lode parameter and the stress triaxiality

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Inuence of the Lode parameter and the stress triaxiality on the failure ofelasto-plastic porous materials

K. Danas a,, P. Ponte Castaeda b,ca Laboratoire de Mcanique des Solides, C.N.R.S. UMR7649, cole Polytechnique, 91128 Palaiseau Cedex, FrancebDepartment of Mechanical Engineering and Applied Mechanics, University of Pennsylvania, Philadelphia, PA 19104-6315, USAcMadrid Institute for Advanced Studies of Materials (IMDEA-Materials), 28040 Madrid, Spain

a r t i c l e i n f o

Article history:Received 20 April 2011Received in revised form 8 February 2012Available online 3 March 2012

Keywords:PlasticityVoid growthPorous materialsLode parameterShear localizationHomogenizationMicrostructure evolution

a b s t r a c t

This work makes use of a recently developed second-order homogenization model to investigate failurein porous elasto-plastic solids under general triaxial loading conditions. The model incorporates depen-dence on the porosity and average pore shape, whose evolution is sensitive to the stress triaxiality andLode parameter L. For positive triaxiality (with overall tensile hydrostatic stress), two different macro-scopic failure mechanisms are possible, depending on the level of the triaxiality. At high triaxiality, voidgrowth induces softening of the material, which overtakes the intrinsic strain hardening of the matrixphase, leading to a maximum in the effective stressstrain relation for the porous material, followedby loss of ellipticity by means of dilatant shear localization bands. In this regime, the ductility decreaseswith increasing triaxiality and is weakly dependent on the Lode parameter, in agreement with earlier the-oretical analyses and experimental observations. At low triaxiality, however, a new mechanism comesinto play consisting in the abrupt collapse of the voids along a compressive direction (with small, butnite porosity), which can dramatically soften the response of the porous material, leading to a suddendrop in its load-carrying capacity, and to loss of ellipticity of its incremental constitutive relation throughlocalization of deformation. This low-triaxiality failure mechanism leads to a reduction in the ductility ofthe material as the triaxiality decreases to zero, and is highly dependent on the value of the Lode param-eter. Thus, while no void collapse is observed at low triaxiality for axisymmetric tension L 1, theductility of the material drops sharply with decreasing values of the Lode parameter, and is smallestfor biaxial tension with axisymmetric compression L 1. In addition, the model predicts a sharp tran-sition from the low-triaxiality regime, with increasing ductility, to the high-triaxiality regime, withdecreasing ductility, as the failure mechanism switches from void collapse to void growth, and is in qual-itative agreement with recent experimental work.

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1. Introduction

Due to its critical technological importance, ductile failure andfracture of metallic materials has been the focus of continuedattention over the last sixty years. The main mechanism for mate-rial failure in ductile solids is the nucleation, growth and eventualcoalescence of voids and micro-cracks as a result of the appliedloading conditions (Garrison and Moody, 1987). It has been knownfor many years that the stress triaxiality, denoted here by XR anddened as the ratio of the mean stress to the von Mises equivalentor effective deviatoric stress, is the critical parameter controllingductile failure at high triaxiality. Thus, large amounts of experi-mental data (Hancock and Mackenzie, 1976; Le Roy et al., 1981;

Johnson and Cook, 1985) have shown a monotonic decrease ofmaterial ductility with increasing stress triaxiality. This is consis-tent with the expected increase in the rate of growth of the voidswith a larger tensile hydrostatic stress component. Nonetheless,recent experimental evidence (Bao and Wierzbicki, 2004; Barsoumand Faleskog, 2007a; Mohr and Ebnoether, 2009; Dunand andMohr, 2010) suggests that a new, different mechanism shouldcome into play at low triaxialities, leading to a substantial reduc-tion of the material ductility with decreasing stress triaxiality. In-deed, in these studies, it has been found that a second loadingparameter, the Lode parameter, L (or equivalently Lode angle, h)also plays a signicant role in ductile failure at low stress triaxial-ities. The Lode parameter is a function of the third invariant of thestress deviator and is used to distinguish between the differentshear stress states in three dimensions (3-D), ranging from axisym-metric tension to biaxial tension with axisymmetric compressionand passing through in-plane shear. The key experimental observa-

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Corresponding author.E-mail addresses: [email protected] (K. Danas), [email protected]

du (P. Ponte Castaeda).

International Journal of Solids and Structures 49 (2012) 13251342

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tions are summarized in Fig. 1, which is taken from the work ofBarsoum and Faleskog (2007a). Specically, Fig. 1a shows thelow-triaxiality regime with increasing ductility as the triaxialityincreases, followed by an abrupt transition to the standard high-triaxiality regime with the opposite trend. Correspondingly,Figs. 1bd show SEM micrographs of the failure surfaces for low,intermediate and high triaxiality. At low stress triaxialities(Fig. 1(b)), the dimples are shallow and elongated suggesting sig-nicant shear plastic strains and void elongation, together withshear localization between voids. At high stress triaxialities(Fig. 1(d)), the dimples are deep suggesting the well-known voidcoalescence mechanism with necking of inter-void ligaments lead-ing to nal rupture. At an intermediate value of the stress triaxial-ity XR 0:7 0:8, a transition between the void shearing andvoid growth mechanisms is observed. In conclusion, these carefulexperimental observations strongly suggest that, void elongation(with signicant changes in shape), which is dependent on thespecic shear stress state (as measured by the Lode parameter),becomes the dominant mechanism leading to the failure of thematerial at low stress triaxialities, and should therefore be ac-counted for in the constitutive modeling of such material systems.

The underlying microstructural mechanism at large triaxialities(i.e., large mean stresses compared to the deviatoric ones) was

identied early on by McClintock (1968) and Rice and Tracey(1969), who related it to the growth of pre-existing voids or/andnucleated micro-voids mainly due to the presence of impuritiesin the material. This knowledge led to the development of thewell-known Gurson (1977) model (and its modications by Tverg-aard (1981)) which is based on a micromechanical analysis of aspherical shell, assumed to remain spherical even for general load-ing conditions. However, while this assumption is entirely consis-tent with the void growth mechanisms observed under purehydrostatic stress states, as already noted, it becomes less ade-quate with the addition of shear loads, since such loads can inducesignicant changes in the void shape. Early studies of the effect oftriaxiality on void growth, accounting for shape changes, as well asits implications for ductile failure, were carried out by McClintock(1968, 1971) and Budiansky et al. (1982) (see also Teirlinck et al.,1988). Building on these early works and on the works of Duvaand Hutchinson (1984) and Lee and Mear (1992), Gologanu et al.(1993, 1994) proposed a model for porous materials with alignedspheroidal voids that are subjected to axisymmetric stress statesaligned with the voids symmetry axis. These Gurson-type modelshave been further developed by Gologanu et al. (1997), Garajeuet al. (2000), Pardoen and Hutchinson (2000), Benzerga (2002),Flandi and Leblond (2005), Monchiet et al. (2007), Benzerga and

Fig. 1. Failure of hot rolled medium-strength steel. (a) Effective plastic strain and failure vs. stress triaxiality. (On the plot, epcf and epnf refer to the critical plastic strains at the

center of a notch and the average plastic strain at the notch, respectively.) The rest of the gures correspond to SEM fractographs showing the rupture modes: (b) a sheardimple rupture mode with inter-void shearing mechanism and elongated voids for stress triaxiality XR 0:47; (c) the transition between shear dimples and void growthrupture for XR 0:85; (d) necking of inter-void ligaments, i.e., void coalescence due to void growth, for XR 1:10. All gures are taken from Barsoum and Faleskog (2007a)and correspond to Weldox 420.

1326 K. Danas, P. Ponte Castaeda / International Journal of Solids and Structures 49 (2012) 13251342


Leblond (2010) and Keralavarma and Benzerga (2010) to accountfor more general loading conditions (and anisotropic matrix behav-ior in the case of Keralavarma and Benzerga (2010)), but still makethe rather strong approximation that the voids remain spheroidal inshape for general triaxial loading histories.

A different class of constitutive models for porous porous visco-plastic materials capable of accounting for more general (i.e., arbi-trary ellipsoidal) pore shape and orientation evolution have beendeveloped by Ponte Castaeda and Zaidman (1994), Kailasamet al. (1997a) and Kailasam and Ponte Castaeda (1997) to deal withcompletely general, three-dimensional loading conditions. Thesemodels make use of the variational linear comparison homogeni-zation method of Ponte Castaeda (1991) (see also Willis, 1991;Michel and Suquet, 1992), togetherwith the estimates of Ponte Cas-taeda and Willis (1995) for porous linear-elastic materials withellipsoidal microstructures (i.e., particulate microstructures con-taining orthotropic distributions of ellipsoidal pores), to generatecorresponding estimates for the dissipation potential of the visco-plastic porous materials. They are supplemented by evolution lawsformicrostructural variables corresponding to the porosity, averagepore shape and orientation, which are obtained from the homogeni-zation analyses in a self-consistent fashion (Ponte Castaeda andZaidman, 1994; Kailasam and Ponte Castaeda, 1998).

The above-mentioned non-linear homogenization methodshave also been extended to include strain hardening elasto-plasticbehavior for the matrix material, and implemented numerically inlarge-scale, structural nite element programs by Kailasam et al.(2000) and Aravas and Ponte Castaeda (2004). While these mod-els are quite general, they tend to give overly stiff predictions athigh triaxialities (i.e., they are quite a bit stiffer than the Gurson-based models), especially for small porosity levels. However, thislimitation has been removed, at least for isotropic matrix systems,in recent work by Danas and Ponte Castaeda (2009a), Danas andPonte Castaeda (2009b), making use of the more accurate sec-ond-order linear comparison homogenization method of PonteCastaeda (2002a), Ponte Castaeda (2002b), and building on ear-lier works by Danas et al. (2008a), Danas et al. (2008b) and Danas(2008). The resulting model, which will be referred to here as theSOMmodel, will be extended to account for strain-hardening, elas-to-plastic behavior for the matrix phase, and will be capable ofhandling general ellipsoidal particulate microstructures and gen-eral three-dimensional loading conditions, including those leadingto pore rotation, while remaining quite accurate at large stresstriaxialities and recovering the Gurson model for purely hydro-static loadings and spherical pores.

Application of the linear comparison constitutive models to var-ious types of loading conditions has revealed the importance ofvoid shape evolution in determining the overall response of plasticporous solids. For example, it was found by Ponte Castaeda andZaidman (1994) that under uniaxial tension the softening inducedby the growth of porosity associated with the Gurson model forideally plastic porous materials is overpowered by pore shapechanges, since the pore elongation in the tensile direction providesa hardening mechanism resulting in overall hardening for the por-ous materialin agreement with numerical simulations (see Kaila-sam et al. (1997b)). However, the evolution of the void shape canalso induce overall softening of the porous material, and in fact itwas shown by Ponte Castaeda and Zaidman (1994), and con-rmed by Danas and Ponte Castaeda (2009b), that a porous rigidideally plastic material could even lose ellipticity by void collapseleading to shear band formation at low triaxialities. It is importantto emphasize that such an effect could not be captured by theGurson model, since at low-triaxiality conditions the source ofthe instability cannot be identied with a void growth mechanism(Yamamoto, 1978). In this context, it should also be mentionedthat Nahshon and Hutchinson (2008) have proposed an ad hoc

modication of the Gurson model in an attempt to account for soft-ening of the material at low stress triaxialities. While, by construc-tion, this modication brings in an effect of the Lode parameter, itis inconsistent with mass conservation, and still fails to account forthe development of morphological anisotropy associated with poreshape changes.

Motivated by the above observations, in the present work, wewill make use of the SOM model to investigate the inuence ofthe Lode parameter (i.e., the different shear stress states) and thestress triaxiality on the overall behavior of porous elasto-plasticmaterials and the possible development of macroscopic instabil-ities (Geymonat et al., 1993) due to the evolution of the underlyingmicrostructure, e.g., void growth and void shape changes. In partic-ular, we will consider two possible failure mechanisms for the por-ous medium: (i) the existence of a limit load (i.e., a maximum pointon the effective stressstrain curve or equivalently zero materialhardening rate) and (ii) loss of ellipticity of the incremental re-sponse leading to localization of deformation in dilatant shearbands, as discussed originally by Rice (1976). It should be notedin this connection that while loss of ellipticity calculations will leadto predictions that are typically on the high side when compared toexperimental results, these arematerial instabilities which can pro-vide useful information about the theoretical load-carrying capacityof the material. In actual experiments, the loading conditions andspecimen geometry will invariably lead to non-uniform elds, suchthat the instabilities nucleate at critical locations in the specimenwhere the local elds are in excess of the applied average elds,leading to progressive failure of the material by propagation ofthe instability into the specimen.

In this work, we will not attempt to model specic experimen-tal conditions, nor structural geometries. However, it is relevant tonote that, based on the earlier variational model of Ponte Casta-eda and Zaidman (1994), constitutive user-material subroutinesfor implementation in nite element codes have been developedby Kailasam and Ponte Castaeda (1997), Kailasam et al. (2000)and Aravas and Ponte Castaeda (2004). Recently, Danas and Ara-vas (2012) have proposed a modication of these earlier modelsin accord with the present SOM model that recovers the sphericalshell solution (i.e., Gursons hydrostatic point) at purely hydro-static loadings, while including all the advanced features of thevariational and second-order methods such as arbitrary ellipsoidalvoid shapes and general loading conditions. In addition, in thiswork we will not address microscopic coalescence criteria (seeBenzerga et al. (1999), Benzerga and Leblond (2010) for detailson this alternative approach to material failure). It should be re-marked in this connection that the incorporation of loss of ellip-ticity predictions into numerical simulations of actual structuralproblems (including crack propagation) is a challenging and stilllargely open problem, which is also beyond the scope of thiswork.

For clarity and simplicity, in our analysis we will considerpurely triaxial loading conditions and initially spherical voids sothat the void orientation vectors remain aligned with the principalloading conditions for the entire deformation process and there-fore do not contribute to the overall material response. Note, how-ever, that for the case of general (non triaxial) loading conditions orinitially anisotropic microstructures (i.e., initially ellipsoidal voidsmisaligned with the laboratory frame axes), the void orientationvectors evolve due to nite deformations and could hence affectthe overall response of the porous material (e.g. Kailasam and Pon-te Castaeda (1997), Kailasam et al. (2000), Aravas and Ponte Cas-taeda (2004), Danas and Ponte Castaeda (2009b)). Theinvestigation of void rotation effects in this context will be leftfor future work, but it should be mentioned that recent numericalstudies by Barsoum and Faleskog (2007b) and Tvergaard (2009)have shown that rotation and elongation of the voids along the

K. Danas, P. Ponte Castaeda / International Journal of Solids and Structures 49 (2012) 13251342 1327


shearing direction can contribute to the localization of deformationand subsequent failure of the material.

The rest of the paper is organized as follows. First, in Section 2,we describe the geometry and loading conditions, and dene thepertinent variables used in this study including the stress triaxial-ity and the Lode parameter. In this section, we also describe themicrostructure and present a brief summary of the SOM model,as well as the characterization of the limit load and localizationconditions used in this study as failure criteria for the porous med-ium. Then, Section 3 discusses the results obtained by the SOM forthe evolution of the stress and the underlying microstructure un-der nite deformations, and compares them with the (isotropic)modied Gurson model of Nahshon and Hutchinson (2008), re-ferred to as MGUR below. Limit load and localization maps are con-structed as a function of the stress triaxiality and the Lodeparameter. A parametric study to investigate the inuence of thematerial hardening rate and initial porosity on the limit load andlocalization maps is also carried out. Finally, we conclude withsome general comments and perspectives for future work.

2. The non-linear homogenization model

This section briey describes the application of the second-order nonlinear homogenization model (SOM) of Danas and PonteCastaeda (2009a) for porous elasto-plastic materials subjected totriaxial loading conditions. We rst dene the stress triaxiality andLode parameters, followed by microstructural variables describingthe volume fraction, shape, distribution and orientation of the voids.Next, buildingon theworkofAravas andPonteCastaeda (2004),wedevelop consistent constitutive relations for the elastic and plasticdeformations of the porous medium, and provide evolution lawsfor the above-mentioned microstructural variables, as well as thestrain hardening law for the matrix material. Finally, expressionsfor the hardening rate and the localization conditions are derived.

2.1. Triaxial loading conditions: Stress triaxiality and Lode parameter

This subsection discusses the loading conditions and the associ-ated stress measures used to distinguish between hydrostatic load-ing and different shear stress states. We consider purely triaxialloading conditions with the principal stresses r1 r11;r2 r22and r3 r33 (rij 0 for i j) being aligned with the laboratoryframe axes, e1; e2 and e3, respectively. This allows for the de-nition of alternative stress measures that are more appropriatefor dilatational plasticity of porous materials. The three alternativemeasures are the hydrostatic (or mean) stress, rm, the von Misesequivalent (or effective) stress, re, and the third invariant of thestress deviator, J3, dened as

rm rkk=3; re 3J2

p

3sijsij=2

q; J3 detsij; 1

where sij rij rmdij is the stress deviator. Using these denitions,we can readily dene the stress triaxiality, XR, and Lode angle, h, orLode parameter,1 L, via the following expressions

XR rmre ; L cos 3h 272

J3r3e

: 2

By denition, the range of values for the XR and L, (or h) are

1 < XR 0, as shown in Fig. 2(b) for XR 1, or downwardsfor XR < 0 (not shown here for brevity). Note also that jXRj ! 1and XR 0 correspond to purely hydrostatic and purely deviatoricloadings, respectively.

2.2. Microstructure

The porous material is composed of two phases. The matrixphase is elasto-plastic and isotropic following a J2-ow rule withisotropic strain hardening described by the yield stress ry as afunction of the accumulated equivalent plastic strain epM . The inclu-sion phase is vacuous and consists of initially spherical voidsdistributed uniformly and isotropically, such that the initial re-sponse of the porous medium is also isotropic. However, due tothe nite deformations considered in this problem the voids evolveinto non-spherical shapes and hence the porous medium becomeslocally anisotropic (i.e., develops morphological anisotropy). Con-sequently, it is necessary to dene microstructural variables thatnot only describe the volume fraction of the voids, as is the casein the models of Gurson (1977) and Nahshon and Hutchinson(2008), but also their shape, distribution and orientation.

According to the schematic representation shown in Fig. 3(a)and at some nite deformation state, we consider that the porousmaterial is characterized by a particulate microstructure consist-ing of ellipsoidal voids (i.e., with semi-axes a1a2a3) aligned in acertain direction as a result of the previously described triaxialloading conditions. In addition, it is assumed (Willis, 1978; PonteCastaeda andWillis, 1995) that the centers of the voids are distrib-uted with ellipsoidal symmetry (see Fig. 3(b)). This description of aparticulate microstructure represents a generalization of the Eshel-by (1957) dilute microstructure to the non-dilute regime. In thiswork, which is based on the model of Danas and Ponte Castaeda(2009a), we will make the simplifying assumption that the ellipsoi-dal shape and orientation of the distribution function is identical tothe ellipsoidal shape and orientation of the voids at each stage ofthe deformation. This is schematically shown in Fig. 3(b), wherethe dashed ellipsoids representing the pore distribution are takento have the same ellipsoidal shape as the actual pores (in white).This assumption has been shown (Danas and Ponte Castaeda,2009b) to provide accurate estimates, especially at small to moder-ate porosities. Nonetheless, it should be mentioned that, in general,the void distribution shape could be different from the void shape,as discussed by Ponte Castaeda and Willis (1995), and this effectcan be accounted for at least approximately (Kailasam et al., 1997a).

Moreover, in the present study we consider purely triaxialloading conditions and initially isotropic materials (i.e., comprising

1 Note that our choice for the Lode parameter L differs from the standard denition,l 2r1 r3 r2=r3 r2, but the two parameters are simply related byL l9 l2=

l2 33

q, and therefore agree for the values 1, 0, and +1.



initially spherical voids). This implies that the orientation of thevoids remains xed and aligned with the triaxial loading condi-tions. Thus, the vectors ni (with i 1;2;3) denoting the orienta-tion of the principal axes of the voids (see Fig. 3(a)) remainaligned with the principal laboratory axes ei. Consequently, theporous medium becomes, at most, orthotropic with nite deforma-tions, with the axes of orthotropy coincidingwith the principal axesof the ellipsoidal voids and the laboratory frame axes, i.e., withni ei. It should be emphasized, however, that the model ofDanas and Ponte Castaeda (2009a) can account for more generalloading conditions, non-spherical initial void shapes and rotationof voids, as has already been shown in Danas (2008) and Danasand Ponte Castaeda (2009b), but such a study is not carried outhere because it will not be needed to describe the effects of interestin this work.

In view of the above hypotheses, the relevant internal variablesdescribing the state of the microstructure in this problem are:

sa fepM ; f ;w1;w2g; 5where epM is the accumulated plastic strain in the undamaged matrixphase, f is the porosity (i.e., volume fraction of the voids), andw1 a3=a1 and w2 a3=a2 are two aspect ratios characterizing theellipsoidal shape of the voids (with a1; a2 and a3 denoting the princi-pal semi-axes of the ellipsoidal voids) and their distribution function.

2.3. Elasto-plastic constitutive relations

The overall strain-rate D in the porous material is decomposedinto its elastic and plastic parts via

D De Dp; 6where De and Dp, respectively, denote the elastic and plastic parts.Note that due to the presence of voids the overall material behavioris compressible (i.e., pressure dependent) implying that the plasticstrain-rate tensor is not deviatoric (i.e., Dpkk0). On the other hand,due to the triaxial loading conditions and the fact that the voids donot rotate during the deformation process, the overall spin as wellas the microstructural spins are identically zero. In addition, in viewof the fact that the pores can carry no loads and following Aravasand Ponte Castaeda (2004), it is assumed that the elastic and plas-tic parts of the strain rate can be estimated by independent, butconsistent homogenization analyses.

Thus, the elastic response of the porous material is described interms of an effective compliance tensor M via

Dekl Mijkl _rij; with Mijkl MMijkl f

1 f Q1ijkl; 7

where _r represents the material time derivative of the stress, whichwill be taken here to be given by the (partial) derivative with re-spect to time, since the stress is assumed to be uniform and the spinis zero, and Q Q w1;w2;ni ei is directly related to the well-known Hill or Eshelby tensor for ellipsoidal microstructures and itsevaluation is detailed in Willis (1981) (see also Danas (2008)). Thefourth-order tensor MM is the compliance modulus of the matrix(metallic) phase and is taken to be isotropic such that

MMijkl 1 mE

12dikdjl dildjk m1 m dijdkl

; 8

where E and m denote the elastic modulus and Poisson ratio,respectively.

On the other hand, the yield condition for the porous materialcan be written in the functional form

Ur; sa r^eqr; f ;w1;w2 ryepM 0; 9where r^eq is a scalar function of the stress tensor and the micro-structural variables, which is detailed in Danas and Ponte Castaeda(2009a) (c.f. Eqs. (25) and (28)), while ry is the effective stress gov-erning ow of the undamaged matrix material and in general de-pends on the accumulated plastic strain epM in the matrix phase.

a b

Fig. 2. Normalized principal stresses 32re fr1;r2;r3g, as a function of the Lode angle h or equivalently the Lode parameter L. Parts (a) and (b) correspond to stress triaxialitiesXR 0 and XR 1, respectively.

( )3e ( )2e( )1e

3a

( )3n

2a

1a( )2n( )1n

a b

Fig. 3. Graphical representation of the microstructure. Part (a) shows the localorientation axes ni with i = 1, 2, 3 of a representative ellipsoidal void with semi-axis a1; a2 and a3. Part (b) shows the a cross-section of the specimen where thewhite ellipsoids denote voids with ellipsoidal shape while the dashed ellipsoidsrefer to the distribution of their centers.



The overall plastic strain rate Dp of the porous material is then ob-tained from the normality rule via

Dpij _KNij; Nij @U@rij

; 10

where _KP 0 is the plastic multiplier, which is determined by theconsistency condition as discussed in subSection 2.5, and Nij is thenormal to the yield surface U. The reader is referred to Danas andPonte Castaeda (2009a) for more detailed expressions for U andNij.

2.4. Evolution equations

Following the work of Ponte Castaeda and Zaidman (1994),Kailasam and Ponte Castaeda (1998), Aravas and Ponte Castaeda(2004) and Danas and Ponte Castaeda (2009a), evolution equa-tions are given in this section for the microstructural variablesepM; f ;w1 and w2 dened in relation (5). Once again, in this workthe orientation vectors remain aligned with the principal loadingdirections, i.e., ni ei (i 1;2;3), during the deformationprocess.

The evolution equation for the accumulated plastic strain in thematrix phase epM is determined by the condition (Gurson, 1977)that the macroscopic plastic work rijDpij be equal to the correspond-ing microscopic plastic work 1 f ry _epM , which implies that

_epM rijDpij

1 f ry _K

rijNij1 f ry : 11

For strain hardening materials, ry is a function of epM , which, in gen-eral, is to be extracted from experimental uniaxial stressstraincurves. In our work, a rather general strain hardening law forryepM will be given in the results section.

Any changes of the pores are assumed to be only the result ofplastic deformations (Aravas and Ponte Castaeda, 2004) whileelastic deformations are considered to have a negligible effect onthe evolution of the voids volume fraction. Noting further thatthe matrix material is plastically incompressible (J2 plasticity),the evolution equation for the porosity f follows easily from thecontinuity equation and reads

_f 1 f Dpkk _K1 f @U@rkk

: 12

We point out that void nucleation is not considered in the aboverelation but can be readily included by proper modication of(12) (e.g., Needleman and Rice (1978); Chu and Needleman(1980); Tvergaard (1990)).

The evolution of the aspect ratios w1 and w2, describing theshape of the voids, is given in terms of the average strain-rate inthe vacuous phase Dv such that

_ws wsn3i n3j nsi nsj Dvij _Kysw r=ry; sa; with ni ei;13

and no sum on s = 1, 2. The average strain-rate Dv in the vacuousphase is estimated via the linear comparison material, as discussedin Danas and Ponte Castaeda (2009a) (see Eq. (76) in that refer-ence). Finally, the associated functions ysw have also been given inDanas and Ponte Castaeda (2009a) (see relation (80) in that refer-ence) and will not be repeated here.

2.5. The consistency condition and the hardening rate

In this subsection, we determine the plastic multiplier _K andhardening rate H by means of the consistency condition(Dafalias, 1985) for continuously applied loading, which in thiscase reads

_U @U@rkl

_rkl @U@epM

_epM @U@f

_f Xs1;2

@U@ws

_ws 0: 14

Substitution of the evolution Eqs. (11)(13), and of @U=@epM dry=depM , in this last relation provides the following expressionfor the plastic multiplier

_K 1H

@U@rkl

_rkl 1HNkl _rkl; 15

where H is the hardening rate dened by

H rijNij1 f rydrydepM

1 f Nkk @U@f

Xs1;2

@U@ws

ysw : 16

The hardening rate is a measure of the overall hardening of the por-ous material. When H > 0, the material is said to harden, whilewhen H < 0, it is said to soften. The critical point when H 0 usu-ally provides the transition from the hardening regime to the soft-ening regime, and can be identied with the maximum stress orlimit load of the material. Clearly, the maximum stress is importantfor stress-controlled boundary conditions, since the material willnot be able to support stresses exceeding the limit load, and thematerial will fail at this point under increasing stress.

By observation of relation (16), we note that the rst two termsof the right-hand side appear also in the Gurson (1977) and theNahshon and Hutchinson (2008) models, and incorporate the ef-fects of the matrix strain hardening and the evolving porosity (ordamage in Nahshon and Hutchinson (2008)) on the overall re-sponse of the porous material. By contrast, in the present model,additional terms appear in (16), due to void shape changes. Thislast term of the right hand side in (16), which comprises the evo-lution of the two aspect ratios w1 and w2, affects the overall hard-ening rate of the porous material in a nontrivial manner. All theseeffects will be investigated in detail in the next section.

2.6. Localization conditions

In this subsection, we summarize the localization conditionscorresponding to the loss of ellipticity of the governing equationsand leading to non-unique solutions, bifurcations and instabilities,as described by Rice (1976). By making use of denition (15), theincremental constitutive relations (6), (7) and (10) describing theoverall elasto-plastic response of the porous material can be writ-ten in the form

_rij LincijklDkl; where Lincijkl Lijkl NpqLpqijLklmnNmnH NrsLrsuvNuv 17

is the effective incremental elasto-plastic modulus of the porousmaterial, and L M1 is the effective elastic modulus of the porousmaterial.

Following Rice (1976), we consider an innite porous mediumwith no initial imperfections, which implies that the trivial solu-tion to this problem is homogenous deformation throughout theinnite region. Then, we look for conditions under which the defor-mation would localize inside a thin band leading to unloading out-side the band, as shown schematically in Fig. 4. This secondsolution to the problem is a discontinuous bifurcation of the uni-form solution and leads to a lower energy state than the uniformone. As already known, the specimen would tend to localize earlierif an initial imperfection were considered. However, the goal of thepresent study is to investigate pure material instabilities leavingaside any geometrical imperfections for a future study whereactual boundary value problems resulting from experimentalgeometries will be investigated.

In any event, the condition for the localization of deformationinside a thin band with normal ni becomes (Rice, 1976; Needlemanand Rice, 1978)



det niLincijklnl Ajk

h i 0; where

2Ajk rjk rjsnsnk nprpqnqdjk njnrrrk: 18When this localization condition is rst met in a program of defor-mation, the difference between the total strain-rate inside and out-side the band can be written as DDij ginj nigj=2, with gi being afunction only of distance across the localization band nixi (with xibeing the position vector). The use of ni and gi provide informationabout the deformation state inside the localization band. For in-stance, in the case that the material is fully incompressible, it canbe shown that gi is perpendicular to ni and parallel to the band tan-gent vector ti which implies that the deformation state in the bandis simple shear, i.e., a shear localization band. In the present study,however, the material is compressible due to the nite porosity, andcan accommodate deformation states other than simple shear in-side the band. In that case, ni and gi are not necessarily perpendic-ular to each other as shown in Fig. 4, which can lead to a nonzeronormal component of the deformation state inside the band, i.e.,niDDijnj 0.

In connection with the above-described localization conditions,it should be emphasized that the (uniform) solutions obtained di-rectly from the constitutive model for the porous material wouldcease to be valid at the point of the instability. Then, a post-bifur-cation analysis would be required beyond this point. Such an anal-ysis should make use of geometrical effects or initial imperfectionsand is outside the scope of the present work, which focuses on uni-form solutions under xed stress triaxialities and Lode parameterloadings throughout the entire deformation history. However, inthe results to be described in the next section, the (uniform) solu-tions will still be shown beyond the onset of said instabilities,mostly because they are suggestive of the mode of the onset ofthe instability. Of course, such solutions are not meant to be repre-sentative of what actually happens beyond the instability. As al-ready known from investigation in other contexts (e.g., failure ofber-reinforced composites), the nal failure mode requires thefull post-bifurcation analysis. More often than not, such failuremodes are inherently different from the mode of the onset of theinstability.

3. Results and discussion

As already mentioned in the previous section, our objective is toinvestigate the effects of the stress triaxiality XR and Lode param-eter L (or Lode angle h) on the macroscopic response and failure ofporous elasto-plastic materials subjected to triaxial loading condi-tions. Given the fact that a maximum stress is expected, in this

work the strain rate D33 will be prescribed, together with the val-ues of XR and L, which will serve to determine all three (principal)stresses, r1;r2 and r3, as well as the evolution of the microstruc-tural variables, the porosity f, and the average aspect ratios, w1 andw2, as functions of time t. However, it will be convenient to use as atime-like variable the total equivalent strain ee

Rt

2D0ijD

0ij=3

qdt,

with D0ij denoting the strain-rate deviator, and to consider the over-all von Mises equivalent stress re instead of the individual stresscomponents in the characterization of the macroscopic response.Because of the special loading conditions imposed, it can be shownthat the maximum on the re versus ee plots will correspond exactlyto a vanishing hardening rate H 0, indicating a possible instabil-ity under stress-controlled loading conditions. In addition, the lossof ellipticity condition will be determined for the material makinguse of the condition (18). For completeness, a comparison will alsobe made between the predictions of the second-order model(SOM) of Danas and Ponte Castaeda (2009a) and the modiedGurson model (MGUR) proposed by Nahshon and Hutchinson(2008). In keeping with standard practice (Barsoum and Faleskog,2007a), the maximum stress (i.e., the locus of points whereH 0) and loss of ellipticity (LOE) conditions will be displayed interms of the total equivalent plastic strain (or effective plastic

strain) epe Rt

2Dpij0Dpij0=3

qdt, with Dpij0 denoting the plastic

strain-rate deviator. In this work, the resistance of the materialto failure by either condition will be referred to as the overall duc-tility. Furthermore, it should be emphasized, that as a consequenceof the very small magnitude of the overall elastic strains, the differ-ence between the overall total strain and the overall plastic strainis very small for all practical purposes. Finally, a parametric studywill be carried out to investigate the inuence of different matrixstrain hardening exponents and initial porosities on the limit loadand LOE maps.

3.1. Material parameters and initial conditions

The Youngs modulus and Poissons ratio of the matrix phase aretaken to be E 200GPa and m 0:3, respectively, and the matrixphase to exhibit isotropic strain hardening following the law2

ryepM r0 1epMe0

N; e0 r0=E: 19

In this expression, r0 and e0 denote the initial yield stress and yieldstrain of the matrix material (i.e., the material with f 0, andN 6 1 is the strain hardening exponent. Typical values for theseparameters are r0 200MPa and N 0:1, which will be usedthroughout this work except in Section 3.4 where a parametricstudy is carried out with N=0.01, 0.05, 0.1 and 0.2.

The matrix phase is taken to be initially unloaded with zeroaccumulated plastic strain epM 0, while the voids are initiallyspherical with w1 w2 1. The initial porosity is taken to bef0 1% except in Section 3.4 where a parametric study is carriedout with f0 0:1%;1% and 5%. It should be noted that the depen-dence of the failure maps on the Youngs modulus, and Poissonsratio has been found to be weak, and for this reason no results willbe reported here for different values of these parameters.

3.2. Stressstrain response and microstructure evolution results

In order to investigate the main effects of the stress triaxialityand Lode parameter on the effective response of the porous

3

2

n

g

n3

21

t

Fig. 4. Graphical representation of a localization band. The gure on the rightshows the local system of coordinates, where n is the normal to the band and t isthe tangent. The angle between n and g provides the deformation inside the band.For instance, if n ? g, the deformations inside the band is a simple shear. However,due to the compressibility of the porous material g is not, in general, perpendicularto n and the deformation inside the band can also have normal components (e.g.niDijnj 0), leading to the formation of a dilatant shear band.

2 It should be noted here that any hardening law for the matrix phase involvingtemperature effects or different non-monotonic strain hardening stages can be readilytaken into account. However, the simple isotropic model will sufce for the purposesof this work.



material, we show results for three representative values of stresstriaxialities, XR 0:1;0:6;1, and four of the Lode parameter,L=1,0.5,0,1 (or Lode angle h 0;20;30;60o, respectively).

Fig. 5 shows plots of (a) the equivalent stress re, (b) the porosityf, and the aspect ratios (c) w1 and (d) w2, as a function of the equiv-alent strain ee, for given values of the Lode parameter and a low va-lue of the stress triaxiality (XR 0:1). The main observation inFig. 5a is that the Lode parameter strongly affects the onset of soft-ening (i.e., maximum load) and localization of the porous material.For axisymmetric tensile loadings (L 1), the stress increases fol-lowing the prescribed strain hardening law of the matrix phase(N 0:1 here). On the other hand, for L 0:5;0, and 1, we ob-serve abrupt drops in re at different levels of the total strain ee,indicating a sudden loss in the load-carrying capacity of the mate-rial. In addition, after the maximum stress re (see inset graph inFig. 5(a) or limit load (black dot on the graph) strong softening ofthe material is observed eventually leading to localization andhence loss of ellipticity (open circle on the graph) of the homoge-nized equations.

With the objective of shedding light on the mechanism leadingto this sharp stress drop, it is necessary to consider the evolution ofthe microstructural variables, f ;w1 and w2, provided in Figs. 5 (b),

(c) and (d), respectively. In part (b), we observe an overall reduc-tion in the porosity f as a function of ee up to the point of the limitload (black dot on the graph), followed by a sharp increase in fshortly after the maximum stress has been achieved. It is clearby Fig. 5(b) that at the strain level at which the limit load and lossof ellipticity occur, the porosity is still very small. Therefore, thecorresponding stress drop observed in part (a) cannot be due tothe increase in the porosity, and the only microstructural variablesthat can possibly affect the overall response of the porous materialare the aspect ratios, w1 and w2. As shown in part (c), w1 can be-come rather large for L 1, but remains below the value of 5for L > 0:5. On the other hand, as shown in part (d), w2 increasesvery fast for all values of L > 1. In particular, for L 1 (corre-sponding to axisymmetric compression along the x2 direction,see Fig. 2(a)),w1 1, whilew2 blows up at a certain critical valueof ee (around 0.6). This means that the voids collapse in the x2 direc-tion, becoming attened cracks (lying in the x1 x3 plane) witha2 ! 0, while the material becomes locally anisotropic (i.e., exhib-its strong morphological anisotropy due to the very signicant voidshape changes). However, since the porosity f remains nite at thiscritical point where a2 ! 0; a1 a3 must tend to innity, sug-gesting coalescence of the voids in the x1 x3 plane.

a

c d

b

Fig. 5. Plots of the SOM estimates for (a) the equivalent stress re , (b) the porosity f, and the aspect ratios (c) w1 and (d) w2 as a function of the equivalent strain ee , for a lowvalue of the stress triaxiality (XR 0:1) and four values of the Lode parameter. The inuence of the Lode parameter is dramatic at low triaxialities mainly due to the extremelysharp evolution of the aspect ratio w2 in (d). The strain hardening exponent is N 0:1 and the initial porosity f0 1%. The inset in part (a) shows a blow up of the regionaround the maximum stress for L 0:5 (or h 20o).



To clarify this failure mechanism further, it is recalled here thatthe aspect ratios serve to denote both the shape of the voids as wellas the shape of their distribution function. Hence, as a2 ! 0 anda1 a3 !1 both the shape of the voids and the shape of their dis-tribution function become extremely at in the x1 x3 plane. Thisobservation together with the fact that the porosity is small but -nite, implies that the pores grow without a bound in the x1 x3plane, eventually linking up to form layers of pores in the solidmaterial, which can be associated with void coalescence in thatplane and subsequent loss of the load-carrying capacity of thematerial in the transverse direction. Such a failure mechanismwould be consistent with the at dimples observed in Fig. 1(b)from the experimental results of Barsoum and Faleskog (2007a)at low stress triaxialities. (Note, however, that the presence ofthe second-phase particles may interfere with the collapse of thevoids, and should be accounted for in situations where the voidsare not pre-existing, but instead nucleate from second-phase par-ticles.) For other values of L with 1 < L < 1, essentially the samemechanism is observed except that in this case the pores alsochange shape in the collapse plane (x1 x3). However, as can beseen in Figs. 5a and d, the effect becomes more pronounced asthe value of L increases from 1 toward 1. At the extreme valueof L 1, the shape of the pores is constrained to remain circular

in the x1 x2 cross-section, and this kinematic restriction preventscollapse of the pores, explaining the lack of a maximum stresspoint and corresponding loss of ellipticity in this case.

Fig. 6 shows plots of re; f ;w1 and w2 as a function of the equiv-alent strain ee, for several xed values of the Lode parameter L andfor a high value of the stress triaxiality (XR 1). The main result isthat the effect of the Lode parameter on the overall mechanical re-sponse of the porous material is negligible, as can be seen inFig. 6(a), since all the re ee curves almost coincide. In particular,they exhibit a limit load at rather low strains and then smooth butsignicant softening as the deformation progresses. Note furtherthat for L 1 no LOE (open circles on the plots) is detected. How-ever, failure of the porous material is not excluded (see the signif-icant drop of the material loading capacity). As already pointed outby Rice (1976), this type of localization analysis based on uniformelds only provides an upper bound for failure while the presenceof more realistic geometries can lead to localization much earlier.

The fact that the stressstrain curve is independent of the Lodeparameter at XR 1 is easily explained by referring to Fig. 6(b),where the increase of porosity is signicant for all values of theLode parameter (L 1;0:5;0;1). In addition, looking at parts(c) and, especially, (d), we note that the void shape still evolvesas a function of ee, but in a much weaker manner than for the

a b

dc

Fig. 6. Plots of the SOM estimates for (a) the equivalent stress re , (b) the porosity f, and the aspect ratios (c) w1 and (d) w2 as a function of the equivalent strain ee , for a highvalue of the stress triaxiality (XR 1) and four values of the Lode parameter. The inuence of the Lode parameter becomes negligible in this case since the response of theporous material is dominated by the signicant evolution of porosity f. The strain hardening exponent is N 0:1 and the initial porosity f0 1%.



previous case of XR 0:1. This indicates that the main softeningmechanism in this high-triaxiality situation (XR 1) is clearlythe evolution of porosity which is found to lead to signicant soft-ening of the effective response of the porous material. Note thatthis void-growth mechanism is expected to eventually lead to(three-dimensional) coalescence of the voids, and failure consis-tent with the deep dimples observed in the micrographs shownin Fig. 1(d) from the experimental results of Barsoum and Faleskog(2007a). Also, it is clear that the dominance of the evolution ofporosity will prevail at larger stress triaxialities XR > 1 not shownhere (but see Danas and Ponte Castaeda (2009b)).

Fig. 7 shows plots of re; f ;w1 and w2 as a function of the equiv-alent strain ee for several values of the Lode parameter(L 1;0:5;0;1) and a moderate value of the stress triaxiality(XR 0:6). As can be observed in part (a), for L 1;0:5;0, thestress curves reach a maximum (limit load) and then smoothly de-crease, leading to overall softening for larger values of the strain ee.On the other hand, the L 1 curve exhibits a sharp decrease of re,albeit less dramatic than the corresponding one for XR 0:1. More-over, it is interesting to note that the limit load occurs at lower eewhen L 1 than when L 0:5 or L 0. In fact, as L increases to

the value of 0, the critical strain ee at which the limit load occursincreases, whereas it decreases again as we increase further L to-ward the value of 1. This non-monotonic dependence on the Lodeparameter L can be understood by considering the evolution of themicrostructure shown in parts (b), (c) and (d) of Fig. 7. In Fig. 7(b),the porosity increases for all values of L, with the weakest growthobserved for L 1 and the strongest for L 1 (reaching relativelyhigh values at this last case). In turn, in part (c), w1 increases sim-ilarly to the previous case of XR 0:1. In Fig. 7(d), consideringL 0:5;0, we nd that w2 does not exhibit the sharp increase ob-served in Fig. 5(d) for XR 0:1 (for the same values of L). This ex-plains the smooth softening (gentle decrease of re) of the porousmaterial observed in the curves of part (a), for L 1;0:5, and0. By contrast, when L 1;w2 increases sharply attaining very highvalues corresponding to void collapse, leading to a sharp drop ofthe stress (similar to the corresponding case for XR 0:1). Thisexample reveals that at moderate values of the stress triaxiality(e.g., XR 0:6) there is a transition from softening induced by voidgrowth for L 1;0:5 to failure induced by void collapse for L 1,while for L 0 the failure mechanism is a combination of both voidshape and porosity effects.

a b

dc

Fig. 7. Plots of the SOM estimates for (a) the equivalent stress re , (b) the porosity f, and the aspect ratios (c) w1 and (d) w2 as a function of the equivalent strain ee , for amoderate value of the stress triaxiality (XR 0:6) and four values of the Lode parameter. The inuence of the Lode parameter is signicant in this case of moderate triaxialityindicating a transition mechanism from void collapse-dominated response for L 1 to porosity-dominated response for L 6 0. The strain hardening exponent is N 0:1 andthe initial porosity f0 1%.



3.3. Limit load and loss of ellipticity failure curves

The purpose of this section is to analyze and summarize theeffect of the stress triaxiality and the Lode parameter on the limitload and LOE failure instabilities. For completeness, the predictionsof the present second-order model (SOM) will be compared andcontrasted with the corresponding predictions of the recently pro-posed phenomenological model of Nahshon and Hutchinson(2008), labeled here as MGUR, which is based on an empirical mod-ication of thewell-knownGurson (1977)model. TheMGURmodelrequires the choice of the parameter kx (see expression (10) in thereferenced publication for the denition of kx) which is directlyrelated to the Lode parameter. For the identication of this param-eter several experiments have been performed indicating a value of

kx between 1 and 3. In our study, we make the choice kx 2:5,without insisting on the quantitative aspects of the results, butrather on their qualitative nature.

Fig. 8 shows plots of the SOM and MGUR predictions for thecritical equivalent plastic strain epe attained at the limit load (i.e.,the maximum in the re ee curve, or equivalently, the criticalhardening rate H 0), as a function of the stress triaxiality XR,for xed values of the Lode parameter L (or Lode angle h). As de-picted in Fig. 8(a), for xed values of L, the SOM predictions clearlyexhibit two regimes, a low-triaxiality regime where the materialductility increases with increasing triaxiality, followed by a second,high-triaxiality regime with the opposite trend. The two regimesare separated by a rather abrupt transition, or high-ductilitypeak, and as already pointed out, in the low-triaxiality regime,

a b

Fig. 9. Loss of ellipticity failure curves as predicted by (a) the SOM model and (b) the MGUR model with kx 2:5, as a function of the stress triaxiality XR and the Lodeparameter L (or h). The critical equivalent plastic strain epe at loss of ellipticity, with localization of deformation into dilatant shear bands taking place, provides an alternativemacroscopic measure of the overall ductility of the material. The strain hardening exponent is N 0:1 and the initial porosity f0 1%.

a b

Fig. 8. Limit load failure curves as predicted by (a) the SOM model and (b) the MGUR model with kx 2:5, as a function of the stress triaxiality XR and the Lode parameter L(or h). The critical equivalent plastic strain epe at the limit load where the hardening rate H 0 provides a macroscopic measure of the overall ductility of the material. Thestrain hardening exponent is N 0:1 and the initial porosity f0 1%.



a b

Fig. 10. Orientation of the localization band dened by the angle u as predicted by (a) the SOM model and (b) the MGUR model with kx 2:5, as a function of the stresstriaxiality XR and the Lode parameter L (or h). The strain hardening exponent is N 0:1 and the initial porosity f0 1%.

a b

dc

Fig. 11. SOM limit load failure curves as a function of the stress triaxiality XR and the strain hardening exponent N 0:01; 0:05;0:1;0:2 for various values of the Lodeparameter: (a) L 1, (b) L 0:5, (c) L 0 and (d) L 1. The initial porosity f0 1%.



the source of the instability is void collapse, while in the second, itis void growth. In addition, the high-triaxiality regime is ratherinsensitive to the Lode parameter, while the low-triaxiality regimeand the transition between the two is strongly dependent on theLode parameter, with the ductility increasing from a value ofL 1 to the value of L 1 (where no void collapse is possibleand therefore no low-triaxiality regime is observed). In this con-nection, the predictions of the SOM model for failure at the limitload appear to be qualitatively consistent with the experimentalresults of Barsoum and Faleskog (2007a), presented in Fig. 1(a).Note, however, that in the results of Fig. 1(a), the stress triaxialityevolves (and is non-uniform) during the deformation process as aconsequence of the complex geometry of the experimental setup,and, hence, comparisons of the SOM results (which involve uni-form elds and xed triaxiality) with the experimental results ofBarsoum and Faleskog (2007a) can only be qualitative in nature.

By contrast, as shown in Fig. 8(b), the MGUR model predictionsexhibit qualitatively different behavior for the limit load failurecurves. As expected from the way in which it was constructed,the limit load curves depend strongly on the Lode parameter, butin a manner that is monotonic with respect to the triaxiality andtherefore does not exhibit the two different regimes and particu-larly the high-ductility peaks predicted by the SOM model andshown by the experimental results in Fig. 1(a). This signicant dif-ference found between these two models is clearly linked to thefact that the SOM model can account for void shape changes andtherefore can capture the void collapse mechanism contrary tothe MGUR model which assumes spherical void shapes duringthe entire deformation process. In addition, the MGUR predictionsfor the limit load exhibit a symmetry of h 30o, implying in partic-ular that the limit loads for L 1 (corresponding to uniaxial ten-sion) and L 1 (biaxial tension with uniaxial compression) areidentical. This result is a direct consequence of the ad-hoc qua-dratic character of the dependence of the MGUR model on the Lodeparameter (see relation (4) in Nahshon and Hutchinson (2008)),and is in sharp contrast with the SOM model which, for low triax-iality, predicts low ductility for L 1, but very high ductility forL 1. It is also worth noting that the MGUR model predicts theexistence of limit loads for negative values of the stress triaxiali-ties. This is also in contrast with the the SOMmodel which predictsthat the hardening produced by the porosity reduction with nega-tive triaxialities completely overwhelms any softening due tochanges in the shape of the voids, and therefore the material con-tinues to harden all the way up to complete void closure.

Fig. 9 shows SOM and MGUR predictions for the critical equiv-alent plastic strain epe at localization of the deformation, or loss ofellipticity (LOE), dened by condition (18), as functions of thestress triaxiality XR, for several values of the Lode parameter L(or Lode angle h). As shown in Fig. 9(a), the SOM predictions forLOE are roughly similar to those for the limit load depicted inFig. 8(a), and also exhibit two sharply separated regimes. However,in addition to the strong dependence in the Lode angle observed inthe low-triaxiality regime, there is also some (smaller) sensitivityin the high-triaxiality regime with the ductility decreasing as thevalue of L is increased from 1. In fact, no LOE is detected for val-ues of L > 0 and XR > 0:5 0:7, but note that the stress drops tozero for sufciently high deformation as a consequence of the con-tinued porosity growth discussed in the previous section, whileother well-known failure mechanisms such as high-triaxiality voidcoalescence are present (see review work of Benzerga and Leblond(2010)).

On the other hand, as shown in Fig. 9(b), the MGUR model pre-dicts LOE only for values of L 6 0, while no LOE is detected for L > 0(for all stress triaxialities XR). Furthermore, contrary to the corre-sponding SOM predictions, no LOE is detected for low triaxialitiesexcept for a small branch for L 0. This is a direct consequence

of the fact that the MGUR model remains isotropic during the en-tire deformation process as a result of no void shape changes, andtherefore completely misses the morphological anisotropy devel-oped due to the signicant void shape evolution in the low-triaxi-ality regime. Finally, it is noted in the context of this gure that forthe special values of L 1 and 1, the MGUR model reduces to theGurson model and note that the predictions for LOE for these twovalues are different (in one case there is LOE and in the other thereis not), which in view of the identical predictions for the limit loadfor these two cases demonstrates the sensitivity of the LOE condi-tion to the pertinent kinematical conditions.

Finally, in Fig. 10, the earlier LOE results are completed bydepicting the orientation of the localization band in terms of theangle u that denes the orientation of the normal to the band nwith respect to the x2 axis (see inset sketches in the plots). TheSOM and the MGUR results are shown in Fig. 10(a) andFig. 10(b), respectively, as a function of the stress triaxiality XRfor several values of the Lode parameter L (or Lode angle h). Thecomplementary angle ug , associated with the vector g, which con-trols the type of deformation inside the band, is found to beug u for both the SOM and the MGUR models. Moreover, notethat the normal to the band n, as predicted by both the SOM andthe MGUR models, lies on the x2 x3 plane. In particular, for thecase when u ug 45o (i.e., n ? g), the state of deformation in-side the band is a simple shear and thus formation of a shear local-ization band is produced. This is the case for LP 0 (or hP 30o) forthe SOM model and L 0 (or h 30o) for the MGUR model. Notethat, in accord with the earlier discussions, the MGUR model pre-dicts no loss of ellipticity for L > 0 and hence no angles are shownfor these cases. On the other hand, for L < 0, we observe for boththe SOM and the MGUR models that the predicted localizationband angle is smaller than 45o and hence the state of deformationinside the band is a combination of shear plus dilatation across theband (in the direction of the normal to the band). The lowest valuefor u is attained in both models for L 1, where the localizationband is found to be at an angle of about 10o. It should be empha-sized that at large triaxialities the SOM and the MGUR models pre-dict very similar localization angles, highlighting once again thefact that the main difference between the models is for low triax-ialities when changes in the shape of the pores become possible.

Fig. 12. SOM loss of ellipticity (LOE) failure curves as a function of the stresstriaxiality XR and the strain hardening exponent N 0:01;0:05;0:1;0:2 for variousvalues of the Lode parameter: L 1;0:5;1. The initial porosity f0 1%.



3.4. Inuence of strain hardening exponent and initial porosity onfailure curves

Making use of the SOM model, a parametric study is carried outto investigate the effect of the strain hardening exponent of thematrix phase and the initial porosity on the limit load and loss ofellipticity (LOE) failure curves. Thus, the following gures showplots of the critical equivalent plastic strain epe attained at the limitload (i.e., maximum in the re ee curve, or equivalently criticalhardening rate H 0, and at loss of ellipticity (LOE), or localiza-tion of deformation (given by (18)), as functions of the stress triax-iality XR and the Lode parameter L (or Lode angle h.

Fig. 11 shows limit load maps for (a) L 1, (b) L 0:5, (c)L 0 and (d) L 1 as a function of the stress triaxiality XR usingdifferent strain hardening exponents for the matrix phase,N 0:01;0:05; 0:1;0:2. (Note that N 0 and N 1 correspond toideally plastic and linear hardening behaviors. respectively.) Thelimit load failure curves are strongly dependent on N for moderateand high triaxialities such as XR > 0:4, as observed in all parts ofFig. 11. By contrast, at low stress triaxialities (XR < 0:35) andL > 1, i.e., Fig. 11(b), (c), (d), the limit load failure curves exhibitnegligible dependence on the strain hardening exponent. This is

due to the fact that at low XR, the limit load occurs in such anabrupt manner due to the very fast void shape changes (observedin the context of Fig. 5) that the hardening of the matrix plays al-most no role on the overall softening mechanism of the porousmaterial. On the other hand, as the triaxiality increases the growthof porosity dominates the limit load mechanism. The porositygrowth however is rather smooth allowing the strain hardeningexponent to play a dominant role on the overall softening of thematerial. In contrast, as shown in Fig. 12, the strain hardeningexponent N has only a negligible effect on the LOE predictions. Thissuggests that once the material enters the softening regime, kine-matics controls the localization mechanism and hence the effect ofN is not important.

Fig. 13 shows the limit load failure curves as a function of thestress triaxiality XR for different initial porosities f0 0:1;1;5%and Lode parameters: (a) L 1, (b) L 0:5, (c) L 0 and (d)L 1. Overall, an effect is observed especially near the transitionfrom the low- to the high-triaxiality regimes, which becomes lesssharp with decreasing porosity. It should also be noted that higherinitial porosities f0 lead to a reduction in ductility, as determinedby the limit load, except in the transition regime, where the oppo-site trend is observed. Finally, Fig. 14 presents LOE critical curves as

a b

dc

Fig. 13. SOM limit load failure curves as a function of the stress triaxiality XR and the initial porosity f0 0:1;15% for various values of the Lode parameter: (a) L 1, (b)L 0:5, (c) L 0 and (d) L 1. The strain hardening exponent is N 0:1.



a function of the stress triaxiality XR for different initial porositiesf0 0:1;1, and 5% and Lode parameters: (a) L 1, (b) L 0:5,(c) L 0 and (d) L 1. The effect of f0 on the LOE failure curvesis non-negligible contrary to the effect of the strain hardeningexponent N shown in Fig. 12. As observed here, higher initialporosity f0 leads to lower critical strains for localization, at leastfor the range of porosities considered in this study.

4. Conclusions and perspectives

In this work, we have investigated the inuence of the stress tri-axiality and the Lode parameter on the failure of elasto-plastic por-ous materials subjected to macroscopically uniform, triaxialloadings. For this purpose, we have made use of a recently devel-oped second-order nonlinear homogenization model (SOM) ofDanas and Ponte Castaeda (2009a), which can account for the ef-fects of void shape and porosity evolution on the overall softening/hardening response of the porous material. Material failure of theporous ductile solid has been modeled by means of two differentmacroscopic criteria: (i) vanishing of the overall hardening rate(H 0), corresponding to the existence of limit load, or maximum

stress in the constitutive response of the material, and (ii) loss ofellipticity of the incremental response of the material correspond-ing to localization of the deformation into dilatant shear bands dueto the compressible overall response of the porous material (Rice,1976).

The main nding of this work is that failure can occur by twovery different mechanisms at high- and low-triaxiality. In agree-ment with well-established results, at high triaxialities, the modelpredicts signicant void growth leading to a softening effect whicheventually overtakes the intrinsic strain hardening of the solidmaterial and produces overall softening. Thus, a limit load isreached at a critical strain that decreases with increasing triaxialityand is found to be independent of the Lode parameter. This limitload point is then followed by a signicant reduction in the load-carrying capacity of the material and loss of ellipticity (at leastfor negative values of the Lode parameter). On the other hand, atlow triaxialities, the model predicts void collapse due to an abruptattening of the initially spherical voids with decreasing porosity,which in turn leads to a sharp drop in the load-carrying capacityof the porous solid. The precise value of the strain at the onset ofthe instability, which determines the overall ductility of the mate-rial, depends on the competition of the hardening produced by the

a b

dc

Fig. 14. SOM loss of ellipticity failure curves as a function of the stress triaxiality XR and the initial porosity f0 0:1;15% for various values of the Lode parameter: (a) L 1,(b) L 0:5, (c) L 0 and (d) L 1. The strain hardening exponent is N 0:1.



reduction of the porosity and the softening due to the change inshape of the pores, and is highly sensitive to the value of the Lodeparameter. Thus, for biaxial tension with axisymmetric compres-sion (L 1), the onset of the limit load instability, as well as theloss of ellipticity shortly thereafter, decreases as the triaxiality isreduced toward zero, while for axisymmetric tension (L 1) novoid collapse is possible and therefore no instability is observedfor small values of the triaxiality. Moreover, for xed, small valuesof the triaxiality (XR < 0:6), the ductility of the porous material de-creases as the value of the Lode parameter increases from 1 to +1.In addition, a sharp transition is observed as the failure mechanismswitches from void collapse to void growth for intermediate valuesof the stress triaxiality (0:3 < XR < 0:7), depending strongly on thevalue of the Lode parameter and leading to high-ductility peaks inthe failure maps. In this regard, the theoretical predictions arefound to be in qualitative agreement with recent experimentalobservations by Barsoum and Faleskog (2007a) and Dunand andMohr (2010), even though it should be emphasized that the stressand deformation elds are not uniform and that the values of thetriaxiality and Lode parameter are not controlled independentlyin these experiments. In this sense, the theoretical predictions pre-sented in this work suggest the critical need for new experimentswith improved control over the uniformity of the stress and strainelds, as well as the loading conditions.

The predictions of the second-order model have been comparedwith the corresponding results of the ad hoc modication of theGurson model, MGUR, proposed by Nahshon and Hutchinson(2008), and signicant differences have been identied. First andforemost, the MGUR model cannot capture void collapse, becausethe voids are assumed to remain spherical throughout the deforma-tion. Because of this, while it is possible to articially soften thematerial response by introducing a dependence on the Lode angle,the failure curves still increase with decreasing triaxiality into thenegative triaxiality regime. In addition, in contrast with the sec-ond-order predictions, the effect of the Lode parameter on the max-imum load is symmetric with respect to the sign of the Lode angle,and does not lead to loss of ellipticity for most values of the Lodeparameter in the low-triaxiality regime. In this connection, it isimportant to emphasize that the relevance of the Lode angle isnot so much through its direct effect on the macroscopic yield sur-face, which is relatively small, but instead through its much moresignicant implications for the evolution of the microstructure,especially when changes in the shape of the voids are allowed. In-deed, this ability to account for the very different and generallystrongly anisotropic evolution of the microstructure of the materialat xed, low values of the stress triaxiality, but with different Lodeparameters ranging from axisymmetric tension (L 1) to biaxialtension with axisymmetric compression (L 1), is the mainadvantage of the SOM model over the models of Gurson (1977),Nahshon and Hutchinson (2008) and Nielsen and Tvergaard (2010).

For completeness, the SOM model has also been used to inves-tigate the possible effects of the matrix strain-hardening exponentN and the initial porosity f0. We have found that the strain-harden-ing exponent N has a signicant effect on the limit load for stresstriaxialities XR > 0:4, and consequently the location of the transi-tion from the void collapse to the void growth mechanisms. In con-trast, it has only a negligible effect on the limit load at low stresstriaxialities, due to the abruptness of the void collapse mechanismin this case, leading to strong material softening. On the otherhand, the strain hardening exponent affects only slightly the lossof ellipticity curves. In turn, different initial porosities f0 have an ef-fect on both the limit load and loss of ellipticity failure curves.Higher initial porosities lead, in general, to lower critical strainsfor the limit load and loss of ellipticity, except for the limit loadcurves in the transition region (0:4 < XR < 0:6), where the oppositetrend is observed.

It should also be emphasized that this work deals only withinstabilities at the material level and that no actual macroscopicgeometries have been considered. Nonetheless, the instability re-sults obtained assuming that macroscopically uniform elds arepresent in a given specimen should correspond to material insta-bilities, and provide a loose upper bound for the resistance of thematerial to ductile failure under more general loading conditions(Rice, 1976). In this connection, it is also relevant to mention thatthe three-dimensional studies of Barsoum and Faleskog (2007b)and Barsoum and Faleskog (2011) and the corresponding two-dimensional studies of Tvergaard (2009) in two-dimensions, seemto suggest that void rotations may somehow be necessary for low-triaxiality failure. However, the results of the presentwork for triax-ial loading conditions (with xed loading axes) show thatwhile voidrotationsmay enhance (or reduce) the ductility of thematerial, voidrotations are not strictly necessary for material instabilities (of themaximum load, or loss of ellipticity type) at low-triaxiality, sincethe basic micro-mechanism of void collapse does not require them.In any event, void rotations can easily be handled by the general ver-sion of the SOMmodel (Danas and Ponte Castaeda, 2009a), and thiswill be pursued in future work. Interestingly, Kailasam and PonteCastaeda (1997) have shown (refer to Fig. 2 in that reference) usingan earlier version of the model (Ponte Castaeda and Zaidman,1994; Kailasam and Ponte Castaeda, 1998) that the effective hard-ening rate of a porous rigid-plastic material subjected to simpleshear can become zero as a consequence of the combined effectsof the changes in shapes and orientation of the voids.

It should also be remarked that the larger issue of how to pro-ceed after (local) loss of ellipticity in the analysis of an actual struc-tural problem is still a largely open issue. However, it is clear thatmore general and reliable models, as well as estimates for their lossof ellipticity, are essential for further progress, as are nite elementimplementations of such models in order to be able to handle thenon-uniform elds that would be expected to develop under actualexperimental conditions. In this latter connection, it should bementioned that such implementations are already available (seeKailasam et al. (2000) and Aravas and Ponte Castaeda (2004))for the earlier variational framework of Ponte Castaeda andZaidman (1994). In addition, a numerical implementation of an im-proved version of the variational framework, which providesmore accurate results for both low and high stress triaxialitieshas been developedand implemented for three-dimensionalexperimental geometriesrecently by Danas and Aravas (2012).

As a nal remark, it should be mentioned that an additionaladvantage in the use of a homogenization approach for porousand other heterogeneous solids is its generality. Thus, for example,the effect of anisotropy in the matrix can be accounted for in astraightforward fashion by treating this phase as a polycrystallineaggregate and using the second-order homogenization method(Liu and Ponte Castaeda, 2004) consistently to estimate the over-all response including both the effects of porosity and crystallo-graphic texture. A rst step in this direction is presented in therecent work of Lebensohn et al. (2011), which opens up the possi-bility of modeling the simultaneous effects of porosity and textureevolution on the overall response and stability of porous polycrys-talline solids, which is expected to be especially important for por-ous low-symmetry metals, such as porous Ti and Mg alloys.

It is also relevant to mention in this connection that the second-order homogenization method has been used successfully to esti-mate loss of ellipticity in porous elastomers (Lopez Pamies andPonte Castaeda, 2007a,b). Although the failure maps are very dif-ferent for this case, comparisons with careful numerical calcula-tions (Michel et al., 2007) show that the model indeed has thecapability of capturing not only the overall macroscopic behavior,but also the possible onset of macroscopic instabilities, such asloss of ellipticity (Geymonat et al., 1993).



Acknowledgments

K.D. would like to acknowledge the support of the EngineeringDepartment, Cambridge University, where parts of this work werecarried out, as well as of the Solid Mechanics Laboratory of theEcole Polytechnique. P.P.C would like to acknowledge partial sup-port by the National Science Foundation under Grant NumberCMMI-0969570.

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lode parameter and the stress triaxiality

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