3d modeling of damage growth and crack initiation using...

Transaction A: Civil EngineeringVol. 17, No. 5, pp. 372{386c Sharif University of Technology, October 2010

3D Modeling of Damage Growthand Crack Initiation Using

Adaptive Finite Element Technique

H. Moslemi1 and A.R. Khoei1;�

Abstract. In this paper, the continuum damage mechanics model originally proposed by Lemaitre(Journal of Engineering Materials and Technology. 1985; 107: 83-89) is presented through an adaptive�nite element method for three-dimensional ductile materials. The macro-crack initiation-propagationcriterion is used based on the distribution of damage variable in the continuum damage model. The micro-crack closure e�ect is incorporated to simulate the damage evolution more realistic. The Zienkiewicz-Zhuposteriori error estimator is employed in conjunction with a weighted Superconvergence Patch Recovery(SPR) technique at each patch to improve the accuracy of error estimation and data transfer process.Finally, the robustness and accuracy of proposed computational algorithm is demonstrated by several 3Dnumerical examples.

Keywords: Damage mechanics; Crack initiation; Crack closure; Adaptive mesh re�nement; WeightedSPR technique.

INTRODUCTION

The fracture of ductile materials is the consequenceof a progressive damaging process and considerableplastic deformation usually precedes the ultimate fail-ure. The numerical prediction of damage evolutionand crack initiation-propagation can be described bythe means of continuum damage approach. Thecontinuum damage mechanics was originally developedto describe the creep rupture. It was �rst introducedby Kachanov [1] to describe the e�ects of an isotropicdistribution of spherical voids on plastic ow. Riceand Tracy [2] analytically investigated the evolution ofspherical voids in an elastic-perfectly plastic matrix.Gurson [3] proposed a model based on the theory ofelasto-plasticity for ductile damage where the (scalar)damage variable was obtained from the considerationof microscopic spherical voids embedded in an elasto-plastic matrix. It was shown that the theory isparticularly suitable for representation of the behavior

1. Center of Excellence in Structures and Earthquake Engineer-ing, Department of Civil Engineering, Sharif University ofTechnology, Tehran, P.O. Box 11155-9313, Iran.

*. Corresponding author. E-mail: [email protected]

Received 28 February 2010; received in revised form 18 May2010; accepted 9 August 2010

of porous metals. Murakami and Ohno [4] proposeda second-rank symmetric tensor for the anisotropicdamage variable in which the de�nition of damagevariable follows from the extension of e�ective stressconcept to three-dimensions by means of the hypothesisof the existence of a mechanically equivalent �ctitiousundamaged con�guration. Lemaitre [5] proposed amicro-mechanical damage model to simulate the phys-ical process of void nucleation, growth and coalescenceusing continuum mechanics. An anisotropic theoryof continuum damage mechanics was developed byChow and Wang [6] for ductile fracture in whichthe anisotropic damage evolution characterized by ageneralized damage characteristic tensor. Lemaitre andChaboche [7] pointed out the fracture as the ultimateconsequence of material degradation process.

The conventional continuum damage descriptionsof material degeneration su�er from the loss of well-posedness beyond a certain level of accumulated dam-age. Peerlings et al. [8] introduced the higher-order de-formation gradient in the constitutive model to improvethe de�ciency of standard damage models. Pardoen [9]proposed an extended Gurson model to encompassboth the low and large stress triaxiality regimes. Acomparison between the Lemaitre and Gurson damagemodels was performed by Hambli [10] in crack growth

Damage Growth and Crack Initiation via Adaptive FE Model 373

simulation and demonstrated that the Gurson damagemodel cannot predict the fracture propagation pathin a realistic way while the Lemaitre model givesgood results. Areias and Belytschko [11] coupled thecontinuum damage constitutive model with the X-FEMformulation to model the crack initiation and propaga-tion. Grassl and Jirasek [12] combined the stress-basedplasticity and strain-driven scalar damage to model thefailure of concrete. Mediavilla et al. [13] simulated theductile damage and fracture in metal forming processesusing a combined continuous-discontinuous approachwhich accounts for the interaction between macroscopiccracks and the surrounding softening material. Atransition from continuum damage to cohesive crackpropagation was developed by Comi et al. [14] inconcrete structures via the X-FEM technique. Theaim of present study is to model the fracture ofductile materials using an adaptive FE mesh re�nementtechnique.

Adaptive �nite element method utilizes the errorestimation to assess the quality of results and mod-ify the mesh during the solution, aiming to achieveapproximate solution within some bounds from theexact solution of continuum problem. An overview ofvarious error estimation techniques was presented byVerfurth [15] for elasticity problems. Error estimatorscan be basically divided into two categories. A priorierror estimation, which is based on the knowledge ofcharacteristics of the solution, provides qualitative in-formation about the asymptotic rate of convergence asthe number of degrees of freedom goes to in�nity [16].A posteriori error estimation employs the solutionobtained by the numerical analysis, in addition to apriori assumptions about the solution [17]. Babuskaand Rheinboldt [18] proposed a residual based methodof error estimation which considers local residuals ofthe numerical solution. The recovery technique isan alternative approach which computes an improvedsolution using a recovery process in which the erroris simply estimated as the di�erence between therecovered solution and numerical solution [19]. Variousrecovery procedures were proposed in the literature inwhich the superconvergent patch recovery method isone of the most e�ective ones.

The Superconvergent Patch Recovery (SPR)method was �rst introduced by Zienkiewicz andZhu [19,20] in linear elastic problems. The techniquewas applied in nonlinear analysis by Boroomand andZienkiewicz [21] in which the strain was recoveredby SPR in elasto-plasticity problems. The SPRmethod was proposed by Wiberg et al. [22] to improvethe accomplishment of equilibrium equations and theboundary conditions applied to the smoothed stresses.A modi�ed SPR method was applied by Gu et al. [23] toimprove the accuracy and stability of the technique byintroducing additional nodal points. Bugeda [24] pro-

posed an enriched SPR method based on the sensitivityanalysis for better estimation of error. An extension ofSPR technique to 3D plasticity problems was presentedby Khoei and Gharehbaghi [25,26]. A modi�ed-SPRtechnique was applied by Khoei et al. [27] for simulationof crack propagation in which the polynomial functionwas replaced by singular terms of analytical solution ofcrack problems in the process of recovery solution. Thetechnique was improved by Moslemi and Khoei [28] andKhoei et al. [29] to estimate a more realistic error inLEFM problems and cohesive zone models by applyingthe weighting function for various sampling points.

In the present paper, an adaptive �nite elementmethod is presented based on the weighted-SPR tech-nique to model the damage of ductile material in 3Dproblems. The Lemaitre damage model is employedand the micro-crack closure e�ect is incorporated tosimulate the damage evolution. The macro-crackinitiation-propagation criterion is used based on thedistribution of damage variable. The plan of the paperis as follows: The Lemaitre damage model and itsimplementation in �nite element context are describedtogether with the crack closure e�ect. The errorestimation based on weighted-SPR method is thenpresented along with the adaptive mesh re�nementand data transfer process. Finally, the robustness andaccuracy of computational algorithm are demonstratedby several numerical examples.

NONLINEAR DAMAGE MODEL

Damage in materials is mainly the process of initiationand growth of micro-cracks and cavities. Continuumdamage mechanics discusses systematically the e�ectsof damage on the mechanical properties of materialsand structures as well as the inuence of externalconditions and damage itself on the subsequent de-velopment of damage. In this study, this nonlinearinteraction is investigated via the Lemaitre damageconstitutive model. In order to describe the internaldegradation of solids within the framework of the con-tinuum mechanics theory, new variables intrinsicallyconnected with the internal damage process need tobe introduced in addition to the standard variables.Variables of di�erent mathematical nature possessingdi�erent physical meaning have been employed in thedescription of damage under various circumstances.The damage variable used here is the relative area ofmicro-cracks and intersections of cavities in any planeoriented by its normal n as [30]:

D(n) =S�S; (1)

where S� is the area of micro-cracks and intersectionsand S is the total area of the cross section as shownin Figure 1. It is assumed that micro-cracks and

374 H. Moslemi and A.R. Khoei

Figure 1. A damaged element presenting the areas S and S�.

cavities are distributed uniformly in all directions. Inthe isotropic case, the damage variable is adoptedas a scalar. The behavior of damaged material isgoverned by the principle of strain equivalence whichstates that the strain behavior of a damaged materialis represented by constitutive equations of the virginmaterial (without damage) in the potential of whichthe stress is simply replaced by the e�ective stress. Bythis assumption the e�ective stress tensor is related tothe true stress tensor by:

�e� =1

1�D�: (2)Consider Y be the thermodynamic variable associatedwith the damage variable D as:

Y =@u@D

= �12"e : De : "e; (3)

where `:' denoted component by component product oftensors. This quantity is called damage energy releaserate and is expanded by using the inverse of the elasticstress-strain law as:

Y = � 12(1�D)2� : [De]�1 : �

= � 12E(1�D)2 [(1 + �)� : � � �(tr�)2]

= � q22E(1�D)2

"23

(1 + �) + 3(1� 2�)�pq

�2#;(4)

where p is the hydrostatic stress and q is the equivalentvon-Mises stress. In Lemaitre damage model theevolution of damage variable is assumed to be givenby:

_D =

(0 "peq � "pD_ 11�D

��Yr

�s "peq > "pD (5)where r and s are material and temperature-dependantproperties, "peq =

p2=3k"pk is equivalent plastic strain,

_ is the plastic multiplier and is equal to the rate ofequivalent plastic strain and "pD is threshold damage

where damage growth starts only at this critical value.As can be seen in above equation, the damage ratedepends on the stress state, plastic strain growthand instantaneous damage variable. The e�ect ofdamage variable on mechanical behavior of material isaccounted in degradation of elastic modulus of materialand its yield surface. Based on the equivalent strainprinciple, this modi�cation can be expressed as:

De� = (1�D)De; (6)

� =r

32ksk

(1�D) � [�0Y +R("peq)]; (7)

where De and De� are the elastic modulus of materialbefore damage and after damage, respectively, � isthe modi�ed yield surface, s is the deviatoric stresstensor and R is the isotropic plastic growth function.In a damaged medium, three distinctive regions canbe recognized (Figure 2): �rstly, the region S0 whereno damage has been occurred, secondly, the region Sdwhere the damage is sub-critical and only degradesthe material, and �nally, the region SC where thedamage variable is reached the critical value DC . If thecritical damage value is satis�ed within an element, theelement fractures and cracks occur.

Finite Element Implementation

The accuracy of the overall �nite element scheme de-pends crucially on the accuracy of particular numericalalgorithm adopted. This section describes a numericalprocedure for integration of the Lemaitre damageelasto-plastic model, presented in preceding section,

Figure 2. Damaged zones in a schematic model.


based on the well-known two-step elastic predictor-plastic corrector method [31]. At each Gauss point,the values of state variables including the stress tensor�n, plastic strain tensor "pn, equivalent plastic strain("peq)n and damage variable Dn are known at the startof interval, and for a given strain increment �", thevalue of variables are desired at the end of interval. Atthe �rst stage of computational algorithm the materialbehavior is assumed to be elastic; the yield surface atthe end of interval can be then evaluated as:

� =r

32ksn+1k

(1�Dn) � [�0Y +R("

peq)n]: (8)

If � � 0, the assumed elastic behavior is correct and thedamage variable and plastic strain remain unchanged.If � > 0, the plastic corrector step must be applied toobtain the updated state variables by simultaneouslyestablishing four equations consisting of the plastic owequation, equivalent plastic strain growth equation,damage growth equation and the yield surface equationas:

"pn+1 = "pn +

�1�Dn+1

r32

sn+1ksn+1k ; (9)

("peq)n+1 = ("peq)n + �; (10)

Dn+1 = Dn + �1

1�Dn+1��Yn+1

r

�s; (11)

� =r

32ksn+1k

(1�Dn+1) � [�0Y +R("

peq)n+1] = 0: (12)

The solution of these four nonlinear coupled equationssimultaneously is a costly computational task. Stein-mann et al. [32] presented that by performing relativelystraightforward algebraic manipulations, the abovesystem can be reduced to a single nonlinear algebraicequation for the plastic multiplier � expressed as:

!(�)� !n + �!(�)��Y (�)

r

�s= 0; (13)

where ! is the integrity variable in contrast to damagevariable and is evaluated by:

!n+1 = 1�Dn+1 = !(�)

=3G�

�qtrialn+1 � �Y (Rn + �) ; (14)

where �qtrialn+1 is the von-Mises equivalent stress obtainedby the elastic predictor step. The damage energyrelease rate is a function of � calculated by:

Y (�) =[�Y (Rn + �)]2

6G+

�p2n+12K

; (15)

where �pn+1 is the elastic predicted hydrostatic pressurewithout damage e�ect. Equation 13 can be solved byan iterative method such as Newton-Raphson method.An initial guess is very e�ective in the rate of conver-gence of the method. In this study, the following initialguess proposed by de Souza Neto et al. [33] is employedas:

�(0) =[�qtrialn+1 � �Y (Rn)]!n

3G: (16)

It was shown that the above initial guess reduces thetotal number of iterations required for convergence,as compared to the usual choice of �(0) = 0. Bycomputing the plastic multiplier, the updated statevariables can be obtained using four basic equations.

Micro-Crack Closure E�ect

From the micromechanical view, the damage can beconsidered as the degradation of material propertiesdue to the evolution of voids and micro-cracks. TheLemaitre damage model discussed in the previoussection su�ers from an important drawback, since thee�ect of hydrostatic stress is captured by the damageenergy release rate Y with equal response in tensionand compression. On the other hand, the micro-cracks which open in tensile stresses may partially closeat a compression stress state. Hence, after havingbeen damaged in tension, the material recovers itssti�ness partially under compression. To remedy thisproblem, several decompositions are proposed, suchas the elastic energy decomposition [34], the Kelvindecomposition of compliance tensor [35], the damagetensor decomposition [36], or more classically basedon the stress tensor decomposition into a positive andnegative part [37-40]. In this study, the stress tensor isdecomposed into the positive and negative parts andthe e�ect of compressive stress tensor is considereda fraction of the e�ect of tensile stress tensor. Inthe process of decomposition, the stress tensor is �rstmapped to the principle directions to form a diagonalmatrix with the principle stresses. This matrix is thendecomposed to tensile and compressive stress tensorsas:

� = �+ + ��: (17)

This decomposition is de�ned mathematically using theMacaulay bracket h i as:

�+ =

24h�1i 0 00 h�2i 00 0 h�3i

35 ;�� = �

24h��1i 0 00 h��2i 00 0 h��3i

35 : (18)


The Macaulay bracket is a scalar function de�ned as:

hai =(a if a � 00 if a < 0

(19)

By decomposition of stress tensor, each componenta�ects the damage energy release rate separately. Thee�ect of tensile component given in previous sectionremains valid, but the compression component has amore moderate e�ect with an experimental reductionfactor of h, which can be determined by the techniqueproposed by Arnold et al. [41]. This value is calledthe crack closure e�ect constant and has the value inthe range of [0 � 1]. At the extreme values, h = 0implies the full crack closure and h = 1 represents thedamage model without crack closure e�ect. Thus, thee�ect of compressive component of stress tensor canbe superposed with the e�ect of tensile component ofstress tensor by modifying Equation 4 as:

Y =� 12E(1�D)2 [(1 + �)�+ : �+ � �htr�i2]

� h2E(1� hD)2 [(1 + �)�� : �� h�tr�i2]:(20)

By modifying the value of damage energy releaserate, the remaining parts of the procedure of originalLemaitre model will not be changed.

ADAPTIVE FE STRATEGY

In numerical analysis of FE solution, it is essential to in-troduce some measures of error and use adaptive meshre�nement to keep this error within prescribed boundsto ensure that the �nite element method is e�ectivelyused for practical analysis. To automate this process,adaptive �nite elements have been implemented toobtain an optimal mesh. Most of the pioneering math-ematical works, such as Babuska and Rheinboldt [18],Zienkiewicz and Zhu [19, 20], Gago et al. [42] and Kellyet al. [43], have been today translated into engineeringusage. Due to the localized material deterioration inthe damaged body problems, many elements will beseverely distorted producing unacceptably inaccuratesolutions and this optimization takes a more importantand necessary role. In order to obtain an optimalmesh, in the sense of an equal solution quality, itis desirable to design the mesh such that the errorcontributions of the elements are equally distributedover the mesh. This criterion illustrates what partsof the discretized domain have to be re�ned/de-re�nedand what degree of mesh �neness is needed to maintainthe solution error within the prescribed bounds. Theplastic deformation of problem necessitates transferringall relevant variables from the old mesh to new one.

In general, the procedure described above can beexecuted in four parts: an error estimation, an adaptivemesh re�nement, an adaptive mesh generator, and themapping of variables.

Error Estimation Using Weighted SPRTechnique

In damage mechanics problems, the damage variableplays an important role in predicting the crack ini-tiation and crack growth. Thus, the error needs tobe estimated based on the damage variable D, i.e.eD = D � _D, with D denoting the exact value ofdamage variable and

_D the damage value derived by

a �nite element solution. Since the exact value ofdamage variable is not available, a recovered solutionis used instead of exact one and then approximate theerror as the di�erence between the recovered values andthat given directly by the �nite element solution, i.e.eD � D� � _D, where D� denotes the recovered valueof damage variable D. In order to obtain an improvedsolution, the nodal smoothing procedure is performedusing the Weighted Superconvergent Patch Recovery(WSPR) technique, which was originally proposed byMoslemi and Khoei [28] to simulate the crack growthin linear fracture mechanics. The objective of recoveryof the FE solution is to obtain the nodal values ofdamage variable D such that the smoothed continues�eld de�ned by the shape functions and nodal values ismore accurate than that of the �nite element solution.

A procedure for utilizing the Gauss quadraturevalues is based on the smoothing of such values by apolynomial of order p in which the number of samplepoints can be taken as greater than the number ofparameters in the polynomial. The recovered solutionof damage variable D� can be obtained as D� = Pa,with a denoting a vector of unknowns assumed as:

a = ha1; a2; a3; a4iT ;and P the polynomial base functions given by:

P = h1; x; y; zi:The determination of the unknown parameters a canbe made by performing a least square �t to the valuesof superconvergent or sampling points. After �niteelement analysis, a patch is de�ned for each vertex nodeinside the domain by the union of elements sharingthe node. At each node of interior patch center, theconnected tetrahedral elements along with their nodesand Gauss points are obtained. In the standard SPRtechnique, all sampling points have similar propertiesin the patch, which may produce signi�cant errors inthe boundaries, particularly the edges of crack. In fact,for elements located on the boundaries or crack edges,


which do not have enough sampling points, we needto use the sampling points of nearest patch. Thesenew sampling points induce unreal value of damagevariables in the patch and overestimate the value oferror, which results in unreasonable mesh re�nementin the boundaries of domain. Moslemi and Khoei[28] proposed the weighted-SPR technique by usingdi�erent weighting parameters for sampling points ofthe patch. It results in more realistic recovered valuesat the nodal points, particularly near the crack tip andboundaries. Hence, if we have n sampling points in thepatch with the coordinates (xk; yk; zk), the function Fneeds to be minimized in this patch as:

F (a) =nXk=1

wk[D�i (xk; yk; zk)�_Di(xk; yk; zk)]2

=nXk=1

wk[P(xk; yk; zk)a� _Di(xk; yk; zk)]2: (21)

In order to incorporate the e�ects of nearest samplingpoints in the recovery process, the weighting parameteris de�ned as wk = 1=rk, with rk denoting the distanceof sampling point from the vertex node which is underrecovery. The minimization of function F with respectto a results in the unknown parameters a as:

a =

nXk=1

w2kPTkPk

!�1 nXk=1

hw2kP

Tk_Di(xk; yk)

i: (22)

Once the components of the vector of unknown param-eters a are determined, the damage variables at nodalpoints inside the patch are computed by using the in-terpolation of the shape functions. These nodal values�D� can be used to construct a continuous damage �eldover the entire domain at the next step, that is, for eachelement, the recovered damage variable is representedas an interpolation of nodal values using the standardshape functions N in �nite element analysis as D� =N �D�. The recovered damage variable obtained by thisrelation can be used to obtain a pointwise error inthe domain. Since the pointwise error becomes locallyin�nite in critical points, such as crack tip, the errorestimator can be replaced by a global parameter usingthe norm of error de�ned as:

keDk = kD� � _Dk=�Z

(D��_D)T (D��_D)d

�1=2:

(23)

The above L2 norm is de�ned over the whole domain

. The overall error can be related to each elementerror by:

keDk2 =mXi=1

keDk2i ;

with i denoting an element contribution and m the

total number of elements. The distribution of errornorm across the domain indicates which portions needre�nement and which other parts need de-re�nement,or coarsening elements. Since the total error permis-sible must be less than a certain value, it is a simplematter to search the design �eld for a new solution inwhich the total error satis�es this requirement. In fact,after remeshing each element must obtain the sameerror and the overall percentage error must be less thanthe target percentage error, i.e.:

� = keDk=k_Dk � �aim = keDkaim=k_Dk;with �aim denoting the prescribed target percentageerror. Hence, the aim error at each element can beobtained as:

(keDki)aim = 1pmk_Dk�aim: (24)

The rate of convergence of local error depends on theorder of elements. Thus, the new element size can beevaluated as:

(hi)new =�

(keDki)aim(keDki)old

�1=p(hi)old; (25)

where h is the average element size and p is the orderof element. To obtain the nodal element size, a simpleaveraging between elements joining a node is used.

Data Transfer Operator

In the nonlinear FE analysis, the new mesh mustbe used starting from the end of previous load stepsince the solution is history-dependent in nonlinearproblems. Thus, the state and internal variablesneed to be mapped from the old �nite element meshto the new one. The process of data transfer canbe carried out in three steps [44,45]. In the �rststep the continuous internal variables are obtained byprojecting the Gauss point components to the nodalvalues using the 3D weighted-SPR method. In thesecond step, the nodal values of internal variables ofold mesh are transferred to the nodes of new mesh. Forthis purpose, we must �rst determine which elementin the old mesh contains the nodal point in the new�nite element mesh. The nodal components in the oldmesh are then transferred to the nodes of new meshby applying the old shape functions of old elementsand the global coordinates of the new nodes. Thecomponents of internal variables at the Gauss pointsof new mesh are �nally obtained by interpolation usingthe shape functions of elements of the new mesh. Thesethree steps are illustrated schematically in Figure 3.

Consider that a state array �oldn =�uoldn ; "oldn ; ("pn)old;�oldn ; Doldn

�denote the values


Figure 3. Three-step procedure of the transfer operator.

of displacement, strain tensor, plastic strain tensor,stress tensor and damage variable at time tn for themesh Mh. Also assume that the estimated error ofthe solution �oldn respects the prescribed criteria,while these are violated by the solution �oldn+1. Inthis case, a new mesh MH is generated and a newsolution �newn+1 is computed by evaluating the stresstensor �newn and the damage variable Dnewn for a newmesh MH at time step tn. In this way, the statearray

_�

new

n = (�newn ; Dnewn ) is constructed, where_�

is used to denote a reduced state array. It must benoted that the state array

_� characterizes the history

of the material and provides su�cient information forcomputation of a new solution �newn+1. The aim is totransfer the internal variables ((�n)oldG ; (Dn)oldG ) storedat the Gauss points of the old mesh Mh to the Gausspoints of new mesh MH. The transfer operator T1between meshes Mh and MH can be de�ned as:

((�n)newG ; (Dn)newG ) = T1[(�n)oldG ; (Dn)oldG ]: (26)

The variables ((�n)oldG ; (Dn)oldG ) speci�ed at Gausspoints of the meshMh are transferred by the operatorT1 to each point of the domain , in order to specifythe variables ((�n)newG ; (Dn)newG ) at the Gauss points ofnew mesh MH. The operator T1 can be constructedby a least-squares method, or a suitable projectiontechnique.

In order to obtain the continuous values of stresstensor and damage variable

_�

old

n = ((�n)old; Doldn ),the Gauss point components (�n)oldG and (Dn)oldG areprojected to nodal points to evaluate the components(�n)oldN and (Dn)oldN . In this study, the projection of theGauss point components to the nodal points is carriedout using the weighted-SPR technique, as described inprevious section. The nodal components of the stresstensor (�n)oldN and the damage variable (Dn)oldN for themesh Mh are then transferred to the nodes of thenew mesh MH resulting in components (�n)newN and(Dn)newN . The components of stress tensor and damagevariable at the Gauss points of the new meshMH, i.e.(�n)newG and (Dn)newG , are �nally obtained by using the

interpolation of the shape functions of the new �niteelements. In this procedure, the local coordinates areused to interpolate the variables from the nodes of meshMh to the nodes of mesh MH. In the case of linearfour-noded tetrahedral element, the local coordinatesare obtained by solving a linear system of algebraicequations. In the case of higher order elements, theproblem becomes nonlinear and the Newton-Raphsoniterative scheme is implemented to obtain the localcoordinates.

NUMERICAL SIMULATION RESULTS

In order to illustrate the accuracy and robustness ofthe proposed adaptive �nite element method in three-dimensional damage mechanics, several problems aresimulated numerically. Two benchmark examples arechosen to evaluate the performance of adaptive FEstrategy for the crack initiation in a cylindrical pre-notched bar and a �nite crack in the 3D rectangularspecimen. In order to tackle 3D damage mechan-ics problems the proposed adaptive FE method isimplemented in the computer software SUT-DAM,which was designed by the senior author in [46-48] foradaptive mesh re�nement of large plasticity deforma-tions and has been extended to adaptive analysis ofdamage growth simulation. The ten-noded tetrahedralelements are employed for the �nite element meshes,and the numerical integration is carried out using fourGauss-Legendre quadrature points. The projection ofthe values of Gauss points to nodal points is carriedout by using the 3D weighted-SPR method. The stresstensor and damage variable are mapped from the oldmesh to new one during the data transfer process. TheLemaitre coupled plasticity-damage model is used inconjunction with the micro-crack closure e�ect to pre-dict the damage growth in specimens. In order to solvethe nonlinear equation systems, the global sti�nessmatrix is formed at the �rst iteration of each loadingincrement, and remains unchanged during iterationsafterwards, i.e. the modi�ed Newton-Raphson iterativeprocedure is employed. In all examples, the results arecompared with those reported by previous researches.


Crack Initiation in a Cylindrical Pre-notchedBar

The �rst example is of a classical tensile test of acylindrical pre-notched bar which has been extensivelyused by various researchers [49-52]. The geometryand boundary conditions of the specimen are shownin Figure 4. On the virtue of symmetry, only one-eighth of the problem is modeled. The specimen issubjected to the tensile prescribed displacement at thetop edge. The bar is constructed by a low carbon steelin a rolled state with the following material properties:E = 210 GPa and � = 0:3. The strain hardeningis considered to be isotropic and its curve has anexponential form governed by:

�Y =620+3300[1�exp(�0:4"peq)] (MPa); (27)where "peq is the equivalent plastic strain. The param-eters of Lemaitre model were calibrated by Benallalet al. [53] for this specimen given as s = 1:0 andr = 3:5 MPa. This specimen was also simulated byde Souza Neto et al. [33] using 2D FE modeling tovalidate the performance of their constitutive modelin damage mechanics. The vertical displacement isapplied incrementally in 600 increments of 0.001 mm.The damage evolution is predicted by Lemaitre modeluntil its value reaches to the critical damage value ofDC = 0:99.

In Figure 5, the distribution of damage contoursare shown at three stages and compared with thosereported by de Souza Neto et al. [33] in two-dimensionalmodeling. It can be seen that the predicted damagecontours are in good agreement with those obtainedin [33]. It is interesting to note that the location ofmaximum damage is not �xed in the model and movesat di�erent stages of loading. It can be observed thatthe maximum damage occurs in the outer part of thespecimen at the �rst stages of loading, however, itmoves toward the center of the bar by increasing theload. This can be justi�ed by the fact that at early

Figure 4. The cylindrical notched specimen; thegeometry and boundary conditions.

Figure 5. The distribution of damage contours at variousload steps. A comparison between the present 3D modeland 2D result reported by de Souza Neto et al. [33].

stages of loading the hydrostatic stress is low and thedamage evolution is a�ected by the plastic ow. Thus,the damage grows in outer layers where the maximumequivalent plastic strain occurs. However, by increasingthe load, the hydrostatic stress increases and its e�ectbecomes dominant. Hence, the damage critical pointmoves toward the center of the bar where the maximumvalue of hydrostatic stress occurs. This phenomenon isalso observed experimentally by Hancock and Macken-zie [54] where the failure of the bar initiates from thecenter of the bar. A comparison of the damage variableevolution at the centre of the specimen between thepresent study and 2D results reported by de Souza Netoet al. [33] is shown in Figure 6. The rate of damagegrowth is proportional to the damage value itself. InFigure 7, the variation of reaction force is plotted withthe prescribed displacement. In order to control the


Figure 6. The damage variable evolution at the centre ofspecimen. A comparison between the present 3D modeland 2D result reported by de Souza Neto et al. [33].

Figure 7. The force-displacement diagram for thecylindrical notched specimen. A comparison between thepresent 3D model and 2D result reported by de SouzaNeto et al. [33].

error of the solution, an adaptive FE mesh re�nement iscarried out to generate the optimal mesh. The weightedsuperconvergent patch recovery technique is used withthe aim error of 5%. This process is carried out attwo steps of 50 and 360, as shown in Figure 8. This�gure clearly presents the distribution of elements onthe specimen with the growth of damage. In Figure 9,the e�ect of adaptive strategy can be observed onthe estimated error. Obviously, the adaptive meshre�nements result in a reduced estimated error andconverge to the prescribed target error.

A Finite Crack in a 3D Rectangular Specimen

The second example illustrates the damage growthin mode-I loading condition of a three-dimensional

Figure 8. Adaptive mesh re�nement technique for thecylindrical notched specimen. a) Initial FE mesh; b)adapted mesh at u = 0:05 mm; and c) adapted mesh atu = 0:36 mm.

Figure 9. The variation of estimated error withprescribed displacement during adaptive mesh re�nement.

Figure 10. A 3D rectangular specimen with �nite crack;the geometry and boundary conditions.

rectangular specimen of 20 � 10 � 2 cm with a �nitecrack, as shown in Figure 10. This example is chosento validate the accuracy of proposed adaptive FEalgorithm for a benchmark problem. The 3D specimenis subjected to the prescribed displacement until the


damage variable reaches to the critical value DC =0:95. This condition was attained for a prescribedisplacement of u = 1:1 mm. This example wasproposed by Chung et al. [55] to verify the abilityof their two-step parallel computing technique. Themodel is made by an aluminum alloy with the materialproperties given in Table 1. The strain hardeningof material is governed by the following exponentialfunction, i.e.:

�Y = h1"peq + (h1 � h0) exp(�m"peq)=m: (28)In Figure 11, the distribution of damage contours areshown at various load steps and compared with thosereported by Chung et al. [55]. Complete agreementcan be observed between two numerical simulations.Obviously, the critical damage point is obtained at thecrack tip and the crack growth is expected to initiatefrom this point. The distribution of damage is alsosymmetric around the crack tip. The magnitude of

Table 1. Mechanical properties and material constants ofaluminum alloy.

Material Constant Parameter Value

Young modulus E 72.4 GPa

Poisson ratio � 0.32

Hardening constant m 25

Initial hardness constant h0 1150 MPa

Ultimate hardness constant h1 1670 MPa

Damage constant 1 s 1.0

Damage constant 2 r 1.5 MPa

Figure 11. a) A comparison of �nal damage contoursbetween the present model and that reported by Chung etal. [55]; and (b-d) damage contours near the crack tip atvarious load steps.

maximum damage obtained by the present simulationand that reported by Chung et al. [55] are plotted inFigure 12. The discrepancy between two numericalsimulations can be related to the di�erent damagemodels used in these simulations, and the calibrationperformed for obtaining the parameters of two damagemodels. In Figure 13, the initial FE mesh is showntogether with the adaptive mesh re�nements obtainedusing the weighted-SPR technique for the damagemodeling. It must be noted that the �nest adaptedFE mesh used in present simulation has 30000 degrees-of-freedom, while the �nite element mesh used inreference [55] was taken more than one million DOFsto achieve an acceptable result. The process of errorestimation is carried out in this example at two stagesand the results are plotted in Figure 14. Obviously,the adaptive mesh re�nement procedure reduces theestimated error considerably.

Figure 12. The evolution of damage variable at crack tipfor the 3D tensile specimen; a comparison between thepresent model and that reported by Chung et al. [55].

Figure 13. Adaptive mesh re�nement model for the 3Dtensile specimen. a) Initial FE mesh; b) adapted mesh atu = 0:4 mm; and c) adapted mesh at u = 0:8 mm.



A Knee-Lever with a Corner Crack

The last example is of a knee-lever connection, which isextensively used for machine systems. The componenthas an initial crack in the symmetric plane and orientedperpendicular to the adjacent surfaces. The crack edgehas a circular shape with a radius of 5 mm. Theconnection is loaded by the prescribed displacement intwo edge holes simultaneously and restrained at themiddle hole. The geometry and boundary conditionsof this component are shown in Figure 15. In Figure 16the location of corner crack is shown in the three-dimensional view. The connection is made of thealuminum alloy AlZnMgCu 1.5 which is a proper choicein lightweight structures for its high ratio of strengthto density. The elasticity modulus of specimen isE = 32:5 GPa and the Poisson ratio is � = 0:3. The

Figure 15. A knee-lever with a corner crack; thegeometry and boundary conditions (all dimensions inmm).

Figure 16. A knee-lever with a corner crack; the crackgeometry.

initial yield stress is assumed to be �Y 0 = 350 MPaand the material hardening is governed by a linearhardening of H = 820 MPa. Schollmann et al. [56]simulated this component to illustrate the capabilitiesof their software in handling complex structures andcrack under fatigue loading. To emphasize on three-dimensional behavior of component, the thickness ofthe specimen is increased here from 15 mm to 40 mm.

Based on the Lemaitre damage model with nomicro-crack closure e�ect, the maximum damage oc-curs at loading points of edge holes because of highcompression stresses at these regions. However, byintroducing this e�ect to the model, the critical damagetransmits to the crack tip region, as demonstrated inFigure 17. As can be observed from this �gure, thecrack tip in the thickness direction illustrates the higherdamage than the crack tip in the upper surface ofthe component. Thus, the crack �rst grows in thethickness direction and then penetrates in the depth ofcomponent. This observation can be veri�ed with thatreported by Schollmann et al. [56]. Figure 18 displaysthe reaction force versus prescribed displacement forthe knee-lever connection. In Figure 19 the maximumdamage variable evolution at di�erent displacementsis plotted. It can be observed that the damagegrowth starts at prescribed displacement of 1 mm andrapidly reaches to its critical value at displacement of2 mm, however, the crack tip region reaches the criticaldamage at displacement of 9 mm. Furthermore, thecritical damage point is obtained at the front cracktip during loading steps, since both the hydrostaticstress and e�ective plastic strain are concentrated onthis point. During the damage growth process, two

Figure 17. The damage contour for a knee-lever with acorner crack.


Figure 18. The force-displacement curve for a knee-leverwith a corner crack.

Figure 19. The evolution of damage variable at crack tip.

Figure 20. The adaptive mesh re�nement strategy for aknee-lever with a corner crack. a) Initial FE mesh, b)adapted mesh at u = 1:15 mm, c) adapted mesh atu = 6:10 mm.

successive mesh re�nements with the aim error of 5%are performed. In Figure 20, the initial and adaptedFE meshes are shown at u = 1:15 and 6:10 mm.The adaptive mesh re�nement is performed at thestart of damage growth and when the estimated errorexceeds 10%. The e�ect of adaptive mesh re�nement


is obvious in the variation of estimated error at thedamage growth process, as shown in Figure 21.

CONCLUSION

In the present paper, an adaptive �nite element methodwas presented for the three-dimensional analysis ofdamage growth and crack initiation. The formulationtakes into account the micro-crack closure e�ect ondamage evolution, to distinguish between the tensileand compressive stresses. The constitutive modelingwas implemented within the framework of continuumdamage mechanics. A simpli�ed version of Lemaitredamage model was employed to estimate the damageevolution. The adaptive �nite element technique wasimplemented through the following three stages: anerror estimation, adaptive mesh re�nement and datatransferring. The error estimation procedure wasused based on the Zienkiewicz-Zhu error estimatorand a weighted superconvergent patch recovery tech-nique was employed. The accuracy and robustnessof proposed computational algorithm in 3D damagemechanics were presented by three numerical examples.The results clearly show the ability of the model incapturing the damage growth and crack initiation intwo benchmark examples and a complex 3D problem.In a later work, we will show how the proposedtechnique can be used in a 3D automatic simulation ofcrack propagation in the fracture of ductile materials.

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BIOGRAPHIES

H. Moslemi received his B.S. in Civil Engineeringfrom Sharif University of Technology, in 2003, and hisM.S. degree in Structural Engineering from Universityof Tehran, in 2005. He completed his Ph.D. in 2010in Computational Mechanics under the supervisionof Professor A.R. Khoei from Sharif University ofTechnology.

Dr Moslemi is currently an assistant professor inthe Civil Engineering Department at Shahed Univer-sity. His academic career also includes positions ina number of Iranian universities such as InternationalCampus of Sharif University of Technology in Kish Is-land and Qazvin Islamic Azad University. His research

interests include Computational Fracture Mechanics,Crack Propagation and Damage Mechanics.

Amir Reza Khoei received his Ph.D. in Civil En-gineering from the University of Wales Swansea inUK in 1998. He is currently a professor in theCivil Engineering Department at Sharif University ofTechnology. He is the editor of Scientica Iranica,Transaction A, Journal of Civil Engineering, a memberof the editorial board of the Journal of Science andTechnology (Sharif University of Technology), andthe co-editor of the 11th International Symposium onPlasticity and Its Current Application in US, 2005.Dr Khoei has selected as a distinguished researcher atSharif University of Technology in 2003, 2005, 2007 and2008, and as a distinguished professor by the Ministryof Science, Research and Technology in 2008. He is thesilver medal winner of Khwarizmi International Award(2005) organized by the Iranian Research Organizationfor Science and Technology.

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