An Assessment of the Gurson

Assessment ofthe Gurson yield

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Engineering Computations:International Journal for

Computer-Aided Engineering andSoftware

Vol. 26 No. 3, 2009pp. 281-301

# Emerald Group Publishing Limited0264-4401

DOI 10.1108/02644400910943626

Received 11 September 2007Revised 14 April 2008

Accepted 14 April 2008

An assessment of the Gursonyield criterion by a computational

multi-scale approachS.M. Giusti and P.J. Blanco

LNCC, Laboratorio Nacional de Computacao Cientıfica, Petropolis, Brasil, and

E.A. de Souza Neto and R.A. FeijooCivil and Computational Engineering Centre, School of Engineering,

University of Wales Swansea, Swansea, UK

Abstract

Purpose – The purpose of this paper is to assess the Gurson yield criterion for porous ductile metals.Design/methodology/approach – A finite element procedure is used within a purely kinematicalmulti-scale constitutive modelling framework to determine estimates of extremal overall yieldsurfaces. The RVEs analysed consist of an elastic-perfectly plastic von Mises type matrix under planestrain conditions containing a single centered circular hole. Macroscopic yield surface estimates areobtained under three different RVE kinematical assumptions: linear boundary displacements (anupper bound); periodic boundary displacement fluctuations (corresponding to periodically perforatedmedia); and, minimum constraint or uniform boundary traction (a lower bound).Findings – The Gurson criterion predictions fall within the bounds obtained under relatively highvoid ratios – when the bounds lie farther apart. Under lower void ratios, when the bounds lie closetogether, the Gurson predictions of yield strength lie slightly above the computed upper bounds inregions of intermediate to high stress triaxiality. A modification to the original Gurson yield functionis proposed that can capture the computed estimates under the three RVE kinematical constraintsconsidered.Originality/value – Assesses the accuracy of the Gurson criterion by means of a fullycomputational multi-scale approach to constitutive modelling. Provides an alternative criterion forporous plastic media which encompasses the common microscopic kinematical constraints adopted inthis context.

Keywords Porous materials, Plasticity, Modelling, Finite element analysis

Paper type Research paper

1. IntroductionOver the past decade, the use of analytical and computational tools for the prediction ofthe constitutive behaviour of materials relying on information at two or more physicalscales has been the subject of increasing interest in academic circles. Particularly,interesting applications of multi-scale concepts include: first, the estimation of theeffective parameters of a pre-defined macroscopic continuum constitutive model aswell as the definition of new phenomenological macroscopic constitutive modelsthrough the analysis of a microscopic representative volume element (RVE) (Hill, 1963;Gurson, 1977; Michel and Suquet, 1992, G�aar�aajeu and Suquet, 1997; Michel et al., 1999,2001; Pellegrino et al., 1997, 1999) and second, the development and use of macroscopicconstitutive models whose generally dissipative behaviour is the result of thehomogenisation of the response of a RVE, without reference to any pre-defined set of

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P.J. Blanco and E.A. de Souza Neto were partly supported by CNPq (Brazilian Research Council)under the grants 140686/2005-3 and 381942/2004-1, respectively. S.M. Giusti was supported byCAPES (Brazilian Higher Education Staff Training Agency). This support is gratefullyacknowledged.

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constitutive equations at the macroscopic scale (Swan 1994; Miehe et al., 1999, 2002;Kouznetsova et al., 2002; Terada et al., 2003).

Within the category of applications classed as (a) in the above, the seminal paper byGurson (1977) deserves special mention. In his paper, Gurson has derived upper boundyield loci estimates for porous ductile metals by means of a semi-analytical methodbased on the study of the collapse of representative perforated rigid-perfectly plasticcells under an assumed mechanism. This work has been followed up by otherresearchers who proposed refinements to the original Gurson model (see, for example,Tvergaard (1981)). Further multi-scale based studies of porous plastic media have beenconducted, among others, by G�aar�aajeu and Suquet (1997) who proposed a generalisationof Gurson’s yield function to account for the presence of rigid inclusions within theporous microstructure, and by Michel et al. (2001) who assessed the influence of therandomness of void distribution on the macroscopic yield strength.

Our main purpose in the present paper is to further investigate Gurson’s planestrain criterion by assessing it against finite element predictions of extremal yieldsurfaces for porous plastic media. The numerical predictions are obtained here within apurely kinematical homogenisation-based multi-scale constitutive framework. SquareRVEs consisting of an elastic-perfectly plastic von Mises type matrix containing acircular void are considered in the computational homogenisation procedure. Thepaper is organised as follows. The kinematical multi-scale constitutive framework isreviewed in section 2. Within this framework, upper and lower bounds of thehomogenised constitutive behaviour correspond respectively to the choices of: first,linear RVE boundary displacements – the most constrained – and, second, minimumkinematical constraint – or uniform RVE boundary tractions. A further prediction canbe obtained under the kinematical assumption of periodicity of the RVE boundarydisplacement fluctuation – typically associated with the analysis of microstructurallyperiodic media. The finite element implementation of the resulting multi-scaleconstitutive models is described in section 3. Here, the linearised system of algebraicequations required by the Newton-Raphson scheme for the solution of the associatedincremental microscopic equilibrium problems is briefly described. The kinematicalconstraints of the RVE are enforced directly upon the finite element-generated spacesof displacement fluctuations and virtual displacements. The main contribution ispresented in section 4. Predictions of extremal yield surfaces are determined within thecomputational homogenisation framework. The original Gurson upper bound forporous plastic media under plane strain is assessed against the numerical results. Amodification of the original Gurson yield function is then suggested which is able tocapture the yield loci of porous metals under the three kinematical assumptionsconsidered. The paper ends in section 5 where some concluding remarks are presented.

2. Homogenisation-based multi-scale constitutive theoryThe starting point of the kinematically-based family of multi-scale constitutive theoriesupon which the present paper relies is the assumption that any material point x of the(macroscopic) continuum is associated to a local RVE whose domain, �� (refer toFigure 1), has a characteristic length, l�, much smaller than the characteristic length, l,of the macro-continuum. At any instant t, the strain tensor at an arbitrary point x of themacro-continuum is assumed to be the volume average of the microscopic strain tensorfield, "�, defined over ��:

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"ðtÞ ¼ 1

V�

ð��

"�ðy; tÞ dV ; ð1Þ

where V� is the volume of the RVE and

"� ¼ rsu�; ð2Þ

where rsu� denotes the symmetric gradient of the microscopic displacement field u�of the RVE.

2.1 Kinematically admissible RVE displacement fieldsBy replacing (2) into (1) and making use of Green’s theorem, it can easily be establishedthat the averaging relation (1) is equivalent to the following constraint on thedisplacement field of the RVE (de Souza Neto and Feijoo, 2006):ð

@��

u� �s n dA ¼ V� "; ð3Þ

where n denotes the outward unit normal field on @�� and

a�s b � 1

2ða� bþ b� aÞ ð4Þ

for any vectors a and b. This constraint requires the (as yet not defined) set � ofkinematically admissible RVE displacement fields to be a subset of the minimallyconstrained set of kinematically admissible microscopic displacements, �

� :

� � �� �

v; sufficiently regularÐ@��

v�s n dA ¼ V� "

( ); ð5Þ

with sufficiently regular meaning that the relevant functions have the sufficient degreeof regularity so that all operations in which they are involved make sense. By splitting

Figure 1.Macro-continuum with a

locally attached micro-structure

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u� into a sum

u�ðy; tÞ ¼ "ðtÞyþ ~uu�ðy; tÞ; ð6Þ

of a homogeneous strain displacement, "ðtÞy, and a displacement fluctuation field, ~uu�,the above constraint is made equivalent to requiring that the space ~� of kinematicallyadmissible displacement fluctuations of the RVE be a subspace of the minimallyconstrained space of kinematically admissible displacement fluctuations, ~ �

�:

~� � ~ �� �

v; sufficiently regularÐ@��

v�s n dA ¼ 0

( ): ð7Þ

Further, it can be trivially established (refer to de Souza Neto and Feijoo, 2006 fordetails) that the (yet to be defined) space ~� coincides with the space of virtualdisplacements of the RVE.

Following the split (6) the microscopic strain (2) can be expressed as the sum

"�ðy; tÞ ¼ " ðtÞ þ rs ~uu�ðy; tÞ; ð8Þ

of a homogeneous strain field (coinciding with the macroscopic, average strain) and afieldrs ~uu� that represents a fluctuation about the average.

2.2 Macroscopic stress, Hill-Mandel principle and RVE equilibriumSimilarly to the macroscopic strain definition (2), the macroscopic stress tensor, �, isdefined as the volume average of the microscopic stress field, ��, over the RVE:

�ðtÞ � 1

V�

ð��

��ðy; tÞ dV : ð9Þ

Another crucial concept underlying models of the present type is the Hill-Mandel principle of macro-homogeneity (Hill, 1965; Mandel, 1971), which requiresthe macroscopic stress power to equal the volume average of the microscopic stresspower for any kinematically admissible motion of the RVE. This is expressed by theequation

� : _"" ¼ 1

V�

ð��

�� : _""� dV ; ð10Þ

that must hold for any kinematically admissible microscopic strain rate field, _""�. Theabove is equivalent to the following variational equation:ð

@��

t � �; dA ¼ 0 ;

ð��

b � � dV ¼ 0 8 � 2 ~�; ð11Þ

in terms of the RVE boundary traction and body force fields denoted, respectively, t andb. That is, the virtual work of the RVE body force and surface traction fields vanish –they are the reaction forces associated to the imposed kinematical constraintsembedded in the choice of ~�.

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With the above at hand, the variational equilibrium statement – the virtual workequation – for the RVE is given byð

��

�� : rs� dV ¼ 0 8� 2 ~�: ð12Þ

Further, we assume that at any time t the stress at each point y of the RVE is deliveredby a generic constitutive functional y of the strain history "t

�ðyÞ at that point up totime t:

��ðy; tÞ ¼ yð"t�ðyÞÞ: ð13Þ

This constitutive assumption, together with the equilibrium Equation (12) leads to thedefinition of the RVE equilibrium problem which consists in finding, for a givenmacroscopic strain " (a function of time), a displacement fluctuation function ~uu� 2 ~�such that ð

�s�

yf½"ðtÞ þ rs ~uu�ðy; tÞ�tg : rs� dV ¼ 0 8 � 2 ~�: ð14Þ

2.3 Characterisation of the multi-scale constitutive modelThe general multi-scale constitutive model in the present context is defined as follows.For a given macroscopic strain history, we must firstly solve the RVE equilibriumproblem defined by (14). With the solution ~uu� at hand, the macroscopic stress tensor isdetermined according to the averaging relation (9), i.e. we have

�ðtÞ ¼ ð"tÞ � 1

V�

ð��

yf½"þrs ~uu��tg dV ; ð15Þ

where denotes the resulting (homogenised) macroscopic constitutive functional.2.3.1 The choice of kinematical constraints. The characterisation of a multi-scale

model of the present type is completed with the choice of a suitable space ofkinematically admissible displacement fluctuations, ~� � ~�

�. In general, differentchoices lead to different macroscopic response functionals. The following choices willbe considered in the assessment of the Gurson model addressed in section 4:

(i) Linear boundary displacements (or zero boundary fluctuations) model:

~� ¼ ~lin � fv; sufficiently regular jvðyÞ ¼ 0 8y 2 @��g: ð16Þ

The displacements of the boundary of the RVE for this class of models are fullyprescribed as

u�ðyÞ ¼ " y 8y 2 @��: ð17Þ

(ii) Periodic boundary fluctuations. This assumption is typically associated with thedescription of media with periodic microstructure. The macrostructure in this case isgenerated by the periodic repetition of the RVE (Michel et al., 1999). For simplicity, wewill focus the description on two-dimensional problems and we shall follow the

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notation adopted by Miehe et al. (1999). Consider, for example, the square or hexagonalRVEs, as illustrated in Figure 2. In this case, each pair i of sides consists of equallysized subsets

�þi and ��i

of @��, with respective unit normals

nþi and n�i ;

such that

n�i ¼ �nþi :

A one-to-one correspondence exists between the points of �iþ and �i

�. That is, eachpoint yþ 2 �i

þ has a corresponding pair y� 2 �i�.

The key kinematical constraint for this class of models is that the displacementfluctuation must be periodic on the boundary of the RVE. That is, for each pair {yþ,y�} of boundary material points we have

~uu�ðyþ; tÞ ¼ ~uu�ðy�; tÞ: ð19Þ

Accordingly, the space ~� is defined as

~� ¼ ~per � f~uu�; suff. reg. j ~uu�ðyþ; tÞ ¼ ~uu�ðy�; tÞ 8 pairs fyþ;y�gg: ð20Þ

(iii) The minimally constrained (or uniform boundary traction) model:

~� � ~��: ð21Þ

It can be shown (de Souza Neto and Feijoo, 2006) that the distribution of stress vectoron the RVE boundary, reactive to the minimal kinematic constraint, satisfies

��ðy; tÞnðyÞ ¼ �ðtÞnðyÞ 8 y 2 @��: ð22Þ

As for the linear boundary displacements assumption, there are no restrictions on thegeometry of the RVE in the present case.

Figure 2.RVE geometries forperiodic media. Squareand hexagonal cells

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3. Finite-element approximationThis section provides a brief description of the computational implementation of multi-scale constitutive theories of the above type within a non-linear finite elementframework. At the outset, we shall assume the constitutive behaviour at the RVE levelto be described by conventional internal variable-based dissipative constitutivetheories, whereby the stress tensor is obtained by integrating a set of ordinarydifferential equations in time (or pseudo-time) for the given strain tensor history.Elasto-plasticity and visco-plasticity are classical, widely used examples of suchspecialisations of (13). In these cases, numerical approximations to the initial valueproblem defined by the constitutive equations of the model are usually obtained byEuler-type difference schemes. For a typical time (or pseudo-time) interval [tn, tnþ1],with known set �n of internal variables at tn, the stress �nþ1

� at tnþ1 is a (generallyimplicit) function of the (prescribed) strain "nþ1

� at tnþ1 (refer, for instance, to Simo andHughes (1998) or de Souza Neto et al. (2008) for a detailed account of procedures of thiskind in the context of plasticity and visco-plasticity). This can be symbolicallyrepresented as

�nþ1� ¼ ��yð"nþ1

� ;�nÞ; ð23Þ

where ��y denotes the integration algorithm-related implicit incremental constitutivefunction at the point of interest, y.

The above leads to the definition of an incremental version of the homogenisedconstitutive function defined in (15), obtained by replacing y with its time-discretecounterpart ��y:

�nþ1 ¼ ��ð"nþ1; ���nÞ � 1

V�

ð��

��yð"nþ1 þrs ~uunþ1� ;�nÞ dV ; ð24Þ

where ���n denotes the field of internal variable sets over �� at time t n and ~uunþ1� is the

displacement fluctuation field of the RVE at t nþ1 – the solution to the time-discreteversion of equilibrium problem (14):

ð�s�

��yð"nþ1 þrs ~uunþ1� ;�nÞ : rs� dV ¼ 0 8 � 2 ~�: ð25Þ

3.1 Finite-element discretisation and solutionWe now focus on the finite element solution of the time-discrete equilibrium problem(25) – a crucial step in the definition of the approximate homogenised constitutivefunctional. Following a standard notation, the finite element approximation to problem(25) for a given discretisation h consists in determining the unknown vector~uunþ1� 2 ~ h

� of global nodal displacement fluctuations such that

hð~uunþ1� Þ �

ð�h�

BT ��yð" nþ1 þB ~uunþ1� Þ dV

( )� � ¼ 0 8 � 2 ~ h

�; ð26Þ

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where ��h denotes the discretised RVE domain, B the global strain-displacement

matrix (or discrete symmetric gradient operator), "nþ1 is the fixed (given) array ofmacroscopic engineering strains at tnþ1; ��y (with upright �) is the functional thatdelivers the finite element array of stress components, � denotes global vectors of nodalvirtual displacements of the RVE and ~h

� is the finite-dimensional space of virtualnodal displacement vectors associated with the finite element discretisation h of thedomain ��.

The solution to the (generally non-linear) problem (26) can be efficiently undertakenby the Newton-Raphson iterative scheme, whose typical iteration (k) consists in solvingthe linearised form,

Fðk�1Þ þKðk�1Þ �~uuðkÞ�

h i� � ¼ 0 8 � 2 ~ h

�; ð27Þ

for the unknown iterative nodal displacement fluctuations vector, �~uuðkÞ� 2 ~ h

� where

Fðk�1Þ �ð

�h�

BT ��yð"nþ1 þB ~uuðk�1Þ� Þ dV ; ð28Þ

and

Kðk�1Þ �ð

�h�

BT Dðk�1ÞB dV ð29Þ

is the tangent stiffness matrix of the RVE with

Dðk�1Þ � d��y

d"

����"¼"nþ1þB~uu

ðk�1Þ�

ð30Þ

denoting the consistent constitutive tangent matrix field over the RVE domain. Inthe above the bracketed superscript denotes the Newton iteration number and thetime station superscript n þ 1 has been dropped whenever convenient for ease ofnotational. With the solution �~uu

ðkÞ� at hand, the new guess ~uu

ðkÞ� for the

displacement fluctuation at tnþ1 is obtained according to the Newton-Raphsonupdate formula

~uuðkÞ� ¼ ~uuðk�1Þ� þ �~uuðkÞ� : ð31Þ

Under the assumption of linear boundary displacements, the solution of problem (27)follows the conventional route of general linear solid mechanics problems – here withthe degrees of freedom (fluctuations) of the boundary fully prescribed as zero. Hence,the finite element implementation of this class of multi-scale models requires nofurther consideration. For the periodic boundary condition and minimallyconstrained models, however, the kinematic boundary conditions of the RVE are non-conventional. The main difference lies in the finite element-generated finitedimensional spaces of admissible fluctuations and virtual displacements whoseconstraints, here, are not simply described in terms of either fully constrained orcompletely free nodal degrees of freedom. For the sake of completeness, this issue isaddressed in the following.

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3.1.1 Periodic boundary fluctuations model. For the periodic boundary displacementfluctuations model, the RVE geometry must comply with the constraints set out in item(ii) listed in section 2.3.1. In this case, it is convenient to assume[1], further, that eachboundary node iþ, with coordinates yi

þ, has a pair i�, with coordinates yi�, as

schematically illustrated in Figure 3. Under this assumption, the space ~ h� of

discretised kinematically admissible nodal displacement fluctuation vectors (withperiodicity on the boundary) can then be defined as

~ h� ¼ v ¼

vi

vþv�

24

35jvþ ¼ v�

8<:

9=;; ð32Þ

where vi;vþ andv� denote the vectors containing, respectively, the degrees of freedom ofthe RVE interior and the portions �þ and �� of the RVE boundary. Here we adopt thedirect approach suggested by Michel et al. (1999) whereby the periodicity constraint isenforced exactly in the discretised space of fluctuations and virtual displacements. Thisis at variance with the treatment adopted by Miehe and Koch (2002) who used aLagrange multiplier method to enforce the discrete space constraint.

By splitting F, K, �~uu� and � in the same fashion as v in the above and taking intoaccount definition (32) as well as the fact that both � and �~uu

ðkÞ� belong to space ~KKh

�, thelinearised Equation (27) takes the form

Fi

Fþ

F�

264

375ðk�1Þ

þkii kiþ ki�

kþi kþþ kþ�

k�i k�þ k��

264

375ðk�1Þ

�~uui

�~uuþ

�~uuþ

264

375ðkÞ8><

>:9>=>; �

�i

�þ

�þ

264

375

¼ 0 8 �i; �þ: ð33Þ

Straightfoward manipulations, considering the repetition of �~uuþ and �þ in the relevantvectors of nodal degrees of freedom, reduce the linearised discrete equilibriumEquation (33) to the following form

Fi

Fþ þ F�

� �ðk�1Þþ

kii kiþ þ ki�

kþi þ k�i kþþ þ kþ� þ k�þ þ k��

� �ðk�1Þ �~uui

�~uuþ

� �ðkÞ( )

��i

�þ

� �¼ 0 8 �i; �þ; ð34Þ

Figure 3.Discretised RVEs for

periodic media

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which, finally, in view of the arbitrariness of �i and �þ, yields the linear system of

algebraic equations for the unknown vectors �~uuðkÞi and �~uu

ðkÞþ ,

kii kiþ þ ki�kþi þ k�i kþþ þ kþ� þ k�þ þ k�

� �ðk�1Þ�~uui

�~uuþ

� �ðkÞ¼ � Fi

Fþ þ F�

� �ðk�1Þ: ð35Þ

3.1.2 Minimally constrained model. A procedure completely analogous to the oneabove is followed to obtain the final Newton-Raphson set of algebraic finite elementequations under the assumption of minimally constrained kinematics (or uniform RVEboundary traction). We then start by defining the discrete counterpart of the minimallyconstrained space of fluctuations and virtual displacements (7, 21):

~ h� � v ¼ vi

vb

� �����ð@�h

�

Nbvb �s n dA ¼ 0

( ); ð36Þ

where vb is the vector containing the boundary degrees of freedom and Nb is theglobal interpolation matrix associated solely with the boundary nodes of thediscretised RVE.

It can be easily established that the integral constraint on vb can be equivalentlywritten in matrix form as

C vb ¼ 0; ð37Þ

where C is the constraint matrix. For a RVE mesh with k interior nodes and mboundary nodes, in the two-dimensional case vb is a vector of dimension 2 m and C isthe 3 2 m matrix given by

C ¼

Ð@�h

�Nkþ1 n1 dA 0 � � �

0Ð@�h

�Nkþ1 n2 dA � � �Ð

@�h�

Nkþ1 n2 dAÐ@�h

�Nkþ1 n1 dA � � �

Ð@�h

�Nkþm n1 dA 0

0Ð@�h

�Nkþm n2 dAÐ

@�h�

Nkþm n2 dAÐ@�h

�Nkþm n1 dA

2664

3775;ð38Þ

where n1 and n2 denote the components of the outward unit normal field along the globalorthonormal basis fe1; e2g and Nj; j ¼ 1; . . . ;m, are the global shape functionsassociated with the boundary nodes. In this case, equation (37) poses three linearconstraints upon the total number of 2m boundary degrees of freedom of the discrete RVE.For three-dimensional RVEs, vb is of dimension 3m and matrix C has dimension 6 3m.

In practice, rather than using global shape functions, matrix C is obtained byassembling elemental matrices which in two dimensions, for an element e with p nodeson the intersection �ðeÞ between the boundary of the element and the boundary of theRVE, read

CðeÞ ¼

Ð�ðeÞ N

ðeÞ1 n1 dA 0 � � �0

Ð�ðeÞ N

ðeÞ1 n2 dA � � �Ð

�ðeÞ NðeÞ1 n2 dA

Ð�ðeÞ N

ðeÞ1 n1 dA � � �

Ð�ðeÞ N

ðeÞp n1 dA 0

0Ð�ðeÞ N

ðeÞp n2 dAÐ

�ðeÞ NðeÞp n2 dA

Ð�ðeÞ N

ðeÞp n1 dA

2664

3775;ð39Þ

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where we have assumed that the nodes of element e lying on �ðeÞ are locally numbered1 to p and N

ðeÞj ; j ¼ 1; � � � ; p, are the associated local shape functions. For example,

a conventional eight-noded bilinear quadrilateral element (of the type employed insection 4), having a single straight side of length lðeÞ with n ¼ e1 and three equallyspaced nodes intersecting the RVE boundary, has

CðeÞ ¼ lðeÞ

1

60

2

30

1

60

0 0 0 0 0 0

01

60

2

30

1

6

26664

37775: ð40Þ

In order to handle constraint (37) upon the discrete space of fluctuations and virtualdisplacements it is convenient to split vb as

vb ¼vf

vd

vp

24

35; ð41Þ

where the subscripts f, d and p stand, respectively, for free, dependent and prescribeddegrees of freedom on the boundary of the discrete RVE. Accordingly, the globalconstraint matrix is partitioned as

C ¼ Cf Cd Cp

� �; ð42Þ

so that the constraint Equation (37) reads

Cf Cd Cp

� � vf

vd

vp

24

35 ¼ 0: ð43Þ

Prescribed degrees of freedom are needed here in order to remove rigid bodydisplacements of the RVE and make the corresponding discrete equilibrium problem(26) well-posed. Trivially, we then prescribe

vp ¼ 0; ð44Þ

where, in two and three dimensions, vp contains, respectively, three and six suitablychosen degrees of freedom. The constraint equation is now reduced to

Cf Cd

� � vf

vd

� �¼ 0: ð45Þ

In two dimensions, the above represents three scalar equations involving 2m–3variables, whereas in the three-dimensional case, we have six scalar equations and3m – 6 variables. Hence, the number of dependent variables – the dimension of vd andof the square sub-matrix Cd – is 3 and 6 for the two- and three-dimensional cases,respectively. The total number of free variables – the dimension of vf and number ofcolumns of Cf – is 2m – 6 and 3m – 12, respectively in two and three dimensions.

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Finally, following a trivial manipulation of (45), vd can be expressed in terms of vf as

vd ¼ R vf ; ð46Þ

where

R � �C�1d Cf : ð47Þ

Note that the dependent degrees of freedom (corresponding to vd) must be chosen suchthat Cd is invertible.

With the above considerations at hand, we can re-define the discrete space (36) offluctuations and virtual displacements of the RVE as

~ h� � v ¼

vi

vf

vd

24

35jvd ¼ R vf

8<:

9=;; ð48Þ

which, for convenience, contains now only the non-prescribed degrees of freedom.The particularisation of the linearised finite element Equation (27) for the present

case is obtained, analogously to (33), by splitting the corresponding vectors andtangential stiffness matrix according to the above partitioning and taking (48) intoaccount. This gives

Fi

Ff

Fd

24

35ðk�1Þ

þkii kif kid

kf i kff kfd

kdi kdf kdd

24

35ðk�1Þ

�~uui

�~uuf

R �~uuf

24

35ðkÞ

8><>:

9>=>; �

�i

�f

R �f

24

35 ¼ 0 8 �i; �f ; ð49Þ

which, after straightforward matrix manipulations taking into account thearbitrariness of �i and �f , is reduced to the final form

kii kif þkidR

kfiþRTkdi kff þkfdRþRTkdf þRTkddR

� �ðk�1Þ �~uui

�~uuf

� �ðkÞ¼�

Fi

Ff þRTFd

� �ðk�1Þ:

ð50Þ

4. Assessment of the Gurson porous plasticity modelIn his landmark paper, Gurson (1977) has proposed macroscopic yield surfaces forporous ductile metals by means of a semi-analytical approach relying on the use of theupper bound theorem of plasticity and the analysis of representative cells of materialcontaining a pore embedded in a von Mises type rigid-perfectly plastic matrix.In Gurson’s procedure, a sufficient number of points on the corresponding macroscopicyield surface in p–q space[2] are firstly obtained by computing numerically the overallstress required to cause plastic collapse of the cell under an assumed mechanism. Themacroscopic (upper bound) yield surface is then approximated by curve fitting of theresulting points. Under plane strain, the Gurson yield surface has been derived for acylindrical cell with a single centered cylindrical void and is expressed by means of the

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293

yield function

� ¼ q� 1

Ceq1þ f 2 � 2f cosh

ffiffiffi3p

p

�Y

!" #( )1=2

�Y ; ð51Þ

where

Ceq ¼ ð1þ 3f þ 24f 6Þ2; ð52Þ

with f denoting the void volume fraction of the cell and �Y the uniaxial yield stress ofthe matrix material.

In what follows we shall derive alternative predictions of yield functions for porousmetals following a methodology similar to that of Gurson. The essential differencebetween Gurson’s approach and present procedure is that, here, the yield surface pointswill be computed by means of finite element simulations of the plastic collapse of theunderlying RVE within the multi-scale constitutive framework described in sections2 and 3. That is, each computed point of the macroscopic yield surface point representsthe homogenised stress state at which plastic collapse of the RVE occurs under aprescribed monotonically increasing proportional macroscopic strain loading and a

particular choice of space ~ h�. This approach has been employed originally by

Pellegrino et al. (1997, 1999) in the estimation of yield surfaces for periodic elasto-plastic composites where, accordingly, the periodic boundary fluctuations constraint isenforced. In the present study, we shall focus the application on square RVEs underplane strain containing a single centered circular void within an elastic-perfectlyplastic von Mises type matrix. In addition to macroscopic yield surfaces predictedunder the periodicity constraint, which correspond to periodically perforated media,we shall obtain upper and lower bound surfaces by adopting, respectively, the linearboundary displacements and minimum kinematical constraint assumptions.

4.1 Computational homogenisation-based methodologyIn order to predict macroscopic yield surfaces as functions of the void ratio of theporous metal, the procedure outlined in the above and further described below will becarried out for RVEs with f ¼ 0.5, 2, 10 and 30 per cent – the same void ratios used inGurson’s calculations. The corresponding finite element meshes adopted – allconsisting exclusively of isoparametric 8-noded bi-quadratic quadrilaterals with(reduced) 2 2-point Gaussian integration quadrature – are illustrated in Figure 4.The material properties chosen for the von Mises matrix are the following: Young’smodulus E ¼ 200 GPa, Poisson’s ratio � ¼ 0:3 and yield stress �Y ¼ 0:24 GPa.

The loading programme imposed on the RVEs consists in prescribing amacroscopic strain path

"ð�Þ ¼ � �"" ð53Þ

parametrised by a monotonically increasing load factor �, where �"" is the unit planestrain tensor defined as

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�"" ¼ �

1ffiffiffi2p 0

01ffiffiffi2p

2664

3775þ ffiffiffiffiffiffiffiffiffiffiffiffiffi

1� �2p 0

1ffiffiffi2p

1ffiffiffi2p 0

2664

3775; ð54Þ

satisfying

k�""k ¼ 1; ð55Þ

and � 2 ½0; 1� is a prescribed parameter defining the direction of " in strain space. Notethat � ¼ 0 corresponds to a pure shear direction whereas � ¼ 1 is the in-planespherical direction. Any other value of �within this range is a combination of both.

For each chosen value of f, finite element simulations are carried out for a number ofvalues of � (or directions in strain space) covering the range [0, 1]. For each value of �,the load factor � is increased monotonically, starting from an unstressed plasticallyvirgin state of the RVE with � ¼ 0, until plastic collapse of the RVE occurs. Thecorresponding macroscopic collapse stress is computed according to Equation (9) asthe volume average of the finite element-predicted microscopic stress field �� at thecollapse state of the RVE. Within the present strain-driven framework, plastic collapseof the RVE is deemed to have occurred when no changes in macroscopic stress areobtained by increasing the load factor. The hydrostatic and von Mises components ofthe macroscopic collapse stress define a yield surface point. Finally, an estimate of the

Figure 4.RVE geometries and finiteelement meshes

Assessment ofthe Gurson yield

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295

functional format of the p-q space yield surface is obtained by curve fitting of the yieldsurface points obtained by this procedure.

4.2 Estimated yield surfacesThe macroscopic yield surface points obtained according to the above methodology areplotted in Figure 5 in the normalised �pp-�qq space, where

�pp � p

�Y

and �qq � q

�Y

: ð56Þ

The results are shown for the four values of void ratio considered. To describe thecorresponding approximate yield surfaces which fit the computed points we proposethe following alternative to the original plane strain Gurson yield function (51):

��� ¼ q� C1 1� C2 sin2 p

2Pm

� � �1

Ceq1þ f 2 � 2fC3 cosh

ffiffiffi3p

C4p

�Y

!" #( )1=2

�Y ;

ð57Þ

where Pm is the value of p at failure obtained with � ¼ 1 and Ci , i ¼ 1; . . . ; 4, are non-dimensional parameters determined so as to provide a best fit to the yield surface point data.The resulting approximate yield surfaces in �pp-�qq space are also shown in Figure 5. For ease ofcomparison, the results for the three RVE kinematical constraints considered are re-grouped

Figure 5.Estimated yield surfaces

for f ¼ 0.5, 2, 10 and30 per cent. Computed

points and approximatefunctional form. (a) Linearboundary displacements;

(b) Periodic boundaryfluctuations; and (c)

Minimum kinematicalconstraint (uniformboundary tractions)

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in Figures 6(a) and (b) for f ¼ 0.5 and 10 per cent, respectively. As expected, the linearboundary displacements condition (the most constraining of the three kinematicalassumptions considered) provides an upper bound for the yield locus of the porous metal.The minimum kinematical constraint, in turn, provides a lower bound. The results under theperiodicity assumption lie between the other two in very close proximity to the lower bound.

In Figure 7 the upper and lower bounds are plotted along the original plane strainGurson yield surface. The approximate surface obtained under the periodicityassumption is omitted due to its proximity to the lower bound. It should be noted thatthe present upper and lower bounds lie farther apart for higher void ratios. For higher

Figure 7.Gurson’s criterion andcomputed lower andupper bounds

Figure 6.Estimated yield surfacesunder different RVEkinematical constraints

Assessment ofthe Gurson yield

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297

void ratios, the Gurson surface lies between the present upper and lower bounds –closer to the upper bound for states near pure hydrostatic stress and closer to the lowerbound for states close to pure shear. For lower void ratios the Gurson surface predicts,for some combinations of hydrostatic and von Mises equivalent stress, a yield limitabove the present upper bound. Under such conditions, the Gurson criterion isexpected to overestimate the yield strength of the porous metal.

4.3 General functional formsThe values of the Ci parameters of the proposed yield function (57), obtainedfor the estimated yield surfaces presented in the above, are given in Table I. In orderto obtain general functional forms of yield function for each type of assumedkinematical constraint under a range of void ratios, we carry out one last curvefitting exercise based on the data of Table I to express the coefficients Ci as functionsof f. Upon close inspection of the data, we find that each coefficient varies as afunction of f following trends that do not depend on the assumed kinematicalconstraint. The functional forms found to be particularly suitable to express C1, C2, C3

and C4 as functions of f are, respectively, as a fourth order polynomial, a logarithmicfunction, an exponential function and a fifth order polynomial. Under the assumptionof linear RVE boundary displacements, for which an upper bound for the homogenisedyield surface is obtained, the following functional form for the coefficients Ci isdetermined:

C1ð f Þ ¼ �435:6f 4 þ 277:9f 3 � 66:4f 2 þ 10:2f þ 0:99

C2ð f Þ ¼ 0:01 lnð0:015f Þ þ 0:1

C3ð f Þ ¼ 6:7 expð�15f Þ þ 0:56

C4ð f Þ ¼ �2604f 5 þ 2524f 4 � 921:2f 3 þ 153f 2 � 9f þ 0:6761:

ð58Þ

Under the assumption of periodicity of displacement fluctuations, we have:

C1ð f Þ ¼ �256:2f 4 þ 183:3f 3 � 48:5f 2 þ 6:12f þ 0:95

C2ð f Þ ¼ 0:08 lnð0:01f Þ þ 0:81

C3ð f Þ ¼ 7:9 expð�16:5f Þ þ 0:76

C4ð f Þ ¼ �3232f 5 þ 3095:6f 4 � 1111f 3 þ 180f 2 � 10:3f þ 0:677;

ð59Þ

Table I.Parameters C1, . . . ;C4

for alternative (57) toGurson’s yield function

0.5% 2% 10% 30%f lin. per. min. lin. per. min. lin. per. min. lin. per. min.

C1 1.04 0.98 0.98 1.17 1.055 1.055 1.58 1.235 1.23 2.05 1.295 1.26C2 0.005 0.02 0.06 0.0188 0.128 0.145 0.035 0.255 0.2415 0.0459 0.345 0.307C3 6.776 8.034 8.08 5.52 6.45 6.45 2.055 2.30 2.32 0.6344 0.81 0.90C4 0.6347 0.63 0.635 0.55 0.535 0.535 0.61 0.615 0.645 0.99 1.01 1.00

Notes: Linear RVE boundary displacements, periodic fluctuations and minimum kinematicalconstraint

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and, finally, for the minimum kinematical constraint (or uniform RVE boundarytraction), which provides a lower bound yield surface, we obtain:

C1ð f Þ ¼ �223:6f 4 þ 170:7f 3 � 48f 2 þ 6:11f þ 0:95

C2ð f Þ ¼ 0:06 ln 0:03fð Þ þ 0:59

C3ð f Þ ¼ 7:9 exp �16:5fð Þ þ 0:82

C4ð f Þ ¼ �3813f 5 þ 3588f 4 � 1264:6f 3 þ 199:7f 2 � 11f þ 0:6853:

ð60Þ

The above functions together with the data of Table I are plotted in Figure 8.

4.4 Mechanisms of plastic collapseThe differences in macroscopic yield strength observed under the assumed RVEkinematic constraints considered arise from the fact that distinct kinematicalconstraints lead in general to different mechanisms being responsible for the plasticcollapse of the RVE. This is illustrated Figures 9 and 10. These show contour plots ofincremental effective plastic strain of the RVE obtained at collapse states in the finiteelement solution for � ¼ 0 (pure shear strain path) and � ¼ 1 (spherical in-plane strainpath), respectively. In both cases, the void ratio is f ¼ 10 per cent and the threekinematical constraints are considered. It is clear that the collapse mechanism

Figure 8.Estimated coefficients forexpression ���

Assessment ofthe Gurson yield

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299

triggered under the linear boundary displacements constraint is quite different fromthe mechanisms under the other two constraints considered. This explains the yieldstrength differences observed previously. Also, it should be noted that the mechanismsassociated with collapse under the periodic fluctuations and minimum constraintassumptions are very similar to each other, justifying the similarity between thehomogenised yield surfaces obtained under these two conditions.

5. Concluding remarksYield surface estimates for porous metals have been obtained by means of a purelykinematical finite element-based multi-scale constitutive modelling approach relyingon the volume averaging of the strain and stress tensors over a representative volumeelement of material. The macroscopic yield surface estimates have been obtainedunder three different kinematical constraints of the RVE: linear boundarydisplacements (an upper bound); periodic displacement fluctuations (corresponding toperiodically perforated media); and, minimum kinematical constraint or uniformboundary traction (a lower bound). An assessment of the classical Gurson criterion hasbeen carried out in the light of the computed bounds. A modification of the Gursonyield function has been proposed which is able to capture the yield loci of porousmetals under the three kinematical assumptions considered. The proposed functionalform can be used in the estimation of bounds for the plastic behaviour of porous metalsunder plane strain.

Figure 10.RVE collapse for

f ¼ 10 per cent and� ¼ 1. Contour plots of

incremental effectiveplastic strain

Figure 9.RVE collapse for

f ¼ 10 per cent and� ¼ 0. Contour plots of

incremental effectiveplastic strain

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Notes

1. This assumption is not necessary, but simplifies considerably the finite elementimplementation of the periodic fluctuations model.

2. p and q here denote, respectively, the hydrostatic component of the stress tensor and thevon Mises equivalent stress.

References

de Souza Neto, E.A. and Feijoo, R.A. (2006), ‘‘Variational foundations of multi-scale constitutivemodels of solid: small and large strain kinematical formulation’’, LNCC Research andDevelopment Report, No. 16/2006, National Laboratory for Scientific Computing, Petropolis.

de Souza Neto, E.A., Peric�, D. and Owen, D.R.J. (2008), Computational Methods for Plasticity:Theory and Application, Wiley, Chichester.

G�aar�aajeu, M. and Suquet, P. (1997), ‘‘Effective properties of porous ideally plastic or viscoplasticmaterials containing rigid particles’’, Journal of the Mechanics and Physics of Solids, Vol. 45No. 6, pp. 873-902.

Gurson, A.L. (1977), ‘‘Continuum theory of ductile rupture by void nucleation and growth: Part I-yield criteria and flow rules for porous ductile media’’, Journal of Engineering MaterialsTechnology Transactions of the ASME.

Hill, R. (1963), ‘‘Elastic properties of reinforced solids: some theoretical principles’’, Journal of theMechanics and Physics of Solids, Vol. 11, pp. 357-72.

Hill, R. (1965), ‘‘A self-consistent mechanics of composite materials’’, Journal of the Mechanics andPhysics of Solids, Vol. 13 No. 4, pp. 213-22.

Kouznetsova, V., Geers, M.G.D. and Brekelmans, V.A.M. (2002), ‘‘Multi-scale constitutivemodelling of heterogeneous materials with a gradient-enhanced computationalhomogenization scheme’’, International Journal for Numerical Methods in Engineering,Vol. 54, pp. 1235-60.

Mandel, J. (1971), Plasticite Classique at Viscoplasticite, CISM Lecture Notes, Springer-Verlag,Udine.

Michel, J.C. and Suquet, P. (1992), ‘‘The constitutive law of nonlinear viscous and porousmaterials’’, Journal of the Mechanics and Physics of Solids, Vol. 40 No. 4, pp. 783-812.

Michel, J.C., Moulinec, H. and Suquet, P. (1999), ‘‘Effective properties of com- posite materials withperiodic microstructure: a computational approach’’, Computer Methods in AppliedMechanics and Engineering, Vol. 172, pp. 109-43.

Michel, J.C., Moulinec, H. and Suquet, P. (2001), ‘‘A computational scheme for linear and non-linear composites with arbitrary phase contrast’’, International Journal for NumericalMethods in Engineering, Vol. 52, pp. 139-60.

Miehe, C. and Koch, A. (2002), ‘‘Computational micro-macro transitions of discretizedmicrostructures undergoing small strains’’, Archive of Applied Mechanics, Vol. 72, pp. 300-17.

Miehe, C., Shotte, J. and Lambrecht, M. (2002), ‘‘Homogenization of inelastic solid materials atfinite strains based on incremental minimization principles. Application to the textureanalysis of polycrystals’’, Journal of the Mechanics and Physics of Solids, Vol. 50 No. 10,pp. 2123-67.

Miehe, C., Schotte, J. and Schroder, J. (1999), ‘‘Computational micro-macro transitions and overallmoduli in the analysis of polycrystals at large strains’’, Computational Materials Science,Vol. 6, pp. 372-82.

Pellegrino, C., Galvanetto, U. and Schrefler, B.A. (1997), ‘‘Computational techniques for periodiccomposite materials with elasto-plastic components’’, in Owen, D.R.J., Onate, E. andHinton, E. (Eds), Computational Plasticity: Fundamentals and Applications, CIMNE,Barcelona, pp. 1229-36.

Assessment ofthe Gurson yield

criterion

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Pellegrino, C., Galvanetto, U. and Schrefler, B.A. (1999), ‘‘Numerical homogenisation of periodiccomposite materials with non-linear material components’’, International Journal forNumerical Methods in Engineering, Vol. 46, pp. 1609-37.

Simo, J.C. and Hughes, T.J.R. (1998), Computational Inelasticity, Springer- Verlag, New York, NY.

Swan, C.C. (1994), ‘‘Techniques for stress- and strain-controlled homogenization of inelasticperiodic composites’’, Computer Methods in Applied Mechanics and Engineering, Vol. 117,pp. 249-67.

Terada, K., Saiki, I., Matsui. K. and Yamakawa, Y. (2003), ‘‘Two-scale kinematics and linearizationfor simultaneous two-scale analysis of periodic heterogeneous solids at finite strains’’,Computer Methods in Applied Mechanics and Engineering, Vol. 192, pp. 3531-63.

Tvergaard, V. (1981), Influence of voids on shear band instabilities under plane strainconditions’’, International Journal of Fracture, Vol. 17, pp. 389-407.

Corresponding author

E.A. de Souza Neto can be contacted at: [email protected]

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