A potential-of-mean-force approach for fracture mechanics of heterogeneous materials using the lattice element method

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A potential-of-mean-force approach for fracture mechanics of heterogeneous materials using the lattice

element method

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

To cite this version:

Hadrien Laubie, Farhang Radjai, Roland Pellenq, Franz-Josef Ulm. A potential-of-mean-force ap- proach for fracture mechanics of heterogeneous materials using the lattice element method. Journal of the Mechanics and Physics of Solids, Elsevier, 2017, 105, pp.116 - 130. �10.1016/j.jmps.2017.05.006�.

�hal-01720479�

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A potential-of-mean-force approach for fracture mechanics of hetero - geneous materials using the lattice element method

HadrienLaubieâ, FarhangRadjaï^b,^c, RolandPellenqâ,^b,^d, Franz-JosefUlmâ,^b,^∗

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

bMSE², UMI 3466 CNRS - MIT Energy Initiative, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge 02139, USA

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

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

Keywords:

Inhomogeneous material Elastic material

Crack branching and bifurcation Crack propagation and arrest Fracture mechanisms Fracture toughness

a b s t ra c t

Fracture ofheterogeneous materials hasemergedas acriticalissue inmanyengineeringapplications,rangingfrom subsurfaceenergy to biomedical applications, and requires arational framework that allows linking local fracture processes with global fracture de- scriptors such as the energy release rate, fracture energy and fracture toughness. This isachieved here by means of a local and a global potential-of-mean-force(PMF)inspiredLatticeElementMethod(LEM) approach.Inthe localapproach,fracture-strength criteria derived from the effective interaction potentials between mass points are shown to ex- hibit a scaling commensurable with the energy dissipation of fracture processes. In theglobal PMF-approach, fracture is considered as a sequence of equilibrium states associatedwith minimum potentialenergy states analogous to Griﬃth’s approach. It is foundthatthisglobal approach has much in common with aGrand Canonical Monte Carlo(GCMC)approach,in whichmass points arerandomly removedfollowing amaximum dissipation criterion until the energy release rate reaches the fracture energy. The duality of thetwo approaches is illustrated throughthe application ofthePMF-inspiredLEM forfracturepropagationinahomogeneouslinearelasticsolidusingdifferentmeans of evaluating theenergy release rate. Finally, by application of the method to a textbook example of fracture propagation in a heterogeneous material, it is shown that the proposed PMF-inspired LEM approach captures some well-known toughening mechanisms relatedtofracture en-ergycontrast,elasticitycontrastandcrackdeﬂection intheconsideredtwo-phaselayered compositematerial.

1. Introduction

Fracture Mechanics dealswiththe fractureresistance ofsolids subjectto load. Inits continuum version, it eitherfol- lows a globalora localapproach using concepts of linearor non-linear elastic fracture mechanics. The globalapproach wassetinstoneby Griﬃth(1921)astheirreversibledissipationofpotential energybymeansoffracturesurfacecreation;

thesecond by Irwin’sstress concentration approach(Irwin,1958) that recognizesthat stress singularitiesatthecracktip

∗ Corresponding author at: MSE ², UMI 3466 CNRS - MIT Energy Initiative, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge 02139, USA.

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

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havesomecharacteristicasymptoticpatternswhichpermit predictionoffracture propagation.Bothapproacheshavebeen extendedsincethe1960sfarintotheregimeofnon-linearbehaviorofsolids(startingwiththeworksofBarenblatt,1962;

Bažant,1984;Dugdale,1960;Hutchinson,1968;RiceandThomson,1974andmanymore).Thesemethods,however,devel- opedforhomogeneousmaterials. Itiswell knownthatthemicrostructureofheterogeneoussolidscan contributetotheir toughness,that isthe increase of their fracture resistance induced by their local micro-texture. Thesemechanisms were notablyobservedinnaturalhierarchicalcompositematerials suchasnacre(see e.g.Kamatetal.,2000) orbones(see e.g.

Launey etal., 2010) and severalmicrostructure-based fracture mechanics models were derived. Theyinclude toughening duetomicrocracks (ShumandHutchinson, 1990), crackbridging by uncrackedstiff ortough inclusion(Bowerand Ortiz, 1991)orlayers(Shaoetal.,2012),crackfrontroughening (GaoandRice,1989)orcrackdeﬂection(FaberandEvans,1983;

HeandHutchinson,1989).Allthesetheoreticalworkswere,however,appliedtospecificinhomogeneousmorphologiesand spatialconfigurations.Recentdevelopmentsinimagingtechniquessuchasmicro-computedtomographynowgiveaccessto thefull microstucture of realmaterials but inorder toassess their failure behavior, new numericaltoolsmust be intro- duced.Different methodshavebeen usedforthenumericalstudyofheterogeneoussolids.The weight-functiontheory of Rice(1985)andBueckner(1987)wasusedbye.g.Démeryetal.(2014).However,thisapproachismoresuitedtothestudy ofplanarcrackpropagation.Direct applicationofthefiniteelement method(FEM)tocomplexmicrotexturesispossiblein theorybutrequirescumbersomeremeshingtechniquesandmaybecometoocomputationallydemanding. Morerecentex- tendedformulationsoftheFEMtechniquesuchastheXFEMareyetnotwelladaptedtocomplexproblemsinvolvingcrack nucleationorcrackbranchingphenomena(Sukumaretal.,2015).Suchrestrictionsareabsentfromrecentvariational-based approaches(Bourdinetal.,2010).Phase-fieldmodels forfracture ofbrittle solidsthatemergedfromthispioneeringwork werepromisinglyappliedtoheterogeneoussolids(seee.g.Hossain etal.,2014;Nguyenetal.,2015).Apersistentquestion yetremains astotheapplicationoffracturemechanicsmethodstodiscretematerialsystems,despitethegrowingnumber ofapplicationsranging frommolecularscale (foran overviewon thetopicsee Brochardetal.,2015 andreferences cited herein)tomeso-scaleofheterogeneousmaterials(forarecentreview,seeBonamyandBouchaud,2011).

Thesimplestdiscretesysteminsolidmechanicsisalattice-typediscretization(Hrennikoff,1941),whosealgorithmicim- plementationwasﬁrst coinedby Topinet al.(2007)astheLatticeElement Method(LEM);which iswhy thispaperwill focuson thismethod.The methodemerged congruently incomputational solid mechanicsandstatistical mechanicsasa truss-beam-typediscretizationofa solidforboth 2-Dand3-D problems,includingstrengthandfractureinvestigations of randommediausingregularorirregularnetworks(HerrmannandRoux,1990),whichwereappliedtoalargerangeofhet- erogeneousmaterialsrangingfromparticlecompositesandcementedaggregatestoconcrete,ceramicsandinterfacecracks inmasonrycomposites(NayfehandHefzy,1978;Hansenetal.,1989;vanMier,1996;SchlangenandGarboczi,1996;Chiaia etal.,1997; SchlangenandGarboczi, 1997;Bolander JrandSaito,1998;vanMier etal., 2002;Lilliu andvanMier, 2003;

Topinetal.,2007;Affesetal.,2012;MohammadipourandWillam,2015;2016).Whiletheseapplicationstoalargerangeof heterogeneousmaterialsystemsevidencethesuitabilityofLEMtoinvestigatelocalrupture,fracturesurfacegeneration,frac- turecoalescence,percolation,andcrackdeﬂection,mostinvestigations employlocallink-failurecriteriabasedonstrength criteriathatrestupontheassumptionthatlinkelementsbreakatagivenstressorforce-momentstrengthcriteria,indepen- dentofthediscretization.Suchlocalstrength-basedapproachesareexpectedtofailtocapturesizeeffectsassociatedwith fracturethatresultfromtheintrinsiccompetitionbetweenbulkdissipation(relatedtostrength)andsurfacedissipationthat deﬁnesfracture.

Withtheselimitationsinmind,thispaperproposesaframeworkthatcantacklethedualityoftheglobalandthelocal fracturemechanicsapproachinthecontextofdiscretesimulations ofsolidsusingLEM.Thestartingpointofthisapproach istherealizationthat LEMcan be viewedasapotential-of-mean-force approach(PMF) akintothe oneemployed by the soft-matterphysics community:anumberofmasspointsdefinedonaregular orirregularlatticeinteractviaeffectivepo- tentialsfromwhichforcesandmomentsderive.ThisPMFapproachtoLEMwasrecentlyproposedforelasticsystems(Laubie etal., 2017a;2017b),showingthattheelasticityinthePMFcontextisbutanevaluationoftheenergycontentofthesys- temaroundthe equilibriumstate definedby thelattice structure,forwhich mostnon-harmonicpotentialsdegenerate to harmonicpotentialscommensurableto theoriginal truss-beamtype formulationusedinclassical LEMapproaches.Onthe otherhand,thePMFapproach putstheLEMonthesamefooting asmolecular approachesthus permittingtoemploy the canonofstatisticalphysics,suchasthermodynamicensembledefinitions,toextendtheLEMapproachasatoolofsolidme- chanicstoporomechanics(Monfaredetal.,2017).ByconsideringthePMF-approachforfracturemodelingofhomogeneous andheterogeneoussystems,thepurposeofthispaperistoextendourearlierdevelopmentsfromareversiblesolidbehavior toirreversiblesolidbehavior.

ThePMF-approach isﬁrst presentedandthen applied tosome complex examples ofinhomogeneous solidsexhibiting texture-relatedtoughening.

2. PMF–fractureapproachforLEM

Consider asolid in itsreferenceconﬁguration discretizedby N=n_xn_yn_z masspointsin thex,y andz directionson a cubiclatticeofunitcellsizea₀ (Fig.1).Eachmasspointi(initialpositionX_i)interactswithaﬁxednumberofneighboring pointsj (18 inthis paper–so-calledD3Q18 lattice– corresponding to acut-off radius rcut=√

2a₀ used forthe neighbor- listdeﬁnitioninPMFapproaches)via theinteractionpotential asafunction ofthetranslationalandrotationaldegreesof

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Fig. 1. (a) Degrees of freedom of a bond element between points i and j , (b) D3Q18 unit cell, (c) simulation box, (d) harmonic (gray curves) and Morse (black curves) interaction potentials and (e) associated gradient (i.e. force/moment).

freedomofthetwomasspoints(δi=x_i−X_i,δj=x_j−X_j,ϑi,ϑj)^:

Ui j=U_{i j}^s+U_{i j}^b, (1)

whereU_{i j}^s =U_{i j}^s δⁿ_j−δⁿ_i

deﬁnesthe two-body‘stretch’ interactions in function ofthe change indistance betweenmass pointsiandj;withδⁿi =δi·e_n^{i j}andδⁿj=δj·e_n^{i j} thedisplacementcomponentsinthebonddirectione_n^{i j}=r_{i j}/l_{i j}⁰ [withr_{i j}= X_j−X_i=l_{i j}⁰e_n^{i j} thevector connecting nodeito node jof rest-lengthl_{i j}⁰=α^a0 (α=1forbonds paralleltothecubiclattice directions;andα=√

2fordiagonalbonds),orientedby unitvectore_n^{i j} inalocalorthonormalbasis(^eⁿ,e_b,et)^{i j}^];^whereas U_{i j}^b=U_{i j}^b

ϑj−ϑi;δ^b_j−δ_i^b;δ^t_j−δ^t_i

considers bending interaction terms associated withrotations, ϑj−ϑi, andtransversal displacements, δ^bj−δi^b=(δj−δi)·e_b^{i j} and δ^tj−δ^ti=(δj−δi)·e^{i j}_t in a local right-handed orthonormal basis associated to bondij.Withthisparameterization,theinteractionforcesandmomentsbetweentwomasspointsiandj thatderivefrom theeffectivepotentialU_ijsatisfyforceandmomentequilibrium,thatis(foradetailedderivation,seeLaubieetal.,2017b):

F_i^j=−∂^Ui j

∂δi

; F_i^j+F_jⁱ=0, (2)

M_i^j=−∂^U^{i j}

∂ϑi

; M_i^j+Mⁱ_j+r_{i j}×F_jⁱ=0. (3) Afterprescribingamechanicalload(forceordisplacement)tothelatticestructure,therelaxedconﬁgurationisobtained by minimizing the potential energy.For thepurpose ofthisstudy, a non-linearconjugate gradientmethod wasused for thenumerical energyminimization:the Fletcher–Reeves–Polak–Ribieremethod.Forsuch discretesystem, thestresses are modeledusingthevirialexpression;whileneglectingthemomentumterm(Christoffersenetal.,1981):

σσσⁱ⁼₂¹_V

i N^b_i

j=1

ri jF_i^j, (4)

withV_i denotingthe volume of theunit cell,andN^b_i representing the numberof node i’s neighboringmass points. The stressinvolumeVcomposedofNmasspointsissimplythevolumeaverageofthelocalstresses;thatis:

σσσ⁼_V¹^N

i=1

Viσσσⁱ^. ⁽⁵⁾

All what ittakes to implementthe LEMapproach is tochoose appropriate expressions for theinteraction potential rep- resentative ofthesolid’s behavior.This hasbeen illustrated byLaubie etal.(2017b) forelastic isotropicandtransversely

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isotropicsolids;andbyMonfaredetal.(2017)forlinearporoelasticsystems.Thefocusofthenextsectionsistoextendthe PMFapproachtofractureproblems.

2.1. LocalapproachtoLEMfracturemechanics

ClassicalapproachestofractureproblemsusingLEMaregenerallybasedonstrengthcriteriathatrestricttheadmissible valuesoftheforcesandmoments(^F_i^j,M_i^j)^toâ^threshold^value^beyond^which^the^bondîjîs^consideredâs^broken;^hence:

f

F_i^j, M_i^j

≤0. (6)

In contrast,in fracture mechanics, an appropriate localfracture criterion should be based on an energy criterion. Ifone considersthattheenergydissipatedwhenthebondbreaksequalstheenergystoredinthebondbetweentheequilibrium state,r₀=l_{i j}⁰,andthecriticalstate,rc=l_{i j}⁰(¹+λ^c),anappropriatelocalfracturecriterionisoftheform:

∀^{i j};

⎧⎨

⎩

^Ui j=U_{i j}

r_{i j}

−U_{i j}

l⁰_{i j}

≤U_{i j}(^r^c)⁻^Ui j(^r⁰)⁼^Gc^bd _{i j} d i j≥0

^Ui j−Gc^bd i j

d i j=0

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whereGc^b=

U_{i j}(^r^c)−U_{i j}(^r0)

/d _{i j} canbeconsideredasthebond’sfractureenergywhichisdissipatedinthelocalfracture surfacecreationd _{i j}=(αâ0)²^situatedⁱⁿ^the^plane^defined^byûnit^normalên^{i j}.

2.1.1. Central–forcePMFfractureapproach

Toillustratethisfracturecriterion,weﬁrstfocusoncentral-forcelattices,inwhichinteractionsaresolelydeﬁnedbythe stretchenergyU_{i j}=U_{i j}^s

δⁿj−δⁿi =λⁿi jl⁰_{i j}

.In the isotropicelastic case, such central-force cubiclattices are well known to restrictthedomainofapplicationtosolidsexhibitingaPoisson’sratioν=1/(^D+1)^(with^D^the^space^dimension)^(see^e.g., Laubieetal.,2017b).Withafocuson linearandnon-linearfracture mechanics,consider,forpurposeofillustrationofthe PMF-approach,theMorsepotential(Morse,1929):

U_{i j}^M=−i j⁰+ i jⁿ

2β²

1−exp

−βλⁿi j

2

, (8)

withβ=

i jⁿ/(²i j⁰)^(to^ensure^that^U_{i j}^M→0forλⁿi j→ +∞)andtheassociatedbondforceaccordingtoEq.(2): F_jⁱ^,ⁿ= i jⁿ

β^li j⁰

exp

−βλⁿi j

exp

−βλⁿi j

−1

, (9)

orequivalently,replacingthelineardilation(orstretch)inthebonddirectionλⁿi j= δⁿj−δiⁿ

/l_{i j}⁰ bytheMorsepotential:

F_jⁱ^,ⁿ= i jⁿ

β^li j⁰

⎧⎨

⎩¹⁺

U_{i j}^M λⁿi j

i j⁰

+sign(λ)

1+U_{i j}^M λⁿi j

⁰i j

1/2⎫

⎬

⎭^. ⁽¹⁰⁾

Herein,−i j⁰ deﬁnesthewell-depthatrestlengthr0=l_{i j}⁰;whileβ^governs^the^elastic^response.Furthermore,aTaylorexpan- sionofU_{i j}^Maroundtheequilibriumpositionforλⁿi j1showsthattheMorsepotentialdegeneratestoaharmonicpotential:

U_{i j}^M=U_{i j}^H,s(λi j)⁺^O λⁿi j

3

, (11)

with:

U_{i j}^H(λi j)=−i j⁰+ⁿi j

2

λⁿi j

2

. (12)

Sucha harmonicpotential expressionis akintoclassical truss theoryemployed incentral-forceLEM formulationsin2-D (Hansen etal., 1989;Topinet al.,2007) and3-D (Affesetal., 2012; Kosteskietal.,2012; NayfehandHefzy,1978).In its turn,theenergyparameterⁿi j whichdefinesthelinearelasticbehavioraround theequilibriumpositionisdefinedby the elasticityofthesolid;forinstanceforanisotropicmaterialdefinedbyYoung’smodulusEandPoisson’sratioν^:

i jⁿ=Ea³₀Fn(ν)^, ⁽¹³⁾

wherethedimensionlessfunctionFn(ν)îs^defined^forêach^bond^direction,ând^takesîntoâccount^the^solid’s^Poisson^ratio, ν^,ând^the^level ôfdiscretizationnofthesolid (foradetailedderivation,andextension totransverseisotropy, seeLaubie etal.,2017b).