(Article) Fracture of rigid solids: a discrete approach based on damaging interface modelling

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(Article) Fracture of rigid solids: a discrete approach based on damaging interface modelling

Claire Silvani, Thierry Désoyer, Stéphane Bonelli

To cite this version:

Claire Silvani, Thierry Désoyer, Stéphane Bonelli. (Article) Fracture of rigid solids: a discrete approach

based on damaging interface modelling. Comptes Rendus Mécanique, Elsevier, 2007, 335 (8), pp.455-

460. �10.1016/j.crme.2007.05.023�. �hal-00199582�

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pp. 455-460, 2007 (DOI : 10.1016/j.crme.2007.05.023).

Fracture of rigid solids: a discrete approach based on a damaging

interface modelling.

ClaireSILVANI(silvani@lma.cnrs-mrs.fr),UniversitédeProvence&LMA(UPR7051-CNRS),31

cheminJosephAiguier,13402Marseillecedex20,France.

Thierry DÉSOYER (thierry.desoyer@ec-marseille.fr), EC-Marseille & LMA (UPR 7051-CNRS),

TechnopôledeChâteau-Gombert,38rueJoliotCurie,13451MarseilleCedex20,France.

Stéphane BONELLI (stephane.bonelli@aix.cemagref.fr), Cemagref, 3275 route de Cézanne, CS

40061,13182Aix-en-Provencecedex5,France.

AbstractWedescribetheprogressiveanddelayedfractureofrigidsolidsbyadiscretemodelling.

Each rigid solid is considered as anassembly of particleswith initial cohesive bonds, the latter

decreasingprogressivelyduring theloading. Adamaging interfacemodelisproposedto describe

thisprogressivephenomenon. Themodel hasbeenimplementedinadiscreteelementcode. The

rstillustrativeexample,whichisactuallyaparametricstudy,dealswiththeprogressivedamage

andsuddenfractureofasingleinterfacesubmittedtoanuniaxialtension. Thesecondexampleis

relatedtothecrushingofanassemblyofrigidsolidsi. e. agranularmediumsubmittedtoan

÷dometriccompression.

Keywords: Granularmedium;rigidsolids;interfaces;damage;fracture.

RésuméNousdécrivonslaruptureprogressiveetdiéréedesolidesrigidesparuneapprochediscrète.

Chaquesolide rigideest représentéparune collectiondeparticules, initialement liées par unecohésion

quipeutprogressivementdiminueraucoursduchargement. Unmodèled'endommagementinterfacial est

proposépourdécrirecettedécroissanceprogressive. Implémentédansuncodedecalculparélémentsdis-

crets,cemodèlepermetdesimulerlarupturediéréedecollectionsdesolidesrigides. Lepremierexemple

illustratif,quiestenfaituneétudeparamétrique,estrelatifàl'endommagementprogressifpuislarupture

d'uneuniqueinterfacesoumiseàunetractionsimple. Lesecondexempleportesurlaruptureetl'attrition

d'unecollectiondesolidesrigidesi. e.d'unmilieugranulairesouscompression÷dométrique.

Mots-clés: Milieugranulaire;solidesrigides;interfaces;endommagement;rupture

1 Introduction

The general frame of this study is that of the progressive (nite cracking velocity) and delayed (with

respecttotheloading)fracture ofrigidsolidsinteractingbycontactandfriction. Anillustrativeexample

ofsuchastructuralproblemisthisofarocklldam,whichcangloballysettleduetothelocalfractureof

rockblocksinthetime,seee. g. DeluzarcheandCambou,[1 ];OldecopandAlonso,[2].

Choiceisheremadetogetnumericallyapproximatedsolutionsofthecontact-frictionpartoftheproblem

byusingthediscreteelementmethodproposedbyJeanandMoreau(seee. g. [3 ],[4 ]). However,dueto

thefactthattherigidsolids(orgrains)whichwillbeallassumedofthesamecharacteristicsizeD^S^can

break,eachofthemisconsideredasanassemblyofrigidparticleswhichwillbealsoallassumedofthe

samecharacteristicsizeD^pD^S^. ^These^particlesâreâssumed^to ^beînitially ^'glued'. ^Fromâ^numerical

pointofview,agrain,i. e. anassemblyofrigidparticles,mustthusbeseenasameshoftherigidsolid,

inwhichacrackcaninitiate(resp. propagate)onlyon(resp. through)thecontactzonesbetweenrigid

grains. Consequently,froma physicalpointof view,these contactzones haveto be consideredas rigid

butbreakableinterfaces.

Strongcohesiveforcesaresupposedtoexistinitiallyontheinterfaces(seee. g. Delenneetal,[5 ]),giving

to themtheirinitial tensile strength. Itis thenassumedthat, whenagiveninterface I characteristic

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areaS ≈(D^p)² ^is^submitted^to ^a^suciently^strong^tensile^force, microcracksand/ormicrocavities,i.

e. damage,initiate,growand,eventually,coalesce, thatleadstothefractureoftheinterface (andso,to

theirreversible vanishingofthecohesiveforces).

Section2ofthispaperisdevotedtothepresentationofathermodynamicallyconsistentdamaginginterface

modelwhere, inagreement withthe generalframe of this study, the evolutionof the damage isat the

sametimes progressive and delayed. Twoillustrative examplesare presentedinSection3. Therstone

is that ofa singleinterface betweentwoparticles submittedto anuniaxialtensile force: the analytical

solutionis given,from whichaparametric studyof the damaginginterfacemodelis done. The second

exampleisrelatedtothecrushingofanassemblyoftwo-dimensionalrigidsolidsi.e. atwo-dimensional

granularmediumduetoan÷dometriccompression:theresultsherepresentedhavebeenobtainedusing

anumericalcodeinwhichthedamaginginterfacemodelhasbeenimplemented.

2 A damaging interface modelling

The (thermo)dynamicsystem considered inthis section is aninterface I ^between ^two ^grains. ^Like^the

grains,I îsâssumed^to^be^rigid^: ^theâreaôf^the^surfaceS ôccupied ^byI îs^then^constant,^whatever^the

forcesactingonare. Furthermore,thedisplacementjump[u]^throughS îsâssumed^to^be^zero^whenever I îs^not^destroyed^(i. ê. ^wheneverS îs^clearly^dened)^;consequently,[u]^cannot^be^consideredâsâ^state

variableofI^.Âctually,ônlyône'mechanical'statevariablewillbeconsideredthere,denotedbyd^(scalar)

andcharacterizingthedamagebymicrocrackingand/ormicrocavitationoftheconstitutivematerialofI^.

Itwillbeassumedthatd∈ 0, _m¹

wherem >0^is^a^material^parameter^whose^physical^meaning^will^be

discussedlateron. Itmustbehereemphasizedthat,assoonasd= 1/m^,I ^is^destroyed^and^the^contact-

frictioninteractionsbetweenthebothgrainshavetobeconsideredonthebasisoftheSignorini-Coulomb

equations(seee. g.[4 ]),whichwillnotbedetailedinthepresentpaper.

Thedamaging interfacemodelis actuallybasedonprevious worksoncontinuumdamage mechanicsby

Marigo, [6 ], wherethe necessaryandsucient conditionfor theintrinsicdissipation to benon negative

is simplygivenby

d˙ ≥ 0^. ^Denoting ^by σ thestresses actinginS^,^assumption^is ^then^made^that σ is homogeneous. Onthe otherhand,it isassumedthat, dueto thedamage,the eective toughsurfaceof

I îs ^notS ^butîtsônlyûndamaged ^part(1−md)S^. Consequently,the stressesare simplylinkedtothe globalforceF (denedinsuchawaythatFN=F. N>0^whenIîs^submitted^toâ^tensile^force)^by:

F = (1−md)Sσ. N (1)

A damage yield surfaceis nextintroduced. Once more, it is clearly inspired by the works by Marigo,

[6 ]. However,for asakeof consistencybetweenthepresentinterfacialdamage modeland theCoulomb-

Signorinione(seealsoCangemietal,[7 ]),whichmust'merge'inthelatteroneassoonasd= 1/m^,^the

damageyieldsurfaceishereexpressedasafunctionofFN ^andF_t=F −FNN,i. e.:

g^d(FN,F_t, d) = FN + 1

µ|F_t| − F0^d(1−md) = 0 ⁽²⁾

where µ ^is^the ^friction ^coecient ^between ^the^both ^grains ^whenI ^is ^destroyed ⁽d= 1/m^), F0^d >0^the

damageyieldwhend= 0^,^andm >0^a'softening'parameter(thegreaterm^,^the^stronger^thesoftening).

Aspreviouslyindicated, Eqn. (2)reducesto theclassical Coulomb'syieldsurface assoonas d= 1/m^.

Asfor the fractureof I^,^which^can ôccur ^suddenly^whenI îs ^suciently^damaged, îtîs ^controlled ^by â

fractureyieldsurface,whichreads:

g^f(FN,F_t, d) = FN + 1

µ|F_t| − F0^f(1−md) = 0 ⁽³⁾

whereF0^f ≥F0^d îs^the^maximal^tensile^forceI ^canûndergo. Ît^must^be^hereêmphasized^that^mechanical

states(FN,F_t, d)^such^thatg^f(FN,F_t, d)>0^cannot^be^reachedî. ê.,âs^soonâsg^f(FN,F_t, d) = 0^,Iîs

destroyedandthat,whateverthereachablemechanicalstate(FN,F_t, d)^is,g^d(FN,F_t, d)≥g^f(FN,F_t, d)

i. e. damage takes place before fracture, apart from the limit case of a perfectly brittle interface

(F0^f =F0^d^),^where^damage^and^fracture^areconcomitant.

Eventually,thedamageevolutionlawisgivenby(η^is^acharacteristictime):

d˙ = 1 η

g^d(FN,F_t, d) F0^d

H⁻(−g^f(FN,F_t, d)) + 1

m−d

δ

g^f(FN,F_t, d)

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whereh.i^denotes^the^MacCauley^brackets^andH⁻^is^the^modied^Heaviside^function⁽H⁻(0) = 0^). ^The

Diracdistributionδîndicates^that, âs^soonâs g^f(FN,Ft, d) = 0^,d˙îs^to ^beûnderstoodâs âdistribution derivative(i. e.d^'jumps'^toîts^maximal^value1/m^).

3 Illustrative examples

3.1 Tension

Apart from the friction coecient µ^, ^four ^material^parameters ^have ^to ^be ^identied^for ^the ^damaging

interfacemodel(seeSection2)tobefullydened: thesofteningparameterm^;^the^damage^yieldF0^d^;^the

fractureyieldF0^f = (1/r)F0^d⁽r≤1^);^thecharacteristic timeη^. ^Theînuenceôfêachôf^these^parameters

onthedamageevolutionishere studied,consideringasingleinterface(surfaceS⁾^submitted^to^a^simple

tensionsuchthatF_t = 0^andσ˙N= ˙FN/S=cst >0^.

Forasakeof convenienceanddueto thefactthat t= (σNσ⁰)/(σ⁰σ˙N)^,^where^the^damage^yield^stress σ0 îs^given^byσ0=F0^d/Sd^will^be^here^consideredâsâ^functionôfσN/σ0 însteadôf^the^timet^. ^Thus,

noticing thatg^d(σN, d)>0âs^soonâsσN/σ⁰ >1−md⁰^,Êqn. ⁽⁴⁾^can^be^rewritten ^(denoting^by d,N0

therstderivativeofd^with^respect^toσN/σ0^):

ησ˙N

σ0

d,N0 −mH(σN

σ0

−1 +md0)d = (σN

σ0

−1)H(σN

σ0

−1 +md0) ⁽⁵⁾

with the initial condition d(σN/σ0 = 0) = d0^. ^The êxact ^solution ôf ^this êquation ^reads ^(whenever

g^f(σN, d) =σN−σf(1−md)<0^,^where^the^fracture^yield^stressσf ^is^given^byσf =F0^f/S^):

d(σN

σ0

) =d0−H(σN

σ0

−1 +md0)

( ησ˙N

σ0m²) exp m σ⁰

ησ˙N

(σN

σ0

−1 +md0)

+ 1

m(1−σN

σ0

)− ησ˙N

σ0m² −d0

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Dependingondierentvaluesofthematerialparameters,thedierentshapesofthissolutionarepresented

onFig.1. Noticethat,duetothefactthat,inEqn.(6),thematerialparameterη^and^the^loading^parameter

˙

σNâresystematicallylinkedbytheirproduct,choicehasbeenactuallymadetoconsiderσ˙Nâsâ^parameter

andη^as^a^constant.

AsshownonFig.1,themainfeaturesofthedamageevolutionare:

• ^the^loading^rateσ˙N ^(or,ⁱⁿânêquivalent^way,^theînverseôf^thecharacteristic timeη⁾âctsôn^both

thepresentdamaged^forânârbitrary^given^loadingσN/σ0^,^the^greaterσ˙N^,^the^smallerdând

thecriticalvalueofthedamage(dc^,^such^thatg^f(σN, dc) = 0⁾^the^greaterσ˙N^,^the^smallerdc^,

• ^theînitial^damaged0^hasînuenceôn^both^the^damage^yield⁽σN0^,^such^thatg^d(σN0, d0) = 0⁾^the

greaterd⁰^,^the^smallerσN0 anddc^the^greaterd⁰^,^the^greaterdc^,

• ^the^softening^parameterm^immediatly^gives^theupper-boundofthedamagerange(sinced∈ 0, _m¹

,

see Section 2) and constrains the present damage d ^for ^an ^arbitrary ^given^loading σN/σ0^, ^the

greaterm^,^the^greaterd^,

• ^the^ratior=F0^d/F0^f =σ0/σf ≤1âctsônlyôn^the^critical^valueôf^the^damage^the^greaterr^,^the

smallerdc^.

Anotherinterestingresultconcerns theultimatephaseofthedamageevolution,i. e. thefractureofthe

interface: thelatterisnottriggeredbyacriticalvalueofthedamage,apriori dened,butdependsathe

sametimes onthematerialparametersandonthe loadingparameter. Froma modellingpointofview,

thisisduetothefactthat thedamaging interfacemodelisactuallybasedontwoyieldsurfaces,onefor

thedamage,the otherfor thefracture; fromaphysical pointof view,thisresultsimplymeansthatthe

fractureoftheinterfacecanbeeither'brittle'(smallvaluesofdc⁾^e. ^g. ^when^submitted^to^high^loading

ratesor'ductile'(greatvaluesofdc⁾^e. ^g. ^for^small^values^of^the^softening^parameter.

3.2 Compression

Wenowconsideranassemblyoftwo-dimensionnalrigidsolids(grains)i.e. atwo-dimensionalgranular

mediumsubmittedtoacompressiveforce|T|ⁱⁿ^÷dometric^conditions^(no^lateraldisplacements). Inthe initialstate,seeFig.2-a,thesample(initialheigth: H= 42cm^;^initial^width: W= 48cm⁾^is^composed^by

75grains(diametersbetween5and6cm^),^each^of^them^beingconstitutedby60to70particles(diameters

D^p^between⁵^and⁶^mm).^More^precisely,^the^numericalsimulationsinvolve4980particles. TheloadingT

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isdenedbyaramp(timerateT˙ =cst6= 0⁾^followed^by^a^constant^value⁽T˙= 0^),ⁱⁿ^order^to^highlight^the

creeplikeresponseofthegranularmedium.Theaxialstrainisdenedby=|U|/H^whereU ^is^the^global

displacementinduced byT ^;^the âxial^stressîs ^denoted^by σ=|T|/eW ^wheree îs ^the^(unit) ^thickness

ofthesample. Anotherimportantparameter,denotedbyν^,^is ^the^ratio^between^the ^present^number^of

brokeninterfaces andthe initialnumberofcohesive contacts. Noticealsothat all the simulationswere

performedwiththediscreteelementcodeLMGC90(seee. g. [8 ])andwithµ= 1, m= 1^,η= 1s^and ^a

timestep∆t= 5.10⁻⁴s^.

Aswehaveatimedependentdamagemodel,theloadingratestronglyinuencesthemechanicalresponse

of the sample. This is clearly shown on Fig.2-b, where t ^is ^scaled ^by ^the ^loading characteristic time

tF =F0^f/ T˙

^. ^For â^given^value ôf σ0^, ^Fig.2-c^shows ^that r înuences^the ^kineticsôf ^the ^creep^phase,

while for agivenvalue of σf^,^see^Fig.2-d, ^this îs^the âmplitudeôf^the âxial^strain^whichîs ^modied^by

r^. ^Noticeêventually^thatνândêvolvesⁱⁿ^the^same^way^during^the^creep^phase: ^the^kineticsîs^mainly

governedbythefractureoftheinterfaces.

4 Conclusion

Mostofthestructuralfailuresareduetothepre-existenceofvariouskindsofmicro-defects(microcracks

and/or microvoids) in the materials, which propagate and eventually coalesce in a macro-crack. The

modellingof thesepropagationand coalescence isanimportantissue. Thediscrete approachpresented

hereisintendedasasteptowardthisissue. Theproposeddamaginginterfacemodelisbasedonareduced

setof veparameters. Theillustrativeexamplesseem toindicatethat thenumerical codeinwhichthe

damagingmodelhasbeenimplementedisanecienttoolforsimulatingtheinitiationandthepropagation

of macro-cracks inrigidsolids, including thetime eect. Examples of applications clearly includedam

engineering: rockllmaterialischaracterizedbydelayedgrainbreakageunderconstantload. Thisisthe

maincauseofthemajorityof post-constructivedisplacementsobservedinhighrocklldams, whichcan

producepipingorcrackingoftheimperviouselement.

Acknowledgements

ThisprojectwassponsoredbytheRégionProvenceAlpesCôted'Azur

References

[1] R.Deluzarcheand B.Cambou, Discretenumerical modellingofrocklldams,Int.J.Numer.Anal.

Geomech.,30(2006)1075-1096.

[2] L.A. Oldecop and E.E. Alonso, Fundamentals of rockll time-dependent behaviour, in: Juca, de

Campos&Marinho(Eds),UnsaturatedSoils,2002,pp.793-798.

[3] M. Jean,Thenon-smoothcontactdynamicsmethod,ComputerMethodsinAppliedMechanicand

Engineering,177(1999)235-257.

[4] J.-J. Moreau, Unilateral contactand dryfriction innitefreedomanalysis, in: J.J MoreauetP.D

Panagiotopoulos(Eds),NonSmoothMechanicsandApplication,chapterCISMCoursesandLectures,

Vol.302,Springer-Verlag,1988,pp.1-82.

[5] J.-Y.Delenne, M.S.El Youssou, F. Cherblanc, J.-C. Benet, Mechanical behaviour and failureof

cohesivegranularmaterials,Int.J.Numer.Anal.Geomech.,28(2004)1577-1594.

[6] J.-J. Marigo, Formulation d'une loi d'endommagement d'un matériau élastique, C. R. Acad. Sci.

Paris,Ser.II292(1981)1309-1312.

[7] L.Cangemi,M.CocouandM.Raous,Adhesionandfriction modelforthebre/matrixinterfaceof

acomposite, in: A.B.Sabir, C.Bohatier,M.G. Fertis,G.T. Tsatsaronis,R.J.Krane,K.M. Abbott

(Eds),Proc. ThirdBiennalJoint ConferenceonEngineering Systems,Designand Analysis(ESDA

96),Montpellier,Vol.1,1996,pp.157-163.

[8] B.Cambouand M.Jean,Micromécaniquedesmatériaux granulaires,HermèsSciencePublications,

Paris,2001.

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0 0.5 1 1.5

Σ_N

Σ₀ 0

0.2 0.4 0.6 0.8 1

d Influence ofΣ _N

0 0.5 1

Σ_N

Σ₀

0 0.2 0.4 0.6 0.8 1

d Influence of d0

0 0.5 1

Σ_N

Σ₀ 0

0.2 0.4 0.6 0.8 1

d Influence of m

0 0.5 1 1.5

Σ_N

Σ₀ 0

0.2 0.4 0.6 0.8 1

d Influence of r

Figure 1: Simple tensionof asingle interface: inuence of theloadingand material parameters

onthedamageevolution. Noticethatonlytheexponentialpartof eachgraph(endingind=dc⁾

correspondstoaregulardamageevolution: thelinearpart(endingonthed^-axis^to^the^maximum

valueofd^,1/m⁾îsônlyânârbitraryrepresentationofthedamagejump[d] = 1/m−dc^,^which^leads

tothefractureoftheinterface. Beyondσ⁰= 0.9M P a^andη= 0.1s^,^the^reference^parameters^are:

σ˙N = 2.3M P a.s⁻¹^,d⁰ = 0^,m = 1^, r= 0.25 ^-^1a ^(up-left): ^inuence ^of ^the^loading^rate, σ˙N = σ˙N,2 ˙σN,4 ˙σN^; ^the ^greaterσ˙N^, ^the ^smallerdc ^- ^1b (up-right): inuence of the initial damage,

d0 = 0,0.2,0.4 ^- ^1c (down-left): inuence of the softening parameter, m = m,2m,4m^; ^the

greaterm^,^the^smallerdc ^-^1d(down-right): inuenceoftheratior=σ0/σf^,r=r, 0.1r,0.001r^;

thegreaterr^,^the^smallerdc^.

Figure 2: 2a(up-left): Samplecomposedbyan assemblyof 75non'glued' grains(initial heigth:

H = 42cm^; ^width: W = 48cm⁾ând ^submitted^to ân ^÷dometric ^loading^; êach ôf ^the ^grainsîs

composed of ≈⁶⁵ ^particles,^initially ^'glued' ^- ^2b(up-right): Axialstrain=|U|/H ^versus^di-

mensionlesstimet/tF ^for²^loading^rates^;σ0= 900kP a^,r= 0.25 ^- ^2c(bottom-left): Axialstrain

=|U|/Hând^ratio^between^the^present^numberôf^brokenînterfacesând^theînitial^numberôf^co-

hesivecontacts,ν^,^versusdimensionless timet/η^; σ0= 900kP a; ˙σ= 2300kP a.s⁻¹;r= 0.25^or0.5

- 2d(bottom-right): Idem 2capartfromσf = 5500kP a; ˙σ= 180kP a.s⁻¹;r= 0.17^or0.5