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European Journal of Mechanics A/Solids 35 (2012) 119e127

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European Journal of Mechanics A/Solids

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

A Linear Complementarity formulation of rate-independent finite-strainelastoplasticity. Part I: Algorithm for numerical integration

Andrea Bassi a, Nikolaos Aravas b,c, Francesco Genna a,*

aDepartment of Civil Engineering, University of Brescia, Via Branze, 43-25123 Brescia, ItalybDepartment of Mechanical Engineering, University of Thessaly, Pedion Areos, 38834 Volos, Greecec The Mechatronics Institute, Center for Research and Technology, Thessaly (CE.RE.TE.TH.), 1st Industrial Area, 38500 Volos, Greece

a r t i c l e i n f o

Article history:Received 28 April 2011Accepted 20 October 2011Available online 3 November 2011

Keywords:Singular or multi-surface plasticity modelsLarge strainsNumerical integration

* Corresponding author. Tel.: þ39 030 3711275; faxE-mail address: francesco.genna@ing.unibs.it (F. G

0997-7538/$ e see front matter � 2011 Elsevier Masdoi:10.1016/j.euromechsol.2011.10.002

a b s t r a c t

A methodology for the numerical integration of rate-independent, elasticeplastic finite-strain models isdeveloped. The methodology is based on the idea of local linearization of the yield surface that wasproposed in Maier (1969), adopted as the basis for an integration scheme in Hodge (1977), and developedfurther in Franchi and Genna (1984, 1987), so far for small-strain problems only. The proposed algorithmis based on the solution of a local Linear Complementarity Problem and is suited particularly for plasticitymodels that involve yield surfaces with singular points (corners, edges, apexes, etc.).

� 2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

The proposed integration scheme has its origins in a work byMaier (1969), who pointed out that, if the elasticeplastic consti-tutive law for a multi-surface plasticity model is written in the formof a local Linear Complementarity Problem (LCP), then, at a singularpoint of a yield surface (corner), each active yield surface could beapproximated locally by its tangent plane. Hodge (1977) was thefirst to propose an integration scheme employing piecewise-linearstress paths between two non-linear yield surfaces, one of whichwas exact and the other approximate, external and homothetic tothe first; the plastic part of the constitutive law was written in theform of a LCP and an exact assessment of the maximum violation ofthe yield condition and of the flow-rulewas furnished. Hodge’s ideaprovides an alternative to the most commonly adopted numericaltreatment of elasticeplastic models that is based on the early workof Wilkins (1964).

It is worth observing that several commonly adopted elasticeplastic material models involve “complex” yield functions, that canbehandledwith somedifficulty by the standard integrationmethodsbased on some variation of a Generalized Trapezoidal Rule. Typicalexamples are the classic multi-surface models, such as the Tresca orMohreCoulomb ones; models involving singular points, such as theclassic DruckerePrager or the ChristofferseneHutchinson ones

: þ39 030 3711312.enna).

son SAS. All rights reserved.

(Christoffersen and Hutchinson, 1979); models deriving from crystalplasticity; piecewise-linear models obtained from the linearizationof non-linear ones in order to reduce the formulation to a LCP even inthe casewhen deformation theories of plasticity need to be adopted.In all these cases, a rate formulation based on a LCP and Hodge’sintegrationmethod appear as the most appropriate tools to describeand solve constitutive problems. To the best of our knowledge, suchan approach has been considered, so far, only for the case of small-strain constitutive problems.

The methodology proposed in this paper is an extension tofinite-strain problems of the scheme used in Franchi and Genna(1984) and Franchi and Genna (1987) for small-strain problems,where the original Hodge algorithmwas improved in order to avoidnumerical problems arising when the stress point approached theexternal surface. Small-strain versions of this method have beendiscussed in Franchi and Genna (1987); Genna (1993, 1994), Gennaand Pandolfi (1994) for several yield functions, and proved to havesatisfactory accuracy characteristics. Here, the method is extendedto the case of large plastic strains.

We focus on rate-independent, ductile materials in which theelastic strains are always infinitesimal and the elastic properties arenot affected by plastic deformation. The solution method is devel-oped only for the special case of associated isotropic plasticity withnon-negative hardening. An extension to anisotropic plasticity isbriefly presented in Appendix A. Extensions to the softening case,for small strains only, have already been developed in the past(Franchi and Genna, 1991), and can be further expanded to thelarge-strain case by adopting the same techniques illustrated in the

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 119e127120

present work. Further extensions, to cover evenmore general cases,are similarly possible. They entail either lack of symmetry of thematrix governing the LCP, or lack of its definition, or both;complications of mathematical nature arise, that are expected to bepossibly dealt with by adopting techniques similar to those pre-sented in Part II of this paper (Bassi et al., 2012), with reference tothe calculation of critical points in the equilibrium path ofboundary value problems, for the class of materials consideredhere.

Indeed, this paper is followedbya Part II, inwhich a LCP approachis used for the solution of large-strain, large-displacement elasticeplastic boundary value problems for the considered class ofmaterials.

Standard notation is used throughout. Boldface symbols denotetensors the order of which is indicated by the context. All tensorcomponents are written with respect to a fixed Cartesian coordi-nate system, and the summation convention is used for repeatedLatin indices. Let a and b be vectors,A and B second-order tensors, Ca fourth-order tensor; the following products are used in the text:a$b ¼ aibi, ðA$BÞij ¼ AikBkj, ðC : AÞij ¼ CijhkAhk. For repeated Greekindices, one of which is underlined, the following summationconvention is adopted

bacahXna¼1

ba ca; (1)

where the value of n is defined in the text. Superscripts t and �1

denote transposition and inversion of a tensor respectively.

2. Rate kinematics in the presence of large elasticeplasticstrains

A systematic formulation of the kinematics of inelastic bodiesundergoing large elasticeplastic deformations has been developedby Kröner (1960), Lee and Liu (1967), Lee (1969), Lee andMcMeeking (1980), Lee (1981), Fox (1968), and Willis (1969). Theformulation is based on a multiplicative decomposition of thedeformation gradient F into an elastic and a plastic part: F ¼ Fe$Fp.When the intermediate configuration defined by Fe is chosen to be“isoclinic” (Mandel, 1971), the velocity gradient L can be decom-posed additively into an elastic part Le ¼ _F

e$Fe�1 and a plastic part

Lp ¼ Fe$ _Fp$Fp�1$Fe�1, so that L ¼ Le þ Lp. The deformation rate D

and the spin W tensors can be written as

D ¼ De þ Dp; W ¼ We þWp; (2)

with De ¼ Les ; Dp ¼ Lps ; We ¼ Lea, and Wp ¼ Lpa , where thesubscripts s and a denote the symmetric and antisymmetric parts ofa tensor respectively.

The plastic spin Wp vanishes in isotropic materials (Dafalias,1985).

3. A general LCP formulation of plasticity models in thepresence of large plastic strains

The constitutive equations arewritten in the current (deformed)configuration G in terms of the so-called Kirchhoff stress tensordefined as s ¼ Js, where J ¼ det½F� and s is the true (Cauchy)stress. The local mechanical state is characterized by the stresstensor s together with a set of state variables sgðg ¼ 1;2;.; yÞ. Forsimplicity, we formulate the problem for isotropic materials, inwhich all state variablesmust be scalar quantities. As is discussed inAppendix A, the proposed methodology can be used also foranisotropic materials with tensor-valued state variables.

The elastic part of the constitutive equation is written in hypo-elastic form:

sV ¼ Ce : De; (3)

where sV ¼ _sþ s$W �W$s is the Jaumann or co-rotational deriva-

tive of the Kirchhoff stress tensor, and Ce is the standard fourth-order isotropic elasticity tensor defined as

Ce ¼ 2GI þ�K � 2

3G�dd; (4)

whereK is the elastic bulkmodulus,G the elastic shearmodulus, d thesecond-order identity tensor having the Kronecker delta dij as Carte-sian components, and I the symmetric fourth-order identity tensorhaving, as Cartesian components, Iijkl ¼ ð1=2Þðdikdjl þ dildjkÞ.

It should be noted that the hypo-elastic Equation (3) statedabove is consistent, to leading order, with hyper-elastic behavior,when the elastic strains are small (Needleman, 1985; Aravas, 1992).The Kirchhoff stress s is used in the constitutive equations, insteadof the true stress s, because s is work-conjugate to D (Hill, 1968,1978) and is known to lead to a symmetric global tangent stiff-ness matrix when a finite element discretization of the equilibriumequations is introduced (McMeeking and Rice, 1975).

Let s ¼ fs1; s2;.; syg be the collection of state variables of themodel; in view of the assumed isotropy, all sg ðg ¼ 1;2;.; yÞ arescalar quantities.

The plastic part of the material behavior is characterizedthrough

� a set of yield functions

4a ¼ 4aðs; sÞ; a ¼ 1;2;.; y; (5)

with y � 1, that define independent, non-redundant constraints at

any state ðs; sÞ, and determine the current elastic domain throughthe inequalities

4aðs; sÞ � 0; a ¼ 1;2;.; y; (6)

the yield functions that satisfy Equation (6) with the equality signare called active, and are grouped into the set L, defined as

L ¼ fa˛f1;2;.; ygj4aðs; sÞ ¼ 0g; y � y; (7)

� a set of differentiable plastic potentials

~4a ¼ ~4aðs; sÞ; a ¼ 1;2;.; y; (8)

that define the plastic flow-rule

Dp ¼ _la ~Naðs; sÞ;�~Na

�ij¼ v~4a

vsij; _la � 0; a ¼ 1;2;.;y: (9)

Isotropy and objectivity require that the functions 4a, ~4a, and ~Na

are isotropic functions of s and s. In view of the isotropy of 4a, thecorresponding consistency condition can be written in the form(Dafalias, 1985)

c a˛L : _4a ¼ Na : sV þ v4a

vsb_sb � 0; Nb ¼ v4b=vs (10)

Note that Nb is also an isotropic function of s and s.Taking into account that

sV ¼ Ce : De ¼ Ce : ðD� DpÞ ¼ Ce :

�D� _la ~Na

�; (11)

we conclude that, at any stress point s at yield,

c a˛L :

_4a ¼ Na :Ce :�D� _lb

~Nb

�þv4a

vsg_sg �0; _la �0; _4a

_la ¼ 0;(12)

where the summation over g covers the number y of state variables.

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 119e127 121

The plasticity model is completed by the evolution equations forthe scalar state variables:

_sg ¼ _lbggbðs; sÞ; g ¼ 1;2;.; y; b ¼ 1;2;.; y; (13)

where ggbðs; sÞ are isotropic functions of s and s.Substituting (13) into (12), we can rewrite the incremental

elasticeplastic constitutive law as follows:

ca˛L :_4a¼Na :Ce :D� _lb

�Na :Ce : ~NbþHab

��0; _la�0; _4a

_la¼0; (14)

where

Hab ¼ �v4a

vsgggb: (15)

Relations (14) constitute a local (i.e., referred to a single materialpoint) Linear Complementarity Problem (LLCP), in which D is the“driving” quantity. For a given deformation rate D and any stresspoint s at yield, the solution of the LLCP (14) furnishes the activeyield surfaces and the corresponding plastic multipliers _la. Thecorresponding plastic deformation rate Dp and Jaumann derivativeof s can be obtained respectively from (9) and (11):

Dp ¼ _la ~Naðs; sÞ; a ¼ 1;2;.;y; and sV ¼ Ce : ðD�DpÞ: (16)

The LLCP (14) is the basis of the integration scheme described inthe next section, as well as for the detection of bifurcation points inlarge-strain elasticeplastic problems discussed in Part II of thiswork (Bassi et al., 2012).

For the special case of a single smooth yield function ðy ¼ 1Þ,which serves also as the plastic potential ð~4 ¼ 4; ~N ¼ NÞ, if thestress state s is on the yield surface and D is such thatN : Ce : D > 0,the solution to the LLCP is simply

_l ¼ N : Ce : DN : Ce : Nþ H11

>0; _4 ¼ 0; (17)

and, in the special case of a perfectly plastic material ðH11 ¼ 0Þ, thecorresponding direction of s

V ¼ Ce : ðD� _lNÞ in the stress space istangent to the yield surface at s, as shown in Fig. 1.

In what follows, in order to simplify the formulation, we makethe following assumptions:

1. the plastic flow-rule is associated with the yield function, i.e.,the plastic potentials coincide with the yield functionsð4a ¼ 4a;

~Na ¼ NaÞ ;2. the material is characterized by reciprocal hardening, i.e.,

Hab ¼ Hba;3. the material is non-softening, i.e., the matrix corresponding to

Hab is positive semidefinite, so that the solution of the LLCP (14)is unique (Maier, 1969; Murty, 1972).

4. The transformation of the constitutive law

We consider an elasticeplastic body which is subjected toboundary displacements and/or forces. In the process of its

N

τ

φ( )=0τ

Fig. 1. Schematic representation of sV.

deformation, the body evolves from a given configuration Gn toa new one Gnþ1. The mechanical state ðFn; sn; snÞ is known at eachcontrol point in the body, and the integration of the incrementalconstitutive law consists of the following problem:

given: ðFn; sn; snÞ in Gn at time tn, and Fnþ1 in Gnþ1 at time tnþ1,determine: the values ðsnþ1; snþ1Þ in Gnþ1.In order to solve this problem, we restate the elasticeplastic

constitutive equations in terms of “rotation-neutralized quantities”.In that case, the constitutive equations involve the usual materialderivative of the “rotated stress”, instead of the Jaumann derivativeof s that appears in the original equations (Govindarajan andAravas, 1994; Ramaswamy and Aravas, 1998).

During the transition from Gn to Gnþ1, the deformation gradientFðtÞ can be written as

FðtÞ ¼ DFðtÞ$Fn ¼ DRðtÞ,DUðtÞ$Fn; tn � t � tnþ1; (18)

where DRðtÞ and DUðtÞ are the rotation and the right stretchtensors associated with the increment of the deformation gradientDFðtÞ. Note that

DFðtÞ ¼ FðtÞ$F�1n ¼ DRðtÞ$DUðtÞ; (19)

DFn ¼ DRn ¼ DUn ¼ d; (20)

DFnþ1 ¼ Fnþ1$F�1n ¼ known: (21)

If it is assumed that the Lagrangian triad associated with DFðtÞ(i.e., the eigenvectors of DUðtÞ) remains fixed over the time interval½tn; tnþ1�, it can be shown that, over that interval, the followingholds:

DðtÞ ¼ DRðtÞ$ _EðtÞ$DRtðtÞ; WðtÞ ¼ D _RðtÞ$DRtðtÞ;sVðtÞ ¼ DRðtÞ$ _sðtÞ$DRtðtÞ; (22)

where EðtÞ ¼ lnðDUðtÞÞ is the Lagrangian logarithmic strain corre-sponding to DFðtÞ (i.e., the Lagrangian logarithmic strain relative tothe start of the increment), and sðtÞ ¼ DRtðtÞ$sðtÞ$DRðtÞ is theso-called co-rotational stress. It should be noted that the afore-mentioned assumption on the Lagrangian triad is less severe thanthe usual hypothesis of “constant strain rate” over the increment.

Using the above results and taking into account the isotropy ofthe constitutive functions involved, we conclude that the elasticeplastic constitutive equations can be written as (Govindarajan andAravas, 1994; Ramaswamy and Aravas, 1998)

ds ¼ Ce : dEe; (23)

dEp ¼ dla Na; (24)

ca˛L :d4a ¼ N a : Ce : dE� dlb

�Na : Ce : Nb þ Hab

�� 0;

dla � 0; d4adla ¼ 0; (25)

where NaðtÞ ¼ DRtðtÞ$NaðtÞ$DRðtÞ ¼ NaðsðtÞ; sðtÞÞ, in view of theisotropy of the functions Na.

It should be emphasized that the constitutive Equations(23)e(25) are formulated in terms of the increments ds, dE, d4b,and dla, whereas “time” t does not appear explicitly. This is becausethe constitutive model is “rate-independent” and the quantity tused above is essentially an arbitrary “loading parameter”.

The transformed constitutive Equations (23)e(25) are formallyidentical to those of infinitesimal-strain problems. Therefore, thenumerical integration of (23), (24), and (25) can be carried out byusing any of the techniques that are available for infinitesimal-strain

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 119e127122

problems. The key operations that characterize the integrationprocedure proposed here, and discussed in the next section, are asfollows:

� given Fn and Fnþ1, DFnþ1 is calculated by applying (21);� the polar decomposition of the deformation gradient incre-ment DFnþ1 allows for the computation of DE ¼ lnðDUðtnþ1ÞÞ;

� the constitutive equations are integrated by using the methodpresented in Section 5, and snþ1 and snþ1 are determined;

� the integration of the constitutive law is completed by thetransformation

snþ1 ¼ DRnþ1$snþ1$DRtnþ1: (26)

As described in Appendix A, the methodology developed above

can be extended easily to anisotropic materials, inwhich the plasticspin Wp does not vanish.

5. The integration scheme

Several methods for the numerical integration of elasticeplasticmodels with a single smooth yield surface are now available in theliterature (e.g., see Simo and Hughes, 1998 and de Souza Neto et al.,2008 for details). Most of these methods are based on the “elasticpredictor-plastic corrector” idea and are designed mainly forsmooth yield surfaces and smooth plastic potentials.

In this section, a methodology for the numerical integration ofelasticeplastic models that involve yield surfaces with singularpoints, such as “corners”, “edges”, “apexes”, etc., is developed. Themethodology is an extension to the case of finite strains and rota-tions of what was proposed originally by Franchi and Genna (1987)and developed further in Genna (1993), Genna (1994), and Gennaand Pandolfi (1994). In particular, for a given DE, we consider theintegration of the constitutive model described by (23), (24), and(25). The proposed method is based on the LLCP discussed in theprevious section and on the idea of a local linearization of the yieldfunction suggested by Hodge (1977).

In the following, for the sake of brevity, the superimposed ^ isomitted from s and Na.

Before presenting the proposed methodology in detail, wediscuss some features of the Linear Complementarity Problem (25).For a given stress state sn at yield, the LCP (25) can be written in anincremental form, as long as the gradients collected into matrix Na

are taken to be constant, as follows

ca˛L :D4a ¼ Najn : Ce : DE� Dlb

�Najn : Ce : Nb

��n þ Hab

��n

�� 0;

Dla � 0; D4aDla ¼ 0; (27)

where

Najn ¼ v4aðs; snÞvs

����s¼sn

(28)

and sn denotes the values of the state variables at the start of theincrement. An effective, engineering methodology for the solutionof problem (27) has been given by Franchi and Genna (1987); it isbased on the Lemke scheme 1 for the solution of LCPs (Lemke,1967), but it exploits the physical meaning of the variablesinvolved. The solution of the LCP provides the active yield functionsand the corresponding Dla. Then, the plastic strain increment, theincrement of the state variables, and the stress increment aredetermined as follows:

DEp ¼ DlaNa

��n; Dsg ¼ Dlb ggb

��n;

Ds ¼ Ce : DEe ¼ Ce :�DE� DlaNajn

�:

(29)

In the above procedure for the evaluation of stress, a forward-Euler scheme is used for the integration of the plastic strain rate.

In the special case of perfect plasticity, the stress incrementdetermined by (29c) is tangent to the yield surface at s ¼ sn (seeFig. 1). Therefore, the stress

snþ1 ¼ sn þ Ds (30)

is outside the yield surface and some kind of a “return” scheme isneeded in order to bring snþ1 back onto the yield surface. A similarsituation appears in the case of hardening as well.

In order to minimize the drift of snþ1 and facilitate its returnonto the yield surface, Hodge (1977) introduced an approximateyield surface 4appr

a ðs; sÞ ¼ 0, which is close to the exact yieldsurface 4aðs; sÞ ¼ 0 and lies just outside it. We use Hodge’s (Hodge,1977) technique here and, during plastic flow, the applied strainincrement DE is subdivided automatically into sub-increments,whose size is always chosen so that the stress state snþ1 lies onor between the exact and the approximate yield surface in stressspace.

For yield surfaces enclosing the zero stress state (i.e.,4að0; sÞ < 0),we use

4appra ðs; sÞ ¼ 4a

�s

1þ h; s�

¼ 0; (31)

where h is a small positive number. The definition (31) implies that

v4appra ðs; sÞvs

¼ 11þ h

v4a

�~s; s

�v~s

�����~s¼ s

1þh

(32)

and

v4appra ðs; sÞvsg

¼v4a

�~s; s

�vsg

�����~s¼ s

1þh

: (33)

5.1. Detailed description of the integration algorithm

Assume that the stress state at the start of the increment sn isinside the current yield domain defined by 4aðs; snÞ � 0ða ¼ 1;2;.; yÞ and DE is the given strain increment. The “elasticpredictor” is determined first:

se ¼ sn þ Ce : DE: (34)

If se is within the current yield domain, we set snþ1 ¼ se andsnþ1 ¼ sn, and the integration is completed.

If se is such that 4aðse; snÞ > 0 for some of the yield functions,then sub-incrementation is used, so that the consecutive changes ofthe stress state are on or between the exact and the approximateyield surfaces. The integration scheme consists in general of threesteps. After the first, which is done only if the starting stress state iselastic, the second and third keep repeating, and require each (i) thesolution of a LCP and (ii) the calculation of the size of a sub-increment of the total strain increment DE, determined asdescribed in the following.

5.1.1. Step 1This step is done only if the starting stress state is elastic. If the

starting stress state lies on the approximate yield surface, theintegration starts with the step 2 described next; if the startingstress state lies between the exact and the approximate yieldsurfaces, or on the exact yield surface, the integration starts withthe step 3 described next.

In step 1 we determine only the fraction x of the strain incre-ment DE that brings the stress state on the approximate yield

A΄A΄΄

A

Β

C

φ ( , Δτ s+ s)=00=)s+s Δ,(φ τ

τ

αα

n appr

Fig. 3. Schematic representation of the integration scheme: sub-increments 2 and 3.

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 119e127 123

surface 4appra ðs; snÞ ¼ 0 by solving for xa the uncoupled set of non-

linear equations

4appra ðsn þ xaC

e : DE; snÞ ¼ 0; a ¼ 1;2;.; y: (35)

The smallest positive of all xa is chosen, i.e., x ¼ min0<xa<1

½xa� (if x ¼ 1

then the elastic predictor of Eq. (34) would be the correct final solu-tion). Then, the stress state

sA ¼ sn þ xCe : DE (36)

is on the current approximate yield surface, as shown schematicallyin Fig. 2, where point A is the image of sA in stress space. Since theresponse is elastic during step 1, the state variables do not change,i.e.,

sA ¼ sn: (37)

At the end of step 1, the fraction xDE of the total strain incrementDE has been applied, and the stress and state variables at the end ofstep 1 are

s ¼ sA and s ¼ sA: (38)

5.1.2. Step 2At the start of step 2, the stress sA is outside the exact yield

surface 4aðs; sAÞ ¼ 0 and on the approximate yield surface.Let A0 be the point in stress space representing the stress state

sA0 ¼ sA=ð1þ hÞ. According to the definition (31) of the approxi-mate yield surface, the stress state sA0 satisfies the exact yieldcondition 4aðsA0 ; sAÞ ¼ 0, as shown in Fig. 2. For later use, weconsider the normals to the approximate and exact yield surfaces atA and A0 :

NaA ¼ v4appr

a ðs; sAÞvs

����s¼sA

; NaA0 ¼ v4aðs; sAÞ

vs

����s¼sA0

; (39)

and note that, in view of (33a),

NaA0 ¼ ð1þ hÞNa

A0 : (40)

Step 2 is a “return-step” and brings the stress state on thelinearized version of the evolving exact yield surface, as shownschematically in Fig. 3.

The stress state moves from point A to point B, which lies on thetangent to the expanding exact yield surface (Fig. 3). An analyticaldescription of the return-step is given in Appendix B. Here we notethat there is no strain increment associated with the return-stepðDE ¼ 0Þ and the driving force for the return is an arbitrarilychosen, positive quantity Dr, used in a LCP of the form

ca˛L : D4a ¼ �Dr � Dlb�NaA : Ce : Nb

A þ Hab

��A

�� 0;

Dla � 0; D4aDla ¼ 0:(41)

In the LCP stated above,Dr > 0 is the driving force for the return;the value of Dr is discussed in the following. The resulting values of

O

A΄

A

φ ( ,τ s)=0αφ ( ,s)=0ταappr

τn

Fig. 2. Schematic representation of the exact and the approximate yield surface instress space.

the Dla’s are all proportional to Dr. The increment of stress Ds andthe increments of state variables Dsa associated with the Dla’s ofthe solution of the LCP are

Ds ¼ �Dlb Ce : Nb

A and Dsa ¼ Dlb gabðsA; sAÞ: (42)

Next, we determine the fraction z of the Dla’s to be used in (42)by requiring that, at the end of the return-step, a stress point B bereached (Fig. 3), lying on the linearized version (tangent) of theevolving exact yield surface. Analytically, z is found so that thequantities

sB ¼ sA � zDlbCe :Nb

A and sajB ¼ sajA þ zDlb gabðsA; sAÞ (43)

are such that sB lies on the linearized form near A0 of the evolvingexact yield surface. The details of the calculation are given inAppendix B, where it shown also that the resulting sB and sB areindependent of the value of Dr used in the LCP (41).

The stress and state variables at the end of step 2 are

s ¼ sB and s ¼ sB: (44)

5.1.3. Step 3Let DE be the remaining part of the total strain increment DE.

The third step starts at point B of Fig. 3 and relies on the solution ofthe following LCP:

ca˛L : D4a ¼ NaB : Ce : DE� Dlb

�NaB : Ce : Nb

B þ Hab

��B

�� 0;

Dla � 0; D4aDla ¼ 0; (45)

that defines the Dla’s, which are proportional to the magnitudeof DE.

Next, we determine the fraction x of DE to be used by requiringthat the stresses and the state variables at the end of the third stepdefined by

sCðxÞ ¼ sB þ xCe :�DE� DlbN

bB

�and

sajCðxÞ ¼ sajB þ xDlbgabðsB; sBÞ(46)

are such that the stress at the end of the third step lies on theevolving approximate yield surface (point C in Fig. 3). This isachieved by solving the set of the uncoupled non-linear equations

4appra ðsCðxaÞ; sajCðxaÞÞ ¼ 0; a ¼ 1;2;.; y; (47)

for xa and choosing the smallest positive value, i.e., x ¼ minxa>0

½1; xa�.If x ¼ 1, the stress state sC defined in (46a) lies inside or on thecurrent approximate yield surface, and the integration is termi-nated. If x < 1, the stress state sC defined in (46a) is on the currentapproximate yield surface, as shown schematically in Fig. 3, wherepoint C is the image of sC in stress space; in this latter case, since wecould not apply the whole strain increment yet, steps 2 and 3 arerepeated in succession until the whole desired strain increment DEis applied.

We conclude this section mentioning that the proposed meth-odology can be used also for smooth yield surfaces defined bya single analytic function, i.e., y ¼ 1 in (5). The main advantage of

Fig. 4. Homogeneous finite biaxial extension.

Fig. 5. Loading-unloading biaxial extension. Path in the s1 � s2 plane.

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 119e127124

the method, however, is the robust treatment of yield surfaces withcorners, which cannot be dealt with easily by traditional numericalintegration schemes (Simo et al., 1988; Simo and Hughes, 1998;Pramono and Willam, 1989).

6. A numerical example

In this section, a simple problem involving homogeneous fieldsis solved. The integration algorithm has been implemented in thefinite element code STRUPL2 (STRUctural PLasticity), developed atthe Politecnico of Milano and the University of Brescia, Italy.

In order to prove the effectiveness of the proposed methodwhen dealing with singular yield surfaces, we use the classicalTresca yield condition, written in the form

4ðs; 3pÞ ¼ seqðsÞ � syð 3

pÞ � 0; (48)

where

seqðsÞ ¼maxðjsI�sIIj;jsI�sIIIj;jsII�sIIIjÞ; syð 3pÞ ¼ s0þh 3

p; (49)

ðsI ;sII ;sIIIÞ are the principal Kirchhoff stresses, s0 is the yield stressin uniaxial tension of the virgin material, h is the isotropic hard-ening modulus, 3p is the equivalent plastic strain defined as (e.g.,see Lubliner, 1990, p. 128)

3p ¼

Z_3pdt; _3

p ¼ 12���Dp

I

��þ ��DpII

��þ ��DpIII

��� ¼ max���Dp

I

��; ��DpII

��; ��DpIII

���;(50)

so that

s : Dp ¼ seq _3p; (51)

and ðDpI ;D

pII ;D

pIIIÞ are the principal values of Dp.

For definiteness, we assume that sI � sII � sIII . If all eigenvaluesare distinct, i.e., sI > sII > sIII , the associated flow-rule gives

DpI ¼ �Dp

III ¼ _l1>0; DpII ¼ 0: (52)

Also, if sI ¼ sII > sIII , the stress state is at a corner of the yieldsurface and the associated flow-rule gives

DpI ¼ _l1>0; Dp

II ¼ _l2>0; DpIII ¼ � _l1 � _l2 < 0: (53)

Finally, if sI > sII ¼ sIII , the stress state is again at a corner of theyield surface and the associated flow-rule gives

DpI ¼ _l1 þ _l2>0; Dp

II ¼ � _l1 < 0; DpIII ¼ � _l2 < 0: (54)

In all cases (52), (53), and (54)

_3p ¼ 1

2���Dp

I

��þ ��DpII

��þ ��DpIII

��� ¼ Xya¼1

_la so that 3p ¼

Xya¼1

la; (55)

where y is the number of active yield surfaces (y ¼ 1 or 2 in thiscase).

This version of the Trescamodel involves only one state variable,i.e., y ¼ 1 and s1 ¼ 3p. Nevertheless, the presence of corners in themodel makes it difficult to handle by means of the traditionalalgorithms, whereas the proposed methodology faces nodifficulties.

In the following, we analyze the problem of a square block withsides of length L0 and parallel to the axes Xi of a Cartesian coordi-nate system, subjected to a homogeneous, plane stress loading-unloading biaxial extension with perfect plasticity (h ¼ 0, Fig. 4).The calculations are carried out for Poisson’s ratio n ¼ 0:3 and30hs0=E ¼ 1=500 ¼ 0:002. The problem is solved by using thefinite element method with a single 4-node square isoparametricfinite element with 2 � 2 Gauss integration stations.

The displacements u1 and u2 shown in Fig. 4 are applied in threestages: first u2 ¼ 0:05 L0 with u1 ¼ 0 is imposed; then u2 is kept

fixed and u1 ¼ �0:05 L0 is imposed; finally, with u1 fixed, u2 isrestored to u2 ¼ 0.

In this case we have that D ¼ _3. Let ð 31; 32Þ and ðs1; s2Þ be theprincipal logarithmic strains and stresses that develop as thematerial deforms. During stage 1, 31 ¼ 0 and 32 increases from zeroto lnð1:05Þ ¼ 0:04879 ¼ 24:40 30. Plastic yielding occurs for thefirst time when

s2 ¼ s0 or 32 ¼�1� n2

� s0E

¼ 0:91 30: (56)

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 119e127 125

The corresponding value of s1 is s1 ¼ n s0 ¼ 0:3 s0 (point A inFig. 5). As 32 increases from 0:91 30 to 24:40 30, the stresses remainat A, where

_3p1 ¼ 0; _3

p2 ¼ _l1>0 so that 31 ¼ 3

e1 ¼ 3

p1 ¼ 0; 3

p2 ¼ l1>0: (57)

The strains at the end of this stage are 31 ¼ 0 and 32 ¼ 24:40 30.During stage 2, 32 is held fixed to 24:40 30 and 31 changes from

zero to lnð0:95Þ ¼ �0:0513 ¼ �25:65 30. The stress point unloadsand moves to the oblique plane (Fig. 5). On that planes2 � s1 � s0 ¼ 0 and _3p2 ¼ �_3p1 ¼ _l2 > 0, which cannot accom-modate the imposed strain; therefore the stress point moves to thecorner s1 ¼ �s0, s2 ¼ 0 (point B in Fig. 5), where it has thekinematical freedom to maintain the imposed strain since

_3p1 ¼ � _l2 � _l3 < 0; _3

p2 ¼ _l2>0 at B: (58)

The strains at the end of this stage are 31 ¼ �25:65 30 and32 ¼ 24:40 30.

During stage 3, 31 is held fixed to �25:65 30 and 32 reduces tozero. The stress point moves down the plane s1 þ s0 ¼ 0, alongwhich

_3p1 ¼ � _l4 < 0; _3

p2 ¼ 0: (59)

This cannot accommodate the desired strain; therefore thestress point ends up at the corner s1 ¼ s2 ¼ �s0 (point C), where

_3p1 ¼ � _l4 < 0; _3

p2 ¼ � _l5 < 0: (60)

The stresses remain at C until thefinal values 31 ¼ �25:65 30 and32 ¼ 0 are reached.

The stress path discussed above is shown in Fig. 5 as determinedby the finite element solution.

7. Closure

An accurate and efficient integration method for the treatmentof multi-surface plasticity models, as well as of yield functionscharacterized by the presence of singular points, has been devel-oped. The methodology is based on the solution of a local LinearComplementarity Problem i.e., a problem at the local integrationpoint level. Details on the implementation of the method in a finiteelement program can be found in Bassi’s Ph.D. Thesis (Bassi, 2002).

The present work is followed by a Part II, where the proposedmethodology is combined with a structural Linear ComplementarityProblem for the solution of boundary value problems that involvefinite plastic strains, and for the detection of limit or bifurcationpoints.

Acknowledgement

Work done within a research project financed by the ItalianMinistry of Education and Research (MIUR).

Appendix A

We formulate the problem for anisotropic rate-independentelastoplasticity as follows. Let s ¼ fs1; s2;.; svg be the collectionof state variables, which can be scalars, vectors, and second-ordertensors. The constitutive equations can be summarized as follows(Dafalias, 1987):

D ¼ De þ Dp; W ¼ We þWp; (A.1)

s+h_sþ s$u� u$s ¼ Ce : De; u ¼ W �Wp; (A.2)

Dp ¼ _la ~Naðs; sÞ; Wp ¼ _laUaðs; sÞ; a ¼ 1;2;.; y; (A.3)

s+a ¼ _lb gabðs; sÞ; a ¼ 1;2;.; v; b ¼ 1;2;.; y; (A.4)

4aðs; sÞ ¼ 0; a ¼ 1;2;.; y; (A.5)

with

s+ah

8<:

_sa þ sa$u�u$sa if sa is a 2nd order tensor;_sa � u$sa if sa is a vector;_sa if sa is a scalar;

(A.6)

where the constitutive functions ~Na, Ua, gab, and 4a are isotropicfunctions of s and s. The mathematical isotropy of the aforemen-tioned functions guarantees the invariance of the constitutiveequations under rigid body rotations. Nevertheless, it should beemphasized that thematerial is anisotropic, because of the tensorialcharacter of the state variables sa.

In the anisotropic model the plastic spinWp does not vanish andthe substructural spin u is different from the average continuumspin W. Also, the hypo-elastic Equation (A.2) and the evolutionequations of the state variables (A.4) are written in terms of a rateco-rotational with the substructure. In the special case whereWp ¼ 0, the above model reduces to the simpler isotropic modeldiscussed in the main body of the paper.

Equations (A.2) and (A.4) can bewritten in terms of the Jaumannrate as

s7 ¼ Ce : D� _la

�Ce : ~Na � s$Ua þUa$s

�; (A.7)

s7a ¼ _lb Babðs; sÞ; (A.8)

where

Babðs;sÞ ¼8<:gabþ sa$Ub�Ub$sa if sa is a 2ndorder tensor;gab�Ub$sa if sa is a vector;gab if sa is a scalar:

(A.9)

In view of the isotropy of 4a, we can write

_4b ¼ Nb : sV þ v4b

vsg* sV

g

¼ Nb :hCe : D� _la

�Ce : ~Na � s$Ua þUa$s

�iþ _la

v4b

vsg*Bga; (A.10)

where

ðnosumongÞ v4b

vsg* s7

g¼

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

v4b

v�sg�ij

�s7

g�ij if sg is a2ndorder tensor;

v4b

v�sg�i

�s7

g�i if sg is a vector;

v4b

vsg_sg if sg is a scalar:

(A.11)

Equation (A.10) can be written as

_4b ¼ Nb : Ce : D� _la�Nb : Ce : ~Na þ Hba

�; (A.12)

where now

Hba ¼ �v4b

vsg*Bga �Nbðs$Ua �Ua$sÞ: (A.13)

Equation (A.12) is the same as (14a), which is the basis of theLLCP approach developed in the paper; anisotropy simply modifiesthe definition of the hardening moduli Hab.

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 119e127126

Following the methodology presented in Section 4, we define also

WðtÞ ¼ DRtðtÞ$WðtÞ$DRðtÞ; (A.14)

and

saðtÞ ¼8<:

DRtðtÞ$saðtÞ$DRðtÞ if sa is a 2nd order tensor;DRtðtÞ$saðtÞ if sa is a vector;saðtÞ if sa is a scalar:

(A.15)

If it is assumed that the Lagrangian triad associated with DFðtÞremains fixed over the time interval tn � t � tnþ1, it can be shownthat over that interval the constitutive equations can be written inthe rotation neutralized form:

_E ¼ _Ee þ _E

p;

_W ¼ _Weþ _W

p; _s ¼ Ce : _E� _laMa;

Ma ¼ Ce : ~Naðs; sÞ � s$Uaðs; sÞ þUaðs; sÞ$s;_Ep ¼ _la ~Naðs; sÞ; Wp ¼ _laUaðs; sÞ;

_sa ¼ _lb Babðs; sÞ;4bðs; sÞ ¼ 0; (A.16)

where Ceijkl ¼ DRpiDRqjDRrkDRslC

epqrs:

The methodology developed in the main text for the simplerisotropic model is then applicable to the anisotropic case as well.

Appendix B

In this Appendix we discuss the analytical determination of thestress state sB at the end of the return-step, which corresponds topoint B of Fig. 3.

We want to determine the fraction za of the Dlb’s determinedfrom the LCP (41) so that the stress state

sB ¼ sA � zaDlb Ce : Nb

A (B.1)

lies on the linearized form (tangent) of the exact yield surface at A0,translated to the evolving exact yield surface 4aðs; sBÞ ¼ 0 (point A00

in Fig. 3), corresponding to the value of the state variables definedby

sg��B ¼ sg

��A þ zaDlb ggbðsA; sAÞ: (B.2)

Taking into account that the exact and the approximate yieldsurfaces are defined for the same values of the state variables, i.e.,sA0 ¼ sA, we conclude that the exact yield surface 4aðs; sÞ ¼ 0 canbe linearized at A0 as follows:

4aðs; sÞx4aðsA0 ; sAÞ|fflfflfflfflfflfflffl{zfflfflfflfflfflfflffl}¼0

þ v4a

vs

�����s ¼ sA0s ¼ sA|fflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflffl}hNa

A0

: ðs� sA0 Þ

þ v4a

vsg

����� s ¼ sA0

s ¼ sA

�sg � sg

��A

�

¼ NaA0 : ðs� sA0 Þ þ v4a

vsg

�����s ¼ sA0s ¼ sA

�sg � sg

��A

� ¼ 0: (B.3)

Next, we take into account that

sA0 ¼sA

1þ h; Na

A0 ¼ ð1þ hÞNaA; (B.4)

and

v4a

vsg

����� s ¼ sA0

s ¼ sA0

¼ v4appra

vsg

�����s ¼ sAs ¼ sA

(B.5)

to find

4aðs; sÞxð1þ hÞNaA :

�s� sA

1þ h

�þ v4appr

a

vsg

�����s ¼ sAs ¼ sA

�sg � sg

��A

�

¼ ð1þ hÞNaA : s� Na

A : sA þ v4appra

vsg

�����s ¼ sAs ¼ sA

��sg � sg

��A

� ¼ 0: (B.6)

We determine za so that the stress state sB defined in (B.1) lieson the above tangent to the evolving exact yield surface, with sgjBas defined in (B.2). Substituting the aforementioned values s ¼ sBand sg ¼ sgjB in (B.2), we find that

ð1þ hÞNaA :

�sA � zaDlbC

e : NbA

�

� NaA : sA þ zaDlb

v4appra

vsg

�����s ¼ sAs ¼ sA

ggb���A¼ 0: (B.7)

Taking into account that

v4appra

vsg

�����s ¼ sAs ¼ sA

ggb���A¼ �Hab

��A; (B.8)

we write (B.7) in the form

ð1þhÞNaA :

�sA�zaDlbC

e :NbA

��Na

A : sA�zaDlbHab

��A ¼ 0; (B.9)

which is solved for za:

za ¼ hNaA : sA

Dlbhð1þ hÞNa

A : Ce : NbA þ Hab

��A

i : (B.10)

Since the Dla’s are all proportional to Dr, Equation (B.10) showsthat the resulting za’s are all proportional to 1=Dr. Therefore, theproduct zaDlb, used in the stress and the state variable calculationsin (B.1) and (B.2), is independent of the chosen Dr.

If there are more than one active approximate yield functions4appra , we carry out the above calculation for all of them, and choose

the smallest positive za, i.e., z ¼ minza>0

½za�.

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