Lecture Notes in Applied and Computational Mechanicsmatperso.mines-paristech.fr/Donnees/data04/489-rolduc11.pdf · Lecture Notes in Applied and Computational Mechanics Prof. Dr.-Ing

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Lecture Notes in Applied and Computational Mechanics

Prof. Dr.-Ing. Friedrich Pfeiffer Prof. Dr.-Ing. Peter Wriggers

Volume 55

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Multiscale Methods inComputational Mechanics

Ekkehard RammRené de Borst •Editors

Progress and Accomplishments

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René de BorstEditors

EindhovenThe Netherlands

Eindhoven University of Technology

[email protected]

Ekkehard Ramm

University of StuttgartInstitute of Structural Mechanics

StuttgartGermany

ISSN 1613-7736 e-ISSN 1860-0816ISBN 978-90-481-9808-5 e-ISBN 978-90-481-9809-2DOI 10.1007/978-90-481-9809-2

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Table of Contents

Preface ix

List of Authors xi

Part 1: Computational Fluid Dynamics

Residual-Based Variational Multiscale Theory of LES Turbulence Modeling 3Y. Bazilevs, V.M. Calo, T.J.R. Hughes and G. Scovazzi

A Posteriori Error Estimation for Computational Fluid Dynamics.The Variational Multiscale Approach 19G. Hauke, M.H. Doweidar and D. Fuster

Advances in Variational Multiscale Methods for Turbulent Flows 39P. Gamnitzer, V. Gravemeier and W.A. Wall

Variational Germano Approach for Multiscale Formulations 53I. Akkerman, S.J. Hulshoff, K.G. van der Zee and R. de Borst

Dissipative Structure and Long Term Behavior of a Finite ElementApproximation of Incompressible Flows with Numerical Subgrid ScaleModeling 75R. Codina, J. Principe and S. Badia

Large-Eddy Simulation of Multiscale Particle Dynamics at High VolumeConcentration in Turbulent Channel Flow 95B.J. Geurts

Part 2: Materials with Microstructure

An Incremental Strategy for Modeling Laminate Microstructures in FinitePlasticity – Energy Reduction, Laminate Orientation and Cyclic Behavior 117K. Hackl and D.M. Kochmann

v

vi Table of Contents

The Micromorphic versus Phase Field Approach to Gradient Plasticity andDamage with Application to Cracking in Metal Single Crystals 135O. Aslan and S. Forest

Homogenization and Multiscaling of Granular Media for DifferentMicroscopic Constraints 155C. Miehe, J. Dettmar and D. Zäh

Effective Hydraulic and Mechanical Properties of Heterogeneous Mediawith Interfaces 179L. Dormieux, L. Jeannin and J. Sanahuja

An Extended Finite Element Method for the Analysis of Submicron HeatTransfer Phenomena 195P. Lee, R. Yang and K. Maute

Part 3: Composites, Laminates and Structures: Optimization

Multiscale Modeling and Simulation of Composite Materials andStructures 215J. Fish

Multiscale Modelling of the Failure Behavior of Fibre-ReinforcedLaminates 233M.V. Cid Alfaro, A.S.J. Suiker and R. de Borst

Improved Multiscale Computational Strategies for Delamination 261O. Allix, P. Gosselet and P. Kerfriden

Damage Propagation in Composites – Multiscale Modeling andOptimization 281E. Ramm, A. Erhart, T. Hettich, I. Bruss, F. Hilchenbach and J. Kato

Computational Multiscale Model for NATM Tunnels: Micromechanics-Supported Hybrid Analyses 305S. Scheiner, B. Pichler, C. Hellmich and H.A. Mang

Optimization of Corrugated Paperboard under Local and Global BucklingConstraints 329T. Flatscher, T. Daxner, D.H. Pahr and F.G. Rammerstorfer

Framework for Multi-Level Optimization of Complex Systems 347A. de Wit and F. van Keulen

Table of Contents vii

Part 4: Coupled Problems and Porous Media

Multiscale/Multiphysics Model for Concrete 381B.A. Schrefler, F. Pesavento and D. Gawin

Swelling Phenomena in Electro-Chemically Active Hydrated Porous Media 405W. Ehlers, B. Markert and A. Acartürk

Propagating Cracks in Saturated Ionized Porous Media 425F. Kraaijeveld and J.M. Huyghe

Author Index 443

Subject Index 445

Preface

For a long time limited computational resources restricted the scale of observationand modeling of physical systems mostly to one scale in time and space. This modusoperandi was accepted although it was well known that the response at the level ofpractical interest is to a large extent determined by processes that occur at scaleswhich are one to several orders of a magnitude smaller, namely the meso, microor even nanoscales. The rapid increase in computer power and the development ofefficient computational methods allow coming closer to the human dream of a con-tinuous model through all spatial and temporal scales. However seeking solutionsusing numerical models simultaneously at the various length scales has proven to bebeyond current computational capabilities. This perception is the starting point forthe development of Multiscale Methods which is currently one of the hot researchtopics all over the world. Multiscale Methods either derive properties at the levelof observation by repeated numerical homogenization of more fundamental physicalproperties several scales below (upscaling), or they devise concurrent schemes wherethose parts of the domain that are of interest are computed with a higher resolutionthan parts that are of less interest or where the solution is varying only slowly.

The present volume contains selected papers presented at the InternationalColloquium on Multiscale Methods in Computational Mechanics in Rolduc, theNetherlands, on 11–13 March 2009 (MMCM 2009). The contributions in the firstpart address multiscale methods in Computational Fluid Dynamics; in particularturbulence modeling applying the Variational Multiscale Method is described. Thesecond part deals with materials having a distinct microstructure such as single- andpoly-crystals, granular materials, aggregates embedded in a matrix, micro-laminatesand nano-composites. Part three focuses on materials and structures and the interac-tion between their behavior on micro/meso- and macro-scales. Failure of compositesand laminates, material modeling of shotcrete in tunnels, buckling of paperboard ondifferent scales as well as material and structural optimization are investigated. Thefourth part concentrates on coupled problems and porous media like concrete andcharged hydrated materials.

The mentioned colloquium was organized by the German–Dutch Research Unit“Multiscale Methods in Computational Mechanics”, in which a group of scientistsin Germany at Universität Stuttgart and TU München and in the Netherlands at TU

ix

x Preface

Delft and TU Eindhoven have joined forces in order to address a variety of problemareas and computational solution strategies for multiscale methods in mechanics ofstructures, materials and flows. This joint project, which serves as a prototype forresearch cooperations within Europe, was financially supported by the German Re-search Foundation (Deutsche Forschungsgemeinschaft, DFG FOG 509), The Neth-erlands Technology Foundation (STW) and The Netherlands Organization for Sci-entific Research (NWO). This support is very much appreciated.

We also would like to thank the authors very much for their valuable contribu-tions. Furthermore our thanks go to Springer for taking over the publication of thisvolume in its LNACM series.

January 2010René de Borst, Technische Universiteit EindhovenEkkehard Ramm, Universität Stuttgart

List of Authors

Acartürk, Ayhan Y.Universität Stuttgart, Institut für Mechanik (Bauwesen), Stuttgart, [email protected]

Akkerman, IdoUniversity of California, San Diego, Department of Structural Engineering, [email protected]

Allix, OlivierL.M.T., ENS de Cachan, [email protected]

Aslan, OzgurMines ParisTech, Centre des Matériaux, Evry, [email protected]

Badia, SantiagoUniversitat Politècnica de Catalunya, International Center for Numerical Methods inEngineering (CIMNE), Barcelona, [email protected]

Bazilevs, YuriUniversity of California, San Diego, Department of Structural Engineering, [email protected]

de Borst, RenéEindhoven University of Technology, Department of Mechanical Engineering,Eindhoven, The [email protected]

xi








xii List of Authors

Bruss, IngridUniversität Stuttgart, Institut für Baustatik und Baudynamik, Stuttgart, [email protected]

Calo, Victor M.KAUST, Earth and Environmental Science and Engineering, Saudi [email protected]

Cid Alfaro, Marcela V.Corus, IJmuiden, The Netherlands

Codina, RamonUniversitat Politècnica de Catalunya, Departament de Resistencia de Materials iEstructures a l’Enginyeria, Barcelona, [email protected]

Daxner, ThomasCAE Simulations & Solutions, Wien, [email protected]

Dettmar, JoachimUniversität Stuttgart, Institut für Mechanik (Bauwesen), Stuttgart, Germany

Dormieux, LucÉcole Nationale des Ponts et Chaussées, Laboratoire des Matériaux et Structures duGénie Civil (LMSGC), Champs-sur-Marne, [email protected]

Doweidar, Mohamed H.Centro PolitCcnico Superior de Zaragoza, Área de Mecánica de Fluidos, Zaragoza,Spain

Ehlers, WolfgangUniversität Stuttgart, Institut für Mechanik (Bauwesen), Stuttgart, [email protected]

Erhart, AndreaDYNAmore, Stuttgart, [email protected]

Fish, JacobRensselaer Polytechnic Institute, Multiscale Science and Engineering Center,Troy, NY, [email protected]









List of Authors xiii

Flatscher, ThomasTechnische Universität Wien, Institut für Leichtbau und Struktur-Biomechanik,Wien, [email protected]

Forest, SamuelMines ParisTech, Centre des Matériaux, Evry, [email protected]

Fuster, DanielCentro Politécnico Superior de Zaragoza, Área de Mecánica de Fluidos, Zaragoza,Spain

Gamnitzer, PeterTechnische Universität München, Lehrstuhl für Numerische Mechanik, München,[email protected]

Gawin, DariuszTU Lodz, Department of Building Physics and Building Materials, Lodz, [email protected]

Geurts, Bernard J.University of Twente, Mathematical Sciences, Enschede; Eindhoven University ofTechnology, Department of Applied Physics, Eindhoven, The [email protected]

Gosselet, PierreL.M.T., ENS de Cachan, [email protected]

Gravemeier, VolkerTechnische Universität München, Lehrstuhl für Numerische Mechanik, München,[email protected]

Hackl, KlausRuhr-Universität Bochum, Lehrstuhl für Allgemeine Mechanik, Bochum, [email protected]

Hauke, GuillermoCentro Politécnico Superior de Zaragoza, Área de Mecánica de Fluidos, Zaragoza,[email protected]










xiv List of Authors

Hellmich, ChristianTechnische Universität Wien, Institut für Mechanik der Werkstoffe und Strukturen,Wien, [email protected]

Hettich, Thomas M.Mahle GmbH, Stuttgart, Germany

Hilchenbach, FrédéricUniversität Stuttgart, Institut für Baustatik und Baudynamik, Stuttgart, [email protected]

Hughes, Thomas J.R.The University of Texas at Austin, Institute for Computational Engineering andSciences, Austin, [email protected]

Hulshoff, Steven J.Delft University of Technology, Aerospace Engineering, Delft, The [email protected]

Huyghe, Jacques M.R.Eindhoven University of Technology, Department of Biomedical Engineering,Eindhoven, The [email protected]

Jeannin, LaurentGaz de France (GDF), France

Kato, JunjiUniversität Stuttgart, Institut für Baustatik und Baudynamik, Stuttgart, [email protected]

Kerfriden, PierreL.M.T., ENS de Cachan, [email protected]

Kochmann, Dennis M.Ruhr-Universität Bochum, Lehrstuhl für Allgemeine Mechanik, Bochum, [email protected]

Kraaijeveld, FamkeEindhoven University of Technology, Eindhoven, The [email protected]










List of Authors xv

Lee, PilhwaUniversity of Colorado at Boulder, Department of Mechanical Engineering,Boulder, [email protected]

Mang, HerbertTechnische Universität Wien, Institut für Mechanik der Werkstoffe und Strukturen,Wien, [email protected]

Markert, BerndUniversität Stuttgart, Institut für Mechanik (Bauwesen), Stuttgart, [email protected]

Maute, KurtUniversity of Colorado at Boulder, Department of Aerospace Engineering Sciences,Boulder, [email protected]

Miehe, ChristianUniversität Stuttgart, Institut für Mechanik (Bauwesen), Stuttgart, [email protected]

Pahr, Dieter H.Technische Universität Wien, Institut für Leichtbau und Struktur-Biomechanik,Wien, [email protected]

Pesavento, FrancescoUniversità degli Studi di Padova, Dipartimento di Costruzioni e Trasporti,Padova, [email protected]

Pichler, BernhardTechnische Universität Wien, Institut für Mechanik der Werkstoffe und Strukturen,Wien, [email protected]

Principe, JavierUniversitat Politécnica de Catalunya, International Center for Numerical Methods inEngineering (CIMNE), Barcelona, [email protected]










xvi List of Authors

Ramm, EkkehardUniversität Stuttgart, Institut für Baustatik und Baudynamik, Stuttgart, [email protected]

Rammerstorfer, Franz G.Technische Universität Wien, Institut für Leichtbau und Struktur-Biomechanik,Wien, [email protected]

Sanahuja, JulienElectricité de France (EDF), Department of Materials and Mechanics ofComponents, Moret-sur-Loing, [email protected]

Schrefler, Bernhard A.Università degli Studi di Padova, Dipartimento di Costruzioni e Trasporti,Padova, [email protected]

Scheiner, StefanThe University of Western Australia, Computer Science and Software Engineering,Crawley, Perth, Western [email protected]

Scovazzi, GuglielmoSandia National Laboratories, Albuquerque, [email protected]

Suiker, Akke S.J.Delft University of Technology, Aerospace Engineering, Delft, The [email protected]

van der Zee, KrisThe University of Texas at Austin, Institute for Computational Engineering andSciences, Austin, USA

van Keulen, FredDelft University of Technology, Department of Precision and MicrosystemsEngineering, Delft, The [email protected]

Wall, Wolfgang A.Technische Universität München, Lehrstuhl für Numerische Mechanik,München, [email protected]










List of Authors xvii

de Wit, Albert J.Dutch National Aerospace Laboratory (NLR), Amsterdam, The [email protected]

Yang, RongguiUniversity of Colorado at Boulder, Department of Mechanical Engineering,Boulder, [email protected]

Zäh, DominicUniversität Stuttgart, Institut für Mechanik (Bauwesen), Stuttgart, [email protected]




The Micromorphic versus Phase Field Approach toGradient Plasticity and Damage with Application toCracking in Metal Single Crystals

Ozgur Aslan and Samuel Forest

MINES ParisTech, Centre des Matériaux, CNRS UMR 7633BP 87 91003 Evry Cedex, France; [email protected]

Abstract ��Author, please supply!��

Key words: ��Author, please supply!��

1 Generalized Continua and Material Microstructure

The mechanics of generalized continua represents a way of introducing, in the con-tinuum description of materials, some characteristic length scales associated withtheir microstructure [24]. Such intrinsic lengths and generalized constitutive equa-tions can be identified in two ways. Direct identification is possible from experi-mental curves exhibiting clear size effects in plasticity or fracture or from full-fieldstrain measurements of strongly heterogeneous fields [13]. The effective propertiesof such generalized continua can also be derived from scale transition and homogen-ization techniques by prescribing appropriate boundary conditions on a representat-ive volume of material with microstructure [5].

The multiplication of generalized continuum model formulations from Cosseratto strain gradient plasticity in literature may leave an impression of disorder andinconsistency. Recent accounts have shown, on the contrary, that unifying presenta-tions of several classes of generalized continuum theories are possible [11, 18]. Oneof them, called the micromorphic approach, encompasses most theories incorporat-ing additional degrees of freedom from the well-established Cosserat, microstretchand micromorphic continua [9] up to Aifantis and Gurtin strain gradient plasticitytheories.

The objective of this chapter is to present this systematic approach for incorporat-ing intrinsic lengths in non-linear continuum mechanical models and to illustrate theefficiency of the method in the case of an anisotropic plasticity and damage model.The so-called microdamage model takes the crystallography of plasticity and fracturein metal single crystals.

The micromorphic approach is exposed in Section 2 together with the closelyrelated phase field approach. Differences and similarities between the micromorphic

2 O. Aslan and S. Forest

framework and the phase field approach are pointed out following the general frame-work provided in [17]. A single crystal plasticity and damage model is explored inSection 3. The finite element implementation and its validation for monotonic crackgrowth are shown in the two last sections.

The notations used are the same as the ones given in [11]. For the sake of con-ciseness, the theory and applications are presented within the framework of smalldeformation.

2 Micromorphic Approach

2.1 Thermomechanics with Additional Degrees of Freedom

One starts from an elastoviscoplasticity model formulation within the framework ofthe classical Cauchy continuum and classical continuum thermodynamics accordingto Germain et al. [16] and Maugin [22]. The material behaviour is characterized bythe reference sets of degrees of freedom and state variables

DOF0 = {u }, ST AT E0 = {ε∼, T , α} (1)

on which the free energy density function ψ may depend. The displacement vectoris u . The strain tensor is denoted by ε∼ whereas α represents the whole set of internalvariables of arbitrary tensorial order accounting for nonlinear processes at work in-side the material volume element, like isotropic and kinematic hardening variables.The absolute temperature is T .

Additional degrees of freedom χφ are then introduced in the previous originalmodel. They may be of any tensorial order and of different physical nature (deform-ation, plasticity or damage variable). The notation χ indicates that these variableseventually represent some microstructural features of the material so that we willcall them micromorphic variables or microvariables (microdeformation, microdam-age, etc.). The DOF and ST AT E spaces are extended as follows:

DOF = {u , χφ}, ST AT E = {ε∼, T , α, χφ, ∇χφ} (2)

Depending on the physical nature of χφ, it may or may not be a state variable. Forinstance, if the microvariable is a microrotation as in the Cosserat model, it is not astate variable for objectivity reasons and will appear in ST AT E only in combinationwith the macrorotation. In contrast, if the microvariable is a microplastic equivalentstrain, as in Aifantis model, it can then explicitly appears in the state space.

The virtual power of internal forces is then extended to the power done by themicromorphic variable and its first gradient:

P (i)(v �,χ φ�) = −∫

Dp(i)(v �,χ φ�) dV

p(i)(v �,χ φ�) = σ∼ : ∇v � + aχ φ� + b · ∇χ φ� (3)

Micromorphic Approach to Gradient Damage 3

where D is a subdomain of the current configuration � of the body. The Cauchystress is σ∼ and a and b are generalized stresses associated with the micromorphicvariable and its first gradient. Similarly, the power of contact forces must be extendedas follows:

P (c)(v �,χ φ�) =∫

Dp(c)(v �,χ φ�) dV, p(c)(v �,χ φ�) = t · v � + ac χ φ� (4)

where t is the traction vector and ac a generalized traction. For conciseness, we donot extend the power of forces acting at a distance and keep the classical form:

P (e)(v �,χ φ�) =∫

Dp(e)(v �,χ φ�) dV, p(e)(v �,χ φ�) = ρf · v � (5)

where ρf accounts for given simple body forces. Following Germain [15], givenbody couples and double forces working with the gradient of the velocity field, couldalso be introduced in the theory. The generalized principle of virtual power withrespect to the velocity and micromorphic variable fields, is presented here in thestatic case only:

P (i)(v �,χ φ) + P (e)(v �,χ φ�) + P (c)(v �,χ φ) = 0, ∀D ⊂ �, ∀v �,χ φ (6)

The method of virtual power according to Maugin [21] is used then to derive thestandard local balance of momentum equation:

div σ∼ + ρf = 0, ∀x ∈ � (7)

and the generalized balance of micromorphic momentum equation:

div b − a = 0, ∀x ∈ � (8)

The method also delivers the associated boundary conditions for the simple and gen-eralized tractions:

t = σ∼ .n , ac = b .n , ∀x ∈ ∂D (9)

The local balance of energy is also enhanced by the generalized micromorphic poweralready included in the power of internal forces (3):

ρε = p(i) − div q + ρr (10)

where ε is the specific internal energy, q the heat flux vector and r denotes externalheat sources. The entropy principle takes the usual local form:

−ρ(ψ + ηT ) + p(i) − q

T.∇T ≥ 0 (11)

where it is assumed that the entropy production vector is still equal to the heat vectordivided by temperature, as in classical thermomechanics according to Coleman andNoll [6]. Again, the enhancement of the theory goes through the enriched power


density of internal forces (3). The entropy principle is exploited according to classicalcontinuum thermodynamics to derive the state laws. At this stage it is necessaryto be more specific on the dependence of the state functions ψ, η, σ∼, a, b on statevariables and to distinguish between dissipative and non-dissipative mechanisms.The introduction of dissipative mechanisms may require an increase in the number ofstate variables. These different situations are considered in the following subsections.

2.2 Non-Dissipative Contribution of Generalized Stresses and MicromorphicModel

Dissipative events are assumed here to enter the model only via the classical mech-anical part. Total strain is split into elastic and plastic parts:

ε∼ = ε∼e + ε∼

p (12)

The following constitutive functional dependencies are then introduced:

ψ = ψ(ε∼e, T , α,χφ,∇χφ), σ∼ = σ∼(ε∼

e, T , α,χφ,∇χφ), η = η(ε∼e, T , α,χφ,∇χφ)

a = a(ε∼e, T , α,χφ,∇χφ), b = b (ε∼

e, T , α,χφ,∇χφ) (13)

The entropy inequality (11) can be expanded as(σ∼ − ρ

∂ψ

∂ε∼e

): ε∼

e +ρ

(η + ∂ψ

∂T

)T +

(a − ρ

∂ψ

∂ χφ

)χ φ +

(b − ρ

∂ψ

∂∇χφ

)·∇χ φ

+σ∼ : ε∼p − ρ

∂ψ

∂αα − q

T· ∇T ≥ 0 (14)

Assuming that no dissipation is associated with the four first terms of the previousinequality, the following state laws are found:

σ∼ = ρ∂ψ

∂ε∼e, η = −∂ψ

∂T, X = ρ

∂ψ

∂α(15)

a = ρ∂ψ

∂ χφ, b = ρ

∂ψ

∂∇χφ(16)

and the residual dissipation is

Dres = Wp − Xα − q

T.∇T ≥ 0 (17)

where Wp represents the (visco-)plastic power and X the thermodynamic force as-sociated with the internal variable α. The existence of a convex dissipation potential,�(σ∼,X) depending on the thermodynamic forces can then be assumed from whichthe evolution rules for internal variables are derived, that identically fulfill the en-tropy inequality, as usually done in classical continuum thermomechanics [16]:

ε∼p = ∂�

∂σ∼, α = ∂�

∂X(18)


Micromorphic Model

After presenting the general approach, we readily give the most simple examplewhich provides a direct connection to several existing generalized continuum mod-els. We consider first cases where φ and χφ are observer invariant quantities. Thefree energy density function ψ is chosen as a function of the generalized relativestrain variable e defined as:

e = φ −χφ (19)

thus introducing a coupling between macro and micromorphic variables. Assumingisotropic material behavior for brevity, the additional contributions to the free energycan be taken as quadratic functions of e and ∇χφ:

ψ(ε∼, T , α,χφ,∇χφ) = ψ(1)(ε∼, T , α) + ψ(2)(e = φ −χφ,∇χφ, T ), with (20)

ρψ(2)(e,∇χφ, T ) = 1

2Hχ(φ −χφ)2 + 1

2A∇χφ · ∇χφ (21)

After inserting the state laws (16)

a = ρ∂ψ

∂χφ= −Hχ(φ −χφ), b = ρ

∂ψ

∂∇χφ= A∇χφ (22)

into the additional balance equation (8), the following partial differential equation isobtained, at least for a homogeneous material under isothermal conditions:

φ =χφ − A

Hχ

�χφ (23)

where � is the Laplace operator. This type of equation is encountered at severalplaces in the mechanics of generalized continua especially in the linear micro-morphic theory [7, 9, 23] and in the so-called implicit gradient theory of plasticityand damage [8, 25, 26]. Note however that this equation corresponds to a specialquadratic potential and represents the simplest micromorphic extension of the clas-sical theory. It involves a characteristic length scale defined by

l2c = A

Hχ

(24)

This length is real for positive values of the ratio A/Hχ . The additional materialparameters HX and A are assumed to be positive in this work. This does not excludea softening material behaviour that can be induced by the proper evolution of theinternal variables (including φ = α itself).

2.3 Viscous Generalized Stress and Phase Field Model

Generalized stresses can also be associated with dissipation by introducing the vis-cous part av of a:


ε∼ = ε∼e + ε∼

p, a = ae + av (25)

The entropy inequality (11) now becomes(

σ∼ − ρ∂ψ

∂ε∼e

): ε∼

e +ρ

(η + ∂ψ

∂T

)T +

(ae − ρ

∂ψ

∂ χφ

)χ φ+

(b − ρ

∂ψ

∂∇χφ

)·∇χ φ

+σ∼ : ε∼p − ρ

∂ψ

∂αα + av χ φ − q

T· ∇T ≥ 0 (26)


σ∼ = ρ∂ψ

∂ε∼e, η = −∂ψ

∂T, X = ρ

∂ψ

∂α(27)

ae = ρ∂ψ

∂ χφ, b = ρ

∂ψ

∂∇χφ(28)


Dres = σ∼ : ε∼p − Xα + av χ φ − q

T· ∇T ≥ 0 (29)

Evolution rules for viscoplastic strain, internal variables, and the additional degreesof freedom can be derived from a dissipation potential �(σ∼,X, av):

ε∼p = ∂�

∂σ∼, α = ∂�

∂X, χ φ = ∂�

∂av(30)

Convexity of the dissipation potential then ensures positivity of dissipation rate forany process.

Note that no dissipative part has been assigned to the generalized stress b sincethen exploitation of second principle does not seem to be straightforward. Instead,the total gradient ∇χφ can be split into elastic and plastic parts, as it will be done inSection 2.4.

Phase Field Model

The dissipation potential can be decomposed into the various contributions due toall thermodynamic forces. Let us assume for instance that the contribution of theviscous generalized stress av is quadratic:

�(σ∼,X, av) = �1(σ∼,X) + �2(av), �2(a

v) = 1

2βav2 (31)

The use of the flow rule (30) and of the additional balance equation (8) then leads to


β χ φ = av = a − ae = a − ρ∂ψ

∂χφ= div ρ

∂ψ

∂∇χφ− ρ

∂ψ

∂χφ(32)

One recognizes the Landau–Ginzburg equation that arises in phase field theories.The previous derivation of Landau–Ginzburg equation is due to Mühlhaus [1] andPeerlings et al. [17]. The coupling with mechanics is straightforward according tothis procedure and more general dissipative mechanics can be put forward.

2.4 Elasto-Plastic Decomposition of Generalized Strains

Instead of the previous decomposition of generalized stresses, the introduction ofadditional dissipative mechanisms can rely on the split of all strain measures intoelastic and plastic parts:

ε∼ = ε∼e + ε∼

p, χφ =χφe +χφp, κ = ∇χφ = κ e + κ p (33)

The objectivity ofχφ is required for such a unique decomposition to be defined. Wedo not require here that

κ e = ∇χφe, κ p = ∇χφp (34)

although such a model also is possible, as illustrated by the gradient of strain theoryput forward in [12]. The Clausius–Duhem inequality then writes(

σ∼ − ρ∂ψ

∂ε∼e

): ε∼

e + ρ

(η + ∂ψ

∂T

)T +

(a − ρ

∂ψ

∂ χφ

)χ φe +

(b − ρ

∂ψ

∂∇χφ

)· κ e

+σ∼ : ε∼p − ρ

∂ψ

∂αα + aχφp + b · κ p − q

T· ∇T ≥ 0 (35)


σ∼ = ρ∂ψ

∂ε∼e, η = −∂ψ

∂T, X = ρ

∂ψ

∂α(36)

a = ρ∂ψ

∂ χφe, b = ρ

∂ψ

∂κ e(37)


Dres = σ∼ : ε∼p − Xα + aχ φp + b · κ p − q

T· ∇T ≥ 0 (38)

Evolution rules for viscoplastic strain, internal variables, and the additional degreesof freedom can be derived from a dissipation potential �(σ∼,X, a, b ):

ε∼p = ∂�

∂σ∼, α = ∂�

∂X, χ φp = ∂�

∂a, κ p = ∂�

∂b(39)


As a result of the additional balance equation (8) combined with the previous statelaws, the type of derived partial differential equation can be made more specificby adopting a quadratic free energy potential for b (modulus A) and a quadraticdissipation potential with respect to a (parameter β). We obtain:

β χ φ = a + β χ φe = div Aκ − div Aκ p + β χ φe (40)

Decompositions of stresses and strains can also be mixed, for instance in the follow-ing way:

ε∼ = ε∼e + ε∼

p, a = ae + av, κ = ∇χφ = κ e + κ p (41)

based on which a constitutive theory can be built.

3 Continuum Damage Model for Single Crystals and ItsRegularization

We present here a constitutive model for damaging viscoplastic single crystal be-haviour aiming at simulating crack initiation and propagation. The micromorphicapproach is then applied to this model in order to obtain a regularized continuumdamage formulation with a view to simulating mesh-independent crack propagationin single crystals.

3.1 Constitutive Equations

In the proposed crystal plasticity model taken from [20], viscoplasticity and damageare coupled by introducing an additional damage strain variable ε∼

d , into the strainrate partition equation:

ε∼ = ε∼e + ε∼

p + ε∼d (42)

where ε∼e and ε∼

p are the elastic and the plastic strain rates, respectively. The flow rulefor plastic part is written at the slip system level by means of the orientation tensorm∼

s :

m∼s = 1

2(n s ⊗ l s + l s ⊗ n s) (43)

where n s is the normal to the plane of slip system s and l s stands for the corres-ponding slip direction. Then, plastic strain rate reads:

ε∼p =

Nslip∑s=1

γ sm∼s (44)

The flow rule on slip system s is a classical Norton rule with threshold.

γ s =⟨ |τ s − xs | − rs

K

⟩n

sign(τ s − xs) (45)


Fig. 1. Illustration of the cleavage and two accommodation systems to be associated to thecrystallographic planes.

where rs and xs are the variables for isotropic and kinematic hardening respectivelyand K and n are material parameters to be identified from experimental curves.

Material separation is assumed to take place w.r.t. specific crystallographicplanes, like cleavage planes in single crystals. The word cleavage is written in amore general sense that its original meaning in physical metallurgy associated withbrittle fracture of non-f.c.c. crystals. In the continuum mechanical model, cleavagemeans cracking along a specific crystallographic plane as it is often observed in lowcycle fatigue of f.c.c. crystals like single crystal nickel-base superalloys. The damagestrain ε∼

d is decomposed in the following crystallographic contributions:

ε∼d =

Ndamage∑s=1

δscn

sd ⊗ n s

d + δs1n

sd

sym⊗ l sd1

+ δs2n

sd

sym⊗ l sd2

(46)

where δs , δs1 and δs

2 are the strain rates for mode I, mode II and mode III crack growth,respectively and Nd

damage stands for the number of damage planes which are fixedcrystallographic planes depending on the crystal structure. Cleavage damage is rep-resented by the opening δs of crystallographic cleavage planes with the normal vectorn s . Additional damage systems must be introduced for the in-plane accommodationalong orthogonal directions l s

1 and l s2, once cleavage has started (Figure 1). Three

damage criteria are associated to one cleavage and two accommodation systems:

f sc = ∣∣n s

d · σ∼ · n sd

∣∣ − Y sc (47)

f si =

∣∣∣n sd · σ∼ · l s

di

∣∣∣ − Y si (i = 1, 2) (48)

The critical normal stress Y s for damage decreases as δ increases:

Y sc = Y s

0 + Hδsc , Y s

i = Y s0 + Hδs

i (49)

where Y s0 is the initial damage stress (usually coupled to plasticity) and H is a negat-

ive modulus which controls material softening due to damage. Finally, evolution ofdamage is given by the following equations:


δsc =

⟨f s

c

Kd

⟩nd

sign(n s

d · σ∼ · n sd

)(50)

δsi =

⟨f s

i

Kd

⟩nd

sign(n s

d · σ∼ · l sdi

)(51)

where Kd and nd are material parameters.These equations hold for all conditions except when the crack is closed (δs

c < 0)and compressive forces are applied (n s

d · σ∼ · n sd < 0). In this case, damage evolution

stops (δsc = δs

i = 0), corresponding to the unilateral damage conditions.Note that the damage variables δ introduced in the model differ from the usual

corresponding variables of standard continuum damage mechanics that vary from 0to 1. In contrast, the δs are strain-like quantities that can ever increase.

Coupling between plasticity and damage is generated through initial damagestress Y0 in (49) which is controlled by cumulative slip variable γcum:

γcum =Nslips∑s=1

|γ s | (52)

Then, Y0 takes the form:Y s

0 = σcn e−dγcum + σult (53)

This formulation suggests an exponential decaying regime from a preferably highinitial cleavage stress value σc

n , to an ultimate stress, σult which is close to but notequal to zero for numerical reasons and d is a material constant.

This model, complemented by the suitable constitutive equations for viscoplasticstrain, has been used for the simulation of crack growth under complex cyclic loadingat high temperature [19]. Significant mesh dependency of results was found [20].

In the present work, the model is further developed by switching from classical tomicrodamage continuum in order to assess the regularization capabilities of a higherorder theory. The coupling of the model with microdamage theory is achieved byintroducing a cumulative damage variable calculated from the damage systems anda new threshold function Y0(δ, γcum):

δcum =Nplanes∑

s=1

δs , where δs = |δsc| + |δs

1| + |δs2| (54)

Y0 = σcn e−dγcum−Hδcum + σult (55)

3.2 Microdamage Continuum

The micromorphic medium introduced by Eringen and Suhubi [10] possesses a fullmicrodeformation field χ

∼, in addition to the classical displacement field u . Con-

taining additional degrees of freedom and balance equations, the micromorphic con-tinuum approach can be considered as the main framework for most generalized


continuum models [11]. Alternative micromorphic variables other than the full straintensor can be chosen [2, 11]. The strain gradient effect can for instance be limited tothe damage strain ε∼

d gradient and more specifically to the damage variable δcum

introduced in the previous model and noted here δ for conciseness.In microdamage theory, the introduced microvariable is a scalar microdamage

parameter χδ:

DOF = {u , χδ} ST RAIN = {ε∼, χ δ,∇ χδ} (56)

The power of internal forces is extended as

p(i) = σ∼ : ε∼ + a χ δ + b .∇ χ δ (57)

where generalized stresses a, b have been introduced. The generalized balance equa-tions are:

div σ∼ = 0, a = div b (58)

The free energy density is taken as a quadratic potential in the elastic strain, damageδ, relative damage δ − χδ and microdamage gradient ∇χδ:

ρψ = 1

2ε∼

e : c∼∼: ε∼

e + 1

2

Ndamage∑s=1

Hδ2s + 1

2χH(δ − χδ)

2 + 1

2A∇ χδ · ∇ χδ (59)

where H, χH and A are scalar material constants. Then, the elastic response of thematerial becomes

σ∼ = ρ∂ψ

∂ε∼e

= c∼∼: ε∼

e (60)

The generalized stresses read

a = ρ∂ψ

∂ χδ= − χH(δ − χδ) , b = A∇ χδ (61)

and the driving force for damage can be derived as

Y s = ρ∂ψ

∂δs= Hδs + χH(δs − χδ) (62)

The damage criterion is now

f s = ∣∣n s · σ∼ · n s∣∣ − Y0 − Y s = 0 (63)

Substituting the linear constitutive equations for generalized stresses into the addi-tional balance equation (58), assuming homogeneous material properties, leads tothe following partial differential equation for the microdamage:

χδ − A

χH�χδ = δ (64)


Fig. 2. Comparison between force vs. displacement diagram of a 1D softening rod for linearand exponential decay.

where the macrodamage δ acts as a source term. Exactly this type of Helmholtzequation has been postulated in the so-called implicit gradient theory of plasticityand damage [8, 14, 25, 26], where the microvariables are called non local variablesand where the generalized stresses a and b are not explicitly introduced (see [7, 11]for the analogy between this latter approach and the micromorphic theory).

The solution of the problem of failure of a 1D bar in tension/compression wastreated in [2]. The characteristic size of the damage zone was shown to be controlledby the characteristic length corresponding to the inverse of

ω = √|H χH/A(H + χH)| (65)

In comparison with the standard strain gradient approaches [14, 25], microdamagetheory eliminates the final fracture problem without any modification to the damagefunction, since there exists no direct coupling between the force stress σ∼ and thegeneralized stresses, a and b . For a better representation of a cracked element, wesuggest an exponential drop both for damage threshold Y0 and the modulus A, sincethe element should become unable to store energy due to the generalized stresseswhen broken (see Figure 2):

Y0 = σcn e−Hδ + σult , b = Ae−Hδ∇ χδ (66)

4 Finite Element Implementation

4.1 Variational Formulation and Discretization

The variational formulation of the microdamage approach can be derived directlyfrom the principle of virtual power (57):

∫�

p(i)dV +∫

∂�

p(c)dS = 0 (67)

Micromorphic Approach to Gradient Damage 13∫

�

(σ∼ : ε∼ + a χ δ + b .∇ χ δ)dV +∫

∂�

(t .u + a χ δ)dS = 0 (68)

Finite element discretization of the displacement field u and the microdamage fieldχδ takes the following form:

u = Nudu, ∇u = Budu,χδ = Nδdδ, ∇χδ = Bδdδ (69)

where du and dδ are the nodal degrees of freedom. Nu and Nδ represent the shapefunctions and Bu and Bδ stand for their partial derivatives with respect to the co-ordinates. In this work we use isoparametric quadratic elements for both types ofdegrees of freedom.

Finally, the discretized equilibrium equations read:∫

�

BTu σ∼dV =

∫�

NuT f dV +

∫�

NuT t dS (70)

∫�

(NδT a + Bδ

T b )dV =∫

�

NδT acdS (71)

4.2 Implicit Incremental Formulation

A fully implicit Newton–Raphson incremental formulation is developed for solving(70, 71). The corresponding time discretization is now introduced. Using the knownvalues of the state variables ε∼

e(t), υs(t) (integrated from υs = |γ s |),δsc,i(t), δ

scum(t)

for the current time step, the values at t + �t are estimated by a straight forwardlinearization procedure.

ε∼e(t + �t) = �t ε∼

e(t + �t)︸︷︷︸�ε∼

e

+ε∼e(t) (72)

υs(t + �t) = �t υs(t + �t) + υs(t) (73)

δsc,i(t + �t) = �t δs

c,i(t + �t) + δsc,i(t) (74)

δscum(t + �t) = �t δs

cum(t + �t) + δscum(t) (75)

The model is implemented into the FE code ZeBuLoN [4], using a θ -method for thelocal integration. In order to calculate the state variable increments, the residuals andtheir Jacobian are written as follows:

Rε∼e = �ε∼

e + �ε∼p + �ε∼

d − �ε∼ (76)

= �εe +Nslip∑s=1

m∼s�υssign(τ s − xs) +

Nplanes∑s=1

�δscn

sd ⊗ n s

d + �δsi n

sd ⊗ l s

di(77)

Rυs = �υs − �t

⟨�s

K

⟩n

(78)


Rδsc

= �δsc − �t

⟨f s

c

Kd

⟩nd

sign(n sd · σ∼ · n s

d) (79)

Rδsi

= �δic − �t

⟨f s

i

Kd

⟩nd

sign(n sd · σ∼ · l s

di) (80)

Rδscum

= �δcum − �

⎛⎝

Nplanes∑s=1

∣∣δsc

∣∣ + ∣∣δs1

∣∣ + ∣∣δs2

∣∣⎞⎠ (81)

[J ] = ∂{R}∂{�ν} = 1 − �t

∂{ν}∂{�ν}

∣∣∣∣t+�t

(82)

where {R}T =[Rεe, Rυs , Rδs

c, Rδs

i, Rδcum

]and ν stands for the internal state vari-

ables to be integrated locally. Then, the Jacobian matrix becomes

[J ] =

⎛⎜⎜⎜⎜⎜⎜⎜⎜⎜⎜⎝

∂Rεe

∂�εe∂Rεe

∂�υs∂Rεe

∂�δsc

∂Rεe

∂�δsi

∂Rεe

∂�δcum

∂Rυs

∂�εe∂Rυs

∂�υe∂Rυs

∂�δsc

∂Rυs

∂�δsi

∂Rυs

∂�δcum

∂δsc

∂�εe∂δs

c

∂�υe∂δs

c

∂δsc

∂δsc

∂δcum

∂δsc

∂δcum

∂δsi

∂�εe

∂δsi

∂�υe

∂δsi

∂δsc

∂δsi

∂δsi

∂δsi

∂δcum

∂δcum

∂�εe∂δcum

∂�υe∂δcum

∂�δsi

∂δcum

∂�δsc

∂δcum

∂�δcum

⎞⎟⎟⎟⎟⎟⎟⎟⎟⎟⎟⎠

(83)

After convergence, the θ -method allows the calculation of the tangent matrix of thebehaviour. R can be decomposed into two parts as

{R} = {Ri} − {Re} (84)

where Re corresponds to the applied load. After the convergence (i.e. {R} ≈ {0}) aninfinitesimal variation can be applied to the residual equation such as

δ{R} = {0} = δ{Ri} − δ{Re} (85)

which can be rewritten in the form:

δ�νint = [J ]−1 δ{Re} (86)

For the calculation of elastic strain increment, the above relation reads

δ�ε∼e = J∼∼

eδ�ε∼, δ�σ∼ = C∼∼: J∼∼

eδ�ε∼ (87)

Note that a consistent tangent matrix can directly be obtained from [C∼∼: J∼∼

e]. A more

comprehensir presentation of this systematic numerical integration method can befound in [3].


Fig. 3. FE mesh of a CT-like specimen created by ZeBuLoN GUI.

Fig. 4. Crack growth in a 2D single crystal CT-like specimen with a single cleavage planealigned through the horizontal axis under vertical tension. Field variable δ. (Left) A =100 MPa·mm2, H = −20000 MPa, χH = 30000 MPa; (Right) A = 150 MPa·mm2,H = −10000 MPa,χH = 30000 MPa.

5 Numerical Examples

As a 2D example, a single crystal CT-like specimen under tension is analysed. Thecorresponding FE mesh is given in Figure 3. Analyses are performed for two differentcrack widths, obtained by furnishing different material parameters which control thecharacteristic length. The propagation of a crack, corresponding stress fields and thecomparison with classical elastic solutions are given in Figure 5. This comparisonshows that the microdamage model is able to reproduce the stress concentration atthe crack tip except very close to the crack tip where finite stress values are predicted.

Another 2D example, namely a plate under uniaxial tension with several cleav-age planes, is investigated (see Figure 6). In order to trigger localization, an initialgeometric defect is created on the left edge. First, a cleavage plane is oriented at 30◦from the horizontal axis. FEA results show that localization path is perfectly match-ing with the cleavage plane and the size of the localization band is controlled by ω

in (65) (Figure 6, left). Second, two orthogonal cleavage planes are placed with anorientation of 45◦ from the horizontal axis representing {111} planes. For the formercase, damage is coupled with plasticity and two localization bands merge togetherand form a straight crack path which can be considered as a type of ductile crack


Fig. 5. Evolution of the crack and the stress fields in a CT-like specimen compared with cor-responding elastic solutions.

Fig. 6. Crack growth in a 2D single crystal block with a single inclined cleavage plane (left)and two orthogonal planes oriented at 45 degrees (middle and right) under vertical tensionwith 10% strain. Field variable δ.

(Figure 6, middle). For the latter case, plasticity is excluded from the calculation andcrack path is allowed to choose its path between the orthogonal planes which resultsin a brittle type of crack propagation (Figure 6, right).

In [2], FEA of a CT-like fracture mechanics specimen under tension was con-sidered. The analysis was done by creating a cleavage plane parallel to the horizontalaxis and the loading was performed from the center of the pin. For a given charac-teristic length (associated with parameters A = 200 MPa·mm2, H = −16000 MPa,χH = 50000 MPa), mesh refinement of the specimen lead to a unique fracture curveand a finite size crack width, as shown in Figure 7 and in the loading curves in [2].


Fig. 7. Crack growth in a CT-like specimen under tension under vertical tension with 8%strain. Field variable δ.

6 Conclusion

The proposed systematic treatment of the thermomechanics of continua with addi-tional degrees of freedom leads to model formulations ranging from micromorphic tophase field models. In particular, a general framework for the introduction of dissip-ative processes associated with the additional degrees of freedom has been proposed.If internal constraints are enforced on the relation between macro and microvariablesin the micromorphic approach, standard second gradient and strain gradient plasticitymodels can be retrieved as shown in [11].

As a variant of micromorphic continuum, microdamage continuum and its regu-larization capabilities for the modelling of crack propagation in single crystals havebeen studied. First, a crystallographic constitutive model which accounts for con-tinuum damage with respect to fracture planes has been presented. Then, the theoryhas been extended from classical continuum to microdamage continuum, respect-ively. It has been shown that the approach can be a good candidate for solving meshdependency and the prediction of final fracture. Analytical fits and numerical resultsshowed that the theory is well suited for FEA and possesses a great potential for thefuture modelling aspects. Comparison with available data on crack growth especiallycyclic loading in nickel-based superalloys, will be decisive to conclude on the abilityof the approach to reach realistic prediction of component failure.

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