Computational Materials Science 82 (2014) 427–434

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Computational Materials Science

journal homepage: www.elsevier .com/locate /commatsci

Using the finite cell method to predict crack initiation in ductilematerials

0927-0256/$ - see front matter � 2013 Elsevier B.V. All rights reserved.http://dx.doi.org/10.1016/j.commatsci.2013.10.012

⇑ Corresponding author. Tel.: +98 311 3915216.E-mail address: mashayekhi@cc.iut.ac.ir (M. Mashayekhi).

M. Ranjbar a, M. Mashayekhi a,⇑, J. Parvizian b, A. Düster c, E. Rank d

a Department of Mechanical Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iranb Department of Industrial Engineering, Isfahan University of Technology, Isfahan 84156-83111, Iranc Technische Universität Hamburg-Harburg, Germanyd Technische Universität München, Germany

a r t i c l e i n f o a b s t r a c t

Article history:Received 23 June 2013Received in revised form 20 August 2013Accepted 7 October 2013

Keywords:Ductile damageFCMCrack initiationHigh-order FEM

In this paper, the Finite Cell Method (FCM) is used to predict the crack evolution in ductile materialsunder small strains and nonlinear isotropic hardening conditions. The FCM is the result of combiningthe p-version finite element and fictitious domain methods, and has been shown to be effective in solvingproblems with complicated geometries for which the meshing procedure can be quite expensive. Thecrack evolution is introduced to the constitutive equations by using the simplified Lemaitre ductile dam-age model. The performance of the method is verified by means of two numerical examples in both 2Dand 3D problems.

� 2013 Elsevier B.V. All rights reserved.

1. Introduction

To use an efficient numerical method is one of the most impor-tant steps in simulating engineering problems. Although the finiteelement method has been the most prevalent method used for sev-eral decades, some modified versions have proved their efficiencyin special applications. For example, the generalized finite elementmethod (GFEM) and extended finite element method (XFEM) canbetter resolve discontinuities and singularities [1]. The modifica-tions may target the approximation space for special problems.For example, XFEM adds special functions to the polynomial An-satz space to simulate the singularity near the crack tip [2].

As a combination of fictitious domain methods and high-orderfinite elements, the FCM is a promising choice for solving problemswith complicated geometries. It replaces the mesh generation dif-ficulties with efforts to integrate accurately and adaptively thestiffness matrix and load vector. Interesting to know is that meshgeneration often takes longer than solving the equations in a gen-eral finite element set; for instance mesh generation takes about14% of the whole engineering process while solving equationstakes only 4% [3]. So, it is worth trying to reduce the mesh gener-ation necessity. The FCM has been first introduced by Parvizianet al. [4] for two-dimensional problems and developed for three-dimensional problems by Düster et al. [5]. It was later successfullyapplied to shell elements and thin-walled structures [6]. The foci of

the recent researches have been on both improving the numericalefficiency of the method, e.g. Abedian et al. [7], and on developingits applications in different problems of mechanical engineering,for instance geometrically nonlinear problems [8]. Düster et al.used this method for heterogeneous and cellular materials [9].The method was developed for elastoplastic material behavior byAbedian et al. [10], and was used in topology optimization prob-lems by Parvizian et al. [11]. The finite cell method can have greatadvantages in the field of biomechanics where geometries are verycomplex, for example in analyzing bones [12–14]. The hp-d adap-tive finite cell method has been developed recently [15,16].

In this paper, the application of FCM in ductile crack initiation isinvestigated. The continuum damage mechanics (CDM) theory isused for this purpose. This theory is based on continuum mechan-ics, and aims to study the failure behavior of materials. Some inter-nal state variables are introduced to the constitutive equations inorder to model the crack bands of the material.

The damage analysis, which was already provided, converted toa continuum damage mechanics framework since about four dec-ades ago [17]. It was first used by Hult [18], then by Gurson [19],Tvergaard and Needleman [20]. Lemaitre [21] developed the the-ory and introduced the continuum damage mechanics. The modelshave been presented up to now in the field of continuum damagemechanics can be categorized into four main classes: creep dam-age, ductile damage, quasi-brittle damage and fatigue damage.Creep failure of metals under uniaxial loads using a scalar internalvariable was proposed by Kachanov [22]. In this model, damagedoes not have any direct physical meaning. Within the theory of

428 M. Ranjbar et al. / Computational Materials Science 82 (2014) 427–434

elastoplasticity, Gurson [19] proposed a model for ductile damagewhere a damage variable is obtained from the consideration ofmicroscopic spherical voids embedded in an elastoplastic matrix.A purely phenomenological model for ductile materials using ascalar damage variable was introduced by Lemaitre [21]. Muraka-mi [23] proposed a model for anisotropic damage in brittle mate-rials. Janson [24] developed a continuum theory to model fatiguecrack propagation which was in good agreement with uniaxialexperiments. Lemaitre [25] presented a general formulation con-sidering low and high cycle fatigue as well as creep fatigue in anarbitrary stress state.

The research in the field of CDM continues either in developingdifferent and suitable constitutive equations, or by developing thenumerical methods for implementing the corresponding models.The constitutive equations sought for are either for special materi-als such as composites [26], glass, elastomers, piezoelectric oradhesive materials, or for special cases of loading such as fatigue[24], creep or impact [27]. Adaptive numerical methods for damagesimulation have also been investigated in e.g. [28]. The numericalbehavior of continuum damage models can be improved usingnonlocal models; localization effect is intrinsically included inthe local constitutive models of continuum damage mechanics.Although these models suffer from mesh sensitivity and localiza-tion problems, they are used widely in various mechanical applica-tions to predict a good location of the crack initiation [29]. Anextensive discussion about the reasons and remedies of localiza-tion can be found in the literature [30].

In this paper, we develop the FCM method for simulation ofdamage. The article is organized as follows: in Section 2 we brieflypresent the Lemaitre continuum damage theory and its numericalimplementation procedure. Section 3 is to introduce the ‘‘FiniteCell Method’’ and its numerical implementation. Then, ‘‘NumericalExamples’’ are presented in Section 4 and followed finally by ‘‘Con-clusions’’ in Section 5.

2. Damage theory

Due to the thermodynamically consistent formulation, theLemaitre constitutive model is very appealing and has been fre-quently used in the literature. In this model, the damage internalstate variable, D, is defined as a measure of degradation of the elas-tic modulus, which is a macroscopic mathematical representationof voids and micro-cracks growth. The extreme values of D are 0for the intact material and 1 for a unit volume element with a nullload-carrying capacity and a complete local rupture. In the simpli-fied Lemaitre damage model, kinematic hardening is excludedfrom the equations which can be used for all engineering applica-tions without load reversal [31].

2.1. Thermodynamic framework

The specific free energy potential is given by [17]:

w ¼ wedðee;DÞ þ wpðRÞ; ð1Þ

where wed and wp are the elastic damage and plastic contribution tothe free energy, respectively. In the present theory, the followingforms are postulated:

qwedðee;DÞ ¼ 12 e

e : ð1� DÞD : ee

qwpðRÞ ¼ qw1ðRÞð2Þ

in which D is the elasticity tensor and w1 is an arbitrary function ofthe single argument R. The corresponding conjugate forces for ee, Dand R are given respectively as:

r ¼ q@w@ee ¼ ð1� DÞD : ee ð3Þ

Y ¼ q@w@D¼ �1

2ee : D : ee ð4Þ

j ¼ q@wp

@R¼ q

@w1

@R¼ jðRÞ ð5Þ

The yield function has a von Mises definition:

uðr;j;DÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffi3J2ðsÞ

p1� D

� ryðRÞ ð6Þ

in which s is the deviatoric stress tensor and ry denotes the currentyield stress of the material computed as:

ry ¼ ry0 þ jðRÞ ð7Þ

in which ry0 is the initial yield stress and j(R) is the isotropic hard-ening function. The dissipation potential R is given as:

R ¼ uþ rð1� DÞðsþ 1Þ

�Yr

� �sþ1

ð8Þ

r and s are experimentally determined material parameters. Theevolution laws are given by the normality rule of the plasticitytheory:

_ep ¼ _c@R@r

ð9Þ

_D ¼ _c@R

@ð�YÞ ð10Þ

_R ¼ � _c@R@j

ð11Þ

where _c denotes the plastic multiplier which meets the Kuhn–Tuck-er conditions:

_c P 0; u 6 0; ; _cu ¼ 0: ð12Þ

2.2. Constitutive equations

The right hand side of the Eq. (3) can be expressed in terms ofdeviatoric and hydrostatic stresses. Therefore, the elasticity lawincluding damage reads as [31]:

s ¼ ð1� DÞ2Gee; p ¼ ð1� DÞkve ð13Þ

where G is the shear modulus, k is the bulk modulus, ee and ve are,respectively, the elastic strain deviator and elastic volumetric strain.Based on Eq. (4), the energy release rate Y is computed as thefollowing:

Y ¼ �q2

6Gð1� DÞ2� p2

2Kð1� DÞ2ð14Þ

where

q �ffiffiffiffiffiffiffiffiffiffiffiffiffi3J2ðsÞ

pð15Þ

The evolution law of stress tensor is given by splitting the strainrate in small deformation theory,

_e ¼ _ee þ _ep ð16Þ

The internal variables, ep, D and R are determined usingEqs. (9)–(11):

M. Ranjbar et al. / Computational Materials Science 82 (2014) 427–434 429

_ep ¼ _c@u@r

ð17Þ

_D ¼ _c1

1� D�Y

r

� �s

ð18Þ

_R ¼ _c ð19Þ

2.3. Time discretization and return mapping algorithm

The first step in the finite element implementation of elasto-plastic equations in a pseudo time interval [tn, tn+1] is the evalua-tion of an elastic trial state. Assuming ep

n , Dn and Rn given fromthe last globally converged Newton–Raphson iteration, the elastictrial state is computed as the following [31]:

rtrialnþ1 ¼ ð1� DnÞD : ee trial ð20Þ

Based on Eq. (6) the yield function can be computed as:

utrial ¼ qtrial

1� Dn� ryðRnÞ ¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi3J2ðstrial

pÞ

1� Dn� ryðRnÞ ð21Þ

If utrial6 0 then the trial state is accepted. It means that the process

is elastic and _c ¼ 0, so the internal variables have no evolution andconsistent tangent matrix corresponds to D. Otherwise, we have torun a return mapping algorithm which is extensively described e.g.in [32]. Straightforward specialization of standard return mappingfor the present constitutive equation leads to the following set ofdiscrete evolution equations:

eenþ1 ¼ ee trial � Dc

ffiffi32

qsnþ1

ð1�Dnþ1Þksnþ1k

Rnþ1 ¼ Rn þ Dc

Dnþ1 ¼ Dn þ Dc1�Dnþ1

�Ynþ1r

� �s

qnþ11�Dnþ1

� ryðRnþ1Þ ¼ 0

8>>>>>>>><>>>>>>>>:

ð22Þ

as a set of nine equations and nine unknown variables: six compo-nents of symmetric strain tensor, Rn+1, Dc and Dn+1. Neto [31] hassimplified this set of equations and reduced them to one equation:

FðDcÞ ¼ 3GDc~qtrial � ryðRn þ DcÞ þ ðDn � 1Þ

þ~qtrial � ryðRn þ DcÞ

3G�YðDcÞ

r

� �s

¼ 0 ð23Þ

in which:

~qtrial ¼ qtrial

1� Dn; � YðDcÞ � ½ryðRn þ DcÞ�

6Gþ

~p2nþ1

2K; ~p ¼ p

1� Dð24Þ

This equation is solved by the Newton–Raphson iteration method tofind Dc. Hence, the state variables will be updated and the stresstensor and consistent tangent matrix can be computed. The com-plete procedure of the time integration is addressed in [31].

The output of iteration in one load step is the stress tensor andtangent operator. The stress tensor can be updated simply aftercalculating Dc. When a Newton–Raphson scheme is used in con-junction with a return mapping algorithm, the tangent operatorshould be consistent with the integration procedure. The deriva-tion of the consistent tangent operator follows only standard appli-cation of consistent linearization concepts [32]. The components ofthe stiffness matrix for the prescribed damage formulation are gi-ven in [31].

3. Finite cell method

The departure point of the FCM from general finite elementmethod is to extend the physical domain by a fictitious domain.It changes the real domain in such a way that only rectangular ele-ments in two dimensional or hexahedral elements in three dimen-sional problems are necessary. The way of domain extension isshown in Fig. 1. The physical domain X is embedded in the ex-tended domain Xe. In Fig. 2 a mesh is shown for the prescribed het-erogeneous material with pores and inclusions. Only rectangularelements cover the extended domain and no effort is necessaryto fill the real domain with elements, which may cause some prob-lems such as aspect ratio of the elements or distorted elementswith small or even negative Jacobian. The non-conforming ele-ments are called cells in the finite cell method.

As now some of the cells only partially intersect with the do-main of computation, an adaptive integration procedure is neededto follow the boundary of physical domain. As shown in [7] thequadtree in 2D and the octree in 3D are efficient methods for adap-tive integration in the FCM. In Fig. 3(a), the procedure of tracing theboundary using a quadtree integration method is shown. In thismethod, each parent cell intersected by the boundary is dividedinto four sub cells with the same integration order. Each childagain is divided to four sub-cells and this procedure continues untila certain accuracy is achieved in the geometry representation. Thepartitioning procedure can be stopped until the area of a subcell issmaller than a specific fraction of the main cell which is namedintegration threshold.

In the integration part, all Gauss points at each sub cells areconsidered. Each Gauss point is characterized by a which dependson its location. We define a = 1 if the Gauss point is in the physicaldomain, and a = 0 if not. A mapping to (n, g) which is the local coor-dinate system of the cell is also necessary for each Gauss point. Themapping procedure is shown in Fig. 3(b). In all numerical integra-tions, the integrant is multiplied by parameter a at the correspond-ing Gauss point. The stiffness matrix of a cell with nsc sub-cells isthen calculated by [7]:

Kc ¼Xnsc

sc¼1

Zs

Zr

BTc ðnðrÞÞaðxðnðrÞÞCBcðnðrÞÞdet Jc det~Jsc

c drds ð25Þ

in which C is the consistent tangent matrix and J denotes the deter-minant of the Jacobian matrix due to the change of variables [5].

4. Numerical examples

A subroutine is added to AdhoC [33] to compute stress and tan-gent stiffness matrices of a damaged material at a Gauss point. Thedirect solver ‘‘PARDISO’’ [34] is employed, being suitable for theFCM due to its numerical robustness.

4.1. 2D example

This section illustrates, using a benchmark example, the use ofFCM for crack initiation prediction in a cylindrical pre-notched barsubjected to a monotonic axial loading. The geometry is describedin Fig. 4(a). This example has been reported in the literaturequite frequently [31,35]. The material follows the hardeningformulation:

ry ¼ ry0 þ ðry1 � ry0Þ½1� expð�xRÞ� ð26Þ

In which ry0,ry1 and x are material parameters and listed inTable 1 in conjunction with other elastic and damage parameters.

Only a quarter of the bar is modeled due to axial symmetricproperties along the xz plane. The geometry and mesh of the modelcan be seen in Fig. 4(b). We are using the p-version of finite

+ =

ΩΩ \e

eΩΩ

Fig. 1. Extended domain in the FCM.

Fig. 2. Mesh generation for the extended domain.

430 M. Ranjbar et al. / Computational Materials Science 82 (2014) 427–434

element method to compute a reference solution. The mesh hasbeen refined to: (a) minimize the localization effect, (b) reachacceptable convergence in the values of damage parameter andvon Mises equivalent stress and (c) reduce the computationalcosts. Based on these requirements we have chosen the elementsize and polynomial degree accordingly. A coarse mesh is selectedwith 90 axisymmetric quadrilaterals while the polynomial degreeis 6. The mesh is refined near the root and center of the specimen,due to strain softening in this region. The global element size is ofthe order of 2 mm while in the softening zone it is reduced to atypical size of 1 mm. A displacement of u = 0.576 mm has been ap-plied in 86 steps. The way how the load is applied is very importantin presence of softening since the problem will not remain ellipticas soon as the damage variable begins to evolve. Considering Eqs.(6) and (8), a non-associated flow rule is used in the Lemaitre dam-age model because the dissipation potential is not the same as theyield function. Therefore, an unsymmetrical solver is necessary tosolve the finite element equations.

Fig. 5 represents damage parameter contours when the edgedisplacement reaches about 0.24 mm. The maximum damageparameter is verified by numerical results available in the litera-ture [31] and a good agreement is observed.

Fig. 3. (a) Quadtree adaptive integration method, (b) map

Now, let us move towards the FCM implementation. The simple2D coarse mesh together with the adaptive subcell refinementintroduced for the integration is shown in Fig. 6. To avoid meshdependency problem and have a fair comparison of the resultsthe mean mesh sizes in both p-FEM and FCM are almost equal.The fictitious domain is penalized by factor a = 10�12 to avoid theill-conditioning of the stiffness matrix and polynomial degree isset to 6. The threshold for adaptive integration, defined in Section 3,is 10�5.

In the FCM algorithm, the stress and consistent tangent matri-ces are calculated first, and then, in the numerical integration pro-cess, will be multiplied by a. A Hooke elastic material is used forthe points within the fictitious domain to avoid expensive compu-tational costs of the return mapping algorithm for the Gauss pointswhich are beyond the yield surface. Actually, the material behaviorin this region is not important due to the null contribution of thecorresponding Gauss points in the stiffness matrix composition.

The damage variable contour plot is shown in Fig. 7. It can beseen that the maximum damage parameter appears near the rootduring the early stages of loading. As the specimen is stretchedmore, the maximum damage variable moves toward the centerand gradually localizes in a very small zone, which is in agreementwith Neto [31]. If the damage behavior is ignored and only plasticbehavior is considered, the softening point appears near the root,while experimental observations show that fracture initiation hap-pens at the center [36]. The reason of occurring maximum damagevariable in the center is the stress triaxiality ratio which is thehighest at the center of the specimen. Ductility decreases as the tri-axiality ratio increases. This phenomenon is captured by theLemaitre ductile damage model. To show the accuracy of FCM re-sults more precisely, in Table 2 the damage parameter in the centerof the specimen is compared with the p-FEM results in differentsteps of loading. Softening behavior of the material due to plasticstrain accumulation is captured in Fig. 8 which is in a good agree-ment with p-FEM.

ping of sub-cells to standard integration coordinates.

Fig. 4. (a) Geometry of the bar, (b) p-version finite element mesh.

Table 1Material properties for notched bar specimen.

E 210 GPam 0.3ry0 620 MPary1 3920 MPax 0.4s 1.0r 3.5 MPa

M. Ranjbar et al. / Computational Materials Science 82 (2014) 427–434 431

4.2. 3D example

A flat rectangular notched bar specimen is considered usingboth finite element and finite cell methods, Fig. 9. The thickness

Fig. 5. Damage variable contour plots at u

of the specimen is 8 mm. Due to symmetry conditions only oneeighth of the specimen is modeled using eight-node, hexahedral,high-order elements with polynomial degree 4. The mesh densityin both FEM and FCM models is equal near the notch root to justifya fair comparison of the results. More than 600 hexahedral ele-ments are used in both p-FEM and FCM. A monotonic load is ap-plied to the specimen under displacement control on the topsurface of the specimen equal to 1.2 mm and the load step size isequal to 0.02 mm. The threshold for adaptive integration is10�4.The material properties of the specimen are listed in Table 3.

The equivalent von Mises stress for both FEM and FCM is pre-sented in Fig. 10 which shows a good agreement between FEMand FCM results. As soon as damage parameter begins to evolve,

= 0.24 mm: (a) p-FEM, (b) Neto [31].

Fig. 6. FCM mesh together with quadtree refinement for integration purposes forthe notched bar.

u=0.05mm u=0.074mm

u =0.249 mm u=0.576mm

Fig. 7. Damage variable contour plots in FCM presentation of the notched bar.

Table 2Damage parameter at the center of the notch in FEM and FCM methods.

Top edge displacement(mm)

Damage value in FEM Damage value in FCM

0.124 0.00296 0.002960.499 0.361 0.3600.517 0.392 0.3910.569 0.538 0.5400.576 0.574 0.577

0 0.1 0.2 0.3 0.4 0.5 0.60

100

200

300

400

500

600

700

prescribed edge displacement (mm)

von

Mis

es S

tres

s (M

Pa)

FCMp-FEM, p=6

Fig. 8. Softening behavior of the material in the center of the specimen.

Fig. 9. (a) Flat rectangular notched bar specimen geometry and dimensions, (b)finite element model of 1/8 specimen using symmetry condition, (c) FCMrepresentation of the geometry.

Table 3Material parameters for the flat rectangular notched bar specimen.

E 200 GPam 0.3ry0 430 MPaPower-law work hardening ry ¼ 998e0:18

p MPas 1.0r 2.8 MPa

432 M. Ranjbar et al. / Computational Materials Science 82 (2014) 427–434

a discrepancy is observed between FEM and FCM results. This canbe due to several sources of error such as: (a) FCM geometryapproximation; this can be reduced by increasing the integration

0 0.2 0.4 0.6 0.8 1 1.2 1.40

100

200

300

400

500

600

700

prescribed surface displacement (mm)

von

Mis

es S

tres

s (M

Pa)

FCM

p-FEM, p = 4

Fig. 10. Equivalent von Mises stress in the middle of xz symmetric plane of the flatbar at the apex of the notch.

Table 4Damage parameter in the middle of xz symmetric plane of the flat bar at the apex ofthe notch, in FEM and FCM methods.

Top surface displacement(mm)

Damage value inFEM

Damage value inFCM

0.1 0.0080 0.00820.2 0.033 0.0340.4 0.071 0.0750.8 0.129 0.1361.2 0.207 0.219

M. Ranjbar et al. / Computational Materials Science 82 (2014) 427–434 433

tolerance, (b) mesh-dependency of the employed local damagemodel; this is more serious especially in 3D problems when thedamage parameter evolves, and (c) the density of the integrationpoints is increasing within the damaged region in FCM comparedto the p-FEM. A combination of such errors causes the discrepancyof the two methods in the softening area. However the error is neg-ligible within the engineering disciplines.

Table 4 shows a comparison of damage value in different stepsof loading; the higher the plastic strain, the higher the error in thedamage parameter. This phenomenon is due to the localization ef-fect which will be more serious by approaching the fracture time.

The damage contours for both p-FEM and FCM are shown inFig. 11. The elements cut by the notch boundary are removed tobetter observe the damage parameter distribution in the criticalzone. The damage parameter reaches to its maximum value nearthe notch which follows the experimental results [37]. For similar

Fig. 11. Damage contour plots in the flat rectan

mesh densities as presented here, p-FEM with conforming meshand FCM with non-conforming mesh yield in similar damagecontours.

5. Conclusions

Employing the FCM, this research predicts the location of thecrack initiation in 2D and 3D problems using the Lemaitre damageconstitutive law in some benchmark problems. The achievementsare advantageous in industrial applications with a complex geom-etry, while FCM provides a solution without relying on a conform-ing mesh. The damage results of the FCM representation of 2D and3D numerical examples are in good agreement with the p-FEM andthe point of crack initiation is confirmed by experimental results.Finally, an extension of the model to a nonlocal damage constitu-tive equation to reduce the localization effects is underinvestigation.

Acknowledgement

The support provided by the Alexander von Humboldt founda-tion (AvH) is gratefully appreciated by the authors.

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