPonts et Chaussees, 1985, p. 251-262.

[NEE 87] NEEDLEMAN A., «A continuum model for void nucleation by inclusiondebonding », Journal of Applied Mechanics, vol. 54,1987, p. 525-531.


[TVE 81] TVERGAARD V., «Influence of voids on shear band instabilities under plane strainconditions », International Journal of fracture, vol. 17, 1981, p. 389-407.

[TVE 82] TVERGAARD V., « On localization in ductile materials containing spherical voids »,International Journal of fracture, vol. 18, 1982, p. 237-252.

[TVE 84] TVERGAARD V., NEEDLEMAN A., « Analysis of the cup-cone fracture in aroundtensile bar », Acta Metallurgica, vol. 32,1984, p. 157-169.


Chapter 11

Numerical Aspects of FiniteElastoplasticity with Isotropic DuctileDamage for Metal Forming

Khemais Saanouni, Abdelhakim Cherouat and Youssef HammiUniversite de Technologie de Troyes, France


Aspects of Finite Elastoplasticity 185

1. Introduction

Displacement-based finite element codes that are industrially utilized for thestatic or dynamic analysis of mechanical structures require accurate and efficientconstitutive equation subroutines. This accuracy concerns both the description ofthe physical phenomena taken into account by those constitutive equations, as wellas their numerical discretization with respect to time and space. This leads, ingeneral, to highly non linear algebraic systems to be solved at both local and globallevels.

At the global level, the spatial discretization of the principle of virtual work (orpower) leads to a non linear system for displacement (or velocity) fields in theform of partial differential equations (PDE). This algebraic system is usuallylinearized to be solved for each load increment by either an implicit iterativeNewton-type strategy, or a dynamic explicit or implicit one. Through linearization,many terms arise which can be classified into two classes: the first contains thematerial non linearity related to the material behavior (stress, internal variables) orthe friction behavior; and the second contains the geometrical non linearities relatedto the finite deformations and rotations as well the evolving contact conditions.Particularly, the derivative of the stress tensor with respect to the total deformationtensor is needed. Generally, this "incremental" stress differential differs from the"continuous" differential given directly by the constitutive equations of rate type. Ithas been shown (see [NAG 82], [SIM 85]) that the use of the incremental stressdifferential consistent with the time discretization scheme of the stress tensor leads toquadratic convergence. For the explicit strategy to solve the system of PDE, only thestress increment at each time step is needed.

The calculation of the stress increment at each time step needs the localintegration of the overall constitutive equations representing the coupled physicalphenomena. There exist various explicit or implicit integration schemes for ordinarydifferential equations (ODE). Experience has shown that implicit time integrationschemes have the advantage of stability and are suitable for application to thoseconstitutive equations involving yield and loading-unloading conditions. Theseconditions are generally modeled by constructing a special procedure as the elasticpredictor and plastic corrector scheme.

When a metallic material is formed by large straining processes such as forging,stamping, hydroforming and deep drawing, it experiences large irreversibledeformations, leading to the formation of high strain localization zones caused by thenucleation and growth of micro defects (voids) generally referred to as ductileisotropic damage. Accordingly, to increase the efficiency and the predictivecapabilities of the virtual forming tools, an accurate theoretical and numericalmodeling of the damage initiation and growth under finite transformations should betaken into account. This can be achieved by using the coupled approach in the sensethat the damage evolution equation is directly incorporated and fully coupled withthe elastoplastic constitutive equations. This kind of approach has been employed by


many authors using damage models based either on Gurson's theory ([GEL 85],[ARA 86], [ONA 88], [BON 91], [BRU 96], among many others), or on ContinuumDamage Mechanics (CDM) in the Kachanov's sense ([MAT 87], [ZHU 92], [SAA99], [HAM 00], [SAA 00], etc. These fully coupled approaches allow the predictionof not only the large transformation of the processed workpiece as largedeformations, rotations, and evolving boundary conditions, but they can also indicatewhere and when the damaged zones can appear inside the formed part during theprocess ([SAA 99], [HAM 00], [SAA 00]).

In the present work, fully coupled constitutive equations accounting for bothcombined isotropic and kinematic hardening as well as the ductile damage in theCDM framework are presented. The particular case of the fully isotropic andisothermal flow considering small elastic strains, large plastic strains, isotropic andkinematic hardening, isotropic damage and the evolving contact with friction isimplemented into ABAQUS/STD. The associated numerical aspects concerning boththe local integration of the coupled constitutive equations as well as the (global)equilibrium integration schemes are presented. For fully implicit resolution strategy,special care is given to the consistent stiffness matrix calculation. The integration ofthe coupled constitutive equations is realized thanks to the backward Euler schemetogether with the asymptotic integration procedure pioneered by Freed and Walker[FRE 86], The efficiency of this integration procedure in the 3D isotropic case isenhanced by reducing the number of the constitutive equations from 14 to 2 asproposed by Simo and Taylor [SIM 85] and widely used since then (see [HAR 93],[DOG 93], [CHA 96], [HAM 00] among many others). The numericalimplementation of the damage is made in such a manner that calculations can beexecuted with or without damage effect, i.e. coupled or uncoupled calculations.

2. Kinematical background

The transformation gradient F between the initial (undeformed and undamaged)and the current (deformed and damaged) configuration is multiplicativelydecomposed so that the following classical definitions are used:

where B is the total Eulerian left Cauchy-Green deformation tensor associated withthe Cauchy stress tensor a; L is the spatial velocity gradient in the currentconfiguration, D and W are respectively the pure strain rate and the material spin


tensors. The superimposed dot () denotes the usual time derivative.

To satisfy the objectivity requirement, the so-called Rotated Frame Formulation(RFF) is used. This leads to the expression of the constitutive equations in a rotatedconfiguration obtained from the current one by an orthogonal rotation tensor Qdefined by [DOG 89]:

from which the classical Jaumann and Green-Nagdhi rotational derivatives can beeasily obtained. On the other hand the objective rotated tensor TQ by the rotation Q

is given by:

Its time and rotational derivatives are related by:

where e^ is the Jaumann derivative (rotational objective derivative) of the elastic straintensor (for simplicity the subscript J will be removed), and Dp is the plastic strain ratetensor defined by the constitutive equations as will be shown in the next section.

Accordingly, for any symmetric second order tensor T, the objective rotationalderivative with respect to the rotating frame is given by:

Consequently, the constitutive equations are formulated in the same way as undersmall strain hypothesis and their generalization to the large strain case is simplyachieved by replacing all the tensorial variables by their rotated correspondingquantities by using Eq. [6].

The second main question posed by the finite transformation aspect is how thetotal strain rate can be decomposed into elastic (reversible) and plastic (irreversible)parts. For metallic materials, dealing with large plastic strain but small elasticstrain, the total Eulerian strain rate tensor decomposition can be approximated by:


3. Coupled constitutive equations for metal forming

3.1. State variables versus effective state variables

Finite thermo-elastoplastic constitutive equations coupled with continuousdamage is developed in the framework of the classical thermodynamics of irre-versible processes with state variables. For the sake of simplicity, this will bepresented hereafter using the classical small strain notations keeping in mind that thegeneralization to the finite strain hypothesis is made according to the RFF formula-tion presented above. This formulation uses a unified yield surface for both plasticityand damage as in [SAA 94]. A more general formulation using two different (butcoupled) yield surfaces can be found in [HAM 00].

Limiting ourselves to the simple first displacement gradient theory, two couplesof external state variables are used, namely: (1) the total strain associated with theCauchy stress tensors (e,o) and (2) the absolute temperature associated with thespecific entropy (T,s). Five couples of internal variables are taken into account: (1)the (small) elastic strain representing the inelastic flow associated with the Cauchystress tensor (ee,a); (2) the normalized heat flux vector associated with the gradientof the absolute temperature (q /T, g = grad(T)); (3) the isotropic hardeningvariables (r, R) representing the size of the yield surface in strain space (r) and stressspace (R); (4) the tensorial (deviatoric) kinematic hardening variables (a, X)representing the displacement of the center of the yield surface in strain space (a)and stress space (X), (5) the isotropic damage variables (D, Y), in Chaboche's sense[CHA78].

Suppose that the current configuration contains some isotropic ductile damagedistribution i.e. a given homogeneous distribution of micro-defects such as voidsand/or micro-cracks ; the concept of the effective stress ([CHA 78], [LEM 85])together with the hypothesis of total energy equivalence [SAA 94] are used to definethe effective state variables by:

where, for simplicity, it has been assumed that the damage effect on the elasticbehavior is the same than on the hardening variables (both isotropic and kinematic).


These effective state variables are used in the state and dissipation potentials toderive the complete set of fully coupled constitutive equations for metal formingprocesses (see [CHA 78], [LEM 85], [LEM 92], [SAA 94] among others).

3.2. State potential: state relations

The Helmoltz free energy \|/(ee, cc,r,D,T) is taken as a state potential. It issupposed to be a convex function of all the deformation-like state variables definedabove and additively decomposed into thermo-elastic/damage and plastic/damagecontributions:

where p is the material density in the current undamaged configuration and thevariable T in the last term *Fpd acts as a simple parameter. In this work, only isotropicphenomena are considered, and have the following state potential:

where K and n are the classical Lame's constants of elasticity, a is the coefficient ofthermal expansion, C is the kinematic hardening modulus, Q is the scalar isotropichardening modulus, T0 is the reference absolute temperature, Cv is the classicalspecific heat parameter and I being the second order unit tensor.

By using the Clausius-Duhem Inequality (GDI) one can easily derive, after somealgebraic manipulations, both the state relations (Eq. [15 to 19]) and the residualinequality (Eq. [20]) defining the volumetric dissipation:

- state relations:

- residual inequality:

Note that in the state relations above, the main material properties are affected bythe damage according to:


- elasticity properties of damaged material:

- kinematic hardening modulus of damaged material:

- isotropic hardening modulus of damaged material:

- thermal expansion of the damaged material:

3.3. Dissipation potentials: complementary relations

The volumetric total dissipation given above (Eq. [20]) should be identicallyverified for any selected dissipative phenomenon. In this equation the force-likevariables namely: CT, X, R,Y are given by the state relations (Eq. [15 to 19]), and theflux variables should be defined by using the generalized standard materials[HAL 75]. This is achieved by introducing both yield functions and dissipationpotentials for each class of dissipative phenomena. As a first approximation, the totaldissipation is additively decomposed into two terms, namely: mechanical dissipation3>m (plasticity, hardening and damage) and thermal dissipation 0th, each of thembeing supposed independently positive or zero:


Each of these dissipations will be analyzed to derive the flux variables associatedwith each selected dissipative phenomenon.

3.3.1. Thermal dissipation: fully coupled heat equation

Classically, the heat equation is derived from Fourier's dissipation potentialwhich is a quadratic scalar function of the force g j. For a thermoelastoplastic mediumwith mixed hardening and damage (strong coupling) the final form of the generalizedheat equation can be written as follows [SAA 94]:

where A(T) stands for the Laplacian of the temperature and the prime (X') indicatesthe derivative of X with respect to temperature. The weak form of the partialdifferential equation Eq. [26] can be discretized with respect to time (FiniteDifference Method) and space (Finite Elements Method) and solved together withthe discretized weak form of the equilibrium problem thanks to a sequential methods.

3.3.2. Mechanical dissipation: fully coupled constitutive equations

In the present case of time independent flow, a yield function in the stress spacef(o, 2L R; D, T) and a plastic potential (non associative theory [LEM 85]) F(a, X, R;D, T) are introduced to derive the constitutive equations for plasticity with damageeffect:

where the temperature dependent material constants a and b are the non linearityparameters for kinematic (a) and isotropic (b) hardening respectively; while S, s andP characterize the ductile damage evolution and the parameter oy represents the

initial size of the plastic yield surface. The notation o-X defines the norm of the

effective stress according to:


where gd is the deviatoric part of the stress tensor.

The generalized normality rule allows the derivation of the complementaryrelations for plasticity, with hardening including the damage effect:

The tensor n represents the outward normal to the yield surface in the stress spacegiven by:

The accumulated plastic strain rate p can be calculated from Eq. [30] using thefollowing norm:

which indicates that the isotropic hardening strain r is not equal to the accumulatedplastic strain p unless the hardening is linear (b = 0) as clearly indicated by the Eq.[32]. The plastic multiplier A, is given by the classical consistency condition appliedto the yield function f : f > 0 , A, > 0, Af = 0. This gives for the fully flow:

giving


where <(.)> stands for the positive part of (.) and HH > 0 is the tangent plasticmodulus given by :

and HT represents the thermal effect given by:

If the variation of the Poisson's ration v with the temperature is neglected,Eq. [39.a] writes under the following simpler form:

Note that, in this unified formulation, a single yield function is taken for bothplasticity and damage, leading to a single plastic multiplier. This restrictive choice isjustified in the case of metal forming where the damage develops only at materialpoints with large plastic deformation. However, for some other materials such asconcrete or composite structures, damage can develop without plasticity and viceversa. In those cases the use of multisurface formulation should be preferred : oneyield function for plasticity with damage effect (coupling) and another one for thedamage yielding [HAM 00].

Finally, the direct time derivative of the stress tensor (Eq. [15]) gives with thehelp of the Eq. [37]

with:

and:


where use has been made of the following notations (i being the fourth order unittensor):

is the classical fourth order symmetric operator of the isotropic elastic propertiesaffected by the damage, and:

is the thermoelastic contribution in the tangent operator.

It is clear from the equations [40] and [41] that the continuous tangentelastoplastic-damage operator is non symmetric for the coupled problem i.e. if Y isnon zero.


In metal forming, the large deformation and damage behavior experienced bymetallic materials are described by nonlinear equilibrium, the coupled thermo-elasto-plastic-damage constitutive equations and the contact conditions with frictionalconstitutive equations presented above. For the sake of simplicity, in thispaper we limit ourselves to solving the equilibrium problem associated withelastoplastic-damaged solids without thermal effect nor the contact/frictionconditions (see [HAM 00] for more details).

4.1. Finite element formulation

The velocity (displacement) based finite element formulation starts with theprinciple of virtual power (work) which states that, among all the kinematicallyadmissible velocity (displacement) fields u'(u'), the solution of the equilibriumproblem minimizes the functional G (weak form) given here in continuous formlimited to the quasi-static case using the classical updated Lagrangian formulation:

where V is the volume of the current configuration, FF is the boundary of the solidwhere external forces F (including the contact forces) are prescribed, f represents the


vector of volumetric applied forces, D* is the virtual strain rate tensor and o is thestress tensor given by the coupled constitutive equations discussed above.

By applying the minimization principle to the spatially discretized form of Eq.[45], one can obtain for the overall structure:

where 5R is called the equilibrium residual vector; Fint and Fext are the internal andexternal force vectors written here using the natural coordinates as:

where V0 and FFo are the volume and its boundary of the reference solid element, Jv isthe Jacobian determinant of the isotropic transformation between global and naturalcoordinates for the solid element, Js is the Jacobian determinant for the surfaceelement, N is the matrix of interpolation functions and B is the matrix of strain (rate)interpolation. Note that the matrices B, N and the Jacobians, Jv and Js are functions ofthe displacements (geometrical non linearities).

The most widely used implicit iterative method to solve the system [46] is theNewton-Raphson method, which consists in linearizing Eq. [46], for the (n+1)* loadincrement and at the iteration (p+1), as follows :

where (u^jis the approximation of the solution at the iteration (p). The currenttangent stiffness matrix Kp

n+l is defined by:

The second term, (K^, / describes the dependence of the external loads on the

geometry and will not be discussed here. The first, {K£+1}"' , represents the variation

of the internal forces with displacements. As shown by Eq. [47], this variation is dueto the stress o (material non linearities) given by the fully coupled constitutiveequations, and the fact that the matrix B as well as the Jacobien determinant Jv aredisplacement dependent (geometrical non linearities). For the sake of simplicity,only the term related to the material non linearities will be discussed hereafter.


4.2. Time integration procedure

In order to calculate the internal forces \Fnp+1}"" and the tangent stiffness

matrix JK^, / , we must first compute updated stresses at the end of the current load

increment. This can be achieved by integrating the overall set of coupled constitutiveequations discussed above. The implicit Euler integration scheme (Backwardmethod) is used since it contains the property of absolute stability and the possibilityof appending further equations to the existing system of nonlinear equations. Let usconsider the system of ordinary differential equations given above (Eq. [30-35])formally represented by y = f(y,t) . The implicit method is defined by (for clarity

the iteration subscript (p) is omitted):

with the abbreviations yn + 1 = y(tn + At) and yn = y(tn) . When applied to the stress

tensor for example, Eq. [51] reads:

Using the elasticity relation (Eq. [15]) and the decomposition of the strain tensorwe get:

where we have incorporated the fact that the plastic strain rate and damage rate onlyoccur if the field condition is satisfied, i.e. during the time interval Atp < At. In the

following, the subscript (n+1) will be omitted and the variables, which do notcontain the subscript (n+1), are computed at tn + At.

For the calculations of hardening variables ot and T; the AI 'AsymptoticIntegration' procedure proposed by Freed and Walker [FRE 86], for a betterintegration of first-order ordinary equations is used. The AI procedure is mainlybased on the fact that the above discussed constitutive equations have the followingform:

where Y denotes here a set of state variables to be considered and A(Y) and()>(Y) are given functions depending on the concerned constitutive equation. One canintegrate Eq. [53] exactly over the time step and obtain the following recursiveintegral equation:


where £ is the parameter of time integration. Freed and Walker [FRE 92] haveconsidered several discretization schemes of this exact solution. We retain here theasymptotic integration scheme at time t + At:

Applied to the kinematic and isotropic hardening evolution equations this gives:

where AX = XAt is related to the accumulated plastic strain increment according toEq. [35].

By using the complete set of constitutive equations we end up with a system of14 nonlinear scalar equations for 15 unknowns : six stresses, six back-stresses forkinematic hardening, one isotropic hardening stress, one isotropic damage variableand the plastic multiplier. The 14 first equations are:

where:

The remaining (15th) equation is given by the yield condition Eq. [27], whichmust be satisfied at the end of each time step.

Before solving iteratively (Newton's method) the above system of 15 equations,it is very helpful to reduce the size of this system by eliminating some equationsamong them. Following an idea originally proposed by Simo and Taylor [SIM 85]and widely used since that, we derive from Eq. [58] and [59] the deviatoric tensorialquantity cf'-X between t and t+At:


where the deviatoric tensor Z at tn is given by :

The multiplication of the yield function (Eq. [27]) by p^-X gives

This implies:

with the notation

Hence, the unknown tensor n is replaced by the tensor Z, which depends only onone scalar unknown, namely AA, as shown by the Eq. [67].

Furthermore, the system of 15 equations is now restricted to two scalar equations,namely:

where the expression of the damage release rate Y (scalar) is given by :

This small system (Eq. [68-69]) is solved iteratively thanks to the Newton-


Raphson numerical integration procedure to compute the two unknowns : AX and D(see [HAM 00] for details). Tables (1) and (2) summarize schematically theproposed stress calculation.

NOTE - For plane stress hypothesis the total strain component e33 is not definedby the kinematics but by a new constraint namely : G3(AX,D,e33J=a33 =0. Thisleads to an additional scalar equation with the new unknown e33 to be determinedtogether with AX and D by solving the three equations Gi, G2 and G3 [HAM 00].

4.3. Consistent elastoplastic-damage tangent operator

As discussed in paragraph 4.1 the quasi-static tangent stiffness matrix for largedeformation is viewed as relating the rate of internal nodal forces to the nodalvelocities. This gives rise to three main contributions: the stress contribution, thecontact/friction contribution and geometry variation contribution. Only the firstcontribution is discussed here (see [HAM 00] for more details). The computation ofthis term needs the calculation of the tangent operator representing the stressvariation with respect to the total strain for each load increment. The continuousform of this operator is given by Eq. [40] including thermal contribution. As reportedby many authors, the equality of the global/local convergence of a Newton-Rahpsonmethod is greatly improved when using a tangent stiffness matrix consistent with thediscretized incrementation of the local constitutive equations ([NAG 82], [SIM 85]).This consistent operator is given here (thermal contribution being neglected) bydifferentiating with respect to the total strain the time discretized expression of thestress as follows:

This needs the calculation of the derivatives of D and AX with respect to the totalstrain e. These are obtained by solving equations [68-69] and the final expression ofthe consistent tangent operator is [HAM 00]:

The above discussed constitutive equations and the corresponding local timeintegration have been implemented in the general-purpose finite element codeABAQUS/STD thanks to the user's material subroutines UMAT for static implicitsolving procedure.


(I) Calculate elastic predictor :

a1™' =(l-Dn)K(e:l)l + 2^(l-Dn)(£-£^

If F<0 set 2 = 2""', X = Xn, R = R n > D = Dnand X = Xn

Else if F > 0 continue with (II) otherwise EXIT(II) Calculate (AX,D) and hence n = Z/||ZJ , according to Table 2.

(III) Calculate stresses with plastic corrector :a" = (l - Dn )b* - 2^1(1 - D)AXn and a = (l - D)K(§ : 1)1 + o"

(IV) Calculate the hardening stresses :a Eq. [56] and r Eq.[57]

Table 1. Computation of the Cauchy and internal stresses

Otherwise start form the beginning

Table 2. Computation of AX and D by Newton-Raphson method ((p) stands foriterations)

5. Numerical examples

5.1. Accuracy contemplation

Let us now investigate some effects of both hardening and damage on thenumerical accuracy of the proposed stress algorithm. The convergence properties ofthe algorithm will be studied in a large domain, covering ranges of relative errorsbetween 1 and 10'7. The selected loading conditions are a two-steps loading pathunder strain control:


- first step: uniaxial strain path with eu = 5% e,2 = 0

- the second step: multiaxial strain path with en = 5% e,2 = 5%

This defines a non-proportional tensile-shear loading path making an angle of90° between the actual normal to the yield surface and the stress increment.

The relative error measure is defined by

where n is the number of increments and yref is the reference solution calculated withn = 100 000 load increments. These increments are constant for each path andequally distributed on the two loading steps. The Newton iterations are stopped for amaximum relative error of 10"10.

Figures 1 and 2 summarize the obtained results. First the equivalent stress error isplotted versus the equivalent (cumulated) plastic strain error (Figure 1) where it isclear that the error is less than 0.001 for n= 1 000 increments. Figure 2 shows thedamage error versus the equivalent plastic strain error which is still lower than 0.001for the same number of iterations. From this figure, it is clear that the convergencerate is linear and better for accumulated plastic strain than for damage.

Figure 1. Equivalent stress versus cumulated plastic strain relative errors


Figure 2. Damage versus cumulated plastic strain relative errors

Figure 3. Modeling of the quarter of the plane strain notched bar I DOG 93]

Figure 4. Stress-strain response in the vicinity of the notch root during the first 4loading cycles

Figure 5. Evolution of the back stress tensor components during the first 4 loadingcycles



5.2. Notched bar under bending cyclic strain conditions

The selected example is a notched bar subjected to a cyclic 4-points bending load,assuming plane strain state condition, already investigated by Doghri [DOG 93]. Thecalculation is achieved using the same mesh used by Doghri with the materialparameters E = 210.0 GPa, v = 0.3, ay = 200.0 MPa, Q = 520.0 MPa, b = 0.26,C = 25 500.0 MPa, a = 81.0, and the boundary conditions shown in Figure 3.

Table 3. Comparison of the iteration number for both ABAQUS and Umat for thecalculation shown in Figures 4 & 5

First we start with a comparison between our model (Umat) without damage andthe standard nonlinear isotropic/kinematic hardening available in ABAQUS/STD.Figure 4 shows that the local material response (of element 166 located at the notchroot) in terms of the first component of the Cauchy stress versus the firstcomponent of plastic strain obtained by our Umat compares well with the one

Number of iterations

Nbloadstep

22222222222222222222

Increm.Number

123456781234567891011121314151617181920

Abaqus

133323332

22

222235111

Umat

133323333

12211222235111

Number of iterations

Nbloadstep

2222222222222222222222222222

Increm.Number

21222324252627282930313233343536373839404142434445464748

Abaqus

1122211222233111222221122223

Umat

1122211222233111222221122223

11

11

111

111111

11111

11

1


obtained by the ABAQUS/STD. The same remark applies to the variation of thethree components of the back stress tensor (kinematic hardening) as shown inFigure 5. Careful examination of Table 3 shows that the consistent tangentoperator proposed in this study gives the same numerical performance as theABAQUS/STD one. Note that, in Table 3, the change between the first and thesecond load steps corresponds to the rotation of the outward normal to the actualyield surface. At that point the proposed Umat needs one additional iterationcompared to ABAQUS/SDT (see the highlighted cells in Table 3). These resultsshow that the proposed stress computation procedure based on a consistent tangentoperator possess a good numerical properties compared to the similar modelavailable in ABAQUS/STD.

5.3. Fracture prediction during hydraulic deep drawing

The last example concerns the hydraulic deep drawing of a spherical box.Starting from a circular thin sheet ( 3.0 mm Thickness and 245.0 mm radius) fixedalong its boundary on a table containing a circular hole (77.0 mm diameter), anincreasing hydrostatic pressure is applied on the top of the system table/sheet givingrise to a vertical displacement of the table (2 mm/s) aiming to maintain the sheet to afixed hemispherical punch of 72.25 mm of radius (Figure 7a). At the initialconfiguration the circular sheet is tangential to the top of the hemispherical punch asshown by Figure 7a. The Aluminum alloy sheet is characterized by the followingmaterial parameters : E = 84.0 GPa, v = 0.3, oy = 120.0 MPa, Q = 600.0 MPa,b = 3.0 (kinematic hardening being neglected), S = 200.0 MPa, s = (3 = 1.0. Figure 6shows the material response of the^ aluminium used in both coupled and uncoupledcases. The contact and friction between the sheet and the table is supposed to be ofCoulomb type with friction coefficient of 0.3. Numerical simulation aims to predictwhere and when damaged zones can be initiated inside the sheet formed during theprocess.

Figure 6. Local response of the used material in both uncoupled and coupled cases


Figure 7. Damaged zones prediction during a hydraulic deep drawing process

Figure 7 shows the comparison between the predicted (Figures 7 b,c) and theexperimentally observed (Figure 7d) damaged zones at the end of the process. Figure7b gives the numerically predicted damaged zones with the uncoupled formulation(i.e. no coupling between the damage and the elastoplastic behavior); while Figure7c gives the same numerical result obtained with the fully coupled formulation. Fromthese figures one can note that only the coupled formulation gives a result close tothe experimentally observed one concerning the fully damaged zones at the end ofthe process. As expected, the uncoupled formulation is unable to predict correctlythe location of the fully damaged zones. This shows the capability of the proposedcoupled approach to predict the damage initiation location (in space and time) duringmetal forming processes. Many other results are available in [HAM 00].

6. Conclusion

The main purpose of this paper is to derive a fully implicit stress algorithm andthe associated consistent tangent operator for a finite elastoplastic constitutiveequations accounting for nonlinear isotropic/kinematic hardening and ductileisotropic damage. A problem-optimized procedure, which reduces the fully nonlinearsystem to only two scalar equations (three equations for the plane stress hypothesis)has been proposed. It has been shown that using an asymptotic integration procedure,in conjunction with the backward Euler method, leads to very good accuracy. Theresults obtained in the prediction of damaged zones for a 3D hydroforming process


have shown the capacity of this coupled approach to optimize metal formingprocesses with respect to damage initiation.

It is worth noting that because the present formulation is local, the results ofcoupled calculations are mesh dependent. A generalization of the present model to adamage gradient formulation is under progress and will be published later.

7. References

[ARA 86] ARAVAS N., « The Analysis of Void Growth that Leads to Central Burst DuringExtrusion », 7. Mech. Phys. Solids, 34, p. 55-79, 1986.

[BON 91] BONTCHEVA N. and IANKOV R., « Numerical Investigation of the Damage Processin Metal Forming », Eng. Frac. Mech., 40, p. 387-393, 1991.

[BRU 96] BRUNEI M., SABOURIN F. and MGUIL-TOUCHAL S., « The prediction of Neckingand Failure in 3D Sheet Forming Analysis Using Damage Variable», Journal dePhysique III, 6, p. 473-482, 1996.

[CHA 78] CHABOCHE, J.L., Description Thermodynamique et Phenomenologique de laviscoplasticite cyclique avec endommagement, These de doctoral, Univ. Paris VI, 1978.

[CHA 96] CHABOCHE J.L. et GAILLETAUD G., "Integration methods for complex plasticconstitutive equations", Comput. Methods Appl. Mech. Eng., 133, p. 125-155, 1996.

[DOG 89] DOGUI, A., Plasticite anisotrope en grandes deformations, These de doctoral es-sciences, Universil6 de Claude Bernard, Lyon 1, 1989.

[DOG 93] DOGHRI I., "Fully implicit integration and consistent tangent modulus in elasto-plasticity", Int. J. Numer. Methods Eng., 36, p. 3915-3932, 1993.

[FRE 86] FREED A.D., WLKER K.P., "Exponential integration algorithm applied to viscoplasticity",NASA TM 104461,3rd Int. Conf. On Comput Plasticity, Barcelona, 1992.

[GEL 85] GELIN J.C., OUDIN J. and RAVALARD Y., « An Imposed Finite Element Method forthe Analysis of Damage and Ductile Fracture in Cold Metal Forming Processes », Annalsof the CIRP, 34(1), p. 209-213, 1985.

[HAL 75] HALPHEN B., NGUYEN Q. S., "Sur les mat6riaux standards g6neralise"s", Journal deMecanique, 14 (39), 1975.

[HAM 00] HAMMI Y., Simulation nume'rique de 1'endommagement dans les proce'de's de miseen forme, These de doctoral, Universite" de Technologic de Troyes, Avril 2000.

[HAR 93] HARTMANN S., HAUPT P., "Slress compulation and consistenl langenl operatorusing nonlinear kinematic hardening models", Int. J. Numer. Methods. Eng., 36, p. 3801-3814, 1993.

[LEM 85] LEMAITRE J. and CHABOCHE J.L., Mecanique des Milieux Solides, Dunod, Paris,French edition 1985, Cambridge Univ. Press, English edition, 1990.

[LEM 92] LEMAITRE J., A course on Damage Mechanics, Springer Verlag, 1992.


[MAT 87] MATHUR K. and DAWSON P., Damage Evolution Modeling in Bulk FormingProcesses, Computational Methods for Predicting Material Processing Defects, Edt;Predeleanu, Elsevier, 1987.

[NAG 82] NAGTEGAAL J. C., "On the implementation of inelastic constitutive equations withspecial reference to large deformation problems", Comput. Methods Appl. Mech. Eng., 33(1982), p. 494-484.

[ONA 88] ONATE E. and KLEIBER M., « Plastic and Viscoplastic Flow of Void ContainingMetal - Applications to Axisymmetric Sheet Forming Problem », Int. J. Num. Meth. InEngng. 25, p. 237-251, 1988.

[SAA 94] SAANOUNI K., FORSTER C. and BEN HATTRA F., « On the Anelastic Flow withDamage », Int. J. Dam. Mech., 3, p. 140-169, 1994.

[SAA 99] SAANOUNI K. and FRANQUEVLLLE Y., « Numerical Prediction of Damage DuringMetal Forming Processes », Numisheet 99, Besanc,on, September, France, p. 13-17, 1999.

[SAA 00] SAANOUNI K., NESNAS K. and HAMMI Y., « Damage modelling in metal formingprocesses », Int. J. of Damage Mechanics, Vol 9, n° 3, p. 196-240, July 2000.

[SAA 00] SAANOUNI K., HAMMI Y., « Numerical simulation of damage in metal formingprocesses », in Continuous Damage and Fracture, Editor A. Benallal, Elsevier, p. 353-363, 2000.

[SIM 85] SIMO J.C., TAYLOR R., "Consistent tangent operators for rate independentelastoplasticity", Comput. Methods Appl. Mech. Eng., 48 (1985), p. 101-118.

[ZHU 92] ZHU Y.Y., CESCOTTO S. and HABRAKEN A.M., « A Fully Coupled ElastoplasticDamage Modeling and Fracture Criteria in Metal forming Processes », /. Met. Proc.Tech., 32, p. 197-204, 1992.

Chapter 12

3D Nonlocal Simulation of DuctileCrack Growth: A Numerical Realization

Herbert Baaser and Dietmar GrossInstitute of Mechanics, Darmstadt University of Technology, Germany


3D Nonlocal Simulation of Ductile Crack Growth 211

1. Introduction

Many structural parts of technical applications require a detailed computationalanalysis of their load-carrying capacity either during the design phase or later on whileoperating in a larger system or in a machine. Today a computational evaluation andsimulation of such structural parts is also able to consider the influence of damage andfailure occurance by continuum damage models implemented in the framework of theFEM. A well-known disadvantage in the numerical treatment of solid mechanics pro-blems, where softening material behavior occurs, is the so-called mesh-dependenceof numerical results. In a considerable number of investigations different methodshave been proposed to overcome mesh-dependence of finite element results. Thecommon idea is the introduction of a characteristic or internal length (scale) intothe constitutive model or its evaluation. We consider four different types of models.COSSERAT models consider in addition to the displacement of a material point also itsrotation as an independent kinematic variable, see [EHL 98]. The internallength introduced by this was shown to determine the width of shear bands especiallyin soil materials, where the additional rotational degrees of freedom are activated du-ring the deformation history. For such shear dominated problems this method seemsto be a suitable regularization technique. Another advanced regularization method isthe introduction of higher displacement gradients as additional degrees of freedom.A consistent formulation in terms of small strains is available, see e.g. [BOR 99]. Atotally different approach to model the failure occurrence is the discrete representa-tion of the actual failure or damage mode. Such models are able to represent failureoccurrence by special finite elements either capable of showing displacement jumpsinternally within the element structure, see [OLI96], or between the element edgesthrough a specific cohesive law defining the stress-strain behavior, see e.g. [HOH 96]or [BAA 97]. Another type of regularizing approaches is known as non-local, basedon a spatial smoothing of certain quantities over the volume or structure of interest,see [BAZ 88]. The main goal of this type of model is an additional evaluation of a vo-lume integral for the internal variables like plastic strain or damage, convoluted e.g. bya bell-like kernel function, see [PIJ 93]. Many authors used this smoothing techniqueas in [LEB 94], [TVE 97] or [BAA 98]. They computed the smoothing of the damageparameter as an additional process applied to the actual result of the local solution ofthe constitutive equations. In this paper we describe a new approach of evaluating thevolume integral for the increment of the damage parameter used. The idea is to com-pute the smoothing during the iteration of the set of constitutive equations, which isadvantageous due to the iterative character of the equation solver applied. In this study20-noded brick 3D elements are used with quadratic shape functions along the ele-ment edges. [MAT 94] have found that this element type well represents localizationin the 3D case. As constitutive model we use the ductile damage model of [ROU 89].An advantage of this model is the description of material softening behavior due to da-mage by the influence of just three material parameters. Nevertheless, there are onlya few articles treating the calibration of the ROUSSELIER parameters to experimentaldata, see [ROU 89] and [LI 94]. A second advantage is related to the numerical im-plementation of the constitutive law by means of an implicit integration scheme. The


type of constitutive equations leads to symmetric tangent material moduli, which isadvantageous in computing and storing the matrix expressions.

2. Nonlinear solution & nonlocal formulation

2.1. Classical iterative solution procedure

In order to approximate the real nonlinear behavior of a structure, the final load,leading to an unknown displacement result u, is usually divided into smaller loadsteps. Thus, for each load level the structural response is computed by finding theactual equilibrium, using an iterative solution procedure. The accumulation of the in-cremental solutions Au of the displacement field results in the total answer u.

Weak Form of Equilibrium

Consistent Linearization

FIG. 1. Classical iterative FEM solution procedure for a given load level

Starting from the weak form of equilibrium <?(u, Su) = 0, where Su indicates thevector of test functions which is identified with the virtual displacements, an iterativesolution method can be constructed to determine for the unknown displacements u at


a given load, with prescribed boundary displacements u. The displacements describethe difference between the current and the initial configuration by u = x — X. Therepresentation of g (u, 8u) = 0 in terms of a TAYLOR series from a known positionu with Ail = u - u leads to a first order approximation of the weak form. Thusexpression is the basis for the global iteration loop to find the increments Au and thusthe new displacements u. This procedure is schematically described in Fig. 1, where# describes the body considered in the current configuration with the volume v andwith tboun as traction vector applied on the boundary dBa. In the next load step thenew solution of the system has to be determined by repeating this iteration.

2.2. Nonlocal formulation

In the usual iterative procedures all quantities are stored locally on the level ofintegration points. However, the nonlocal approach which is proposed here acts onthe level of internal quantities on the integrations points. [BAZ 88] have shown that anonlocal treatment of the damage quantities leads to mesh-insensitivity. For this pur-pose we apply the nonlocal smoothing integral to the increment A/? of ROUSSELIERSdamage quantity

where the kernel function, chosen as

for an arbitrary position vector p is responsible for the nonlocal smoothing of the lo-cal quantity A/?'0"' with respect to the reference state X. Formally the characteristiclength lc describes the standard deviation of the normal distribution. For a discus-sion of the factor k depending on the spatial dimension of the problem considered see[BAZ 88]. Following their arguments k = (6^/n)1/3 ~ 2.2 for the 3D case conside-red here. For the numerical implementation the volume integral [1] is applied for eachintegration point considering just the surrounding by a sphere of radius 3lc, which issufficient for a tolerable integration error in this application. A possible crossing ofsymmetry lines or planes during the integration loop is respected by accounting forthe virtual contributions of the volume integration. But the whole computation of therespective influence of the neighbourhood of each integration point can be calculatedonce as a preprocessing procedure and takes just a few minutes for the problems con-sidered. The main issue of the new solution algorithm is the assembling of the globalstiffness matrix and the global residual vector. In Fig. 1 the global equilibrium itera-tion is illustrated, with the use of the residual p(u, <5u) and the actual system stiffnessT>g(u, 6u). In the FE representation the quantity #(11,6u) results in the residual vector


and the system stiffness T>g(a, 6u) in the global stiffness matrix, respectively. Duringthe assembling procedure for these terms the actual increments for the local descrip-tion of the constitutive behavior are computed. The numerical procedure for this kindof solution is described in the Appendix (see eqn. [23]), where the calculation of thelocal solution of the constitutive equations is demonstrated. In contrast, the new ideaconsists of a global computation of the increments Ax in [23] by a NEWTON-like ite-ration scheme. The specific algorithmic treatment of this iteration is described shortlyin Section [4.2] and in an earlier version in [BAA 00]. Due to the iterative characterof the new solving scheme it is possible to modify the actual solution of the incrementof the damage parameter /? by a nonlocal approach. This modification is not possiblein the traditional iterative solution procedure, where the evaluation of the constitutiveequations is treated in each integration point violating the yield condition, withoutconsidering its spatial position. On the contrary, with this new approach a "commu-nication" between the integration points is enabled due to eqn. [1], where the localincrement of the damage parameter is modified depending on its location. The descri-bed nonlocal modification of the increment A/3 of the damage parameter influencesthe solution convergence of the set of constitutive equations, but this intervention actsin a moderate way so that convergence is obtained. In contrast to the method describede.g. in [BAA 98], we here operate in small steps, which influences the solution of theconstitutive equations by the nonlocal character, but guarantees the global quadraticconvergence behavior.

3. Finite strain plasticity and damage model

3.1. Finite strain plasticity

In elastic-plastic solids under sufficiently high load finite deformations occur,where the plastic part of the strains usually is large compared with the elastic part.The description of finite plastic deformations in conjunction with damage modelsis often carried out using the additive decomposition of the elastic and plastic strainrates, [TVE 89]. Here however, we use the framework of multiplicative elastoplasti-city which is widely accepted in plasticity. Its kinematic key assumption is the multi-plicative split of the deformation gradient

into an elastic and a plastic part, providing the basis of a geometrically exact theoryand avoiding linearization of any measure of deformation. As a further advantage, fastand numerically stable iterative algorithms, proposed and described by [SIM 92], canbe used. In the following, only a brief summary of the algorithm in the context of aFE-implementation is given.

An essential aspect of [3] is the resulting additive structure of the current logarith-mic principal strains within the return mapping scheme as


Here, e; = In \i (i = 1,2,3) and A? are the eigenvalues of an elastic trial state,described by the left CAUCHY-GREEN tensor b^. The elastic strains eel are definedby HOOKE'S law and the plastic strain corrector AepZ can be derived by the normalityrule of plastic flow. The elastic left CAUCHY-GREEN tensor can be specified with thedecomposition [3] as

which clearly shows the "connection" between the elastic and plastic deformationmeasure by the occurance of the plastic right CAUCHY-GREEN tensor Cp/ = F^ • Fp/.By means of the relative deformation gradient (see [SIM 92])

which relates the current configuration x to the configuration belonging to the previoustime step at £n-i> an elastic m'a/-state is calculated for the current configuration attime tn

with frozen internal variables at state £n-i- If the condition $ < 0 (see eqn. [10]) isfulfilled by the current stress state T, this state is possible as is the solution. If, onthe other hand, $ < 0 is violated by the trial-state, the trial stresses must be projectedback on the yield surface $ = 0 in an additional step. This "return mapping" procedureis used as the integration algorithm for the constitutive equations described in Section3.2. It should be mentioned that the algorithmic treatment in terms of principal axeshas some advantages concerning computational aspects like time and memory saving.Based on this, the integration procedure of the constitutive equations for large and forsmall deformations is very similar, [ARA 87].

3.2. The Rousselier damage model

First, some notations and characters, which will be used in this description of theconstitutive law and later on in the algorithmic setting, are specified. Following theideas of [ARA 87] we decompose the stress and strain tensors in scalar values, whichis of great advantage for the numerical implementation. Thus, we write the KlRCH-HOFF stress tensor r as weighted CAUCHY stress tensor as follows

where J := detF = ^ = ^ and p = — \Tij6 ij defines the hydrostatic pres-

sure, q = \l\tijiij the equivalent stress and tij = r^ 4- p6ij are the components ofthe stress deviator. In this notation an additional important quantity is the normalized


stress deviator nr = ^t. The second order identity tensor 1 is defined as the KRONE-CKER symbol by its components (5y in the Cartesian frame. In an analogous way theplastic strain rate can be written as

where A£P and A£? describe scalar rate quantities which are defined later on. Theconstitutive model used in this study is the damage model proposed by [ROU 89].Here, the yield function taking ductile damage processes into account may be writtenas

where cr* represents the material hardening in terms of a power law, and the last partof [10] describes the damage (softening) behavior by the function B((3) and an expo-nential assumption. Furthermore, E is YOUNGS modulus, a$ the initial yield stress, Nthe material hardening exponent, and D and cr\ are damage material parameters. Notethat the formulation of [10] in quantities p and q as functions of the KlRCHHOFF stresstensor r is comparable to the original formulation F = ^- + ... = 0 in [ROU 89],

because we can write ^^ = ^^ J = ^^ = q for po = 1 in the case of 8 = 0,P P Q P O * r U / - >

see also comments from the authors on eqn. (17) in [ROU 89]. The function of B(j3)is the conjugate force to the damage parameter /?, defined by

Here, the initial void volume fraction /o is the third damage depending materialparameter used in this constitutive set of equations. The set of constitutive equationsis complemented by the evolution equations for the plastic strain ep

elqv and the damage

parameter /?. The macroscopic plastic strain rate ep is determined by the classicalassociated flow rule

Note that epl coincides with the plastic increment Aepi for the algorithmic settingin [4] written in principal axes. The last bracket on the right hand side of [12] showsa further advantage of this formulation following [ARA 87], since it is very easy todetermine the derivatives of $ with respect to the scalar quantities q and p. One cansee with [9] and [12] that


These two equations allow the algebraic elimination of A if

is fulfilled. Thus, the increment of the plastic strain can be expressed by the two scalarquantities A£P and Aeg. Then the equivalent plastic strain e%l

qv can be incrementeddirectly by Aeq. The evolution equation for the damage parameter fi is given by

which is obviously dependent on the deviatoric part of the strain rate Aeg and theactual hydrostatic pressure p. With this the whole set of constitutive equations is com-pleted.

The evaluation of the material model for a given load level requires the solutionof the three equations [10], [14] and [15] for the unknowns Aep, Aeq and A/?, res-pectively. Classically, this evaluation would be done pointwise at the local level of theintegrations points by an implicit EULER backward integration rule, which is descri-bed in the Appendix. In Section 2.2 we discussed a new approach, which allows fornonlocal formulation; its algorithmic treatment is shown in Section 4.2.

The exact linearization of the set of equations follows the description in [ARA 87].At this point we just mention the starting point of the linearization

where C is the elastic modulus defined by the LAME constants. The variational ex-pression for [16] is found as

Some extended algebraic manipulations on [17], as described in [ARA 87], lead to theexpressions <5Aep and <5Ae9 and finally to

which is the material modulus for the implicit integration procedure at the end of theconsidered time interval [£, t + Ai].

4. Finite element formulation

4.1. 3D-element

The starting point is the weak form of equilibrium g(u, <5u), see Fig. 1, formulatedin the current configuration, where u is the displacement and ti are the prescribed


tractions acting on the loaded boundary dBa of the body in the current configurationB. Linearization with respect to the current deformation state, and rearrangement leads(with dv = J dV) to the following representation of the element stiffness

where J = det F and BQ denotes the reference configuration. As for the global resi-

FlG. 2. 20-noded solid element

duum vector resulting from g(u, 6u) the elementwise results from [19] are assembledto the global stiffness matrix K. For further explanations on the implementation of theconsistent linearization of the algorithm used, see [SIM 92] and the modifications in[REE 97] concerning the determination of the eigenvalue decomposition.

The discretization chosen in this paper is based on a 20-node-displacement ele-ment formulation with shape functions A^, (i = 1,2,..., 20), so that quadratic func-tions describe the element edges. Fig. 2 shows such an element in an arbitrary confi-guration. As in [MAT 94] we use a 2 x 2 x 2-integration scheme, which means anunderintegration with respect to the quadratic shape functions Ni. It shall be pointedout that again no hourglassing modes were detected like in the case of an 8-node-displacement element formulation and a 1 x 1 x 1-integration scheme, see [BAA 98].

4.2. The algorithmic treatment of the new approach

As described above, the iterative solution procedure used for solving nonlinearproblems by the finite element method requires an evaluation of the constitutive equa-tions on the level of integration points. This is known as the lowest level of iterationin contrast to the global load/time incrementation and the subsequent global iterationfulfilling the weak form of equilibrium.

In our new approach the originally local evaluation of the set of constitutive equa-tions on each integration point is shifted from the lowest level to a global solutionwhile assembling the system stiffness matrix and the right hand side residual vector.


Instead of solving iteratively for each integration point on the element level, we as-semble a large system of equations where the set of equations from each integrationpoint enters blockwise along the main diagonal of the new system matrix, see also[BAA 00].

The global iteration scheme to find the current equilibrium requires in every itera-tion loop the assembling of the (global) system matrix and the right hand side vector.Parallel to this assembling procedure we solve the new global system of constitutiveequations by an NEWTON-like iteration scheme. Due to the blockwise structure ofthe resulting system matrix it is possible to invert this matrix block-by-block andreassemble the resulting 3 x 3-submatrices for the use in the NEWTON-like itera-tion scheme. The character of a full JACOBIAN matrix for a real NEWTON scheme islost because of the coupling of the quantities A£P, Aeg and A/? through [1]. But thiscoupling is very weak in comparison to the dependence of the constitutive equations[20]-[22] in the local evaluation. So the rate of convergence is nearly quadratic asassumed for a full NEWTON scheme, see also time consumption in Tab. 2. Havingdetermined the nonlocal results, the global stiffness and right hand side for the globalequilibrium iteration can be assembled following the same scheme as for the classicalsolution technique.

5. Example and results

5.1. Model of a CT specimen

As an example we examine a three dimensional model of a CT specimen discreti-zed by 20-node solid elements as shown in Fig. 3. Due to symmetry just a quarter of

FIG. 3. Model of CT specimen

the structure is modeled, where the edge lengths used are shown in the right hand part of


Tab. 1. The loading is applied by the displacement up of the marked nodes and therespective nodes located in the middle of each element edge length, see Fig. 2 andFig. 3. For simplicity applying an uniform numerical postprocessing procedure andfor getting easier comparable results we just take the results of the marked nodes intoaccount and sum up their reaction force to get a global answer of the structure. Thisprocess can also be justified by the long distance between the loaded nodes andthe zone of interest in front of the crack tip. So the resulting reaction forces on theloaded nodes are uniformly distributed and just scaled by a factor resulting from thequadratic shape functions along the loaded element edges. Here, we distinguish twodifferent mesh types named by the typical element edge length e in front of the cracktip. The first mesh is characterized by e = 1.0 mm, the other one by e — 0.5 mm.The set of material parameters used is shown in Tab. 1, where the first four parameters

E I v I cr0 I N II D I ai I /Q II fF III r I / I b I hImJOOO | 0.2 | 460 | 7 I 3 | 300 | 0.01 || 0.19 ||| 6 | 5 | 3 | 5~

TAB. 1. Material parameters, stress dimension [MPa], and geometry in [mm]

can be obtained by simple tensile tests, and D, a\ and /o are responsible for the da-mage representation of the constitutive model. Ductile crack growth is represented byreaching the threshold value of fp = 0.19 by the local value of the volume fraction/ expressed in terms of the damage parameter /? by / = -~, see Eqn. [11], in anyintegration point. This final value fp as threshold for the evolution of / is not urgentin this formulation using the ROUSSELIER damage model and its damage parameterft, but in order to get comparable results to the use of e.g. the GURSON damage mo-del, see e.g. [BAA 98], it is very attractive. The maximum distance of the respectiveintegration points from the initial crack tip defines the actual crack growth Aa. Toshow the effect of the nonlocal regularization technique introduced here, the charac-teristic length scale is fixed to lc = 1.0 mm for the two different mesh types. So lc

corresponds to the coarse discretization with e = 1.0 mm.

5.2. Results

The response of the nodal reaction force vs. the applied displacement up is plottedin Fig. 4. It can be seen from the curves (1) and (3) that the 3D model behaves inthe usual way depending on the discretization for the traditional local approach withlc = 0.0 mm. The finer the used FE mesh is, the earlier the onset of strain localizationis reached. Afterwards the negative slope of the respective curves gets steeper withincreasing refinement of the discretization.

In contrast to these classical mesh sensitive results, the responses of the nonlocalapproaches show a different behavior for the two types of discretization. Although thetypical element edge length e is varied from e = 0.5 mm to e = 1.0 mm, the curves(2) and (4) characterizes nearly the same behavior of the regularized calculations with


FlG. 4. Reaction force vs. end displacement for different discretization

the identical internal length lc — 1.0 mm, while the onset of global softening can beobserved for all curves at a displacement of about up — 0.38 mm on the loadednodes. Fig. 5 represents the maximum values for the void volume fraction / in thefirst integration point in front of the initial crack tip in the center of the specimenvs. the number of load steps. These values are plotted for the cases (2), (3) and (4)to show the effect of the regularization technique. In addition, the threshold value offp = 0.19 is marked by a dashed line indicating the transition of this value by thedifferent calculations at different load steps. The crack advance Aa, which corres-ponds to this maximal value of / in the front of the crack tip, is marked by dots in theplot. This shows that the local analysis (3) responses with a larger crack advance ofAa ~ 1.7 mm at load step 90, while the nonlocal representations (2) and (4) for bothdiscretization result in a crack growth of about Aa ~ 0.8 mm (the accurate positionof the respective integration points varies due to the different element edge lengths infront of the crack tip) at this load step 90. So, these regularized results show nearlythe same crack advance, although different discretizations are used. Tab. 2 shows

DOF Time : Equiv. Inv. Time : gbl. NLoc"e = 1.0 mm 850 0.08 sees. 0.61 sees.

e = 0.5 mm 7545 8.5 sees. 10.1 sees.

TAB. 2. Comparison of computational costs iterating the global equilibrium at loadlevel 71

a comparison of time consumption for calculations with two different discretizationse = 1.0 mm and e = 0.5 mm and the resulting degrees of freedom. The third columnshows the time for inverting the FE system matrix by an advanced GAUSS eliminationduring the iteration of the equilibrium state on an IBM RS6000/397 workstation. Wi-


FIG. 5. Maximum values of f = -^- in the center of the specimen

thin the considered load level of step 71 about seven iteration steps are needed to reacha converged situation. This time consumption is independent from the local or nonlo-cal evaluation procedure. The last column represents the duration for solving the set ofconstitutive equations on the global level within the proposed nonlocal approach. Theclassicaly local solution needs —in contrast to that— a vanishing time to assemblethe FE stiffness matrix, indeed without nonlocal smoothing. So one can see that thetime consumption for solving the nonlocal system increases obviously with increasingnumber of integration points, but the amount of time is in tolerable limits respectingthe advantage of a nonlocal evaluation. These first results for the application of a newnonlocal solution approach demonstrate the capability of this technique in smoothingthe classically local results on the level of integration points to a nonlocal distributionof the damage parameter j8, which is responsible for the softening behaviour of astructure. In this sense it can be seen that the applied modification of the classical FEsolution approach, shifting the level of iteration of the constitutive equations to amore global, spatially dependent one, affects the structural response of a loaded spec-imen. Keeping these results in mind, attractive future work could be a modification ofthe classical FE solution approach with respect to a useful possibility to enable a"communication" between the integration points depending on their spatial position.

6. Summary

In this contribution we show a study of a 3D simulation of CT specimen using theROUSSELIER damage model combined with a 3-dimensional finite element formula-tion based on 20-node-solid elements. The main attention is put to a new algorithmic


approach to the iteration of the constitutive equations applying a nonlocal regulariza-tion to avoid mesh sensitive results.

The first results for this treatment are shown here and underline the algorithmicapplicability of the new approach. The additional time consumption for the nonlo-cal evaluation is in tolerable limits, although some further investigation concerninga faster realization of the new approach have to follow. Studies especially about thecalibration of the model to experimental data are in preparation.

7. Bibliography

[ARA 87] ARAVAS N., « On the numerical Integration of a Class of pressure-dependent Plas-ticity Models », International Journal for Numerical Methods in Engineering, vol. 24,1987, p. 1395-1416.

[BAA 97] BAASER H., HOHE J., GROSS D.,« Ductile crack growth analysis using the Gursondamage model », KOSINSKIW., DE BOER R., GROSS D., Eds., Problems of Environmentaland Damage Mechanics, n° ISBN 83-906354-1-0, Warszawa, Poland, 1997, p. 139-147.

[BAA 00] BAASER H., TVERGAARD V., « A New Algorithmic Approach treating NonlocalEffects at Finite Rate-independent Deformation using the ROUSSELIER Damage Model »,submitted to Computer Methods in Applied Mechanics and Engineering, , 2000, see alsoDCAMM Report 647, TU Denmark, Lyngby.

[BAA 98] BAASER H., GROSS D., « Damage and Strain Localisation during Crack Propaga-tion in Thin-Walled Shells », BERTRAM A., SIDOROFF F., Eds., Mechanics of Materialswith Intrinsic Length Scale, n° ISBN 2-86883-388-8, Magdeburg, Germany, 1998A, EDPSciences, p. 13-17, Journal de Physique IV, 8.

[BAZ 88] BAZANT Z., PIJAUDIER-CABOT G., « Nonlocal Continuum Damage, LocalisationInstability and Convergence », Journal of Applied Mechanics, vol. 55, 1988, p. 287-293.

[BOR 99] DE BORST R., PAMIN J., GEERS M., « On coupled gradient-dependent plasticityand damage theories with a view to localization analysis », European Journal of Mechanics-A/Solids, vol. 18, 1999, p. 939-962.

[EHL 98] EHLERS W., DIEBELS S., VOLK W., « Deformation and Compatibility for Elasto-plastic Micropolar Materials with Applications to Geomechanical Problems », BERTRAMA., FOREST S., SIDOROFF F., Eds., Mechanics of Materials with Intrinsic Length Scale,Magdeburg, Germany, 1998, p. 120-127.

[HOH 96] HOHE J., BAASER H., GROSS D., « Analysis of ductile crack growth by means ofa cohesive damage model », International Journal of Fracture, vol. 81, 1996, p. 99-112.

[LEB 94] LEBLOND J., PERRIN G., DEVAUX J.,« Bifurcation Effects in Ductile Metals withNonlocal Damage », ASME Journal of Applied Mechanics, vol. 61, 1994, p. 236-242.

[LI 94] Li Z., BiLBY B., HOWARD I., « A study of the internal parameters of ductile damagetheory », Fatigue & Fracture of Engineering Materials & Structures, vol. 17, n° 9, 1994,p. 1075-1087.

[MAT 94] MATHUR K., NEEDLEMAN A., TVERGAARD V.,« Ductile failure analyses on mas-sively parallel computers », Computer Methods in Applied Mechanics and Engineering,vol. 119, 1994, p. 283-309.


[OLI 96] OLIVER J., « Modelling strong Discontinuities in Solid Mechanics via Strain Softe-ning Constitutive Equations : Part I and II », International Journal for Numerical Methodsin Engineering, vol. 39, 1996, p. 3575-3623.

[PIJ 93] PIJAUDIER-CABOT G., BENALLAL A.,« Strain localization and bifurcation in a non-local continuum », International Journal of Solids and Structures, vol. 30, n° 13,1993,p. 1761-1775.

[REE 97] REESE S., WRIGGERS P., « A material model for rubber-like polymers exhibitingplastic deformation : computational aspects and a comparison with experimental results »,Computer Methods in Applied Mechanics and Engineering, vol. 148, 1997, p. 279-298.

[ROU 89] ROUSSELIER G., DEVAUX J.-C., MOTTET G., DEVESA G., « A Methodologyfor Ductile Fracture Analysis based on Damage Mechanics : An Illustration of a LocalApproach of Fracture », Nonlinear Fracture Mechanics, vol. 2, 1989, p. 332-354.

[SIM 92] SiMO J., « Algorithms for Static and Dynamic Multiplicative Plasticity that preservethe classical Return Mapping Schemes of the infinitesimal Theory », Computer Methods inApplied Mechanics and Engineering, vol. 99, 1992, p. 61-112.

[TVE 89] TVERGAARD V., « Material Failure by Void Growth to Coalescence », Advances inApplied Mechanics, vol. 27, 1989, p. 83-151.

[TVE 97] TVERGAARD V., NEEDLEMAN A., « Nonlocal Effects on Localization in a Void-Sheet », International Journal of Solids and Structures, vol. 34, n° 18, 1997, p. 2221-2238.

Appendix : Iteration of the set of constitutive equations

The unknowns Aep, Aeq and A/3 are determined by a NEWTON-like iteration scheme. Theelastic trial-stress state is determined as described in Section 3.1, where also the quantities ptr

and qtr are obtained as described below [8]. The "return mapping" iteration described here iscarried out for integration points violating the yield condition [10], either on the local level oras shown in Sections 2.2 and 4.2 with the new global algorithm.

For having a better condition of the iteration system the function r\ is weighted by YOUNGSmodulus, so that a dimensionless expression results as for TI and ra.Evaluate for i, j — 1,2,3

where Ax is either calculated directly on each integration point by Arti = J^1 r, or calculatedusing a global iteration scheme where the nonlocal smoothing [1] is applied after every iterationstep as described in the previous sections. So one gets [xi, x%, ZS]T = [Aep, Aeq, A/?]T. Thequantities ej^r and /3tr are used to store the accumulated history of the actual integration pointin addition to the tensorial quantity needed in [7].

Chapter 13

On the Theory and Computation ofAnisotropic Damage at Large Strains

Andreas Menzel and Paul SteinmannChair of Applied Mechanics, Department of Mechanical Engineering, University ofKaiserslautern, Germany


Anisotropic Damage at Large Strains 227

1. Introduction

The main objective of this work is the development of a phenomenological,geometrically nonlinear formulation of anisotropic tensorial second order continuumdamage.

In fact, it is the evolution of microscopic internal structure of materials bynucleation and growth of distributed microcracks or microvoids which in turn leadsto the deterioration of the mechanical properties of the material. Especially the shape,orientation and evolution of these microdefects show a significant dependence onthe direction of stress and strain. Obviously, the nature of damage is anisotropic andthus a continuum damage theory should provide sufficient freedom to capture theseanisotropic damage effects.

The appropriate choice of the physical nature of mechanical variables describingthe damage state of a material and their tensorial representation is since long underdiscussion, for an overview see e.g. Lemaitre [LEM 96] or Krajcinovic and Lemaitre[KRA 87] and the literature cited therein. Following the attempts of Betten [BET 82]and Murakami [MUR 88], the well-known concept of deformed and reference,or rather undeformed, macroscopic configurations of a material body within thegeometrically nonlinear continuum theory is supplemented by the concept of fic-titious undamaged microscopic configuration. Nevertheless, in strong contrast tothe classical approaches mentioned above, the present damage theory, fully outlinedin Steinmann and Carol [STE 98] and further exploited in Menzel and Steinmann[MEN 00], is based on the notion of a second order damage metric tensor and itseffects on the stored strain energy. Thereby, as the fundamental assumption, thestorage of strain energy due to either nominal or effective strains is measured byeither the damage or the energy metric based on the hypothesis of strain energyequivalence between microscopic and macroscopic configurations, see e.g. Sidoroff[SID 81]. The framework of standard dissipative materials, as proposed by Halphenand Nguyen [HAL 75] is strictly applied. Another approach to formulate anisotropicdamage based on the introduction of an internal second order damage tensor similarto structural tensors has been given in Menzel and Steinmann [MEN 99].

2. Anisotropic hyper-elasticity based on a fictitious configuration

The reference and spatial configuration of the body of interest are denoted byBQ C E3 and Bt C E3. Let <p(X,t) : B0 x R+ -> Bt represent the non-linearmap of material points X G BQ onto spatial points x = <p(X,t) G Bt. In terms ofconvected coordinates 9l(x) and Ql(X) the natural and dual base vectors are givenby the derivatives 0; = d^x € TBt, g{ = 6*0* € T*Bt, G, = d&X € TB0

and Gl = dx®1 € T*BQ. Now, the spatial and material metric tensors followstraightforward - g* = gijg

i <g> g*, g* = g*Jg. <g> 9j, Gb = GijGi ® Gj,

G" = Gli Gi <8> GJ - and in addition we introduce the mixed-variant identity-tensors

9^ — 9i ® gl and G*5 = G, <8> G\ On this basis, the linear tangent map of the directmotion reads F^ = dx<P = dgi<f> <8> d\0l = g{ <8> Gl € GL+ and, for notational


simplicity, the gradient of the inverse motion 3>(x,£) = (p~l : Bt x 1+ -> BO readsf^ — dx& — dQ*3> <S> dxQ

l = Gi <g> gl € GL+. Several kinematic tensors can beintroduced, e.g. the Finger tensors b" = F^ • G" • [F^Y = Gij g{ <8> #., or the rightCauchy-Green tensor Cb = [F^f • 0b • F^ = gij Gi <8> Gj which enters the definitionof the Green-Lagrange strain tensor E* = [C — G ]/2. A graphical representationis given in Figure 1 and in view of a detailed outline on non-linear kinematics werefer to the work of Marsden and Hughes [MAR 94].

2.1. The fictitious configuration

Now, in addition to the physical and material space we introduce a fictitiousisotropic configuration with natural tangent space TBo and corresponding dual spaceT*BQ. In analogy to the intermediate configuration within the multiplicative de-composition of elasto-plasticity, the fictitious configuration is generally incompati-ble. Mathematically speaking, we have a non-vanishing Riemann-Christoffel tensorwhich means that the conditions of compatibility are not fulfilled and the correspond-ing direct fictitious linear tangent map - denoted by F - takes the interpretation as anon-holonomic Pfaffian, see e.g. Haupt [HAU 00]. Nevertheless, within the proposedmultiplicative composition F allows the interpretation as pre-stretch and defines thefictitious base vectors Gi G TBo and G1 G T*Bo which are obviously not derivablefrom position vectors. Next, the fictitious metric and identity-tensors follow as

The linear tangent maps due to the direct and inverse fictitious motion read

see Figure 1 for a graphical representation. Without loss of generality usual push-forward and pull-back operations in terms of the fictitious linear tangent map hold,e.g. E* = [F13]4 • E* • F* = [9ij - Gtj\/2 & ® Gj .

2.2. Energy metric tensors

In the sequel we incorporate a contra-variant energy metric tensor which readswithin the fictitious and undeformed setting as follows

whereby the push-forward operation

is implied. As a key idea, the fictitious energy metric tensor is chosen equal to thefictitious contra-variant metric tensor and replaces this metric within the construction


Figure 1. Non-linear point map (f> and linear tangent maps F and F^

of the free Helmholtz energy function i/V Consequently, due to the principle of strainenergy equivalence the relation

holds, which means that the free Helmholtz energy remains invariant under the actionofF*.

NOTE 2.1 - Note that eq. (5) includes all assumptions of the proposed framework. Inparticular isotropy is included if the energy metric tensor A^ is spherical whereas

u u

otherwise anisotropy is considered. Since the relation A = G is incorporated, thefictitious configuration is isotropic and thus standard isotropic constitutive equationcan be applied to model anisotropic material behaviour.

NOTE 2.2 - For conceptual simplicity we focus here on the composition F^ • F anddo not consider the spatial fictitious tangent space TBt-

2.3. Hyper-elasticity

Since the fictitious configuration is isotropic three (basic) invariants in terms ofL u

E and A enter the formulation. Application of the standard pull-back operationsEb = [F^ • Eb • F* and A* = /"" • A* • [/"]* renders two corresponding sets ofinvariants

with n = 1,2,3. Thus the usual hyper-elastic framework yields e.g the second Piola-Kirchhoff stress tensor


and the Hessian E E £B — d^>Ê\>^Q takes a similar format compared to standardisotropy.

2.4. Representation of the energy metric tensor

In order to demonstrate the nature of the energy metric tensor we choose thefollowing ansatz for the fictitious contra-variant vectors & with respect to theanisotropic reference configuration BQ (with AN^,AN^ 6 S"2)

With this representation in hand the two-field tensor F = Gi <S> G* reads

U tl

Now, straightforward computations due to eq.(4) with A = G render the symmetricenergy metric tensor

whereby the abbreviated notations /30 = o%, fi\ = 2 a0 a\ + o%, 02 = 2 aQ a2 + ot\,/3s = ai ct2 have been applied.

u

NOTE 2.3 - The rank one tensors A\ )2 allow similar interpretation to structuraltensors. Indeed, the incorporation of eq.(10) into eq.(6) yields a set of invariantswhich can be expressed as a function of the set of invariants for general orthotropy asgiven e.g. by Spencer [SPE 84].

NOTE 2.4 - Within geometrically linear orthotropic hyper-elasticity based on struc-tural tensors up to nine independent material parameters are included, see e.g.Spencer [SPE 84J. Contrary, the formulation based on the fictitious configuration in-corporates four independent parameters (thereby the two Lame constants are takeninto account, and thus the additional isotropic parameter ctQ is not independent.)This underlines that we deal with a reduced formulation. For the sake of clar-ity, we consider the constant tangent operator of a material of linear St.-VenantKirchhofftype ^k£» = d2

EÊ^0 = A A8 <g> A* -I- /z[A t t® A8 + A8® A*].Now, referring to a Cartesian frame Ci, we choose without loss of generality A =@Q I + /?i e\ <8> e\ + 02 €2 <8> 62- Then, the relevant coefficients of the Hessian read


Obviously, we deal with a sub-class of rhombic symmetry.

2.5. Numerical examples

For the following numerical examples a non-linear material of Kauderer-type isapplied, see Kauderer [KAU 49]. Within the general representation

compare Ogden [OGD 97], the coefficients of the implemented constitutive equationread:

The subsequent material parameters are chosen: K = 8.333 x 104, G = 3.8461 x 104,/ôi _ Kdev = 0.5 anci Kvoi _ ^dev _ g.25. In view of the energy metric tensorthe following spherical coordinates define the orthogonal unit-vectors AN\^,\ti\ - 5/6 TT, tff = 1/6 TT, $\ = 1/3 TT and tf2, = 1/2 TT, see the Appendix, and theadditional scalars to compute F read a0 = 1.0, ai = 0.25 and a2 = 0.5.

2.5.1. Simple shear

For the homogeneous simple shear deformation (F = I + ê\ <g> e2 with I =dij €i <8> GJ ) Figures 2 and 3 highlight the anisotropic behaviour of the applied material,compare Appendix. The anisotropy measure 6 shows a strong dependence on the shearnumber 7. Typically, the stereographic projection with respect to the stress and straintensors underline their non-coaxiality. Finally, the plots of the determinant of theacoustic tensor show a different shape for the anisotropic and isotropic (ai = a? = 0)setting. They are given at 7 = 0.25 in normalised form with respect to the linearisotropic case with det(g)lin'iso = G2 [3/4 G + K}.

2.5.2. Cook's problem

A three-dimensional version of Cook's problem has been discretised with 16 x16 x 4 enhanced eight node bricks (Q1E9) as advocated by Simo and Armero[SIM 92]. Figure 4 shows the reference geometry (L = 48, HI = 44, H2 = 16,T = 4), boundary conditions and a deformed mesh at \\F\\ = 1.28 x 105 whichis the amount of the conservative resultant force of a continuous shear stress inBQ . Furthermore, Figure 5 highlights the displacement of the mid point node at thetop corner for the anisotropic meterial and in addition for an isotropic setting withisoA" = \pl + [/3o + 0i]2 + [0o + 02]

2]1/2 G". Obviously the anisotropic case resultsin a non-vanishing component ^3 which indicates the "out-off-plane" deformation.


Figure 2. Anisotropy measure 6 and stereo-graphic projection due to the principaldirections of strain E : o and stress S : • with respect to a Cartesian frame

Figure 3. Determinant of the acoustic tensor for 7 = 0.25 within the anisotropicand the isotropic setting

For more detailed background information on non-linear finite elements we refer e.g.to Oden [ODE 72]. Note that the proposed framework results in an efficient numer-ical setting. Practically, we end up with similar costs compared to isotropic hyper-elasticity, since the metric tensor G^ of the standard formulation is replaced by A.Contrary, the classical approach based on the incorporation of a structural tensoryields numerous additional terms in the computation of the stress tensor and espe-cially of the tangent operator which ends up with tremendous numerical costs.

3. Anisotropic damage based on a fictitious configuration

Here, as the key idea, the energy metric tensor is introduced as an internal variableand denoted as damage metric tensor in the sequel. Then the fictitious linear tangentmap F is no longer constant and allows the interpretation as damage deformationgradient. The fictitious configuration remains isotropic and un-damaged but thestandard reference configuration BQ as well as the spatial one Bt can be damaged andanisotropic.


Figure 4. Three-dimensional Cook's problem and deformed mesh at\\F\\ = 1.28 x 105

Figure 5. Load-displacement curve of the mid point node at the top corner for theanisotropic and isotropic setting

3.1. Standard dissipative materials

We adopt an additive decomposition of the free Helmholtz energy

incorporating a scalar-valued hardening variable K. Based on the theory of stan-dard dissipative materials, see Halphen and Nguyen [HAL 75], the local form of theClausius-Duhem inequality for the isothermal case with respect to BQ"reads

Within the standard argumentation of rational thermodynamics the nominal stress, thedamage stress and the hardening stress are obtained by


Next, an admissible elastic cone is introduced

whereby $ is a convex function and Y = YQ + H(K). Moreover, associated evolutionequations DÂ" = A<9Ab$, Df/c = — Ad#$ = A are applied. The underlying con-strained optimisation problem yields the Kuhn-Tucker conditions and the Lagrangemultiplier A can be computed via the consistency condition. Nevertheless, for thesake of demonstration, in the sequel the hardening contributions are assumed to beconstant.

NOTE 3.1 - Standard pull-back operations yield e.g. S* = /"" • 5tt • [ff and Ab =[F ]* • A • F which allow the interpretation as effective stress measures.

3.2. Construction of the damage function

The specific form of the damage function significantly affects the evolution ofanisotropic damage. Therefore, we will especially focus on the evolution of theeigendirections of the damage metric tensor.

The most general form is of course based on the set of ten invariants in terms ofthe damage stress Ab and the damage metric A* itself

Nevertheless, two selected representations seem to be natural and will be highlightedin the sequel, compare Schreyer [SCH 95].

The direct formulation introduces the damage rate negative proportional to a sym-metric, positive semi-definite second order tensor 2S"(A"). Obviously, the simplestchoice 22* = A* results in

and the dissipation inequality reads V = A Y > 0. Indeed, the damage rate and thedamage metric itself are coaxial but since the damage metric could be non-sphericalwe denote this type of damage as quasi isotropic.

The formulation based on conjugate variables constructs the damage rate as alinear map of the damage stress via a symmetric, positive semi-definite fourth ordertensor 4s"(A"). Again, a simple choice 4S" = A* ® A* ends up with the quadraticform

and the reduced local form of the Clausius-Duhem inequality reads T> = 2 A Y > 0.Obviously, the damage rate and the damage metric are no longer coaxial which moti-vates the terminology anisotropic damage.


NOTE 3.2 - Eqs.(19,20) can alternatively be motivated via the following quadraticform

whereby for the fourth order tensor 4S the structure of the tangent operator of linearelasticity has been adopted. Now, due to the central idea of the proposed framework,the contra-variant metric tensor is replace by the damage metric

Then the first term, incorporating the scalar rji, represents the quasi isotropic damagefunction (p\ of eq.(19) and the second term, incorporating the scalar 772, representsthe anisotropic damage function (pi ofeq.(20).

NOTE 3.3 - In contrast to isotropy the incorporation of an in-elastic potential to-gether with the application of associated evolution equations within an anisotropicsetting is an assumption since the obtained rate equations represent reduced formsof the most general tensor functions in terms of all appropriate arguments, see e.g.Betten [BET 85].

NOTE 3.4 — The two introduced types of damage functions and the character ofthe initial damage metric tensor AQ define a general classification of the couplingof hyper-elasticity and damage. In particular, the following four categories areobtained:

i| a

(1) isotropic hyper-elasticity (A\ = /?0 G*) & quasi isotropic damage ((p\)(2) isotropic hyper-elasticity (A\ = /?0 G*) & anisotropic damage ((p?)(3) anisotropic hyper-elasticity (AQ ^ /?o G*) & quasi isotropic damage ((pi)(4) anisotropic hyper-elasticity (A0 ^ /?o G*) & anisotropic damage ((ft)

Assuming a material of St. Venant-Kirchhojf type category (1) is directly related tothe classical [I — a] damage formulation via A^ = foG^ = [I — d]2 G". In thiscase flo represents three equal eigenvalues, which degrade for increasing damage,e.g. characterised by d. Moreover, note especially that formulations within category(2) become anisotropic within the purely elastic domain for unloading after damageevolution has taken place.

3.3. Numerical examples

In the sequel we apply a compressible Mooney-Rivlin material of the form

whereby the principal invariants ° G Ji,2,3 are expressed in terms of the basic invari-


ants EA I\,2,3, to be specific

In order to define the initial damage metric tensor we chose the specific formatAj = 70 G" + Y%=i TC AN\ ® ANC Co™?*1"6 Svendsen [SVE ]. Moreover, thefollowing material parameter have been taken into account c\ = 10, c^ = 20, \p — 5,70 = 1-0, 71 = 0.5, 72 = 0.25, Y = 10 and the orthogonal unit-vectors AN\^ aredefine by spherical coordinates tfj = 2/3 TT, 0} = 1/3 rr, tf\ = 4/3 TT, 0} = 1/6 TT(compare Appendix). For quasi isotropic damage (<î) an exponential integrationscheme is available whereas for anisotropic damage evolution (<,02) an implicit Eulerbackward scheme is applied. We do not focus on numerical aspects here since theyare discussed in detail in Menzel and Steinmann [MEN 00]. Within the followingfinite element setting the tangent operator has been evaluated numerically as e.g.outlined in Miehe [MIE 96].

3.3.1 Simple shear

We consider again the homogeneous deformation of simple shear (F = I +761 ® 62) and take anisotropic damage (1^2) into account. Figure 6 shows the dif-ferent degradations of the eigenvalues of the damage metric tensor and highlights thenon-coaxiality of stress and strain, compare Appendix. In addition the stereographicprojection of the principal damage directions are given which evolve during the defor-mation process.

Figure 6. Degradation of the eigenvalues Aî,2,3 and stereo-graphic projection dueto the principal directions of strain E : o, stress S : • and the damage metric A : *with respect to a Cartesian frame


3.3.2 Strip with a hole

To give a three-dimensional finite element example a strip with a hole is discretisedby 64 x 2 enhanced eight node bricks (Q1E9) as advocated by Simo and Armero[SIM 92]. The geometry of the specimen is defined by a length of 12, a width of4, a thickness of 0.5 and a radius of 1. One end is totally clamped while the otherend is subject to displacement conditions in a longitudinal direction. Figure 7 showsthe deformed mesh and the anisotropy measure S for a maximal longitudinal stretchof A = 1.5 for the purely hyper-elastic solution (Y —> oo). In addition the load-displacement curve of the mid node at the un-clamped end underlines the anisotropicbehaviour since the displacement components u\ and us would be identical to zerowithin an isotropic setting. Now, incorporating damage evolution the degradation isconcentrated (un-symmetrically) at the boundary of the hole. In this context Figure 8highlights the smallest eigenvalue A AI of the damage metric tensor and the anisotropymeasure S within quasi isotropic damage ((p\). Moreover, Figure 9 visualises the samecontents for anisotropic damage (</>2)-

Figure 7. Anisotropy measure S and deformed mesh for a maximal longitudinalstretch A = 1.5 and load-displacement curve of the un-clamped mid point node atthe free end within pure hyper-elasticity (Y -> oo)

4. Outlook

The proposed thermodynamically consistent framework for anisotropic materialsat large strains results in a manageable numerical setting. Nevertheless, concerningfuture work, localisation has somehow to be taken into account. Furthermore, thecoupling to finite strain plasticity is an outstanding issue and finally the constructionof specific damage functions for engineering material as well as the correspondingidentification of material parameters are important areas constituting future research.

Appendix: Visualisation of Anisotropy

For three-dimensional examples especially it is not a trivial task to visualiseanisotropy. In the sequel three different propositions are made.


Figure 8. Smallest eigenvalue A\i of the damage metric and anisotropy measure 6for quasi isotropic damage (<pi) at \\F\\ = 114

Figure 9. Smallest eigenvalue A\i of the damage metric and anisotropy measure 6for anisotropic isotropic damage (ipz) at \\F\\ = 65.

In case that stress and strain tensors are not coaxial we deal with an anisotropicmaterial. This motivates the introduction of the anisotropy measure

Within the method of stereographic projection the eigenvectors of a symmetricsecond order tensor - which allow interpretation of being elements of the unit-sphereS2 - are projected onto the equatorial plane by viewing from the south pole. Mathe-matically speaking, this method represents the homomorphism 5O(3) —> SU(2), seee.g. Altmann [ALT 86].

Determinant of the acoustic tensor: Incorporating the common wave equationansatz into the incremental equation of motion for F^ = const yields (see e.g. Antman[ANT 95])

with whereby


the spatial unit-vector rfi € 52 can be defined by two spherical coordinates $1>2.Referring to a Cartesian frame Si, one of three possible parametrisations to definen — nl €i 6 S2 reads

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[SPE 84] SPENCER A. J.M. « Constitutive theory of strongly anisotropic solids ». InSPENCER A. J.M., Ed., Continuum Theory of the Mechanics of Fibre-ReinforcedComposites, number 282 in CISM Courses and Lectures. Springer, 1984.

[STE 98] STEINMANN P. AND CAROL I. « A framework for geometrically nonlinearcontinuum damage mechanics ». Int. J. Engng. Sci., 36:1793-1814,1998.

[SVE ] SVENDSEN B. On the modeling of anisotropic elastic and inelastic materialbehaviour at large deformation. Preprint.

Chapter 14

On the Numerical Implementation of aFinite Strain Anisotropic DamageModel based upon the LogarithmicRate

Otto Timme Bruhns and Christian Ndzi BongmbaInstitute of Mechanics, Ruhr-University Bochum, Germany


Finite Strain Anisotropic Damage Model 243

1. Introduction

Materials undergoing finite deformation suffer a loss in load carrying capacity cau-sed by the nucleation, growth, and coalescence of microdefects which is generally re-ferred to as damage. Save the simplest, but nontrivial, special case where the materialis initially isotropic and the microdefects are voids, any damaged material is aniso-tropic. The first damage model was presented by KACHANOV [KAC 58]. This modelis generally accepted and forms the fundamental basis of Continuum Damage Me-chanics (COM). Several generalizations of this one-dimensional model of the three-dimensional regime exist. Here, we mention isotropic damage models of the type des-cribed in [LEM 90] and anisotropic models based upon ideas introduced in [COR 79],and [MUR 88]. For an extensive bibliography, we refer to the review article [KRA 89]and the books [LEM 96, KRA 96] on COM.

The isotropic damage model is limited to the case where the material is initiallyisotropic and all the defects are voids. Moreover, as shown by Ju [JU 90], this modelfails to predict the effects of damage on Poisson's ratio. Assuming isotropic condi-tions, one can easily show that the anisotropic damage model predicts only the changein a single material parameter. And, there is even a more fundamental problem withthis model, since it is usually developed using a symmetric effective stress measurea. Generally, cr is derived by the symmetrization of a stress measure denoted in[MUR 88] by cr*, which is a function of the damage variable, and considered to beunsymmetric. To derive cr*, finite deformation kinematics is used. In fact, cr* is afirst Piola-Kirchhoff type stress measure, a two-point tensor, and represents a bilinearmap between two different vector spaces - the current and the fictitious undamagedconfiguration. One can therefore not expect cr* to be symmetric, since symmetry willrequire that we identify the two vector spaces. We mention here the fact that the nomi-nal stress tensor, also a two-point tensor, plays a prominent role in non-linear elasticityand elastoplasticity [OGD 84, HIL 78]. We note also that using the work-conjugacynotion introduced by HILL [HIL 78], any stress measure (and its work-conjugate strainmeasure) can be used in formulating non linear constitutive equations. Moreover, notethat one can derive, using finite deformation kinematics, a symmetric stress measure (asecond Piola-Kirchhoff type tensor) that is defined in the fictitious undamaged confi-guration, and that none of the effective stresses employed in the literature is equal tothis stress measure. The ad hoc symmetrization of cr* and damage models based uponthe effective stress & are therefore questionable.

Within a phenomenological framework, the anisotropy induced by damage is in-distinguishable from the so-called microstructural or initial anisotropy. In general,microstructural anisotropy is modelled by the introduction of so-called structural ormaterial tensors [DOY 56, BOE 87]. BOEHLER [BOE 87], for example, uses the fol-lowing set as material tensors:

where (711,712,713) denote orthonormal vectors. Apparently, the set in Equation [1]can be used as a basis for the representation of any second order tensor.


In this paper, we present an anisotropic damage model and show its numericalimplementation. Based on the above mentioned fact about structural tensors and ani-sotropy, we make use of an interpretation of the damage parameter as an evolvingstructural tensor to model damage-induced anisotropy. We assume the damage pa-rameter to be a second order, symmetric, and positive semi-definite tensor. On theirreversible thermodynamics side, the damage parameter is treated as an internal statevariable. With these interpretations, damage modelling effectively reduces to formula-ting thermodynamically consistent evolution equations, finding critical values for thedamage variable and determining material constants. The framework of [XIA 00] iscapable of effectively incorporating anisotropic material behaviour, and is thereforegoing to be adopted here as basis for the damage model; i. e. we use the spatial loga-rithmic strain and its work-conjugate stress measure. Also, we use a corotating frameand all our rate constitutive equations are formulated using the logarithmic rate.

An outline of this paper is as follows. In Section 2, we briefly review the kine-matic foundations of finite deformation. Strain and stress measures are presented inSection 3. In Section 4, we describe briefly the logarithmic rate and the logarithmicspin, and discuss the advantages of the logarithmic rate over other objective corota-tional time derivatives. In Section 5, within the framework of [XIA 00], we specifyour damage model. The Gibbs and dissipation potentials, and the evolution equationsare specified. Using thermodynamics with internal state variables, restrictions on thematerial parameters are derived. From Gurson's flow potential (which serves here as amicromechanical basis), we determine the material parameters in our Hill-type yieldcondition. In Section 6, the material moduli for computing the stiffness matrix and astep-by-step summary of the integration of the constitutive laws are given. We alsoshow that our model is kinematically consistent.

2. Basic kinematic quantities

Let X denote a particle of a body B undergoing finite deformation. Let X haveposition vectors X in the reference and undeformed configuration Boand x in thecurrent configuration B. The motion that takes X into x is denoted by x(X,t}. Thetwo-point tensor F = Gradx(X,t) with Jacobian J = detF > 0 is called the defor-mation gradient. From the polar decomposition theorem, we obtain

where R is the rotation and U and V are the right and left stretch tensors, respectively.The tensors defined as

are referred to, respectively, as the right and left Cauchy-Green tensors. Let A^ (a =1, • • • ,n) denote the eigenvalues of V, \a = A^ those of B, Bff their corresponding


orthonormal eigenprojections, and n the number of distinct eigenvalues. Then, B hasthe spectral representation

3. Strain and stress measures

Following [XIA 00], we use the following form for Hill's generalized strain mea-sures:

Here, 6ij is the Kronecker delta, and 1 denotes the second order identity tensor. Letv denote the velocity of X. The spatial velocity gradient is given by L = gradu =FF~1 and has the following unique additive decomposition:

The symmetric tensor D is called the stretching, and the antisymmetric tensor W thevorticity.

where Ca — P?BaR. The scale function f(Xff), a smooth monotonic increasingfunction with the property /(I) = /'(I) — 1 = 0, is given by

where for m = 0 the limiting process is understood. Note that [6] include most com-monly used strain measures; Hencky's logarithmic strain measures H and h are ob-tained by setting m = 0; i. e. / = In A:

The spatial Hencky strain h is a forward rotation of the material Hencky strain H,and vice versa:

Let a- denote the Cauchy stress tensor, and r the Kirchhoff stress tensor, cr and rare related by r = Jcr. Again, following [XIAOO], we denote the work-conjugatestress measure to H and h in the sense of HILL [HIL 78] independent of any material


symmetry by II and TT, respectively; i. e. the stress power per unit reference volumecan be written as

7T

where G is a time-differentiable, objective, second order, Eulerian tensor, and £1 is ao

time-dependent skew-symmetric, second order tensor chosen such that G is objective.For a general discussion of objective corotational rates and their defining spins, werefer to [XIA 98a] and the references cited therein. Well-known examples of objective

o

corotational rates are the Zaremba-Jaumann rate G} for Q = W, and the Green-O

Naghdi or polar rate GR for tt = 17R = RRJ. For the logarithmic rate (log-rate)o

G1'08 of G, we use the the logarithmic spin (log-spin)

owhere (»)R denotes the Green-Naghdi rate of (•), see Section 4. The following ex-pression for the Eulerian stress measure it is taken from [XIA 00]:

The Lagrangian stress measure II is a back-rotation of TT, and vice versa.

4. Log-spin and Log-rate

Inelastic material behaviour has to be formulated in rate or incremental form. Ina Eulerian setting, this requires the use of objective corotational time derivatives. Thegeneral form for an objective corotational time derivative is

in Equation [12]. The time-dependent rotation tensor .RLo8 that defines the log-spin isreferred to as the logarithmic rotation (log-rotation) and is the solution of the tensordifferential equation

The log-rate was recently proposed by XIAO, BRUHNS & MEYERS [XIA 97, XIA 98a,XIA 98b] and has since been successfully used in formulating constitutive equations[BRU 99, XIA 00]. The log-rate has two major advantages over any other objectivecorotational rate. First, the log-rate of the spatial Hencky strain h is equal to the stret-ching D:


Actually, the validity of Equation [15] was the prime motivation for the derivationof the log-spin and the log-rate. Second, the hypo-elastic model based upon the log-rate is self-consistent; i. e. it is exactly integrable to deliver an (hyper)elastic relation[BRU99].

5. Constitutive equations

The damage variable T> is assumed to be a second order, symmetric, and positivesemi-definite tensor. It is a member of the set a of internal state variables that alsoincludes the hardening variables K and a:

The Gibbs potential 3> is assumed to be a function of TT and a, and the stretching Dto have the additive decomposition

where De denotes the elastic part of D and Dei the elastic-inelastic part. Then, star-ting from the Clausius-Duhem inequality and using a standard procedure due to Co-LEMAN & NOLL [COL 63], we derive the (hyper)elastic constitutive equation

and the dissipation inequality

where p0 is the density in the reference configuration BQ. Using Equation [15], andintroducing the complementary hyperelastic potential £ as £ = PQ$, we obtain thefollowing exactly-integrable rate form of the elastic constitutive law:

To separate elastic from inelastic deformations, we use, following KRONER [KRO 60]and LEE [LEE 69], the multiplicative decomposition of the deformation gradient:

where Fe denotes the elastic part and Fl the damage-plastic or simply the inelasticpart. Note that [21] introduces an intermediate configuration which is assumed to bestress free and is unique only to within an arbitrary rotation. To relate the two decom-positions [17] and [21], we use

where here sym(») denotes the symmetric part of (•). We note that [22] representsthe only natural, and direct relation between the two decompositions [17] and [21]and that such a relation does not exist if we assume, as widely done, that Fe is asymmetric tensor [XIA 00].

Let the function F(?r,a) = 0 denote the yield criterion in stress space. Then, usingF, we define the elastic domain as the set E^ := {(7r,a) : F(7r,a) < 0} . Taking


5.1. Particular form of the Gibbs and dissipation potentials

Following LEHMANN [LEH 89], we assume that the Gibbs potential takes the ad-ditive decomposition

where the elastic part 3>e is assumed to depend only on the stress, and the damagevariable and the inelastic part <!>' only on the hardening variables. Using the interpre-tation of P as a structural tensor, it follows that <£e is an isotropic tensor functionof TT and 'D. Using the representation theorem of tensor functions, we postulate thefollowing quadratic function in TT for E:

with

and

Here, I denotes the symmetric fourth order identity tensor, and 771-7/4 are assumed tobe material constants. The additive split of the compliance tensor D> into an isotropic orundamaged part DP and a damage-induced part ^ is motivated by micromechanicalresults given in [BUD 76]. With [24] and for isotropic damage, the elastic law takesthe form

where k,p depend on the damage variable. The elastic law in the form [27] is due toHencky and is widely used in finite elastoplasticity [ANA 86, SIM 92, SCH 95].

For the inelastic part <J>! of the Gibbs potential, we use

where Hkin is the kinematic hardening modulus. The conjugate forces to X>, a, andK are respectively,


the back stress tensor X as the centre of the yield surface (F — 0), and the damageparameter as an evolving structural tensor, we derive the following anisotropic or Hill-type yield condition:

where #1 = <fe = 1.5 and qi = I are material parameters [TVE 82], and (»)D refersto the deviator of (•). In its original form, FQ is a function of tr, so that Gurson'sflow condtion as given here in [36] is a further modification. Gurson's model has beenextensively and successfully used by many researchers to model, amongst many otherthings, localization, shear banding and macrocracking.

To put [32] on a sound micromechanical basis, its material parameters are thereforedetermined from [36]. To this end, we consider a material that is isotropically damagedunder one-dimensional conditions; i. e. we take T2 = fl, and assume a loading casewhere only one normal component, say, TTH = TT of the stress tensor is different fromzero. In this case, / can be regarded as the void volume fraction. From [32] and [36],the yield stresses are, respectively,

Here a3,05,07,ag; 60 and 61; 7r0(the initial yield stress) are material constants and /3denotes the determinant of T>. The fourth order tensor A is called the anisotropy tensor,and its additive decomposition into an isotropic or undamaged part A° and a damage-induced part Ad is partly motivated by the additive decomposition of the compliancetensor. Note that A° is an orthogonal projection that maps a second order tensor into3/2-times its deviatoric part, and that without damage, Equation [32] reduces to theflow condition of classical J-2 isotropic plasticity.

Perhaps the most generally accepted and widely used isotropic damage model isthe micromechanical model of GURSON [GUR 77]. The most important ingredient ofthis model is the flow condition

with the weighting functions w<2. and w\ given by


Figure 1. Weighting functions w\ andwi

Figure 2. Flow conditions FQ and F. Left: isotropic case with f = 0.10. Right:anisotropic case, where only the component T>n = 0.10 is different from zero

Considering [36] as exact, and taking the critical value of / as 0.5 [HIL 65], the ma-terial parameters

for [32] were chosen to get the best fit between w\ and u>2 as shown in Figure 1. Thetwo-dimensional case with the set of parameters from [40] is shown in Figure 2. Forthe isotropic case, there is virtually no difference between the two flow conditions.Both flow conditions are inside the initial or undamaged yield surface. For the sim-plest case of anisotropic damage also shown in Figure 2, the flow condition [32] liesbetween the initial yield surface and Gurson's yield surface and predicts, as expected,a decrease in flow stress in ~K\ -direction with virtually no change in the TT^-direction.

of the Kirchhoff stress appears naturally in the rate form of the weak form of momen-tum balance [SIM 98, Equation (7.2.21)]. To obtain the material moduli C11 which are


5.2. Evolution equations and thermodynamic restrictions

The evolution equations for the elastic-inelastic strain and the hardening variablesare of the associative plasticity type:

The plastic multiplier A is determined from the consistency condition F = 0 andsatisfies the loading/unloading condition

Based on the characteristics of the damage variable mentioned above, we postulate thefollowing evolution equation:

where /?i,/?2 > 1 are material parameters and (•)+ denotes the positive projectionof (•). Substituting the evolution equations into the dissipation inequality, we obtainthe following thermodynamic restrictions on the material constants 773 and 774 and thefunction m:

Note that [46] is consistent with the fact that for a given stress, the complementaryhyperelastic potential E of a damaged material is greater than that of an undamagedmaterial, and that these inequalities simply state (or ensure) that damage causes softe-ning.


6.1. Material moduli

In standard displacement-type finite elements, the starting point of the solutionprocess with the Newton-Raphson iteration scheme is the discretized form of the weakform of momentum balance. The Lie derivative

where Op are the elastoplastic tangent moduli. SlMO & TAYLOR [SIM 85] showedthat to retain the quadratic rate of asymptotic convergence of Newton's method, theelastoplastic tangent in Equation [53] should be replaced by the consistent tangent ma-trix. For our model, the consistent tangent matrix is currently computed numericallyusing the perturbation method as described, for example, in [KOJ 87].

where R°+i and H1^8 refer to the log-rotation at time £n+i and tn, respectively. Nu-merical solution of the tensor differential equation [14] yields an exponential map for


needed in computing the material part of the stiffness matrix, we proceed as follows.Using the basis-free expression for fiLog from [BRU 99], we obtain the following re-

lation between TL and rLog:

Substituting Equation [48] into the rate form of the weak form of momentum balance,we obtain the material moduli C" as

6.2. Integration of the constitutive equations

To integrate the constitutive equations from Section 5, we use the operator splitmethod [SIM 85, SIM 98]. Tables 1 and 2 contain a step-by-step summary of the in-tegration algorithm. In the following, we explain our notation, and comment on someaspects of the integration. First, we note the fact that in the elastic predictor step, andconsistent with the use of the log-rate, the tensor internal variables are forward-rotatedwith -Ru°8, the relative log-rotation:


Table 1. Integration of the constitutive equations

jRy°8, and this is evaluated as described in [SIM 98]. Second, we mention the factthat for moderate strains the log-rotation /ZLog approximately equals the transpose ofthe rotation tensor R from the polar decomposition of F. Third, note that with thehelp of Lemma A from [XIA 00] all quantities related to the decomposition [21] canbe uniquely determined; from the stress update algorithm (Table 2), we obtain thestress 7rn+i. Then using [18], the hyperelastic form of the constitutive equation, wedetermine Ve:

The elastic stretching £>e is taken from the rate form of the elastic law. Then, with theelastic spin tensor given by

and from a tensorial differential equation similar to [14] for .Re, we determine theelastic rotation tensor JRe. From the polar decomposition and the multiplicative de-composition, Fe and Fl follow. The model presented is therefore regarded as kine-matically consistent, since Fe and F1 and their related kinematic quantities can be


Table 2. Integration of the constitutive equations

uniquely determined without the use of ad hoc assumptions about Fe. Note also thatwith [56] our model satisfies the objectivity requirement in a general sense [NAG 90].For details, we refer to [XIA 00].

6.3. Example

To demonstrate the performance of our model, we consider the plane-stress exten-sion of a perforated strip. This is a classical example, and here the geometry, boun-dary conditions, and material parameters are chosen as in [SIM 98] page 189. Fordamage evolution, the material parameters are taken as /^ = 0.10 and /^ = 0.0; i.e. weare considering only isotropic damage. In Figure 3, we have plotted the reaction-displacement curves for the computation without damage, and with damage. Damagecauses softening of the material. On the structural level, this softening is reflected inthe reduction of the total reaction forces and is clearly, as shown in Figure 3, capturedby the model. Qualitatively, these results are in a good agreement with other resultsreported in the literature, see for example [SOU 92].


Figure 3. Reaction versus displacement curves

7. Summary and conclusions

A finite anisotropic damage model was presented. The elastoplasticity frameworkof [XIA 00] was used. Thus, the model can be regarded as kinematically, and self-consistent. It was shown that using a dual interpretation of the damage parameter as 1)an evolving structural tensor and 2) an internal variable, damage modelling reduces toformulating thermodynamically consistent evolution equations, finding critical valuesfor the damage variable and determining material constants. We think that the formerinterpretation of the damage parameter, since it ties progress in damage modellingwith that in the modelling of so-called strongly anisotropic materials, will prove use-ful in future. On the numerical implementation side, a closed form expression for thematerial moduli was given. Thus, the implementation of the model in existing finiteelement codes can be readily accomplished. More numerical examples are presentedelsewhere.

Acknowledgements

We thank Dr. H. Xiao, Dr.-Ing. A. Meyers and Dipl.-Ing. H. Schiitte for their inter-est and the fruitful discussions. This paper is abstracted from work conducted withinthe Collaborative Research Center SFB 398 at the Ruhr University Bochum. Financialsupport was provided by the Deutsche Forschungsgemeinschaft (DFG).

8. References

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[BOB 87] BOEHLER J. P., « Representation for isotropic and anisotropic non-polynomial ten-sor functions », BOEHLER J. P., Ed., Applications of Tensor Functions in Solid Mechanics,n° 292 CISM Courses and Lectures, Springer-Verlag, 1987, p. 31-53.


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[BUD 76] BUDIANSKY B., O'CONNEL R. J., « Elastic moduli of a cracked solid », Interna-tional Journal of Solids and Structures, vol. 12, 1976, p. 81-97.

[COL 63] COLEMAN B. D., NOLL W., « The thermodynamics of elastic materials with heatconduction and viscosity », Archive for Rational Mechanics and Analysis, vol. 13, 1963,p. 167-178.

[COR 79] CORDEBOIS J. P., SIDOROFF F., « Damage Induced Elastic Anisotropy », BOEH-LER J. P., Ed., Mechanical Behavior ofAnisotropic Solids, n° 295 CNRS, Martinus NijhoffPublishers, 1979, p. 761-774.

[DOY 56] DOYLE T. C., ERICKSEN J. L., Nonlinear Elasticity, N° 4 Advances in AppliedMechanics, Academic Press, New York, 1956.

[GUR77] GURSON A. L., « Continuum theory of ductile rupture by void nucleation andgrowth: Part I - Yield criteria and flow rules for porous ductile media », Journal of En-gineering Materials and Technology, vol. 99, 1977, p. 2-15.

[HIL 65] HILL R., « A self-consistent mechanics of composite materials », Journal of theMechanics and Physics of Solids, vol. 13, 1965, p. 213-222.

[HIL 78] HILL R., « Aspects of invariance in solid mechanics », YlH C.-S., Ed., Advancesin Applied Mechanics, vol. 18, p. 1-75, Academic Press, New York, 1978.

[JU 90] Ju J. W., « Isotropic and anisotropic damage variables in continuum damage mecha-nics », Journal of Engineering Mechanics, vol. 116, n° 12, 1990, p. 2764-2770.

[KAC 58] KACHANOV L. M., « On the time to failure under creep conditions », Isw. ANSSSR.Old Techn. Nauk, vol. 8, 1958, p. 26-31, Zitat aus Jansson & Stigh (1985).

[KOJ 87] KOJIC M., BATHE K.-J., « The effective-stress-function algorithm for thermo-elasto-plasticity and creep », International Journal of Numerical Methods in Engineering,vol. 24, 1987, p. 1509-1532.

[KRA 89] KRAJCINOVIC D., « Damage Mechanics », Mechanics of Materials, vol. 8, 1989,p. 117-197.

[KRA 96] KRAJCINOVIC D., Damage Mechanics, Applied Mathematics and Mechanics,North-Holland, Amsterdam, 1996.

[KRO 60] KROENER E., « Allgemeine Kontinuumstheorie der Versetzungen und Eigen-spannungen », Archive for Rational Mechanics and Analysis, vol. 4, 1960, p. 273-334.

[LEE 69] LEE E. H., « Elastic-plastic deformation at finite strains », Journal of Applied Me-chanics, vol. 36, 1969, p. 1-6.

[LEH 89] LEHMANN T., « Some thermodynamical considerations on inelastic deformationsincluding damage processes », Acta Mechanica, vol. 79, 1989, p. 1-24.

[LEM 90] LEMAITRE J., CHABOCHE J.-L., Mechanics of Solid Materials, Cambridge Uni-versity Press, Cambridge, 1990.

[LEM 96] LEMAITRE ].,A Course on Damage Mechanics, Springer-Verlag, Berlin, 2 edition,1996.

[MUR 88] MURAKAMI S., « Mechanical Modeling of Material Damage », Journal of AppliedMechanics, vol. 55, 1988, p. 280-286.

[NAG 90] NAGHDI P. M., « A critical review of the state of finite plasticity », Journal ofApplied Mathematics and Physics (ZAMP), vol. 41, 1990, p. 315-394.


[OGD 84] OGDEN R. W., Non-linear Elastic Deformations, Ellis Horwood Limited, Chiches-ter, 1984.

[SCH 95] SCHIECK B., STUMPF H., « The appropriate corotational rate, exact formula forthe plastic spin and constitutive model for finite elastoplasticity », International Journal ofSolids and Structures, vol. 32, n° 24, 1995, p. 3643-3667.

[SIM 85] SIMO J. C., TAYLOR R. L., « Consistent tangent operators for rate-independentelastoplasticity », Computer Methods in Applied Mechanics and Engineering, vol. 48,1985, p. 101-118.

[SIM 92] SIMO J. C., « Algorithms for static and dynamic multiplicative plasticity that pre-serve the classical return mapping schemes of the infinitesimal theory », Computer Methodsin Applied Mechanics and Engineering, vol. 99, 1992, p. 61-112.

[SIM 98] SIMO J. C., HUGHES T. J. R., Computational Inelasticity, vol. 7 de Interdiscipli-nary Applied Mathematics, Springer, New York, 1998.

[SOU 92] DE SOUZA NETO E., PERIC D., OWEN D. R. J., « A Computational model forductile damage at finite strains », OWEN D. R. J., ONATE E., HlNTON E., Eds., Compu-tational Plasticity: Fundamentals and Applications, Pineridge Press, 1992, p. 1425-1441.

[TVE 82] TVERGAARD V., « Ductile fracture by cavity nucleation between larger voids »,Journal of the Mechanics and Physics of Solids, vol. 30, n° 4, 1982, p. 265-286.

[XIA 97] XIAO H., BRUHNS O. T., MEYERS A., « Logarithmic strain, logarithmic spin andlogarithmic rate », Acta Mechanica, vol. 124, 1997, p. 89-105.

[XIA 98a] XIAO H., BRUHNS O. T., MEYERS A., « On objective corotational rates and theirdefining spin tensors », International Journal of Solids and Structures, vol. 35, n° 30,1998, p. 4001^014.

[XIA 98b] XIAO H., BRUHNS O. T., MEYERS A., « Strain rates and material spins », Journalof Elasticity, vol. 52, 1998, p. 1-41.

[XIA 00] XIAO H., BRUHNS O. T., MEYERS A., « A consistent finite elastoplasticity theorycombining additive and multiplicative decomposition of the stretching and the deformationgradient », International Journal of Plasticity, vol. 16, n° 2, 2000, p. 143-177.


Chapter 15

Ductile Rupture of Aluminium SheetMaterials

Jacques BessonEcole des Mines de Paris Centre des Materiaux, Evry, France; and Institute ofMaterials Research, GKSS Geesthacht, Germany

Wolfgang Brocks, Olivier Chabanet and Dirk SteglichInstitute of Materials Research, GKSS Geesthacht, Germany


Ductile Rupture of Aluminium Sheet Materials 261

1. Introduction

A realistic assessment of the residual strength of cracked sheet materials (steelin pipeline or ships, aluminum in aircraft structures, zirconium in nuclear fuelcomponents) requires methods to experimentally characterize crack growth resistanceas well as numerical simulation tools capable of predicting crack initiation andpropagation. Modeling can be done following two different strategies. On the onehand, the global approach to failure uses macroscopic rupture parameters such as theCrack Tip Opening Angle (CTOA) [GUL 99], the energy dissipation rate [TUR 92],the J-integral [RIC 68] or the J—Q parameters [O'D92]. In general, thesequantities suffer from a lack of transferability of fracture data from specimen to actualstructures. On the other hand, the local approach to fracture provides a solutionto the transferability problem by describing the degradation of the material usingmicromechanical state variables (e.g. void volume fraction, nucleated porosity, etc).These models have been applied successfully to predict crack growth for thick walledcomponents of structural steels [SUN 88, GUL 00] where a high stress triaxialitytriggers the growth of voids as the main failure mechanism. Their application to thinwalled high strength aluminum alloys faces some specific problems: (i) the stresstriaxiality ratio in sheets is much lower than in thick structures whereas models ofductile damage have been established for high triaxialities; (ii) the fracture plane oftenshifts from a normal to a 45° inclined orientation to the applied load (Figure 2), (iii)rolled sheets generally show an anisotropic behavior with respect to both plastic hard-ening and void nucleating particles. However, fracture surfaces still show dimpleswhich might result from growth of voids nucleated at second phase particles. Becauseof this, the application of models of ductile tearing seems to be promising.

In this work, the plastic and rupture behavior of thin aluminum sheets ischaracterized using smooth, blunt and sharp notched specimens. A 3D finiteelement modeling of fracture of small samples is performed using the Rousseliermodel [ROU 87]. The emphasis is put on the simulation of slant fracture. As thisapproach does not appear suitable to model crack extension in large structures, a2D simulation using a cohesive zone model [NEE 90] is adopted.

2. Material and experiments

In this study, sheets of 1.73 mm thickness of a 2024 Al-alloy are investigated.The surface of the sheets is protected by a layer of 1050 aluminum whose thicknessvaries between 50 and 80 //m. The sheets are heat treated (T351 treatment). Severaltypes of precipitates are present in the material [HER 98]: (i) coarse secondary phasescontaining iron and copper (size: 5 to 30 //m), (ii) Al^M^Cu dispersoides (size: 20to 500 nm), (iii) Al2(Cu,Mg) strengthening precipitates (size: a few nm).

The samples used to characterize the plastic and fracture behavior of the materialare shown in Figure 1. They include: smooth tensile bars, notched bars with U or V-notches and Kahn cracked specimens. All tests were performed with the load applied


Figure 1. Geometries of the specimens: (a) smooth tensile, (b) V-notch plate, (c) V-notch plate (notch radius: 0.25 mm), (d) Kahn specimen (notch radius: 0.06 mm,ASTM B-871-96). Gray dots indicate measurement points for the notch openingdisplacement A<5

along the rolling direction (L); cracks propagate along the transverse direction (T).The short transverse direction will be referred to as S. Due to bending, stable crackgrowth is observed on Kahn samples whereas unstable rupture is obtained on the otherspecimens. Post-mortem measurements on tensile specimens outside the neckingregion have shown that the thickness reduction along the S—direction is larger thanalong the T—direction evidencing an anisotropic plastic behavior.

Fig. 2 shows the fracture surface of a V-notched specimen. Fracture is initiated atthe middle of the notch where the fracture surface is normal to the loading direction(zone ® in Figure 2b). This region forms a small triangle. Outside this zone, slantfracture in the L—S plane is observed. A closer examination of the fracture surfacesshows void growth around the iron/copper particles. Void growth is more pronouncedin the flat fracture region than in the slant fracture zones ((D and ® in Figure 2b).Between these voids, small dimples, probably initiated on dispersoides or Al2(Cu,Mg)precipitates, are observed. Similar features were observed on Kahn specimens.

On tensile and U-notched specimens, the macroscopic fracture surface is entirelyslanted (Figures 4 and 5). An example of the microscopic fracture surface is shown inFigure 3. The rupture surface exhibits flat and smooth areas where small dimples


Figure 2. Fractography (V-notch plate): (a) Macroscopic view showing the inclinedcrack path, (b) Macroscopic view showing (L direction) the flat crack initiation region(triangle), (c) Microscopic fracture faces corresponding to different regions of thecrack path (1,2 and 3; locations are indicated on (b))

cannot be seen. This indicates a possible change in the fracture mechanisms betweencracked or severely notched samples and specimens containing smooth geometricaldefects. The proportion of smooth fracture zones increases in the case of the tensilespecimens.

Figure 3. Fracture surface of smooth (a) and U-notched specimens (b) exhibiting flatfracture zones

The total effective porosity ft appearing in eq. 1 is defined as the sum ft — / + fn-Considering that nucleation starts when a given amount of plastic strain has beenreached, An will be written as: An = A0H(p — p0) where H (•) is the Heavy sidefunction and AQ and p0 are parameters to be adjusted. Different values for AQ and powill be used depending on the specimen type.


3. Continuous damage mechanics: Rousselier model

3.1. Constitutive equations

Continuous damage mechanics (COM) can be used to describe the rupturebehavior of the material. In this study, the Rousselier model has beenemployed [ROU 87]. The original model is extended to account for plastic anisotropyof the matrix material and damage nucleation. A similar extension was usedin [GRA 00] for the Gurson model. In that case the yield surface is expressed as:

cTjfcjfc is the trace of the stress tensor a_ and an is the Hill equivalent stress defined by:

where Sij are the components of the stress deviator. The coefficients hss... are used todescribe the plastic anisotropy. Using the deformation measurements and consideringthat the yield limit in the T-direction is 10% smaller than in the L-direction [HER 98,SIE99] one gets: h^ = 0.79, h^ = 1.16, hss = 1.68. It was assumed that: /IST =h^ — hSL = 1; this assumption has however little importance as shear stresses aST, cr-ând <TSL remain small. R(p) represent the plastic hardening of the undamaged material(shown in Figure 4) and p the von Mises accumulated plastic strain.

D and a\ are material parameters related to void growth. D = 2 and a\ —260 MPa were used and correspond to the values recommended in [ROU 87]. Theplastic strain rate tensor is obtained using the normality rule as: ep = \d$lda_ whereA is the plastic multiplier. The evolution of the void volume fraction / is given bymass conservation:

The initial porosity /o is taken equal to the volume fraction of coarse particleswhich are assumed to debond at the onset of plasticity: /0 = 0.12% [HER 98]. Usingthese parameters, ductility is however overestimated. Based on the observation thatsmaller precipitates can play a role in the fracture process, strain controlled nucleationwas used in the model. The nucleated effective porosity /„ is given by


3.2. Computational procedures

FE calculations are performed using the software Zebulon [BES 98]. Constitutiveequations are integrated using a fully implicit scheme which allows the calculation ofthe consistent tangent matrix [SIM 85]. An updated Lagrangian formulation is used;the Jauman stress rate is used to define the objective stress rate; constitutive equationsare expressed in the associated corotational frame [LAD 80].

In order to reduce the number of degrees of freedom, mixed 2D/3D meshes areused. Geometrical symmetries are also accounted for. 20 nodes (3D) bricks withreduced integration (8 Gauss points) are used to model regions of crack extension.The 2D parts are freely meshed using 6 or 8 nodes plane stress elements (see 4.2.1).In order to reduce the number of elements, computations exploit the initial geometricalsymmetries of the various specimens. This symmetry is however lost due tolocalization as evidenced by the fracture surfaces. Accounting for this effect wouldrequire twice as many elements for a given mesh size.

The material is considered as broken as ft reaches 85%. The behavior isthen considered as elastic with a very low Young's modulus (1 MPa) as proposedin [LIU 94]. An element containing more than 5 broken Gauss points is automaticallyremoved by checking this condition after each time increment.

3.3. Results and discussion

3.3.1. Smooth and U-notched specimens

Simulated macroscopic responses and fracture modes for smooth and U-notchedspecimens are compared with experimental results in Figures 4 and 5 (F: force, S0

initial minimum cross section, LQ, SQ: initial gauge length, AL: elongation, A5: notchopening displacement (see Figure 1)). All simulations were carried out using AQ~ 0.5andpo = 0.35.

Smooth specimens: The macroscopic response of the specimen and the fractureplane are well reproduced. Failure occurs after the onset of necking. The fracturesurface corresponds to a plane whose normal lies in the L—S plane. The angle 9between the normal and the L—direction is about 45°. The simulation was alsoperformed assuming an isotropic plastic behavior; in this case the normal also liesin the L—S plane with a smaller value for 9 which is close to localization angle underplane stress in an isotropic incompressible material (35.26°) usually observed in theL—T plane. The fact that localization still occurs in the L—S plane is likely relatedto the relatively small width of the specimen. Note however that due to the plasticanisotropy, e^ < e^ which favors localization in the L—S plane [RUD 75].

U-notched specimen: Rupture of U-notched specimens is very similar to failurein smooth specimens. One single fracture plane is observed which also makes a 45°

Figure 4. Simulation of the rupture of tensile specimens. Left: Macroscopic response:(I) no damage, no necking, (2) no damage, necking, (3) damage and necking. Thearrow indicates the experimental rupture. The photograph shows the experimentalfracture surface (9 w 45°). Right: Values of the plastic deformation at Gauss pointsshowing the localization into slant fracture

angle in the L—S plane. Macroscopic behavior as well as fracture plane are wellreproduced by the FE simulation (Figure 5).

NOTES - For both samples, FE simulations were carried out using variousvalues for both AQ and po as well as different mesh sizes. Very similar results areobtained using more than 5 elements to mesh the S—direction (10 are used in Figures 4and 5). The influence of po and A0 is limited for values of po larger than 0.3. Thisindicates that the onset of failure of smooth and U-notched specimens is essentiallycontrolled by necking and void growth. Accounting for rapid damage by nucleation isonly needed to model final failure.

3.3.2. V-notched and Kahn specimens

In V-notched and Kahn specimens, the sharp notch generates strong stress andstrain gradients so that a higher mesh size sensitivity is to be expected. Transition fromflat to slanted fracture is experimentally observed. In this work, the emphasis was puton the modeling of this 3D phenomenon. Preliminary plane strain calculations haveshown that a slanted crack path can be obtained provided a sufficiently fine mesh isused. In addition, elements should have a square shape at the onset of damage andstrain localization. 3D meshes were designed according to these results.

Figure 6 shows the crack path obtained with 7 elements in the S-direction usingvarious values of A0 and p0 = 0.2. It is shown that increasing AQ leads to an increase



Figure 5. Simulation of the rupture of U-notched specimens. Left: Macroscopicresponse (the arrow indicates the experimental rupture). Right: Values of the plasticdeformation at Gauss points showing slant fracture

of the flat fracture area. High values of AQ generate a highly damaged zone whichremains limited to one element so that crack deflection is not possible. A similar trendis obtained with increasing PQ. In this case the low hardening rate of the materiallimits the size of zones where p is higher than PQ .

Figure 7 (calculation done with AQ = 0.5 and p0 = 0.2) indicates the values of thestress triaxiality ratio T ahead of the crack tip after the initiation of the flat crack area.Values as high as 1.6 are obtained showing that plane stress conditions are not met;in that case the maximum value of the stress triaxiality is equal to %. However, thismaximum value remains smaller than values obtained under plane strain which lie inthe range 2.5—3.0. Figure 7 also shows details of the flat to slanted fracture transition.

Figure 6. Effect of AQ on the flat to slanted fracture transition (PQ = 0.2)


Figure 7. Left: Gauss point value of damage (ft) and stress triaxiality (T) at crackinitiation. Right: Final crack path showing details of the flat to slanted fracturetransition. 10 elements are used in the S—direction, AQ = 0.5, po = 0.2

The calculation qualitatively reproduces the experimental results shown in Figure 2but overestimates the size of the flat region. The zone of normal fracture is smallerwhen using a smaller mesh size (e.g. Figure 6) but still larger than what is experimen-tally observed (Figure 2).

Simulations of the Kahn specimen are shown in Figure 8. Calculations with 5,7 and10 elements along the S—direction using AQ — 0.5 and po = 0.2 were performed.Decreasing the element size leads to a smaller crack growth resistance as it is expecteddue to the softening behavior of the material; however simulations overestimate theexperimental data. For a given element size, meshing one half of the specimen (i.e. notaccounting for the initial symmetries) would also reduce the simulated crack growthresistance. Faster crack growth can also be obtained by modifying the void nucleationparameters. As an example, Figure 8 shows the macroscopic response obtained forPo = 0.15. Although fractographic examinations have shown that failure mechanismschange with increasing defect severity, such a low value is unlikely to be realistic. Another fitting strategy would be to increase po and AQ (i.e. delay onset of nucleationand increase nucleation rate); this however leads to flat fracture as the mesh is thentoo coarse to correctly capture the deformation and damage fields.

Mesh size plays a critical issue in modeling crack advance. It is often considered asa material parameter which needs to be adjusted [ROU 87, GUL 00]. Using nodularcast iron having the same volume fraction of nodules but different particle sizes, ithas been shown that ductility decreases with decreasing interparticle spacing andthat this effect can be modeled using meshes of decreasing sizes [STE 98]. In this


Figure 8. Left: Force—notch opening curves using 5, 7 and 10 elements to mesh thethickness direction (S) using AQ = 0.5 and PQ — 0.2. 7* corresponds to 7 elementswithpo = 0.15. Points are experimental data. On the photograph, the normal fracturezone is within the white dashed line. Right: Simulated crack path using 10 elementsin the S—direction

study, fractographies have shown (Figure 2) that the mean distance between iron/copperparticles is in the range 10—20 //m which is significantly smaller than the meshsize used in the present calculations (= 80/zm). However, using such a small meshsize would require a much larger computational effort (both CPU time and memorysize). Note also that the use any non-local damage model would indeed require asmaller mesh size than the mesh size needed to fit the experiments using a localtheory. In order to reproduce the experimental load—displacement curves on Kahnspecimens, nucleation parameters can be changed to obtain a faster damage kinetic; inthat case the normal to slant fracture transition cannot be modeled so that the benefitof 3D calculations is lost.

4. Cohesive zone model for plane stress state

In the previous section, it has been shown that the 3D-modeling of crack growthalong a slanted path requires a large number of elements. This approach does notappear as suitable in the case of large panels; in this case a simplified 2D-modelinghas to be developed. Several possibilities are however possible: (i) adjust Rousselierparameters in the plane stress case, (ii) use CTOA or CTOD, (iii) describe the crackpath as a damageable interface. In all cases the actual stress state at the crack tip willnot be accurately described (for instance the stress triaxiality will be underestimated).


It becomes indeed impossible to represent the slanted crack path. It is therefore likelythat the associated material parameters will be geometry dependent [SIE 99].

where e = exp(l). 6C and crmax are two adjustable parameters of the model. crmax

represents the maximum stress carrying capacity of the interface. 8C is a characteristiclength. The function ern (5W) is shown in Figure 9a. The fracture energy per unitsurface F is given by:

In practice, it will be considered that the interface is broken (i.e. an = 0) when6n > 6C so that the actual fracture energy is reduced by 4.6%.

4.2. Computational procedures

4.2.1. Plane stress elements

Plane stress [along the z-direction] conditions are usually taken into accountby rewriting the material constitutive equations so that the condition crzz = 0 isenforced. However, this method requires modifying the implementation of everymaterial behavior. Another strategy, which requires more computational time butless programming, testing and maintenance efforts, relies of the implementation ofplane stress elements [BES 97, BES 98]. These elements have the usual nodal degreesof freedom (displacements) and additional degrees of freedom (el

zz) representingthe transformation gradient along the z-direction at each integration point (i =1.. .number of Gauss points): e\z = &W/WQ (WQ initial thickness, Au; thicknessvariation). The principle of virtual work is written in the current (end of timeincrement) configuration in order to derive the reactions associated with the differentdegrees of freedom. It can then be shown that the reaction R\ corresponding toeach additional degree of freedom is equal to Rl

e = crzzul where a/ is the volume

associated to the corresponding Gauss point. In the absence of prescribed values for

4.1. Cohesive zone model

In the present investigation, a simplified treatment of crack growth is proposedbased on a cohesive zone model. This type of model describes the material damageand separation along a surface [NEE 92]. For all specimens, the crack is assumed tostay in the initial plane of symmetry. In that case, only normal separation is to beconsidered. The mechanical behavior of the interface is described using a constitutiverelation relating the normal component of the stress vector on the interface (crn) to thenormal opening of the interface (<£„). The following relation will be used in this study:


Figure 9. (a) Traction-separation law (eq. 5). (b) Cohesive zone elements withsurrounding plane stress elements

elzz or applied values of azz (e.g. pressure field), the force equilibrium requires that

Rle = 0, so that the plane stress condition is enforced at each Gauss point.

4.2.2. Cohesive zone elements

Cohesive zone elements are placed along the crack path and embedded in asurrounding of eight nodes large deformation plane stress elements with reducedintegration (4 Gauss points). CZ elements have 6 nodes and 2 integration points. Dueto symmetry, surface integrals have to be evaluated on the line of symmetry (Figure9b). The virtual work for each CZ element is written as (/' denotes the end of the timeincrement):

where <5* denotes a virtual normal displacement. The surface integral is computedaccounting for the deformation of the symmetry plane but also for the sheet thicknessreduction. This is done by associating each Gauss point of the CZ elements with thenearest Gauss point of the continuum plane stress elements whose thickness reductionis then used to evaluate the surface integral. Tests have shown the importance ofaccounting for thickness reduction. In the case of constant thickness CZ elementsa slight increase of the cohesive strength or a slight modification of the crack tipconstraint can result in the localization of the plastic deformation in the above lyingcontinuum elements with no crack advance. The calculation is then equivalent to thereference elastoplastic large deformation computation without CZ.

Figure 10. Simulation of the Kahn specimen using the CZM: (a) Force vs. notchopening for different mesh sizes: 1: 100 x 100 //in2. 2: 200 x 200 pm2, 4: 400 x400 /mi2; p: no crack growth, (b) Crack extension vs. notch opening (200 x 200 /zm2

mesh). Contour map shows the stress (0-^2) in the loading direction. (Points representexperimental data)

4.3. Results

The load—notch opening curve was adjusted using the following parameters:0max = 720 MPa, 6C = 80 /mi. Results of the simulations shows a good agreementfor both the macroscopic response (Figure lOa) and crack advance (Figure lOb). Themacroscopic response is also compared to the response obtained without CZ. Inthat case failure occurs by plastic necking. Calculations were performed with anelement size of 200/mi x 200/mi and using a coarser (400/mi x 400/mi) and finer(100/mi x 100/mi) mesh. Results obtained using 200/mi and 400/mi differ onlyslightly whereas a faster crack advance is obtained with 100/mi. This is due to thelarger deformations that are computed in that case at the crack tip so that necking (andtherefore stresses) is overestimated. However, mesh size dependence remains smallerthan in the case of CDM (section 3).

5. Summary

Ductile tearing of Aluminum 2024 sheets has been studied. The following resultswere obtained:

Experiments. Experiments were conducted on smooth and moderately notchedspecimens as well as severely notched samples. The plastic behavior is anisotropic.



Fractography shows that failure mechanisms change between these two sets ofspecimens with an increase of void growth in the second case. The crack path isslanted for moderately notched specimens whereas a transition from normal to slantfracture is observed on the second set.

3D modeling using COM: 3D modeling using an extension of the Rousselier modelto plastically anisotropic material was used to represent the experimental results.Strain controlled nucleation was also modeled. On the one hand, failure of tensileand U-notched specimen is essentially controlled by necking and results depend onlyslightly on the nucleation parameters. In that case, experimental results can be welldescribed. On the other hand, nucleation and mesh size play a very important rolein the case of V-notched and Kahn specimens. The main features of ductile tearing(in particular the normal to slant fracture transition) are qualitatively described butthe load is overestimated. Adjusting the nucleation parameters in order to obtain theexperimental load leads to a flat crack path. This is attributed to the relatively coarsemesh needed to perform the 3D calculations. It is believed that using a finer meshwould allow correct description of the load and the crack path and to use materialparameters closer to those obtained on moderately notched specimens. Simulationsshow that stress triaxiality levels reached ahead of the crack tip are much higher thanthose obtained under plane stress conditions.

2D modeling using CZM: In order to overcome the difficulties encountered usingthe 3D CDM modeling, a simplified model using CZ elements modified to account fornecking in the plane stress case has been adopted. Good results are then obtained onKahn specimens; however, based on the results presented in [SIE 99] the transfer-ability of these parameters to other cracked geometries is questionable.

Acknowledgements:

This work was performed during the sabbatical leave of JB at GKSS whichis acknowledged for financial support and hospitality. Materials for the studywere provided by Pechiney which also financially supported OC. Many thanks toDr. J. Heerens and his team for carrying out the experiments and to V. Heitmann forthe SEM examinations.

6. References

[BBS 97] BESSON J., FOERCH R., "Large scale object-oriented finite element code design",Computer Methods in Applied Mechanics and Engineering, vol. 142, 1997, p. 165-187.

[BES 98] BESSON J., FOERCH R., "Application of object-oriented programming techniquesto the finite element method. Part I- General concepts", Revue europeenne des elementsfinis, vol. 7, num. 5, 1998, p. 535-566.

[GRA 00] GRANGE M., BESSON J., ANDRIEU E., "An anisotropic Gurson model to representthe ductile rupture of hydrided Zircaloy-4 sheets", Int. J. Fracture, vol. 105, num. 3, 2000,p. 273-293.


[GUL99] GULLERUD A., DODDS R., HAMPTON R., DAWICKE D., "Three dimensionalmodeling of ductile crack growth in thin sheet metals: computational aspects andvalidation", Eng. Fracture Mechanics, vol. 63, 1999, p. 347-373.

[GUL 00] GULLERUD A., GAO X., DODDS JR R., HAJ-ALI R., "Simulation of ductile crackgrowth using computational cells: numerical aspects", Eng. Fracture Mechanics, vol. 66,2000, p. 65-92.

[HER 98] HERMANN G., Dechirure ductile de toles minces d'un alliage Aluminium-Cuivre,report, 1998, Pechiney CRV.

[LAD 80] LADEVEZE P., Sur la theorie de la plasticite en grandes deformations, report, 1980,Rapport interne No. 9, LMT, ENS Cachan.

[LIU 94] Liu Y., MURAKAMI S., KANAGAWA Y., "Mesh-dependence and stress singularityin finite element analysis of creep crack growth by continuum damage mechanicsapproach", Eur. J. Mech, A/Solids, vol. 13, num. 3, 1994, p. 395-417.

[NEE 90] NEEDLEMAN A., "An analysis of tensile decohesion along an interface", J. Mech.Phys. Solids, vol. 38, 1990, p. 289-324.

[NEE 92] NEEDLEMAN A., "An analysis of decohesion along an imperfect interface", Int. J.Fracture, vol. 40, 1992, p. 1377-1397.

[O'D92] O'Dowo N., SHIH C., "Family of crack-tip fields characterized by a triaxialityparameter-II. Fracture Applications", J. Mech. Phys. Solids, vol. 40, num. 8, 1992, p. 939-963.

[RIC 68] RICE J., "A path independent integral and the approximate analysis of strainconcentration by notched and cracks", J. Appl. Mech., vol. 35, 1968, p. 379.

[ROU 87] ROUSSELIER G., "Ductile fracture models and their potential in local approach offracture", Nuclear Engineering and Design, vol. 105, 1987, p. 97-111.

[RUD 75] RUDNICKI J., RICE J., "Conditions for the localization of deformation in pressure-sensitive dilatant materials", J. Mech. Phys. Sol., vol. 23, 1975, p. 371-394.

[SIE 99] SlEGMUND T., BROCKS W., "Prediction of the work of separation and implicationsto modelling", Int. J. Frac., vol. 99, 1999, p. 97-116.

[SIM 85] SlMO J., TAYLOR R., "Consistent tangent operators for rate-independentelastoplasticity", Computer Methods in Applied Mechanics and Engineering, vol. 48,1985,p. 101-118.

[STE98] STEGLICH D., BROCKS W., "Micromechanical modelling of damage and fractureof ductile materials", Fatigue Fract. Engng Mater. Struct., vol. 21, 1998, p. 1175-1188.

[SUN 88] SUN D., SlEGELE D., VOSS B., SCHMITT W., "Application of local damagemodels to the numerical analysis of ductile rupture", Fatigue and Fracture of EngineeringMaterials and Structures, vol. 12, 1988, p. 201-212.

[TUR 92] TURNER C., "A re-assessment of ductile tearing resistance. Part I: The geometrydependence of J-R curves in fully plastic bending. Part II: Energy dissipation rate andassociated Jft-curves in fully plastic bending.", Fracture behavior and design of materialsand structures, ECF 8, 1992, p. 933-968.

Chapter 16

On Identification of Small Defects byVibration Tests

Yitshak M. RamDepartment of Mechanical Engineering, Louisiana State University, USA

George Z. VoyiadjisDepartment of Civil and Environmental Engineering, Louisiana State University,USA


Identification of Small Defects 277

1. Introduction

There is a wealth of literature associated with diagnostic criteria, procedures, andmethods for damage detection in structures and systems. With technologicalprogress, high-speed machinery and transport carriers are prone to fatigue andcatastrophic failure. There is thus an increasing interest in developing reliablediagnostic methods allowing the detection of structural defects at an early stage.Following the early work of Cawley and Adams [CAW 79], a variety of methods foridentifying the existence of damage have been proposed, see e.g. [GAS 98], [HEA91], [LIA 92], [RAT 00], [REY 96], and [REY 00].

If the damage exceeds a certain significant level then it can be detected bychanges in the spectral properties. It is a matter of debate, however, whetherdamage can be identified at an early stage, at which the physical parameters of thesystem are only slightly altered from their nominal undamaged values. An inherentdifficulty associated with repeatability of modal testing results conducted inindependent laboratories prevents easy confirmation or rejection of published resultsand procedures associated with damage detection. It appears that the fundamentalproblem of determining whether there exist measurable quantities that are highlysensitive to small changes in the physical parameters has not been appropriatelyaddressed yet. In this context an analytical example demonstrating the possibility ofidentifying small damages in the theoretical model framework of vibrating systemsmay be of more significant importance than merely displaying experimental results.In this paper we furnished such an analytical example, and through it address thefundamental problem of identifying small defects in structures by means of vibrationtests.

For simplicity consider a conservative vibrating system modelled by:

with symmetric positive definite mass matrix M , and non-negative definite stiffnessmatrix K. It is well known that the eigenvalues of the system [2] are continuousfunctions of the elements in the system matrices, see e.g. [PAR 80]. If theeigenvalues are distinct then the mode-shapes are continuous functions of thephysical parameters as well. Changes in the physical parameters of the system arerelated by bounds to changes in the orientation of the eigenvectors, as shown in[RAM 93]. Hence small changes in the physical parameters may produce only smallvariation in the spectral data. The measurable data in modal test however are rationalfunctions, e.g.,


where H(JCO} is the frequency response function and P(JCO) and Q(JCO) are

functions depending on the physical properties of the system. Hence continuity of COwith respect to the coefficients of P(jcd) and <2(jto) does not necessarily imply

continuity of co with respect to the elements of H(JCO).

We present in Section 2 an example demonstrating that a certain measuredfunction is highly sensitive to changes in the physical parameters of the system.Physical interpretation of this result is given in Section 3. In Section 4, whileapplying the result to groove identification in a vibrating rod, we find that thefrequency of excitation required in determining the location of the groove isextremely high. We therefore conclude that the problem of whether a small damage ina realistic structure can be identified by a dynamic test is still subject to debate.

2. Damage in a discrete model of a uniform vibrating rod

Figure 1. Mass-spring system

Consider the n-degree-of-freedom system shown in Figure 1, which consists ofmasses mi and springs of constants kf, / = l,2,...,n. Such a system represents adiscrete lumped-parameter-model, or finite difference model, of an axially vibratingrod. Suppose that the harmonic excitation f ( t ) = sin(cot) is applied to they-th degreeof freedom. Then, the motion of the system is described by the set of ordinarydifferential equations

where:

and ey is the j-th unit vector of dimension n. The system [2] has a particularsolution of the form

where h; is a constant vector. Substituting [5] in [2] gives

The ^-equations defined by [6] can be assembled as follows

where I is the identity matrix and

We thus have

The matrix H(a>) is called the Frequency Response Function (FRF) matrix. Itselement hy (co) represents the steady state amplitude of the harmonic response in thej'-th degree-of-freedom due to a unit sinusoidal excitation applied to the y'-th degree-of-freedom. Hence, the elements of H(O;) can be determined by simple vibration tests.

Figure 2. Systems description: (a) pure system, and (b) damaged system



Consider now the uniform system of unit parameters mi=ki =1, 1 = 1,2,...,100of dimension n = 100, shown in Figure 2(a). A harmonic exciting force withfrequency ft) = 2 is applied to the system at they-th degree-of-freedom. Suppose thatthe constant of the p-th spring is reduced due to a damage to kp = 0.99 , as shown in

Figure 2(b). Let /i,,(2) and hu(l) be the collocated frequency-response-functions atj=l,2,...,n of the pure system and the damaged system, respectively. Thesefunctions are plotted in Figure 3(a) for the case where p = 25, i.e. the damage isapplied to the 25-th spring. Similar graphs for the cases when the damage is applied tothe 50-th and 75-th degrees of freedom are shown in Figures 3(b) and 3(c),respectively. It is apparent that the damage and its location is clearly observable by

discontinuity in slope of hu(2). In contrast, damaged applied to the 5-th spring

cannot be unambiguously identified as shown in Figure 3(d). Such a result can begrasped by intuition. The fifth spring is too close to the support and hence cannot beexcited easily by the harmonic force.

Figure 3. Damage identification


It should be note that other harmonic forces, of different frequencies, may notprovide clear identification of the damage and its location. For example, Figures 4(a)and 4(b) display the functions hn (1.5) and ha (1.5) for the case where the damage is

located at the p = 50 spring and the exciting frequency is (0 = 1.5. The two

functions associated with the pure and damaged systems look almost identical.Figures 4(c) and 4(d) display these frequency-response-functions for the case wherethe exciting frequency is co = 2.5 . Here a variation between the two functions at thedamage location p = 50 is observed. This variation is small, however, and cannot be

considered as a reliable criterion for damage detection for realistic systems. Theseintriguing results deserve further considerations.

3. The interpretation

Denote the stiffness matrix of the undamaged system shown in Figure 2(a) by K ,and let K be the stiffness matrix of the damaged system (Figure 2(b)). Then thestiffness matrix of the undamaged system shown in Figure 2(a) is:

and the stiffness matrix of the damaged system (Figure 2(b)) is:

Both systems share the same mass matrix M = I. Define a diagonal matrix:


Figure 4. Excitations by various frequencies


The physical interpretation of equation [15] is that hu is the static deflection ofthe y'-th node due to a collocated unit static load applied to if, as shown in Figure5(a). Note that the spring configuration of the system of Figure 5(a) is similar to thatof the undamaged system shown in Figure 2(a), but with an additional spring ofconstant k = 2 attached between the n-th node and the ground.

Following a similar process we find that for the damaged system the equation

Then, since D = D ', we have by [6]

which implies that

For co = 2 equation [14] takes the following explicit form:

holds, or explicitly:


The element hit in equation [15] represents the static deflection of the^-th nodedue to a collocated unit static load applied to it, as shown in Figure 5(b). When theapplied force is in the close neighbourhood of the damaged the two springs ofconstant 28 became dominant resulting in large change in the response.

The systems of Figure 5 can be represented by using equivalent springs as shownin Figure 6, where

Figure 5. Static models for (a) pure system, and (b) damaged system

It thus follows that

In a similar manner, using:


Figure 6. Equivalent static models for (a) pure system, and (b) damaged system

The response ^(2) in Equation [19] describes a smooth function in j, while

hjj (2) features derivative discontinuity at j = p , which allows identification of the

damage and its location. In fact, the plots in Figures 3 and 4 are precisely h^ (2) and

h^ (2) given by equations [19] and [21].

4. Damage identification in an axially vibrating rod

Consider an axially vibrating uniform rod of length L, modulus of elasticity E,density p and cross-sectional area A , which is fixed at one end, jc = 0, and free tooscillate at the other end, x = L. The axial vibrations of this rod are governed by thedifferential equation and boundary conditions

we obtain from Figure 6(b):


A discrete mass-spring system model of dimension n , with equal length elementsof length h = L/n , leads to the eigenvalue problem:

and where K is given by [10].

Figure 7. An axially vibrating uniform rod with groove at x = a

A corresponding damaged rod with groove at x = a is shown in Figure 7. Leta = (p - i)h for some integer 1 < p < n . Let the groove width be w , and let Al < A

be the cross-sectional of the rod at the groove position. Then the eigenvalue problemassociated with the damaged rod takes the form

given by [11],

where


In order to identify the location of the groove in this system using the methodpresented in Section 2 it is required to excite the system with the frequency

For steel rod p=7800 kg/m3, E = 1.962 xlO11 N/m2, of length L = l m, with

A = 0.01 m2, AI - 0.0095 w = 0.01 m, n = 50, and p = 25 , we obtain: 8 = 0.0256,

rj =0.9750, and co =5.0154xl05 rad/s. The functions h^co) for the uniform rod,

and hjj(a)) associated with the damaged rod, are shown in Figure 8. Although the

damaged position is observable, its effect is less drastic than that obtained in Section2 for the chain of mass-spring system. The reason is that the groove in the rod affectsboth the stiffness and the mass of the p-th element. In practice it may be difficult toexcite the rod with the high frequency CO required. With current progress inmaterials research, however, such excitations using piezoelectric materials are notentirely beyond futuristic expectations.

Figure 8. Groove identification in an axially vibrating rod

and:


5. Conclusions

There are procedures in the literature for damage identification, accompanyingexperimental results. The inherent difficulty associated with the repeatability ofmodal analysis testing forms a barrier in achieving a scientific consensus regardingthe applicability of these procedures when slightly damaged systems are considered.In the theoretical arena the fundamental problem regarding the existence ofmeasurable quantities, which are sensitive to small changes in the physicalparameters of the system, is still open. In an effort to address this issue we havepresented an analytical example showing that under certain circumstances smallchanges in the stiffness of a uniform chain of mass-spring system can be detected.The example was restricted by the need for imposing a special frequency ofexcitation. It is well known that the frequency response of a damped system issmoother than that associated with its conservative counterpart. Hence damage in adamped system may less identifiable. In the context of realistic components, such asidentification of a small groove in an axially vibrating steel rod, the method requiresexcitation with very high frequencies. We may thus argue that the fundamentalquestion whether small defects in structures can be detected by vibration tests is stillsubject to debate.

6. References

[CAW 79] CAWLEY P., ADAMS R.D., "The location of defects in structures frommeasurements of natural frequencies", Journal of Strain Analysis, vol. 4, 1979, p. 49-57.

[GAS 98] CASSER A., LADEVEZE P., PERES P., "Damage modelling for a laminated ceramiccomposite", Materials Science and Engineering A-Structural Materials PropertiesMicrostructure and Processing, vol. 250, 1998, p. 249-255.

[HEA 91] HEARN G., TESTA R.B., "Modal analysis for damage detection in structures", ASCEJournal of Structural Engineering, vol. 117 n° 6, 1991, p. 3042-3063.

[LIA 92] LIANG R.Y., Hu J., CHOY F., "Theoretical study of crack-induced eigenfrequencychanges on beam structures", ASCE Journal of Engineering Mechanics, vol. 118, 1992,p. 384-396.

[PAR 80] PARLETT B.N., The Symmetric Eigenvalue Problem, Englewood Cliffs, Prentice-Hall, 1980.

[RAM 93] RAM Y.M., BRAUN S.G., "Eigenvector error bounds and their applications tostructural modification", AIAA Journal, vol. 31, 1993, p. 759-764.

[RAT 00] RATCLIFFE C.P., "A frequency and curvature based experimental method forlocating damage in structures", ASME Journal of Vibration and Acoustics, vol. 122,2000, p. 324-329.

[REY 96] REYNIER M., NADJAR B., "Identification of defaults in beams structures using statictests", Mecanique Industrielle el Materiaux, vol. 49, 1996, p. 28-31.


[REY 00] REYNIER M., NADJAR B., "Damage identification of flexible beam structures usinglarge displacements", Inverse Problems in Engineering, vol. 8, 2000, p. 251-281.


Chapter 17

Multi-scale Non-linear FE2 Analysis ofComposite Structures: Damage andFiber Size Effects

Frederic Feyel and Jean-Louis ChabocheONERA, DMSE-LCME, France


Multi-scale Non-linear FE Analysis 293

1. Introduction

The inelastic analysis of structural components working under complex and severeenvironments, especially under high temperature cyclic loading conditions, has nowan increasing impact on structural design. Most often these computations of inelas-tic response and stress redistributions serve to predict damage development and thecomponent lifetime, either in uncoupled or in coupled damage simulations.

The great reduction in computational costs and the considerable improvementsmade in parallelisation techniques and substructuring methods open new possibilitiesfor improved numerical simulations. Several directions for enlarged representative-ness of non linear structural analyses can be considered. Let us summarise them inthree words: time, size and scale:

- in the "time" direction, we have the capability to compute the component onits whole life, incorporating coupled inelastic and damage effects, really important fortaking into account non-stationary material evolutions, especially under cyclic load-ing conditions. The "cycle jump technique" for instance [SAV 78, LES 89, DUN 94,NES 00] has proved to be particularly efficient to enable integration of non linearevolutions incrementally at two time scales : the real time increments within eachcomputed cycle, and increments in number of cycles (external "time") to treat largenumbers of cycles. We have also the possibility to introduce "multiphysics" coupledanalyses;

- in the "size" direction, we can perform the non linear analysis of large sizethree-dimensional finite element models , with very large numbers of degrees of free-dom (we expect quite soon IMDoF for components treated in viscoplasticity), in or-der to improve the geometrical complexities of the structure as well as its complicatedloading conditions. In this domain the progress of finite element codes in terms of par-allel treatments is exceptionally powerful and promising [QUI 96, ROU 94, FAR 91,DUR 97,BJ0 86].

- for the "scale" direction, we can consider the "multiscale structural analyses",in which the material constitutive behaviour itself is built up from in-situ numericalcomputations. The present paper considers developments in this third class of meth-ods.

Multiscale modelling of structural components can be considered at two differentlevels :

(i) - the "sequential multiscale analyses", successively using micromechanics mod-els, analytical or numerical, to deliver the material constitutive responses on whichmore or less macroscopic models are identified and the finite element structural anal-ysis itself, still based on these macroscopic constitutive equations.

(ii) - the "integratedmultiscale analyses", in which the micromechanical local be-haviors and criteria are incorporated directly into the finite element structural analysis.There is no more need for a macroscopic constitutive model that reproduces the typicalresponses of the micromechanics analysis. In fact, the numerical analysis of the lower


scale delivers, in situ and in real time, the appropriate material response to its specificoverall loading, ie the overall strain for each local material Representative VolumeElement in the structure, at each Gauss Point.

Two such numerical approaches of the structural inelastic analyses were developedand exploited recently:

- for polycrystalline metallic components, treated in 3D, the polycrystalline ag-gregate models, with uniform stress within each grain and crystal plasticity constitu-tive laws at the level of average slip systems inside each grain. With between 40 and1000 grains for each local RVE, it leads to 1000 to 10000 state variables at each GaussPoint of the overall finite element model. Examples of such applications are given in[FEY 97b], but there are also other attempts in the literature [PIL 90, CAI 94]. Im-proved such models as "multicrystalline aggregates", with 3D third order stress fieldsredistributions within the grains are presently studied, but only at the level of a singleRVE [QUI 99, BAR OOa, BAR 00b], not at the component level. A similar methodwas proposed independently in a different context, by Smit et al. [SMI 98].

- for composite structures a two-level imbricated finite element methodology,called FE2 [FEY 98, FEY 00], was proposed and applied to MMC's. This methodsolves the local stress equilibrium and constitutive equations, inside each RVE treatedat the microstructural level by periodic homogenization and a unit-cell finite elementmodel, as well as the overall stress equilibrium at the structural level. In that case thenumber of internal state variables for each macroscopic Gauss Point can be increasedby one or two orders of magnitude compared with the previous case.

The present paper discusses the conditions of application of this second class ofmethods for problems related with long fiber SiC/Ti MMC's used in the context of"bling" components (or "bladed rings"), as candidates to replace turbine and/or com-pressor discs in future aircraft turboengines. In that case the approach is exploited asa 2D or 2D| problem (generalised plane strain). We first recall the main lines of theFE2 multiscale approach (section 2), including some details about its implementationand use. In section 3 some recent improvements made about "relocalization" tech-niques are discussed, that take into account the "material lengthscale" effect inducedby the presence of a "coarse grain" microstructure. Section 4 presents some results ona schematic bling, treated in cyclic elasto-viscoplasticity, the same constitutive equa-tion being used at the macroscopic level for the pure Titanium alloy part and for thematrix inside the unit cell of the microstructural level. Moreover, additional recent ex-ploitations of the FE2 method include damaging effects through cohesive zone modelsused at the lower scale, at the fibre-matrix interface. Some examples are also givenon the use of the specific relocalisation technique in order to obtain stress, strain andplastic strain local fields in the extreme cases where the microstructure (fibre size) isextremely large (significantly larger than the wavelength of the overall homogenisedsolution).

Multi-scale Non-linear FE2 Analysis 295

2. The FE2 method

2.1. Periodic homogenization theory: summary

As with other homogenization theories, the main objective of the periodic homog-enization theory is to access the mechanical properties of a homogenous mediumwhich exhibits the same mechanical response as a given heterogenous medium.

This theory is based on a hypothesis to simplify the analysis:- the macroscopic and microscopic scales are supposed to be separated. In other

words, the characteristic size of all heterogeneities / is supposed to be small enoughrelative to the macroscopic length L : 77 = l/L « 1,

- the spatial distribution of all heterogeneities is supposed to be periodic.

A point in the heterogenous structure can then be located using two spatial coor-dinates : a macroscopic coordinate x (which is also the coordinate of that point inthe homogeneous structure) and a microscopic coordinate y which is the location ofthat point around the heterogeneity (whose size goes to zero). As r; = l/L « 1, itis possible to perform an asymptotic expansion of the stress and displacement fieldswith respect to r?:

The smaller the heterogeneities are, the smaller 77 is and the smaller the higher orderterms in the previous developments are. Writing the micromechanical equilibriumdiva = 0 and using the Hill-Mandel macrohomogeneity lemma leads to the definitionof the macroscopic stress and strain tensors :

where < A > denotes the spatial average of \(y) over the microscopic unit cell.This also induces the definition of elastic stress and strain localization tensors (in theabsence of interface discontinuities):

Another important result of the theory is that the displacement field u can be splitinto a periodic part v and a macroscopic contribution:


Figure 1. Principle ofFE2 models

2.2. FE2 principle

The "FE2 method" refers to a class of models which belongs to the more generalmultiscale model class. We suppose in this paper that a "displacement f.e. formula-tion" is used, ie that (from a macroscopic point of view) the model gives the stressesat time t knowing the strain and the strain rate at that time. As soon as relevant me-chanical scales are chosen, FE2 models are constructed using three main ingredients :

1. a modeling of the mechanical behavior at the lower scale (the RVE),2. a localization rule which determines the local solutions inside the unit cell, for

any given overall strain,3. a homogenization rule giving the macroscopic stress tensor, knowing the mi-

cromechanical stress state.

In the case of FE2 models (see Figure 1), a finite element computation is usedto model the microscopic behavior of the RVE. Any localization / homogenizationscheme can be used, but we focus in this paper on the use of the periodic homog-enization because one of the current application of FE2 models is to access the me-chanical behavior of long fiber SiC/Ti metal matrix composites. Using that theory,the homogenization rule is nothing but a spatial averaging of the microscopic stressdistribution : E = (ff)cell. The localization rule is obtained using relation (1) whichleads to a set of linear equations to be imposed on each pair of nodes on the sides ofthe RVE.

This class of models is called "FE2" (or also "imbricated finite element") becauseis requires the simultaneous computation of the mechanical response at two differentscales : the macroscopic scale (which is the scale of the whole structure) and the un-derlying microscopic representative volume element at each macroscopic integrationpoint.

Macroscopic phenomenological relations are completely useless, even in non lin-ear cases. The mechanical behavior arises directly from what happens at the micro-scopic scale, phenomenological constitutive equations being written only at that scale.


Figure 2. Finite elements : evaluation of material response in the case of a phe-nomenological model (left) and in the case of an FE2 model (right) combining twofinite elements scales

3. Implementation

3.1. General algorithm

From an implementation point of view, FE2 models follow the classical frame-work of internal variables models, and are very easy to implement if modernprogramming techniques are used [FOE 96, BES 97].

At each macroscopic Gauss Point, such models allow computation of the stresstensor at time t knowing: (i) the strain and strain rate at that time and (ii) the mechanicalhistory since t = 0. In classical phenomenological models, mechanical history istaken into account by the use of some internal variables. In the case of FE2 models,the internal variable set is constructed by assembling all microscopic data required bythe lower finite element computation. This includes, of course, microscopic internalvariables used to describe dissipative phenomena, but also all other useful quantitiesrequired by the finite element procedure.


Figure 2 compares the integration of the macroscopic constitutive equations in thecase of a phenomenological model and in the case of an FE2 model. The integrationof the phenomenological relations (using, for instance Runge-Kutta or Theta-method)is replaced by a finite element evaluation of the microscopic mechanic cell response.

3.2. Computation of the tangent stiffness matrix

FE2 models are used in classical finite element codes, based upon a Newton-Raphson algorithm to handle all non linearities. For optimum performances, one has tocompute the tangent stiffness matrix for all material models, and not only to computethe stress response. This matrix can be written as :

where all A in the right member denotes the increment of the quantity between time tand time t + At.

In the case of FE2 models, this computation depends of course on the homogenei-sation theory used, and on its finite element implementation. In this subsection wewant to present this computation in the case of the periodic homogenization theory, inthe particular framework of the ZeBuLoN finite element code.

Periodic homogenization theory is implemented in ZeBuLoN through the use ofspecific elements named "periodic elements". These elements are classical ones, ex-cept that some degrees of freedom are added corresponding to the E (average ormacroscopic strain) components. The unknown displacements are the non-periodicpart v of the total displacement u on the cell: u = v -f E x x.

The deformation tensor e is computed by derivation of the previous expression :

(Vs is the symmetric gradient operator)

The B matrix (symmetric gradient of the shape functions after discretization) isroughly the same as usual, except that a new part comes from degrees of freedomassociated with E (we suppose that these degrees of freedom are at the end of thewhole degrees of freedom list):

Bstd denotes the "classical" symmetric gradient of the shape functions.

Let us recall that the previous finite element discretization is at the microscopiclevel (ie at the cell scale). The macroscopic stiffness matrix to be computed is then :


(remember that E has associated degrees of freedom, and thus as-

sociated reaction give E to be multiplied by the volume of the cell). The assembledtangent stiffness matrix at the cell scale can be written as (J± represents the tangentmatrix given by all microscopic phenomenological constitutive equations):

leading to

Kmacro is then nothing but a condensation of the previous matrix onto the degreesof freedom associated with E (at most, 6 degrees of freedom shared across the wholemicroscopic mesh) inserted at the end of the list. That is

with:

This condensed matrix is then very easy to compute, and avoid the use of otherapproximative methods, such as the perturbation method.

3.3. Parallel computing

Let us suppose that one has to compute a macroscopic structure whose finite ele-ment discretization involves K integration points in the region where FE2 modelling isused. If the microscopic discretization of the representative volume element requires kintegration points, the cost in terms of global internal variables is equivalent to K x k,and increases very fast with the size of the structure to be studied. It is then neces-sary to use a powerful technique to solve such a big problem which is usually stronglynon-linear.

Parallel computing is of great interest in this area. It has been shown that the useof parallel computing can be associated with any sophisticated non-linear behaviorprovided that this behavior relies on the local state assumption [FEY 98, FEY 97a,FEY 97b]. Parallel computing is also used to compute structures using FE2 behaviormodels. The attention of the reader is focused on the fact that parallel computing pro-cedures are fully independent of the kind of constitutive equations, and therefore thatthe use of FE2 models with parallel computing do not require any extra development.


FE2 models have been implemented in the finite element code ZeBuLoN, jointlydeveloped at Onera and Ecole des Mines de Paris. This finite element code uses theFETI method [FAR 91] to split the computation of a structure into a number of sub-domains. In FE2 modelings, the main bottleneck is the local stage, ie the computa-tion for each macroscopic integration point of the corresponding microscopic finiteelement step. Parallel computing allows one to distribute these computations onseveral processors.

4. Relocalization

The classical first order treatment using periodic homogenization relies on a strongassumption : it is supposed that the heterogeneities in the structure to be computed aresmall enough (compared to the size of the structure and to the mechanical loadings tobe applied on that structure) so that macroscopic and microscopic scales are separated.

From a practical point of view it is however difficult to define precisely what "smallenough" means. From our experience it seems that fibers (ie heterogeneities) maybe relatively big, as soon as appropriate relocalisation techniques are used in order tocompute actual mechanical fields from the homogeneous solution given (at the macro-scopic scale) by the FE2 computation. The purpose of this section is to explicit thistechniques applied to the FE2 models. Nothing is new from a theoretical point ofview, but it seems that other authors never used this technique (usually because theyare interested only in the homogenous result), although it can be used to obtain alsothe actual mechanical fields.

The goal is to compute e(x), for any x, without any reference to y (cell coordinate)because all cells have to be mapped at their real locations.

An "interpolated-mapping" of the results obtained by FE2 methods is used to com-pute actual fields (Figure 3). Let us suppose that all heterogeneities remain elastic andthat the surrounding medium is also elastic. For each macroscopic integration pointwhose spatial coordinate is Xi, and for each position inside the underlying unit cell,the instantaneous strain tensor (for instance) is equal to

The "interpolated-mapping" relocalization technique is nothing but a macroscopic in-terpolation of results coming from all microscopic computations. Let 7(x) be a me-chanical component to be interpolated inside a macroscopic element; 7(x) can becomputed using the shape functions of the finite element containing x :


Figure 3. Principle of the "interpolated-mapping" relocalization technique

Ni(x) are the shape function of the current element, 7(xi) are the values of 7 atthe nodes surrounding the point x. This relation can also be applied to microscopicquantities as soon as that the unit cell, resized and translated to its real location andsize, is mapped onto the macroscopic mesh. For instance :

This kind of relation can be generalized and extended in non-linear cases like in plas-ticity or viscoplasticity.

One major advantage of using FE2 techniques is that the required estimationsof A (which are very difficult to estimate in non-linear cases) to compute relocatedcomponents have already been implicitly computed during the FE2 computation. An-other advantage is that the computation of relocalized values can be made a posterioriin a post-processor, since it only requires already computed informations. It is thenpossible to restrict this extra-computation to critical zones of the structure.

Note that we restrict ourselves (for practical reasons) to the average on a singlemacroscopic element: in a more general framework it would be necessary, for a givenmacroscopic node, to take into account the contribution of all elements connected to


this node. Results presented at the end of this paper will show however that this restric-tion is not limiting. From a practical and programing point of view, the relocalisationis performed by three steps :

1. map the unit cell onto the global mesh at its real location,2. extract the microstructural information from all integration points near x,

compute nodal values,3. compute the relocalized component using equation (2).

It is relatively easy to understand why this technique leads to continuous macro-scopic fields. Let us consider the situation shown in Figure 4: the goal is to computerelocalized fields for two near points A and B. This computation will involve micro-scopic datas extracted from the macroscopic integration points (plain circles). Due tothe periodic homogenization theory, values extracted from microscopic results con-cerning point A (right side of the microscopic cell) and point B (left side) are equal.Because A and B have the same spatial coordinates, the macroscopic reinterpolationleads to equal values for all mechanical fields.

Figure 4. Illustration of the continuity of all mechanical fields computed using the"interpolated-mapping" technique : thanks to the periodic homogenization theoryf ( A ) — f ( B ) for all couples of point A andB

5. Application to the computation of a bling disk

All critical aeronautical components are subjected to specific weight optimization:new materials are studied in order to increase the performance-to-weight ratio. Longfiber SiC/Ti composites have been developed on this principle. SiC fibres are orderedperiodically inside a titanium matrix (see Figure 5).

Engine manufacturers are currently considering the possible replacement of somemetallic turbine disks by rings whose central part are reinforced by such composites(see Figure 6). This part is a "bling" or bladed ring. A major problem is to be able to


Figure 5. Typical microstructure of a long fibre SiC/Ti composite. General layout(left) and detail of a single fibre (right)

compute such structures. FE2 models are especially interesting for that purpose. TheRVE of the microstructure consists in a fiber surrounded by a matrix part with a fibrevolume fraction of 22 %. It is assumed that the average radius of the part is largeenough so that the curvature of the fibres can be neglected. Generalized plane strain isthen assumed as the condition applied to the unit cell.

5.1. Mesh and boundary conditions

Figure 7 shows macroscopic and microscopic meshes as well as macroscopic bound-ary conditions. Mechanical loading consists in imposing an increasing centrifugalforce. The homogeneous part is made of titanium and the reinforced kernel is made ofSiC/Ti composite (the titanium in the composite is supposed to be the same as the oneused in the homogeneous part).

Figure 6. An experimental "bling"


Figure 7. Macroscopic mesh, boundary conditions and domain decomposition (leftmicroscopic mesh (right)

Figure 8. Macroscopic stress EH (top). Microscopic deformation (displacementsx 10) £n at point 1 (bottom,left) and 2 (bottom,right)


This computation was calibrated to fit in the memory of our cluster (the cluster ismade of four PC Linux PIII with 512 Mb of RAM in each machine). The computationwas distributed on these 4 processors and lasted about 2 hours.

5.2. Results

As explained in section 2, FE2 modeling works by computing in real time themacroscopic and the microscopic scales. It is then possible to analyze mechanicalresults at these two scales simultaneously. Figure 8 shows for instance the macroscopicstress distribution in the 12 direction (Si2) at the end of the loading. On the samefigure microscopic results are presented for two macroscopic points (denoted (1) and(2) at the macroscopic scale) at the same time. It is then possible to make a linkbetween a macroscopic shear (£12 ^ 0) and the specific shape of the correspondingunit cell (point 1).

5.3. Case of a coarse grain structure

5.3.1. Undamageable disk

The relocalization technique presented in section 4 has been applied to this exam-ple. It is here supposed that the fibre are not so small. Figure 9 shows the completemesh of the disk (ie a mesh of the actual structure, including all heterogeneities). Itwill serve to obtain a complete reference solution with "coarse grains", from whichthe FE2 method and the associated relocalization procedure will be validated. Theline plotted horizontaly on that mesh highlights the points where results are plottedand compared.

Figure 11 shows the stresses Sn and E22 at time t = 7.5s. Figure 10 showsthe corresponding cumulated viscoplastic strain to emphasize that non-linearities arestrong at that time. The plain curves in Figure 11 are obtained using a computation onthe real structure whereas the dashed curves are obtained via the FE2 modelling fol-lowed by the relocalization operations exposed before in this paper. The comparisonis fairly good, except on the edge of the structure where some side effects appear dueto the loss of periodicity (nothing is currently done at present to handle this effect).Figure 12 shows the same comparison for the inelastic strain in the 11 direction (en),whereas Figure 13 shows the contour of this field.

5.3.2. Damageable disk

The main damage mechanism in such long fiber composite is a debounding be-tween each fiber and the surrounding matrix. The computation presented in the pre-vious section has been run again, taking into account damage at each fiber/matrixinterface using a Needleman-Tvergaard debounding model ([NEE 87]).


Figure 9. Complete mesh of the actual structure to serve as a reference to the FE2

validation

Figure 10. Cumulated viscoplastic strain at time t = 7.5s


Figure 11. Comparisons between the reference stresses (plain) and the relocalizedstressed (dashed) at time t = 5s fern top, 022 bottom)


Figure 12. Comparisons between the reference inelastic strain (plain) and the relo-calized inelastic strain (dashed) at time t = 5s in the 11 direction (E^)

The presence of potential discontinuity surfaces modify the homogeneization rulesinto a slightly different form :

in the previous equation, [aj denotes the jump of the quantity a across the interface Fand n the interface normal at a given position x along both sides of the interface, {a}denotes the symmetric part of tensor a.

The second term in the homogenization rule of a turns out to be zero, because inthe debounding case normal stresses along both sides of all interfaces are null. Thehomogenization rule for a is then not modified by the presence of a debounding mech-anism at fiber/matrix interface. It is then still possible to use the implicitely computedlocalization tensor A, and therefore to use the relocalization technique presentedbefore in this paper.

The extra term in the E relation does not vanish, but it can be proved that it is stillpossible to define a localization tensor linking E and e. The relocalization techniquepresented before in this paper is then still valid.

The FE2 computation shown in section 5.3.1 has been run again taking into accountdebounding at fibre/matrix interface. Mechanical fields were then relocalized. Figure14 shows the an component at the end of the computation. One can observe thatstresses are relaxed in fibers due to the partial failure of some interfaces. In these ».


Figure 13. Relocalized e\\ at time t = 7.5s in the reinforced region


Figure 14. Relocalized a\\ at time t = 6.76s in the reinforced region, taking intoaccount damage at fiber/matrix interface


Figure 15. Relocalized spatial distribution of damage D at time t = 6.76s in thereinforced region


regions, stresses are transfered to the surrounding matrix. This contour map can becorrelated with Figure 15 showing the damage values D along all interfaces at thesame time (this last map has been obtained the same way, although it is not clearwhether or not it is possible to use this relocalization scheme for an interfacial field;but this gives a good idea of the spatial damage distribution).

6. Concluding remarks

The FE2 methodology presently developed offers several interesting capabilitiesfor the inelastic and damage analysis of structural components, especially for com-posite systems that can be considered as quasi- periodic at the microstructural level:

- the usual macroscopic constitutive and damage equations that serve to redis-tribute the overall stress fields are no longer needed. All the physics of the processes iscontained in the microscale constituents and in their finite element discretisation andthe material response to any overall strain control (at each Gauss point) is delivered in"real time", taking into account the whole history of local state variables.

- for applications to bling components made in SiC/Ti MMC's, the local consti-tutive equations were involving the cyclic thermo-elasto-viscoplasticity of the matrixand the damage at the fibre-matrix interface, using debonding models. The mate-rial parameters of the matrix constitutive equations were deduced from tests made on apure matrix [BAR 95] and the debonding models by combining micromechanical tests(push-out) and tension-compression transverse tests.

- efficiency of the method is greatly improved by the massive parallel computa-tional capabilities (independently of the parallel solution strategies used for the macro-scopic structural analysis);

- in order to deliver correct stress and strain fields at the lower scale, applica-tion of a specific relocalisation procedure was presented in the extreme case of a very"coarse grain" microstructure, in comparison with the structural size and the macro-scopic solution wavelength. Such a method was designed consistently with FE2, basedon the classical finite element solution interpolating techniques, considering all localstresses, strains and displacement components in the unit cell as the internal state vari-ables associated with each Gauss Point.

- This technique was shown to give adequate results in several examples, espe-cially for the bling analysis. The fibre size has been greatly enlarged over the actualsize, in order to be able to compare to the exact reference solution in which all themicrostructure has been meshed in the component. The comparisons are extremelygood except near the boundary of the composite region in the bling.

Some problems or difficulties have still to be treated, with the final objective ofa really powerful numerical methodology. The following axes of improvements arecurrently under examination:


- this relocalisation method must also be generalised in cases where edge effectsplay a role, due to the abrupt loss of periodicity near structural boundaries or in theregion near the (fictitious) interface between the composite substructure and the purematrix part. Such a situation is specific of our present application to the SiC/Ti MMC'sreinforced bling component; A method used recently in a similar context [KRU 98],but from taking overall constitutive equations for the homogeneous equivalent medium(in place of FE2), is assumed to offer good potentialities;

- another way to treat the coarse grain microstructure could be to enrich the unitcell periodic homogenisation, building an overall constitutive equation in the frame-work of Generalised Continuum Media (with material couples or higher order gradi-ents). Recent researches in this area could also provide interesting procedures[FOR 98];

— after solving these various problems it is expected to have a method with thepotential capability to predict the crack initiation at both the local (micro) level andat the macroscopic level. This capability will have to be checked by the treatment ofsome specific examples;

- moreover, the generalisation of FE2 method should also be examined for mi-crostructures with a lower degree of organisation and periodicity. A first attempt couldbe studied by introducing a more or less pronounced distortion in the fibre spatial ar-rangement. This is a long term objective for multiscale numerical methods.

AcknowledgmentsThe authors are very grateful to P. Suquet (CNRS-LMA Marseille, France) for a lot

of stimulating discussions especially regarding relocalization techniques in periodichomogenization.

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Index

3D damage modelling 93 et seq3D ductile crack growth, nonlocal

simulation 2093D modelling of necking 155

aluminium sheet materials, ductilerupture 259 et seq

anisotropic damage modelling 93 etseq

damage model, finite strain 241 etseq

damage modelling, 3Dcoupling effects of damage

and frictional sliding 93et seq

damage and sliding model 96numerical ductile rupture 165

hyper-elasticity 227sheet-metals, necking-failure

criterion and 141anisotropy, visualisation of 237axially vibrating rod, damage

identification 284

Banabic yield criterion 156bling disk, computation of a 302

closed mesocrack lips, frictionalsliding on 99

coarse grain structure, FE2 analysis305

cohesive zone model 270plane stress state 269

composite structures, multi-scalenon-linear FE2 analysis of

damage and fiber size effects 291et seq

concrete structures, transientresponse of 111 et seq

coupling damage and friction 111energy dissipation 111

consistent elastoplastic-damagetangent operator 199

constitutive damage models 168continuous damage mechanics,

Rousselier model 264Cook's problem 231coupling damage effect, friction and

sliding 93 et seq, 1013D anisotropic damage modelling

93 et seqcrack growth

damage and 114ductile, 3D nonlocal simulation

209cross-tool, deep-drawing with 159

damage analyses, nonlocalanisotropic model, finite strain

241modelling, 3D

coupling effects of damage,frictional sliding and 93 etseq

numerical ductile rupture165 et seq

sliding model and 96


consistent elastoplastic tangentoperator 199

coupling, friction andenergy dissipation, transient

response of concrete 111et seq

crack growth and 114criteria, evolution laws and 115ductile fracture and

numerical modelling 21implicit gradient-

enhanced formulation19 et seq

fiber size effects andmulti-scale non-linear FE2

analysis of compositestructures 291 et seq

gradientformulation 1, 3 et seqkey ingredients 4

identification, axially vibrating rod285

inelasticity and friction coupled to116

isotropic ductile, metal forming forfinite elastoplasticity 183et

seqmechanics, continuous

Rousselier model 264mesocrack growth, unilateral

behaviour and 96model

finite strain plasticity 214Gurson-Tvergaard 130Rousselier, the 215simple nonlocal 85"symmetric" isotropic

local 78nonlocal version 80

modelling, non local 63models

constitutive 168Gerson's 168Gologanus' 168

nonlocal 41adaptive analysis 39

numerical aspects of 75numerical aspects of 75parameter identification 136sheet metal forming

prediction of neckingphenomenon 147

sliding coupling and 101uniform vibrating rod 278

damageable disk, FE2 analysis 305deep-drawing

cross-tool 159hydraulic, fracture prediction 205

deformationlarge

analysis 33hyperelasto-plasticity model

23small

analysis 32elasto-plasticity model 22

discretized form, coupled problem 5disk, FE2 analysis

bl ing, computation of a 302damageable 305undamageable 305

dissipative materials, standard 233ductile crack growth, 3D nonlocal

simulation of 209 et seqdamage 27

isotropic, metal formingfinite elastoplasticity 183

et seqnumerical modelling

fracture and implicitgradient-enhancedformulation 19

rupturealuminium sheet materials

259 et seqnumerical, anisotropic

damage applied to 165et seq

EfiCoS, finite element code 119elasticity-based local approach,

fracture to 55 et seq

elastoplastic-damage tangentoperator, consistent 199

elasto-plasticityfinite

isotropic ductile damage,metal forming for 183 etseq

formulations 22model, small deformation and 22

energy dissipation, transient responseof concrete structures

coupling damage, friction and 111et seq

mechanical 191metric tensors 228, 230potentials 190

Gibbsand 248thermal 191transient response of concrete

structurescoupling damage, friction and

111 et seqerror estimator, the 42evolution laws, damage criteria and

115

failure, sheet metal form, analysis127

FE2 method 295multi-scale non-linear analysis of

composite structuresdamage and fiber size effects

291principle 296

FE simulations, necking predictionand 158

FEM solution procedure, given loadlevel 212

fiber size effects, damage and multi-scale non-linear FE2 analysis ofcomposite structures 291 et seq

finite elastoplasticityelement

code:EfiCoS 119implementation 66

Index 319

isotropic ductile damage, metalforming for 183 et seq

strainanisotropic damage model

241 et seqplasticity, damage model

and 214fracture

ductile damage andnumerical modelling 21

local approach to 58elasticity-based 55 et seq

prediction, hydraulic deep drawing205

frequency response function (FRF)279

frictioncoupled to damage, inelasticity and

116coupling damage and 111 et seq

frictional slidingclosed mesocrack lips on 99coupling effects of damage and

3D anisotropic damagemodelling 93 et seq

Gibbs, dissipation potentials and 248given load level, FEM solution

procedure 212Gologanu's model 168gradient damage

enhanced formulation, implicitnumerical modelling of

ductile damage 19formulation 1, 3key ingredients 4

Gurson's model 168Gurson-Tvergaard damage model

130

hydraulic deep drawing, fractureprediction 205

hyper-elasticity 227, 229anisotropic 227

hyperelasto-plasticity model, largedeformation 23


implicit gradient-enhancedformulation 19

numerical modelling, ductiledamage of 19

inelasticity, friction coupled todamage and 116

integrated multiscale analyses 293"interpolated-mapping" relocalization

technique 301isotropic ductile damage, metal

forming forfinite elastoplasticity 183 et seq

isotropic "symmetric" damage modellocal 78nonlocal version 80

large deformationanalysis 33hyperelasto-plasticity model 23

localisation, mesh sensitivity and 61

material moduli 251mechanical dissipation 191mesh sensitivity, localisation and 61mesocrack

growth 106unilateral behaviour and,

damage by 96lips, closed

frictional sliding on 99metal forming

coupled constitutive equations for188

isotropic ductile damage 183 etseq

finite elastoplasticity 183 et seqmodel non-locality of 45multi-scale analyses

integrated, sequential 293non-linear FE2 analysis, composite

structures ofdamage, fiber size effects and

291 et seq

necking3D modelling of 155

failure criterion, anisotropic sheet-metals 141

phenomenon, prediction ofsheet metal forming,

damage in 147 et seqsimulations and prediction of 155

FE 158

one-dimensional bar, uniaxial tension9

parallel computing 299periodic homogenization theory 295perturbation technique 149plane stress

elements 270state, cohesive zone model 269

plastic limit-load model, voidcoalescence criterion 139

potential, sliding criteria and 117yield locus 152

plasticity 214finite strain, damage model and214

relocalization 300technique, "interpolated-mapping"301

Rousselier damage model 215, 264continuous mechanics 264

rupture, ductilealuminium sheet materials 259 et

seqnumerical, anisotropic damage

165 et seq

seismic case study 122sequential multiscale analyses 293shear, simple 236sheet materials, aluminium

ductile rupture 259 et seqmetal forming

damage inprediction of necking

phenomenon 147 et seqfailure in

numerical analysis 127 et seqmetals, anisotropic

necking-failure criterion141

simple nonlocal damage model 85sliding coupling, damage and 101 et

seqcriteria, plastic potential and 117frictional 108

closed mesocrack lips 99coupling effects of damage

3D anisotropic damagemodelling 93 et seq

model, anisotropic damage and 96small defects, identification of

vibration tests 275 et seqstandard dissipative materials 233strain 214,241,245

finiteanisotropic damage model

241 et seqplasticity, damage model

and 214stress measures and 245

stressloading, uniaxial 121measures, strain and 245plane

elements 270state, cohesive zone model

269

Index 321

"symmetric" isotropic local damagemodel 78,80

nonlocal version 80

tangent operator, consistentelastoplastic-damage 199

stiffness matrix, computation of298

thermal dissipation 191time integration procedure 196two-dimensional panel, uniaxial

tension 14

undamageable disk 305uniaxial stress loading 121

tensionone-dimensional bar 9two-dimensional bar 14

uniform vibrating rod, damage 278unit cells tensile test 173

vibrating rodaxially, damage identification 285uniform, damage in 278

vibration tests, identification of smalldefects 275

void coalescence criterion, plasticlimit-load model 139

Documents

Numerical Modelling in Damage Mechanics_edited by Kh. Saanouni _ All937572640Limit