A Gurson-type layer model for ductile porous solids with isotropic and kinematic hardening

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To cite this version :

Léo MORIN, Jean-Claude MICHEL, Jean-Baptiste LEBLOND - A Gurson-type layer model for

ductile porous solids with isotropic and kinematic hardening - International Journal of Solids and

Structures - Vol. 118–119, p.167-178 - 2017

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A

Gurson-type

layer

model

for

ductile

porous

solids

with

isotropic

and

kinematic

hardening

Léo

Morin

a,b,∗

_,

_Jean-Claude

_Michel

a

_,

_{Jean-Baptiste}

_Leblond

c

a Laboratoire de Mécanique et d’Acoustique, CNRS, UPR 7051, Aix-Marseille Univ, Centrale Marseille, 4 impasse Nikola Tesla, CS 40 0 06, 13453 Marseille

Cedex 13, France

b PIMM, Arts et Métiers-ParisTech, CNAM, CNRS, UMR 8006, 151 bd de l’Hôpital, 75013 Paris, France

c Sorbonne Universités, Université Pierre et Marie Curie (UPMC), CNRS, UMR 7190, Institut Jean Le Rond d’Alembert, F-75005 Paris, France

Keywords:

Porous ductile solids Isotropic hardening Kinematic hardening Sequential limit-analysis

a

b

s

t

r

a

c

t

TheaimofthisworkistoproposeaGurson-type modelforductile poroussolidsexhibitingisotropic and kinematic hardening. The derivationisbased ona“sequential limit-analysis” of ahollow sphere madeof arigid-hardenable material. Theheterogeneity ofhardeningis accounted for bydiscretizing thecellintoafinitenumberofsphericallayersineachofwhichthequantitiescharacterizing harden-ingareconsideredashomogeneous.Asimplifiedversionofthemodelisalsoproposed,whichpermits toextendthepreviousworks ofLeblond etal. (1995)andLacroixetal. (2016)forisotropic hardening tomixedisotropic/kinematichardening.Themodelisfinallyassessedthroughcomparisonofits predic-tionswiththe resultsofsomemicromechanicalfiniteelementsimulations ofthesamecell.First, the numericaland theoreticaloverallyield lociarecomparedforgivendistributions ofisotropicand kine-maticpre-hardening.Thenthepredictionsofthemodelareinvestigatedinevolutionproblemsinwhich bothisotropicandkinematichardeningparametersvaryintime.Averygoodagreementbetweenmodel predictionsandnumericalresultsisfoundinbothcases.

1. Introduction

The failure of metals,whose impact on the integrity of engi-neering structures neednot be stressed,isone ofthe most chal-lengingproblems facedby thescientific andindustrial communi-ties. Indeedits analysisandmodellingare complextasksbecause itisa multiscaleproblem.Variousmechanisms,atthemicroscale (e.g.structuresofdislocationsorgrainboundaries)andmesoscale (e.g. hard precipitates orvoids), can induce damage leading ulti-mately to macroscopiccracks. In particular, a difficultbut essen-tialtaskconsists inprovidingpredictive micromechanically-based modelsthatpermittoaccountforbothmonotonicandcyclic load-ings.

In the case of ductile materials considered in this paper, fail-ure essentially takes place in three steps (see e.g. Benzerga and Leblond,2010;Pineauetal.,2016;Benzergaetal.,2016forrecent reviewsofthetopic):(i)thenucleationofvoids,(ii)their growth, change of shape and rotation, and finally (iii) their coalescence leadingtofinalfailure.Themodellingofthesemechanismsstarted withthepioneeringworkofGurson(1977),whocombined

homog-∗ _{Corresponding author.}

E-mail address: leo.morin@ensam.eu (L. Morin).

enization and limit-analysisof a hollow sphere made of a rigid-ideal-plastic isotropic material to derive a model of ductile ma-terials incorporating voidgrowth. Thismodel has beenextended in various directions to account for phenomena not included in its original version, notably void shape effects (Gologanu et al., 1993;1994;1997;Garajeuetal.,2000;MadouandLeblond,2012a; 2012b;2013;Madouetal.,2013),plasticanisotropy(Benzergaand Besson, 2001; Monchiet et al., 2008; Keralavarma and Benzerga, 2010;Morinetal.,2015c) andvoidcoalescence(Thomason, 1985; Tekoglu et al., 2012; Benzerga and Leblond, 2014; Morin et al., 2015b).Ithasmet,inbothitsoriginalandimprovedforms, consid-erablesuccessinthereproductionofexperimental testsoffailure ofductilematerialsundermonotonicloadingconditions.

The failure of ductile metals under cyclic loadings is less wellunderstoodandmastered.Experiments(Schmidtetal.,1991; Kobayashi et al., 1992) have shown that the strain to fracture is considerably lower, for a given load, if it is reached under cyclicconditionsratherthanmonotonically.Thisreductionof duc-tility is commonly attributed to an effect of gradual increase of the mean porosity (volume fraction of voids) during each cy-cle termed the ratcheting of the porosity. This phenomenon was firstevidencedinmicromechanicalfiniteelementsimulations per-formed by Gillesetal. (1992)underconditions ofconstant

over-all triaxiality (in absolute value), and later confirmed by several authors (Devaux et al., 1997; Besson and Guillemer-Neel, 2003; BrocksandSteglich,2003;RaboldandKuna,2005;Steglichetal., 2005;Mbiakop etal., 2015;Lacroixetal., 2016).As explainedby

Lacroixetal. (2016),the ratcheting of theporosity is fundamen-tallytiedto twofeatures ofthematerial behaviour,namelystrain hardeningandelasticity.Theeffectofelasticity,althoughimportant inthe contextoftheratchetingoftheporosityundercyclic load-ings,willnotbeconsideredinthispaper,itsstudyandmodelling beingpostponedtoafuturepaper.Weshallthusfocusexclusively ontheeffectofstrainhardening,asafirststeptowardacomplete modellingofductileruptureundercyclicloadingconditions.

Devaux et al. (1997), among other things, showed that

Gurson (1977)’s classical model does not predict the effect of ratcheting of the porosity under cyclic loadings, but a stabiliza-tionoftheevolutionoftheporosityrightfromthefirstsemi-cycle. TheexplanationofthisshortcomingofGurson(1977)’smodellies inthecrudemodellingofstrainhardeningwithinthismodel,and morespecificallyinthefactthatthesame“averageyieldstressof thematrix” appearsinboththe “square” andthe“cosh” terms of theyieldfunction.A fewtheoretical workshavetriedto improve the modelling ofstrain hardening effects within Gurson (1977)’s model:

• Forisotropichardening,Leblondetal.(1995)proposeda heuris-tic extension of Gurson (1977)’s yield function involving dis-tinct “averageyieldstressesofthematrix” inthe“square” and “cosh” terms. Their approach wasbased on some approximate analyticalsolution totheproblemofa hollowspheremadeof somerigid-hardenablematerialandsubjectedtosomearbitrary loading. Theirmodel,which accountsforthe heterogeneity of isotropichardeningwithinthematrix,hasnotablypermittedto qualitatively reproducetheratchetingoftheporosityobserved in micromechanicalfinite element simulations of porous cells subjectedtocyclicloadingsunderconditionsofconstant over-all triaxiality(inabsolutevalue)(Leblondetal.,1995). Froma quantitative point of view, however, the comparison was not fully satisfactory. This obviously arose from the fact that the model was not a priori designed for cyclic loadings, since it was basedon an assumption of positively proportional strain-ingwhichisinadequateforsuchloadings.Thishypothesiswas relaxed by Lacroix et al. (2016), at the expense of introduc-tionofradialdiscretizationofanunderlyingmesoscopichollow sphereandcalculationandstorageofthehardeningparameters ineach ofthesphericallayers thusdefined. Thisresultedina muchimprovedagreementofmodelpredictionsandresultsof micromechanicalfiniteelementsimulations.

• For kinematic hardening, Mear and Hutchinson (1985) intro-ducedamacroscopicbackstressinGurson(1977)’smodel. How-ever, this backstress was not linked to some heterogeneously distributedmicroscopiccounterpartinthematrix,andits evo-lution equation was complex: it wasadjusted in such a way thattheisotropic(Gurson)andkinematicmodelsyielded iden-ticalpredictionsforproportionalloadings.Theaimofthis kine-maticmodelwasnottodealwithcyclicloadings butratherto evidence the impact of the local curvature of the yield locus uponmacroscopicstrainlocalization.Thisaspectwasexamined indetailbyBeckerandNeedleman(1986).

The importance of the effect of strain hardening upon duc-tile failure - and especially the ratcheting of the porosity under cyclicloadings -actsasa strongincentivetodevelop models ac-counting better for the heterogeneous distribution of hardening intheplasticmatrix.Inparticular,theincorporation ofkinematic hardening appears to be necessary in order to deal with cyclic plasticity (Chaboche, 1991). The development of a Gurson-type, micromechanically-basedmodelaccountingforboth isotropicand

kinematichardening,withpossiblycomplexevolutionlawsforthe parametersgoverningthelattertypeofhardening,seemsofgreat interest to reproduce the failure of ductile metals under cyclic loadings.

Theaimofthispaperistoderivesuchamodel.Itisorganized asfollows:

• Section 2presentsthe basicingredients ofthe theoretical ap-proach and notably the “sequential limit-analysis” approach, which extends the methods and results of classical limit-analysistomaterialsexhibitingstrainhardening.

• Section 3 is devoted to thederivation of a Gurson-like “layer model” extendingthat developedby Lacroixet al.(2016), ac-countingforbothisotropicandkinematichardening.

• Section 4 finally compares the predictions of the theoretical modeltothe resultsofsome newmicromechanical finite ele-mentsimulations.

2. Position of the problem

In the major part of the paper, no hypothesis is made what-soeveron themagnitudeof thedisplacements andstrains,anda largedisplacement-largestrainformulationisused; d represents thelocalEulerianstrainrateand

σ

thelocalCauchystresstensor. 2.1. Geometry,material,admissibilityconditionsofvelocityfields

In order to derive the overall constitutive law of the porous medium, we consider, following Gurson (1977), a spherical “el-ementary cell”



containing a concentric spherical void

ω

. The porosity(voidvolumefraction)isdefinedby

f= vol

(

ω

)

vol

()

=

a3

b3, (1)

where a isthe void’sradius andb thecell’s external radius. The spherical coordinates and associated local orthonormal basis are denotedr,

θ

,

ϕ

and(e r,e _θ,e ϕ).

The material isassumed tobe rigid-plastic(no elasticity)and exhibitamixed,isotropicandkinematichardening;itisthus sup-posedtoobeythefollowingcriterion:

φ

(

σ

(

x

))

=

(

σ

(

x

)

−

α

(

x

)

)

2_eq− ¯

σ

2

₍

_x

₎

_{≤ 0,}

_∀

_x∈

_

₋

_ω

_{, (2)}

where

σ

¯ isthecurrentyieldstressand

(

σ

−

α

)

2

eqisdefinedby

(

σ

−

α

)

2 eq= 3 2

(

σ

−

α

)

:

(

σ

−

α

)

. (3) Inthisexpression,

σ

=

σ

−1

3

(

tr

σ

)

I (where I isthe second-order

unit tensor) is the deviatorof

σ

, and

α

is a traceless backstress tensorduetokinematichardening.ThePrandtl–Reussflowrule as-sociatedtothecriterionviathenormalitypropertyreads

d=

_λ∂φ

˙

∂σ

(

σ

)

=3

λ

˙

(

σ

−

α

)

, (4)

where

λ

˙ ≥ 0 is the plastic multiplier. The evolution equations of the yield stress

σ

¯ and the backstress

α

characterizing kinematic hardeningwillbepresentedinduetime.

The spherical cell is subjected to conditions of homogeneous boundarystrainrate:

v

(

x

)

=D· x,

∀

x∈

∂



, (5)

where v denotes the local velocity, x thepresent position-vector and D someoverallstrainratetensor.

The velocitymust verifythe propertyofincompressibility im-posedbytheabsenceofelasticityandtheplasticflowrule:

2.2. Principlesoflimit-analysis

Limit-analysiscombinedwithHill–Mandelhomogenizationisa convenient frameworkto deriveconstitutive equationsforporous ductilesolids.Itpermitstoeffectivelyoperatethescale transition byevidencingtheeffectsofmicrostructuralfeaturesatthe macro-scopicscale.

Classical limit-analysis is limited to rigid-ideal-plastic materi-als withina smalldisplacement -smallstrain (linearized) frame-work. Under such assumptions the macroscopic yield locus can be determined using the classical upper-bound theorem (see e.g.

Benzerga and Leblond, 2010). The macroscopic stress and strain rate tensors



and D being defined as the volume averages of their microscopiccounterparts

σ

and d ,thefundamental inequal-ityenunciatedbythetheorem,



:D ≤

(

D

)

, (7)

leadstotheparametricequationoftheyieldlocus



=

∂

_∂

D

(

D

)

, (8)

where D isarbitraryand independentof



in inequality (7), but tiedtoitthroughthemacroscopicconstitutivelawinEq.(8).The macroscopic plastic potential

(D ) in Eqs.(7) and(8) is defined by:

(

D

)

= inf

v∈K(D)

(

1− f

)



π

(

d

)

−ω, (9)

where the notation



_.

_−ω stands forvolume averaging over the sound volume



−

ω

. In this definition the set K

(

D

)

consistsof those velocityfields v which arekinematicallyadmissible with D andverifythepropertyofincompressibility,and

π

(d )isthe micro-scopic plasticpotential definedforanytraceless d by theformula

π

(

d

)

=sup

σ∗_∈C

σ

∗_{:d, (10)}

whereC isthemicroscopicconvexdomainofreversibility. Sequentiallimit-analysis(Yang,1993;Leu,2007)heuristically ex-tends themethodsandresultsofclassicallimit-analysisby incor-poratingtheeffectsofstrainhardeningandgeometricchanges.The ideais,still disregardingelasticity,toconsiderahardenable mate-rialasthesequenceofdifferent,successiverigid-idealplastic ma-terials. At a given instant, a hardenablematerial without elastic-itybehaves, ifthehardeningandthe geometryare considered as fixed, like a rigid-ideal plastic material with some pre-hardening modifyingitsyieldcriterionandflowrule.Aninstantaneous limit-loadcanthusbedeterminedusingtheclassicallimit-analysis the-orems.Inordertoaccountforchangesofthestrainhardeningand geometry, the local hardening parameters and present positions arethen updatedapproximatelyusingthetrialvelocity fieldused inthelimit-analysis,integratedinasmalltimestep.

3. A Gurson-type model accounting for isotropic and kinematic hardening

In this section we will use sequential limit-analysis to derive a Gurson-type model incorporating both isotropic andkinematic hardening.Hardeningwillbeintroducedlocallyinthecriterionas a fixed pre-hardening,andthe resultinginstantaneous loads pro-moting plastic flow of the cell will be evaluated. The hardening parametersinthematrixwillthenbeconsideredtoevolvea pos-teriori according to the trial velocity field adopted in the limit-analysis.

3.1. Macroscopicplasticpotential

In order to derive a Gurson-type model accounting for both isotropic and kinematic hardening,we first need to evaluate the

macroscopicplasticpotentialforaninitialarbitrarydistributionof thequantitiescharacterizinghardening,thatis

σ

¯

(

x

)

and

α

(x ). 3.1.1. Velocityfields

To approximately calculate the plastic potential, we consider

Gurson(1977)’strialincompressiblevelocityfield1

v

(

x

)

=b3

r2Dmer+D· x, (11)

whereDm=1₃trD denotes themean overallstrain rateand D =

D − DmI the deviatoric overall strain rate tensor. The associated

microscopicstrainratereads

d

(

x

)

=D+b3

r3Dm

(

−2erer+eθeθ+eϕeϕ

)

. (12)

3.1.2. Expressionoftheapproximateplasticpotential

Usingequations(2)and(10),thevalueofthemicroscopic plas-ticpotential

π

(d )iscalculatedtobe,foranytraceless d ,

π

(

d

)

=

σ

¯deq+

α

:d, (13)

wheretheequivalentstrainratedeq isdefinedby

deq=



2

3d:d. (14)

Theapproximatemacroscopicplasticpotentialthusreads,adopting thetrialvelocityfielddefinedbyEq.(11)andtheassociatedstrain rate d givenbyEq.(12):

(

D

)

= 1 vol

()

 −ω

π

(

d

)

d



= 1 vol

()

 −ω

(

σ

¯deq+

α

:d

)

d



=

iso

₍

_D

₎

₊

kine

₍

_D

₎

_, (15)

where the “isotropic” and “kinematic” contributions

iso_{(D ) and}

kine_{(D )to}

_{(D )aregivenby}

⎧

⎪

⎨

⎪

⎩

iso

₍

_D

₎

₌ 1 vol

()

 −ω

σ

¯deqd



kine

₍

_D

₎

₌ 1 vol

()

 −ω

α

:dd



. (16)

Some further approximationsare needed in orderto evaluate theseintegralsanalytically.Onenecessaryassumptionwillbethat thelocalhardeningparametersaredistributedinacertain,specific waywithinthematrix.Thecellisthusconsideredtobecomposed ofafinitenumberNofphasesdistributedinconcentricspheresof radii2_(seeFig.1)

a=r1<...<ri<...<rN+1=b. (17)

Thephasecontainedwithintheinterval[ri,ri+1]isdenotedPi.The distribution of the quantity

σ

¯ characterizing isotropic hardening willbeconsideredashomogeneousineachphasePi.The distribu-tionofthequantity

α

characterizingkinematichardeningwillalso be considered as“homogeneous” in each Pi, in a moreelaborate sensespecifiedbelow.

Itshouldbenotedthatthiskindofapproachisbasicallysimilar, inthecontextofductilerupture,tothatofHerveandZaoui(1993), inthecontextofEshelby’sinclusionproblem.

1 The component proportional to D

m of this velocity field reproduces the exact solution for a hollow sphere made of plastic material (with or without hardening) and subjected to a hydrostatic loading.

2 The thickness of the phases is not necessarily uniform and may be chosen ar- bitrarily.

Fig. 1. Hollow sphere: definition of some geometric parameters.

3.1.3. Calculationoftheisotropiccontribution

iso

Theexpression(16)1oftheisotropicpart

iso(D )ofthe

macro-scopicplasticpotentialreads

iso

₍

_D

₎

₌ 1 vol

()

b a

4

π

r2

σ 

¯ deq

(

r

)

S(r)dr, (18)

wherethesymbol



.

_S(r) denotesanaveragevalueoverthesphere S(r)ofradiusr.

InordertoevaluateanalyticallytheintegralinEq.(18),weuse

Gurson(1977)’s classicalapproximation detailedby Benzerga and Leblond (2010) and Leblond and Morin (2014); the potential is evaluatedas:

iso

₍

_D

₎

1 vol

()

 b a 4

π

r2

_σ

_¯



_

_d2 eq

(

r

)

S(r)dr = 1 vol

()

 b a 4

π

r2

_σ

_¯



D2 eq+ 4b6 r6 D 2 mdr. (19)

Inordertocalculatethepotential

iso_{(D ),weintroducethe}

fol-lowingapproximationonthespatialdistributionoftheyieldlimit ¯

σ

:

A1:Ineachphase Pi,theyieldlimit

σ

¯ =

σ

¯iisconsideredas

uni-form.

The expression(19)oftheisotropicpart

iso_{(D ) ofthe}

macro-scopicplasticpotentialcanthusbesimplifiedinto:

iso

₍

_D

₎

N i=1

iso i

(

D

)

, (20)

wherethe“partial” isotropicpotential

iso

i

(

D

)

inphasePiisgiven by

iso i

(

D

)

= 3

σ

¯i b3 ri+1 ri r2



D2 eq+ 4b6 r6 D 2 mdr, (21)

orequivalentlyuponuseofthechangeofvariableu=b3_/r3_,by

iso i

(

D

)

=

σ

¯i b3_/r3 i b3_/r3 i+1



D2 eq+4D2mu2 du u2. (22)

Definenowthe“localvolumefraction” fias

fi=

_r i b



3 . (23)

Theexpressionofthepartialisotropicpotentialthenbecomes

iso i (D)=

σ

¯i 1/ fi 1/ fi+1



D2 eq+4D2mu2 du u2 =

σ

¯i



−



D2 eq u2 +4D2m+2Dmln

2Dmu Deq +



1+4D2mu2 D2 eq





u=1/ fi u=1/ fi+1 . (24)

3.1.4. Calculationofthekinematiccontribution

kine

Inordertocalculatethekinematicpart

kine_{(D ) ofthe}

poten-tial,weintroducethe followingapproximationonthespatial dis-tributionofthebackstress

α

:

A2:InphasePi,thebackstress

α

isoftheform:

α

=

α

i_=Ai

1+Ai2

(

−2erer+e_θe_θ+eϕeϕ

)

, (25)

where A i

1 is asecond-ordertraceless tensorandAi2 ascalar, both

uniform within Pi. (Note that the term Ai2

(

−2e r er+e θ eθ+

e _ϕ eϕ

)

is not uniform since the vectors e r, e _θ, e ϕ vary within

P_i). It should be noted that the formconsidered in Eq. (25)has beenchoseninordertobecompatiblewiththemicroscopicstrain rategivenbyEq.(12).Thisisareasonablechoicesincetherateof the backstressdependson the microscopic strain rate, regardless ofthemodelconsideredforkinematichardening.

The expression (16)2 of the kinematicpotential

kine(D ) then

becomes

kine

₍

_D

₎

₌ 1 vol

()

b a 4

π

r2

_

_α

_:d

S(r)dr = _vol1

_()

N i=1 ri+1 ri 4

π

r2

_

_α

i_:d

S(r)dr. (26)

Letusfirststudythequantity



α

i_{: d}

S(r):



α

i_:d

S(r)=Ai1:D+Ai2 b3 r3Dm



b:b

S(r) where b=−2erer+e_θe_θ+eϕeϕ. (27) In this expression the property



b

S(r)=0 has been used

(Gurson,1977).Onethengetsuponcalculationof



b : b

S(r):



α

i_:d

S(r)=Ai1:D+6Ai2 b3 r3Dm. (28) Itfollowsthat

kine

₍

_D

₎

₌ N i=1 3 b3 ri+1 ri Ai1:Dr2dr + N i=1 ri+1 ri 18Ai 2Dm dr r =A1:D+3A2Dm, (29) where

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

A1= N i=1 Ai 1

(

fi+1− fi

)

A2= N i=1 2Ai2ln



fi+1 fi



. (30)

3.2. Macroscopicyieldcriterion

The macroscopic yield criterion is given by the parametric equation



=

∂

(

iso+

kine

)

∂

D

(

D

)

=



iso₊

_

kine_{, (31)}

wherethe“isotropic” and“kinematic” contributions



iso_and

_

kine

tothestress



aredefinedby

⎧

⎪

⎪

⎨

⎪

⎪

⎩



iso₌

∂

iso

∂

D

(

D

)



kine₌

∂

kine

∂

D

(

D

)

. (32)

Isotropiccontribution. Sincetheisotropiccontributiontothe plas-tic potential

iso_{(D ) dependsonly on D}

m and Deq, the isotropic

stress



iso_reads



iso₌

∂

iso

∂

Dm

∂

Dm

∂

D +

∂

iso

∂

Deq

∂

Deq

∂

D = 1 3

∂

iso

∂

Dm I+

∂

iso

∂

Deq 2D 3Deq =



iso mI+



isoeq 2D 3Deq, (33)

where the mean and equivalentisotropic contributions



iso

m and



iso

eq tothestressaredefinedby

⎧

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎩



iso m = 1 3

∂

iso

∂

Dm = N i=1



iso m,i



iso eq =

∂

iso

∂

Deq = N i=1



iso eq,i (34) with

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩



iso m,i= 1 3

∂

iso i

∂

Dm = 2 3

σ

¯ i



_ln

₂

_ξ

_u+



₄

_ξ

2_u2₊₁



u=1/ fi u=1/ fi+1



iso eq,i=

∂

iso i

∂

Deq =

σ

¯ i



−



4

ξ

2₊ 1 u2



u=1/ fi u=1/ fi+1 ,

ξ

=Dm Deq. (35)

Eqs.(34)and(35)provideaparametricrepresentationofthe quan-tities



iso

m and



eqiso,

ξ

actingasaparameter.

Kinematic contribution. The calculation of the kinematic stress



kine_{isstraightforwardandyields}



kine₌

∂

kine

∂

D =A1+A2I. (36)

Summary. By Eqs.(31),(33)and (36), the macroscopicstress in-ducingplasticflowisgivenby



−

(

A1+A2I

)

=



misoI+



isoeq 2D 3Deq.

(37)

Separating the mean and deviatoric parts in thisexpression and calculating the “von Mises norm” of the latterpart, one obtains themacroscopicyieldlocusintheparametricform





m− A2=



miso

(

ξ

)

(



− A1

)

eq=



eqiso

(

ξ

)

,

(38)

where



iso

m,



eqiso, A 1andA2aregivenbyEqs.(34),(35)and(30).

3.3. Flowrule

Sincethepropertyofnormalityoftheflowruleispreservedin thehomogenizationprocedure(seeGurson,1977),itreads

D=

_

˙

∂



∂

 ()

,



˙



= 0 if

(

)

<0

≥ 0 if

(

)

=0, (39)

where D denotestheplasticEulerianstrainrate,

_

˙ theplastic mul-tiplierand



(



)themacroscopicyieldfunction.

It is worth noting that writing the flow rule (39) explicitly is more difficult than it seems at first sight, since the yield lo-cusisdefinedbytheparametricequations(38),thecorresponding yieldfunction



havingnoexplicitexpression.Morinetal.(2015a) haveanalyzedthisproblemforaparametriccriterion ofthe form

(38)andshownthattheflow rulemaybe rewritten,witha suit-able re-definitionofthe plasticmultiplier beingnow denoted



˜˙, inthefollowingparametricform:

D=

_

˜˙



−1 3 d



iso eq d

ξ

(

ξ

)

I+ d



iso m d

ξ

(

ξ

)

3 2



_{− A1}



iso eq

(

ξ

)



, (40)

where



isthedeviatorof



.

3.4.Evolutionofhardeningparameters

Isotropichardening. Letg1 denotethefunctionprovidingthelocal

yield limit

σ

¯ of the matrix asa function ofthe local cumulated plasticstrainp:

¯

σ

=g1

(

p

)

. (41)

Weintroducethe followingapproximationonthe evolutionof isotropichardening:

A3:The“meanyieldlimit” ¯

σ

iinphasePiisthensupposedto

de-pendonsomeaveragevalue ¯pi_{ofthecumulatedplasticstraininthis}

phase,throughtheformula

¯

σ

i_=g

1

(

¯pi

)

. (42)

Thecumulatedplasticstrain ¯pi_{istakenonthemid-surfaceofthe} sphericalshelloccupiedbyphase P_i,thatisattheradial position ¯ri= 12

(

ri+ri+1

)

;itsexpressionreads

¯pi₌ t 0 ¯ dieq

(

τ

)

d

τ

, (43) whered¯i

eq istheaverageequivalentplasticstrainrateatthe

posi-tion ¯riassociatedtoGurson’strialvelocityfield: ¯ dieq=



deq

S(¯ri)≈





d2 eq

S(¯ri)=



D2 eq+4 b6 ¯r6 i D2 m, (44)

whereGurson’sclassicalapproximationhasagainbeenused. Inparticular,fora power-lawisotropichardening,thefunction g1(p)isgivenby

g1

(

p

)

=

σ

¯0+hpm, (45)

where

σ

¯0 is theinitial yield stress,h theisotropic hardening

pa-rameterandmthehardeningexponent.

Kinematichardening. Let g 2 denotethefunctionprovidingthe

lo-calrate

α

˙ ofthebackstress

α

asafunctionofthisbackstressitself, thestrainrate d ,thecumulatedplasticstrainpandtheequivalent plasticstrainratedeq:

˙

α

=g2

(

α

,d,p,deq

)

. (46)

Notethat thefunction g 2 mustbehomogeneousofdegree1with

respecttothestrainrate d since onlyrate-independentbehaviors areconsideredinthelimit-analysistheory.

Weintroducethe followingapproximationonthe evolutionof kinematichardening:

A4:Therate

α

˙i ofthebackstress

α

i inphasePiisthensupposed

to depend locally on this backstress itself, theplastic strain rate d i

andsomeaveragevalues ¯pi_,d¯i

eq ofthecumulatedplasticstrain and

equivalentplasticstrainrateinthisphasethroughtheformula

˙

α

i=g2

(

α

i,di,¯pi,d¯eqi

)

. (47)

Weagaintakethestrainrated i_{onthemid-surfaceofthespherical} shelldefiningphaseP_i:

di

₍

_θ

_,

_ϕ

₎

_=d

₍

_¯r

i,

θ

,

ϕ

)

∀

(

θ

,

ϕ

)

, (48)

andtheaveragevalues ¯pi_,d¯i

eq ofthecumulatedplastic strainand

equivalentplasticstrainratearedefinedbyEqs.(43)and(44), re-spectively.

Eq.(47)isbasedontheimplicitassumptionthattheform pos-tulated for

α

˙i iscompatible withthat postulated for

α

i,Eq.(25). Thisisnotthecaseforallpossiblefunctionsg 2.Howevertheforms

(25)and(47)arecompatibleforArmstrongandFrederick(1966)’s quite(thoughnotfully)generalkinematichardeninglaw,

g2

(

α

,d,p,deq

)

=c

(

p

)

d− k

(

p

)

α

deq, (49)

wherec(p)isthe“kinematichardeningslope” andk(p) the“strain recovery parameter” (which may both depend on p). For such a

lawcombinationofequations(25)and(47)leadstothefollowing evolutionequationsfortheparameters A i

1andAi2:

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

˙ Ai 1=c

(

¯pi

)

D− k

(

¯pi

)

Ai1



D2 eq+4 b6 ¯r6 i D2 m ˙ Ai 2=c

(

¯p i

₎

b3 ¯r3 i Dm− k

(

¯pi

)

Ai2



D2 eq+4 b6 ¯r6 i D2 m, (50)

whereEqs.(12)and(44)havebeenused.

3.5.Evolutionofgeometricalparameters

Theevolutionoftheporosityisclassicallygivenby

˙

f=

(

1− f

)

trD. (51)

The proposed approximate evolution equation of the internal radiireads ˙ ¯ri= ¯ri fi Dm. (52)

Notethat thisequationaccountsforthegeometrychangesdueto thesolehydrostaticpartoftheloading.

3.6.AsimplifiedversionofthemodelanditslinkwithLeblondetal. (1995)’smodel

3.6.1. Leblondetal.(1995)’smodelforisotropichardening

Leblond et al. (1995) proposed to replace Gurson (1977)’s heuristic approach to isotropic hardening effects in porous duc-tile materials (briefly recalled in Appendix A) with some mi-cromechanicalapproachbasedonsomeapproximateanalytical so-lution to the problem of a hollow sphere made of some rigid-hardenable material and subjected to some arbitrary loading. The criterion they obtained was a heuristic extension of that of

Gurson(1977)involvingdistinct“averageyieldstressesofthe ma-trix” in the “square” and “cosh” terms of the yield function:



2 eq



2 1 +2fcosh



3 2



m



2



− 1− f2_{=0, (53)}

where



1 and



2 are macroscopic internal variables connected

tothe spatialdistribution of thelocal yield stress

σ

¯.These vari-ables govern the yielding in purely deviatoricand purely hydro-staticloadings,respectively:

• Forapurelydeviatoricloading(



m=0),thevalue



eqLPDofthe

overallyieldstressisgivenby



LPD

eq =

(

1− f

)

1. (54)

• For a purely hydrostaticloading (



eq=0), the value



mLPD of

theoverallyieldstressisgivenby



LPD

m =−

2

3



2lnf. (55)

In theoriginalversion ofthemodel(Leblondetal.,1995), the expressionsof



1 and



2 wereobtained intwo steps,usingthe

approximatesolutionofthehollowhardenablesphereproblem re-ferredto above.First,



1 and



2 wereexpressed asfunctionsof

the distribution of the local yield stress, using estimates of the overallyieldstressesofthehollowsphereunderpurely deviatoric andpurelyhydrostaticloadings.Second,thedistributionofthe lo-calyieldstressresultingfromthepreviousmechanicalhistorywas obtainedinan analyticalformundertheassumptionofpositively proportionalstraining.Althoughthemodeldidprovideabetter de-scriptionofisotropicstrainhardeningeffectsinplasticporous ma-terialsthanthat ofGursonrecalledinAppendixA,itspredictions

for cyclic loadings were found to only qualitatively capture the ratchetingoftheporosityobserved innumericalmicromechanical simulations (Devaux etal., 1997), the quantitative agreement re-mainingmediocre.Theexplanationwas,ofcourse,theinadequacy ofthehypothesisofpositivelyproportionalstraininginthecaseof cyclicloadings.

In an improved version of the model (Lacroix et al., 2016), thehypothesisofpositivelyproportionalstrainingwasdropped.In the absence ofanyspecific hypothesis onthe evolution of strain in time, theanalytical calculation of the distribution ofthe local yieldstressresultingfromthepreviousmechanicalhistorywasno longer possible andnumerical integrationbecame necessary. The sphericalcellwasthereforediscretizedradially-exactlylikeinthe layermodeldevelopedherewhichextendstheideasoasto incor-porate kinematic hardening. Using ournotations, theexpressions of



1 and



2 weregivenbyLacroixetal.(2016)as

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩



1= 1 1− f N i=1 ¯

σ

i

₍

_f i+1− fi

)



2=− 1 lnf N i=1 ¯

σ

i_ln



fi+1 fi



. (56)

3.6.2. Asimplifiedversionofthemodel

Itisworthnotingthatinthecaseofpurelyisotropichardening, thelimit-loadsgivenbytheparametriccriterion(38)coincide ex-actlywiththepredictionsofLeblondetal.(1995)’smodel,forboth purelyhydrostaticandpurelydeviatoricloadings.Indeed:

• Hydrostaticloading.Inthiscase, theparameter

ξ

goesto infin-ity.Thelimit-loadthenreads



iso m

(

ξ

→∞

)

= 2 3 N i=1 ¯

σ

i_ln



fi+1 fi



=



LPD m . (57)

• Deviatoricloading.Inthiscase,theparameter

ξ

isequaltozero. Thelimit-loadthenreads



iso eq

(

ξ

=0

)

= N i=1 ¯

σ

i

₍

_f i+1− fi

)

=



eqLPD. (58)

Anaturalideais thentoreplace theparametricformgivenby

Eq.(38)for



iso

m

(

ξ

)

and



isoeq

(

ξ

)

bythesimplerexplicityield

func-tionproposedbyLeblondetal.(1995).Wethusproposea simpli-fiedversionofthemodelinvolvingtheexplicitcriterion

(

iso eq

)

2



2 1 +2fcosh



3 2



iso m



2



− 1− f2_{=0, (59)} orequivalentlybyequations(38):

(



− A1

)

2eq



2 1 +2fcosh



3 2



m− A2



2



− 1− f2_{=0, (60)}

where



1,



2, A 1 andA2 aregivenby Eqs.(56)and(30),

respec-tively.

Itis worthnotingthat thissimplifiedformcoincides withthe fullparametricforminthecaseofahomogeneousdistributionof theisotropichardeningparameter

σ

¯ butaheterogeneous distribu-tionofthe kinematichardeningparameter

α

;indeedin thiscase the parametric expressions (38) of



iso

m

(

ξ

)

and



eqiso

(

ξ

)

exactly

yield,uponeliminationof

ξ

,thecriterion(59),with



1=



2=

σ

¯.

4. Assessment of the model

The modelwill nowbe assessed by comparing its predictions totheresultsofsome micromechanicalfiniteelementsimulations oftheelementarycellconsideredinitsderivation.

Fig. 2. Spherical mesh used in the finite element calculations ( f = 0 . 01 ).

4.1. Yieldsurfaceswithinitialpre-hardening 4.1.1. Descriptionofthesimulations

First, inorderto studythemacroscopic criterion,we consider allinternalparameters(geometryandhardening)asfixed.Wethus solve the limit-analysis problem for a given, fixed pre-hardening (resulting from some given prestraining), without any geometry update,usingthefiniteelementmethod(FEM).

The calculations are performed with a homemade code us-ing 2D axisymmetric meshes subjectedto conditionsof homoge-neous boundary strain. Fig. 2 shows the mesh used for a void volume fraction f=0.01. Eight-node quadraticelements subinte-gratedwith2× 2Gausspointsareused.Themeshcontains3540 elementsand10,859nodes(21,718degreesoffreedom).This dis-cretization is adequate for the numerical calculations envisaged, further meshrefinement makingno appreciabledifference tothe results.Axisymmetricloadingsareconsidered:



11=



22=0,



33



=0,and



i j=0otherwise.TwoLodeangles(denoted

θ

L)are con-sidered:

θ

L=0correspondingto



33−



11>0and

θ

L=

π

corre-spondingto



33−



11<0.Thesimulationsareperformedby

solv-ing anelastic-plasticevolutionproblem, thelimit-loadbeing con-sideredasreachedwhen theoverallstress componentsnolonger evolve(Micheletal.,1999).

4.1.2. Isotropicpre-hardening

We consider two cases with isotropic pre-hardening (and no kinematicpre-hardening:

α

=0 everywhereinthematrix):

• Case 1. Hardening isassumed to be more important nearthe cavity;theyieldstressissupposedtovarylinearlywithrfrom thevalue

σ

¯

(

r=a

)

=1.5

σ

¯0tothevalue

σ

¯

(

r=b

)

=0.5

σ

¯0.

• Case 2. Hardeningis assumed tobe moreimportant nearthe cell’s boundary; the yield stress is supposed to vary linearly withrfromthevalue

σ

¯

(

r=a

)

=0.5

σ

¯0tothevalue

σ

¯

(

r=b

)

=

1.5

σ

¯0. Note that this caseis probably unrealistic in practice

sincehardeningwilllikelyconcentratenearthevoids’s bound-ary; itis however interesting to consider itin order to study therobustnessofthemodel.

Fig.3comparestheyieldsurfacesassociatedtothetheoretical modelofSection3.2anditssimplifiedversionofSection3.6with N=10 phases,3 _{Gurson (1977)’s model without pre-hardening}

and the finite element results, for a porosity f₌0_.01. Since

Gurson(1977)’smodeldoesnotaccountforthedifferentvaluesof themacroscopicyieldstressesinthe“square” and“cosh” termsof the yield criterion (resulting fromthe heterogeneous distribution

3 In all the simulations, the thickness of the phases has been chosen uniform for simplicity.

ofthelocalyieldstresswithinthematrix),twovaluesofGurson’s unique “overall yield stress” ¯



(see Appendix A) are envisaged,

¯



=



1 and



¯ =



2,thesequantitiesbeinggivenbyEq.(56).

Somecommentsareinorderhere:

• In the two cases considered, the model developed is in very good agreementwiththefiniteelement resultsforall the tri-axialitiesconsidered;inparticular,thehydrostaticpointis per-fectlyreproduced.

• Inbothcases,thefullandsimplifiedmodelsyieldverysimilar results:thesimplifiedmodelwithan explicitformoftheyield criterionthusseemstobeaviablealternativetothemore com-plexone.

• The comparisonwithGurson(1977)’s modelhighlightsthe in-fluence ofthe heterogeneous distributionof pre-hardening on the strength of the porous material: this model involving a single overallyield limit



¯ can captureeitherthe hydrostatic point or the deviatoric point, depending on the choice made forthisoverallyieldlimit,butnotboth.4

• Finally, it is worth noting that the finite element results are slightly sensitive to the Lode angle

θ

L (small dissymmetry of the numerical results with respect to the horizontal axis), in contrast to the model developed that does not account fora Lodeangledependency;thisisduetoGurson(1977)’s approx-imation (leadingfrom Eqs.(18) to(19) above)that erases the Lode angle dependency in the macroscopic plastic potential (Cazacuetal.,2013;LeblondandMorin,2014).

4.1.3. Kinematicpre-hardening

We now consider three cases with kinematic pre-hardening (andnoisotropicpre-hardening:

σ

¯ =

σ

¯0=Cst.everywhere).Inall

casesthelocalbackstress

α

isoftheform

α

=

α

1

(

r

)

+

α

2

(

r

)(

−2erer+e_θe_θ+eϕeϕ

)

, (61) wherethetracelesstensor

α

1(r)andthescalar

α

2(r) dependonly

onr.

• Case 1. Pre-hardening is assumedto affect the sole deviatoric stress; the non-zero components of

α

1 are:

α

1(11)=

α

1(22)=

−

α

1(33)/2=− ¯

σ

0/6and

α

2isnil.

• Case2. Pre-hardeningisassumedtoaffectthesolehydrostatic stress;

α

2 is supposed to vary linearly fromthe value

α

2

(

r=

a

)

=− ¯

σ

0/6tothevalue

α

2

(

r=b

)

=0,and

α

1isnil.

• Case 3. Pre-hardening is assumed to affect both the hydro-static and deviatoric stresses; the nonzero components of

α

1

are

α

₁₍₁₁₎₌

α

_1(22)=−

α

_1(33)/2_{=− ¯}

σ

0/4,and

α

2issupposedto

vary linearly from the value

α

2

(

r=a

)

=− ¯

σ

0/4 to the value

α

2

(

r=b

)

=0.

Fig. 4 compares the yield surfaces associated to the the-oretical model developed in Section 3.2 with N₌30 phases,

Gurson(1977)’smodelwithoutpre-hardening(with

σ

¯ =

σ

¯0

every-whereinthematrix),5_{andthefiniteelementresults,foraporosity}

f₌0_.01.(Inthiscase,thesimplifiedmodelexactlycoincideswith themorecomplete one sincethere isnoisotropic pre-hardening, soitspredictionsneednotbeshown).

In all three cases considered, the model developed is in very good agreement with the finite element results for all the tri-axialities considered; in particular, the model reproduces the “translatory motion” of the yield surface perfectly, in contrastto 4 The reason why _{Gurson (1977) ’s model fails so utterly to reproduce the numer-} ical results is that the variation of ¯σ considered here in the matrix is very impor- tant; for a more moderate variation the results would have been more acceptable.

5 _{Gurson (1977) ’s criterion, which does not account for kinematic hardening, is} represented with the sole purpose of evidencing the influence of pre-hardening.

Fig. 3. Yield surfaces for isotropic pre-hardening: theoretical model given by Eq. (38) (Present model), simplified model given by Eq. (60) (Simplified model), Gurson’s model (Gurson) and finite element results (FEM). (a) Case 1, (b) Case 2 (see text).

Gurson(1977)’s modelwithoutkinematic hardening.As expected fromthetheoreticalformofthecriterion,theterm A 1affectsonly

thepositionoftheyieldsurfaceinthe“deviatoricdirection” while thetermA2 affectsonlyitspositioninthe“hydrostaticdirection”.

4.2.Evolutionproblems

4.2.1. Descriptionofthesimulations

Wenowwishtocomparethemodel’spredictionstotheresults ofmicromechanical finite element simulations of evolution prob-lems.Thehardeningisnowsupposedtoevolvelocallywithinthe matrixbutgeometrychangesarestill disregarded.Themotivation forthischoiceistocomplywiththemodel’shypothesisof spher-icalvoids,inorder toassessthe soleevolution ofthehardening; themodification ofthe geometrywouldinduce some voidshape effectsthat are disregarded by the model andthus would intro-ducesomediscrepancies.

Again, the simulations are performed using 2D axisymmetric meshessubjected to conditionsof homogeneous boundary strain (seeFig.2). Themacroscopicstress tensorisconstrainedtobe of theform



=C

(

t

)



0_,

_

0₌



0 11 0 0 0



0 11 0 0 0



0 33



. (62)

Inthisexpression,C(t)isatime-varyingscalarand



0_isaconstant

tensorwhosecomponents



0

11 and



330 aregivenby



0 11=cos

β

− 1 3sin

β

,



0 33=cos

β

+ 2 3sin

β

, (63)

wheretheangle

β

dependsonthestresstriaxialityT:

β

=argtan

1

T



. (64)

Onlythevalue

θ

L=0oftheLodeangleisconsidered, correspond-ingto



33−



11>0.Thesimulationsareperformedincrementally,

forproportional loadingpaths -implyinga constantstress triaxi-ality(Micheletal., 1999).Finallythemacroscopicstrain tensoris denoted E .

Weinvestigatethecasesofisotropichardeningandlinear kine-matic hardening. Isotropic hardening is supposed to follow the power-lawdefinedby Eq.(45),and kinematichardening the law definedbyEq.(49)withc=Cst.andk=0(nostrainrecovery).

The values of the material parameters considered are as fol-lows:Young’smodulus, E=10,000GPa, Poisson’sratio,

ν

=0.25; isotropic hardening parameters,

σ

¯0=100 MPa,h=400 MPa and

m=0.2; kinematic hardening slope, c=400 MPa. Note that the value chosen forYoung’s modulus isextremely highsothat elas-ticity is negligible inthe simulations, in agreement withthe hy-pothesismadeinthederivationofthemodel.

Theresponseisinvestigatedforthreestressstates:(i)pure hy-drostaticloading,T_=+∞; (ii)hightriaxiality,T₌2_.333;(iii)low triaxiality,T=1/3.Weareinterestedin:

1. Macroscopicquantities.Westudytheevolutionofthestress am-plitudeC(t)versusthe“measureofdeformation” E:



0_.

2. Microscopic quantities - isotropic hardening. For this type of hardening we study the distribution of the local equivalent von Misesstress

σ

eq (at the end ofthe simulation forwhich

E :



0_{=0.05).Thisvariablereducestothelocalyieldstress}

_σ

_¯

inthe modelwhich assumesthat the matrix isplastic every-where; but thisis not true in the finite element calculations sincethematerialisnotnecessarilyentirelyplastic.

3. Microscopic quantities - kinematic hardening. For that type of hardening we study the distribution of the local equivalent backstress

α

eq=



3

2

α

:

α

(at the end of the simulation for

which E :



0_=0.05).

4.2.2. Isotropichardening

Fig. 5 compares the strain-stress predicted by the model de-velopedwithN=10 phases,to thatpredictedby Gurson(1977)’s classical model includinga heuristic modellingof isotropic hard-ening(seeAppendix A),andtheresultsobtainedbythefinite el-ement method.Notethat thepredictions ofthesimplified model are not represented since they are almost indistinguishablefrom thoseofthefullmodelforthethreecasesconsidered.

The presentmodel’spredictions are globallyvery closeto the numerical results in the three cases considered. In the interme-diate case (T=2.333) the model slightly overestimate the finite element results.It isworth notingthat the verygood agreement observed at very low triaxiality (T=1/3) is due in part to the factthatthegeometryisnotallowedtoevolveinthesimulations, which ensures that the void remains spherical. Gurson (1977)’s

Fig. 4. Yield surfaces for kinematic pre-hardening: theoretical model given by Eq. (38) (Present model), Gurson’s model without hardening (Gurson) and finite element results (FEM). (a) Case 1, (b) Case 2, (c) Case 3 (see text).

Fig. 5. Stress-strain curve in the case of isotropic hardening: predictions of the present model (Present model), Gurson (1977) ’s model with a heuristic modelling of isotropic hardening (Gurson) and finite element results (FEM).

predictionsare ingood agreementwiththe numericalresultsfor lowandmoderatetriaxialitiesbutbecomepooratveryhigh triax-iality.ThishighlightsthedetrimentaleffectofGurson’shypothesis thattheoverallyieldstressesunderpurelyhydrostaticandpurely deviatoricloadingsmaybe relatedtoa singleaverageyieldstress ofthematrix.

ThedistributionofthelocalequivalentvonMisesstressis rep-resented in Fig. 6 for the present model and the finite element simulations.Thepresentmodelgloballyaccuratelyreproduces the heterogeneousdistributionofhardening.Inparticular,thefinite el-ementresultsrevealthatthehypothesismadeinthemodelofan essentially radial variation of the yield limit

σ

¯ is acceptable. In the caseof a pure hydrostaticloading, T=+∞, this approxima-tionbecomesexact.Inthecasesofloworhightriaxiality,T=1/3 and2.333, the maps show that the hypothesis of essentially ra-dialdependenceisacceptableinalargedomainfarfromthevoid’s boundary, and that the stress level predicted is correct. The hy-pothesis is no longer verified near the void’s boundary, but the angular average of the finite element values is qualitatively well predictedbythemodel.

Fig. 6. Distribution of the local equivalent von Mises stress σeq at E : 0 = 0 . 05 in the case of isotropic hardening. (a) T = 1 / 3 , (b) T = 2 . 333 , (c) T = + ∞ . Top: finite element results, bottom: predictions of the present model.

Fig. 7. Stress-strain curve in the case of kinematic hardening: predictions of the present model (Present model) and finite element results (FEM).

4.2.3. Kinematichardening

Fig.7comparesthestrain-stresscurvepredictedbythepresent modelwithN₌30phasestotheresultsobtainedbythefinite el-ementmethod.

In thiscase, thepredictionsofthemodelcoincidealmost per-fectlywiththefiniteelementresults,emphasizingthatthe “macro-scopicbackstresses” A1 andA2 aresufficient tocapturetheeffect

oftheheterogeneousdistributionofkinematichardening.Againit

isworthnotingthatthegeometryisnotallowed toevolveinthe simulations,whichensuresthatvoidshapeeffectsaredisregarded. Thedistributionofthelocalequivalentbackstress

α

eqpredicted

by the modelis comparedin Fig.8 tothe finite elementresults. Thepresentmodelgloballyaccuratelyreproduces thedistribution ofkinematic hardening. Again, thefinite element resultsalso re-vealthatthehypothesismadeinthemodelaboutthedistribution of

α

in the phases(Eq.(25)) is acceptable; the commentsmade above for isotropic hardening also apply to kinematic hardening. Note, however, that for kinematic hardening, unlike for isotropic hardening,formula(25)predicts somedependenceof

α

upon the sphericalangle

θ

,whichisconfirmedbythefiniteelementresults, seecase(b).

4.3. Remarks

Itshouldbenotedthattheverygoodresultsobservedforyield surfacesaswellasevolutionproblemshavebeenachievedwitha reasonable numberof layers (N=10 forisotropic hardening and N₌30forkinematichardening).Thesediscretizationswerefound tobesufficienttoaccuratelydescribeboththemacroscopicresults andthedistributionsofmicroscopichardening,furtherrefinement makingnoappreciabledifferencetotheresults.

Itshouldbenotedtoothatthemodeldeveloped,withthe num-beroflayersconsidered,doesnotrequireasignificantlylargerCPU timethanGurson’soriginalmodel.Thisisduetothefactthatthe integrationoverthelayersrepresentsonlyasmallnumberof oper-ationsinthelocalprojectionalgorithm,representingitselfa mod-estpartoftheglobaliterativealgorithm.However,thenewmodel requires more memory than Gurson’s original model in order to storetheinternalvariablesinthelayers.

Fig. 8. Distribution of the local equivalent backstress αeq at E : 0 = 0 . 05 in the case of kinematic hardening. (a) T = 1 / 3 , (b) T = 2 . 333 , (c) T = + ∞ . Top: finite element results, bottom: predictions of the present model.

5. Conclusion

The aim of this paper was to develop a model for ductile porousmaterialsaccountingforbothisotropicandkinematic hard-ening. This model differs (although it is inspired) from that of

Gurson (1977) including only isotropic hardening, andalso from its extension to kinematic hardening due to Mear and Hutchin-son(1985),inthatitisnolongerbasedonapurely heuristicand macroscopicapproach,butonsome detailedanalysisoftheeffect oftheheterogeneousdistributionofmicroscopichardening param-etersnearthevoids.

An approximate yield criterion was derived by performing a “sequential limit-analysis” of a hollow sphere made of a rigid-hardenable matrix.Toapproximately account forthe heterogene-ityof hardening, thecell wasdiscretized into a finitenumberof sphericallydistributedphasesinwhichthequantities characteriz-ing hardeningwereconsideredashomogeneous.Themacroscopic yield locus was characterized by an overall criterion expressed in a parametric form, wherein the heterogeneous local harden-ingparameterswereaccountedforthroughmacroscopicvariables. A simplified version of the model involving an explicit form of the overall criterion wasthen proposed. This version reduces to

Lacroixetal.(2016)’smodelin thecaseofpurely isotropic hard-ening.

Themodelwasassessednumericallyusingmicromechanical fi-niteelementsimulations.First,overallyieldlociwereinvestigated forbothisotropicandkinematicpre-hardening:averygood agree-mentwasobservedbetweenthenumericalresultsandthe predic-tionsofthemodel.Thenthesepredictionswereassessedon evolu-tionproblems:theywerefoundtobeinverygoodagreementwith the finiteelement results, withregard to boththe overall

stress-straincurvesandthedistributionsofisotropicandkinematic hard-eningparameters.

Futuredevelopmentsoftheworkwillinclude:

• Numerical micromechanicalsimulations for cyclicloadings in-cluding geometry changes, with some comparisons with the predictions of the present model. Such studies are desirable in ordertounderstandthe effectofmixedisotropic/kinematic hardening on theratcheting of theporosity mentioned in the Introduction.

• Finite element implementationof the model and applications tonumericalstudiesoftheductileruptureofactualtest speci-menssubjectedtocyclicloadings.

• Derivationofamodelincludingvoidshapeeffects,inorderto investigatecyclicductileruptureunderconditionsoflowstress triaxiality.

Acknowledgments

FruitfuldiscussionswithP. SuquetandD.Kondo aregratefully acknowledged.

Appendix A. Gurson’s heuristic approach to isotropic hardening effects in porous ductile materials

Gurson (1977)’s original model wasobtained for a rigid-ideal plastic material obeying von Mises’s criterion and the associated Prandtl-Reuss flow rule, from an approximate limit-analysis of a hollow sphere made of such a material and subjected to condi-tionsofhomogeneousboundarystrainrate.Themacroscopicyield