Disorder-induced stiffness degradation of highly disordered porous materials

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Disorder-induced stiffness degradation of highly disordered porous materials

Hadrien Laubie, Siavash Monfared, Farhang Radjai, Roland Pellenq, Franz-Josef Ulm

To cite this version:

Hadrien Laubie, Siavash Monfared, Farhang Radjai, Roland Pellenq, Franz-Josef Ulm. Disorder-

induced stiffness degradation of highly disordered porous materials. Journal of the Mechanics and

Physics of Solids, Elsevier, 2017, 106, pp.207 - 228. �10.1016/j.jmps.2017.05.008�. �hal-01720439�

Disorder-induced stiffness degradation of highly disordered porous materials

Hadrien Laubie

^a

, Siavash Monfared

^a

, Farhang Radjaï

^{b, c}

, Roland Pellenq

^{a, b, d}

, Franz-Josef Ulm

^{a, b, ∗}

aDepartment of Civil and Environmental Engineering, Massachusetts Institute of Technology, Cambridge, MA 02139, USA

b^MSE2, UMI 3466 CNRS - MIT Energy Initiative, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge 02139, USA

cLMGC, UMR 5508 CNRS - Université de Montpellier, 163 rue Auguste Broussonnet, 34090 Montpellier, France

dCINaM, CNRS - Aix Marseille Université, Campus de Luminy, 13288 Marseille Cedex 09, France

Keywords:

Inhomogeneous material Porous material Elastic material Microstructures Stress concentrations

a b s t r a c t

Theeffectivemechanicalbehaviorofmultiphasesolidmaterialsisgenerallymodeled bymeansofhomogenizationtechniquesthat accountforphasevolumefractionsand elas-ticmoduliwithoutconsideringthespatialdistributionofthe differentphases.By meansofextensive numericalsimulationsofrandomlygeneratedporous materialsusingthelat-tice elementmethod,the roleof localtexturalproperties ontheeffective elasticproper-tiesofdisorderedporous materialsisinvestigatedandcomparedwith differentcontinuummicromechanics-basedmodels.Itisfoundthat thepronounceddisorder-inducedstiffnessdegradationoriginates fromstressconcentrationsaroundporeclustersinhighlydisorderedporous materials.Weidentifyasingledisorderparameter, ϕ^sa, whichcombinesameasureofthespatialdisorder ofpores (theclusteringindex,sa )withtheporevolumefraction(theporosity, ϕ^)toscalethe disorder-inducedstiffnessdegradation. Thus,weconcludethatthe classicalcontinuum micromechanicsmodelswith onesphericalporephase,due totheirunderlyinghomogeneity assumptionfallshortofaddressingtheclustering effect, unless additionaltextureinformation isintroduced,e.g.informofthe shiftofthe perco-lationthresholdwithdisorder,orotherfunctional relationsbetweenvolume fractionsandspatialdisorder;asillustratedhereinforadifferentialschememodelrepresentativeofatwo- phase(solid–pore)composite modelmaterial.

1. Introduction

What is theeffect of disorder on the effective elastic behavior of porousmaterials? – The question isof some relevance forporousmaterials whoseeffectivemechanicalbehavior defiesclassicaldescriptors basedoncontinuummicromechanics theory(forareview,seee.g.Suquet,1997;Zaoui,2002).Indeed,basedprimarilyuponEshelby’sinclusionproblem(Eshelby, 1957) and the assumption of scale separation, the consideration of composite materials as an assembly of (interacting) monodispersesphericalinclusionsexhibitingcharacteristicmorphologiesfrommatrix-inclusion(Mori andTanaka,1973) to

∗ Corresponding author.

E-mail addresses: [email protected] (H. Laubie), [email protected] (S. Monfared), [email protected] (F. Radjaï), [email protected] (R. Pellenq), [email protected] (F.-J. Ulm).

Fig. 1. Two-dimensional porous media: (a) ordered system, (b) type 1 disorder (pores confined in their unit cell), (c) type 2 disorder with λ= R app/R (here λ^{> 1 ).}

poly-crystals andgranular(Hill, 1965), fails to address explicitlymesoscale textureeffects. Such texture effects originate oftenfromthematerialmanufacturing process(suchasinhomogeneousprecipitationincementhydrationDel Gadoetal., 2014;Ioannidouetal.,2016;Masoeroetal.,2012)ormaterialmaturationprocesses(suchasbiologicallymediatedinorganic- organictissuegrowth(HellmichandUlm,2002)orthediagenesisoforganic-rich,naturallyoccurringporousgeocomposites (MonfaredandUlm,2016)).

Some studies have addressed the mesotextural effects associated with the non-spherical shape of inclusions seen in somegeo-orbio-compositeswheretheload-bearingphasecanbemodeledasanarrangementofrandomlyorientedsingle crystals that can havedifferent aspect ratios.These studies includefinite element analysis(see e.g. Meille andGarboczi, 2001)ortheoreticalanalysisusingself-consistentschemesadaptedtothespecificcaseofnon-sphericalinclusions(seee.g.

Fritschetal.,2006;2009;2010;2017;Sanahujaetal.,2010).

Weaddresshereanothermesotextureeffectofporousmaterials:thelocalfluctuationofporosityaroundameanvaluein materials havingamatrix(solid)/inclusions(pores)morphology.While porousmaterialsystemswithboth smallandlarge fluctuationshavebeenstudiedboththeoreticallyandexperimentally,asearchoftherelevantliteraturewasnotconclusive infindingacomprehensiveinvestigationthat bridgesthetwoasymptotesforawiderangeofporosities,andthusremains tobedeveloped.Thisisinshortthefocusofthispaper.Infact,forsmalllocalfluctuations,analyticalexpressionsbasedon asymptoticexpansionforisotropicporousmaterialsshowadisorder-inducedcomplianceincrease(i.e.stiffnessdegradation) (GˇarˇajeuandSuquet,2007).Ontheother endofthespectrum,experimentalresultsbyLobbandForrester(1987)ofhighly disordered 2-D porous systems (obtained by perforating square metal sheets withholes at random positions that could overlap or miss each other by any amount, thus exhibiting large porosity fluctuations), confirm the persistent effect of disorderontheeffectiveelasticity;andhighlighttheneedtobridgethegapbetweensmallandlarge porosityfluctuation systems.

Thiswillbeachieved,inthispaper,byfirstgeneratingalargerangeofdisorderedporousmaterialsbymeansofcanonical MonteCarlomovesonhard-disks/spheresexhibitingdifferentporosities.Theelasticityofthesesystemsistheninvestigated bymeans oftheLatticeElementMethod,withthefocusonidentifyinganappropriate ‘order’or‘disorder’ parameterthat is ableto consistentlyscale disorder–induced stiffnessdegradation.Finally, we suggesta simple methodto integratethis scalingintoaconventionalcontinuum-basedmicromechanicsmodel.

2. Materialsandmethods

Thefocusofthecurrentinvestigationistheelasticityofporousmaterialsystemsexhibitingdifferentlevelsofdisorder.

Thisrequiresthegenerationofalargerangeofporousmaterialsampleswithcontrolled disorder.Thisisachievedhereby considering deviations froma periodic arrangementof pores in2-D and 3-D, takenasreference, andby quantifying the disorder-inducedstiffnessdegradationwithrespecttothereferenceelasticityoftheorderedsystem.

2.1. Poroussamplegeneration.Disordercharacterization

Aperiodicporosityarrangement(periodicityl)forboth2-Dand3-Dporousmaterialsystemsisconsideredasreference.

In2-D,thisisachievedinformofNdisk-shapedporesofradiusR(Fig.1(a))placedinasquareplate(matrixphase)ofsize Lx=Lz=L=√

Nl,andthicknessLy=dL; while,in3-D, sphericalporesofradiusRareplacedina cubicmatrixofsize Lx=Ly=Lz=L=√³

Nl.Theporosity

ϕ

^{ofthesesystemsistunedbyvaryingthesizeRoftheporesand/ortheir numberN:}

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

ϕ

^disk = N

π

_R

L

²

=

π

_R

l

²

ϕ

^sphere = N

4 3

π

_R

L

³

−6

N

v (

^R

)

=⁴₃

π

_R

l

3

−6

v (

^R

)

^,

(1)

withv(R)adimensionlessfunctiondefinedby(seeAppendixA):

v (

^R

)

=

⎧ ⎨

⎩

0, ifR≤l/2

π

24

_2R

l −1

²

_4R

l +1

, otherwise.

(2)

Disorderisintroducedintothisperiodicarrangementbydifferentmethods:

• Type 1: Randommoveswithin a unit cell (Fig.1(b)). Pores(disks orspheres ofradius R) of theordered systems are movedtorandompositionsatadistance0<d<l/2−Rcontainedwithintheiroriginalunitcell(definedbytheperiod- icityl).ThenumberNofporesiskeptfixedandtheporosityincreasesastheporeradiusRisincreased.

• Type2: Monte–CarlomovesatconstantnumberN ofpores ofrespectivelyhard-disksin2-D andspheresin 3-Dwith anapparentporeradiusRapp=

λ

^R.With

λ

∈[1,l/(2R)],thegeneratedsystemscontainonlynon-overlappingporesofan overallporositythatincreaseswiththeporeradius,R.ThistypeofMonte–Carlogeneration–atconstantN– includesas a subset thetype 1 generationmethod. Butitis moregeneralasthe poremovement isnot confined tothe unit cell (seeFig.1(c)),thuspermittinglargertexturedeviationsfromthequasi-orderedsystem,

λ

=l/(^2R),tohighlydisordered systems,

λ

=1.

• Type3:GrandCanonicalMonteCarlo(GCMC)insertionwithvariablenumberNofporesoffixedapparent poreradius Rapp=

λ

^R,where

λ

∈[0,1]definesanarbitrarydegreeofimpenetrability(SmithandTorquato,1988):GCMCwith

λ

=1 corresponds tothehard-disk/hard-sphere porousmodel;whileGCMCwith

λ

=0correspondstofullypenetrabledisks andspheres (overlappingpores).IntheseGCMC-based generations,the porosityis definedbythe numberNof pores, eventuallycorrectedforthelevelofoverlapping.

Themethods thusdescribedgeneratealarge rangeofdifferentdisorderedmicrostructures, evenatsamemeanporos- ity.Toillustrate thisrangeofdisorder, differentdescriptors classicallyemployed inthe characterizationof microstructure (Torquato,2002)areused.Oneclassicaldescriptorofatwo-phase(solid-porosity)microstructureisthetwo-pointprobabil- ityfunctionoftheporephase,S₂(r),whichdefinestheprobabilitythattwo pointsseparatedbyadistancerarebothina pore:

S2

(

^r=

| |

^r2−r1

| | )

=

^I

(

^r1

)

^I

(

^r2

)

^{, (3)}

where the characteristicfunction I(^ri)=1 ifr_i is in the pore, and I(^ri)=0 otherwise; whereas angular brackets denote ensembleaverage.Whiletheone-pointprobabilityfunctiondefinesthemean-porosity,

ϕ

^;i.e.

S₁

(

^ri

)

=

^I

(

^ri

)

⁼

ϕ

, (4)

thetwo-pointprobabilityfunction,S₂(r),exhibitstheasymptoticproperties(SmithandTorquato,1988):

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

S2

(

⁰

)

=

ϕ

rlim→∞S2

(

^r

)

=

ϕ

²

dS2

dr

r=0

= −

ϕ

lc

, (5)

withlc the meanchord length(lc=2Rin thecaseofdisk-shaped orsphericalpores).The two-point probability function offourdifferentsystems,exhibitingthesamemeanporosity,areshowninFig.2withtheirrespectivemicrostructure.For quasi-orderedsystems (Fig.2 (a),generatedwith

λ

=l/(^2R)^{), clearpeaksappearwitha fixed} periodicity.As disorderin- creases(Fig.2(b)and(c),

λ

→1),long-rangepeaksprogressivelydisappear.When overlappingofdisks/spheresisallowed (Fig.2(d),

λ

=0),thetwo-pointprobabilityfunctionisalmostflatafteraquasi-lineardecreasefrom

ϕ

^to

ϕ

^2.

The second descriptor employed is the Probability Distribution Function (PDF) of the local porosity throughout the sample, f_ϕ. It is obtainedby measuring the local porosity

ϕ

^{a in a} square-shaped control volume of side-length a (here, a=

π

^R2/

ϕ

^in2-Danda=

³

4/3

π

^R3/

ϕ

^in3-Dfor

ϕ

≤

π

^{/6).Fig.3displaysthethusobtainedPDFsforthefourmicrotex-}

turesofthesameaverageporosity,

ϕ

=

ϕ

^a

,consideredbefore.Whilequasi-orderedsystems(Fig.3(a),

λ

=l/(^2R)^{) exhibit} a narrowdistribution with a clearpeak centered around the average porosity, theprobability densityfunction broadens asdisorderincreases(Fig.3(b)and(c),

λ

→ 1).Withfurther disorderintroduced byoverlapping disks/spheres(Fig.3(d),

λ

=0), thePDFexhibitsatwo-peakstructurereminiscentofasolid-porespacephaseseparationakintodemixing,witha narrowpeakaround

ϕ

^a →0(solid), anda diffusepeak aroundtheaverage porosity.Analogous tothemixingindexused todetermine the degreeof mixtureof particulatematerials (see e.g. Lacey,1954), thisspread ofthe localporosity

ϕ

^{a is}

captured–infirstorder– byaclusteringindexs_a,definedasthestandarddeviationoftheporosity;thatis,inacontinuum form:

sa=

( ϕ

^a−

ϕ )

²

=

ϕ

a²

−

ϕ

^a

^{2, (6)}

Fig. 2. Two-point probability functions and associated microstructures of four systems of increasing disorder ( ϕ= 0 . 36 ): (a) type 2 disorder with λ= l/ (2 R ) (quasi-ordered system), (b) type 2 disorder with λ= (l/ (2 R )+ 1 )/ 2 , (c) type 2 disorder with λ= 1 and (d) type 3 disorder with λ= 0 (overlapping disks).

Note the asymptotic values: S 2(r → 0)= ϕ= 0 . 36 and S 2(r → ∞ )= ϕ²= 0 . 13 .

Fig. 3. Probability Density Function (PDF) of the local porosity ( ϕ^a) and associated microstructure for the same four systems ( Fig. 2 ). The vertical line corresponds to the average porosity ( ϕ).

Fig. 4. (a) Degrees of freedom of a link element between points i and j , (b) D3Q18 unit cell, (c) simulation box.

or,inadiscreteformasthecorrectedstandarddeviationoftheporosity:

sa

_i^Na

=1

ϕ

aⁱ−

ϕ

2

Na−1 , (7)

where

ϕ

^{istheaverage sampleporosity;}

ϕ

aⁱ isthe localporosityaround arandomly chosen pointi,i ≤Na withNa large enoughtofullysamplethespecimenstudied.Thus,s_a=0correspondstoaperfectlyorderedsystem,exhibitingnovariabil- ityinthe localporosity;whereas largevaluesofsa correspondtosegregatedsystemsinwhichsome degreeofclustering existsintheporeconfiguration.Theclusteringindexincreaseswiththewidthoftheprobabilitydensityfunctions.Bywayof example,theclusteringindexesofthePDFsshowninFig.3,increasefroms_a=0.05forthequasi-orderedsystem(Fig.3(a)) tosa=0.09andsa=0.14fornon-overlappingdisks(Fig.3(b)andc),andreachesthehighestvalue,sa=0.22,foroverlap- ping disks (Fig.3(d)). Forpurpose ofcompleteness, we note that theclusteringindex iscloselyrelated tothe coarseness C=s_a/(¹−

ϕ

)^{introducedbyTorquatoandcoworkers(LuandTorquato,1990;Torquato,2002).Theyprovedthatthecoarse-} nessCwasdirectlyrelatedtothetwo-pointprobabilityfunctionS₂(r)(seeAppendixBinthecaseoffullypenetrabledisks).

TheclusteringindexsadefinedbyEqs.(6)and(7)willbeofsomehelpindelineatingtheeffectofdisorderontheelasticity propertiesofthegeneratedporoussamples.

2.2.Effectivestiffnessmeasureusingthelatticeelementmethod

Thesecondtoolemployedinthisinvestigationoftheeffectofdisorderonmechanicalpropertiesofporousmaterialsis theLatticeElementMethod(LEM)(seee.g.HerrmannandRoux,1990;Topinetal.,2007andreferencesherein).Thegen- eratedporousmesostructuresarediscretizedinanumberofmasspointsrepresentingthesolidortheporedomain.Much akintoPotential ofMeanForce (PMF)approachesused inSoftMatter Physics(see e.g. Masoero etal., 2012), thesemass pointsinteractwiththeirnearest neighborsthrough effectiveinteraction potentials(Laubieetal.,2017a).Inthisapproach (Fig.4), thereferenceconfiguration consistsofN=n_xn_yn_z masspointson acubic lattice (ofunit cell sizea₀),having six degreesoffreedom:threetranslations

δ

^{andthree rotations}

ϑ

^{.Each masspoint i (referencepositionx}i) interacts witha fixednumberofneighboringpointsj (amaximumof18inthisstudy,corresponding toa cut-off radiusr_{cuto f f}=√

2a₀ in PMF-approaches)viathepotential:

Ui j=U_{i j}^s+U_{i j}^b, (8)

whereU_{i j}^s isastretchtermandU_{i j}^babendingterm.Withafocusontheelasticityofthesamplesclosetotheirequilibrium configuration,theinteractionpotentialsinthesoliddomainareapproximatedbyharmonicexpressions:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎩

U_{i j}^s = 1 2

i jⁿ

δ

ⁿj−

δ

iⁿ

l⁰_{i j}

²

U_{i j}^b = 1 2

i j^t

δ

^bj−

δ

^bi

l⁰_{i j} −

ϑ

i^t

²

+

δ

^tj−

δ

^ti

l_{i j}⁰ +

ϑ

i^b

²

+

δ

^bj−

δ

i^b

l_{i j}⁰ −

ϑ

i^t

ϑ

i^t−

ϑ

^tj

+

δ

^tj−

δ

^ti

l_{i j}⁰ +

ϑ

i^b

ϑ

^bj−

ϑ

i^b

+1 3

ϑ

^bj−

ϑ

i^b

2

+

ϑ

i^t−

ϑ

^tj

2

. (9)

Hereinl⁰_{i j}=

r_{i j}

(withr_{i j}=x_j−x_i=l_{i j}⁰en) is the distancebetweensolid masspointsi and j inthe referenceconfigura- tion,whilethesolid’senergyparameters

i j^n,t∼a³₀Es arecalibratedtorecoverthedesiredeffective(ormacroscopic) elastic

behaviorofthe(homogeneous)solidphase(

ϕ

=0;withYoung’smodulusE_sandPoisson’sratio

ν

s),followingtheprocedure outlinedinLaubieetal.(2017a).Ifonlythestretchtermisconsidered(setting

^ti j=0),thevalueofthePoisson’sratioofthe compositeisdefinedbythegeometriclimitvalueofthecubiclattice;thatis

ν

lim=1/(^D+1)^{(withDthespace}dimension).

ThebendingtermisrequiredtomodelmaterialswithlowerPoisson’sratios,

ν

≤

ν

lim.Theconjugatedforcestotranslational degreesoffreedomsderivefromthepotential:

F_i^j=−

∂

^{Ui j}

∂

δ

i

. (10)

Thestressmeasureateachmasspointisobtainedusingthevirialexpression:

σ σ σ

iii= 1 2V_i

N^b_i

j=1

r_{i j}F_i^j, (11)

withV_i=a³₀ thevolume ofthe unit cell,andN^b_i the numberofpoint i’sneighboring masspoints. The total(or average) stressinvolumeV=(ⁿx−1)(ⁿy−1)(ⁿz−1)^a3₀ ^is:

σσσ

⁼_V^1N

i=1

Vi

σ σ σ

iii. (12)

Finally, all links belonging to the pore region have zero energy parameters,

ⁿ_{i j}^,t=0; and thus zero-forces (F_i^j=0) and stresses(

σ σ σ

iii=0).ForanextensionoftheLEMapproachtolinearporomechanics,seeMonfaredetal.(2017).

Inordertomeasuretheeffectivestiffness(intension)ofthedifferentsystemsstudied,adisplacementisprescribedon the boundaries;that is

δ

−=−

δ

/2ex on masspointsonsurfacex=0 ofthestructure, and

δ

+=

δ

/2ex onmass pointson surfacex=L=(^nx−1)^a0.The lateralboundariesofthestructureareforce free.Afterrelaxation, i.e.afterminimization of thepotentialenergy,Epot=minδi,ϑi

linkskl

U_kl ,withrespecttoboththetranslationalandrotationaldegreesoffreedom,the effectivestiffnessisobtainedfrom:

E_{e f f}=

σ

^xx

δ

^{L , (13)}

where

σ

^xx

=ex·

σσσ

·ex istheaxialstressand

δ

L. 2.3. Continuummicromechanicssolutions

The discrete solutions will be bench-marked against continuum micromechanics solutions that explicitly address the effectofdisorderon elasticityofheterogeneousmaterials;byconsidering localporosity fluctuationsoftheform(Gˇarˇajeu andSuquet,2007):

ϕ

t

(

^x

)

=

ϕ

+t

δ

ϕ

(

^x

)

, (14)

with

ϕ

^t(^x)

⁼

ϕ

^and

δ

ϕ

=0.ThestandarddeviationofthelocalporosityisobtainedfromanapplicationofEq.(6)(except thatthereisnoobservationwindowsizeadefinedhere):

σ

_ϕ²⁼

( ϕ

^t−

ϕ )

²

=t²

δ

²_ϕ

. (15)

Thetwobenchmarkmodelshereinconsideredare(1)theasymptoticexpansionmodelofGˇarˇajeuandSuquet(2007),con- sideringsmallfluctuationst1;and(2)adifferentialschememodelinspiredbyNorris(1985)whereacompositeporous solidisbuiltincrementallybyaseriesofadditionofdifferentphaseshavingdifferentelasticmoduli.

Tosimplifythepresentation,aN-phasecompositewithphasei(concentrationi, ^N

i=1i=1)occupyingthevolumei

withi∈

{

¹,...,N

}

^isconsidered.Thelocalporosityishomogeneousinthesubvolumes,i.e.

ϕ

^t(^x)=

ϕ

+t

δ

ϕ⁽ⁱ⁾forx∈i. 2.3.1. Mori–Tanaka-basedasymptoticexpansionmodel

The asymptoticexpansion modelofGˇarˇajeu andSuquet(2007) departs fromtheclassical referencesolutionof an or- deredporousmaterialmorphology(

σ

ϕ=0)^;thatis,thematrix–poreinclusionmorphologyexemplifiedbytheMori–Tanaka scheme¹(superscriptMT):

C^MT=

(

¹⁻

ϕ )

^Cs:

(

¹⁻

ϕ )

^I+

ϕ (

^I−S

)

⁻¹

−1

, (16)

1Although the Mori–Tanaka scheme was originally derived for random microstructures, it captures fairly well the behavior of periodic (ordered) systems (see Section 3.1 ).

whereIisthefourth-orderunittensorandC^sisthesolid’sstiffnesstensor,andSistheEshelbytensor,whichfor2-Dcylin- dricalporesinanisotropicmatrix(Dormieuxetal.,2006)andplane-stressconditionspermitsthefollowingspecificationof theYoung’smodulus:

E_MT^2D

( ϕ )

Es = 1−

ϕ

1+2

1−

ν

s²

ϕ

^{, (17)}

andfor3-Dsphericalporesinanisotropicmatrix(Dormieuxetal.,2006):

E_MT^3D

( ϕ )

Es = 1−

ϕ

1+^(1+ν₂^s₍⁾₇⁽₋¹³₅⁻_νs¹⁵₎^νs)

ϕ

^{, (18)}

withEsand

ν

^{sthesolid’sYoung’smodulusandPoisson’sratio,}respectively.Thefollowingclassicalisotropicelasticityequa- tionswerehereinused:

⎧ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎩

E_MT^2D

( ϕ )

^{= C2}11^D,MT

( ϕ )

1−

C₁₂^2D,MT

( ϕ )

C₁₁^2D,MT

( ϕ )

2

E_MT^3D

( ϕ )

^{= C3}11^D,MT

( ϕ )

1− 2

C³₁₂^D,MT

( ϕ )

2

C₁₁^3D,MT

( ϕ )

2

+C³₁₁^D,MT

( ϕ )

^C12^3D,MT

( ϕ )

. (19)

Thus,forreference,theelasticenergyw˜(

ϕ

,

)^{oftheorderedsystem(}

σ

ϕ=0)isaquadraticfunctionofthestraintensor

^:

˜

w

( ϕ

,

)

= 1

2

^:CMT

( ϕ )

^:

. (20)

Inreturn,whenfluctuationsinporosityareconsidered(

σ

ϕ=0),thiselasticenergybecomes²:

˜˜

w

(

^t,

¯

)

=

^w˜

( ϕ

t,

t

)

^{. (21)}

Theasymptoticexpansionofthisenergyintreads(GˇarˇajeuandSuquet,2007):

˜˜

w

(

^t,

¯

)

=w˜

( ϕ

,

¯

)

+t² 2

δ

²_ϕ

∂

^2w˜

∂ϕ

²

⁽

^¯

)

−

∂

^2w˜

∂ϕ ∂ ⁽

^¯

)

^:H:

∂

^2w˜

∂ϕ∂ ⁽

^¯

)

+O

(

^t3

)

, (22) whereHisafourth-order tensor.Foratwo-phase system(N=2)with

δ

ϕ⁽¹⁾=−

ϕ

/1 and

δ

ϕ⁽²⁾=

ϕ

/2 so that

δ

ϕ

= 1

δ

ϕ⁽¹⁾+2

δ

ϕ⁽²⁾=0and

δ

_ϕ²

=(

ϕ

)²/(12),GˇarˇajeuandSuquet(2007)showedthattheH-tensorreducestoH=

δ

_ϕ²

S:

C^MT

−1

withStheEshelbytensoroftheordered(matrix–inclusion)referencesystem,asemployedinEq.(16).Usingthis result,whilereplacinginEq.(22)t²

δ

²ϕ

bytheporosity standarddeviation

σ

ϕ² accordingtoEq.(15),theelasticenergyof thedisorderedsystemisrecastintheform:

˜˜

w

( σ

ϕ,

¯

)

= 1

2

¯^{:Ce f f}

( σ

ϕ

)

^:

^¯+O

( σ

_ϕ³

)

, (23)

withC^efftheeffectivestiffnesstensor:

C^{e f f}

( σ

ϕ

)

^=CMT+

σ

_ϕ²

2

∂

^2CMT

∂ϕ

^{2 −2}

∂

^CMT

∂ϕ

^:S:

C^MT

−1

:

∂

^CMT

∂ϕ

. (24)

Whileexpression(24)issomewhat involving,inthat itdoesnot permitsimpleclosed-formsolutions, itisreadilyimple- mentedforthe2-Dand3-Dmatrix–inclusionmorphologies,givenbytheMori-Tanakareferencesolution(16)andthecorre- spondingEshelbytensorexpressions.GˇarˇajeuandSuquet(2007)presentedthisimplementationthroughtheconsideration ofsphericalpores inan incompressibleisotropic matrix.Theextension ofthis solutiontospherical pores inacompress- ibleisotropicmatrixisgiveninaclosedforminAppendixC.Similarderivationwere performedforthecaseofcylindrical pores,butdonotpermitsimpleclosed-formsolution.Providedsuch solutionforC^eff(

σ

ϕ),theeffectiveYoung’smodulusis obtainedfrom(19),analogoustoEqs.(17)and(18).

Fig.5displayssampleoutput,E_eff/Es,fortheasymptoticexpansionmodelwith1=2=1/2andPoisson’sratio

ν

^s= 1/(¹+D)ⁱⁿD-dimensionasafunctionoftheporosityfordifferent

σ

ϕvalues(in2-D,Fig.5(a)andin3-D,Fig.5(b)).While someof the

σ

ϕ valuesmaywell bebeyondthe rangeofvalidityofthe model,t1,thefigures clearlyhighlightthat a localvariabilityof theporosity lowersthe effectivestiffnessaswell asthe predictedpercolationthreshold.This effectis particularlypronouncedathighporosity.

2From now on, the dependence in ϕis dropped to simplify the notations. All the functions are implicitly evaluated at ϕt= ϕ.

Fig. 5. Dimensionless effective Young’s modulus: E e f f(ϕ)/E sas function of the porosity in 2-D (a) and 3-D (b). Values obtained using the asymptotic development (24) (lines) and the differential scheme Eq. (26) (symbols) for σϕ = 0 (i.e. Mori–Tanaka scheme), σϕ = 0 . 1 , σϕ= 0 . 2 and σϕ= 0 . 3 .

2.3.2. Differentialschememodeltocaptureporosityfluctuations

Thesecondmodelhereinconsideredisbasedonthedifferentialscheme(Norris,1985),inwhichaN-phasemodelisbuilt incrementally.Startingfromahomogeneouselasticmedium(bulkandshearmoduli,K₀ andG₀,volumefractionc₀(⁰)=1), i=1,N phases(of bulkandshearmoduli,K_i andG_i,volumefractionc_i(t))aresuccessivelyadded,whilekeepingthetotal volume constant,such that c0(^t)+c(^t)=1 withc(^t)=^N

i=1

c_i(^t)^{.The effectivemoduli(K}eff,G_eff) areobtainedfromsolving thecoupleddifferentialequations:

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

K˙^{e f f} = ^N

i=1

Ki−K_{e f f}

˙ ci+c˙ c_i

1−c

Pi

G˙^{e f f} = ^N

i=1

Gi−G_{e f f}

˙ ci+c˙ c_i

1−c

Qi

, (25)

wherethedotdenotestimederivation,whilecoefficients(P_i,Q_i) dependonthephasemorphologyasspecifiedlateron.It isreadilyunderstood,thatthesolutionofthecoupleddifferentialequation(25)dependsonthepathchosen forc_i(t).The pathusedhereissuchthatthec_issatisfyc_i(^t)=it fort∈[0,1].ThusEq.(25)reduceto:

⎧ ⎪

⎪ ⎪

⎨

⎪ ⎪

⎪ ⎩

(

^1−t

)

^{K˙e f f = N}

i=1

K_i−K_{e f f}

iP_i

(

^1−t

)

^{G˙e f f = N}

i=1

G_i−G_{e f f}

iQ_i

. (26)

Last,thecoefficientsP_iandQ_idependonthemorphologyoftheinclusions;namely:

• Forsphericalinclusionsina3-Dmatrix(Norris,1985):

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

P_i^3D = Ke f f+K K_i+K Q_i^3D = G_{e f f}+G

G_i+G

, (27)

withK andGgivenby:

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

K = 4 3G G = Ge f f

6

9Ke f f+8Ge f f

K_{e f f}+2G_{e f f}

, (28)

• Fordisk-shapedinclusionsina2-Dmatrix(ThorpeandSen,1985):

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

P_i^2D = Ke f f+Ge f f

K_i+G_{e f f}

Q_i^2D = 2

(

^Ke f f+G_{e f f}

)

^Ge f f

K_{e f f}G_{e f f}+

(

^Ke f f+2G_{e f f}

)

^Gi

. (29)

Usingthedifferentialschememodelthusdefined,onecanbuildacompositewithanintrinsicvariabilityinporosity,by consideringasinitialconditionsthehomogeneoussolidresponse,(^K0,G₀)=(^Ks,Gs),andaddingeachphaseasasolid-pore compositeofporosity

ϕ

i=

ϕ

+t

δ

⁽_ϕⁱ⁾,withbulkandshearmodulievaluatedbytheMori–Tanakaschemewithporosity

ϕ

i∈ [0,1];thatis(^Ki,G_i)=(^KMT(

ϕ

i),G_MT(

ϕ

i))^{forthe2-Dor3-D}configuration.

Bywayofexample,consider(1)a2-phasecompositewith

ϕ

1=

ϕ

−

ϕ

/1and

ϕ

2=

ϕ

+

ϕ

/2,whichsatisfies

ϕ

^t

=

ϕ

^and

ϕ

^t−

ϕ

=0, whileexhibitinga porosity standard deviation

σ

ϕ=

(

ϕ

^t−

ϕ

)²

=(

ϕ

)

(¹/1+1/2)^{;and(2) a} 4-phase composite with

ϕ

1=

ϕ

−2

ϕ

/1,

ϕ

2=

ϕ

−

ϕ

/2,

ϕ

3=

ϕ

+

ϕ

/3 and

ϕ

4=

ϕ

+2

ϕ

/4, satisfying

ϕ

^t

=

ϕ

,

ϕ

t−

ϕ

=0, and

σ

ϕ=

ϕ

4/1+1/2+1/3+4/4.For these differentconfigurations, the system of Eq.(26) is integratednumerically(Wolfram Research,Inc, 2016) fromt=0→1to determine theeffective Young’s modulus forthe 2-Dandthe3-Dsystemfrom:

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

E_{e f f}^2D = 4K_{e f f}

(

^t=1

)

^Ge f f

(

^t=1

)

Ke f f

(

^t=1

)

+Ge f f

(

^t=1

)

E_{e f f}^3D = 9K_{e f f}

(

^t=1

)

^Ge f f

(

^t=1

)

3K_{e f f}

(

^t=1

)

+G_{e f f}

(

^t=1

)

. (30)

Notethatthecondition

ϕ

i∈[0,1]limitstherangeofaccessibleaverageporosity

ϕ

^atagiven

σ

ϕ value.

Fig. 5 displays sample output, E_eff/Es, for the differential scheme model for a symmetrical 2-phase composite (1= 2=1/2).Asymmetrical4-phasecomposite(1=2=3=4=1/4)andanasymmetrical2-phasesolid(1=0.8and 2=0.2)givethesameresults.Twopointsdeserveattention:(i)atsame

σ

ϕ value,allmodelsprovidethesameeffective stiffness vs. porosity response irrespective of the choice of symmetry and number of considered phases; (ii) the E_eff/Es

response provided by the differential scheme model is strictly identical with the response of the asymptoticexpansion model.ThisismostlikelyduetotheMori–Tanakamatrix-inclusionmorphologyassumedinbothmodels.

3. Results

3.1. Validation:elasticityoforderedsystems

ThenumericalLEM resultsareherevalidatedforthe orderedsystemagainst referencemicromechanicssolutions. This willpermit us inthe sequel to addressthe impact ofdisorder on elasticitywithrespect to theelasticity ofthe ordered systems,i.e.disorder-inducedstiffnessdegradation.

In the simulations, ordered systems are referred to as the periodic arrangement of disk-shaped pores of radius R or square-shapedporesofside-lengtha ,in2-D;andto theirperiodic 3-Danalogs,i.e.sphericalporesofradius R,orcube- shapedporesofside-lengtha.Theporosity

ϕ

^{ofthesesystemsistunedbyvaryingsize(R,a)andnumbers(N)ofthepores}

inthesimulationboxofsizeL,andporeperiodicityl.Thecriticalporosityatpercolation,

ϕ

^{c,– thatistheporosity above}

whichtheeffectivestiffnessvanishes,isobtainedforR=l/2fordisks,a=l forsquares,R=l/√

2forspheresanda=lfor cubes:

⎧ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎨

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎪ ⎪

⎩

ϕ

c^disk =

π

4 0.785

ϕ

c^square = 1

ϕ

c^sphere =

π

12

(

¹⁵−8√

2

)

0.965

ϕ

c^cube = 1

. (31)

Simulations were carried out by considering in the potential calibration different solid Poisson’s ratios (

ν

s=1/3 and

ν

^s=0.1in2-D;

ν

^s=1/4and

ν

^s=0.1in3-D).TheLEMdiscretizationwasnx=ny=221,nz=2,i.e.nxnynz=97,682mass pointsin2-D,andnx=ny=nz=61,i.e.nxnynz=226,981masspointsin3-D.

Alarge numberofsampleswasgeneratedbyvaryingnumber(N)andsize(R,a) ofthepores (N∈{4,16,25,100,121, 400,484}in2-D;N∈{8,27,64,..125,216,1000}in3-D).However,asFig.6shows–informofaplotofthedimensionless effectiveYoung’s modulusE_eff/Es vs. porosity

ϕ

^{– the ordered uniaxialelasticityresponse is}insensitive toboth numberof poresandPoisson’sratio.

Fortheordered system, theLEM simulationresults show afair amountofconsistency withthecontinuum microme- chanicssolutions(Fig.6).Specifically,atthelow porositylimit(

ϕ

1),thediluteapproximationsobtainedfromthe2-D and3-DMori–Tanakasolutions,Eqs.(17)and(18),comparefairlywellwiththesimulationresults,showingalineardecay

Fig. 6. Dimensionless effective Young’s modulus: E e f f(ϕ)/E sas function of the porosity. (a) 2-D systems with disk-shaped pores ( • : ν= 1 / 3 , ◦: ν= 0 . 1 ), square-shaped pores ( ), dilute asymptot from Eq. (32) (gray broken line), Mori–Tanaka homogenization scheme from Eq. (17) (black broken line) and large porosity analytical solutions from Eqs. (37) (disks: gray solid line, squares: black solid line). (b) 3-D systems with spherical pores ( • : ν^{= 1 / 4 , ◦:} ν^{= 0 . 1 ),} cubic pores ( ), dilute asymptot from Eq. (33) (gray broken line), Mori–Tanaka homogenization scheme from Eq. (18) (black broken line) and large porosity analytical solutions from Eqs. (37) (spheres: gray solid line, cubes: black solid line).

Fig. 7. Ordered system under uniaxial strain, (a) schematic stress map, the darker the higher the stress is, (b) zoom in between two neighboring pores and (c) equivalent geometry.

ofE_eff/EsconsistentwithaTaylordevelopmentfor

ϕ

1(Dayetal.,1992);thatis³:

⎧ ⎪

⎪ ⎨

⎪ ⎪

⎩

E²_MT^D

Es 1−

ϕ

1+2

ϕ

^ϕ=11−3

ϕ

+O

ϕ

²

(

³²

)

E³_MT^D

Es 1−

ϕ

1+

ϕ

^ϕ

=11−2

ϕ

^+O

ϕ

²

.

(

³³

)

Thesediluteapproximationsholdirrespectiveoftheporeshape.

Beyondthedilutesituation(Fig.6),asmentionedbyothers(see e.g.Drach etal., 2016), wefindthat theMori–Tanaka micromechanicsmodelpredictsreasonablywellthestiffnessvs.porositybehaviorforawiderangeofporosity;–exceptclose tothepercolationthresholdforwhichtheMori–Tanakaschemepredicts

ϕ

c^MT=1.ThepoorperformanceoftheMori–Tanaka schemeathighporositiesisattributedtothestressconcentrationinnarrowbandsbetweentheporesathighporosities(see Fig.7forthe2-Dcasewithdisk-shapedpores),thatcannotbecapturedwithmean-fieldaveraging.

Totestthishypothesis,theapproachsuggestedbyDayetal.(1992)isfollowed.Zonesofstressconcentrationsareconsid- eredasasuccessionofinfinitesimallysmallelasticbeamelementsoflengthdxandsectionS(x)whichdependsonthepore geometry.TheapproachisschematicallysketchedinFig.7.Thebeamsfollowauniaxialstress-strainrelation,

σ

=Es

ε

,which isrewrittenintermsoftheinfinitesimaldisplacementdu=

ε

^{dxandtheforceF}=

σ

^S(^x)^asdu=Fdx/(^EsS(^x))^.Byintegration oneobtains^u=F/Es

dx/S(^x)^{.Theeffective}constitutiverelationinthisinter-poreregionreducesto

σ

=F/S_{e f f} =E_{e f f}^u/l withE_eff theeffectivestiffnessandS_eff theeffectivearea ofinfluence,i.e.ldin2-D andl² in3-D.Theeffectivestiffnessin

3The approximation in Eq. (32) is valid for small values of νs; whereas the approaximation in Eq. (33) is valid for ν∈ [0, 0.5], and is exact for ν= 1 / 5 and ν= 1 / 3 .