a micro-mechanical damage model based on gradient plasticity: algorithms and applications

INTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERINGInt. J. Numer. Meth. Engng 2002; 54:399–420 (DOI: 10.1002/nme.431)

A micro-mechanical damage model based on gradientplasticity: algorithms and applications

Jian Chen and Huang Yuan∗;†

Laboratory for Material Behaviours; Paul Scherrer Institute; CH-5232 Villigen PSI; Switzerland

SUMMARY

As soon as material failure dominates a deformation process, the material increasingly displays strainsoftening and the �nite element computation is signi�cantly a�ected by the element size. Withoutremedying this e�ect in the constitutive model one cannot hope for a reliable prediction of the ductilematerial failure process. In the present paper, a micro-mechanical damage model coupled to gradient-dependent plasticity theory is presented and its �nite element algorithm is discussed. By incorporatingthe Laplacian of plastic strain into the damage constitutive relationship, the known mesh-dependenceis overcome and computational results are uniquely correlated with the given material parameters.The implicit C1 shape function is used and can be transformed to arbitrary quadrilateral elements. Theintroduced intrinsic material length parameter is able to predict size e�ects in material failure. Copyright? 2002 John Wiley & Sons, Ltd.

KEY WORDS: micro-mechanical damagemodel; non-local damagemodel; gradient plasticity; plastic straingradient; ductile materials; material internal length; �nite element method; C1 continuity

1. INTRODUCTION

It is known that conventional continuum mechanics treats mathematical continua, that is,solids in continuum mechanics consist of mathematical points and do not contain micro-structures. It follows that the stress state is determined by the deformation history at thissingle material point. Although the conventional continuum mechanics is quite su�cient formost applications, there are experimental evidences indicating that under certain circumstancesthe material micro-structures must be taken into account in a suitable way.Recently, material modelling including micro-structure characteristics has been extensively

discussed. A variety of models incorporating material length scales has been proposed. Aifan-tis [1] suggested a simple form of plasticity depending on plastic strain gradients which istermed gradient plasticity theory. A �nite element algorithm of the gradient plasticity hasbeen published by de Borst and co-workers [2; 3] and later extended to �nite strains by

∗Correspondence to: Huang Yuan, DaimlerChrysler Aerospace, MTU Aero Engines, Dachauer strasse 665,D-80995 Munich, Germany

†E-mail: [email protected] 14 July 2000

Copyright ? 2002 John Wiley & Sons, Ltd. Revised 20 August 2001

400 J. CHEN AND H. YUAN

Mikkelsen [4; 5] and Ramaswamy and Aravas [6] using slightly di�erent interpolation meth-ods. Works in References [2; 3] showed that the Aifantis model containing only the secondorder of the plastic strain gradient gave mesh-independent predictions of brittle material failure.Fleck and Hutchinson [7; 8] considered an ‘asymmetric’ strain gradient plasticity theory

in which additional high-order strain tensor and the work conjugated moment stress tensorenter the material model and governing equations. From the viewpoint of application theFleck–Hutchinson strain gradient plasticity theory is similar to Cosserat-type continuum whichintroduces length scale by additional degrees of freedom. A problem of Cosserat continuum isthat in tension-dominated problems, rotations are small and the e�ect of the high-order strainbecomes insigni�cant within the localization band [2]. The shear band width is not uniquelycorrelated with the material length scale. It remains open whether this model does generate aunique solution for strain softening.Failure in ductile metals is characterized by the micro-void nucleation, growth and coales-

cence mechanism. The damage model (GTN model), originally introduced by Gurson [9] andlater modi�ed by Tvergaard [10; 11] and Tvergaard and Needleman [12], is attractive in that itis not derived from purely heuristic arguments but from micro-mechanical analysis. The yieldfunction of the GTN model accounts for voids in terms of one single internal variable, the voidvolume fraction or the porosity. This model is popular in materials mechanics community toanalyse and to predict failure of ductile metallic materials. However, a well-known problem isthat strain localizations and so material failure are restricted in the single element layer of an�nite element model, due to involved strain softening in the material failure process, resultingin a zero dissipated energy as the element size becomes vanishingly small. The �nite elementsimulations show an inherent mesh sensitivity in ductile crack propagation simulations. Theseobservations imply that there is a need for such a micro-mechanical approach to incorporatethe intrinsic material length parameter into the constitutive relation.Non-local forms of the GTN model in which the delocalization is related to the damage

parameter were developed by Leblond [13], Tvergaard and Needleman [14] and others. Inthese works, the porosity is treated non-locally by averaging the porosity value in an assumedneighbouring region. From a numerical point of view, such an approach is similar to those that�t a constant element size with the material micro-structure [15–17], in which the size of a cellelement is chosen to be representative of the mean spacing between voids. It follows that eachcell element contains a single void at the initial volume fraction. Growth and coalescence ofthe void is related to the stress and strain averaged over the cell element. In comparison withthe treatments in References [13; 14], the cell element method is simpler for �nite elementcomputations. Its application is, however, restricted in small-sized specimen due to increasingcomputational e�orts.Recently, Ramaswamy and Aravas [6] suggested a gradient treatment of the porosity of the

GTN model. In their study, e�ects of void di�usion, interaction and coalescence have beenconsidered. In their model, the �rst and second derivatives of the porosity enter the evolutionequation. Variations of the porosity are controlled by a di�usion equation.All these e�orts are concentrating on non-local treatment of the damage indicator and

assuming that the material length scale is only related to damage evolution, which is certainlycontradictory to the known experimental observation of signi�cant size e�ects in materialplasti�cations [7].In the present paper, Aifantis’ gradient plasticity theory is used to describe matrix behaviour

and so coupled into the GTN damage constitutive model [9; 11]. Introducing gradients of

Copyright ? 2002 John Wiley & Sons, Ltd. Int. J. Numer. Meth. Engng 2002; 54:399–420

MICRO-MECHANICAL DAMAGE MODEL 401

plastic strain into the constitutive model leads to material deformations and failure relatedto the vicinity of the material points. In this sense, such a model is non-local. On the otherhand, due to implementation of the Laplacian of plastic strain in the micro-mechanical damagemodel, see Equation (16), the C1 continuous interpolation is unavoidable for a consistent �niteelement formulation. We use an implicit C1 continuous Hermitian interpolation proposed byPetera and Pittman [18] for plastic strain. The element which is used by Pamin [3] andMikkelsen [4] can be transformed from rectangular into arbitrary quadrilateral. Numericalexamples show that using this �nite element formulation the mesh-dependence of damagelocalization is removed. The material length scale may have the potential to predict sizee�ects in material failure.

2. REVIEW OF GRADIENT-DEPENDENT PLASTICITY

Plastic deformations arise from the accumulation of dislocations. From the study of dislocationmotions, it is clear that the stress and strain state of a material point is in�uenced by distortionsin its neighbourhood, that is, plastic deformations are generally non-local. As suggested byAifantis [1], the �ow stress depends on both plastic strain and its gradients as

�y( ��p;∇2 ��p)= ��( ��p)− g∇2 ��p (1)

where ��( ��p) denotes the yield stress measured in uniaxial tensile tests and g is a positivecoe�cient with the dimension of force. In this paper, g is assumed as g=�0l2 with �0 as theinitial yield stress and l as an intrinsic material length scale characterizing micro-structuresof the material. Then, the strain gradient is introduced into the constitutive equation and itfollows

�(�y( ��p;∇2 ��p))=�( ��( ��p))− g∇2 ��p (2)

where �( ��( ��p)) is conventional yield stress measure, e.g. von Mises yield condition, ��p is theequivalent plastic strain. The constitutive equation (2) in the gradient plasticity is a partialdi�erential equation, whereas in the classical plasticity the yield stress �y is uniquely deter-mined by the equivalent total plastic strain alone. During plastic �owing, the stress point mustremain on the yield surface, that is,

�(�y( ��p;∇2 ��p))=0 (3)

which can be re-written as

n� − hp ��p + g∇2 ��p = 0 (4)

with n= @�=@�, −hp= @�=@ ��p, g= @�=@∇2 ��p. � is Cauchy stress tensor. () denotes the ma-terial derivative of the corresponding �eld variable. Repeated index denotes summations over1–3. For g=0; the classical �ow theory of plasticity can be retrieved. Using von Misesyielding condition one can derive ��p = �, where � is the plasticity multiplier.Let V and V p denote the volume occupied by the body and the plastic part volume

of the body, respectively. Let S be the surface bounding the volume V and Sp be theso-called elastic–plastic boundary surface of V p. Following M�uhlhaus and Aifantis [19] as



well as de Borst and M�uhlhaus [2], the generalized variation can be expressed as

�(�; u; ��p; �u; ��)=∫V∇��u dV +

∫S

�t�u dS +∫V p�(�y( ��p;∇2 ��p))�� dV +

∫Sp

@ ��p

@n�� dS

(5)

The solution is obtained as soon as the generalized variation � vanishes,

�(�; u; ��p; �u; ��)=0 (6)

Neglecting body forces, it follows two basic weak form equations as

∫V�u∇� dV =0 (7)

∫V��(�y( ��p;∇2 ��p)) dV =0 (8)

with an additional boundary condition @ ��p=@n=0 on the plastic border Vp. The equationsabove build the fundamentals of the �nite element method for the gradient plasticity theory.Note that we take the total stress and total plastic strain in the integrand which is equivalentto the rate variational equation in References [2; 19] under in�nitesimal assumptions.

3. GRADIENT-DEPENDENT CONTINUUM DAMAGE MODEL

Based on analysis of a single cell containing a central spherical void, Gurson [9] proposed amicro-mechanical damage model, which was modi�ed by Tvergaard and Needleman [12], sothat the yield function of the micro-mechanical damage model (GTN model) can be written as

�(q; p; f; �y)=q2

�2y+ 2q1f cosh

(3q2p2�y

)− (1 + q21f2)=0 (9)

where the constants q1 and q2 were introduced by Tvergaard [12] to bring predictions of themodel into closer agreement with full numerical analysis of a periodic array of voids. p andq are the mean normal and deviatoric part of the average macroscopic Cauchy stress �. �y isyield stress of the matrix material, and f is the void volume fraction, i.e. the porosity. Theevolution equation for the void volume fraction is written as

f= fgrowth + fnucleation (10)

The void growth is described by

fgrowth = (1− f)�pkk (11)

where �pij is the plastic strain rate tensor. A strain-controlled nucleation law suggested by Chuand Needleman [20] is

fnucleation =A ��p (12)



where the parameter A is chosen so that the nucleation strain follows a normal distributionwith mean value �N and standard deviation SN. A can be expressed as:

A=fN

SN√2�exp

[−12

(��p − �NSN

)2](13)

where fN is the volume fraction of void nucleating particles. Based on the assumption of theplastic �ow normality, the macroscopic plastic strain increment is evaluated from

Up = � @�@� (14)

The equivalent plastic strain ��p of the matrix material is assumed to vary according to equiv-alent plastic work expression,

(1− f)�y ��p = �Up = �@�@� � (15)

The matrix material is assumed to satisfy von Mises yield condition. If f=0, this conditionbecomes the conventional von Mises yield condition.In the GTN model one only considers that the material failure process is modelled by nu-

cleation, growth and coalescence of the micro-voids. The conventional continuum constitutiverelation, which is originally suitable for macroscopic analysis, is assumed to be valid for thematrix material at the microscopic level. It is an obvious shortcoming in this model [21].According to recent knowledge, the matrix at the microscopic level may have signi�cantly

di�erent features from that in the macroscopic cases. Discussions on the intrinsic materiallength make it necessary to introduce a material length into the constitutive equation of thematrix. Strain gradients may signi�cantly change the matrix strength. From this backgroundwe postulate the matrix strength depending on the strain gradients. In the frame of gradientplasticity suggested by Aifantis [1], we re-write the yield function of the GTN model as

�(q; p; f; �y)=q2

�2y ( ��p;∇2 ��p)+ 2q1f cosh

(3q2p

2�y( ��p;∇2 ��p)

)− (1 + q21f2)=0 (16)

In the above equation the actual yield stress of the matrix, �2y ( ��p;∇2 ��p), is a function of gradi-

ents of plastic strains, represented by ∇2 ��p. Should material failure be accompanied with highplastic strain gradients, e.g. near a crack tip, the matrix will be strengthened locally to preventstrain localization. Such consideration is consistent with known experimental observations incomposite materials [7].

3.1. Governing equations of �nite element method

In gradient plasticity theory we have two governing equations, (7) and (8), which are validfor the damage model by substituting the new yield function (16). Due to the complicatedconstitutive relation between the plastic multiplier � and e�ective plastic strain ��p in Equation(16), we have to discretize Equation (15) as the third governing equation for the �nite elementformulation, ∫

V� ��p[(1− f)�y ��p − �@�@� �

]dV =0 (17)



The variational equations (7), (8) and (17) build the fundamentals of the �nite elementalgorithm for the non-local micro-mechanical damage model within the gradient plasticity.The basic unknowns in the equations are the displacement vector u, the equivalent plastic

strain ��p as well as the plastic multiplier �. The integral expressions will be converted intoalgebraic equations by using suitable interpolation functions. We take the following interpo-lations for the �eld variables:

u(x) = [N(x)]unode (18)

�(x) = [N1(x)]�internal (19)

��p(x) = [H(x)]�node (20)

where [N(x)] is the standard 8-nodal serendipity interpolation function for displacement,[H(x)] is the C1-continuous implicit Hermitian interpolation function for plastic strain, whichwill be discussed in detail separately. [N1(x)] is the interpolation function for the plasticmultiplier. As suggested by Pinsky [22] and Simo et al. [23], the �eld of �(x) is only to beL2() and discontinuous across the element boundaries. This means that the vector �internal isan internal degree of freedom vector for each individual element. The interpolation function[N1(x)] is de�ned as

[N1(x)]= [h1(xj); h2(xj); h3(xj); h4(xj)] (21)

where

hi(xj)=

{0 if i �= j1 if i= j

(22)

and xj=(x(�j; �j); y(�j; �j)) are Gauss points in spatial co-ordinates. In this paper, the ele-ment takes 2× 2 Gauss point integration. Equations (22) may be thought of as de�ning anorthogonal discontinuous element basis function which assumes a value of 0 or 1 over thequadrants of the biunit square domain of the isoparametric co-ordinates. So using Equation(22), the internal vector �internal denotes the value of � on the four Gauss points of eachelement.Considering �nite strain assumptions, equilibrium equation (7) at current con�guration can

be simply written as [24] ∫V��∇v dV =0 (23)

where v is the velocity tensor. To solve the non-linear di�erential integro-equation, one usesthe Newton method. Setting �= �0 + � dt, the equation above can be converted into∫

V��∇v dt dV = �f (24)



with the residuum force

�f=−∫V�0�∇v dV (25)

In the expression, �0 stands for total stress tensor of the last increment step in the currentcon�guration. � is the rate of Cauchy stress tensor and ∇v denotes the gradient rate ofdeformation. The objective stress rate at current con�guration, the Jaumann stress rate, �∗,which is more suitable to use in constitutive relations, is de�ned as

�∗= �+ �D+D�D− �∇v (26)

D is the rate of deformation tensor which is de�ned as

D= 12(∇v+ v∇) (27)

Then, Equation (24) becomes∫V[�∗�D− 1

2��(2DDT −∇v · v∇)] dV = �f (28)

Substituting Equations (18), (19), (20) into the three governing Equations (28), (8), (17), thethree equations can be re-written as

∫VBT�∗ + �

[(@N@x

)Tv∇− 2BTD

]dV = F� (29)

∫VNT1

(@�@� �+

@�@�y�y +

@�@ff)dV = F� (30)

∫VHT[(1− f)�y ��p − ��@�@�

]dV =0 (31)

In the equations above B is the strain–displacement relation matrix. The last parentheses in(29) are a part of geometric sti�ness matrix [24]. The residuum force F is expressed by

F� =−∫VBT�0 dV (32)

F� =−∫VNT1�

0 dV (33)

with �0 as residuum of the yield function � from the last iteration step.

3.2. Hermitian interpolation for equivalent plastic strain

Owing to the higher order di�erentiation of the equivalent plastic strain �� p, the conventional�nite element technique based on the C0 interpolation [25] becomes inapplicable. A robust



computational algorithm is essential for the validation and application of such a complexconstitutive model.Ramaswamy and Aravas [6] introduced the C0 element by using the Gauss theorem in

integration. Such a formulation assumes a vanishing normal derivative of the plastic strain atall boundaries, @�=@n=0. The C0 element formulation is attractive for general robust �niteelement computation. Such an algorithm is, however, only useful for the original gradientplasticity model by Aifantis [1] as in Equation (2). As soon as the gradient terms are non-linear in the constitutive equation, e.g. Equation (16), the C0 formulation is not applicable.Pamin [3] designed a series of elements for the gradient plasticity model. The most reliable

type is the element with the 8-nodal serendipity interpolation of displacement and 4-nodalHermitian interpolation of plastic strain with 2× 2 Gaussian point integration. Mikkelsen [4; 5]extended this element type to �nite strain computations and simulated necking of uniaxialtension tests of ductile metallic materials. In both formulations, C1 continuity is based onthe local co-ordinate system. Generally speaking, one needs the second order of Jacobian toextend the local C1 continuity to the global co-ordinate system. Without introducing the secondderivatives, the Hermitian shape function can only satisfy C1 continuity in a rectangular, oreven a square element. As a result the �nite element computation may fail to converge dueto strong distortion of the element shape.In the present work we are designing an implicit C1 continuous interpolation function for

complex gradient plasticity model. For this purpose, the method suggested by Petera andPittman [18] is adopted, which will be discussed in the following paragraphs.Let ��p(�; �)= ��p(x(�; �); y(�; �)), where (�; �) are local reference co-ordinates and (x; y) are

global co-ordinates. In a local co-ordinate system

��p = �H(^; W)T ·�‘ (34)

where �H(^; W) is the Hermitian shape function in local co-ordinates and

�‘=[: : : ; ��pI ; ��

p; �I ; ��

p; �I ; ��

p; ��I ; : : :]

T =[: : : ; �� pI ;

@�� pI@�;@�� pI@�;@2 �� pI@�@�

; : : :]T

(I =1; 2; 3; 4) (35)

denotes the unknown variables as shown in Figure 1. �‘ represents the vector of nodaldegrees of freedom for the plastic strain �eld in local co-ordinates. The derivatives of ��p canbe obtained by

��p; � = �HT; � ·�‘ (36)

��p; � = �HT; � ·�‘ (37)

��p; �� = �HT; �� ·�‘ (38)

��p; �� = �HT; �� ·�‘ (39)

��p; �� = �HT; �� ·�‘ (40)



Figure 1. The �nite element introduced in the present work: (a) in local co-ordinates; and (b) inglobal co-ordinates.

The gradient vector of ��p and the Laplacian of ��p are given by

∇��p(�; �) = �QT ·�‘ (41)

∇2 ��p(�; �) = �PT ·�‘ (42)

where �Q and �P are derivatives of shape function �HT in local co-ordinate system (�; �). Toobtain C1 continuity in the global co-ordinate system, all �eld variables must be transferredto the (x; y) system. We denote

�g = [: : : ; ��pI ; ��

p;xI ; ��

p;yI ; ��

p; ��I ; : : :]

T (I =1; 2; 3; 4) (43)

as the vector of nodal degrees of freedom in global co-ordinates. Note that second-ordermixed derivative is not transformed according to Petera and Pittman [18]. It turns out thatthe fourth degree of freedom, ��p; ��I , is not related to the global co-ordinate system, althoughthis degree of freedom is necessary to make the plastic strain �eld C1 continuous [18]. The�eld variables are formally expressed as

�‘=C ·�g (44)



with

C=

......

.... . . . . .

...... 1 0 0 0

...... 0 x; �I y; �I 0

...... 0 x; �I y; �I 0

...... 0 0 0 1

.........

.... . . . . .

...

(I =1; 2; 3; 4; dimension=16) (45)

After lengthy mathematical manipulations we obtain the interpolation formula in the globalsystem and the Laplacian of plastic strain as

��p(x; y) =HT ·�g (46)

∇��p(x; y) =QT ·�g (47)

∇2 ��p(x; y) =PT ·�g (48)

To avoid discontinuity in the derivatives of x and y with respect to � and �, care must betaken in designing the mesh topology. It means that the local co-ordinates of adjacent ele-ments must be parallel to one another [18]. For mathematical proof of such a formulationas well as details of such an interpolation the reader is referred to the work of Petera andPittman [18]. Our computational analysis in micro-indentation con�rms that such an interpo-lation gives numerically reliable results [26; 27].

3.3. Integration of the constitutive equations

In �nite element method the solution is achieved incrementally with the integration of thegoverning equations. To solve the non-linear governing equations we discuss an implicitmethod following the algorithm of Aravas [28]. In the time interval [tn; tn+1], the stress �n+1is at �rst calculated as

�n+1 =RT�nR+De · (U−Up)= �en+1 −De ·Up (49)

where �n is the known stress state of the previous step, U the known strain increment, Ris the rotation tensor and De the elasticity matrix. �en+1 =RT�nR + De ·U is the elastictrial stress. The plastic strain increment at time tn+1 is given by

Up =�(@�@�

)t=tn+1

=�(− @�3@p

I+@�@qn)t=tn+1

(50)



where nt=tn+1 = nn+1 = nen+1 =3Se=(2qe), Se is the deviatoric stress of �e. Furthermore, we

introduce the notation

�p =−�(@�@p

)t=tn+1

(51)

�q =�(@�@q

)t=tn+1

(52)

then the expression of the incremental plastic strain is given by

Up = 13�pI+�qn

en+1 (53)

On substituting Equation (50) into Equation (49), we �nd

�n+1 = �en+1 − K�pI − 2G�qnen+1 =−(pe + K�p)I+ (qe − 3G�q)nen+1 (54)

where K;G are the elastic bulk and shear moduli. De�ning

p=pe + K�p (55)

q= qe − 3G�q (56)

and on substituting Equations (55) and (56) into Equations (51) and (52), we �nd

�q =2�qe

�2y + 6G�(57)

�p =−3f��y

sinh[3(pe + K�p)

2�y

](58)

Since in the time step [tn; tn+1], �, ��p and qe are known, then Equation (57) is known.Equation (58) can be solved by using Newton iteration method. With known �q; �p andnen+1; �n+1 are determined by Equation (54).

3.4. The tangent sti�ness matrix

In Newton iteration method one must provide the tangent matrix of the non-linear algebraicequations to obtain the new incremental solution. In the �nite element method the sti�nessmatrix must be renewed after each iteration when we take the full Newton method [24]. FromEquation (49), we get

d�= @�@U dU+

@�@�d�+

@�@�y(hp d ��p − g∇2 d ��p) (59)



By substituting the equation above into (29)–(30) we re-write the governing equations of the�nite element method as

Kuu Ku� KuUK�u K�� K�UKUu KU� KUU

du

d�

d�

=

Rloadf�fU

(60)

where

Kuu =∫V

[BT[De]B− �

(2BTB−

(@N@x

)T @N@x

)]dV

Ku[ =∫VBTm�H dV

KuU =∫V[BTm�y(hpH − gP)] dV

K[u =−∫VNT1

(@�T

@� · @�@U +

@�@f

1− f1 +Up

@Up@U

)B dV

K[[ =−∫VNT1

(@�T

@� ·m� + @�@f1− f1 +Up

@Up@�

)N1 dV

K[U =−∫VNT1

(@�T

@� ·m�y +@�@f

1− f1 +Up

@Up@�y

)(hpH − gP) dV

KUu =∫VHTTuB dV

KU[ =∫VT�HTN1 dV

KUU =∫V[TUhp + (1− f)�y]HTH − gTUHTP dV

f[ =∫VNT1 �( ��; �Up;∇2 �Up) dV

fU =∫VHT[(1− f)�y�Up −�

(� · @�@�

)]dV

Rload =∫VBT� dV +

∫@vNTt dV

The expressions of the derivatives are summarized in Appendix A.Under �nite strain assumptions the Laplacian should be calculated in the current con�gu-

ration. As shown by Mikkelsen [4; 5], an exact evaluation of the Laplacian of plastic strainneeds the second derivative of displacements. This makes the C 0 interpolation for displacement



�eld ine�ective. In our study increment of ∇2 �Up, i.e. (∇2 ��p), is calculated incrementally incurrent con�guration. Since Laplacian is a scalar, we add all increments of ∇2 ��p and de�neit as the total Laplacian. It avoids the C1-continuous interpolation for displacement. Such anaccumulation which is accurate under the in�nitesimal rotation conditions, will slightly a�ectthe �nal gradient values. The numerical examples show that the choice is suitable.

3.5. Plastic loading=unloading conditions

In the process of plastic loading and unloading, the Kuhn–Tucker conditions

�¿0; �(��p;∇2 ��p)60; ��(��p;∇2 ��p)=0 (61)

must be ful�lled. Since the yield condition is enforced for an element globally, care shouldbe taken in the numerical handling.The plastic multiplier vector �internal denotes the value of � on the four Gauss points of

each element. It is an internal vector of an element. At the integration points, if �¡0, theGauss point is elastic and � is forced to zero. It follows that the second governing equationbecomes trivial. On the other hand, if �¿0, the point is judged plastic and the governingequation

∫V N

T1 �(��

p;∇2 ��p) dV =0 is satis�ed. The condition ��(��p;∇2 ��p)=0 is achieved atall plastic integration points.In the 8-nodal C1-continuous Hermitian interpolation, vanishing of the global residual vec-

tor,∫V H

T[(1 − f)�y �� p − �@�=@��] dV; follows that the integrand (1 − f)�y ��p − �@�=@��approaches zero at all plastic Gauss points in the element. Thus, the classical Kuhn–Tuckerconditions (61) are ful�lled. The integral formulation is equivalent to the discrete condi-tion. The discrete Kuhn–Tucker condition suggested by Ramaswamy and Aravas [6] can beavoided.

3.6. Boundary conditions

Introducing additional gradients into the governing equations, one needs to set more boundaryconditions to maintain uniqueness of the �nite element equations. Except for the conventionaldisplacement and load boundary conditions, we must formulate additional boundary conditionsfor the plastic strains.M�uhlhaus and Aifantis suggested to introduce

@��p

@n=0 (62)

as an additional boundary condition for all plastic boundaries. For C1 element, this conditionalone is not enough to avoid the singular sti�ness matrix. Pamin [3] added

@2 ��p

@n @m=0 (63)

as the additional boundary condition to suppress the system singularity. In the equation aboven and m denote the normal and tangent vector of the plastic boundary, respectively. Accordingto the analysis of Pamin [3] this condition assures the correct rank of the sti�ness matrix.



4. VERIFICATION OF THE C1 FINITE ELEMENT ALGORITHM

To examine the feasibility of the implicit Hermitian interpolation for the equivalent plasticstrains, we consider a tension-dominated specimen with a central circular hole (Figure 2), inwhich strain localization in the shear band takes place at the onset of strain softening. Theradius of the hole is R=0:1B. Furthermore, the dimension of the specimen is characterizedby H=B=2. Three di�erent meshes with 125, 500 and 825 elements, respectively, are used.The specimen is only loaded at the upper edge by a given uniform vertical displacement.The gradient-dependent von Mises yield condition without void growth are taken. Plane

strain conditions and in�nitesimal deformation assumptions are applied. Elasticity modulus isset to E=300�0. Poisson’s ratio is =0:3. The stress–strain relation is assumed to be bi-linearcharacterized by a negative tangent coe�cient hp, that is,

��=�0 + hp ��p

with hp=−0:9�0. The material contains strain-softening as soon as it acquires plasticity. Theyield stress is �y= �� − g∇2 ��p =�0 + hp ��p − �0l2∇2 ��p. The material length scale parameter isset to l=

√0:002B and

√0:004B, respectively.

The overall stress–strain diagram for the three �ne element meshes is plotted in Figure 3. Itshows, without gradient in�uence, that the specimen discretized with the �ner element meshloses strength more quickly than that with the coarser mesh. By setting the material lengthscale parameter di�ering from zero, we can see that the mesh-dependence is removed anddi�erent meshes give the same strength curve. The material strength is controlled by materialparameters, such as l, not a�ected by the element size.The principal strain contour distributions are shown in Figure 4. It is clear that the width of

the shear band is determined by l. At the centre of the shear band, where intensive shearingoccurs, the ∇��p becomes negative, thus the gradient term will arise from the �ow stress atthe centre of the shear band. The shear band width is uniquely correlated by l.

Figure 2. Finite element meshes for a tensile specimen with a centred hole. Owing to sym-metry only a quarter of the specimen is discretized. To study mesh-dependence in strain soft-ening, three meshes with 125, 500 and 825 elements, respectively, are used. The specimen is

loaded only at the upper edge.



Figure 3. Overall stress–strain curves for thecentre-holed specimen with three di�erent �-nite element meshes. The computations are con-ducted using gradient plasticity without damage.The material length parameter l=0,

√0:002B

and√0:004B, respectively.

Figure 4. E�ects of the element size on principalstrain distributions in the centre-holed specimen

(�I =0:1− 1:0) with l=√0:004B.

With the present example we con�rm that the numerical results using di�erent interpolationmethods coincide with the known analytical prediction by de Borst and co-workers [2; 3].The present �nite element algorithm is suitable to analyse material failure process.

5. SHEAR BAND ANALYSIS IN COMBINING WITH DAMAGE

Strain localizations are observed only when the material possesses strain softening, which canbe introduced either by the unstable stress–strain relation or caused by material damage, forinstance, void growth. In this section, we consider shear band evolution in ductile damageprocess using the non-local damage model introduced in the present paper.We consider a rectangular unit cell under plane strain loading conditions with an initial

length A0 and width B0. The unit cell represents a material with a doubly periodic array ofsoft spots, containing initial porosity, as has also been studied by Tvergaard and Needleman[14]. The soft spot is located at the bottom left corner of the cell and the area of the softspot is (0:1A0)× (0:1B0). Symmetric boundary conditions are applied on all edges,

ux =0 at x=0 (64)

uy =0 at y=0 (65)



Figure 5. Overall stress–strain curves in shear band analysis with the damage model. Three di�erent�nite element meshes are used with l=0 and 0:02B0, respectively. The results with l=0 correspond to

the GTN model.

uy =U1 at y=B0 (66)

ux =U2 at x=A0 (67)

where ux and uy are the displacement components in the x and y direction, respectively. U1,U2 are the given displacements. In this analysis, only U1 is prescribed. U2 is not given butretains the form of the boundary. The initial porosity distribution in the soft spot is speci�ed tobe 0.05 and strain-controlled nucleation is assumed in the whole domain except the soft spotwith fN =0:04, �N =0:3, SN =0:1 in Equation (17). Young’s modulus is E=300�0, Poisson’sratio =0:3, q1 = 1:5, q2 = 1. Finite strains are taken into account. The stress–plastic strainrelation is assumed to be a power law,

��0=(3G�0��p)N

(68)

where N =0:1; G is shear modulus. The yield stress is determined by

�y= �� − g∇2 ��p =�0

[(3G�0��p)N

− l2∇2 ��p]

Figure 5 shows the overall stress–strain curves with di�erent material lengths. For l=0 thecomputation corresponds to the GTN model. Results are shown for A0=B0 = 1:5 using threeuniform meshes consisting of 10× 10; 20× 20 and 40× 40 quadrilateral 8-nodal elements.It is obvious, without the strain gradient regulation, the post-localization response is verysensitive to the mesh resolution. The �ner are the elements, the lower the stress levels will



Figure 6. Mesh distortions in shear band analysis with the Gurson model. Three di�erent �nite elementmeshes with 10× 10; 20× 20 and 40× 40 elements are used: (a) without strain gradient regulator, l=0,overall strain �yy=B0=B=0:2; and (b) with strain gradient regulator, l=0:02B0 (where B0 stands for

the initial specimen height), overall strain �yy=B0=B=0:25.

be (Figure 5). Under gradient plasticity all three di�erent meshes show a numerically uniquesolution.Detailed information can be found from the mesh distortions, as shown in Figure 6. Without

the gradient regulator (l=0), the shear band develops within a single arrow of elements, thatis, the shear band width is as narrow as an element size. Strains localize. For a �ner mesh,one needs less energy and so less applied load to reach the given local damage state, whichis related to the material porosity. After introducing the gradient term into the constitutiveequation, the local strain state is a�ected by its local variations. Figure 6(b) shows that themesh distortions with l=0:02B0 are independent of element sizes. Shear band is uniquelydescribed by the material parameter l and the applied loading condition.Figure 7 displays distributions of the void volume fraction f and the equivalent plastic strain

across the shear band with l=0:04B0. Using the conventional GTN model, the variations ofthe void volume fraction and the equivalent plastic strain are restricted in a band as narrowas an element size. Within the frame of gradient plasticity the curves are characterized by theparameter l.The strength of the specimen increases with l, as plotted in Figure 8. By smoothing the

plastic strain distribution, the material becomes stronger, which slows down the developmentof micro-voids and so the damage zone.



Figure 7. Variations of the porosity and plastic strain from shear band analysis using the Gursonmodel with l=0:04B0. The overall mean strain �yy=B=B0 = 0:25: (a) void volume distribution

versus x at y=0:4B0; and (b) e�ective plastic strain distribution versus x at y=0:4B0.

Figure 8. E�ects of the intrinsic material length scale parameter l from the shearband analysis using the Gurson model.

6. FAILURE OF A CENTRE-HOLED TENSILE PANEL

We consider a rectangular specimen with a central hole (Figure 2) introduced in the lastsection using the present damage model. Due to symmetry only one-fourth of the specimen



Figure 9. Overall stress–strain curves with three�nite element meshes for the centre-holed spec-imen using the Gurson model, with l=0

and 0:02B0, respectively.

Figure 10. E�ects of the intrinsic mate-rial length scale parameter l on failureanalysis of the centre-holed specimen using

the Gurson model.

is to be discretized as shown in Figure 2. The specimen is subjected to tensile loading alongthe Y direction under plane strain conditions. The initial porosity is f0 = 0:05 and the voidnucleation is not considered. We adopt the same values of Young’s modulus E, Poisson’s ratio and the stress–strain relation as in the shear band analysis. To examine the mesh sensitivityat material failure, e�ects of rapid void coalescence is taken into account. The void volumefunction f is replaced by f∗, which is de�ned as [12]

f∗=

f if f6fc

fc +f∗u − fcff − fc (f − fc) if f¿fc

(69)

The onset of void coalescence is assumed to begin at a critical void volume fraction, fc,with f∗

u being the value of f∗ at zero stress, i.e. f∗u = 1=q1. As f→ff , f∗ →f∗

u and thematerial loses loading capacity. We assume q1 = 1:5, q2 = 1, ff = 0:388, fc = 0:15. To preventnumerical di�culties occurring after material failure, the �nal �ow stress is limited to 10per cent of the initial yield stress after void coalescence. It means that the material does notlose all loading capacity.The overall stress–strain diagrams using three di�erent �nite element meshes are plotted

in Figure 9. For the conventional GTN model with l=0 the critical loading point for voidgrowth is a�ected by the mesh size. For the �ner mesh the specimen reaches the criticalpoint earlier than that in the coarser mesh. The deviation is proportional to strain gradients.One may expect much stronger mesh-dependence in crack analysis. The mesh sensitivity isremoved by adding the gradient regulator, if the element size is smaller than that the material



length needs. As shown in Figure 9, for l=0:02B the mean stress-mean strain curves with500 and 825 elements are unique. The coarse mesh shows slight mesh-dependence due tovery coarse elements.Figure 10 shows the in�uence of the gradient parameter on the stress–strain curve using

the 500 elements mesh. With an increase in the value of the material length scale parameter,the material strength increases signi�cantly and the material failure is delayed. The in�uenceof gradient plasticity in Figure 10 represents some kinds of size e�ects in material failure,as observed in experiments [29]. For a given material, that is, for a given intrinsic materiallength l, the material strength varies with the specimen geometry: the smaller specimens havehigher strength than the larger ones.The present damage model based on the gradient plasticity provides a tendentious predic-

tion about size e�ects in material failure. To obtain a quantitative agreement, much detailedcomputations and experimental e�orts are needed.

7. CONCLUSIONS

The GTN damage model is introduced into the gradient plasticity. Deformations and so failureof a material point is related to its vicinity by an additional length parameter of the constitutivemodel. In this sense material failure is described non-locally. Furthermore, a �nite elementalgorithm for the damage model under �nite strain conditions has been carried out. Bothdisplacement vector, u, and equivalent plastic strain, ��p, are taken as the basic unknownsin the �nite element formulation, and the plastic multiplier � as an internal unknown ofan element. The algorithm has been implemented in the commercial �nite element programABAQUS, using the user–element interface, and veri�ed by comparing with known examples.Using the implicit Hermitian interpolation function, the rectangular Hermitian element can

be transformed into the arbitrary quadrilateral one. The results presented in this paper con�rmthat the algorithm is suitable for computing the strain-softening problem. Shear band analysisshows that the width of shear band is uniquely determined by the material length scaleparameter l. Should the element size be signi�cantly larger than the shear band width, theconsequent computational results are a�ected by the mesh.Computational analysis of ductile material failure of a tensile panel shows that the develop-

ment of material failure is characterized by the material length parameter. Prediction of spec-imen failure does not depend on arti�cial �nite element size. The fact that increase of thematerial strength will delay the computational material failure prediction is consistent withthe known size e�ects in ductile materials. The gradient plasticity has the potential to give amore reliable and more accurate prediction of material failure.

APPENDIX A: EXPRESSIONS IN STIFFNESS MATRIX AND RESIDUAL VECTOR

@�@U =D

e − KI@�p@U − 2Gne @�q

@U − 2G�q @ne

@U (A1)



m� =−K @�p@�

I − 2G@�q@�

ne (A2)

m�y =−K @�p@�y

I − 2G@�q@�y

ne (A3)

Tu =

[−Bu3IT@�@U + BfAu +

4q�2yneT@�@U

]�− �y��pAu (A4)

T� =−[−Bu3ITm� + BfA� +

4q�2yneTm�

]�−

(p@�@p+ q

@�@q

)− �y��pA� (A5)

T� = (1− f)��p − �y��pA�y −�[−Bu3ITm�y + B�y + BfA�y +

4q�2yneTm�y −

4q2

�3y

]

(A6)

Au =1− f1 +�p

@�p@U (A7)

A� =1− f1 +�p

@�p@�

(A8)

A�y =1− f1 +�p

@�p@�y

(A9)

Bu =3fq1q2�y

sinh(3q2p2�y

)+9pfq1q222�2y

cosh(3q2p2�y

)(A10)

B�y =−3pfq1q2�2y

sinh(3q2p2�y

)− 9p2fq1q22

2�3ycosh

(3q2p2�y

)(A11)

Bf =3q1q2�y

sinh(3q2p2�y

)(A12)

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a micro-mechanical damage model based on gradient plasticity: algorithms and applications

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