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European Journal of Mechanics A/Solids 35 (2012) 128e137

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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 II: Calculation of bifurcation and limit points

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

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

a r t i c l e i n f o

Article history:Received 28 April 2011Accepted 20 October 2011Available online 10 November 2011

Keywords:ElastoplasticityLarge strains and large displacementsBifurcationsLimit points

* 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.2011.10.003

a b s t r a c t

A methodology for the numerical solution of discretized boundary value problems that involve rate-independent, elastic-plastic finite-strain models is developed. The formulation is given in terms of a struc-tural Linear Complementarity Problem. A methodology for the determination of bifurcation and limit pointsalong an equilibrium path is described. The proposedmethod is suited particularly for plasticitymodels thatinvolve yield surfaces with singular points (corners, edges, apexes, etc.).

� 2011 Elsevier Masson SAS. All rights reserved.

1. Introduction

Part I of this work (Bassi et al., 2012) illustrates amethodology forthe numerical integration of rate-independent, elastic-plastic finite-strain constitutive models starting from the rate equations writtenas a Linear Complementarity Problem (LCP). This approach makes itpossible to deal without any difficulty with “complex” yield func-tions, exhibiting singularities.

In the present work, the corresponding large-strain, large-displacement elastic-plastic boundary value problem is considered,with specific focus on the determination of critical points along theequilibrium path.

Already in the case of a small-strain formulation, the commonlyadopted numerical approach, based on the construction of a globaltangent stiffness, and the use of NewtoneRaphson’s method tosolve the global equilibrium system, causes difficulties in thecalculation of even the simplest critical point in the equilibriumpath, i.e., the plastic collapse load. Specific and expensive tech-niques have been developed to overcome the necessity of definingthe collapse load as associated to the lack of convergence of New-toneRaphson’s method, such as the arc-lengthmethod (Riks,1979).These difficulties are mainly related to the choice of formulatingand solving the problem as a finite-step one, instead of adopting theactual rate formulation.

: þ39 030 3711312.enna).

son SAS. All rights reserved.

In the large-strain, large-displacement case the situation is evenmore complex from the numerical viewpoint. The choice ofadopting methods that are not based on actual rate equationsmakes it difficult to asses the level of accuracy given by a specificsolution procedure. The arbitrary subdivision of the loading historyinto finite steps may lead to inadequate choices and cause either anexcessive computational effort, or lack of accuracy, or lack ofconvergence. In the course of a standard analysis, it is impossible todetect exactly any critical point, either limit or bifurcation ones.Several methods have been proposed to determine bifurcationpoints, but they are usually difficult to be correctly applied, andrequire specific procedures, extraneous to the main analysis. Thesemethods do not provide all the possible post-bifurcation paths, andeven the determination of the post-bifurcation path to be followedimplies some arbitrary choice to be made. Also, in most cases thedetection of a bifurcation has to be performed by arbitrarilydefining an imperfection in the studied body: without it, thenumerical solution tends to follow the fundamental path, oftenphysically meaningless. If imperfection analyses are performed, theobtained results often depend on the assumed imperfection, so thatparametric studies may be required.

Again, all these problems are mainly related to the choice ofabandoning the rate equations, in favor of the finite-step ones, easierto use.

The presence of “complex” yield functions, vector-valued, withcorners or other singular points, makes the situation even morecomplex.

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 128e137 129

The technique described in Part I of this work to integrate theconstitutive equations, extended to consider also equilibrium andcompatibility, so as to formulate the complete set of equations fora boundary value problem, allows one to overcome all thesedifficulties, even though the involved computational effort canbecome quite high. Nevertheless, formulating and solving a large-strain, large-displacement elastic-plastic boundary value problemin terms of the rate equations appears to be the theoreticallysoundest way, and in any case the only way to obtain all therequired results without adding any arbitrary ingredient, andwithout the need to recur to special solution techniques. Signifi-cant results can be obtained for cases in which no analysis waspreviously possible, such as, for instance, large-strain plane stressproblems governed by a vector-valued yield function such as theclassic Tresca one.

Following the approach described in Part I, the constitutive lawsarewritten for general multi-surface plasticitymodels, in the form ofa LCPusing the formulations of Koiter (1953); Koiter (1960) andMaier(1969) (see Bassi, 2002 and Bassi et al., 2012 for further details). Theformulation of the corresponding boundary value problem, named“Structural Linear Complementarity Problem” (SLCP), is basedessentially on Maier’s work (Maier, 1969, 1970, 1971) and is anextension to large-strains of the small-strain formulations andimplementations of Franchi and Cohn (1980); Franchi and Genna(1984); Genna (1988, 1994); Feriani et al. (1996).

The necessary and sufficient conditions stated in Franchi et al.(1998) for the occurrence of a critical e bifurcation or limit e pointare implemented into a finite element code.

The following standard notation is used throughout. Boldfacesymbols denote tensors, the order of which is indicated by thecontext. All tensor components are written with respect to a fixedCartesian coordinate system, and the summation convention isused for repeated Latin indices. Let a and b be vectors, A and Bsecond-order tensors, C a fourth-order tensor; the followingproducts are used in the text: a$b ¼ aibi, ðb$AÞi ¼ bj Aji,ðA$bÞi ¼ Aij bj, ðA$BÞij ¼ AikBkj, (AB)ijpq ¼ Aij Bpq, and (C:A)ij ¼ CijpqApq. Direct matrix notation is also employed: a matrix with entriesAij is denoted as [A], a “row” vector and a “column” vector aredenoted respectively as PaR and {a}, and det½,� indicates the deter-minant of a matrix or a second-order tensor. The superscriptt indicates the transpose of a tensor.

2. The large-strain elastic-plastic problem

Let G0 be the reference stress-free configuration of an elastic-plastic body with volume V0 and surface S0. The body is sub-jected to body forces per unit mass b in V0, nominal tractions t onS0t, and displacements u0 on S0u, where S0t W S0u ¼ S0 andS0t X S0u ¼ B.

The position of a material point in G0 is denoted by X. The cor-responding position in the deformed configuration G at a certaintime t is denoted by x. The motion of the body can be thendescribed by the functional relationship x ¼ x(X, t), with x(X,0)¼ X; the difference u(X, t)¼ x(X, t)� X defines the displacementfield. The components of the deformation gradient F associatedwith the motion are defined as Fij(X, t) ¼ vxi(X, t)/vXj.

The velocity at time t of a material particle isvðX; tÞ ¼ vxðX; tÞ=vt ¼ _xðX; tÞ, where _½,� indicates the materialderivative. The velocity field can be written in an Eulerian formv ¼ v(x, t) and the components of the velocity gradient L are definedby Lij(x, t) ¼ vyi(x, t)/vxj; the deformation rate D is defined as thesymmetric part of L and the spinW is its antisymmetric part.

Let s ¼ J s be the Kirchhoff stress tensor, where J¼ det[F] and s

is the true (Cauchy) stress. The Jaumann or co-rotational derivativeof the Kirchhoff stress tensor is defined as

sV ¼ _sþ s$W �W$s: (1)

The weak form of the rate equilibrium equations, known as the“Principle of Virtual Velocities” (PVV), can be written as (Hill, 1959;McMeeking and Rice, 1975)ZS0t

v*$_t dS0 þZV0

r0v*$ _bdV0 ¼

¼ZV

1J

hD* : s

V � L* : ðD$sþ s$D� L$sÞidV ;

(2)

where v* is a virtual velocity field that vanishes on S0u, L* is thecorresponding velocity gradient, D* is its symmetric part, and V isthe volume of the continuum in the deformed configuration.

The same class of materials described in Part I is considered. Theplastic part of a given constitutive law is defined by an additivedecomposition of the deformation rate into an elastic and a plasticpart:

D ¼ De þ Dp (3)

and by a set of yield functions 4a ¼ 4a(s, s) and plastic potentials~4a ¼ ~4aðs; sÞ, where s is a collection of scalar state variables anda ¼ 1, 2,.,y. The same rate formulation of Part I is adopted, whichleads to the following form for the yield condition:

if {4} ¼ {0} then

��_4� ¼

�½N�t ½Ce��~N�þ ½H�

�� _l�� ½N�t ½Ce�fDg � f0g;� _l� � f0g; P _4R� _l� ¼ 0; (4)

where ½Ce� is the elasticity matrix, [N] is the 6� y matrix of thegradients of the yield functions, [H] is the y� y hardening matrix(see Bassi et al., 2012 for details), and y denotes the number of thecurrently active yield modes. The Hessian matrix ½N�t ½Ce�½~N� þ ½H� isa y� y matrix (non-symmetric in general).

3. The discretized rate problem

The finite element discretization is developed by introducing aninterpolation of the velocity field within the volume Ve of eachfinite element:

yiðxÞ ¼XNe

j¼1

ajðxÞvje;i or fyðxÞg ¼ aðxÞfyeg; (5)

where ajðxÞ are suitably chosen shape functions, Ne the number ofnodes in the element, and v

je;i the i -th component of velocity at

node j. The velocity gradient L and the deformation rate D arewritten in matrix form as

fLðxÞg ¼ ½BLðxÞ�fveg and fDðxÞg ¼ ½BDðxÞ�fyeg; (6)

where [BL] and [BD] contain the appropriate spatial derivatives ofthe shape functions.

In the following we formulate the discretized rate problem in theform of a LCP for the entire structure. This problem is formally similarto the Local Linear Complementarity Problem (LLCP) formulated inPart I of this work and used for the integration of the constitutiveequations locally at a material point. The LCP for the entire structureuses the values of the plastic multipliers f _lg of Eq. (4) at the inte-gration stationsof themesh as theprimaryunknowns. In this respect,the present formulation is substantially different from the “tradi-tional”finite element formulationsof the rate problem,whichuse the

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 128e137130

nodal velocities (or displacement increments) as the primaryunknowns.

The structural LCP is formulated as follows. Substituting theinterpolation (5) in the matrix form of the PVV (2) and usinga Gauss quadrature rule for the numerical evaluation of the inte-grals involved, one reaches the following discrete form of the PVV:

½K��yN�� �~G�t� _lI� ¼ �

_F�; (7)

where {yN} is the global vector of nodal velocities, f _lIg the globalvector that contains weighted values of _l at all integration points inthe problem, [K] the instantaneous global elastic stiffness matrix,and ½~G� a global matrix that assembles quantities evaluated at theintegration points (see Bassi, 2002 for details).

The plastic part of the constitutive law (4) is assembled over theplastic zone Vp (with V ¼ Vel W Vp), and is written in a global form asfollows:

�� _F�¼�½G��yN�þ�~U�n _l

Io�f0g;n_lIo�f0g; P _FR

n_lIo¼0; (8)

(again, see Bassi 2002 for details).The complete discretized global problem has the form

½K��yN�� �~G�t� _lI� ¼ �

_F�; (9)

�n_Fo

¼ �½G��yN�þ �~U�� _lI� � f0g;� _lI� � f0g; P _FR� _lI� ¼ 0 in Vp (10)

and determines the unknowns {yN}, f _Fg, and f _lIg at a givenequilibrated configuration G.

The problem can be stated in the following equivalent formu-lation. Eq. (9) is solved for {yN} :�yN

� ¼ ½K��1��

_F�þ �

~G�t� _lI��; (11)

which is then substituted into (10) to yield

�� _F�¼ �~A�� _lI���

_c��f0g;� _lI��f0g; P _FR� _lI�¼0 in Vp; (12)

where�~A� ¼ �~U�� �G�½K��1�~G�t ; �

_c� ¼ ½G�½K��1� _F�: (13)

Matrix ½~A� and vector f _cg have dimension n � n and n � 1respectively, where n is the total number of active yield modes.

Problem (12) is a structural LCP (shortly SLCP) in the unknownsf _Fg and f _lIg, and defines, through (11), the incremental behavior ofthe whole body; the driving force of the problem is f _cg, which isproportional to f _Fg.

It is worthy of note that the SLCP (12) is formally identical tothe LLCP that governs the constitutive problem, discussed indetail in Part I of this work (Bassi et al., 2012). The SLCP is solvedby using the same technique used in the LLCP described in Part I;this involves the use of approximate yield surfaces that introducea suitable tolerance parameter h and the “a posteriori” lineari-zation of the yield surfaces (Hodge, 1977). The solution method-ology is discussed in detail in the section that follows.

We note that the nodal velocities defined in (11) can be writtenin the form�yN

� ¼ �yN;el

�þ ��yN;p

�;

with�yN;el

� ¼ ½K��1� _F�; ��yN;p

� ¼ ½K��1�~G�t� _lI�: (14)

The nodal velocities {yN,el} define through (5a) a compatiblevelocity field computed on a purely elastic structure; similarly,n�yN;p

ocan be used to define a compatible plastic velocity field.

The nodal velocities of an element can be written in the form

fyeg ¼nyele

oþn�ype

o; (15)

so that

fDg ¼ ½BD�fyeg ¼ �Del

�þ ��Dp�;

with�Del

� ¼ ½BD�nvele

o;

��Dp� ¼ ½BD�

�vpe�:

(16)

In view of their definition, the quantities {Del} and f�Dpg are bothcompatible deformation rates. It is emphasized though that {Del}and f�Dpg are different from {De} and {Dp} of Eq. (3), which are notcompatible in general.

In the following, for simplicity, we assume that thematerial obeysthe “plastic normality rule” so that ~4a ¼ 4a, ~Na ¼ Na, and theinteraction between the yielding modes exhibits reciprocity(Hab¼Hba). Then thematrices ½~A� and ½~U� are symmetric and the tildee½,� can be removed. We assume also that the material is non-soft-ening, i.e., the matrix corresponding to Hab is positive semi-definite,so that the solution of the LLCP (4) is unique (Maier, 1969; Murty,1972).

4. Solution of the boundary value problem

This section summarizes the algorithm developed for the solu-tion of the elastic-plastic boundary value problem. The method-ology for the determination of bifurcation or limit points along theequilibrium path is presented in Section 5.

The solution is developed incrementally. Since the material israte-independent, the solutiondepends only on the “loading path”, isindependent of the rate of loading, and does not depend on “time”explicitly. For convenience, we introduce a time-like loadingparameter t, and the external actions are imposed over the interval[t0, tf] in such a way that for t ¼ t0 the structure is in its referenceconfiguration G0 and for t ¼ tf the final given value of the externalloads is reached.

We consider the increment [tn, tnþ1]; let {DF} be the corre-sponding load increment. The solution algorithm involves thefollowing three steps within the increment:

1. computation of the displacement increments,2. updating of the geometry and calculation of the stresses snþ1

and state variables snþ1 at the end of the increment,3. “equilibrium correction”.

Each of the three steps is described next.

4.1. Step 1

The computation of the displacement increments is based onthe SLCP (12), that can bewritten over the increment [tn, tnþ1] in thefollowing differential form:

�fdFg ¼ ½A�ndlI

o� fdcg � f0g;n

dlIo� f0g; PdFR

ndlI

o¼ 0 in Vp:

(17)

The SLCP stated above is formally similar to the LLCP used in PartI of this work (Bassi et al., 2012). Nevertheless, unlike the LLCP, itssize may become quite large (it is related to the extension of Vp),and the calculation of each row/column of the governing matrix [A]is expensive, because it requires the solution of a linear system ofequations governed by the instantaneous elastic matrix [K]. Thereare also some differences in the solution technique.

1 We are not interested, here, in problems of elastic stability.

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 128e137 131

The integration of the SLCP is performed by adopting a piv-oting scheme analogous to the Lemke Scheme 1 (Lemke, 1967),where now the physical meaning of all the variables isexploited (Franchi and Genna, 1984, 1987). This requiresa sequence of sub-increments within the increment [tn, tnþ1], sothat only one new Gauss point enters the plastic zone in eachsub-increment. In this way, the dimensions of [G], [U], and [A]increase by one in every sub-increment, and only one columnand one row of matrix [A] need to be updated per sub-increment, which makes the calculations very efficient. Theelastic stiffness [K]n, used in all sub-increments, is calculated byevaluating the spatial integrals using the geometry at the startof the increment Gn.

A local linearization technique of the yield surface is adopted,identical with that described in Part I of this work (Bassi et al., 2012).This allows one to deal, during each sub-increment, with constantgradients to the yield surface; the response of the system, duringeach sub-increment, is therefore linear, and the SLCP (17) can bewritten in the following incremental form

�fDFgi ¼ ½A�inDlI

oi� fDcgi � f0g;n

DlIoi� f0g; PDFRi

nDlI

oi¼ 0 in Vp;

(18)

where

½A�i ¼ ½U�i � ½G�i½K��1n ½G�ti ; fDcgi ¼ ½G�i½K��1

n fDFgi: (19)

In (18) and (19) the subscript i indicates the i th sub-increment.The driving force in the SLCP (18) is {DF}i, which is the fraction of

{DF} used in the ith sub-increment. Within each sub-increment,approximate yield surfaces are introduced, the yield surface islinearized locally as described in Part I of this work (Bassi et al.,2012), and the corresponding new matrix of the normals to theyield surface [N] and the new Hab are determined and used in thenext sub-increment for the determination of [U]i, [G]i and [A]i. Thecalculation of stress within each sub-increment is carried out as in“small-strain” problems.

The solution of the SLCP (18) determines the plastic zone in thestructure and defines the increment of the plastic multipliers at theintegration points {DlI}i. The corresponding nodal displacementincrements associated with the sub-increment are determinedfrom the incremental form of (11) as follows:

�DuN

�i ¼ ½K��1

n

�fDFgi þ ½G�ti

nDlI

oi

�: (20)

When the sum of all sub-increments {DF}i equals the desiredvalue of {DF}, the end of the increment is reached; at that point, thegeometry is updated and the stresses and state variables arecalculated accurately by using an “incrementally objective” algo-rithm, as described in Step 2 below.

At the first sub-increment of the first load increment, matrix [A]does not exist, since V¼ Vel. At the end of the first increment, if a partof the body has experienced yielding, some rows/columns of matrix[A] have been computed. At the beginning of the next increment, thecurrent matrix [A] is recomputed, using the new instantaneouselastic stiffness [K], referred to the updated geometry. We note thatmatrix [A] is initially positive definite; as the solution develops, if [A]becomes semi-definite, stability tests can be performed in order todetect and identify limit or bifurcation points on the equilibriumpath, as described in Section 5.

The above scheme is similar to that adopted in earlier small-strain formulations (Franchi and Cohn, 1980; Franchi and Genna,1984, 1987; Genna, 1994; Feriani et al., 1996); in Genna (1988) itwas employed to perform shakedown analyses.

4.2. Step 2

At the end of each increment, the nodal displacement increments

nDuN

o¼

Xi

nDuN

oi

(21)

are known, where the summation is over the sub-incrementsassociated with the increment.

The nodal displacement increments fDuNg are used for thecalculation of the new geometry Gnþ1 of the structure and thedeformation gradient Fnþ1 at all Gauss points at the end of theincrement.

Then, the methodology described in Part I of this work is usedfor the calculation of the stresses snþ1 and the state variables snþ1 atthe end of the increment.

4.3. Step 3

The desired load at the end of the increment is {F}nþ{DF}.Nevertheless, the numerical solution is based on the PVV (2),which isequivalent to the rate of the equilibrium equations (as opposed toequilibrium itself). Therefore, the calculated stresses snþ1 are not inequilibriumwith {F}n þ {DF} in general.

The nodal forces {F}nþ1 that are in equilibriumwith the stressessnþ1 in the structure are determined by assembling the corre-sponding element load vectors {fe}nþ1 defined as

ffegnþ1 ¼Z

Ve;nþ1

½BD�tnþ1fsgnþ1dV : (22)

The aforementioned {F}nþ1 is used as the value of the loads atthe start of the next increment.

5. Determination of bifurcation and limit points

Bifurcation points on an equilibrium path of elastic-plastic prob-lems can be determined by using Hill’s theory (Hill, 1958, 1959).According to this theory, in a discretized problem, a sufficientcondition for uniqueness of solution is that the tangent stiffnessmatrix of the corresponding “linear comparison solid” is positivedefinite (Hutchinson and Miles, 1974).

An alternative approach to the elastic-plastic bifurcationproblem has been put forth by Maier (1971), Corradi (1978), andFranchi et al. (1998). In this approach, the necessary and sufficientconditions for the occurrence of a bifurcation or a limit point in theequilibrium path are identified with the degeneracy of the solutionof the rate problem. In particular, Franchi et al. (1998) used the SLCPapproach and stated the specific conditions that distinguish amongbounded bifurcations, unbounded bifurcations, limit points, andunloading points.

In the following, we summarize the approach of Franchi et al.(1998) for the determination of bifurcation and limit points. Weconsider the discretized problem discussed in Sections 3 and 4. LetG be the deformed configuration that corresponds to loads {F}. Weimpose on G a load rate f _Fg and examine whether the rateboundary value problem has a unique solution. The response of thebody is determined from the solution of the SLCP (12), with [A]defined as in (13a).We assume that the elastic stiffness matrix [K] ispositive definite1; therefore, the solution of the SLCP exists and is

Table 1Determination of bifurcation and limit points.

dA ¼ det[A]Eq. (45)

_FNy¼ �Wz f _zg

>0 solution d!cf _cg¼0 >0 any solution d!cf_cg LOCAL UNLOADING¼0 ¼0 �{0} multiple solutions

UNBOUNDED BIFURCATION¼0 ¼0 k{0} multiple solutions

BOUNDED BIFURCATION¼0 <0 k{0} solution d!cf_cg FALSE MECHANISM

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 128e137132

unique if and only if matrix [A] is positive definite (Maier, 1971).Therefore, if [A] is positive definite, f _lg is unique and the nodalvelocities {yN} defined by (14) are also unique.

As mentioned in Section 4, matrix [A] is computed in such awaythat only one yield mode at a time is activated; therefore [A], f _cg,f _Fg, and f _lg can be partitioned as follows (Franchi et al., 1998):

�A� ¼ �

An� ¼

� ½An�1� fangPanR an;n

;

_c�

¼�

_cn�1�

_cn

�; (23)

n_Fo

¼n

_Fn�1

o_Fn

�;

n_lo

¼(n

_ln�1

o_ln

)(24)

where ½An�1� is positive definite and the nth mode has been acti-vated last. Assume now that the reduced SLCP

�n_Fo¼ ½An�1�

n_lo��

_c��f0g;

n_lo�0; P _FR

n_lo¼ 0; (25)

in the 2(n � 1) variables subspace has the solution

� _F� ¼ f0g; � _l� ¼ ½An�1��1� _cn�1

� � f0g: (26)

Uniqueness is lost when [A] becomes positive semi-definite.When that happens, [A] becomes singular and, using Schur’sformula for the determinant of partitioned matrices (Cottle, 1974),one has:

det½A� ¼�an;n � PanR½An�1��1fang

�det½An�1� ¼ 0

0 an;n � PanR½An�1��1nan

o¼ 0: (27)

As soon as [A] becomes singular, one of the following threealternative situations may occur (Franchi et al., 1998):

1. the equilibrium path has reached a limit point. In this case twofurther conditions must be satisfied:� the existence of a “plastic collapse mechanism”, i.e., theoccurrence of a compatible plastic multiplier vector

�_z� ¼

�½An�1��1fang

1

�with ½An�1��1fang �

0�;

(28)

so that the nodal velocities defined by (14) become

fyzghnyN

o¼ �

�yN;p���f _lg¼f _zg ¼ ½K��1½G�t� _z�; (29)

with

�� _F� ¼ ½A�� _z� ¼ f0g; (30)

andD¼ Dp, i.e., the total deformation rate is fully plastic at allGauss points, Dp being now compatible. Vector {yz} in (29)defines the so-called plastic collapse mechanism.If the inequality (28b) is not satisfied, the rate problem admitsyet a solution; this situation is called a “false mechanism” andcan be dealt with as discussed in Franchi and Genna (1984)and Franchi et al. (1998);

� the positiveness of the second-order external work associ-ated to the plastic collapse mechanism, i.e.,

Wz ¼ P _FR��yN;p

� ¼ P _FR½K��1½G�t� _z� ¼ P _cR�_z� ¼� �

¼0 <0 �{0} solution ecf _cg LIMIT POINT

¼ _cn � PanR½An�1��1 _cn�1 >0: (31)

The above condition is equivalent to _Fn ¼ �Wz < 0. Themechanical meaning of inequality (31) is the following: theexternal actions are applied so as to “activate” the plasticcollapse mechanism, i.e., they are not “orthogonal” to {yz}(Franchi et al., 1998);

2. in the neighborhood of the current configuration G there isa set of adjacent configurations equilibrated under the sameexternal actions, i.e., there is an infinite number of incrementalprocesses that start at G and are consistent with the same f _Fg.In this case, a bifurcation point has been reached. The plasticmultiplier vector (28a) defines now a mechanism such that thecorresponding second-order external work Wz vanishes. Thisimplies that the external actions are “orthogonal” to {yz}, i.e.,that the applied loads do not “activate” such mechanism. Inparticular, it can be shown that the SLCP (12) has a degeneratesolution of degree one of the form

n_l1

o¼

n_ln�1

o0

�;

n_F1

o¼

f0g0

�: (32)

Then, other solutions are of the form

� _l� ¼ � _l1�þ a�_z� � f0g; � _F

� ¼ f0g; (33)

where vector f _zg is given by Eq. (28), and a is a non-negativereal number. Two sub-cases have to be distinguished:

(a) the inequality (28b) is satisfied, so that the mechanismvector f _zg has all non-negative components and any non-negative a in (33a) defines a solution to the SLCP (12). Itshould be noted that

� _lN�h

� _l����a/N

(34)

is one of the possible solutions, in such a way that in theneighborhood of G there exists an unbounded set ofadjacent configurations, equilibrated under the sameexternal actions. In this case, the structural system is saidto undergo an unbounded bifurcation;

(b) the inequality (28b) is not satisfied. Then, it can be shownthat a second degenerate solution isn

_l2

o¼

n_l1

oþ a

�_z�;

n_F2

o¼ f0g; (35)

_ _

where the j th components of fl2g and fF2g are equal tozero, a and j being determined from the solution ofa minimization problem (Franchi et al., 1998). In general,any convex combination of these solutions

n_lo¼ x

n_l1

oþð1�xÞ

n_l2

oð0�x�1Þ;

n_Fo¼f0g; (36)

Fig. 1. Compressed elastic-plastic column: generic mesh.

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 128e137 133

is a solution of the SLCP (12). In this case, there existsa bounded set of incremental processes starting at G thatcorrespond to the same f _Fg and that never imply anindefinite development of plastic strains. The systemundergoes a bounded bifurcation;

3. an unloading point has been reached, namely a point along theequilibrium path at which local elastic unloading takes place. In

0 0.025 0.05 0.07Nominal

0

1

2

3

4

5

Nor

mal

ized

for

ce F

* [-

]

Plane s

Fig. 2. Compressed elastic-plastic c

this case, Wz < 0 : this means that the external actions tend to“inactivate” the mechanism {yz}.

The limit point and the unbounded bifurcation point appear tobe defined by very similar conditions: they both occur when all thecomponents of f _zg are non-negative, so that the system mayundergo a free motion according to the kinematics defined by {yz}.Nevertheless, in the case of the limit point Wz> 0, so that equilib-rium is impossible and the mechanism {yz} is “activated”;conversely, if an unbounded bifurcation occurs,Wz ¼0, i.e., the loadrate is “orthogonal” to the mechanism, that is not necessarilyactivated, even if it may develop with an arbitrary amplitude.

When a bifurcation occurs, the post-bifurcation path is providedby the mechanism vector f _zg of Eq. (28), together with Eq. (33) andsubsequent ones.

It is worth noting that, when a bifurcation occurs, the condition(30) is always satisfied, in such a way that in all possible bifurca-tions no unloading takes place.

Table 1 summarizes the alternatives that can take place when[A] becomes semi-definite. Further details concerning the imple-mentation of the illustrated uniqueness tests can be found in Bassi(2002).

6. Numerical examples

The procedure for the solution of the elastic-plastic boundaryvalue problem described in Section 4, as well as the methodologyfor the detection of bifurcation and limit points described in Section5, have been implemented into the finite element code STRUPL2(STRUctural PLasticity (Franchi and Genna, 1984)), developed at thePolitecnico of Milano and the University of Brescia, Italy.

In this section we present two structural applications of themethodology for the calculation of bifurcation points that wasdeveloped in Section 5.

6.1. Compressed elastic-plastic column

We study the response of a non-slender elastic-plastic column ofwidth B0 and length L0¼ 10B0 (Fig.1). The column is clamped at both

5 0.1 0.125 0.15strain [-]

Homogeneousvon Mises, ABAQUSvon Mises, STRUPL2Tresca, STRUPL2

tress

olumn. Plane stress solutions.

Fig. 3. Compressed elastic-plastic column. Plane stress von Mises solution forDua ¼ 0.15L0.

2 Amongst all the possible post-bifurcation paths, the one determined by thesolution f _l2g defined in (35) is chosen automatically (see (Bassi, 2002) for details).

3 The estimates for the critical load due to EngessereShanley (Shanley, 1947) andvon Kármán (von Kármán, 1910) are obtained from Euler’s formula for the bucklingload of elastic slender columns, if the elastic Young’s modulus is replaced by theso-called tangent modulus Et and reduced modulus Er respectively.

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 128e137134

ends and the horizontal displacement of the midpoint of the top andbottom surfaces of the column is set to zero as shown in the rightpart of Fig. 1. The column is compressed by prescribing a uniformdownward displacement ua at its upper side. The top downwardsdisplacement ua is increased monotonically by imposing equaldisplacement increments of magnitude Dua ¼ 0.0075 � 10�2L0.

Calculations are carried out for both the von Mises and Trescayield criteria with associated flow rules. In particular, the yieldfunctions are of the form

4ðs; 3pÞ ¼ seðsÞ � syð 3

pÞ � 0; syð 3pÞ ¼ s0 þ h 3

p; (37)

where h is a constant hardening modulus,

se s� ¼

ffiffiffiffiffiffiffiffiffiffiffiffi32s : s

rand _3

p ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi23Dp : Dp

r(38)

in the von Mises model, and

seðsÞ ¼ maxðjsI � sIIj; jsI � sIIIj; jsII � sIIIjÞ and

_3p ¼ 1

2 ��Dp

I

��þ ��DpII

��þ ��DpIII

��� ¼ max ��Dp

I

��; ��DpII

��; ��DpIII

��� (39)

in the Tresca model, where ðsI ; sII ; sIIIÞ are the principal Kirchhoffstresses, and ðDp

I ;DpII ;D

pIIIÞ are the principal values of Dp.

The material parameters used in the calculations for bothmodels are: Young’s modulus E ¼ 700s0/3 ¼ 233.33s0, Poisson’sratio n ¼ 0.35, and h ¼ 175s0/3 ¼ 58.33s0.

Both plane stress/strain solutions are computed using STRUPL2.For comparison purposes the ABAQUS (Hibbitt et al., 2011) generalpurpose finite element program was used for the solution ofa column with a geometric imperfection as described in thefollowing. Four-node isoparametric finite elements are used in allcalculations, that are carried out with a finite element mesh having10 elements across the column width and 50 along its height.Calculations with finer meshes gave essentially the same results.

A “perfect” column (i.e., with no initial geometrical imperfec-tions) was used with STRUPL2, and the bifurcation path was deter-mined by using themethodology described in Section 5. In STRUPL2,a 2 � 2 Gauss integration scheme is used for the calculation of theelastic stiffness matrix; the calculation of stress and the plasticitycontrol are performed only at the centroid of the elements.

Since no elastic-plastic bifurcation analysis is available in ABA-QUS, an initial geometric imperfection was introduced into allmeshes run with it; all nodes in the column mid-section were dis-placed to the right by an amount d ¼ 10�4B0 in the undeformedconfiguration, and the mesh was constructed by connecting withstraight lines the mid-side nodes to the corresponding nodes on thetop and bottom surfaces of the column. The calculationswere carriedout by using the ABAQUS CPS4 (plane stress) and CPE4 (plane strain)elements. A 2 � 2 Gauss integration is used in the CPS4 elements forall calculations; a “selective integration” scheme is used in the CPE4,in which the volumetric terms of the stiffness are integrated usinga single Gauss point at the element centroid. It is noted that theTresca plasticity model does not exist in the ABAQUS materialslibrary. Nevertheless, ABAQUS provides a general interface, so thata particular constitutive model can be introduced as a “usersubroutine” (UMAT); the Trescamodel was implemented in ABAQUSfor plane strain problems via UMAT by using an algorithm similar tothat presented by Peri�c and de Souza Neto (1999) and Borja et al.(2003) (see also de Souza Neto et al., 2008).

6.1.1. Plane stress solutionsThe homogeneous solution in this case is the uniaxial compres-

sionfield indicated by the solid line of Fig. 2 (linear hardening curve);

in that figure, U* ¼ ua/L0 is the nominal axial compressive strain andF* ¼ R/(s0A0) is the normalized axial force, where R is the verticalreaction force at the lower clamped end of the column and A0 is theinitial cross-sectional area.

Fig. 2 shows also the STRUPL2 bifurcation solutions for the vonMises and Tresca models, as well as the ABAQUS vonMises solutionfor the imperfect column. STRUPL2 detects bounded bifurcationpoints, stable in Shanley’s sense (Shanley, 1947), as follows

U*b ¼ 0:02070 and F*b ¼ 1:81 for von Mises; (40)

U*b ¼ 0:02055 and F*b ¼ 1:80 for Tresca: (41)

These two results are essentially coincident, since the homo-geneous solution before bifurcation is a uniaxial compression stressfield, common in both models. After bifurcation, the stress statebecomes inhomogeneous, and the von Mises and Tresca solutionsdiffer; STRUPL2 predicts that the Tresca post-bifurcation load isslightly lower than the von Mises one.

Both analyses performed with STRUPL2 follow after bifurcationa non-symmetric equilibrium path2. Fig. 3 shows the deformedfinite element mesh for the von Mises model after bifurcation fora vertical displacement Dua ¼ 0.15L0.

As shown in Fig. 2, the STRUPL2 and ABAQUS solutions forvon Mises agree well, especially at the early stages afterbifurcation.

For materials that exhibit linear hardening, an analytical planestress estimate for the buckling load (bifurcation in Shanley’s sense(Shanley, 1947)) is available, and can be written in the followingnon-dimensional form3

0 0.025 0.05 0.075 0.1 0.125 0.15Nominal strain [-]

0

1

2

3

4

5

Nor

mal

ized

for

ce F

* [-

]

Plane strain

Homogeneous

von Mises

Tresca

Fig. 4. Compressed elastic-plastic column: plane strain solutions. The dashed lines are ABAQUS solutions with the imperfection. The lines with symbols on them are STRUPL2bifurcation solutions for a perfect column.

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 128e137 135

F*EShRES ¼ 4p2 I0

2Et;

1 ¼ 1þ 1(42)

s0 A0 A0 L0 s0 Et E h

where I0 is the initial cross-sectional moment of inertia, so thatI0=ðA0 L20Þ ¼ B20=ð12L20Þ. In the current application we have thatF*ES ¼ 1:535. The bifurcation load calculated by STRUPL2 ðF*bx1:80Þis slightly different from the analytical result because this last refersto a non-slender columnbut is obtained adopting all the assumptionsholding for a slender beam. Thenumerical solutionof course does notinclude any beam-type assumption.

6.1.2. Plane strain solutionsFig. 4 shows results analogous to those of Fig. 2, obtained now

for the case of plane strain. The force-displacement curves areorganized as before, and exhibit the same features as the planestress case, with the obvious differences in terms of the numericalvalues of the first yield force and initial hardening slope.

The linear hardening curves correspond to the homogeneousuniaxial compression solutions for the vonMises and Tresca models.

The bifurcation loads predicted by STRUPL2 are

U*b ¼ 0:0229 and F*b ¼ 2:32 for von Mises; (43)

U*b ¼ 0:0219 and F*b ¼ 1:90 for Tresca: (44)

It is interesting that the predicted displacements at bifurcationU*b differ by less than 5% in the two models; the difference in the

bifurcation loads F*b is more than 20% and reflects the corre-sponding difference in the homogeneous solution (the plane strainuniaxial compressive stress is higher in the von Mises solution).

6.2. Necking in uniaxial plane strain tension

We analyze the problem of plane strain tension of a rectangularblock with aspect ratio L0/B0 ¼ 3, where 2L0 is the length of thespecimen and 2B0 its width, and look for possible bifurcations.

The calculations are carried out for the Tresca plasticity modeldescribed in Section 6.1; the material parameters used in the calcu-lations are: Young’s modulus E ¼ 500s0, Poisson’s ratio n ¼ 0.3, andh ¼ s0.

Because of symmetry, only one quarter of the specimen isanalyzed, i.e., a rectangular block with dimensions B0�L0 isconsidered.

The solutions were developed by using STRUPL2. For compar-ison purposes ABAQUS solutions were also developed for a spec-imen with a geometric imperfection as described in the following.The elements used are the same as those described in Section 6.1. A15 � 90 finite element mesh is used in the computations.

A “perfect” specimen (i.e., with no initial geometrical imperfec-tions) was used with STRUPL2 and the bifurcation path was deter-mined by using the methodology described in Section 5. In theABAQUS solution, the size B0 of the lower section of the rectangularmesh was reduced by d ¼ 10�8B0 in the undeformed configuration,and the mesh was constructed by connecting with straight lines thebottom-side nodes to the corresponding nodes on the top surface ofthe specimen.

The specimen is stretched by prescribing a uniform upwarddisplacement ua at its upper side. The displacement ua is increasedmonotonically by imposing equal displacement increments ofmagnitude Dua ¼ 0.001667L0.

Fig. 5 shows the homogeneous solution, the STRUPL2 bifurca-tion solution, and the ABAQUS solution for the imperfect specimen.The STRUPL2 and ABAQUS solutions agree well, especially at theearly stages after bifurcation. In Fig. 5 the normalized force F* andnominal strain U* are defined as

F* ¼ Fs0 A0

and U* ¼ uaL0; (45)

where F is the tensile force on the specimen and A0 the undeformedcross-sectional area of the specimen.

The bifurcation solution predicts the formation of a neck asshown in Fig. 6. The “upper blue” part in Fig. 6 shows the extent ofelastic unloading in the specimen.

6.3. Concluding remarks

Figs. 2, 4, and 5 show that ABAQUS and STRUPL2 furnish slightlydifferent post-bifurcation paths. Apart from the effects of theutterly different basic formulations, this is essentially due to thefact that ABAQUS requires a pre-defined imperfection, in order to

0 0.1 0.2 0.3 0.4 0.5Nominal strain U* [-]

0

0.2

0.4

0.6

0.8

1

1.2

Nor

mal

ized

for

ce F

* [-

]

ABAQUSSTRUPL

Tresca

Homogeneous

Fig. 5. Plane strain tension. The dashed line is the ABAQUS solution with the imperfection; the line with symbols is the STRUPL2 bifurcation solutions for a perfect specimen.

A. Bassi et al. / European Journal of Mechanics A/Solids 35 (2012) 128e137136

predict an equilibrium path different from the fundamental one,whereas STRUPL2 starts from a perfect mesh. Any special choice ofthe initial imperfection will have effects both on the predictedbifurcation point, in ABAQUS, and on the post-bifurcation path.

Finally, a comment is in order about the computational effec-tiveness of the proposed method. We recall that the adopted inte-gration scheme computes in a fully automatic way the step-size,whereas ABAQUS and all the similar finite element codes requirethe user-defined subdivision of the load into finite steps, that can be

Fig. 6. Plane strain tension. Deformed finite element mesh of the ABAQUS solutionafter the neck is formed for a vertical displacement Dua ¼ 0.4333L0.

possibly made automatic on the basis of the number of iterationsrequired by the solution of the system of global equilibrium. Fora comparable step-size, the method proposed in this work has a costcomparablewith that ofmore traditionalmethods, and itmight evenbe cheaper, since, at each loading step, it requires at most 3 solutionsof the system of equations governing the incrementally elasticresponse of the structure. The average step-size required by thetechnique described in this work, however, is very often muchsmaller than the average step-size adopted by ABAQUS, and therelevant cost can be higher by an order of magnitude, in some cases.Although it is obvious that this can be a limitation, in practical situ-ations, we feel that it is a reasonable price to pay if onewants to havea complete and accurate solution of a problem that might otherwisenot be solved at all, or solved only in a very poor way.

Acknowledgments

Work done within a research project financed by the ItalianMinistry of Education and Research (MIUR). The Authors areindebted to Professor Alberto Franchi, of the Politecnico of Milano,Italy, for several useful discussions.

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