European Journal of Mechanics A/Solids 49 (2015) 137e145
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European Journal of Mechanics A/Solids
journal homepage: www.elsevier .com/locate/ejmsol
A computational homogenization of random porous media: Effect ofvoid shape and void content on the overall yield surface
Younis-Khalid Khdir a, b, Toufik Kanit a, b, Fahmi Zaïri a, b, *, Moussa Naït-Abdelaziz a, b
a Univ Lille Nord de France, F-59000 Lille, Franceb Universit�e Lille 1 Sciences et Technologies, Laboratoire de M�ecanique de Lille (LML), UMR CNRS 8107, F-59650 Villeneuve d'Ascq, France
a r t i c l e i n f o
Article history:Received 5 February 2014Accepted 2 July 2014Available online 10 July 2014
A computational homogenization study of three-dimensional cubic cells is presented to estimate theoverall yield surface of random porous media covering a wide range of stress triaxiality ratios. Therepresentativity of the overall yield surface estimates is examined using cubic cells containing randomlydistributed and oriented voids with different volume fractions and shapes (spherical and oblate/prolate).The computational results are compared with some existing Gurson-type yield criteria. The extension ofthe Gurson-Tvergaard model recently proposed by Fritzen et al. (2012) for spherical cavities is shown ingood agreement with the computational results for all investigated porosities.
Over the last four decades, the mathematical development ofyield criteria for the plastic porous solids has been widely investi-gated (Rice and Tracey, 1969; Gurson,1977; Tvergaard, 1982; Koplikand Needleman,1988; Sun andWang,1989; Ponte Castaneda,1991;Gologanu et al., 1993,1994,1997, 2001; Zuo et al., 1996; Garajeu andSuquet, 1997; Faleskog et al., 1998; Ma and Kishimoto, 1998;Corigliano et al., 2000; Pardoen and Hutchinson, 2000; Zhanget al., 2000; Negre et al., 2003; Kim et al., 2004; Wen et al., 2005;McElwain et al., 2006; Monchiet et al., 2008; Zaïri et al., 2005,2008a; Besson, 2009; Laiarinandrasana et al., 2009; Li and Karr,2009; Nielsen and Tvergaard, 2009; Vadillo and Fernandez-Saez,2009; Zadpoor et al., 2009; Dunand and Mohr, 2011; Li et al.,2011; Mroginski et al., 2011; Fei et al., 2012; Shen et al., 2012; Yanet al., 2013) essentially because of the role of porosities regardingthe ductile fracture process, these voids being the consequence ofmanufacturing processes. The mathematical derivations of thesecriteria are generally based upon the continuum-based micro-mechanical framework, for which the starting point is the micro-structural representation of the porous medium. The non-trivialityof the theoretical problem leads to define a basic unit cell
et Technologies, Laboratoire50 Villeneuve d'Ascq, France.
ïri).
served.
containing one centered void for the material volume used torepresent the microstructure. The unit cell is an elementary volumeelement consisting in a hollow sphere or cylinder subjected to auniform macroscopic strain rate at its external boundary. Gurson(1977) proposed the most widely used micromechanics-basedyield criterion to analyze plastic porous solids containing spher-ical voids. The Gurson model is based upon the following as-sumptions: isotropy, incompressibility and rigid-plasticity for thelocal yielding of the surrounding matrix material which obeysto the vonMises criterion. The resulting macroscopic yield criterionof Gurson (1977) for the porous medium is hydrostatic pressure-dependent, integrates the volume fraction of porosities as amodel parameter and accounts for a possible void growth driven bythe local plastic deformation of the surroundingmatrix material. Aspointed out by Tvergaard (1982), the Gurson model gives an upperbound of the macroscopic yield stress as a function of the meanstress for a periodic arrangement of voids. In order to improve itsagreement with two-dimensional finite element simulation resultson a periodic unit cell, Tvergaard (1982) proposed to introduceheuristic parameters in the Gurson yield criterion. These adjustableparameters have no direct physical meaning but may be correlatedto interaction effects between voids. The extension of the Gursonmodel by Tvergaard (1982), known as the Gurson-Tvergaard (GT)model, was thenceforth widely used by many researchers to checkits capability to capture the poroplastic behavior of many engi-neering porous materials. In very useful background papers,Benzerga and Leblond (2010) and Besson (2010) reviewed thevarious extensions of the Gurson model based upon enhanced
Fig. 1. Examined porous media at f ¼ 0.05, 0.13 and 0.23: (a) spherical (a ¼ b ¼ c), (b) oblate (b ¼ c and b/a ¼ 2.5) and (c) prolate (b ¼ c and a/b ¼ 2.5) pores; N z 200 pores.
Table 1Loading parameters used in the simulations.
Y.-K. Khdir et al. / European Journal of Mechanics A/Solids 49 (2015) 137e145138
micromechanical approaches or upon phenomenological general-izations to take into consideration the void shape or the matrixmaterial features such as isotropic/kinematic hardening, visco-plasticity, compressibility and anisotropy. Using micromechanicalapproaches, Ponte Castaneda (1991) and Sun and Wang (1989)proposed, respectively, upper and lower bounds for the overallyield surface of porous media. Using the variational techniqueintroduced by Ponte Castaneda (1991), Garajeu and Suquet (1997)proposed another upper bound which overcomes the well knownbasic drawbacks of the Gurson criterion at low stress triaxialityvalues. The effect of void shape on the macroscopic yield responseof porous materials was investigated by several authors (Gologanuet al., 1993, 1994, 1997, 2001; Yee and Mear, 1996; Son and Kim,
2003; Siruguet and Leblond, 2004; Flandi and Leblond, 2005; Liand Huang, 2005; Li and Steinmann, 2006; Monchiet et al., 2006,2008; Gao et al., 2009; Keralavarma and Benzerga, 2010; Linet al., 2010; Lecarme et al., 2011; Scheyvaerts et al., 2011; Zaïriet al., 2011; Danas and Aravas, 2012; Madou and Leblond, 2012;Monchiet and Kondo, 2013).
Y.-K. Khdir et al. / European Journal of Mechanics A/Solids 49 (2015) 137e145 139
Although the mathematical developments have reached a highdegree of sophistication, the resulting yield criteria generallyinvolve a certain number of parameters with no physical signifi-cance. Thatmay be explained by the fact that thesemicromechanics-based models consider as material volume element an elementaryvolume element containing a single void. Because the voids arediluted in the matrix material, the interactions between voids areneglected. Moreover, this microstructural representation of theporous material implies periodicity. However, to be statisticallyrepresentative, the material volume element should contain suffi-cient information about the porous microstructure, in particularthe void distribution. This last decade, the material response ofporous media was also investigated using computational micro-mechanics. This approach is emerging as a powerful tool to bring abetter understanding of void distribution effects and interactionphenomena on the mechanical behavior of random porous media.The main advantage of the computational homogenization is itsability to directly compute the mechanical fields on the randomporous media by representing explicitly the microstructure fea-tures such as shape, orientation and distribution of voids. Althoughmany studies were dedicated to the development of yield criteriafor plastic porous media, it seems that only few works have beendevoted to three-dimensional computational homogenizationinvolving multiple voids. To our knowledge, only Bilger et al. (2005,2007), Fritzen et al. (2012, 2013) and Khdir et al. (2014) used thisapproach to estimate the overall yield surface of porous materials.Their computations were limited to spherical voids. The calcula-tions of Bilger et al. (2005, 2007) were performed on the basis ofthree-dimensional Fast Fourier Transform. The pore clustering ef-fect on the overall material response was the key point of theirinvestigations. Fritzen et al. (2012, 2013) assumed the randomporous media as a volume of porous material which is periodicallyarranged. The results highlighted by Fritzen et al. (2012) led them toextend the GT yield criterion in order to overcome the analytical/numerical discrepancies. Khdir et al. (2014) focused their in-vestigations on the porous materials containing two populations ofvoids. Their results showed that, for an identical fraction of po-rosities, there is no significant difference between a double and asingle population of voids.
In this contribution, a computational homogenization of randomporous media, including spherical and oblate/prolate spheroidalvoids, is presented in order to determine their overall yield surfacewhile still studying the representativity of the computational
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Eeq
Σ eq/σ
o
S O P12
.
.
.
9
(a)
Fig. 2. Overall (a) von Mises equivalent and (b) hydrostatic stresses as a function of the ovf ¼ 0.23 and all the loading cases given in Table 1; N z 200 pores.
results. Notably, the computational investigations performed in thisstudy can account for the complex coupling existing between voiddistribution, void shape and external loading mode. The first aim isto compare the simulation results with some Gurson-type yieldcriteria. The second aim is to verify the extension of the GT yieldcriterion provided by Fritzen et al. (2012) in the case of randomporous media containing non-spherical voids.
The present paper is organized as follows. We first presentin Section 2 the investigated microstructures and the compu-tational homogenization method. Section 3 is devoted toresults and discussions. Concluding remarks are finally given inSection 4.
2. Computational homogenization
2.1. Porous microstructures
The porous media considered in the computations are made ofperfectly-plastic matrix obeying to the commonly used isotropicvon Mises yield criterion, the yield stress being constant and equalto 290 MPa. The plastic flow is assumed perfect in order to disre-gard hardening effects in the investigation and to compare thesimulation results with the most common analytical models. Thematrix material is sufficiently stiff in order to overcome any yieldstrain effects.
The porous media are represented by three-dimensional cubiccells containing a large number of pores, in order to assure that thestudied material volume element is sufficiently large compared toporosities. The voids are randomly distributed and oriented inspace in the cubic cell. Moreover, they are identical and non-overlapped. The question of the void content effects is examinedin this work. The volume fraction of N spheroidal voids inside acubic cell of volume V is given by:
fspheroidal ¼43Npabc
V(1)
where a is the polar radius along the y axis of the spheroidal voidand, b and c are the equatorial radii along the z and x axis,respectively (see Fig. 1).
The void shape effects are examined in this work which con-stitutes a noteworthy difference with respect to existing literature(Bilger et al., 2005, 2007; Fritzen et al., 2012, 2013; Khdir et al.,
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08Eeq
Σ m/σo
S O P9
.
.
.
2
1
(b)
erall von Mises equivalent strain for spherical (S), oblate (O) and prolate (P) pores at
Y.-K. Khdir et al. / European Journal of Mechanics A/Solids 49 (2015) 137e145140
2014). Fig. 1 presents the designed porous microstructures. Thecases of spherical (a ¼ b ¼ c), oblate (b ¼ c and b > a) and prolate(b ¼ c and a > b) pores are examined. For each shape, three voidvolume fractions f are studied. The finite element method waschosen for the numerical computations using Zebulon software. Astandard small-strain approximation was used for the simulations.The mesh size used was fine enough to represent accurately the
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
0 25 50 75 100 125 150 175 200N
Σ eq /
<Σ e
q>
(a)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
0 25 50 75 100 125 150 175 200N
Σ eq /
<Σ e
q>
(c)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
0 25 50 75 100 125 150 175 200N
Σ eq /
<Σ e
q>
(e)
Fig. 3. Asymptotic overall von Mises equivalent and hydrostatic stresses as a function of the naverage (dashed line) and the standard deviations (colored area) are calculated for N ¼ 50
geometry of the porosity and to ensure the overall responseconvergence.
2.2. Boundary conditions
The porous media being hydrostatic pressure-dependent, theboundary conditions imposed to the designed representative
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
0 25 50 75 100 125 150 175 200N
Σ m /
<Σm
>
t
(b)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
0 25 50 75 100 125 150 175 200N
Σ m /
<Σm
>
(d)
0.8
0.85
0.9
0.95
1
1.05
1.1
1.15
1.2
0 25 50 75 100 125 150 175 200N
Σ m /
<Σm
>
(f)
umber of pores for spherical (aeb), oblate (ced) and prolate (eef) pores at f ¼ 0.23. The.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.005 0.01 0.015 0.02 0.025 0.03 0.035Eeq
Σ xx/σ
o or Σ
yy/σ
o or Σ
zz/σ
o
SOP
0
0.1
0.2
0.3
0.4
0.5
0 0.025 0.05 0.075 0.1 0.125 0.15Eeq
Σ xy/σ
o or Σ
yz/σ
o or Σ
xz/σ
o
SOP
(b)(a)
Fig. 4. RVE isotropy for spherical (S), oblate (O) and prolate (P) pores at f ¼ 0.23: (a) overall tensile stress-strain responses in the three orthogonal directions and (b) overall shearstress-strain responses in three perpendicular planes; N z 200 pores.
Fig. 5. Distribution of the accumulated plastic strain for spherical pores at f ¼ 0.23 and three different loading cases: (a) a ¼ 1, b ¼ 0, (b) a ¼ 1, b ¼ 0.25, (c) a ¼ 0, b ¼ 1.
Y.-K. Khdir et al. / European Journal of Mechanics A/Solids 49 (2015) 137e145 141
element should involve awide range of stress triaxiality ratios to beexplored. The stress triaxiality parameter T ¼ Sm/Seq is defined asthe ratio of the overall hydrostatic stress Sm and the overall vonMises equivalent stress Seq, respectively, given by:
Sm ¼ 13trðSÞ and Seq ¼
ffiffiffi32
rðS0 : S0Þ1=2 (2)
where S is the macroscopic (ensemble-volume average) stresstensor and S′ denotes its deviatoric part.
Fig. 6. Distribution of the accumulated plastic strain for oblate pores at f ¼ 0.23 and th
Stress or strain-driven boundary conditions are usually employedin the literature. In this paper, due to its computational robustness,the following mixed boundary conditions were imposed:
E11ðtÞ ¼ t _ε0ðaþ bÞE22ðtÞ ¼ t _ε0ð�aþ bÞE33ðtÞ ¼ t _ε0bS12ðtÞ ¼ S13ðtÞ ¼ S23ðtÞ ¼ 0
(3)
in which the values assigned to shear components of the overallstress tensor are zero. The terms a and b, introduced to control the
ree different loading cases: (a) a ¼ 1, b ¼ 0, (b) a ¼ 1, b ¼ 0.25, (c) a ¼ 0, b ¼ 1.
Fig. 7. Distribution of the accumulated plastic strain for prolate pores at f ¼ 0.23 and three different loading cases: (a) a ¼ 1, b ¼ 0, (b) a ¼ 1, b ¼ 0.25, (c) a ¼ 0, b ¼ 1.
0
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1Σm/σo
Σ eq/σ
o
Fig. 8. Overall von Mises equivalent stress vs. overall hydrostatic stress for sphericalpores at f ¼ 0.23 and all the loading cases given in Table 1; N z 200 pores (The filledcircles designate the numerical yield points).
Y.-K. Khdir et al. / European Journal of Mechanics A/Solids 49 (2015) 137e145142
diagonal components of the overall strain tensor E, are two loadingparameters, _ε0 >0 is a prescribed deformation rate and t is thesimulation time. The stress triaxiality is indirectly assigned by thetwo measures of stress, given by Eq. (2), which are definedimplicitly by the mixed boundary conditions through the twoloading parameters a and b. The different values of a and b used toobtain different stress triaxiality ratios are listed in Table 1.
3. Results and discussion
3.1. Asymptotic stress response
The asymptotic response of the ideally plastic porous micro-structures was systematically examined by plotting the overall vonMises equivalent and hydrostatic stresses as a function of theoverall von Mises equivalent strain. These two measures of stressare plotted in Fig. 2 in the case of a porosity value of 0.23 and for thethree void shapes. This figure shows that the porous microstruc-tures are subjected to a stationary response beyond a certain strain.Similar observations on large volume computations have beenpointed out by Fritzen et al. (2012, 2013) and by Khdir et al. (2014).In order to define the numerical yield points, the overall stresses atthe end of the simulation are considered. Except at very high meanstresses, it can be observed that the overall response is not affectedby the void shape. This is particularly true at low hydrostaticpressure (or high equivalent stress) values. Note that the resultsobtained with the two other porosities (0.05 and 0.13) give similartrends.
3.2. Representativity
The size of the volume element is conditioned by the number ofporosities which should be chosen large enough to ensure thatthe volume element is representative. This representativity wasinvestigated in terms of the mechanical responses by Huet (1990),Drugan and Willis (1996) and Kanit et al. (2003). These authorshave studied the effects of the volume element size on the elasticstiffness. More recently, Khdir et al. (2013) have investigated theseeffects on the elasticeplastic response. In the case of elasticeplasticcomposites, made of two phases with highly contrasted properties,Khdir et al. (2013) have shown that the minimum size of the vol-ume element in the yield and post-yield region must be greaterthan the minimum size required in the elastic domain. This ques-tion which arises in three-dimensional computational homogeni-zation has to be systematically accounted for.
Several volume elements with different sizes (i.e. containingdifferent number of pores) are simulated for a porosity of 0.23, andthe mechanical representativity of the computational results areexamined. The overall stationary stresses are plotted as a function
of the number of pores in Fig. 3 for the three shapes. Fig. 3a, c and ecorrespond to the loading path 1 in Table 1 characterized by (a ¼ 1,b ¼ 0) for which the deviatoric component exhibits the higheststationary value, whereas Fig. 3b, d and f correspond to the loadingpath 9 in Table 1 characterized by (a ¼ 0, b ¼ 1) for which thehydrostatic component takes its highest stationary value. The sta-tionary stresses are normalized with respect to the average value ofcomputational results of several realizations containing 50 pores.All computed data are found within or close to the colored areadefined by the standard deviations. The stationary values forN ¼ 200 are close to the averages of N ¼ 50 pores (dashed line), thelargest difference being about 7%.
The computations are performed using the largest cubic cells(containing N ¼ 200 voids) in order to assure the mechanicalrepresentativity of the numerical yield surfaces. These cubic cellsare successively stretched in the orthogonal directions and theresults are reported in Fig. 4a for the three void shapes. It can beobserved that identical overall mechanical responses are obtainedwhich is, for isotropy, a necessary condition but not sufficient. Toensure this property the cubic cells must also be subjected tosimple shear loading. The corresponding results presented inFig. 4b show that the overall shear responses are the same in threeperpendicular planes. Then, Fig. 4 shows that, when a sufficientnumber of pores are randomly distributed and oriented in thevolume element, an isotropic response is obtained at themacroscopic
Y.-K. Khdir et al. / European Journal of Mechanics A/Solids 49 (2015) 137e145 143
scale. The found isotropy proves that this large volume element isrepresentative enough of the random porous medium, whateverthe void shape.
3.3. Local plastic strain fields
The local plastic strain fields can be observed in Figs. 5e7 atdifferent triaxiality ratios for the three void shapes. The porosity of0.23 is chosen to illustrate this distribution because a more diffuseplastic strain is observed compared to the other void volumefractions. The observations are presented at the end of the pre-scribed loading. The poreepore interactions and the triaxiality ef-fects on the local fields are illustrated in the figures for threeparticular cases: The cases (a ¼ 1, b ¼ 0) and (a ¼ 0, b ¼ 1) corre-spond to the lowest and highest triaxiality ratios, respectively, andthe case (a ¼ 1, b ¼ 0.25) to an intermediate one.
3.4. Comparison between numerical results and analytical criteria
In this subsection, only the case of spherical pores is analyzed.The common representation of the overall yield surface, plottingthe overall von Mises equivalent stress as a function of the overallhydrostatic stress, is adopted to illustrate the computational data.Fig. 8 shows an example of the relationship between these twomeasures of stress for a given porosity value of 0.23. The normal-ization is performed with respect to the matrix yield stress s0.The hydrostatic pressure dependency of the macroscopic yieldresponse of the porous material is clearly pointed out in the figure.Although an asymptotic response is reached by the “perfectly-plastic” cubic cell (see Fig. 2), the relationship between the twomeasures of stress does not describe a straight line but a rathercomplex non-proportional path, as also observed by Fritzen et al.(2012). The yield points are obtained at the end of the simula-tions and are highlighted by filled circles in Fig. 8. The computedyield data strongly point out the convexity of the overall yieldsurface.
The computed yield data are compared with some existinganalytical models in Fig. 9 for the three considered void volumefractions. Besides the commonly used Gurson model, otheranalytical models are selected (see Table A.1 of Appendix A). It canbe observed that the computed data satisfy the Garajeu and Suquet(1997) (denoted as GS) upper bound and the Sun and Wang (1989)(denoted as SW) lower bound. The GS model is identical to theGurson model (denoted as G) around the normalized hydrostaticstress axis and to the Ponte Castaneda (1991) (denoted as PC)modelaround the normalized equivalent stress axis. It can be observedthat the GS model overestimates the numerical data for highnormalized hydrostatic stress, but becomes closer when decreasingthe mean stress. All the computed data satisfy the SW lower boundbut it is found that the SW model is close to the numerical dataaround the normalized equivalent stress axis at the lowest voidcontent. The PC yield criterion provides too stiff predictions aroundthe normalized hydrostatic stress axis. The divergences with themodel decrease when the void content increases. The G criterionoverestimates the numerical data, the difference between the twosolutions increasing with the void content. The GT model usingthe calibrated parameters of Tvergaard (1982), see Table 2, un-derestimates the numerical yield surface. For the lowest voidcontent, the GT model is close to the numerical surface, especially
Fig. 9. Comparison between some existing analytical models and the simulation re-sults for spherical pores at (a) f ¼ 0.05, (b) f ¼ 0.13, (c) f ¼ 0.23; N z 200 pores (G:Gurson, GT: Gurson-Tvergaard, PC: Ponte Castaneda, GS: Garajeu-Suquet, SW: Sun-Wang).
0
0.2
0.4
0.6
0.8
1
1.2
0 0.5 1 1.5 2 2.5 3Σm/σo
Σ eq/σ
o
S
O
P
Fritzen et al. (2012)
f = 0.23 f = 0.13 f = 0.05
Fig. 10. Simulation results for spherical (S), oblate (O) and prolate (P) pores and,comparison with the Fritzen et al. (2012) model; N z 200 pores.
Y.-K. Khdir et al. / European Journal of Mechanics A/Solids 49 (2015) 137e145144
around the normalized equivalent stress axis. For the highest voidcontent, the GT model becomes a lower bound.
Table A.1Gurson-type yield criteria used in Fig. 9.
G yield criterion(Gurson, 1977)
FðS; f Þ ¼ S2eq
s20
þ 2f cosh�32Sm
s0
�� 1� f 2 ¼ 0 (A.1)
GT yield criterion(Tvergaard, 1982)
FðS; f Þ ¼S2eq
s20
þ 2q1f cosh�32q2
Sm
s0
�
� 1� q3f2 ¼ 0 (A.2)
3.5. GT model for random porous media
To improve its agreementwith the computational results, the GTmodel can be calibrated using our computed data. The followingexpressions of the GT model parameters are found:
q1ðf Þ ¼ 1:69� f ; q2 ¼ 0:92; q3 ¼ q1ðf Þ2 (4)
The model captures all the computed data in a very satisfactorymanner as shown in Fig. 10. We obtain the same expressions asthose found by Fritzen et al. (2012). This calibration can becompared with the values (reported in Table 2) usually obtained bycalibration on two-dimensional finite element simulation resultsusing plane stress, plane strain or axisymmetric “periodic” unit cellmodels.
Interestingly Fig. 10 shows that there is no significant effect ofthe void shape on the overall yield response. This result doesnot support the micromechanics-based analytical yield criteriadeveloped in the literature (e.g. Gologanu et al., 1993, 1994, 1997,2001; Monchiet et al., 2006, 2008; Madou and Leblond, 2012;Monchiet and Kondo, 2013). The theoretical developments
Table 2Different values of GT model parameters (q3 ¼ q21).
References q1 q2
Gurson (1977) 1.0 1.0Tvergaard (1982) 1.5 1.0Koplik and Needleman (1988) 1.25 1.0Zuo et al. (1996) 1.4 1.0Faleskog et al. (1998) 1.46 0.93Ma and Kishimoto (1998) 1.35 0.95Corigliano et al. (2000) 1.08 0.99Zhang et al. (2000) 1.25 1.0Negre et al. (2003) 1.5 1.2Kim et al. (2004) 1.5856 0.909McElwain et al. (2006) 1.31 1.16Nielsen and Tvergaard (2009) 2.0 1.0Vadillo and Fernandez-Saez (2009) 1.46 0.931Dunand and Mohr (2011) 1.0 0.7Fei et al. (2012) 1.8 1.0Yan et al. (2013) 1.55 0.9
considering unit cell representations, periodicity is assumed andvoid shape dependence of the overall yield response is thus ex-pected. From a micromechanics-based analytical point of view, theoverall response of a random porous medium is evaluated fromthat of a unidirectional unit cell averaged over all orientations(Zaïri et al., 2008b). Shape dependence is thus preserved in themicromechanics-based analytical models. If the large volumecomputations performed in this contribution show no significanteffect of the void shape on the volume average behavior, this couldbe a consequence of the cubic cell microstructure in which thepores are (simultaneously) randomly distributed and oriented inspace. However, this statement must be verified for a larger rangeof shape ratios.
4. Concluding remarks
The overall yield surface of plastic porous media was investi-gated via computational micromechanics. The computationalresults were investigated in terms of representativity andwere related to some existing Gurson-type yield criteria. Theoverall yield surfaces were found nearly the same for all investi-gated shapes of voids (spherical, oblate and prolate) providedthat they are randomly distributed and oriented in a largevolume element. Further computations are however required toconfirm the independence of the overall yield surface vis-�a-vis thevoid shape.
Appendix A. Some existing Gurson-type yield criteria forplastic porous media
The mathematical expressions of some existing yield criteria forplastic porous materials, which have been used in the main body ofthe present paper for the purpose of comparison with the numer-ical predictions, are recalled in Table A1.
PC yield criterion(Ponte Castaneda,1991) FðS; f Þ ¼
�1þ 2
3f�S2eq
s20
þ 94fS2m
s20
� ð1� f Þ2 ¼ 0 (A.3)
GS yield criterion(Garajeu and Suquet,1997) FðS; f Þ ¼
�1þ 2
3f�S2eq
s20
þ 2f cosh�32Sm
s0
�
� 1� f 2 ¼ 0 (A.4)
SW yield criterion(Sun and Wang,1989) FðS; f Þ ¼S2
eq
s20
þ f�2� 1
2ln f
�cosh
�32Sm
s0
�
� 1� f ð1þ ln f Þ ¼ 0 (A.5)
Seq is the overall von Mises equivalent stress, Sm is the overallhydrostatic stress, s0 is the yield stress of the matrix material and fis the void volume fraction.
Y.-K. Khdir et al. / European Journal of Mechanics A/Solids 49 (2015) 137e145 145
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