Mechanics of Materials 73 (2014) 58–75

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Mechanics of Materials

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Elastoplastic mechanics of porous materialswith varied inner pressures

http://dx.doi.org/10.1016/j.mechmat.2014.02.0050167-6636/� 2014 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel./fax: +86 10 67396333.E-mail address: qsyang@bjut.edu.cn (Q.-S. Yang).

Lian-Hua Ma a, Qing-Sheng Yang a,⇑, Xiao-Hui Yan a, Qing-Hua Qin b

a Department of Engineering Mechanics, Beijing University of Technology, Beijing 100124, Chinab Research School of Engineering, Australian National University, Canberra, ACT 0200, Australia

a r t i c l e i n f o

Article history:Received 3 May 2013Received in revised form 22 February 2014Available online 11 March 2014

Keywords:Porous materialMicromechanicsHomogenizationInner pressureElastoplastic behavior

a b s t r a c t

A micromechanics model and a computational homogenization method were developed toexamine the macroscopic elastoplasticity and yield behavior of closed-cell porous materi-als with varied inner gas pressures. For the uniaxial stress–strain relation of the porousmaterial, the micromechanics model coincides well with the numerical homogenization,especially for the case of relatively low porosity and gas pressures. The effects of the com-bination of the different gas pressures on the uniaxial stress–strain curve, the nominalPoisson’s ratio, yield surface and initial yield strength of the material are systematicallyinvestigated. The multiple gas pressures can induce the tension–compression asymmetryof the uniaxial stress–strain curves and the nominal Poisson’s ratio of nonlinear deforma-tion. In particular it is shown that when the multiple gas pressures coincide, the yieldsurface of the porous material with inner gas pressures can be simply obtained from thatof the porous material without inner pressures by a shift along the negative direction of thehydrostatic stress axis. However, when the multiple pressures are different, in addition to atranslation along the hydrostatic axis, the yield surface undergoes a change in shape andsize, and the maximal equivalent stress is lowered by a difference in gas pressures.Furthermore, the multiple gas pressures have a significant effect to reduce the yieldstrength of the closed cell porous materials.

� 2014 Elsevier Ltd. All rights reserved.

1. Introduction

It is well known that porous materials are widely usedin engineering structures and have attracted significantattentions for their unique mechanical properties. The por-ous material to be considered here is a porous metal withcellular structure formed by introducing air bubbles intomolten metal (i.e., aluminum). Due to their specificstrength, elastic resilience and energy absorbing proper-ties, the porous metal foams are currently being consid-ered for applications in lightweight, packaging andinsulating structures (Ashby et al., 2000). The mechanical

properties of porous materials, in particular their resis-tance to elastoplastic deformation, strongly depend ontheir microstructural geometry, topology and porosity.Generally, there are two types of geometries for suchmaterials, i.e. closed-cell foams with sealed pores, andopen-cell foams in which the pores construct an intercon-nected network. For the closed-cell porous materials, theyare attractive by many researchers for their interestingmechanical integrity. During fabrication of these closed-cell materials, a large volume fraction of gas-filled poresare formed. The gas pressures are usually varied pore bypore. The presence of multiple micro-sized pores filledwith gas pressures has a prominent effect on the effectivemechanical properties of such closed-cell materials.Therefore, it is essential to examine, from a theoretical

L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75 59

and numerical point of view, the macroscopic elastoplas-ticity, especially the effect of multiple inner gas pressureson the effective mechanical properties of the materials. Inthe present study we restrict our attention to the macro-scopic elastoplastic properties and yield behavior of theporous materials with multiple closed pores filled withvaried gas pressures.

During the past decades, a significant number of theoret-ical and numerical investigations have been carried out toexplore effects of micro-defects on effective mechanicalbehaviors of ductile voided materials (Bilger et al., 2005;Gurson, 1977; Hsu et al., 2009; Le Quang and He, 2008;Moorthy and Ghosh, 1996; Paul and Ramamurty, 2000). Itshould be pointed out, in all these previous works, the poresor cracks were supposed to be pressure-free. As mentionedabove, multiple gas-filled pores are formed in some closed-cell porous metals in processing of microstructure fabrica-tion. Besides, some other porous materials, such as porousceramics (Doltsinis and Dattke, 2001), electronic packagingmaterials (Guo and Cheng, 2002), semisolid Al alloypowders (Chino et al., 2004) and porous uranium dioxide(Vincent et al., 2009), have also large numbers of pores withinitial pressure due to the evaporation of the humidity orgas expansion in the pores. For this kind of gas-filled porousmaterials, it has been concluded from experimental investi-gations that the gas pressure produced within closed poresinevitably affects the shape, size and distribution of pores aswell as the overall mechanical properties of the closed-cellfoams (Lankford Jr and Dannemann, 1998; Sugimura et al.,1997; Yamamura et al., 2001).

A few works incorporating the effect of initial gas pres-sure over the pore surface were reported with different ap-proaches. As a relatively early work in this filed, Ozguret al. (1996a,b) developed a specialized finite element pro-gram to model the elastoplastic properties of internallypressurized closed-cell metal composites. Their resultsshowed that the inner pressure increased compressivestiffness and delayed the onset of plastic yielding of the cellwalls. More recently, Öchsner and Mishuris (2009) usedthe finite element method to numerically investigate apressurized cubic unit cell and found that the internalgas pressure might significantly affect the initial yield con-dition as well as the macroscopic behavior of the unit cell ifthe gas pressure is clearly less than a half of the initial yieldstrength of the matrix material. Xu et al. (2010) numeri-cally simulated the elastoplastic deformation of closed-cellfoams with two varieties of pores distribution, i.e. body-centered and face-centered cubic arrangements of pores.It has been found that the inner gas pressure resulted inan asymmetric deformation for the closed-cell foams un-der tension and compression. Different effects of gas pres-sure on initial yielding of the two kinds of foams aresystematically addressed. Numerical modeling is helpfulto determine the constitutive behavior of the closed-cellporous material, and in some extent it can be used to ob-tain relatively accurate results. However, this approach isrestricted to limited porous solids with the specific porespace geometry only, and its computational efficiency isdependent on the numerical method used. In the aspectof theoretical investigation, Ashby et al.(2000) analyzedthe effect of internal pressure on elastic modulus and the

initial yield stress, and further suggested related semi-empirical relationships. Guo and Cheng (2002) extendedthe classic Gurson’s model by introducing vapor pressureeffects. On the basis of the modified model, they demon-strated the effects of the internal void pressure on the mac-roscopic plastic and void growth behavior of packagingmaterial treated as a closed cell porous solid with pressur-ized gas. Based on mean-field approximation, Kitazonoet al. (2003) developed a micromechanics model incorpo-rating gas pressure to analyze the macroscopic elastic–plastic behavior of closed-cell metal foams. Because theadopted first moment of stress-based mean-field approxi-mation only concerns the averaged stress and strain, andcannot estimate the local heterogeneous deformation(Hu,1996; Qiu and Weng, 1995), the extended micromechanicsmodel fails to predict gas pressure effect on the yield stressof porous materials. To overcome this disadvantage, Zhanget al. (2009) developed a second-order moment microme-chanics model correlating the macroscopic stress with lo-cal second-order stress moment in the matrix. Basedupon this feature, the important effects of inner pore pres-sure on the nonlinear macroscopic constitutive behaviorsof a closed cell porous metal were precisely examined.

It should be noted that, in the theoretical framework ofthe previously developed micromechanics model, the gaspressure was supposed to be identical in all closed voids.In such a case, the porous material was reasonably simpli-fied as a solid containing a single gas-filled void, which canbe referred to as the single-inclusion micromechanicalmodel. However, as mentioned above, different gas pres-sures may be produced in multiple micro-size closed poresduring the fabrication of closed-cell porous metals. An ac-tual example of pressurized closed-cell porous structurewith multiple pore pressures is that of potassium gas bub-bles in tungsten (Kim and Welsch, 1990), the porous mate-rial was produced by ion-injecting potassium into tungstenand then annealing it at high temperature (2300 �C) toform the pores. Under the condition of the annealing tem-perature, the gas pressures in multiple pores range from 5to 200 MPa (Ozgur et al., 1996a). The multi-inclusionmicromechanics model, which is essentially realistic froma practical point of view, and the effects of different gaspressures in multiple closed pores on macroscopicmechanical behaviors of porous materials were not consid-ered in the existing literatures with the exception ofVincent et al. (2009) and Julien et al. (2011). In Vincentet al’s work, two rigorous upper bounds for the effectiveporo-plastic behavior of a class of porous materials con-taining two populations of voids of different size subjectedto inner pressures, were derived respectively usingGurson-like approach (Gologanu et al., 1994) and the var-iational method of Ponte Castañeda (1991). A N-phasemodel, inspired by Bilger et al.(2002), was also proposedwhich matches the best of the two bounds at low and hightriaxiality of stress. Further, Julien et al. (2011) extendedthe N-phase micromechanical model to include the caseof a viscoplastic matrix. It is noted that their investigationswere performed within a consistent framework of poro-mechanics, and the attentions were focused on the boundsfor the effective flow surface and the multi-scale dissipa-tion potentials of the biporous materials.

60 L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75

With the consideration of the inner gas pressures whichare constantly undergoing variations due to pore volumechanges during deformations, in the present work, we ex-tend the micromechanics model and develop a homogeni-zation method to examine the macroscopic elastoplasticityof closed-cell porous materials with possibly different gaspressures.

2. Micromechanics theory

In many engineering applications, the closed-cell por-ous material with inner pressure usually exists as a pre-stressed material. In this section, we first generalize theclassical Hill’s lemma to incorporate the effects of multiplepressures on the overall mechanical behavior, and estab-lish a new micromechanical model to derive the initialaveraged strain fields. Then we present the relation be-tween the local stress moment in matrix and the macro-scopic stress for specific closed-cell materials utilizingthe extended second-order moment model.

2.1. Configuration evolutions and Hill’s lemma for RVE

The Representative Volume Element (RVE, Nemat-Nas-ser and Hori, 1999) of a closed-cell material consists of iso-tropic solid matrix, denoted by the subscript 0, and a largenumber of closed pores filled with possibly different pres-sures, pr, (r ¼ 1; . . . ; n; n is the number of pores), as shownin Fig. 1.

For the RVE of a multiple pressurized closed-cell mate-rial, there are three configurations defined firstly by Zhanget al. (2009), i.e. the reference configuration, the initialconfiguration, and the current configuration. Here, webriefly describe the related configuration evolution as fol-lows. We first assume that each of multiple pressures p�ris held on a cluster of gas molecules such that its volumejust fits a pore, then the imaginary reference configurationis achieved by embedding such gas phases into the voidcells of the material. Note that at this stage no stress existsin the solid matrix. The successive initial configuration canbe obtained by hypothetically releasing the holding pres-

Fig. 1. The RVE of a closed-cell porous material with multiple gaspressures.

sures on the gas clusters. At this configuration, thepressure in each of voids decreases to p0

r ; and the solid ma-trix is pre-stressed in order for the whole material to beself-equilibrated, in which the material does not experi-ence any external load. It should be pointed out that, in thisconfiguration, the initial compatible ‘‘residual’’ elastic fieldin the RVE is produced entirely by gas pressures. When wefurther exert an external load F on the initial configuration,the RVE will deform to a new equilibrium configuration,which is referred to as the current configuration. In thiscase, the current gas pressure in the rth pore is character-ized by pr. Note that the introduction of the fictitious refer-ence configuration is essential in this study, which can beused to derive the local average strains and the initial mac-roscopic strains of the material.

As is well known, the equation of state for the gas insideeach pore, treated as an ideal gas here, shows a nonlinearrelationship between gas pressure and volumetric varia-tion. However, the gases in pores of closed cell porous sol-ids undergo very small volume changes due to theconstraint of the solid matrix with a large bulk modulusrelative to the gas pressures. From the viewpoint ofmechanics, the gases in pores can be considered as isotro-pic ‘‘elastic solids’’, though the gas phases in the closed cellare not made of solid material. Making use of Gay-Lussac’slaw, the stress–strain relation of the gas inside each poreunder constant temperature, can be given as,

rr ¼ �p�r ð1� evÞI ð1Þ

where I is the isotropic unit tensor, and ev is the gas volu-metric ‘‘strain’’ from the reference configuration to the cur-rent configuration. It can be easily observed from thisconstitutive equation that the current gas pressure p�r canbe regarded as the bulk modulus kr of the rth gas phase.On the other hand, the shear modulus of the gas phasecan be defined as lr = 0, due to the pressurized gas notbeing able to resist shear deformation. Therefore, the stiff-ness tensor of the gas phase can be expressed byðCijklÞr ¼ p�r dijdkl (dij is the Kronecker delta). Throughoutthe present study, Einstein’s summation convention overthe repeated indices is used.

From the analysis of configuration evolution above, theexisting traction force P�r at the each solid/gas interface,can be treated as an interfacial force causing a stress jump,where P�r ¼ �p�r I ¼ P�r I ðr ¼ 1 � nÞ: For the material inpractical engineering, the gas pressures already exist be-fore applying the external force F at the initial configura-tion. It is convenient to present the principles of virtualwork during the configuration evolution process in twostages, i.e. the first stage being from the reference configu-ration a to the initial configuration b, and the second beingfrom the initial configuration b to the current configurationc. It is noted that, the traction forces P�r at solid/gas inter-faces keep constant in the second stage once they are line-arly achieved as ramp loads in the first stage. Therefore, theprinciples of virtual work in the series of configurationevolutions, are derived as follows,

Xn

r¼1

Zsr

in

P�r � urpdS ¼Z

Vr : ep dV ða! bÞ ð2Þ

L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75 61

Zsout

F � uFdSþ 2Xn

r¼1

Zsr

in

P�r � urFdS ¼Z

Vr : eF dV ðb! cÞ

ð3Þ

where Sout is the outer boundary of RVE, Srin the inner

boundary of the rth closed pore, and V the volume of RVEconstructed with matrix and multiple gas phases. urp andep represent respectively the displacement of the innerboundary and the strain in the RVE induced by initial gaspressures, urF and uF are respectively the displacementsof the inner boundary and the outer boundary inducedby external forces, and eF is the external force-inducedstrain.

Making use of Eqs. (2) and (3) and noting thate ¼ ep þ eF; we have

Zsout

F � uFdSþ 2Xn

r¼1

Zsr

in

P�r � urFdSþXn

r¼1

Zsr

in

P�r � urpdS

¼Z

Vr : edV ð4Þ

From Eq. (4), one obtains the Hill condition of a porousmaterial with multiple gas pressures,

�r : �eF � 2Xn

r¼1

frP�r I : �erF �

Xn

r¼1

frP�r I : �erp ¼ hr : ei ð5Þ

where �r and �eF , respectively denote the macroscopic stressand strain of the RVE caused by external load; �erp and �erF

are the strain tensor of the rth gas phase respectively in-duced by initial inner pressures and external forces; fr isthe volume fraction of the rth pore, and hXi is the volumeaverage over the whole RVE of the quantity X. Making useof Betti’s reciprocal theorem and replacing �

Pnr¼1 frP

�r I : �erF

in Eq. (5) by �r : �ep, we obtain

�r : �eF þ 2�r : �ep �Xn

r¼1

frP�r I : �erp ¼ hr : ei ð6Þ

where �ep is the macroscopic strain of the RVE caused by in-ner pressures. It is noted from Eq. (6) that the initial mac-roscopic strain �ep and averaged strains �erp in multipleinclusions caused by initial gas pressures play an impor-tant role in the macroscopic mechanical behaviors of theporous material. The averaged strain fields induced by ini-tial inner pressures can be determined using the microme-chanical model developed in this work. Then the fieldfluctuation method can be employed as a further proce-dure to correlate the local second-order stress momentwith the macroscopic stress incorporating the effect ofmultiple pore pressures.

2.2. A new micromechanical model for pressure-inducedstrain field

For a porous material with identical pressure in eachpores, one can easily evaluate the gas pressure-inducedmacroscopic strain and gas inclusion strain using thewell-known composite spheres assemblage model (Hashinand Shtrikman, 1963). However, for the multiple gas-filledporous solids with different pressure in pores, it is difficultto analytically obtain the averaged strain fields of the

matrix and the inclusion induced by multiple gas pressuresusing the Hashin’s composite spheres assemblage model.To bypass this difficulty, in this section, we introduce theeigenstrains to generate inner pressures in the pores. Thena micromechanics model is developed for calculating theinner pressures-induced average strain fields of thematerial.

A closed cell porous material with multiple gas pres-sures can be thought of as a composite with residual stres-ses or internal stresses. The equivalent inclusion method(Eshelby, 1957) is usually used to evaluate residual stres-ses of a composite (Takao and Taya, 1985; Hu and Weng,1998). Herein, this method is used in conjunction withthe Mori–Tanaka model (Mori and Tanaka, 1973; Benven-iste, 1987) to predict the averaged stress and strain fieldsinvolved in porous materials with gas pressures. Fig. 2illustrates a replacement procedure of the RVE of the por-ous material, based upon the equivalent inclusion method.In order to characterize the initial stresses in solid matrixand multiple gas phases of the RVE of the porous material,the zero-stress reference configuration defined above isfirst introduced. By releasing the tractions at the solid/gas interfaces in the reference configuration, we obtainthe initial configuration of such material with multiplegas pressures, as shown in Fig. 2(a). It is well accepted thatthe gas pressure in each void can be treated as an internalstress which is imaginarily produced by prescribing aneigenstrain ~ep

r in each gas phase, as depicted in Fig. 2(b).Further, with the use of the equivalent inclusion method,the inhomogeneity (gas phase) can be replaced with thematrix material, and another eigenstrain ~e�r , denoted asthe equivalent eigenstrain, is accompanied, as given inFig. 2(c). From this replacement procedure, the same stressfield inside and outside the gas phases can be found in thethree equivalent models characterized by Fig. 2(a)–(c),respectively.

For the sake of simplicity, all the pores in the materialare assumed to be spherical. Utilizing the equivalent inclu-sion method, the initial inner stress P0

r in the rth gas phaseof the material can be expressed as,

P0r ¼ Cr : ð~e0 þ ~er � ~ep

r Þ ¼ C0 : ð~e0 þ ~er � ~e�r � ~epr Þ ð7Þ

where P0r is the initial gas stress related to initial gas pres-

sure by P0r ¼ �p0

r I, ~e0 is the initial average strain in the ma-trix induced by initial gas pressures, and ~er is referred to asthe average perturbing strain in the rth gas phase. C0 and Cr

are the stiffness tensor of the solid matrix metal and therth gas phase, respectively.

According to Eshelby (1957), Takao and Taya (1985),the strain ~er can be represented by

~er ¼ S : ð~e�r þ ~epr Þ ð8Þ

where S is the fourth-order Eshelby’s tensor depending onthe pore shape and elastic properties of the solid matrix.Making use of Eq. (7), the gas phases with multiple pres-sures of the RVE in the initial configuration can be re-placed by the solid matrix subjected to two eigenstrains~e�r and ~ep

r .

Fig. 2. Representation of the replacement procedure of the RVE of a porous material with multiple gas pressures, based upon the equivalent inclusionmethod. The multiple gas pressures in the initial configuration of the RVE a porous material (a), can be produced by prescribing eigenstrains ~ep

r in gas phases(b), and the gas phases can be further replaced with the matrix material, by introducing equivalent eigenstrains ~e�r (c). The same stress field inside andoutside the gas phases can be found in the three models, equivalent to each other.

62 L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75

In the initial configuration, the self-equilibrium condi-tion of the multiple gas-filled porous material leads to,

Xn

r¼1

frP0r þ 1�

Xn

r¼1

fr

!C0 : ~e0 ¼ 0 ð9Þ

Substitution of Eq. (7) into Eq. (9) yields,

~e0 þXn

r¼1

frð~er � ~e�r � ~epr Þ ¼ 0 ð10Þ

And combining Eqs. (8) and (10), we finally obtain

Xn

r¼1

fr~er ¼ ðS�1 � IÞ�1: ~e0 ð11Þ

The initial strain of the solid matrix ~e0 can be then derivedfrom Eq. (9) as

~e0 ¼ �Pn

r¼1 frC�10 : P0

r

1�Pn

r¼1 frð12Þ

Under the inner pressure, the induced initial macroscopicstrain of the material with respect to the reference config-uration can be written as,

�e0p ¼ 1�

Xn

r¼1

fr

!~e0 þ

Xn

r¼1

frð~e0 þ ~erÞ ð13Þ

Substituting Eqs. (11) and (12) into Eq. (13), and usingsome algebraic operations, the initial macroscopic strain�e0

p can be simply expressed by

�e0p ¼ðC0 : S� C0Þ�1 :

Pnr¼1 frP

0r

1�Pn

r¼1 f rð14Þ

It can be seen from Eq. (14) that �e0p is determined by the

initial gas pressures and the properties of the solid matrixonly, being independent of the stiffness of the gas inclusions.Because the closed cells are supposed to be spherical, thenon-zero components of the Eshelby’s tensor Sijkl are given as

S1111 ¼ S2222 ¼ S3333 ¼7� 5v0

15ð1� v0Þ;

S1122 ¼ S2233 ¼ S1133 ¼5v0 � 1

15ð1� v0Þ;

S1212 ¼ S2323 ¼ S3131 ¼4� 5v0

15ð1� v0Þð15Þ

where v0 is the Possion’s ratio of the solid matrix. FromEqs. (14) and (15), the volumetric strain of the isotropicmaterial induced by initial pressures can be obtained as

�e0pm ¼

ð3k0 þ 4l0ÞPn

r¼1 frp0r

4 1�Pn

r¼1 fr� �

l0k0ð16Þ

where k0 and l0 are respectively the bulk and shear moduliof the solid matrix. Correspondingly, for the ensemble-vol-ume average of the volumetric strains within the multiplegas inclusions caused by initial gas pressures, we have

Xn

r¼1

fr�erv ¼Xn

r¼1

frAp0r ð17Þ

where

A ¼ 34

1þ v0 þ 2Pn

r¼1 fr � 4v0Pn

r¼1 fr

l0ð1þ v0Þð1�Pn

r¼1 frÞ

� �

Because the interactions between gas-filled pores areignored in the present micromechanics model, the gaspressure-induced volumetric strain of the rth inclusiontakes the form

�erm ¼ Ap0r ð18Þ

According to the ideal gas law, the initial gas pressurep0

r can be related to the introduced interfacial traction p�rin the following form,

p�r ¼ p0r ð1þ �ermÞ ð19Þ

Combining Eqs. (18) and (19) yields,

�erm ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4Ap�r

p� 1

2ð20Þ

Using Eqs. (19) and (20), the pressures-induced macro-scopic volumetric strain of the material can be rewrittenas,

�e0pm ¼

3k0 þ 4l0

2l0k0 1�Pn

r¼1 fr� �Xn

r¼1

frp�rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1þ 4Ap�r

pþ 1

ð21Þ

Recalling that the gas cannot resist shear deformationand the closed cells are assumed to be spherical, the aver-aged shear strains induced by gas pressures are zero, andthe corresponding averaged normal strains are isotropic.Thus the gas pressures-induced averaged strain over the

L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75 63

multiple gas-filled pores and the corresponding macro-scopic strain tensor can, respectively, be expressed asfollows,

�erp ¼13

�ermI; �ep ¼13

�e0pmI ð22Þ

Based on the results of Eqs. (6) and (22), we can furthercorrelate the local second-order stress moment with themacroscopic stress and initial gas pressures by utilizingthe field fluctuation method.

2.3. Effective elastoplastic properties

For a composite containing multiple inclusions, manymicromechanical models can be used to evaluate the effec-tive elastic properties, e.g. Hashin–Strikman variationalmethod (Hashin and Shtrikman, 1963), self-consistentmethod (Hill, 1965a), and Mori–Tanaka method (Moriand Tanaka, 1973; Benveniste, 1987). These models maybe extended to derive the macroscopic nonlinear elasto-plastic properties of composites. Using the concept of lin-ear comparison material (Ponte Castañeda, 1991), thenonlinear macroscopic properties of the composite can becharacterized by the instantaneous secant moduli whichare strain dependent.

For the nonlinear composite modeled as a linear com-parison material, the Hill condition Eq. (6) is still valid.Let �eF ¼M : �r and e ¼m : r: Here M is the macroscopiccompliance tensor of the composite, and m is the localcompliance tensor. Thus Eq. (6) can be rewritten as

�r : M : �rþ 2�r : �ep �Xn

r¼1

frP�r I : �erp ¼ hr : m : ri ð23Þ

It is obvious from Eq. (23) that the mechanical energy ofthe material include three major parts: the ones arisenfrom the external load, from the multiple gas pressure,and from their interaction. Now, under a constant macro-scopic stress �r; let the local compliance tensor m undergoa small perturbation dm. This leads to a perturbation of thelocal stress field, dr: Thus, we have

�r : dM : �rþ 2�r : d�ep �Xn

r¼1

frP�r I : d�erp ¼ hr : dm

: ri þ 2hdr : m : ri ð24Þ

Since, under a constant applied stress, the volume aver-age of the local stress perturbation vanishes, utilizing theHill condition the second term of the right side of inEq. (24) vanishes. Thus one can obtain the relationshipbetween the second moment of local stress and the macro-scopic stress.

The matrix of porous material is taken as isotropic elas-toplastic solid and the gas inclusions are treated as linearelastic solid. For a small perturbation of the shear modulusl0 of the matrix, Eq. (24) becomes

�r : dM : �rþ 2�r : d�ep �Xn

r¼1

frP�r I : d�erp

¼ 1�Xn

r¼1

fr

!d

12l0

� �hr0 : r0iV0

ð25Þ

where V0 ¼ 1�Pn

r¼1fr� �

V is the volume of the matrix, andr0 denotes the local deviatoric stress of the matrix. Usingthe equivalent stress of the matrix (Qiu and Weng, 1992),i.e. �r2

e0 ¼ 32 hr0 : r0iV0

, we can rearrange Eq. (25) as

�3l20 �r :

dMdl0

: �rþ 2�r :d�ep

dl0�Xn

r¼1

frP�r I :

d�erp

dl0

!

¼ 1�Xn

r¼1

fr

!�r2

e0 ð26Þ

It is recognized that the introduced equivalent stress ofthe matrix is based on the energy average of the localdeformation field, and is capable of accounting to some ex-tent for the local stress variation in the matrix. In Eq. (26),the macroscopic stress, �r; is correlated with the averagedequivalent stress of the matrix, �re0; from which the secondmoment of local stress may be obtained under any externalload. For the macroscopically isotropic material, the defor-mation can be decomposed into volume deformation andshape deformation,

�3l20 �

1�K2

�r2m

d�Kdl0� 1

2�G2�r0ij �r

0ij

d�Gdl0þ 2�rm

d�e0m

dl0þXn

r¼1

frp�rd�erm

dl0

!

¼ 1�Xn

r¼1

fr

!�r2

e0

ð27Þ

where �K and �G are respectively macroscopic bulk and shearmoduli of the material. According to the definition ofequivalent stress from the average of the elastic distor-tional energy of the composite, �r2

e ¼ 32

�r0ij �r0ij, Eq. (27) canbe further formulated as

3l20

�r2m

�K2

d�Kdl0þ 1

3�r2

e�G2

d�Gdl0� 2�rm

d�e0m

dls�Xn

r¼1

frp�rd�erm

dlr

!

¼ 1�Xn

r¼1

fr

!�r2

e0 ð28Þ

When the matrix undergoes a plastic deformation, theinvolved material properties are described by the instanta-neous secant moduli of the nonlinear material modeled asa linear comparison material. Indeed, the secant materialproperties of the nonlinear material are heterogeneousdue to the heterogeneous stress and strain field in such aporous material subjected to inner gas pressures andexternal loads. Therefore, without the information of thelocal stress or strain, it is not realistic to obtain the exactinstantaneous secant moduli of the heterogeneous mate-rial. As a simplified treatment, we introduce the widelyused assumption (Hu, 1996; Ponte Castañeda, 1991,1996; Ponte Castañeda and Suquet, 1997; Qiu and Weng,1992, 1993, 1995; Tandon and Weng, 1988; Zhang et al.,2009) and assume that the secant modulus of the nonlin-ear matrix is uniform, and it can be expressed as a functionof an averaged equivalent stress of the matrix. Herein theaveraged equivalent stress �re0 of the solid matrix inEq. (28) is introduced as an effective tool for determininga uniform secant modulus within the matrix.

64 L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75

As a special case, for the material subjected to uniaxialloading with a macroscopic stress �r; �rm ¼ �r=3 and �re ¼ �r;the Eq. (28) can be simplified as

l20

13

�r2

�K2

d�Kdl0þ

�r2

�G2

d�Gdl0� 2�r d�e0

mdl0� 3

Xn

r¼1

frp�rd�erm

dl0

!

¼ 1�Xn

r¼1

fr

!�r2

e0 ð29Þ

where the macroscopic moduli, �K and �G, can be estimatedby several well-developed micromechanics methods, e.g.the Hashin–Strikman variational method (Hashin andShtrikman, 1963), the self-consistent method (Hill,1965a), the general self-consistent method (Christensenand Lo, 1979), and the Mori–Tanaka method (Mori and Ta-naka, 1973; Benveniste, 1987; Norris, 1989). As a widelyused micromechanics model, Mori–Tanaka approach hasproven to be relatively accurate in predicting the effectiveproperties of composites. This model originally proposedfor two-phase composites can be readily extended to mul-tiphase composites. For the material with multiple spheri-cal inclusions under investigation, the Mori–Tanakaestimate of the effective stiffness tensor �C can be givenas (Qu and Cherkaoui, 2006),

�C ¼ 1�Xn

r¼1

fr

!C0 þ

Xn

r¼1

frCr : Ar

" #1�

Xn

r¼1

fr

!Iþ

Xn

r¼1

frAr

" #�1

ð30Þ

where the strain concentration tensor Ar for the rth inclu-sion is

Ar ¼ ½Iþ S : C�10 : ðCr � C0Þ�

�1 ð31Þ

In the derivation of the Mori–Tanaka result for multi-phase composite, an important assumption is that S isequal to the corresponding Eshelby tensor for a singleinclusion in a matrix, which means that the interactionamong inclusions is not fully taken into account in multi-phase composite. For convenience to derive the explicit re-sults of the macroscopic bulk and shear moduli �K and �G,the fourth-order isotropic tensors �C, C0, Cr and S can be gi-ven in the following concise notations (Hill, 1965b; Qu andCherkaoui, 2006):

�C ¼ ð3�K;2�GÞ; C0 ¼ ð3k0;2l0Þ Cr ¼ ð3kr;2lrÞ;S ¼ ð3c0;2d0Þ ð32Þ

with kr ¼ p�r ; lr ¼ 0; c0 ¼k0

3k0þ4l0and d0 ¼ 3k0þ6l0

15k0þ20l0. Use

of Eq. (32) in Eqs. (30) and (31) leads to the final explicitexpressions for �K and �G after a straight forward calculation,i.e.,

�K ¼ k0 þ3k0 þ 4l0Pn

r¼1frðp�r�k0Þ4l0þ3p�r

� ��1� 3

;

�G ¼ l0 �5Pn

r¼1 frl0ð3k0 þ 4l0Þ6Pn

r¼1 frðk0 þ 2l0Þ þ 9k0 þ 8l0

ð33Þ

As remarked by Norris (1989), the Mori–Tanaka predic-tion of multiphase composite with spherical inclusions ofsofter (stiffer) phase is coincide with the Hashin–Shtrink-man upper (lower) bounds for the bulk and shear moduli(Hashin and Shtrikman, 1963).

In Eq. (29), the pressure-induced volumetric strain ofthe whole RVE and the rth inclusion, �e0

m and �erm; were ob-tained by the developed micromechanical model describedin Section 2.2. With the help of the derived results for �e0

m ;�erm; �K and �G, one can correlate the macroscopic stress �rand the averaged equivalent stress �re0 in matrix. If thecomposite is linear elastic, the macroscopic stress andthe averaged von Mises stress in matrix (�re0) are directlyrelated by Eq. (29). When the composite is nonlinear (e.g.due to elastoplastic deformation of the matrix), the moduliof the composite and constituent phase in Eq. (29) shouldbe substituted by their corresponding instantaneoussecant moduli. Recalling the assumption made above, thesecant moduli of the metal matrix are only functions ofthe averaged equivalent stress, given in what follows.

For an elastoplastic metal material, the effective stressre and strain ep

e in the plastic state are supposed to followthe modified Ludwik equation

re ¼ ry þ hðepeÞ

n ð34Þ

where ry, h and n (0 6 n 6 1) are the tensile yield stress,strength coefficient and work-hardening exponent, respec-tively. The secant elastic modulus at a given plastic statecan be given by

Es0 ¼

11E0þ ep

e

ryþhðepe Þ

n

ð35Þ

where E0 is the elastic modulus of metal without any de-fects. From the isotropic property and plastic incompress-ibility, the secant bulk and shear moduli and the secantPoisson’s ratio can be written as

k0s ¼ k0 ¼E0

3ð1� 2v0Þ; l0s ¼

3k0E0s

9k0 � E0s;

v0s ¼3k0 � 2l0s

2ð3k0 þ l0sÞð36Þ

where m0 is the Poisson’s ratio of the matrix material. It canbe seen from Eq. (32) that the plastic state of the matrixcan be characterized by ep

e ; or any of E0s, l0s, and v0s undera monotonic, proportional loading.

From the solution procedure suggested by Zhang et al.(2009), �re0 can be treated as known quantity, and thenthe macroscopic stress �r can be determined using Eqs.(28), (33), (16), and (18). For the case of uniaxial loading,the relationship between the matrix’s effective stress �re0

and the applied uniaxial stress �r can be established byEq. (29), and the only non-zero stress �r causes longitudinalstrain �e1 and transverse strain �e2 ¼ �e3: It is noted that, dueto the constraint of the matrix, the void volume variationfrom the reference configuration to the initial configura-tion can be regarded as very small, and this implies thatthe difference between p�r and p0

r is negligible. For conve-nience, p0

r ; instead of p�r , is used for calculation in the pres-ent work. Once the uniaxial stress �r is determined by a

L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75 65

given �re0; the corresponding real macroscopic strains, i.e.the strains from the initial configuration to the currentconfiguration, can be obtained as

�e1 ¼�r�Esþ ð�eps � �ep0sÞ

1þ �ep0s; �e2 ¼ �e3 ¼

� �vs �r�Esþ ð�eps � �ep0sÞ1þ �ep0s

ð37Þ

where �Es ¼ 9�Gs �Ks3�Ksþ�Gs

and �v s ¼ 3�Ks�2�Gs2ð3�Ksþ�GsÞ

are the effective secant

elastic modulus and secant Poisson’s ratio of the material,respectively. �eps is the normal strain induced by the innerpressures at the current configuration, and �ep0s denotesthe residual normal strain at the initial state. �eps and �ep0s

can be determined by the second equation in Eq. (22)where the shear modulus l0 should be substituted by theinstantaneous secant shear modulus of the matrix atdifferent states. It can be observed from Eq. (37) that theexternal load and the inner pressure have a coupling effecton the deformation of the nonlinear material. The innerpressures in pores play an important role in determiningthe effective elastoplastic properties of the porous mate-rial. To characterize the longitudinal and transverse defor-mations of the material from the initial configuration tothe current configuration, the nominal Poisson’s ratio isdefined as,

�v ¼ ��e2

�e1¼ �

�e3

�e1ð38Þ

The overall stress–strain relationship of the porousmaterial can be obtained by means of the present micro-mechanical model. It is noted that this micromechanicalmodel can analytically capture macroscopic elastoplasticproperties of the porous material with inner pressures,but cannot directly characterize microscopic inhomoge-neous deformations. To solve this problem, we develop ahomogenization theory which can account for the contri-bution of the inner pressures to numerically examine theeffective and local elastoplasticity of the closed-cell porousmaterials in the following section.

3. Computational homogenization for effectiveelastoplasticity

For porous materials composed of a periodic array ofmicroscopic unit cells with gas pressures, an efficient ap-proach to predict their effective elastoplasticity is thehomogenization theory based on the concept of unit cell(Bensoussan et al., 1978; Sanchez-Palencia, 1980; Suquet,1987), which can characterized both the homogenizedmacroscopic behavior and the microscopic stress andstrain distributions in materials. This approach thereforehas been successfully applied to various analyses ofmechanical properties of porous materials in which thepores may be reasonably assumed to be periodically dis-tributed in matrix (Ohno et al., 2002; Okumura et al.,2004). However, the pressures inside pores have not beenincorporated in the conventional homogenization modelused in porous materials. The internal residual stress ini-tially produced in materials may become an important is-sue especially for the porous material with pressurized gasinside their closed cells. A method is therefore requiredwhich enables the homogenization theory to deal effi-

ciently with the effect of inner gas pressures. In this sec-tion, a homogenization theory is developed specially fortime-independent elastoplastic porous materials with in-ner gas pressures.

3.1. Theoretical framework

In this section, a model is developed for evaluating theinitial gas pressure-induced macroscopic strain and thevariation of gas pressure with the deformation of the unitcell when subjected to external loading. The main proce-dure for deriving this model is to incorporate the small-scale gas pressure information into macroscopic mechan-ical behaviors. For this purpose let’s consider a heteroge-neous porous material with closed cell porous material inthe reference configuration, occupying a region X andhaving a boundary as depicted in Fig. 3a. For the reasonsstated below, it can be reasonably assumed that the mac-roscopic structure can be obtained by the periodic transi-tion in the three spatial directions of a regular pattern,usually called RVE or unit cell. In this case, the RVE is athree-dimensional cubic cell containing a random andhomogeneous dispersion of the closed pores. The poredistribution within the RVE can be generated using therandom sequential adsorption algorithm (Rintoul andTorquato, 1997), which ensures a random, isotropic andhomogeneous distribution of pores. In the case of polydis-perse sphere pores, the multiple pores sizes are deter-mined from the porosity and the number of pores in theRVE. The radius of each pore is generated randomly be-tween the maximum and minimum pore radius. Poresare added sequentially until the desired porosity isreached. As for all homogenization theories, the mechan-ical problem is divided into two scales: the structure’sscale and the unit cell’s scale, called ‘‘macroscopic scale’’and ‘‘microscopic scale’’ and the related coordinates aredenoted by x and y, respectively. In real heterogeneousmaterials, the scale of a unit cell is typically small andcan be represented by a ratio defined by y = x/g, whereg is a small scale parameter. A typical section of periodicmicrostructures and the associated unit cell are depictedin Fig. 3b and c, respectively.

Let Y be the open subset of a space occupied by a basicunit cell, which can be divided into a solid matrix phaseY0 and a closed gas phase Yr. According to the assumptionmade above, the volume of Y, |Y|, is much smaller thanthat of X. The stress and strain in Y are called micro-stress r and micro-strain e, respectively. Then, macro-stress R and macro-strain E are obtained by averaging rand e over,

R ¼R

Y0rdY0 þ

Pnr¼1 frrr

jYj ;

E ¼R

Y0edY0 þ

Pnr¼1 frer

jYj ð39Þ

where rr and er are the pore stress and strain in the rthgas phase of the RVE. It is noted that rr and er are uni-form in the rth closed cells because of the inherent prop-erty of gas. The pore stress rr and the corresponding gaspressure pr can be related by rr ¼ �pr . As the geometry of

Fig. 3. Representation of the microstructure and unit cell of the closed cell porous materials.

66 L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75

the material is periodic, the fields r and e are also Y-peri-odic. The local strain field, e, is then the superposition ofan average field E and a fluctuating part ~e resulting fromthe perturbed part of a periodic displacement filed ~u.Similarly, the local stress field, r, is also the sum of anaverage field R and a perturbed stress ~r. The fields Eand R are respectively the strain and stress fields thatwould occur if no pores were in the material and thefields ~e and ~r are the responding correcting parts, whichrepresents the presence of the multiple pores. Then, fol-lowing decompositions hold true,

r ¼ Rþ ~r; e ¼ Eþ ~e; u ¼ E � y þ ~u ð40Þ

and in consequence of the periodicity of ~u, the volumeaverage of ~r and ~e over the RVE are both zero,

ZY

~rdY=jYj ¼ 0;Z

Y

~edY=jYj ¼ 0 ð41Þ

Eq. (41) indicates that the perturbed fields do not con-tribute to macroscopic properties. The local fields r and eexisting in the RVE can be solved by two approachesdepending on whether we are working with prescribedstress R or strain E. In the present study, the stresses ap-proach is employed to solve the local stress and strainfields by finite element method. The incremental macro-scopic constitutive law of any composite with periodicstructure can be obtained. For example, in the adoptedstresses approach, by imposing given initial gas pressuresp0

r in the rth closed cell and macroscopic surface tractionon the boundary of the unit cell and finding the corre-sponding microscopic stress/strain and macroscopic strain.It is expedient to compute the macroscopic strain E interms of displacements along the boundary @Y (S) of theunit cell. The elastic localization problem has the followingproblem for the unit cell can be stated as follows: given R,find u and r such that

divr ¼ 0 in Y0

r � n anti� periodic on @Y; r � n ¼ �prn on @Yr

r ¼ f ðeðuÞÞ in Y0; pr ¼ p0r1þdvr=vr

in Yr

R ¼ 1jYjR@Y y � tdS; E ¼ 1

jYjR@Y

12 ðu� nþ u� nÞdS

8>>>><>>>>:

ð42Þ

where the explicit dependence on y of the microscopicvariables was dropped for sake of brevity. In Eq. (42), n isnormal on the boundary @Y of Y, or the inner normal onthe boundary @Yr of the closed cells, and the antiperiodicityof r � n on @Y means that r � n has opposite values on oppo-site sides of @Y. The fourth and fifth equations representthe microscopic constitutive law of solid matrix and gasphases, respectively. u is the microscopic displacementfiled of the solid matrix and dvr/vr is the volume strain ofthe gas in the rth close cell. y denotes the coordinates ofa point within the unit cell or on its boundary, andt ¼ r � n is the boundary traction. The symbol � denotesdyadic tensor product.

As seen from Eq. (42), the inner gas pressures, in termsof the volume strain of closed cells, are prescribed andequal to the interface stresses on the boundary @Y of therth close cells. As shown in what follows, the contributionof inner gas pressures to the virtual work for the structurewithout gas phases, will be the first step of the homogeni-zation method.

A fundamental difference between the present homog-enization theory and conventional one of the unit cellproblem is due to their formulation for the virtual workof the structure. For typical elastoplastic porous structureswithout inner gas pressures, the virtual work equationtakes the following form,

ZYr : dedY ¼

Z@Y

du � tdS ð43Þ

3y

1y 2y

1p 2p 3p

4p5p

6p

7p

8p9p

10p

3y

1y 2y

1p 2p 3p

4p5p

6p

7p

8p9p

10p

Fig. 4. Representation of the unit cell of the closed cell porous material.

L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75 67

As for the porous material studied in this work, the in-ner gas pressures can produce the deformation of the over-all structure, and they can also undergo variations whenthe material is subjected to external load. It is recognizedthat the gas volume variations are induced by the pressurechange of the closed cells during the deformation. Accord-ing to the ideal gas law, the volume �Vr of compressible gasinside the rth closed cell is a function of the gas pressure,pr; the temperature, T; and the gas mass, mr,�Vr ¼ �Vrðpr ; T;mrÞ. The volume, �Vr ; derived from the gaspressure and temperature should equal the actual volume,Vr, of the rth closed cells. In finite element method this canbe achieved by augmenting the virtual work expression forthe structure with the following constraint equation,

Vr � �Vr ¼ 0 ð44Þ

To account for the coupling behavior between the over-all deformation and the inner pressures exerted by thecontained gas on the boundaries of the closed cells, theaugmented virtual work equation incorporating the gaspressures for the RVE can be given as,Z

Y0

r : dedY0 �Xn

r¼1

prdVr �Xn

r¼1

dprðVr � �VrÞ

¼Z@Y

du � tdS ð45Þ

where the first term is the virtual work done by the struc-ture without the gas phase, the second term is the virtualwork contributed by multiple gas pressures. The negativesigns imply that an increase in the pore volume releasesenergy from the gas phases, and dp in the last term ofthe left side can be regarded as a Lagrange multiplier en-forced the constraint Vr � �Vr ¼ 0: This augmented expres-sion represents a mixed formulation where the structuraldisplacements and gas pressure are primary variables.For the gas inside the rth closed cell, the correspondingvolume-pressure compliance under constant temperature,can be written as

d�Vr

dpr¼ �mr

qR

pR þ pA

ðpr þ pAÞ2 ð46Þ

where qr is the density of the gas in the rth pore, which isalways dependent on the pressure pr. pR and pA denote thepressure at the predefined reference density qR, and theambient pressure, respectively.

Making use of the augmented virtual work principle,the coupling between the structural deformation and theinner fluid pressure in closed cavity has been developedto study the problems of fluid-filled structures, using ahydrostatic fluid element from the commercial finiteelement analysis package–ABAQUS (2010). Ozguret al.(1996b) employed a specialized finite element pro-gram to describe the macroscopic response of closed-cellmetal composites which confine a hydrostatic fluid in theirinterior. More recently, Öchsner and Mishuris (2009) mod-eled the elasto-plastic behavior of porous metals withinternal gas pressure by a developed user subroutinewhich allows for the calculation of the actual pore pressureand volume. Ma et al.(2011) applied the hydrostatic fluidelement to derive the effective elastic properties of fluid-

filled closed cell materials, and comparisons with the the-oretical prediction confirmed the reliability of the fluidelement.

3.2. Numerical model

For the gas-filled structures under consideration, themicroscopic and macroscopic stress/strain fields can becalculated within the framework of homogenization the-ory integrated with the finite element formulation derivedfrom the virtual work Eq. (45). Here, the specific computa-tional procedure employed in this study is briefly de-scribed. Using the concept of the three configurations ofthe RVE defined in Section 2, we can perform the homoge-nization analysis in two steps. Firstly the initial configura-tion is obtained by assuming an initial pore pressure inevery closed cell and associated boundary conditions onthe faces of the RVE. In this case, the initial inner gas pres-sures in the closed cells create a self-equilibrated stressfield in the periodic unit, and the initial macroscopic straincan be obtained using Eq. (39). On the basis of the attainedinitial configuration, we can then proceed to prescribe themacroscopic surface traction on the RVE to reach a newequilibrium state, i.e. the current configuration. The localstress/strain fields within the unit cell are determinedand the corresponding macroscopic strain is also givenby Eq. (39). Taking into account configuration evolutions,the overall stress–strain relationships with respect to theinitial configuration are finally derived.

It should be mentioned that periodic boundary condi-tions (PBC) should be applied to the faces of the 3D RVE,either from the reference configuration to the initial con-figuration, or from the initial configuration to the currentconfiguration, because the relative displacement betweenopposite faces of the RVE is required to be constant or uni-form to maintain the homogeneous state of deformation ofthe periodic material subjected to inner gas pressures andexternal tractions. Let y1, y2 and y3 be the Cartesian coordi-nates corresponding to axes parallel to the RVE edges andwith origin at the lower left back corner of the RVE asshown in Fig. 4. u(y1, y2, y3) is the displacement vector ata point with coordinates (y1, y2, y3). The typical PBCs can

68 L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75

be expressed in terms of the macroscopic strain E, whichstands for the relative strain between opposite faces ofthe unit cell, according to

uð�y1; y2; y3Þ � uð0; y2; y3Þ ¼ E � �y; 0 6 y2 6 �y2; 0 6 y3 6 �y3

uðy1; �y2; y3Þ � uðy1;0; y3Þ ¼ E � �y; 0 6 y1 6 �y1; 0 6 y3 6 �y3

uðy1; y2; �y3Þ � uðy1; y2;0Þ ¼ E � �y; 0 6 y1 6 �y1; 0 6 y2 6 �y2

ð47Þ

where �y ¼ ð�y1; �y2; �y3ÞT is the vector containing the threeedge lengths of the 3D RVE along y1, y2 and y3 directions,respectively. It is pointed out that, for the RVE subjectedto both inner gas pressures and external tractions, themacroscopic strain E is unknown in advance, thereby thePBCs in Eq. (47) may not be directly applied to the bound-ary of the RVE at the reference configuration. To eliminatethe macroscopic strain in Eq. (47), an alternative form ofthe PBCs of RVE subjected to inner pressures and externaltractions has been given by Ma et al. (2011).

As the first step in numerical simulations, the periodicRVE was generated by splitting the spherical voids inter-secting the RVE faces into the appropriate number of partswhich were copied to the opposite faces. In addition, theinterior multiple voids without intersecting the RVE faceswere allowed to be distributed randomly. As for the meshgeneration of the RVE, three faces of the cube were firstmeshed with triangles, and then the meshes were copiedto the opposite sides. For illustration purposes only, themesh of one face of the RVE is shown in Fig. 5. Dependingon the triangular surface meshes generated in advance, thesolid volume of the 3D RVE was meshed with tetrahedrons.These treatments for the geometry and mesh generation ofthe RVE ensure that node positions on opposite faces wereidentical in order to apply PBCs. Similar operations for thegeneration of periodic unit cells with randomly distributedinclusions can be found in Pierard et al. (2007).

In our computational homogenization performed inABAQUS (2010), the solid matrix was finely discretizedwith tetrahedron elements associated with the elastoplas-tic material model. Note that the void surface should notbe too close to the faces, edges and corners of the RVE to

Fig. 5. Representation of the mesh of one face of RVE with multiple voids.

prevent the presence of distorted elements during mesh-ing. In addition, the mechanical response of the gas-filledvoids was simulated by using hydrostatic fluid elementsthat cover the boundaries of multiple voids. In regionswhere the solid elements were used to model the bound-aries of the multiple gas-filled voids, with the idealgas property, the hydrostatic fluid elements share thenodes at the void boundaries with the tetrahedron ele-ments. In conjunction with the PBCs enforced by constraintequations, the finite element model of the periodic RVEwas used to model the effective elastoplastic behavior ofthe porous material with multiple gas-filled voids.

4. Results and discussions

As an application of the presented micromechanicsmodel and the computational homogenization approach,in this section, the macroscopic elastoplastic behavior ofclosed-cell porous metal with different inner pressureswas investigated to discuss the effect of the pressures.

As indicated in the introduction, the realistic metalfoam may contain different gas pressures in the closedcells, resulting from their fabrication process. Since boththe micromechanics and homogenization approaches areparticularly established for the porous materials with theperhaps more practical multiple gas-filled closed cells,we focus on the effects of multiple gas pressures on themacroscopic elasto-plastic properties. As an illustrativeexample, the matrix of the porous aluminum is modeledas an isotropic elastoplastic material obeying the vonMises criterion with isotropic hardening as expressed byEq. (34). The adopted material parameters (Qiu and Weng,1992) of the matrix are listed in Table 1. For simplicity, theporous aluminum with two populations of closed cells ofdifferent sizes with two different gas pressures, is typicallyconsidered in this example. For one kind of the closed cells,the porosity and initial gas pressure are f1 and p1, respec-tively. And the other kind of the closed cells has the poros-ity f2 and initial gas pressure p2. Thus the relative density ofporous metal is 1 � f1 � f2.

4.1. Theoretical and numerical comparison upon uniaxialloading

Our micromechanics model is developed for closed cellporous materials with the randomly distributed gas-filledcells, and that needs to be verified by the computationalhomogenization approach based on a periodic unit cellwith randomly distributed pores. As a simple comparisonbetween the micromechanics model and the computa-tional homogenization model, in the present work, themacroscopic elastoplastic behavior of foam aluminummaterial with two different inner pressures under uniaxialload is first examined analytically and numerically. With

Table 1The material parameters of aluminum matrix.

E0 (MPa) m0 ry (MPa) h (MPa) n

68,300 0.33 250 173 0.455

(a)

(b)

Fig. 6. The theoretical and numerical comparisons of the uniaxial stress–strain curves of a porous metal (a) with p1 = 0.01ry, p2 = 0.02ry and (b) withf1 = f2 = 0.1.

L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75 69

the use of the developed micromechanics model and thecomputational homogenization approach, the theoreticaland numerical comparisons of the uniaxial stress–straincurves of the porous metal are shown in Fig 6(a) and (b).Throughout the figures in this paper, the arrow indicatesthe direction along which the various gas pressures andporosities increase. For the porous metal with the fixed ini-tial gas pressures of p1 = 0.01ry and p2 = 0.02ry, with theincreases of the porosities, the tensile stress–strain curvesare lowered, while the compressive stress–strain curvesare elevated, as shown in Fig. 6(a). In other words, the lar-ger porosity of the material can weaken the overallmechanical properties. When we fix the porosity asf1 = f2 = 0.1, the uniaxial stress–strain curves with differentgas pressures are shown in Fig. 6(b). With the increase ofthe two gas pressures, the tensile stress–strain curves

drops dramatically, and the initial macroscopic yield stressreduces. This implies that the gas pressures make the ten-sile deformation of porous material more easily; however,the compressive deformation of the porous material is notobviously influenced by gas pressures. Therefore, the pres-ence of initial inner pressures results in the asymmetry ofthe tension–compression curve. It can be observed fromFig. 6(a) and (b), the micromechanics results are very wellin agreement with those of the computational homogeni-zation approach.

The theoretical and numerical comparisons of the nom-inal Poisson’s ratio of the porous metal are shown inFig. 7(a) and (b), where both the porosity and the gas pres-sures have very prominent effects. During both tension andcompression with the gas pressures of p1 = 0.01ry andp2 = 0.02ry, the increase of the porosity causes the nominal

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.050.28

0.30

0.32

0.34

0.36

0.38

0.40

0.42

0.44

0.46

0.48

0.50

Micromechanics... Homogenization

f 1=0.20, f 2=0.10

f 1=0.10, f 2=0.05

f 1=0.02, f 2=0.01

Uniaxial strain

Pois

son'

s rat

io

-0.05 -0.04 -0.03 -0.02 -0.01 0.00 0.01 0.02 0.03 0.04 0.050.24

0.26

0.28

0.30

0.32

0.34

0.36

0.38

0.40

0.42

0.44

0.46

0.48

0.50

0.52

Micromechanics... Homogenization

p1=0, p2=0p1=0.2σy , p2=0.4σy

p1=0.3σy , p2=0.6σy

Uniaxial strain

Pois

son'

s rat

io

(b)

(a)

Fig. 7. The theoretical and numerical comparison of the nominal Poisson’s ratio of a porous aluminum (a) with p1 = 0.01ry, p2 = 0.02ry and (b) withf1 = f2 = 0.1.

70 L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75

Poisson’s ratio to decrease, as shown in Fig. 7(a). For thecase with the fixed porosity, during tension, the gas pres-sures cause the nominal Poisson’s ratio to decrease,whereas during compression, the higher gas pressure ele-vates the nominal Poisson’s ratio. Thus, the effect of gaspressures is asymmetric with respect to tension and com-pression in both Fig. 6(b) and Fig. 6(b) (and the asymmetryis more prominent for the nominal Poisson’s ratio). Theseresults are similar to the numerical results reported earlier(Öchsner et al., 2004, Zhang et al., 2009). For the theoreticaland numerical comparisons of the nominal Poisson’s ratioof the porous metal, when the porosity is relatively low,the micromechanics results agree well with the numericalresults. Nevertheless, for the case of the high porosity orhigh initial gas pressures, the micromechanics results arehigher compared to the numerical ones, especially whenthe strain is large. The overestimation of the theoreticalmodel may be attributed to the approximation of linear

elasticity of the gas phase and the inaccuracy of themicromechanics model when dealing with the high poros-ity case. Moreover, in our micromechanics model, it is as-sumed that the secant modulus only depend on theaverage effective stress of matrix. Actually, due to the localinhomogeneous stress field, each point of matrix has dif-ferent secant modulus when the plastic deformation ofmatrix occurs. Although the averaged equivalent stress isbelow the yield limit of matrix, the local equivalent stres-ses at some points may equal or exceed the value of theyield stress of the matrix. For this reason, when comparedto the computational homogenization, the micromechan-ics model overestimate the initial macroscopic yield stressand the nominal Poisson’s ratio of the porous material, asshown in Figs. 6 and 7.

With the use of a very large three-dimensional unit cellwith many randomly distributed voids, the computationalhomogenization method is relatively accurate to simulate

L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75 71

the elatoplastic behaviors of porous materials with multi-ple gas pressures; however, the large computational ex-pense limits its application for a large amount ofcomputation. From the theoretical and numerical compar-ison upon uniaxial loading, the developed micromechanicsmodel is an effective and acceptable analytical approach topredict the macroscopic elastoplastic properties with lesscost. Therefore, in what follows, we will use our microme-chanics model to investigate the effects of the different gaspressures on yield surface and yield strength of the porousmetal.

4.2. Effect of gas pressures on yield surface

For porous materials with relatively low porosity, anapproximate yield condition of porous materials wasproposed by Gurson (1977). This yield function was laterimproved by Tvergaard (1981, 1982). It might be worth-while to note that the Gurson’s model and its modifiedforms only can be only used for the porous materials with-out the initial gas pressures inside voids. Guo and Cheng(2002) and Xu et al. (2010) added the inner gas pressureto the Gurson’s model, however, in their models, the gaspressures were assumed to be identical. Compared to theprevious models, the yield function Eq. (28) derived fromour micromechanical model is more general, and it is validfor multiple gas-filled closed cell porous materials withstrain hardening effect and anisotropic properties. In thissection, the yield function Eq. (28) is used to examine theeffect of the multiple initial pressures on yield surface ofthe porous aluminum.

At the gas pressures of p1 = 0.05ry and p2 = 0.1ry, thevariation of the initial yield surface with respect to theporosity is given in Fig. 8, where the initial yield surfaceswith gas pressures are shown in solid lines, and thosewithout gas pressure are given in dotted lines. It is shownthat the yield surface become smaller with the increase of

-1.6 -1.2 -0.8 -0.40.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

f 1=0

.4,f 2=0

.

f 1=0

.3,f 2=0

.15

f 1=0

.2,f 2=0

.1

e yσ σ

σ

f 1=0

.1,f 2=0

.05

Fig. 8. The effect of the porosities on the yield surface of a porous

the porosities at the certain gas pressures. This implies thatthe larger porosity make the yielding of porous materialmore easily. At the certain porosities, the variations ofthe initial yield surface with respect to the different com-bination of the two gas pressures are given in Fig. 9. It isreadily seen that the yield surface of the porous materialis considerably influenced by initial gas pressures. Withoutthe gas pressures, the yield surface shows a tension–com-pression asymmetry. When the gas pressures are identical,the yield surface of the porous material is simply obtainedfrom that of the porous material without pressures by ashift along the hydrostatic axis, with a shifting distanceequals to the identical gas pressure. For this case of identi-cal gas pressures, the prediction of our model partly shownin Fig. 9, is consistent with the Zhang’s report (2009). Forthe case of multiple gas pressures under investigation,the different combination of the two gas pressures not onlyshift the yield surface in the pressure-free case along thenegative direction of the hydrostatic stress axis, but alsoa change in shape and size of the surface can be found.The maximal equivalent stress which can be sustained bythe material is lowered by a difference in gas pressures.A similar effect has also been observed by Dormieuxet al. (2006) and Vincent et al. (2009). In particular, theshift distance along the hydrostatic axis is the porosity-weighted average of the two inner gas pressures, as moreclearly revealed in Fig. 9(b) and (c).

4.3. Effect of gas pressures on initial yield strength

The initial yield strength is usually used to characterizethe strength of a material. When the local metal matrix of aporous material reaches its yield strength, the strength ofthe overall composite is reached. In the present analysis,we assume that the strength of the matrix is determinedonly by the averaged von Mises stress and independentof the hydrostatic stress. Again, the porous aluminum with

0.0 0.4 0.8 1.2 1.6

2

m yσ

p1=0.05σy, p2=0.1σy

aluminum at the gas pressures of p1 = 0.05ry and p2 = 0.1ry.

-1.0 -0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.6 0.80.0

0.1

0.2

0.3

0.4

0.5

0.6

f1=0.1, f2=0.2

e yσ σ

m yσ σ

p 1=0, p 2

=0p 1

=0.2σ y, p 2

=0.05σ y

p 1=0.5σ y

, p 2=0.05σ y

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.4 0.60.0

0.1

0.2

0.3

0.4

0.5

f1=0.2, f2=0.2

e yσ σ

m yσ σ

p1=0, p2=0

p 1=0.05σ y

, p 2=0.35σ y

p 1=0.2σ y

, p 2=0.2σ y

p 1=0.1σ y

, p 2=0.3σ y

-0.8 -0.6 -0.4 -0.2 0.0 0.2 0.40.0

0.1

0.2

0.3

f1=0.2, f2=0.3

e yσ σ

m yσ σ

p1=p2=0p1=p2=0.3σ

y

p 1=0

.6σy, p 2

=0.1σ

y

(a)

(b)

(c)

Fig. 9. The effect of the combination of the two gas pressures on the yield surface of a porous aluminum with the porosity of (a) f1 = 0.1 and f2 = 0.2,(b) f1 = 0.2 and f2 = 0.2, and (c) f1 = 0.2 and f2 = 0.3.

72 L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

p 1=0

.3σy, p

2=0

.6σy

p 1=0

.2σy, p

2=0

.4σy

p 1=0

.1σy, p

2=0

.2σy

y yσ σ

1 21 f f− −

p 1=0

, p2=0

Relative density

0.0 0.2 0.4 0.6 0.8 1.0-1.0

-0.9

-0.8

-0.7

-0.6

-0.5

-0.4

-0.3

-0.2

-0.1

0.0

0.1

p1=0.3σ

y, p2=0.6σ

yp1=0.2σ

y, p2=0.4σ

yp1=0.1σ

y, p2=0.2σ

y

y yσ σ

1 21 f f− −

p1=0, p

2=0

Relative density

0.0 0.2 0.4 0.6 0.8 1.00.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

p 1=0

.3σy, p

2=0

.6σy

p 1=0

.2σy, p

2=0

.4σy

p 1=0

.1σy, p

2=0

.2σy

y yσ σ

1 21 f f− −

p 1=0

, p2=0

Relative density

0.0 0.2 0.4 0.6 0.8 1.00

1

2

3

4

5

6

p 1=0

.3σy, p

2=0

.6σy

p 1=0

.2σy, p

2=0

.4σy

p 1=0

.1σy, p

2=0

.2σy

y yσ σ

1 21 f f− −

p 1=0

, p2=0

Relative density

(a)

(b)

(c)

(d)

Fig. 10. The effect of gas pressures and relative density on the initial yieldstrength upon (a) uniaxial tension, (b) uniaxial compression, (c) equi-biaxial tension, and (d) equi-triaxial tension.

L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75 73

two gas pressures is taken as an example, and the porosityof the two kinds of closed cells are assumed to be identical.The effects of the combination of the two gas pressures onthe initial yield strength are given in Fig. 10, as the relativedensity and gas pressures are varied. All strengths inFig. 10 are normalized by the uniaxial tensile yield strengthof the matrix, ry. It can be observed from Fig. 10(a) that thegas pressures reduce the initial tension yield strength,which approximately linearly decreases with the decreaseof the relative density. At the initial configuration of theporous material, the larger inner pressures require thesmaller extra tensile stress to produce the yielding ofmaterial. With nonzero gas pressures, at a certain smallrelative density, the initial yield strength becomes 0, whichimplies that the initial gas pressures can cause self-yield-ing of the material without any external load. Differentfrom the uniaxial tension, the initial compression yieldstrength varies nonlinearly with the relative density ofthe material, as shown in Fig. 10(b). For the case withoutgas pressures, the yield compression yield strength in-creases monotonously with the decrease of the relativedensity. For the case with gas pressures, there exists a crit-ical relative density below which the magnitude of the ini-tial compression yield strength slightly decreases with thedecreasing relative density. However, the magnitude of theinitial compression yield strength increases sharply withthe increasing relative density, when the relative densityexceeds the critical value. In addition, for the case of thevery high or very low relative density, the gas pressureshave little influence on the initial compression yieldstrength of the material.

Upon equi-biaxial tension and equi-triaxial tension, thecorresponding yield function needs to be reformulated. Forthe porous material subjected to equi-biaxial stress �r, wehave �rm ¼ 2�r=3 and �re ¼ �r, Eq. (28) can be written as

l20

43

�r2

�K2

d�Kdl0þ

�r2

�G2

d�Gdl0� 4�r d�e0

mdl0� 3

Xn

r¼1

frp�rd�erm

dl0

!

¼ 1�Xn

r¼1

fr

!�r2

e0 ð48Þ

For the porous material subjected to equi-triaxial stress �r,we have �rm ¼ �r and �re ¼ 0, Eq. (28) can be simplified as

l20 3

�r2

�K2

d�Kdl0� 6�r d�e0

mdl0� 3

Xn

r¼1

frp�rd�erm

dl0

!

¼ 1�Xn

r¼1

fr

!�r2

e0 ð49Þ

With different combinations of the gas pressures, the rela-tionships between the initial tension yield strength and therelative density of the porous material upon equi-biaxialand equi-triaxial tension are given in Fig. 10(c) and (d),respectively. It is indicated that the relationship for equi-biaxial tension is very similar to those of uniaxial tension,that is, the yield strength decreases almost linearly withrelative density, whereas such relationship is strongly non-linear for equi-triaxial tension. It should be noted that, asrevealed in Fig. 10(a), (c) and (d), the critical relative

74 L.-H. Ma et al. / Mechanics of Materials 73 (2014) 58–75

density causing the self-yielding of the porous material isidentical for uniaxial tension, equi-biaxial and equi-triaxialtension.

It is important to point out that, in our micromechanicsmodel and computational homogenization approach, thebending and buckling of cell walls are not taken into ac-count, however, in reality, these deformation mechanismsmay dominate in the porous materials with very highporosity, under compressive loads. In addition, the effectof temperature on the macroscopic elastoplastic propertiesof the porous materials is excluded. Thus, these two issuesconstitute the main disadvantages of the current study.The improvements of the micromechanics model and com-putational homogenization approach will be made in fu-ture work.

5. Conclusions

In this paper, a micromechanical model and a computa-tional homogenization approach are developed to investi-gate the macroscopic elastoplasticity of closed-cellporous materials with multiple inner pressures. From thecase study of a porous aluminum, the macroscopic elasto-plastic stress–strain relations, nominal Poisson’s ratio,yield surface and yield strength of such a porous materialwith the combination of different inner pressures are dis-cussed in detail. Our main conclusions can be summarizedas follows:

(1) For the uniaxial macroscopic elastoplastic proper-ties, simple comparisons between the theoreticaland numerical results have been made. The microm-echanics model matches well with the numericalhomogenization for the case of relatively low poros-ity and gas pressures. Nevertheless, when the poros-ity is high or when the initial pore pressure is high,the micromechanics model overestimates theresults, especially of the nominal Poisson’s ratio ofthe porous material.

(2) Upon uniaxial loading, the inner gas pressures mayinduce macroscopic tension–compression asymme-try of the porous material. The multiple gas pres-sures cause downshifts of the macroscopic tensilestress–strain curves of the porous material, whilethey have little effect on the compressive stress–strain curves. Also, the gas pressures can producetension–compression asymmetry of the nominalPoisson’s ratio of nonlinear deformation.

(3) The higher porosity corresponds to the smaller yieldsurface. When the multiple gas pressures coincide,the yield surface can be obtained from that of theporous material without inner pressures by a shiftalong the negative direction of the hydrostatic stressaxis. However, when the multiple pressures are dif-ferent, in addition to a translation along the hydro-static axis, the yield surface undergoes a change inshape and size, and the maximal equivalent stressis lowered by a difference in gas pressures.

(4) The yield strength of the porous material decreaseswith the decrease of the relative density or with

the increase of the initial gas pressures. If the criticalvalues of the initial gas pressures and the relativedensity are reached, the self-yielding of the closed-cell porous material will occur.

Acknowledgements

This work was supported by the Natural Science Foun-dation of China under Grant # 11172012 and the NationalKey Technology R & D Program under Grant #2012BAK29B00.

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