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Nonlocal Modeling of Material’s Failure, NMMF 2007 Proceeding of an International Workshop (2007) 201-218 How does the crack know where to propagate? - A X-FEM-based study on crack propagation criteria unther Meschke 1 and P. Dumstorff 2 1 Institute for Structural Mechanics, Ruhr-University Bochum, Universit¨ atsstrasse 150, 44780 Bochum, Germany, [email protected] 2 Siemens Power Generation, 45478 M¨ ulheim an der Ruhr, Germany Abstract The reliability of numerical analyses of cracked structures crucially depends on the correct prediction of the crack path and, consequently, on the criterion used for the determination of the crack propagation direction. This paper investigates, using the Extended Finite Element Method (X-FEM) to simulate crack propa- gation in the framework of the Finite ELement Method, three different crack propagation criteria proposed in the literature and one criterion proposed by the authors with regards to their influence on the direction of crack propagation. Two local criteria include an averaged stress criterion and the maximum circum- ferential stress criterion based on the Linear Elastic Fracture Mechanics. Two global criteria include a global tracking criterion proposed by [1] and an energy based X-FEM formulation recently proposed in [2, 3]. A representative numerical benchmark example, characterized by Mixed-Mode fracture, is used to study the predicted crack trajectory for the different crack propagation criteria. Keywords: Extended Finite Element Method, Variational formulation, Crack propagation, Fracture, Cohesive cracks, Quasi-brittle materials, Crack interface

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Page 1: Meschke Crack Propagation Criteria X FEM

Nonlocal Modeling of Material’s Failure, NMMF 2007Proceeding of an International Workshop (2007) 201-218

How does the crack know where to propagate? -A X-FEM-based study on crack propagation criteria

Gunther Meschke1 and P. Dumstorff2

1 Institute for Structural Mechanics, Ruhr-University Bochum, Universitatsstrasse150, 44780 Bochum, Germany, [email protected] Siemens Power Generation, 45478 Mulheim an der Ruhr, Germany

Abstract

The reliability of numerical analyses of cracked structures crucially depends onthe correct prediction of the crack path and, consequently, on the criterion usedfor the determination of the crack propagation direction. This paper investigates,using the Extended Finite Element Method (X-FEM) to simulate crack propa-gation in the framework of the Finite ELement Method, three different crackpropagation criteria proposed in the literature and one criterion proposed by theauthors with regards to their influence on the direction of crack propagation.Two local criteria include an averaged stress criterion and the maximum circum-ferential stress criterion based on the Linear Elastic Fracture Mechanics. Twoglobal criteria include a global tracking criterion proposed by [1] and an energybased X-FEM formulation recently proposed in [2, 3]. A representative numericalbenchmark example, characterized by Mixed-Mode fracture, is used to study thepredicted crack trajectory for the different crack propagation criteria.

Keywords: Extended Finite Element Method, Variational formulation, Crack propagation,Fracture, Cohesive cracks, Quasi-brittle materials, Crack interface

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G. Meschke und P. Dumstorff

1 Introduction

The process of cracking in quasi-brittle materials such as cement paste, brickworkor concrete is characterized by the formation of microcracks which eventually co-alesce and form a propagating macrocrack. The realistic modeling of the processof crack opening and propagation is a prerequisite for reliable prognoses of thesafety and the durability of concrete, reinforced concrete and masonry structures.Numerical modeling of cracks and structural analysis of cracked structures datesback to the early 1970’s. Up to the mid of the 1990’s, one focus of research incomputational failure analysis was laid on the development of continuum-basedmodels (see the reviews contained e.g. in [4, 5, 6, 7]) in a local (fracture-energybased) or non-local setting. At the same time another focus was laid on thedevelopment of numerical models for a discrete representation of fracture usingLinear Elastic Fracture Mechanics and, partially, cohesive crack models (see, e.g.,[8, 9]).

To overcome the limitations of continuum-based models (e.g. the need to providea very fine resolution of the damage zone if nonlocal models are used) and ofclassical discrete fracture models (e.g. the need for re-meshing as cracks evolve),more recently attempts have been made to represent cracks as embedded discon-tinuities within finite elements. These formulations can generally be categorizedinto element-based formulations, generally denoted as Embedded Crack Models(see [10, 11, 12, 13, 14], among others) and nodal-based formulations, e.g. theExtended Finite Element Method (X-FEM) (see [15, 16, 17]). For a compara-tive assessment of both approaches we refer to [18, 19]. With the exception ofthe “rotating” crack formulation of the Strong Discontinuity Approach [14], thetopology of crack segments is held fixed once they are signaled to open.

The reliability of numerical analyses of cracked concrete structures crucially de-pends on the correct prediction of crack trajectories. In most of the existingmodels for cohesive cracks, the opening of new cracks is based upon a localstress-based criterion, such as the principal stress, evaluated at the crack tip,while the direction of crack propagation is determined according to the princi-pal axes of stresses evaluated either locally within the cracked element or in thevicinity of the crack tip by means of a non-local averaging procedure [17, 20, 21].To investigate the influence of the chosen model and of selected parameters onthe predicted crack trajectory, the main focus of this paper lies on the numericalassessment of crack propagation criteria and their effects on the predicted crackpath by means of comparative numerical analyses based on the Extended FiniteElement Method. Four different crack propagation criteria proposed in the liter-ature are investigated: a principal stress criterion based on the averaged stressesaround the crack tip [17], a maximum circumferential stress criterion based onthe stress intensity factors determined according to Linear Elastic Fracture Me-chanics [22, 16, 23] and a global tracking algorithm [1, 24]. In addition to thesethree criteria a global energy-based crack propagation criterion recently proposedin [25, 3] is investigated. In this variational formulation, the orientation as well asthe length of new crack segments are included as an additional global unknowns

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in the variational formulation. These additional degrees of freedom are solved forsimultaneously with the regular degrees of freedom. A comparative assessmentof the aforementioned local and global approaches including the energy-basedExtended Finite Element model is provided in [2].

2 Variational formulation of cracked bodies

Assuming quasi-static and isothermal conditions, a body whose domain Ω con-tains an existing curved cohesive crack and a new crack segment, creating anew crack surface Ac, which propagates with a kinking angle θc relative to theexisting segment is considered. The total area of the current configuration of thecrack is denoted as Γc.

The total energy of this body at a certain loading stage t consists of the internalenergy U , the external work Wb and Wt of the body forces b and the surfacetractions t?, respectively, and the surface energy of the crack Wc:

Ψ(u, Ac, θc) = U −Wb −Wt +Wc, (1)

with

U =

Ω\Γc

∫ εt

0σ(ε)dε dV,

Wb =

Ω

∫ ut

0bdu dV, Wt =

Γσ

∫ ut

0t? du dA. (2)

The total energy Ψ depends on the displacement field u, the crack angle θc andthe surface Ac of the created crack segment.

The displacement field of a body containing a crack can be decomposed into acontinuous part u and a discontinuous part u

u(x) = u(x) + u(x) ∀ x ∈ Ω with u(x) = Sc(x) u(x), (3)

where u and u are continuous functions in the domain Ω and Sc is the Signfunction. Inserting the displacement field (3) into (2)1 and assuming linear elasticbehavior of the uncracked material yields:

U = 12

Ω\Γc

ε : C : ε dV

= 12

Ω\Γc

(∇su : C : ∇su+∇su : C : ∇su+ 2Sc∇su : C : ∇su) dV.(4)

For cohesive cracks, the surface energy Wc is given as

Wc =

Γc

∫ [[u]]t

0t([[u]]) d[[u]] dA, (5)

3

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G. Meschke und P. Dumstorff

with the separation-dependent residual traction vector t([[u]]) acting along theprocess zone of the crack.

Among all possible deformed configurations of the body containing one (or more)crack(s) extended by one (or more) new crack segment(s), the actual one, as-sociated with safe equilibrium of the cracked body will lead to a minimum ofΨ(u, θc, Ac) [26]. This is equivalent to the stationarity condition:

δΨ(u, θc, Ac) =∂Ψ

∂uδu+

∂Ψ

∂θcδθc +

∂Ψ

∂AcδAc = 0. (6)

Inserting Equation (1) and (2) into (6) leads to the conditions:

Ω\Γc

σ : δε dV −∫

Γσ

t?δu dA+

Γc

t∂ [[u]]

∂uδu dA = 0,

Ω\Γc

σ :∂ε

∂θcδθc dV +

Γc

t∂ [[u]]

∂θcδθc dA = 0,

Ω\Γc

σ :∂ε

∂AcδAc dV +

Γc

t∂ [[u]]

∂AcδAc dA = 0. (7)

In the implementation of the model (see also Section 3.4.3), the derivatives withrespect to the angle θc and the surface Ac of new crack segments are computednumerically [3].

3 Crack Growth Models

In this section four different criteria for the determination of the crack growthdirection are investigated. Two criteria are based on local information from thevicinity of the individual crack tip and two criteria have a global character: aprincipal stress criterion based on the averaged stresses around the crack tip [17],a maximum circumferential stress criterion based on the stress intensity factorsdetermined according to Linear Elastic Fracture Mechanics [22, 16, 23], a globaltracking algorithm [1, 24] and a global energy-based crack propagation criterion[2, 3].

The main purpose of this paper is the investigation of the four criteria for thedetermination of the crack growth direction. Therefore, the same crack propaga-tion criterion is used for all crack growth models investigated in Section 4. To thisend, an energy based crack initiation criterion is used for all analyses. According

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to this criterion the crack propagation condition is formulated as follows

∂Ψ

∂Acr

> 0 ; no crack propagation

= 0 ; stationarity condition

< 0 ; crack propagation

, (8)

where Acr is the area of a new crack segment. More information on this crackpropagation criterion is given in [27, 9].

3.1 Averaged Stress Criterion

According to the averaged stress criterion proposed in [17], the principal axis ofthe averaged stress tensor σm, corresponding to the maximum principal stress, istaken as the normal vector ns of the crack extension. The motivation for using anonlocal stress quantity instead of the local stresses is to improve the reliabilityof the computed stresses in the vicinity of the crack. To this end, an averagedstress tensor σm is computed

σm =

∫wσ dV (9)

using a Gaussian weight function w defined as:

w =1

(2π)3/2 l3exp

(− r2

2l2

). (10)

In (10) r is the distance from the crack tip and l is the parameter which deter-mines how fast the weight function decays from the crack tip.

3.2 Linear Elastic Fracture Mechanics

Although cohesive crack models are generally not compatible with the underlyingassumptions of Linear Elastic Fracture Mechanics (LEFM), a maximum circum-ferential stress criterion based upon the evaluation of the stress intensity factorsKI and KII has been used in combination with cohesive crack models by variousauthors [22, 16, 23]. The use of this criterion is motivated by the assumptionthat the size of the investigated structure has a minor influence on the crack path.This assumption is corroborated by experimental investigations in [28] and [29].Therefore, this approach has been included in the present comparative evalua-tion in order to assess its suitability in comparison with other criteria specificallydesigned for cohesive cracks.

In this paper the crack is assumed to grow in the direction of the maximum cir-cumferential stress. This direction is determined by means of the stress intensity

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G. Meschke und P. Dumstorff

factors KI and KII as

θcr = 2arctan

1

4

KI

KII±√(

KI

KII

)2

+ 8

. (11)

According to this formula the maximum kinking angle is limited to 70.53 forpure Mode-II fracture.

3.3 Global Tracking Algorithm

The so-called Global Tracking Algorithm has been proposed by [1, 24] in the con-text of the Strong Discontinuity Approach and has been modified recently by [30].Nevertheless the implementation into the Extended Finite Element Method isstraightforward. In this paper only the basic idea of the method is presented.For a more detailed description we refer to [1, 24]. In contrast to the previouslydescribed criteria the Global Tracking Algorithm does not need to be evaluatedfor each individual crack but traces all possible discontinuity paths at once.

The basic idea of this method is to construct a function ϑ whose iso-lines runperpendicular to the directions of the principal stresses (i.e. the directions ofpotential crack normal directions) in all integration points of the investigatedstructure. To this end a stationary anisotropic heat conduction-like problem issolved with the anisotropic conductivity being defined by the principal axes ofthe maximum stresses. The isothermal lines of the calculated temperature fieldrepresent all possible crack paths [1, 24].

3.4 Variational Extended Finite Element Method

This Subsection contains a concise description of the variational Extended FiniteElement formulation as proposed in [3] for 2D analyses of cohesive cracks. Thecrack propagtion is exclusively governed by energy minimization without usingany additional hypotheses. In this energy-based X-FEM model, the orientationas well as the length of new crack segments are included as an additional globalunknowns in the variational formulation and computed from finding a stationarypoint of the total energy functional according to (6). These additional degreesof freedom are solved for simultaneously with the regular degrees of freedom.

3.4.1 Enhanced displacement approximation

If a crack fully crosses an element, the Sign function according to Equation (3)is used to represent the discontinuous displacement field across the crack. Us-ing standard bi-linear finite element shape functions as a partition of unity, theapproximation of the displacement field within this element is - according to

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Equation (3) - given by a continuous part u and a discontinuous part u

u = u+ u = u+ Scu with u ≈nr∑

i=1

Ni uri and u ≈

4∑

i=1

Ni usi , (12)

where nr is the number of nodes used for the spatial discretization of the regulardisplacement field (superscript r). In the present implementation, hierarchicalhigher order shape functions, allowing to chose polynomials in the range betweenp = 1 to p = 4, are used for the approximation of the regular displacements. Forthe displacements u enriched by the Sign-function (superscript s) four low order(bi-linear) shape functions are used.

In the vicinity of a crack tip, crack tip enhancement functions are used. As inthe previous case, the approximation within elements located in the vicinity ofthe crack tip consists of a continuous part u and a discontinuous part u:

u = u+ Scu, with

u ≈nr∑

i=1

uriNi +

4∑

i=1

3∑

k=1

utkiNiFk and u ≈

4∑

i=1

ut4iNiF4 (13)

where ScF4 = F4 for notation consistency and again low order shape functionsare used for the enriched part (superscript t). Details of the crack tip functionsemployed for cohesive cracks F1-F4 are contained in [3].

3.4.2 Traction-separation law for Mixed-Mode conditions

The traction-separation law suitable for cohesive cracks in quasi-brittle materialssubjected to general Mixed-Mode conditions follows the formulation introducedby [31] and later employed in [20]. It accounts for dissipative interface mecha-nisms in Mode-I and Mode-II conditions using an equivalent traction separationlaw.

The relation between the traction vector t and the displacement jump [[u]] rep-resenting its energetic conjugate variable is given in the general format

t = T [[u]] =(Te − Td

)[[u]] , (14)

where Te = T eI is an initial isotropic (elastic) stiffness corresponding to the

initial uncracked situation and Td denotes a damage tensor governing the degra-dation of the stiffness T.

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G. Meschke und P. Dumstorff

In the present model, the relation (14) is formulated in terms of an equivalenttraction teq and an equivalent crack opening [[u]]eq, respectively

teq = (T e − T d) [[u]]eq , (15)

with [[u]]eq defined as:

[[u]]eq =

√[[u]]2n + β2 [[u]]2s. (16)

The subscripts n and s refer to normal and shear components, respectively. Theparameter β governs the ratio between the strength and stiffness in mode I andmode II opening. For both modes of fracture the same fracture energy Gc isassumed in the model (see [31, 3] for more details).

3.4.3 Finite Element Formulation

According to the proposed energy-based crack propagation model [3] the angleθc and the length rc of new crack segments are introduced as additional degreesof freedom in the discretized structural model. Accordingly, the stationarity con-dition (6) and its linearization is recast in vector- and matrix-form, respectively,in the format

δΨ =

δu

δrcδθc

·

ru

rr

r?u

0

0

= 0,

∆δΨ =

δu

δrcδθc

·

kuu kur kuθ

kru krr krθ

kθu kθr kθθ

∆u

∆rc∆θc

, (17)

where r?u is the vector of external forces. The vectors of internal forces areidentified from (6) as

ru = ∂U/∂u+ ∂Wc/∂u,

rθ = ∂U/∂θc + ∂Wc/∂θc,

rr = ∂U/∂rc + ∂Wc/∂rc. (18)

The computation of the components of the vectors rr, rθ, and of the submatriceskur, kuθ, krr, krθ and kθθ requires the determination of the first and the secondderivatives of the total energy (1) with respect to the crack angle θc and thesegment length rc. Due to the complexity involved in the analytical determinationof these derivatives, they are computed by means of numerical differentiation.

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A X-FEM-based study on crack propagation criteria

Details of the numerical determination of these components are contained in[32, 3].

The incremental-iterative solution of boundary value problems using the proposedenergy-based Extended finite element method is characterized by applying thetotal load in a sequence of incremental steps [tn, tn+1] and enforcing the thestationarity condition (171)

δΨn+1(u, rc, θc) = 0 (19)

by solving (19) simultaneously for the three unknowns u, rc and θc. The formatof (19) and its algorithmic formulation and linearization is identical to multifieldproblems. The starting value of the crack angle θc,n+1 is always taken accordingto the previous converged state θc,n.

4 Comparative numerical analyses

In this section the performance of the four crack growth criteria described aboveis investigated by means of re-analyses of a Mixed-Mode fracture benchmarktest [28]. For all analyses, the interface law as described in Section 3.4.2 hasbeen employed. A square shaped double edge notched specimen made of mortarwhich has been tested by Nooru-Mohamed [28] is used as a benchmark for theassessment of the different crack propagation criteria in a Mixed-Mode dominatedsituation. In [28] mortar and concrete specimens of different sizes were testedunder different loading conditions.

The geometry, the loading and boundary conditions of the specimen as well asthe material parameters used for the interface law described in Section 3.4.2 arecontained in Figure 1. The uniaxial tensile strength ftu and the fracture energyGf were estimated using the experimentally determined values of the tensilesplitting strength and the compressive strength. The load sequence is definedas follows: first the specimen is subjected to a shear force which is increasedup to Fs = 10 kN. Subsequently, a displacement-controlled tensile axial loadFn is applied. A relatively coarse finite element mesh consisting of 435 elementsassuming plane stress conditions as shown in Figure 2a was used.

Figure 2 shows a comparison of the experimentally obtained crack path and thenumerical results obtained from the four investigated crack growth criteria. Therange of crack paths observed in the experiments is indicated in light grey colorin Figures 2c-2f. It should be noted, that, as was shown in [32, 3], the chosendegree of polynomial approximation of the regular displacement field has only aminor influence on the crack trajectories.

The computed crack paths using the averaged stress criterion (Figure 2b), theGlobal Tracking Algorithm (Figure 2d) and the maximum global strain energyrelease rate (Figure 2e) are located well within the experimentally determinedrange. Only the crack path based on Linear Elastic Fracture Mechanics (Fig-ure 2c) is not located within the range of experimental observations. The cur-

9

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G. Meschke und P. Dumstorff

200 mm

25

5

200mm

Fn, un

Fn, un

Fs

Fs

E: 30000.00 [N/mm2]

ν: 0.20 [−]

Gf : 0.11 [N/mm]

ftu: 3.00 [N/mm2]

β: 2.00 [−]

α0: 0.001 [mm]

t: 50.00 [mm]

Figure 1: Mixed-Mode fracture test: Geometry and material parameters

vature of the crack observed in the experiments is perfectly reproduced by allnumerical simulations independent of the crack growth criterion.

The load displacement curves computed using the four different crack growthdirection criteria are shown in Figure 2b. With the exception of the LEFM crite-rion, all load displacement curves are more or less identical. The relatively largedeviation of the numerical results obtained from the LEFM-based criterion isassociated with the misprediction of the crack trajectory using this criterion. Itshould be noted, that all calculated load displacement curves deviate from theexperimental observations. This is a consequence of the obvious overestimationof the fracture energy and of the tensile strength as also reported in [33].

4.1 Influence of interface law

In this subsection, the influence of the shear transfer model, i.e. the influence ofthe value of the coupling parameter β on the distribution of the total energy usingthe X-FEM model is investigated. Ψ with a variation of the crack propagationangle θc is investigated.

To this end, only one single loading step is considered. Starting from an initialcrack segment, a new crack segment with constant length rc associated with

10

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A X-FEM-based study on crack propagation criteria

a) b)X-FEM c)

X-FEM d)

X-FEM e)

X-FEM f)

Experiment

Displacement u [mm]0 0.02 0.04 0.06 0.08 0.1

Force

Fn[kN]

0

5

10

15

20

c) d)

e) f)

Figure 2: Numerical analysis of Mixed-Mode benchmark:fracture test: a) Discretization,b) u-P curves, crack based on c) averaged stress criterion, d) LEFM, e) VariationalX-FEM, f) Global Tracking Algorithm

11

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G. Meschke und P. Dumstorff

θc

Angle θc [deg]

Surfaceenergy

Wc

β = 2.0β = 1.5β = 1.0

3020100-10-20

Angle θc [deg]

TotalenergyΨ β = 2.0

β = 1.5β = 1.0

3020100-10-20

Figure 3: Numerical analysis of the Mixed-Mode fracture test: Geometry, distributionof surface energy of interface traction forces Wc and of total energy Ψ [Nm] versus thecrack propagation angle θc within a range −22, 9o < θc < 31, 5o

an increment of the prescribed load is assumed to propagate from the existingcrack tip at different angles θc within a range of θc=−0.4 rad (=-22,9) toθc=0.55 rad (=31,5) and increments of 0, 05 rad (=2,86) (see Figure 3).

For each position of the new segment an equilibrium state was computed andthe respective values of the total energy Ψ and of the surface energy Wc wererecorded. Figure 3 contains the respective diagrams of the energies Ψ and Wcversus the crack kinking angle θc. As is noted from Figure 3b, the energy func-

12

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A X-FEM-based study on crack propagation criteria

Angle θc [deg]

SurfaceenergyW

c

β = 2.0β = 1.5β = 1.0

6040200-20

Angle θc [deg]

Total

energyΨ

β = 2.0β = 1.5β = 1.0

6040200-20

Figure 4: Numerical analysis of the Mixed-Mode fracture test: Distribution of surfaceenergy of interface traction forces Wc and of total energy Ψ [Nm] versus the crackpropagation angle θc within a range -23o < θc < 70o

tional is not convex for all investigated situations. For β = 2.0 and β = 1, 5, thefunctional Ψ is locally convex with a minimum at ∼ 1.9o and ∼ 5.8o, respec-tively. This range of the crack kinking angles corresponds well with experimentalobservations. The maximum curvature of Ψ within the investigated range de-creases with decreasing shear transfer capacity. For β = 1, i.e. assuming a verysmall shear transfer capacity along the interface, no stationarity point is existingwithin the investigated range of crack kinking angles. Consequently, using theproposed energy-based crack model, which is based on the simultaneous solu-tion of δΨ(u, θc, rc) no converged solution could be found for β = 1. Also, forβ = 1.5 convergence problems have been observed and the analysis could not becompleted.

In the interface model described in Section 3.4.2 the same fracture energy isassumed for Mode I and Mode II failure. Obviously, if the energy stored anddissipated by the transfer of shear stresses within the crack is reduced, thereexists a lower energy level for the crack kinking angle θc > 31, 5 outside of theinvestigated range, in which the crack would be aligned along a direction corre-sponding to an almost perfect Mode II situation as is demonstrated by Figure 4which shows the energy distribution for a larger range of θc (-23

o < θc < 70o)computed for the three values of β. Indeed, as expected, a minimum energylevel exists at θc ∼ 60o for all three analyses. This second energy minimumcorresponds to a Mode II-dominated crack propagation. It is assumed, that thissecond energy minimum would not appear if a more realistic interface model,characterized by different values of the fracture energy in Mode I and Mode IIcrack opening, is used.

13

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G. Meschke und P. Dumstorff

X-FEMExperiment

Displacement u [mm]0 0.1

Force

Fn[kN]

0

20

4a

X-FEMExperiment

Displacement u [mm]0 0.1

Force

Fn[kN]

0

20

4b

X-FEM

Experiment

X-FEM

Experiment

0 0.1

Force

Fn[kN]

0

20

4c

LP Crack path Load-displacement curve

Figure 5: Numerical analysis of the Mixed-Mode fracture test: using 3 different loadingpaths (a): Fs = 5 kN, b): Fs = 10 kN, c): Fs = 27.5 kN): Left hand side: Computedcrack paths; Right hand side: Load displacement diagrams

4.1.1 Crack paths for different loading paths

This Subsection contains numerical results from re-analyses of the tension-shearbenchmark for all three different loading scenarios (loading paths 4a, 4b and 4c)

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A X-FEM-based study on crack propagation criteria

documented in [28], characterized by different levels of shear force applied prior topulling apart the fixing frames. In addition to the loading case already discussedin Subsection 4.1, characterized by a shear force Fs = 10 kN, two additionaltests characterized by Fs = 5 kN and Fs = 27, 5 kN are investigated numericallyby means of the energy-based X-FEM model. For all three analyses described inthis section biquadratic shape functions were used for the approximation of theregular part of the displacement field.

Figure 5 contains the crack trajectories and the corresponding load-displacementcurves from all three tests obtained from the analyses. The experimentally ob-served crack paths are included in grey color. In contrast to re-analyses of thisbenchmark reported in the literature [34, 35] identical parameters have beenused for all three tests without any fitting to the individual tests. The overallagreement of the crack trajectories is reasonable for all tests.

5 Concluding remarks

In this paper the influence of the chosen crack propagation criterion on thepredicted crack trajectory has been investigated by means of comparative anal-yses using the Extended Finite Element Method (X-FEM). Two local and twoglobal crack growth criteria have been emplyed in this numerical study: a prin-cipal stress criterion based on the averaged stresses around the crack tip [17], amaximum circumferential stress criterion based on Linear Elastic Fracture Me-chanics [22, 16, 23], the Global Tracking Algorithm proposed by [36] and arecently proposed variational formulation of the X-FEM based on the minimiza-tion of the total energy [25, 3]. An interface law considering dissipative couplingwithin cracks in Mixed-Mode problems has been formulated in the framework ofdamage mechanics and employed in all analyses.

It has been demonstrated, that the criterion based on the minimum of the totalenergy does not need any material-specific assumptions regarding the determi-nation of the crack direction. It is, in contrast to the LEFM-based criterion, fullyconsistent with the assumption of cohesive cracks. It was shown, that the sheartransfer capacity represented by the interface model has a considerable influ-ence on the distribution of the energy for varying crack kinking angles θc and,consequently, on the location of the minimum of this (locally convex) distribu-tion. Physically unrealistic assumptions for the interface law are reflected by thefact that local convexity of the energy distribution Ψ(θc) may be lost within theexpected range of (physically plausible) crack kinking angles. A detailed investi-gation of the influence of the material parameter β on the calculated crack pathis provided in [3].

References

[1] J. Oliver, A.E. Huespe, E. Samaniego, and E.W.V. Chaves. On strate-gies for tracking strong discontinuities in computational failure mechanics.

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G. Meschke und P. Dumstorff

In H.A. Mang, F.G. Rammerstorfer, and J. Eberhardsteiner, editors, FifthWorld Congress on Computational Mechanics (WCCM V). Online publica-tion, 2002.

[2] P. Dumstorff and G. Meschke. Crack propagation criteria in the frameworkof X-FEM-based structural analyses. International Journal for Numericaland Analytical Methods in Geomechanics, 31:239–259, 2007.

[3] G. Meschke and P. Dumstorff. Energy-based modeling of cohesive andcohesionless cracks via X-FEM. Computer Methods in Applied Mechanicsand Engineering, 196:2338–2357, 2007.

[4] G. Hofstetter and H.A. Mang. Computational Mechanics of Reinforced andPrestressed Concrete Structures. Vieweg, Braunschweig, 1995.

[5] R. de Borst, N. Bicanic, H.A. Mang, and G. Meschke, editors. Computa-tional Modelling of Concrete Structures, Proceedings of the EURO-C 1998Conference. Balkema, Rotterdam, Badgastein, 1998.

[6] R. de Borst. Some recent issues in computational failure mechanics. Inter-national Journal for Numerical Methods in Engineering, 52:63–95, 2002.

[7] H.A. Mang, G. Meschke, R. Lackner G., and J. Mosler. Computationalmodelling of concrete structures. In Comprehensive Structural Integrity,volume 3, chapter 9, pages 1–67. Elsevier, 2003.

[8] A.R. Ingraffea and V. Saouma. Numerical modelling of discrete crack prop-agation in reinforced and plain concrete. In G. Sih and A. DiTommaso,editors, Fracture Mechanics of Concrete: structural application and numer-ical calculation, pages 171–225. Martinus Nijhoff, 1985.

[9] M. Xie and W.H. Gerstle. Energy-based cohesive crack propagation model-ing. Journal of Engineering Mechanics (ASCE), 121(12):1349–1358, 1995.

[10] J.C. Simo, J. Oliver, and F. Armero. An analysis of strong discontinuitiesinduced by strain-softening in rate-independent inelastic solids. Computa-tional Mechanics, 12:277–296, 1993.

[11] J. Oliver. Modelling strong discontinuities in solid mechanics via strainsoftening constitutive equations. Part 1: Fundamentals, Part 2: Numericalsimulation. International Journal for Numerical Methods in Engineering,39:3575–3623, 1996.

[12] M. Jirasek and T. Zimmermann. Embedded crack model: Part 1: Basicformulation. part 2: Combination with smeared cracks. International Journalfor Numerical Methods in Engineering, 50:1269–1305, 2001.

16

Page 17: Meschke Crack Propagation Criteria X FEM

A X-FEM-based study on crack propagation criteria

[13] F. Armero and K. Garikipati. An analysis of strong discontinuities in mul-tiplicative finite strain plasticity and their relation with the numerical sim-ulation of strain localization in solids. International Journal of Solids andStructures, 33(20):2863–2885, 1996.

[14] J. Mosler and G. Meschke. 3D modeling of strong discontinuities in elasto-plastic solids: Fixed and rotating localization formulations. InternationalJournal for Numerical Methods in Engineering, 57:1553–1576, 2003.

[15] N. Moes, J.E. Dolbow, and T. Belytschko. A finite element method for crackgrowth without remeshing. International Journal for Numerical Methods inEngineering, 46:131–150, 1999.

[16] N. Moes and T. Belytschko. Extended finite element method for cohesivecrack growth. Engineering Fracture Mechanics, 69:813–833, 2002.

[17] G.N. Wells and L.J. Sluys. A new method for modelling cohesive cracks usingfinite elements. International Journal for Numerical Methods in Engineering,50:2667–2682, 2001.

[18] M. Jirasek and T. Belytschko. Computational resolution of strong disconti-nuities. In H.A. Mang, F.G. Rammerstorfer, and J. Eberhardsteiner, editors,Fifth World Congress on Computational Mechanics (WCCM V). Online pub-lication, 2002.

[19] P. Dumstorff, J. Mosler, and G. Meschke. Advanced discretization meth-ods for cracked structures: The Strong Discontinuity approach vs. the Ex-tended Finite Element Method. In Computational Plasticity 2003. CIMNE,Barcelona, CD-ROM, 2003.

[20] S. Mariani and U. Perego. Extended finite element method for quasi-brittle fracture. International Journal for Numerical Methods in Engineering,58:103–126, 2003.

[21] J. Mergheim, E. Kuhl, and P. Steinmann. A finite element method forthe computational modelling of cohesive cracks. International Journal forNumerical Methods in Engineering, 63:276–289, 2005.

[22] D.A. Cendon, J.C. Galvez, M. Elices, and J. Planas. Modelling the fractureof concrete under mixed loading. International Journal of Fracture, 103:293–310, 2000.

[23] G. Zi and T. Belytschko. New crack-tip elements for X-FEM and applica-tions to cohesive cracks. International Journal for Numerical Methods inEngineering, 57:2221–2240, 2003.

[24] J. Oliver, A.E. Huespe, E. Samaniego, and E.W.V. Chaves. Continuumapproach to the numerical simulation of material failure in concrete. Inter-national Journal for Numerical and Analytical Methods in Geomechanics,28:609–632, 2004.

17

Page 18: Meschke Crack Propagation Criteria X FEM

G. Meschke und P. Dumstorff

[25] P. Dumstorff and G. Meschke. Modelling of cohesive and non-cohesivecracks via X-FEM based on global energy criteria. In D.R:J. Owen, E. Onate,and B. Suarez, editors, Computational Plasticity 2005, (Complas VIII),Barcelona, 2005. CIMNE. 565-568.

[26] K. C. Le. Variational principles of non-linear theory of brittle fracture me-chanics. Journal of Applied Mathematics and Mechanics, 54:543–549, 1990.

[27] Z.P. Bazant and J. Planas. Fracture and size effect in concrete and otherquasibrittle materials. CRC Press, Boca Raton, Florida, 1998.

[28] M.B. Nooru-Mohamed. Mixed-mode Fracture of Concrete: an ExperimentalApproach. PhD thesis, Technische Universiteit Delft, 1992.

[29] E. Schlangen. Experimental and numerical analysis of fracture processes inconcrete. PhD thesis, Technical University Delft, The Netherlands, 1993.

[30] C. Feist and G. Hofstetter. Mesh-insensitive strong discontinuity approachfor fracture simulations of concrete. In Numerical Methods in ContinuumMechanics (NMCM 2003), 2003.

[31] G.T. Camacho and M. Ortiz. Computational modelling of impact damagein brittle materials. International Journal of Solids and Structures, 33:2899–2938, 1996.

[32] P. Dumstorff. Modellierung und numerische Simulation von Rissfortschrittin sproden und quasi-sproden Materialien auf Basis der Extended FiniteElement Method. PhD thesis, Institute for Structural Mechanics, RuhrUniversity Bochum, 2005. in german.

[33] C. Feist. A Numerical Model for Cracking of Plain Concrete Based on theStrong Discontinuity Approach. PhD thesis, University Innsbruck, 2004.

[34] P. Patzak and M. Jirasek. Adaptive simulation of quasibrittle failure. InN. Bicanic, R. de Borst, H. Mang, and G. Meschke, editors, ComputationalModelling of Concrete Structures, EURO-C 2003, pages 109–118. Swets &Zeitlinger, Lisse, 2003.

[35] M. Cervera and M. Chiumenti. Smeared crack approach: back to the orginaltrack. International Journal for Numerical and Analytical Methods in Ge-omechanics, pages 541–564, 2006. in press.

[36] J. Oliver and A.E. Huespe. On strategies for tracking strong discontinuitiesin computational failure mechanics. In Online Proceedings of the FifthWorld Congress on Computational Mechanics (WCCM V), 2002.

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