Classical and sequential limit analysis revisited

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Jean-Baptiste LEBLOND, Almahdi REMMAL, Léo MORIN, Djimédo KONDO - Classical and

sequential limit analysis revisited - Comptes Rendus - Mecanique - Vol. 346, p.336-349 - 2018

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Classical

and

sequential

limit

analysis

revisited

Jean-Baptiste Leblond

a

,

∗

,

Djimédo Kondo

a

,

Léo Morin

b

,

Almahdi Remmal

a

,

c a_{SorbonneUniversités,UniversitéPierre-et-Marie-Curie(UPMC),CNRS,UMR7190,InstitutJean-LeRond-d’Alembert,Tour65-55,4,place} Jussieu,75252Pariscedex05,France

b_{ArtsetMétiers-ParisTech,CNAM,CNRS,UMR8006,Laboratoire“Procédésetingénierieenmécaniqueetmatériaux”,151,boulevardde} l’Hôpital,75013Paris,France

c_{AREVA,TourAreva,1,placeJean-Millier,92084ParisLaDéfensecedex,France}

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Keywords:

Classicallimitanalysis Sequentiallimitanalysis Hill’sapproach Drucker’sapproach Elasticity Strainhardening

Largedisplacementsandstrains

Classical limitanalysisappliestoidealplasticmaterials,andwithinalinearizedgeometrical frameworkimplyingsmalldisplacementsandstrains.Sequential limitanalysiswasproposed asaheuristicextensiontomaterialsexhibitingstrainhardening,andwithinafullygeneral geometrical framework involving large displacements and strains. The purpose of this paperis tostudyand clearlystatethe preciseconditions permitting suchanextension. Thisis doneby comparing the evolution equations of the full elastic–plastic problem, theequationsofclassicallimitanalysis,and thoseofsequentiallimitanalysis.The main conclusionisthat,whereasclassicallimitanalysisappliestomaterialsexhibitingelasticity – intheabsenceofhardeningandwithinalinearizedgeometricalframework–,sequential limitanalysis,tobeapplicable, strictlyprohibitsthepresence ofelasticity–although it tolerates strain hardening and large displacements and strains.For a givenmechanical situation,therelevanceofsequentiallimitanalysisthereforeessentiallydependsuponthe importanceoftheelastic–plasticcouplinginthespecificcaseconsidered.

1. Introduction

Theclassicaltheoryoflimitanalysisneedslittleintroduction.Thisveryusefultheoryishoweverbasedontherestrictive assumptionsofidealplasticityandlinearizedgeometricalframework–smalldisplacementsandstrains.

An essentiallyheuristicproposalwasmadeby Yang[1]toalleviatetheseunhappy restrictions.Theproposed extended framework, namedsequentiallimitanalysis, incorporates the effects ofboth strain hardening andgeometric changes, and hasbeendevelopedincloseconjunctionwithnumericalmethods.Itsgoalistoaddressthefollowingthreequestions,the answerstowhichfullydefinetheevolutionofthestructureintime:

(1) Foragivendistributionofhardeningparameters andagivengeometricalconfiguration,whendoesglobalplasticflowof thestructureoccur –thatis,whatarethenecessaryconditionsontheloadgoverningthisflow?

(2) Supposingsuchaflowtakesplace,howdoesitoccur –thatis,whatisthemodeofdeformationofthestructure? (3) Againassuming a deforming structure,howdothedistributionofhardeningparametersandthegeometricalconfiguration

changeintime?

*

Correspondingauthor.

Inordertoanswerquestions(1)and(2),theprincipleofsequentiallimit analysisconsistsinconsideringahardenable materialinlargestrain asadiscretesequenceofdifferent,successiveidealplasticmaterialsoccupyingdifferent,successive geometricalconfigurations.Inthisapproach,atagivendiscretizedinstant,thehardeningandthegeometryareconsidered asmomentarilyfixed.Thehardenablematerial,consideredwithinafullygeneralgeometricalframework,thenbehaveslike an idealplasticmaterial,considered within alinearized geometricalframework, therole oftheprevious hardening being merelytomodifytheinstantaneouslocalyieldcriterionandtheplasticflowrule,andthatofthepreviousgeometrychanges to fixthepresentgeometrical configuration.An“overall yield locus”,supplemented by an“overall associated plasticflow rule”,canthenbedeterminedusingthestandardtheoremsofclassicallimitanalysis.

Toanswer question (3), the evolutions ofthe distribution ofhardening parameters and thegeometrical configuration arededucedaposteriori,byapproximatelyupdating thesequantitiesusingthetrialvelocityfieldusedinthelimitanalysis, integratedwithinatimestep.

Theproposalhasbeenreceivedwithsomeenthusiasmandsequentiallimitanalysishasbeenusedforvarious applica-tions;seeforinstancetheworks[2–6],toquotejustafew.

However,themechanicalfoundationsofsequentiallimitanalysishaveremainedvagueandtosomelargeextentunclear, andthismaypromptdoubtsabouttheoverallvalidityofthemethod.Howcanoneignore,evenmomentarily,theevolutions ofthehardeningparameters andthegeometry,whereas inrealitythey changeconstantly?Clearly, inordertosettlesuch aquestion,itisnecessarytoscrutinize theconnectionbetweentheevolutionequationsofthefull elastic–plasticproblem andthesimplerequationsofsequentiallimitanalysis.

Itwillalsobe necessarytoinvestigatetheconnection withtheequationsofclassicallimit analysis,becausesequential limitanalysisusestheresultsandmethodsofclassicallimitanalysis,whichisjustifiedonlyprovidedtheequationsofboth theoriesare analogous.Re-examining the equationsofclassicallimit analysiswillalsobe usefultore-assess, andremind thereaderoftheroleplayedbyelasticityinthistheory.IndeedalthoughDruckeretal.[7]haveclearlyshownthatclassical limit analysisperfectly toleratesthe presence ofelasticity, thememory oftheir work seems tohave somewhat fadedin time.

Thepaperisorganizedasfollows.

•

Section2isdevotedtoashortsummaryoftheclassicalworkstheclassicalworksofDruckeretal.[7]andHill[8].The discussionconcentrates onthose resultsofthetheory actuallyused inits practicalapplication. Thediscussion ofthe roleofelasticityfollowstheworkofDruckeretal.[7].

•

Section3thendiscussesthefoundationsofsequentiallimitanalysis,exploitingan analogyofitsequationswiththose ofclassicallimitanalysisandusingsome,ifnotall,elementsofSection2.Specialattentionisagainpaidtotheroleof elasticity.Thepreciseconditionsofapplicabilityofsequentiallimitanalysisaremadeclear.

•

Finally Section 4 considers, as a typical example, the case of porous ductile materials, which involves all possible complexities: elasticity, plasticity, strain hardening, andlarge strains.Using the results of Section 3, we discussthe applicabilityofsequentiallimitanalysistothederivation,throughhomogenization,ofmicromechanically-basedmodels.

2. Classicallimitanalysis

2.1. Hypothesesandnotations–equationsoftheproblem

Weconsiderabody



madeofsomeelastic–idealplastic materialobeyingtheplasticflowruleassociatedwiththeyield criterionthroughHill’snormalityproperty.Theelastic–plasticevolutionproblemisconsideredwithinalinearizedgeometrical framework,sothatitsequationsarewrittenusingtheinitialpositionvectorX,thelinearizedstraintensor



,andtheCauchy stresstensor

σ

.Withthesehypothesestheseequationsreadasfollows(disregardingbodyforcesforsimplicity):

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

divX

σ

=

0 equilibrium

˙



≡

1

2[gradXu

˙

+ (

gradXu

˙

)

] definition of



˙

˙



= ˙



e

_{+ ˙}

_

p _{additive decomposition of}

_

_˙

˙



e

₌

_S

_{: ˙}

_σ

_{elasticity law}

˙



p

= ˙λ

∂f

∂σ

(

σ

)

plastic flow rule obeying the normality property

f

(

σ

)

≤

0

, ˙λ

≥

0

,

f

(

σ

)˙λ

=

0 Kuhn–Tucker’s complementarity conditions

B

.

C

.

boundary conditions

(1)

In theseequations,u denotes

˙

thevelocity vector (u is thedisplacement vector),



˙

e theelastic strain ratetensor,



˙

p the plastic strain ratetensor, S the elastic compliance tensor, f

(

σ

)

the yield function (depending onlyon

σ

in theabsence ofhardening), and

˙λ

theplastic multiplier.Theboundary conditions,symbolizedby the letters“B.C.”, neednot be stated explicitlyhere; itwill sufficeto say that they dependlinearly upon afinite numberof loadparameters Q1

,

Q2

,

. . . ,

QN,

collectivelydefininga“loadvector” Q.Theconjugatekinematicparametersq1

,

q2

,

. . . ,

qN,collectivelydefininga“kinematic

Classicallimitanalysisstudiesloadingpathsendingatlimitloads defined,followingDruckeretal.[7],bythetwo condi-tions

˙

Q

=

0

; ˙

q

=

0 (2)

whichmeanthatwhentheloadisreached,itbecomesstationary,whilethestructurekeepsondeforming.1

2.2. AlemmaofDruckeretal.[7]

Although Hill’s[8]classical presentationoflimit analysiswas basedon theassumption oftheabsence ofelasticity,it hasbeenclearlyshownbyDruckeretal.[7]thatthishypothesisisbynomeansnecessary.Thisstemsfromthefollowing lemma.

Lemma2.1(Druckeretal.[7]).WhenanarbitrarylimitloadQ isreached,theelasticstrainrate



˙

eandthestressrate

σ

˙

arezeroat everypointofthestructure.

Proof. If a limit loadis reached, combining condition (2)1, the definition of conjugate parameters, and the principle of

virtualworkappliedtothestressrate field

{ ˙

σ

}

andthestrainratefield

{˙



}

,onegets 0

= ˙

Q

·˙

q

=





˙

σ

: ˙



d



Onegetsfromthere,usingthedecompositionofthestrainrate(1)3,theelasticitylaw(1)4 andtheplasticflowrule(1)5:

0

=





˙

σ

: ˙



e_d

_

₊





˙

σ

: ˙



p_d

_

₌





˙

σ

:

S

: ˙

σ

d



+





˙

σ

: ˙λ

∂

f

∂

σ

(

σ

)

d



But, inthe absence ofany hardening parameter, f depends onlyon

σ

; thus

σ

˙

:

∂f

∂σ

(

σ

)

=

dtd

[

f

(

σ

)

]

≡ ˙

f , so that thelast

integral mayberewrittenas



_

˙λ ˙

f d

.

Now atevery pointthereare twopossibilities:either

˙λ =

0 (elasticbehavior),2 _or

˙λ >

0 (plasticbehavior)butthen

˙

f

=

0 (theyieldcondition issatisfiedandremains so).Inboth cases,

˙λ ˙

f

=

0.Itfollows thattheintegral



_

σ

˙

: ˙λ

∂f

∂σ

(

σ

)

d

=





˙λ ˙

f d



iszero,sothattheprecedingequationyields





˙

σ

:

S

: ˙

σ

d



=

0

ButthecompliancetensorS ispositive-definite,sothat thequantity

σ

˙

:

S

: ˙

σ

isnon-negativeateverypoint.Therefore,its integralover



canonlybezeroifitisuniformlyzero,whichimpliesthat

σ

˙

=

0 andtherefore



˙

e

₌

_{0 ateverypoint.}

₂

WithLemma2.1,atalimitload,equations(1)oftheproblemmaybere-writteninthefollowingform:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙

q

=

0 divX

σ

=

0

˙



≡

1₂[gradXu

˙

+ (

gradXu

˙

)

]

= ˙



p

˙



p

_{= ˙λ}

∂f ∂σ

(

σ

)

f

(

σ

)

≤

0

,

˙λ ≥

0

,

f

(

σ

)˙λ

=

0 B

.

C

.

(3)

wheretheconditionQ

˙

=

0 hasbeenleftoutbecauseitdoesnotplayanyrolebeyondthejustificationoftheeliminationof

˙



e_{fromtheequations.}

1 _Condition(2)

1appliestomostloadingpathsbutrulesoutspecialones.Forinstance,forsomepathsQ movesalongthesetoflimitloads;thenQ never˙

vanishes.Alimitloadmayalsobereachedonlyasymptotically,sothatQ never˙ trulyvanishes,althoughitcontinuouslydecreasesinmagnitude.

2 _{Thisoccurrenceisperfectlypossibleatsomepoints;astructuredoesnotnecessarilyhavetobeentirelyplasticforalimitloadtobereached.Simple}

systemsofbarsprovideelementaryexamples.Another,moreelaborateoneissuppliedbythecoalescenceofvoidsinporousductilesolids,whereinthe plasticstrainratesuddenlyconcentrateswithinthinlayers,theregionsbetweentheselayersundergoingelasticunloading.

Fig. 1. Illustration of the proof of consequence 1 of Hill’s[8]lemma.

2.3. AlemmaofHill[8]

Weare nowina positiontoestablisha classicallemma ofHill[8]without resortingtohisunnecessary hypothesis of absenceofelasticity.

Theset K ofpotentiallysustainableloads isdefinedasconsistingofthose loadsQ∗ forwhichthere existsastress field

{

σ

∗

}

which isboth statically admissible withQ∗ (it obeysthe equilibrium equations andthe boundary conditions with thegivenvalue ofQ∗)andplasticallyadmissible(itsatisfiesthecondition f

(

σ

∗

)

≤

0 ateverypoint). Then weobtainthe followinglemma.

Lemma2.2(Hill[8]).LetQ denotealimitload,q the

˙

rateofthekinematicvectoritgeneratesalongtheloadingpathconsidered,and Q∗

∈

K anarbitrarypotentiallysustainableload.Then

(

Q

−

Q∗

)

·˙

q

≥

0 (4)

Proof. Let

{

σ

}

and

{˙



}

denotethestressandstrainratefieldsgeneratedbytheloadQ,and

{

σ

∗

}

thestressfieldassociated withthe loadQ∗ inits definitionasa potentially sustainable load. Then, bythe principle ofvirtual work applied tothe stressfield

{

σ

}

− {

σ

∗

}

andthestrainratefield

{˙



}

,plusLemma2.1,

(

Q

−

Q∗

)

·˙

q

=





(

σ

−

σ

∗

)

: ˙



d



=





(

σ

−

σ

∗

)

: ˙



pd



But

(

σ

−

σ

∗

)

: ˙



p

_≥

_{0 ateverypointbyHill’sprinciple(equivalenttothenormalitylaw).Theresultfollows.}

₂

Remarks.2.1.Inequality (4)standsasakindof“overall equivalent”ofHill’slocalprinciple.Thereishoweverasignificant differencearisingfromthepresenceofelasticity.Hill’sprincipleappliestoanyplasticallyadmissiblestress tensor

σ

,even ifitgivesrisetoapurelyelasticbehavior;indeedif f

(

σ

)

<

0,then



˙

p

=

0,sothat

(

σ

−

σ

∗

)

: ˙



p

=

0.Incontrastinequality

(4)standsforloadsQ thatcannotbeexceeded,butsaysnothingaboutloadsthatcan;indeedifQ canbeexceeded,



˙

e_hasno

reasontobezeroatevery point,sothatthequantity

(

Q

−

Q∗

)

·˙

q differsfrom



_

(

σ

−

σ

∗

)

: ˙



pd,andnothingcanbesaid aboutitssign.Thisisanillustrationofthesubtletiesbroughtbyelasticityinlimitanalysis.

2.2.Theonlydifferencebetweentheproofofinequality(4)givenhereandthatprovidedbyHill[8]liesinthe justifica-tionoftheequality



˙

= ˙



p_{,basedonDrucker’sLemma 2.1here,andonthemereneglectofelasticityin[8].}

2.4. ConsequencesofHill’s[8]lemma

Consequence2.1.AlllimitloadsnecessarilylieontheboundaryofthesetK ofpotentiallysustainableloads.

Proof. Fig. 1illustratestheelementarygeometricreasoningleadingtothisconclusion,inthesimplecasewherethevector

Q ofloadparametersconsistsoftwocomponentsonly.Obviously,alllimitloadslieinK .Assumethattheabovestatement is wrong andthat one of them, Q,does not lie on theboundary of thisset; it then liesin its interior.Therefore, there existsanopenball

B(

Q

,

η

)

ofcenterQ andradius

η

>

0 entirelycontainedinsideK .ByLemma2.2,foreveryQ∗

∈

B(

Q

,

η

),

(

Q

−

Q∗

)

·˙

q

≥

0,whereq denotes

˙

therateofthekinematicvectorgeneratedby Q.ButwhenQ∗ spans

B(

Q

,

η

),

thevector

Q

−

Q∗ spansallthedirectionsofthe N-dimensionalspace;therefore,theinequalityX

·˙

q

≥

0 holdsforallvectorsX inthis space.Applying ittothepair

(

X

,

−

X

),

oneseesthatinfact X

·˙

q

=

0 foreveryX, whichimpliesthatnecessarilyq

˙

=

0.But thiscontradictscondition(2)2.

2

This result, offundamental importance inlimit analysis, ascribes tothe set K ofpotentially sustainable loads,defined

apriori asa purelymathematical object,the essentialphysicalrole ofbeingtheset ofactually sustainableloads:all load pathsgovernedbytheequationsoftheelastic–plasticproblemmustendonitsboundary,notatsomeinteriorpoint.

ThecleareststatementofthesecondconsequenceisprobablythatprovidedbyMandel[9]:

Consequence2.2.(i)ThesetK isconvex;(ii)ifQ isalimitload(necessarilylyingontheboundaryofK ),thecorrespondingrateq of

˙

thekinematicvectorisdirectedalongsomelocalexteriornormaltoK .

The geometricalproofisveryclassical(see,e.g.,Salençon’s[10] textbookonplasticityandlimitanalysis)andneednot berecalled.Notethatproperty(i)doesnotinfactbringanythingnew,sincetheconvexityofK wasobviousfromthestart fromits definition,the linearityoftheequilibriumequationsandthe boundaryconditions,andtheconvexity ofthe local reversibilitydomain

{

σ

∗

|

f

(

σ

∗

)

≤

0

}

.

Assume that an explicit analytic expression of the set K has been found in the form of an inequality of the form

F

(

Q

)

≤

0 forsome “overallyield function” F .Ifthisfunctionis regularatthepoint Q (theboundaryof K admits alocal tangenthyperplaneandthereforeasinglelocalnormal),property(ii)abovemeansthatthereexistsascalar

˙

–the“overall plasticmultiplier”–suchthat

˙

q

= ˙

∂

F

∂

Q

(

Q

) ,

˙ ≥

0 (5)

If F isnot regularatthepoint Q (theboundaryof K admitsa localconeofnormals),thisequation mustbereplacedby themoregeneralone

˙

q

∈ ∂

F

(

Q

)

(6)

where

∂

F

(

Q

)

isthesub-differentialofF atthepointQ (Moreau[11]).

2.5. Practicalproceduresofclassicallimitanalysis

The results ofSubsection2.4 show that,in orderto studylimit loadsandthe corresponding overall directionsof the flow, itsuffices to determinethe set K ofpotentially sustainableloads, whichdepends upon theconstitutive lawof the materialonlythroughitsyieldcriterion.

The static orinterior approach (Hill[8]) providesthe mostintuitive,ifnot the mostpractically efficient,wayofdoing so. It simply consistsindetermining some elements of K by exhibitingstress fields staticallyadmissible withthem and plasticallyadmissible.IfQ denotessuchanelement,theconvexityofK impliesthatthewholesegment

[

0

,

Q

]

iscontained insidethisset,sothatthelimitloadlyingonthestraighthalf-lineoriginatingfromtheoriginandpassingthroughQ must

liebeyond thispoint(whencetheterminology“interiorapproach”).

Thelessintuitivebutmoreefficientkinematic orexterior approachreliesonanotherresultofHill[8].Thisresultmakes useofthefunction

π(˙



)

definedforany



˙

bytheformula

π(˙



)

≡

sup

σ∗|f(σ∗)≤0

σ

∗

: ˙



(7)

This function is the support function (the Legendre–Fenchel transform of the indicator function) of the convex set

{

σ

∗

|

f

(

σ

∗

)

≤

0

}

.(Itisoftencalledtheplasticdissipation.Thisterminologyisslightlyinadequateinthat



˙

inequation(7)is

apriori justanarbitrarysymmetricsecond-ranktensor,not theplasticpartofanactualstrainratetensor.Theinadequacy willbeseentobecomemorepronouncedinsequentiallimitanalysis,seeSection 3below.)

Thefundamentalinequalityoflimitanalysis (Hill[8]).IfQ∗denotesanarbitraryelementofK ,q an

˙

arbitraryrateofkinematic parameters(unconnectedtoQ∗),and

{

v

}

anarbitraryvelocityfieldkinematicallyadmissiblewithq (compatible

˙

withthegivenvalue ofthisrate),generatingastrainratefield

{˙



}

,then

Q∗

·˙

q

≤





π(˙



)

d



(8)

Proof. TheresultisasimpleconsequenceofthedefinitionofK ,theprincipleofvirtualworkandthefactthat

σ

∗

: ˙



≤ π(˙



)

forevery

σ

∗ respectingtheinequality f

(

σ

∗

)

≤

0.

2

Inequality (8) provides “upper estimates”, in a vectorialsense, of the limit loads (whence the terminology “exterior approach”),providedthatoneisabletoexhibitsuitablevelocityfields.(Thisgenerallyrequiresmorethansimplysatisfying the requirement of kinematic admissibility, because the function

π(˙



)

is often infinite – making inequality (8) void of information–exceptforsomespecific strainrates;forinstance,foryield criteriaindependentofthemeanstress,



˙

must betraceless,thatisthevelocityfieldmustbeincompressible,inorderfor

π(˙



)

tobefinite.)

3. Sequentiallimitanalysis

3.1. Newhypotheses,notations,andevolutionequations

Wenowcometothemoregeneralproblemofabody



madeofsomeelastic–plasticmaterialsubjecttostrainhardening,

andenvisagedwithinageneralgeometricalframework involvinglargedisplacementsandstrains.Thecollectionoflocal(scalar, vectorialortensorial)hardeningparametersiscollectivelydenotedby

χ

,andthepresentpositionvectorbyx.Theequations oftheevolutionproblemarewrittenusingaEuleriansetting,onthecurrentconfiguration,andinvolvetheEulerianstrain rated andtheCauchystresstensor

σ

:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

divx

σ

=

0 equilibrium

d

≡

1₂[gradxv

+ (

gradxv

)

] definition of d

d

=

de

+

dp additive decomposition of d

de

=

S

:

D_Dtσ hypoelasticity law

dp

_{= ˙λ}

∂f

∂σ

(

σ

,

χ

)

plastic flow rule obeying normality

f

(

σ

,

χ

)

≤

0

, ˙λ

≥

0

,

f

(

σ

,

χ

)˙λ

=

0 Kuhn–Tucker’s complementarity conditions

Dχ

Dt

= . . .

evolution of hardening parameters

˙

x

=

v evolution of geometry

B

.

C

.

boundary conditions

(9)

Intheseequations,v denotesthevelocity,de_{theelasticstrainratetensor,d}p _{theplasticstrainratetensor,andthesymbol} D

Dt representssome objectivetime-derivative (forinstancethat ofJaumann). Incomparisontosystem(1),system(9)

en-compassestwonewequations,(9)7 and(9)8,providingtheevolutionsofthehardeningparametersandthegeometry.The

precise expressionof D_Dtχ inequation (9)7 isnot specified becauseitis immaterialhere, butinpracticethe value ofthis

derivativeisdeterminedbythoseoftheplasticstrainratedp _andthevector

_χ

_itself.

3.2. Conditionsforreductiontotheequationsoflimitanalysis

Ingeneral,thereisnowaytoreducethegeneralevolutionequations(9)tothesimplersystemofequations(3)oflimitanalysis,for atleasttworeasons:

•

inthepresenceofstrainhardeningand/orgeometrychanges,theloadvectorQ hasnoreasontoeverreachastationary value,sothattheconditionQ

˙

=

0 formingthebasisofDruckeretal.’s[7]proofofLemma 2.1ofSubsection2.2,isnever satisfied;

•

evenifthisconditionweresatisfied,Druckeretal.’s[7]proofwouldbreakdown,becauseinthepresenceofhardening

˙

σ

:

∂_∂σf

(

σ

,

χ

)

differs from

˙

f by the term

χ

˙

:

_∂∂_χf

(

σ

,

χ

);

hencethe integral



_

σ

˙

: ˙λ

_∂∂_σf

(

σ

)

d cannot be expressed as





˙λ ˙

f d

.

However, in some cases it is physically reasonable to completely disregard elasticity. (It is difficult to state general conditionsensuringthepossibilityofsuchasimplification,sinceitssoundnesssimultaneouslydependsonthematerial,the geometry,andtheloading;eachindividual casewarrantsaspecific discussion.Anexamplewillbeconsideredindetailin Section4.)ThismeansconsideringthelimitS

→

0 inequations(9);thesystemthenreducesto

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙

q

=

0 divx

σ

=

0 d

≡

1₂[gradxv

+ (

gradxv

)

]

=

dp dp

_{= ˙λ}

∂f ∂σ

(

σ

,

χ

)

f

(

σ

,

χ

)

≤

0

,

˙λ ≥

0

,

f

(

σ

,

χ

)˙λ

=

0 Dχ Dt

= . . .

˙

x

=

v B

.

C

.

(10)

wheretheadditionalconditionq

˙

=

0 simplystatesthatweareinterestedonlyinloadsQ generatinganactualdeformation ofthestructure. Now consider,within system(10), thesubsystemconsistingonly ofthefirstfiveequations plusthe last one:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

˙

q

=

0 divx

σ

=

0 d

≡

1₂[gradxv

+ (

gradxv

)

]

=

dp dp

= ˙λ

_∂∂_σf

(

σ

,

χ

)

f

(

σ

,

χ

)

≤

0

,

˙λ ≥

0

,

f

(

σ

,

χ

)˙λ

=

0 B

.

C

.

(11)

If considered over a finite time interval, this subsystem is incomplete, because it does not specify how the hardening parametersandthegeometryevolveintime,whereassuchfeatureshaveanexplicitimpact(forthehardeningparameters), oran implicitone (forthegeometry),upontheequations.However, assumethatthefulldistributionofhardeningparameters andthefullpresentconfigurationareknownatsomegiveninstant,andwritesystem(11)atthissoleinstant.This systemisthen completeinitselfanddoesnotrequiretheexpressionsof D_Dtχ andx,

˙

sincethesederivativesdonotentertheequations.

Moreover system(11), viewed in thislight, is completely analogous to the system (3) of equationsof classical limit analysis, withthecorrespondencesx

↔

X,v

↔ ˙

u,d

↔ ˙



,dp

↔ ˙



p_{;intheformersystemthehardeningparametersandthe}

geometryplay theroleofmereparameters.Thisjustifies–iftherestrictivehypothesisofabsenceofelasticityisacceptable–the theoryofsequentiallimitanalysis: that isthe useof theresults andmethods ofclassical limit analysis, thedistribution of hardening parametersandthe geometrybeingconsidered asgivenandfixed, their rolebeingmerely tomodifythe yield criterionandprescribethepresentconfiguration.

It isworthstressing, however,that thewording“sequentiallimitanalysis”,although employedinall papersusingthis theory–seetheselectioncitedintheIntroduction–,isparticularlyunhappyinthatitgivestheerroneousimpressionthatit focusesonlimitloads.Ofcourse,inrealitysuchloadsdonotexistinthepresenceofhardeningand/orgeometricchanges.

TheloadsQ consideredandstudiedbysequentiallimitanalysisaresimplythosewhich–intheabsenceofelasticity–generatean actualdeformationofthestructure,q

˙

=

0,andhavenothingtodowithnon-existentlimitloads.

3.3. Proceduresofsequentiallimitanalysis

Theanalogybetweentheequationsofsequentialandclassicallimitanalysisjustmentionedjustifies–intheabsenceof elasticity–thefollowingdefinitionsandresults:

•

onedefinesasetK_G,{χ}consistingofthoseloadsQ∗ forwhichthereexistsastressfield

{

σ

∗

}

staticallyadmissiblewith

Q∗ andsatisfyingthecondition f

(

σ

,

χ

)

≤

0 ateverypoint.Thesubscripts

G

and

{

χ

}

hereunderlinethedependenceof thissetuponthefullgeometrysymbolizedbytheletter

G

,andthefullfield

{

χ

}

ofinternalparameters;

•

the loadsQ generating anactual deformation ofthestructure attheinstantconsidered, q

˙

=

0, necessarilylie onthe boundaryofthesetK_G,{χ};

•

assume that anexplicit inequality F_G,{χ}

(

Q

)

≤

0 definingthe set K_G,{χ} hasbeenfound.The rateq of

˙

thekinematic vectorgeneratedbyQ isthenoftheform

˙

q

= ˙

∂

FG,{χ}

∂

Q

(

Q

) ,

˙ ≥

0 (12) or

˙

q

∈ ∂

F_G,{χ}

(

Q

)

(13)

dependingonwhetherthefunction F_G,{χ}isregularornotatthepointQ.

The set K_G,{χ} may be determined using both static and kinematic approaches. In the static approach, one merely determineselementsofthissetbyexhibitingstressfields

{

σ

∗

}

satisfyingtheconditionsrequestedbyitsdefinition.

Inthekinematicapproach,onedefinesafunction

π

χ bytheformula

π

χ

(

d

)

≡

sup σ∗_|f(σ∗,χ)≤0

σ

∗

:

d (14)

where thesubscript

χ

underlinesthe dependenceupon thevector ofhardeningparameters. Thisfunctionis thesupport function(theLegendre–Fencheltransformoftheindicatorfunction)oftheconvexset

{

σ

∗

|

f

(

σ

∗

,

χ

)

≤

0

}

,where

χ

actsasa parameter.(Thewordingplasticdissipation shouldbeavoidedforthisfunctionevenmorethanforthefunction

π

ofclassical limit analysis, since the true plastic dissipation should include some contribution from the evolution of the hardening variable

χ

,absentintheright-handsideofthedefinition(14).)Then,theinequality

Q∗

·˙

q

≤





holds for all elements Q∗ of K_G,{χ}, all rates q of

˙

the kinematicvector (unconnectedto Q∗), andall velocity fields

{

v

}

kinematicallyadmissiblewithq,

˙

generatingastrainratefield

{

d

}

.

3.4. Updateofhardeningparametersandgeometry

Evenwhentheabsenceofelasticitywarrantsapplicabilityofsequentiallimitanalysis, thistheory,atthepresentstage ofitspresentation, remains incompleteinthat itcannot,by definition,sayanythingabouttheevolutionofthehardening parametersandthegeometry,sincethe expressions(10)6 and(10)7 of D_Dtχ andx are

˙

leftout inthesetofequations(11)

defining it. (Note that it says nothing either aboutthe possible reduction of the full field

{

χ

}

of hardening parameters, definedeverywherein

,

toafinitenumberofquantitiesapproximately“summarizing”thisfield;thisveryimportantpoint istheobjectofactiveresearches,see,e.g.,MichelandSuquet[12].)

Sequentiallimitanalysismustthereforebecompletedwithsomeadhoc procedureaimedatdeterminingsuchevolutions. Withinakinematicapproach,anaturalwayofdoingsoistoassumethatthetrialvelocityfieldsuseddefinean“acceptable approximation” ofthetrueone. Thesefieldscanthen beusedin equation(10)7 to define theevolutionofthegeometry,

in a direct manner, and inequation (10)6 to define the evolution of the hardening parameters, upon calculation of the

correspondingstrainratefield.

Itmustbestressedthat althoughthisprocedureisquitenatural, itdoesnotrelyonanyvariationalsetting,sothatthe judgment onthequalityofthetrialvelocityfieldsisan essentiallysubjective matterandthere isnoobjectivecontrolon thequalityoftheapproximateevolutionsofthehardeningparametersandthegeometryobtained.Thismarksanimportant difference with the results of limit analysis, the quality of which is guaranteed, or atleast controlled, by virtue of the variationalsetting.

3.5. Assessmentofsequentiallimitanalysis

Theadvantagesofsequentiallimitanalysisaretwo-fold:

•

itextendstheresultsandmethodsofclassicallimitanalysissoastoincorporatetheeffectsofstrainhardeningand/or large displacements and strains. This is important because thesetwo effects have a systematic (for the former) or frequent(forthelatter)impactuponplasticityproblemsofpracticalsignificance;

•

its use,withamere transposition,ofthe well-establishedprocedures ofclassical limit analysis, makesit easy touse andpowerful. Thekinematicapproach,inparticular,isversatileandefficient,thankstotheminimumrestrictionsput onthepossibletrialvelocityfields.

Butthetheoryalsohasshortcomings:

•

incontrasttoclassicallimitanalysis, itis,strictly speaking,applicableonlyinthecompleteabsenceofelasticity.Such a hypothesis isof course neverrigorously satisfied.Inthe absence ofgeneralconditions ensuring its soundness,the approximatevalidity, orcompleteinvalidityofthetheory mustthusbe judgedineachspecific case,anddependson whethertheelastic–plasticcouplingplaysaminor,ormajorroleinthesituationconsidered;

•

its variational setting is incomplete andits procedures and results must be supplemented with adhoc estimates of theevolutionsofthehardeningparametersandthegeometry.Thequalityoftheseestimatescannotbeguaranteedby variationalarguments.

Withtheseadvantagesanddrawbacks,sequentiallimitanalysisprovidescompleteanswerstoallthreequestionsraised intheIntroduction,thuspermittingtopredictthefullevolutionintimeofrigid plasticstructures:

•

theanswer toquestion (1),pertainingto themechanicalconditionspromoting overallplasticflow, liesinthe overall yield criterion, depending on the instantaneous distribution ofhardening parameters and instantaneous geometrical configuration;

•

theanswertoquestion(2),pertainingtothedeformationmodecorrespondingtosomeloadobeyingthiscriterion,lies intheoverallflowruleobeyingthenormalityproperty;

•

the answerto question (3),pertaining to theevolution in time of thedistribution of hardening parameters andthe geometricalconfiguration,liesinthecomplementaryprocedurebasedonthetrialvelocityfieldsofthelimitanalysis. Inparticular,whencombinedwithhomogenizationofsome“representativevolumeelement”,thetheoryisabletoprovide a fullset ofconstitutiveequationsforheterogeneous,rigid plastic materials;thespecific exampleofporousplasticsolids willnowbeconsidered.

4. Example:thecaseofporousductilematerials

Inproblems oflarge strain plasticity,elastic strainsare oftenmuch smaller thanplastic ones, sothat it isreasonable todisregard elasticityandapplysequentiallimit analysis.Butthere areexceptions;probablythemostnotableoftheseis

theproblemofcavitationinanelastic–plasticmediumsubjectedtosomehighly hydrostaticloading.Insuchasituation,a sphericalcavitymayspontaneouslyappearandsubsequentlygrowthroughtheplasticflowofthesurroundingmaterial,as a resultoftherelaxationofthe elasticenergystoredthere;see,e.g., Tvergaardetal.[13]. Inthiscasetheelastic–plastic couplingisofparamountimportance,andsequentiallimitanalysisisclearlyinapplicable.

Inthe sequelweshallconsiderthe caseofthequasi-static ductileruptureofmetals,whichisespeciallyinteresting in that (i)itinvolvesallpossiblecomplexities: elasticity,plasticity,strain hardeningandlargedisplacementsandstrains;but (ii)theimportanceoftheelastic–plasticcouplingiseminentlyvariableandwarrantsadetaileddiscussion.

4.1. Gurson’s[14]model

ThemostclassicalconstitutivemodelformaterialssubjectedtoductileruptureisthatofGurson[14].Whendiscussing it, one mustcarefully distinguishbetweenthe derivationproposed byGurson himselfand alternative,possibly improved ones.

•

Gurson’s[14]originalderivation consistedoftwosteps.

In a first step, heperformed an approximate (classical) limit analysis ofa typical representative volume element in a porous plastic medium: namely a hollow sphere, made of an ideal plastic material obeying the von Mises yield criterionandthePrandtl–Reussassociatedplasticflowrule,andsubjectedtosomearbitraryloadingthroughconditions ofhomogeneousboundarystrainrate(Hill[15],Mandel[16]).Theapproximateoverallyieldcriterionheobtainedread

F

(,

f

)

≡



2 eq

¯

σ

2

+

2 f cosh

3 2



m

¯

σ

−

1

−

f2

≤

0 (16)

where



denotestheoverallstresstensor, f theporosity(voidvolumefraction),



eq theoverallvonMisesequivalent

stress (vonMises normof the deviatorof



),



m the overall mean stress (1/3 ofthe trace of



), and

σ

¯

the yield

stress ofthesound(unvoided)materialinuniaxial tension.Also, theoverallflow rulefollowedthenormality rule,in agreementwiththegeneralpropertyrecalledinSubsection2.4:

Dp

= ˙

∂

F

∂

(,

f

) ,

˙ ≥

0 (17)

whereDp_{denotestheoverallplasticstrainrateand}

˙

_{theplasticmultiplier.}

In a second step,Gurson proposed a complete constitutive model forporous plastic materials by combiningseveral elements.Twooftheseweretheoverallyieldcriterion(16)andtheplasticflowrule(17)derivedfrom(classical)limit analysis. Otherelementsincludedunconnected,heuristicmodelingsoftheeffectsofelasticity,isotropichardeningand geometrychanges.

◦

Elasticitywasincorporatedbysupplementingtheoverallplasticstrainratewithsomeoverallelasticstrainrategiven bysomehypoelasticitylaw(disregardingtheinfluenceofporosity).

◦

Isotropic hardeningwasintroducedbyassumingthatitseffectcouldbesummarizedthroughasingle internalvariable. Gurson thus heuristically replaced, in his criterion (16) for ideal plastic materials, the constant yield stress

σ

¯

by some “averageyieldstress”,stilldenoted

σ

¯

;heassumedthisaverageyieldstresstobe connectedtosome“average equivalentstrain”

¯

throughtherelation

σ

¯

=

σ

(

¯

),

where

σ

(

)

denotesthefunctionprovidingthestrain-dependent yieldstressofthesoundmaterial;andheproposedthefollowingevolutionequationfor

¯

:

(

1

−

f

)

σ

¯

d¯

dt

=  :

D

p ₍₁₈₎

Thebasisofthisequationwasaheuristicidentificationoftheplasticdissipationsintherealheterogeneousmaterial,



:

Dp,andin afictitious“equivalent”, homogeneous,porous material withequivalent strain

¯

andyield stress

σ

¯

,

(1

−

f

)

σ

¯

d_dt ¯.

◦

Finally geometricchangeswereintroducedessentiallythroughthefollowingapproximateevolutionequationforthe porosity:

˙

f

= (1

−

f

)

tr Dp (19)

resultingfromtheneglectofelasticityplustheplasticincompressibilityofthesoundmatrix.

•

Whetherthisderivationcanbeimproved,byusingtheelementsexpoundedinSections2and3,willnowbedebated. Thefirstremarkisthatclassicallimitanalysiscanbeofnousehere.Indeed,itsuseofalinearizedgeometrical frame-workrulesoutgeometryupdatesandthereforeanyevolutionoftheporosity–sincethisevolutionisrelatedtochanges inthevolumeofthevoids.Thusitcouldonlyjustifya“small-displacement,small-strain”versionofGurson’s[14]model ofnopractical interest,sincethewholepurposeofthemodelisto depictthegradualdegradation andlossof stress-carryingcapacityofthematerialarisingfromtheprogressiveincreaseoftheporosity.

However,inmostproblemsofductileruptureundermonotoneloadings,elasticityplaysaminorrole,sothatsequential limitanalysismaybe invokedtojustifysome aspectsofthemodelconsidered inits rigid-plasticlimit.3 Such aspects includethe overall yield criterion (16), the plasticflow rule(17) obeying the normality property, andthe evolution equation (19) of the porosity. (Note, however, that in the absence of elasticity, this evolution equation becomes a rigorousconsequenceofplasticincompressibilityofthematrix,sothatthehelpofsequentiallimitanalysisisnotreally neededhere.)

Ontheotherhand,sequentiallimitanalysiscannot beusedtojustifyGurson’s[14]approachofstrainhardeningeffects inporousductilematerials,andinparticularhisevolutionequation(18)fortheinternalvariable

¯

.Indeed,the identi-ficationofplasticdissipationsitresteduponhadnothingtodowith,andinfactwasincompatiblewithsequentiallimit analysis.Gursonnevertried,inthespiritofsequentiallimitanalysis,toaccountfortheeffectofthedetailed heteroge-neousdistributionofhardeningupontheoverallcriterion andplasticflowrule;nordidheusethetrialvelocityfields ofhislimitanalysistospecifytheevolutionintimeofthisdistribution.

4.2. Latermodels

AnimprovedvariantofGurson’s[14] modelingofstrainhardeningeffectsinporousplasticmaterialswasproposed by Leblondetal.[20].TheseauthorsfirstevidencedvariousshortcomingsofGurson’smodelinthehardenablecase–notably itsincompatibilitywiththeexactsolutiontotheproblemofahollowspheremadeofrigidhardenable materialandsubjected tohydrostatictension.Theyshowedthat themain drawbackinthismodelwas theuseofa single internal variable

σ

¯

to describe(isotropic)hardeningeffects.Theythenproposedthefollowingslightmodificationofthecriterion:

F

(,

f

)

≡



2 eq



2₁

+

2 f cosh

3 2



m



2

−

1

−

f2

≤

0 (20)

where



1 and



2 denote distinct internal variables connected to the overall yield stresses under purely deviatoric and

purely hydrostaticloadings,respectively. Usingagainthesolutiontothehollowsphereproblemasabasis,they proposed thefollowingexpressionsforthesequantities:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩



1

=

1 b3

₋

_a3 b3



a3

σ







S(r)



d

(

r3

)



2

=

1 ln

(

b3

_/

_a3

₎

b3



a3

σ







S(r)



d

(

r3

)

r3 (21)

wherea andb aretheinner andouter radii ofthesphere(connected throughthe relation f

=

a3

_/

_b3_),

_σ

₍

₎

_denotesthe

function providingthestrain-dependent yield stressof thesoundmaterial likeabove,and





_S(r) istheaverage value of

theequivalent cumulatedstrain

overthesphericalsurface

S(

r

)

ofradiusr. Acomplexexpression ofthisaveragevalue completing the model,which needs not be given here, was also provided, assuming proportional strainingat the global scale.

Theconnectionsofthisworkwithsequentiallimitanalysiswereasfollows:

•

Leblondetal.’s[20] expressions of thelimit loadsunderpurely deviatoricandpurely hydrostaticloadings, implicitly contained in equations (20) and (21), were based on homogenization through consideration of the problem of the hollowsphere,considered asatypical representativevolume elementintheporous material.Thederivation ofthese equationsdidnot,however,resorttosequentiallimitanalysis;

•

ontheotherhand,Leblondetal.’s[20]approximateexpressionoftheaveragevalue





_S(r)wasderived,intheabsence

ofan exactsolution to thehollow sphere problemfora general, non-hydrostatic loading,by using the trialvelocity fieldsofGurson’s[14]limitanalysis.Thisapproachwasquiteinlinewiththeupdateofinternalparameterscompleting theproceduresofsequentiallimitanalysis(seeSubsection3.4),althoughthiswasnotmentionedbytheauthors. Leblond etal.’s [20] model was recently refined by Lacroix et al.[21] through introduction andradial discretization, ateach “macroscopic materialpoint”, ofan underlyinghollow microsphere,andnumerical calculationof theintegralsin equation (21) basedon thisdiscretization. The interval ofintegration

[

a3

_,

_b3

_]

_{intheseintegrals was thus divided into N}

sub-intervals,

[

a3

=

r3₀

,

r3₁

]

,

[

r₁3

,

r3₂

],

. . . ,

[

r3_N−1

,

r3_N

=

b3

]

, andthe average values





i of

inthe various intervals

[

r3i−1

,

r3i

]

werecalculated(andstoredasinternalvariables)usingtheevolutionequations d dt





i

=

4b 6

¯

r6 i

(

Dpm

)

2

+ (

Dpeq

)

2

,

r

¯

i

≡

1 2

(

ri−1

+

ri

)

(22)

where Dpm denotestheoverallmeanplasticstrainrate(1/3 ofthetraceofDp)andDpeq theoverallequivalentplasticstrain

rate(vonMisesnormofthedeviatorofDp).4 Bypermittingtorelax Leblondetal.’s[20]hypothesis ofproportionalglobal straining,thisapproachledtoamuchimproveddescriptionofthedistributionofhardeningaroundvoids,andthustomore accurateevaluationsoftheparameters



1and



2.

The relations betweenLacroix etal.’s [21] model and sequential limit analysiswere the same asfor Leblondet al.’s

[20]model:thejustificationsofthecriterion(20)andtheexpressions(21)oftheparameters



1 and



2 werenodifferent

fromthose intheearlierwork andthusmadeno referencetosequentiallimit analysis;butthe evolutionequations(22)

oftheaveragevalues





i were againobtainedfromGurson’s[14]trialvelocity fields,that isusing,evenifimplicitly,the

procedureofupdateofinternalparameterscompletingsequentiallimitanalysis.

Even more recently, Morin et al. [22] used Lacroix et al.’s[21] idea of radial discretization of an underlying hollow microsphereinexplicitconjunction –forthefirsttime –withsequentiallimitanalysis,todefineextensionsofLacroixetal.’s

[21] modelto various typesof hardening: linearkinematic,nonlinear kinematic(Armstrong andFrederick’s [23] model), mixed isotropic/kinematic–a complex modelofcyclicplasticity of Chaboche [24] includinga memoryofthe maximum amplitude of previous cycles is even currently considered, with applications to ductile rupture under cyclicloadings in view.5

Consider,forinstance,thecaseofmixedisotropic/kinematichardening,witha localbackstress(centerofthelocal re-versibilitydomain)denoted

α

.Morinetal.[22]assumedthedistributionofthisbackstressineachsphericallayer

[

r3_i−1

,

r3_i

]

tobeapproximatelyoftheform(withclassicalnotationsforsphericalcoordinates)

α

i

(

r

, θ,

ϕ

)

=

A₁i

+

Ai₂

(

−2e

r

⊗

er

+

eθ

⊗

eθ

+

eϕ

⊗

eϕ

)

(23)

whereAi

1 andAi2 denoteasecond-ordertracelesstensorandascalar,respectively,bothuniformwithinthelayer

[

r 3 i−1

,

r

3 i

]

.

Withthishypothesis,andusingthekinematicapproachofsequentiallimitanalysiswithGurson’s[14] trialvelocityfields, theyobtainedthefollowingapproximateoverallyieldcriterion:

F

(,

f

)

≡

(

−

A1

)

2 eq



2₁

+

2 f cosh

3 2



m

−

A2



2

−

1

−

f2

≤

0 (24)

whereA1 andA2 areasecond-ordertracelesstensorandascalargivenby

⎧

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎩

A1

=

N



i=1 Ai₁r 3 i

−

ri3−1 b3 A2

=

N



i=1 2 Ai₂ln



r_i3 r3_i−1



(25)

Morinetal.[22]alsoobtainedtheevolutionequationsofthequantitiesAi₁andA₂i characterizingkinematichardeninginthe layers, by usingtheprocedure ofupdate ofinternal parameterscompleting sequentiallimit analysis, againwithGurson’s

[14] trialvelocity fields.Inthe simplestcaseof linearkinematic hardening,governed atthe localscale by theevolution equation

Dα Dt

=

hd

p ₍₂₆₎

4 _{NotethattheimplementationofthemodeldidnotintroduceunreasonabledemandswithrespecttoCPUtimeandstoragecapacity,sincetheintegrals}

tobecalculatednumericallywerepurelyradial,nonumericalintegrationoverthesphericalvariablesθandφbeingrequested.

5 _{ItisalsoworthmentioningheretheveryrecentpaperofPauxetal.[25],inwhichtheauthorsalsousedsequential-limitanalysisasahelptodefinea}

constitutivemodel,intheslightlydifferentcontextofplasticityofporoussinglecrystals.Theseauthors’approachrestedonacombinationofideasborrowed fromtheworkofLeblondetal.[20],ontheonehand,andsequentiallimitanalysisofahollowsphere,ontheotherhand,withoutresortingtoLacroixet al.’s[21]radialdiscretizationofthisspherefollowedbynumericalintegration.

where _DtD denotes some objectivetime-derivative andh a materialparameter, the evolution equationsofthe Ai₁ and Ai₂ read

⎧

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎩

DA₁i Dt

=

h

(

D p

₎



˙

Ai₂

=

hb 3

¯

r_i3D p m (27)

where

(

Dp

)

denotesthedeviatorofDp.Theseequationscompletethemodel.

Globally,thecombinationofthe conceptof theunderlyingmicrosphere andthe powerfulmethodsof sequentiallimit analysisrevealedveryefficienttoderivemodelsforporousplasticmaterialshavinginternalvariablesobeyingvarious evolu-tionlaws.Finiteelementmicromechanicalsimulationsofthebehaviorofahollowspheremadeofrigid-hardenablematerials obeyingvarioushardeninglaws,andsubjectedtovariouscyclicloadings,arecurrentlyunderprogressinordertoassessthe qualityofthepredictionsofMorinetal.’s[22]modelintheabsenceofelasticity.

However,thismodel,beingexplicitlyderived fromsequentiallimitanalysis, issubjectedtothebasicrestrictionofthis theory to materials devoid of elasticity. Unfortunately, it so happens that for problems of ductile rupture under cyclic loadings,forwhichthemodelwasspecificallydesigned,elasticityeffectsarenot negligible.Thiswasnotedforthefirsttime byDevauxetal.[26],whoidentifiedelasticity andstrainhardening asthe twofactorsresponsiblefortheeffectofratcheting oftheporosityundercyclicloadings–thatisthegradualincreasewiththenumberofcycles,underconditionsofconstant absolutevalue of the triaxiality,of theporosity averaged overone cycle. The impact of elasticityupon the ratcheting of theporosity was fullyconfirmedlater, notably intheworksof Mbiakopetal.[27] andCheng etal.[28], elasticitybeing estimatedtoberesponsibleforroughly50% oftheeffect,theother 50% beingattributabletostrainhardening.

Thus, in spiteof its qualities, Morinet al.’s[22] modelis still incomplete in that, by its very construction, it cannot claimtodescribetheeffectofelasticityupontheratchetingoftheporosityundercyclicloadings.Refinedhomogenization theoriesunderdevelopment fornonlinear heterogeneoussolids(Lahellec andSuquet[29]) may,inthefuture,leadto rig-orousdescriptionsoftheelastic–plasticcouplinginporousplasticsolids,andinparticularofthiseffect.Inthemeantime, however,thereisnoother solutionthantointroducethiscouplinginaheuristicwayinthemodelconsidered–beitthat ofMorinetal.[22]derivedfromsequentiallimitanalysis,orsomeotherone.

It is worth noting that an interesting first step in this direction has been very recently made by Cheng et al. [28]

(independentlyofMorinetal.’s[22]workandmodel).Theirtreatmentwasbasedonahypothesisof“separationofphases”, purely elasticon theone hand,purely plastic(devoidofelasticity) onthe other hand.Therates oftheporosity inthese twotypesofphases,

˙

fe intheelasticones, f

˙

p intheplasticones,were evaluatedseparately.Thisisapproximatebecause in reality presenceof plasticity doesnot mean absence ofelasticity;thus

˙

fe hasno reasonto vanishduring theplastic phases. However, inspite oftheerrors introduced by thishypothesis, themodel seems promisingin that itseems to at least qualitatively capture the mechanismresponsible forthe influence ofelasticity upon the ratcheting of the porosity under cyclic loadings. This mechanism consists of a slight asymmetry introduced by elasticitybetween the tensileand compressive parts of each cycle: during the tensilepart, the volume of the voids first increases a bit reversibly during theinitialelasticphase, whichfacilitatestheirsubsequentirreversiblegrowththroughtheplasticflowofthesurrounding matrix,whereas during thecompressive part, thisvolume first decreases a bitelastically, making the subsequentplastic shrinkageofthevoidsmoredifficult.

5. Concludingsummary

ThemainpurposeofthispaperwastoexamineandpreciselystatetheconditionsofapplicabilityofYang’s[1]sequential limitanalysis,whichstandsasaheuristicextensionoftheclassicaltheoryoflimitanalysisdevelopedby Druckeretal.[7]

andHill[8],incorporatingtheeffectsofstrainhardeningandgeometrychanges.Thetreatmentwasbasedonastudyofthe connectionsbetweentheequationsofthefullelastic–plasticevolutionproblemandthesimpleronesofbothclassicaland sequentiallimitanalysis.Particularemphasiswasputontheroleofelasticity.

Inafirststep,webriefly re-examinedthederivationoftheresultsandmethodsofclassicallimitanalysis,starting from the equationsof theelastic–plastic evolution problemandthe definitionof limit loads.Although not original, thisshort re-expositionwasdeemednecessaryinviewoftheusemadebysequentiallimitanalysis,ourmainobjectofstudy,ofthe resultsandmethodsofthemoreclassicaltheory.Itwas alsoan occasionto re-statethatclassical limitanalysis, although intrinsicallylimitedtoidealplasticmaterialsconsidered withinalinearizedgeometricalframework,perfectlytoleratesthe presenceofelasticity.ThiswasdeemedusefulinviewofthefactthatHill’s[8]treatmentofthesubject, basedonneglect ofelasticityfromthestart,seems tohaveintimeovershadowed,to someextent,that ofDruckeretal.[7],incorporating elasticity.

Wethenexaminediftheequationsofsequentiallimitanalysiscouldbededucedfromthoseoftheelastic–plastic evolu-tionproblem,nowconsideredinthemostgeneralcontextofahardenablematerialandalargedisplacements/largestrains framework.Suchadeductionprovedimpossiblewithoutanyadditionalhypotheses,showingthatthemethodsofsequential limitanalysisarenotjustifiedingeneral.However,intheabsenceofelasticity,thoseequationsoftheevolutionproblemthat donotpertaintotheevolutionofthehardeningparametersandthegeometrypreciselyreducetothoseofsequentiallimit

analysis, whichare themselvesequivalentto thoseofclassicallimitanalysisconsidering thefulldistributionofhardening parameters andthegeometricalconfigurationasknownandfixed.Thisfullyjustifiestheuseofthemethodsofsequential limitanalysis,derivedfromthoseofclassicallimitanalysis,forrigid plasticmaterials,evenifsubjecttostrainhardeningand geometricalchanges.Theresultsderivedfromthesemethodsmust,however,becompletedwithevolutionequationsforthe hardening parameters andthegeometry;theseevolutionequationscan approximatelybe deducedfromthetrialvelocity fieldsusedinthe limitanalysis, butwithoutthesafeguard ofavariationalsettingguaranteeingthequality oftheresults. Withthisadditionalprocedure,thetheoryiscompleteandfullyanswersthethreequestionsraisedintheIntroduction,for

rigid plasticmaterials.

Ofcourse,inreality,materials always involvesome elasticity,butinlarge-strain plasticityproblems,plasticstrainsare oftenmuchlargerthanelasticones;followingtheprecedingdiscussion,forsuchproblems,theuseofsequentiallimit anal-ysisiseithertotallyunjustifiedorapproximatelyjustified,dependingonwhethertheelastic–plasticcouplingisimportantor minor.Adiscussionofthisquestionwasfinallypresented,inthecontextofthehomogenization-basedderivationofmodels forporousductilematerials.Formonotoneloadings,elastic–plasticcouplingisgenerallyminor,sothatelasticitymay reason-ablybeneglected.Withthishypothesis,sequentiallimitanalysisisapplicable,andjustifiesGurson’s[14]classicalmodel– exceptforhisindependent,purelyheuristicmodelingofstrain hardeningeffects.Forcyclicloadings,theelastic–plastic cou-plingisimportant,beingresponsibleforroughlyhalfoftheeffectofratchetingoftheporosityunderconditionsofconstant absolutevalueofthetriaxiality[26–28].AveryrecentmodelofMorinetal.[22],explicitlybasedontheuseofsequential limit analysis, permits todescribe the impact ofstrainhardening upon thisphenomenon accurately,forvarious isotropic, kinematicormixedhardeninglawsofthematrix.Thismodel,however,suffersfromthebasicrestrictionofsequentiallimit analysistorigidplasticmaterials,andthereforeneedscompletioninthefuturebysomemoreorlessheuristicmodelingof theimpactofelasticity upontheratchetingoftheporosity.

Compliancewithethicalstandards

Theauthorsdeclaretheyhavenoconflictofinterest.

Acknowledgements

Jean-Baptiste Leblond acknowledges financial support from the “Institut universitaire de France” under grant éOTP L10K018.

References

[1]W.H.Yang,Largedeformationofstructuresbysequentiallimitanalysis,Int.J.SolidsStruct.30(1993)1001–1013. [2]L.Corradi,N.Panzeri,Atriangularfiniteelementforsequentiallimitanalysisofshells,Adv.Eng.Softw.35(2004)633–643.

[3]S.Y.Leu,Analyticalandnumericalinvestigationofstrain-hardeningviscoplasticthickwalled-cylindersunderinternalpressurebyusingsequentiallimit analysis,Comput.MethodsAppl.Mech.Eng.196(2007)2713–2722.

[4]S.Y.Leu,R.S.Li,ExactsolutionsofsequentiallimitanalysisofpressurizedcylinderswithcombinedhardeningbasedonageneralizedHolderinequality: formulationandvalidation,Int.J.Mech.Sci.64(2012)47–53.

[5] D.Kong,C.M.Martin,B.W.Byrne,Modellinglargeplasticdeformationsofcohesivesoilsusingsequentiallimitanalysis,Int.J.Numer.Anal.Methods Geomech.41(2017)1781–1806,https://doi.org/10.1002/nag.2700.

[6] X.P.Yuan,B.Maillot,Y.M.Leroy,Deformationpatternduringnormalfaulting:asequential limitanalysis,J.Geophys.Res.,SolidEarth122(2017) 1496–1516,https://doi.org/10.1002/2016JB013430.

[7]D.C.Drucker,W.Prager,M.J.Greenberg,Extendedlimitanalysistheoremsforcontinuousmedia,Q.Appl.Math.9(1952)381–389. [8]R.Hill,Onthestateofstressinaplastic-rigidbodyattheyieldpoint,Philos.Mag.42(1951)868–875.

[9]J.Mandel,Coursdemécaniquedesmilieuxcontinus,Gauthier-Villars,Paris,1966.

[10]J.Salençon,Calculàlaruptureetanalyselimite,Pressesdel’ÉcolenationaledesPontsetChaussées,Paris,1983.

[11]J.J.Moreau,FonctionnellesConvexes,publicationoftheConsiglioNazionaledelleRicherche,RomaandtheFacoltadiIngegneriadiRoma“TorVergata”, 2003.

[12]J.-C.Michel,M.Suquet,Amodel-reductionapproachinmicromechanicsofmaterialspreservingthevariationalstructureofconstitutiverelations, J. Mech.Phys.Solids90(2016)254–285.

[13]V.Tvergaard,Y.Huang,J.W.Hutchinson,Cavitationinstabilitiesinapowerhardeningelastic–plasticsolid,Eur.J.Mech.A,Solids11(1992)215–231. [14]A.L.Gurson,Continuumtheoryofductilerupturebyvoidnucleationandgrowth:PartI–yieldcriteriaandflowrulesforporousductilemedia,ASME

J.Eng.Mater.Technol.99(1977)2–15.

[15]R.Hill,Theessentialstructureofconstitutivelawsformetalcompositesandpolycrystals,J.Mech.Phys.Solids15(1967)79–95.

[16]J.Mandel,Contributionthéoriqueàl’étudedel’écrouissageetdesloisd’écoulementplastique,in:Proceedingsofthe11thInternationalCongresson AppliedMechanics,Springer,Munich,FRG,1964,pp. 502–509.

[17]G.Perrin,J.-B.Leblond,Analyticalstudyofahollowspheremadeofporousplasticmaterialandsubjectedtohydrostatictension–applicationtosome problemsinductilefractureofmetals,Int.J.Plast.6(1990)677–699.

[18]J.-B.Leblond,G.Perrin,Aself-consistentapproachtocoalescenceofcavitiesininhomogeneouslyvoidedductilesolids,J.Mech.Phys.Solids47(1999) 1823–1841.

[19]G.Perrin,J.-B.Leblond,Acceleratedvoidgrowthinporousductilesolidscontainingtwopopulationsofcavities,Int.J.Plast.16(2000)91–120. [20]J.-B.Leblond,G.Perrin,J.Devaux,AnimprovedGurson-typemodelforhardenableductilemetals,Eur.J.Mech.A,Solids14(1995)499–527. [21]R.Lacroix,J.-B.Leblond,G.Perrin,Numericalstudyandtheoreticalmodellingofvoidgrowthinporousductilematerialssubjectedtocyclicloadings,

Eur.J.Mech.A,Solids55(2016)100–109.

[22]L.Morin,J.-C.Michel,J.-B.Leblond,AGurson-typelayermodelforductileporoussolidswithisotropicandkinematichardening,Int.J.SolidsStruct. 118–119(2017)167–178.