Volume averaging based integration method in the context of XFEM-cohesive zone model coupling

an author's

https://oatao.univ-toulouse.fr/26926

https://doi.org/10.1016/j.mechrescom.2020.103485

Nikolakopoulos, Konstantinos and Crete, Jean-Philippe and Longère, Patrice Volume averaging based integration

method in the context of XFEM-cohesive zone model coupling. (2020) Mechanics Research Communications, 104.

103485. ISSN 0093-6413

Volume

averaging

based

integration

method

in

the

context

of

XFEM-cohesive

zone

model

coupling

Konstantinos

Nikolakopoulos

a,b

_,

_{Jean-Philippe}

_Crete

b,∗

_,

_Patrice

_Longere

a

a ISAE-SUPAERO, ICA, Université de Toulouse, Toulouse, France b Laboratoire Quartz, SUPMECA, Saint-Ouen, France

Keywords: X-FEM

Cohesive zone model Numerical integration Volume averaging scheme Incompressibility

a

b

s

t

r

a

c

t

The main issue of the extended finite element method (XFEM) is the numerical integration of the system of equilibrium equations. Indeed, in order to have a correct displacement jump vector, the integration needs to be achieved on both sides of the discontinuity and thus requires the existence of integration points on both sides of the discontinuity. A volume averaging based integration method is developed in the present work alleviating this constraint and applied to XFEM coupled with cohesive zone model in a three-dimensional formulation. Moreover, unlike other widely used integration methods, the proposed method does not require the a priori knowledge of the position of the discontinuity inside the finite element nor the projection of the state variables.

1. Introduction

Themain issueoftheextendedfiniteelementmethod(XFEM) is the numerical integration of the system of equilibrium equa-tions, inparticular when XFEM is coupled witha cohesive zone model (CZM). Indeed, CZM is generally governed by a relation-ship between the displacement jump vector across the disconti-nuityandthecohesivetractionforcesthatpreventthecrackfrom opening.If thereis no integrationpoint on one ofthe two sides ofthecrack,classicGaussquadraturedoesnotprovideanaccurate enough integration leading to wrongcalculations of the stiffness matrix,theinternalforcesandthenthedisplacementjumpvector. Inliteraturethereexistseveralapproachesaimingatpalliatingthis deficiency.

The mostcommonlyused technique is thesubdivision of ele-ments,withexamplesin[1–3],whereintheelementsub-volumes V+ and V−, separated by the crack, are triangulated into subdo-mains, viz. triangles (ifR2_{) or tetrahedra(if R}3_{), each containing}

one ormoreGauss points.Thestandard Gaussquadratureisthen applied fortheintegrationoftheweak forminevery subdomain. This methodshould notbe confusedwiththe meshmodification techniques,see[4],asthesaidpartitioningisrealizedonlyforthe integration and thus does not introduce any supplementary de-greesoffreedom.Theelementsubdivisionbasedtechniqueiswell

∗ _{Corresponding author.}

E-mail address: jean-philippe.crete@supmeca.fr (J.-P. Crete).

adaptedforelastic materials orforstructures withinitial defects. Itimpliestheprojectionofstate variablesonthenewintegration points. This procedure can be both computationally cumbersome andleadstoerrors,especiallyinthecaseofelasto-plastic materi-alswhoseresponseispath-dependent.

Toaddresstheaforementioned issue,Elguedjetal.[5]instead proposedto raisesignificantly thenumberofGausspoints(64 in 2D),asa meansto ensurethat there isalways atleast one inte-grationpointonbothsidesofthediscontinuity(ifthe discontinu-itycrossesthe element near a node, the discontinuityis slightly shiftedwithoutsubstantialmodificationofthediscontinuitypath). Withthistechnique,thereisnoprojectionrequirementbutthe ex-tensionin3Drequires512integrationpoints,increasingdrastically thecostandthetimeofcalculation.

Martinetal.[6]proposedthesubstitutionofthediscontinuous Heavisidefunction, intheelementscut by thediscontinuity,bya newcontinuousenrichment function. Using thistechnique, there isnoissueofprojectionofthestatevariablesandnomodification ofthecrack path,fora numberofGauss pointof27 fora linear hexahedron, which also leads to an increase in the computation time. Moreover, the authors do not discuss the use ofthe B-Bar approach(see [7]) toprevent volumetric lockingin thecaseof a stronglynonlinearbehaviormaterial.

The volume averaging based integrationmethod developed in thepresentworkis inspiredfromtheapproach developedin Be-lytschko [8]. The method allows for alleviatingthe need for the existence ofintegration pointson bothsides ofthe discontinuity andis applied to XFEM coupled withcohesive zone model in a

three-dimensionalformulation.Moreover,unlikeotherwidelyused integrationmethods,theproposed methoddoesnot requirethea prioriknowledgeofthepositionofthediscontinuityinsidethe fi-niteelementnortheprojectionofthestatevariables.

In Section 2 the formulation of the X-FEM and the coupling withacohesivezonemodelarerecalled.Thenumericalintegration oftheequilibriumequations,includingtheB-Barapproach,is de-tailedinSection3.Twoapplications,viz.withandwithoutinitial discontinuity,arepresentedinSection4,inviewofcomparingthe proposedtechnique withclassicalapproaches.Concludingremarks aregiveninSection5.

2. X-FEM– CZMcoupling

The eXtendedFiniteElementMethod [1,9]has becomeone of themostwidely used methods forthe simulation ofcrack prop-agationin engineeringstructures. It is based on the idea of em-bedding the crack within the finite element. The kinematics of thecrack is accountedforby enrichingthe regular displacement fieldofthefiniteelementwithadditionaldegreesoffreedom.One ofits mostimportantadvantagesis that thecrack canpropagate independentlyof the meshing, without theneed for the a priori knowledge ofthe crackpath. XFEM hasdemonstrated its perfor-mancesforbrittle and quasi-brittle materials, see[3,10].Now, in the case of strongly non-linear elastoplastic materials subject to ductile damage, this method may lead to unrealistic results, see

[11].Indeed,theactivationoftheenricheddofsinanelementleads toa sudden (tractionfree)crackopening.Yet, inthesematerials, thefailureshould begradual. Totacklethisproblem,theconcept ofcombiningXFEMwithacohesivesegment[2]isadoptedinthe presentwork.

2.1.X-FEMformulation

In this work, the “shifted basis” formulation of the X-FEM

[12]without singularfunctionsis used.The discretized total dis-placementfieldu(x)accordinglyreads

u

(

x

)

= n Niai+  m



H

(

x

)

− Hj



Njbj (1.1)

whereNi isthe ithstandard FEshape function,n the numberof nodes,a_i theithstandarddisplacementdegreeoffreedom,m the numberofenrichednodesandbjthejthadditionaldegreeof

free-domassociatedtothejthnode.HistheHeavisidefunction,which equalsto₊0.5ifapointislocatedatthe“positive” sideofthe dis-continuityand−0.5elseandHjthevalueoftheHeavisidefunction

ateachnode.

ThecouplingbetweenXFEMandCZM[13]consistsininserting acohesivesegmentinthestrongdiscontinuityinordertoforma “cohesivestrongdiscontinuity”. TheXFEMformulationisadopted andthesystemtobesolved reducestothefollowinglinearform:

[K]



da db



=−

⎧

⎨

⎩

∫ Ve BT

σ

_dV ∫ Ve B∗T

σ

dV+∫ D NT_T locd



⎫

⎬

⎭

, (1.2) where [K]=

⎡

⎣

V∫e BT_Dt_{BdV ∫} Ve BT_Dt_B∗_dV ∫ Ve B∗TDt_{BdV ∫} Ve B∗TDt_B∗_dV+4∫ D NT_C locNd



⎤

⎦

(1.3) Where

σ

isthestress vector,Dt_{theelastic-plastictangent}

op-erator,Tloc thecohesivetractionsvector,Cloc thecohesivestiffness

matrix,Bthespatialderivativesmatrixoftheshapefunctions,B∗ thespatialderivativesmatrixofthe(H(x)− Hj)Njfunctionsand



D

thediscontinuity.Thesubscriptlocreferstothelocalframeofthe localizationband.

Fig. 1. Bi-linear cohesive law.

2.2. AdoptedCZMapproaches

AscanbeseeninEqs.(1.2)and(1.3),thecouplingofCZMwith theXFEMimpliestheincorporationofthecohesivetractions vec-tor Tloc and the cohesive stiffness matrix Cloc, in the finite

ele-mentformulation.Themethodofcalculationoftheabovephysical quantities,via thetractionseparationlaw(TSL),stronglydepends on the material andengineering application under consideration

[14].Inthepresentwork,twodifferentapplicationsareconsidered (Sections4.1and4.2),leadingtotheuseoftwodifferentcohesive laws(Sections2.2.1and2.2.2).

Both lawsare activated assoon asthe deterioration criterion ismet.Their commonbasisis inspiredfromthe approachin Ca-manho[15]wheretheevolutionofthesofteningbranchofthe co-hesivelawisgovernedbyade-cohesionvariableD _∈ [0,1],where 0denotes ahealthyelement and1 afullydamaged element.The argument



ofthefunctionD=f(



)isanequivalentdisplacement that takes into account the displacement jump vector

δ

, with

δ1

thenormalcomponentand

δ2,3

theshearcomponents(expressed inthelocalframeofthelocalizationband):



=







δ

1





2+

δ

₂2+

δ

2₃,where





δ

1





=max

(

0,

δ

1

)

(1.4)

Forthesakeofsimplicity,contactisnottreatedhere.

The evolution law of the de-cohesion variable is assumed

hereinoftheform

D=



_{0, i f}

_

_≤

_0

−0 c , i f

0

<



<

0

+



c 1, i f else (1.5)



0 isthe equivalentdisplacement at thedegradation occurrence

and



0+



cthecriticalvalueoftheequivalentdisplacementatthe

completefractureoftheelement.

2.2.1. Pre-existingdiscontinuityinthestructure

Adhesivelyjoinedassembliesandlaminate composites contain pre-existingdiscontinuities.Thenumericalanalysisofsuchbonded structures at themeso–scale isgenerally conductedby means of cohesive interfaces/elements whose behavior obeys a TSL gener-allycharacterizedbyalinearascendingbranch,possiblyaplateau, andthenalinearornonlinearsofteningbranch.Dependingonthe authors,thecriteriaforthelossof(initial)linearity(deterioration onset) andultimate failure (complete de-cohesion) are expressed mostlyasacombinationoftwomaterialparameters,e.g.interms of critical displacement andcritical traction force [16]or critical traction force andenergyreleaserate [17].For simplicity,we are hereconsideringabi-linearcohesivelawasdepictedinFig.1.This cohesivelawwillbeusedlaterinSection4.1.

The cohesive tractions vector corresponding to Fig. 1 is ex-pressedas:

Fig. 2. Linear/plateau cohesive law.

with

Cloc

(

i,j

)

=

(

1− D

)

Pk

(

i,j

)

(1.7)

wherePk(i,i)=Ecoh,thevalueofthecohesivelawslope.Theother

components ofthe matrix are zero,leadingto a non-couplingof thenormalandsheareffects.

Thedeterioration andcomplete failureonsetsare heredefined by critical displacements. The cohesive law accordingly requires threeindependentconstantquantities,viz.Ecoh,



0,



c.

2.2.2. Nopre-existingdiscontinuityinthestructure

Incontinuousbodies initiallyexempt ofdefects,thecrack for-mationisgenerallyprecededbyastrain/damagelocalization possi-blyinduced bymicrovoid/microcrack coalescence.Theprogressive deteriorationofthemechanicalpropertiesofthelocalizationband containing material maybe described via a cohesive law. In this casetheelasticresponseistheoneofthebulkmaterial[18],and thecohesivelawpossessesonlyaplateauandasofteningbranch. Onceagain,criteriaareneededtotriggerthecohesivelawand ul-timatefracture.Aparticularattentionmustbe paidtoensurethe stresscontinuityatthecohesivelawonset.Forexample,theinitial (maximum)tractioncan beafixed valueoravalue thatis deter-mined in the element at the onset as a function of the current stressstate.

In Fig. 2 is depicted an example of a plateau/linear softening cohesive law;asused inthiswork inthecaseofno pre-existing discontinuityinSection4.2.

Theinitial tractionatlocalizationoccurrenceintheelementis calculated throughthe useof astress tensor

σ

˜ averaged over all Gausspointsoftheelementasfollows

T₀=



σ

˜ n



loc (1.8)

T0 is thus not a constant quantity andis different for each

ele-ment since itresults fromits currentstress state atthemoment oflocalization. It mayalsobe definedarbitrarily orresultfroma bifurcationanalysis.

Thecohesivetractionsvectorisexpressedas:

Tloc=

(

1− D

)

T0 (1.9)

Thetangentmatrixofthecohesivelawinthesofteningregime isobtainedby Cloc

(

i,j

)

=

∂

Tloc

(

i

)

∂

δ

j =−

∂

D

∂

δ

j T0

(

i

)

=− 1



c

δ

j



T0

(

i

)

(1.10) withCloc(i,1)=0if

δ1

<0

The cohesive law accordingly requires two independent con-stantquantities,viz.



0,



c.

As soon as the critical value of the de-cohesion variable D is reachedthecohesivetractions areautomaticallysettozero, lead-ing to a traction free crack. The value Dc at failure can thus be

lower than 1depending onthe material behavior. Inthe present workitistakenequalto0.5.

3. Numericalintegration

Inthe following are outlined the three integration techniques consideredinthepresentwork.

3.1. StandardGaussintegration

ItisremindedthatthediscretestandardGaussintegration con-sistsincomputing I= nint  i=1 f



H



ξ

_i



,

ξ

_i,...



wiJ



ξ

_i



(1.11)

whereI can be thestiffness matrixor the internal forcesvector,

nintthenumberofGausspoints,

ξ

ithevectorcontainingthelocal

coordinatesoftheGausspoint,witheweightoftheGausspoint,J

thedeterminantoftheJacobianandfafunctionofthepositionof theintegrationpointwithrespecttothediscontinuity.

3.2.Sub-divisiontechnique

As wasstated inSection 1 the subdivision ofhexahedral ele-ments intotetrahedra isthe mostwidely used technique for the numerical integration of elementscrossed by discontinuities. Af-terthesubdivision,thefieldvariables areprojectedfromthe cur-rentintegrationpointsontonewonesbelongingtothetetrahedra, andthenastandardGaussintegrationruleisappliedoneach sub-element.

3.3.Volumeaveragingintegration(VAI)

Thevolumeoftheelementcrossedbythediscontinuityissplit into two sub-volumes V+_{and V}− _{separated by the discontinuity,}

seeFig.3left.Theproposedvolumeaveragingintegrationmethod makesuseofamodifiedquadraturerulethatperformsthe integra-tiontwiceoverthestandardGausspointsoftheelementbymeans ofanaveragingofthecontributionsofthetwosub-volumesV+_and

V−: I= V− Ve nint  i=1 f



−0.5,

ξ

i,...



wiJ



ξ

i



+V+ Ve nint  i=1 f



+0.5,

ξ

_i,...



wiJ



ξ

_i



(1.12)

Accordingto(1.12),thecontributionofagivensub-volumetoI

consistsinasumoverallintegrationpointsoftheelementwhile assigningthe value +0.5(V +) or −0.5 (V −) to H in (1.11) and weightingtheresultbytheratioofthesub-volumeoverthetotal volumeoftheelement.VAIprincipleisdepictedinthevisual rep-resentationofacutelementwith8integrationpoints(redmarks) inFig.3.

Itisnoteworthythatthismethodhasbeenoriginallyproposed byBelytschko [8]andhasbeenveryrecentlyappliedby Jinetal.

[19] within an embedded-band based approach. The authors do

not,however,addressthecasewherethebandcrossestheelement insuchawaythatone sub-volumedoesnotcontainany integra-tionpoints,a casethat willbepreciselyaddressedinthepresent work.TheVAItechniqueisappliedhereinadifferentcontext,i.e. to XFEM coupled witha cohesive strong discontinuity and com-paredtootherintegrationmethods.

Theproposedschememustbecompleted bytheuseofthe B-Bar approach in order to deal with incompressibility. Indeed, in ordertoalleviate volumetriclockinginnearly-incompressible me-diatheB-barapproachisemployed(seeHughes[7]).ThematrixB

Fig. 3. Visual representation of the “volume averaging” integration scheme.

Fig. 4. Single lap specimen dimensions [22] .

containingthespatialderivativesoftheshapefunctionscanbe de-composedintoadeviatoricandadilatationalcomponent,i.e.Bdev

andBdil respectively: B=



_B dil − − − Bdev



(1.13) The dilatational part is responsible for an over-stiff response. Topalliatethisdeficiency,thecomponentBdil in(1.13) isreplaced

by a new B¯dil component in a way that volume growth is each

integrationpoint isreplaced by a value averaged over all points: ¯ B=



_¯ Bdil Bdev



⇒ (1.14) ¯ Bi=

⎡

⎢

⎢

⎢

⎢

⎢

⎣

Ni,x+N˜¯i,x N˜¯i,y N˜¯i,z ˜ ¯

N_i,x N_i,y+N˜¯_i,y N˜¯_i,z ˜ ¯ Ni,x N˜¯i,y Ni,z+N˜¯i,z N_i,y N_i,x 0 Ni,z 0 Ni,x 0 Ni,z Ni,y

⎤

⎥

⎥

⎥

⎥

⎥

⎦

(1.15) where ˜ ¯ Ni,x= 1₃

(



Ni,x



− Ni,x

)

(1.16)



Ni,x



istheaveragedvalueofNi,x overallintegrationpoints.

For theenriched nodes,the B-bar matrixtakes theform B¯∗_{i =}

(

H

(

x

)

− Hi

)

B¯i.Formoredetailsabouttheimplementationofthe

B-barapproachintheframework ofABAQUSthereadercanreferto Shietal.[20].

4. Application

The methodology coupling XFEM and CZM outlined in

Section 2.1 has been implemented asa userelement subroutine (UEL)inthecommercialfiniteelementcomputationcodeABAQUS. InordertoverifytheabilityofVAItoperformanaccurate integra-tioninthe caseofembeddedcohesive strongdiscontinuities,two different applications are taken into consideration: (i) the single lapadhesivejointinFig.4and(ii)thetensilespecimeninFig.7.

In both numerical tests the infinitesimal strain theory is as-sumedandthetetrahedralelementscreatedduringthesubdivision stagehaveone integrationpoint.As shownin2D inSeabraetal.

Fig. 5. Visualization of the two different single lap models.

[21],bychoosingasinglepointintegrationrule,more incompress-ible modesare reproducible,whichcan alleviate volumetric lock-ing.

Inthe sequel, thedeveloped ‘volumeaveraging integration’or VAItechniquedescribedinSection3.2isnotablyassessedwith re-specttothesubdivisiontechniqueabove.

4.1. Singlelapjointtest– Pre-existingdiscontinuity

Thesinglelapspecimen,seeFig.4,usedinthisworkwastaken fromTsai[22].Twodifferentfiniteelementmodels werecreated, see Fig. 5. The first model, designated SL.1, contained an explic-itlygivensurfaceofzero-thicknessABAQUScohesiveelements be-tweentheadherents,seeFig.5left.

For the second model, designated SL.2, the discontinuity was introducedasthelevel-setofacohesivepre-crackcrossingthe ele-mentsinawaythatoneofthecreatedsub-volumesdoesnot con-tainanyintegrationpoints,seeFig.5right.Thischoiceismadein orderto verifytheaccuracy ofthe proposedscheme inthemost extremeconditions.

Both models were discretized using a structured

three-dimensional meshing of full integration hexahedral elements

(C3D8ABAQUSelements)of1mmlength,thereforefor“SL.1 the lengthofeachcohesiveelementisalsoof1mm.

Thematerialconsideredfortheadherentsisarateand temper-atureindependentmildsteelwithE=210GPaandv=0.33.The plasticflowconditionisdefinedviatheVonMisesyieldcriterion:

f



σ

,

σ

Y



=

σ

eq−

σ

Y=0 (1.17)

where

σ

istheCauchy stresstensorand

σ

eqtheequivalentstress

definedby

σ

eq=



3 2s:s; s=

σ

− 1 3Tr

σ

I (1.18)

Table 1

Hardening law & cohesive laws properties.

R0 ( MPa ) R∞ ( MPa ) k β

400 300 4.4 1

E_{coh (MPa)} 0 (mm) 0 + c (mm) Dc ε¯pc

Bi-lin. 10 0.1 1 0.5 –

Lin/plat. – 0.1 2 0.2

Fig. 6. Single lap test reaction force graph.

wheresisthedeviatoricpartofthestresstensorandItheidentity tensor.Theyieldstress

σ

Yin(1.17)accountsforisotropichardening

andtakestheformofVocelaw:

σ

Y= R0+R∞[1− exp

(

−k

ε

¯p

)

]β (1.19)

where R0,R∞,k,

β

are material constants describing the isotropic

hardening behavior (see Table 1) and

ε

¯p _{the cumulated plastic}

strain.

The linear cohesive law used in thispart of the study is de-scribedinSection2.2.1 andissimilartothecohesivelaw govern-ingthebehavioroftheABAQUScohesiveelements.Theproperties of the cohesive law are indicated in Table 1. Since the cohesive crack isinserted in thestructure fromthe very beginning ofthe analysis, thelaw is activatedwithin the elementscontainingthe meso–crackfromthefirsttimeincrement.

Inthefollowingareconductedfournumericalsimulations(see

Fig.5forSL1andSL2):

• zero-thicknessABAQUScohesiveelementsSL.1

• 8-GausspointstandardintegrationtechniqueSL.2

• sub-divisionintegrationtechniqueSL.2

• volumeaveragingintegration(VAI)techniqueSL.2

SL.1isusedhereasareferencesolutionsinceABAQUScohesive elements are a tool of known accuracy for crack propagation in interfaces.

Thenumericalresultsforthefournumericalsimulationsare su-perimposedinFig.6intermsofreactionforcevs.displacement.

According to Fig. 6, the standard Gauss quadrature (8 GPs) is unable toprovideacorrectpredictionofthereactionforce (lead-ing to0Nall alongtheloading),asexpected.Ontheother hand, the VAI and subdivision schemes give the same response asthe cohesivezonemodel.

4.2. Tensiletest– Nopre-existingdiscontinuity

For the tensile test the specimen of Fig. 7 is used. It is dis-cretized using a structured three-dimensional meshing of com-pleteintegrationhexahedralelements(C3D8ABAQUSelements)of 0.5× 0.5× 0.5mm3_{(intheareaofinterest).}

The material takenintoconsideration hereisthe sameasthe oneoftheadherentsinSection4.1,see(1.17-1.19)andTable1.

Thecriticalvalue tobeattainedbyan elementforlocalization (activationofXFEMenricheddegreesoffreedom)istentativelyan arbitrarycriticalcumulatedstrain

ε

¯_cp,seeTable1,givingthevalue ofT₀ in(1.8).

Fig. 7. Traction specimen dimensions.

Fig. 8. Tensile specimen cut by 0 °, 45 ° and 54.7 ° cracks (cumulated plastic strain distribution).

The linear/plateau cohesive law presented in Section 2.2.2 is thenusedtodescribethemeso–crackdeteriorationeffect.

Inthefollowingareconductedthreenumericalsimulations:

• elasto-plasticresponse(FEMonly)

• sub-divisionintegrationtechnique

• volumeaveragingintegration(VAI)technique

Forthe last configurations involvingthe formation and devel-opmentofacohesivestrongdiscontinuity,threethrough-thickness orientations(withrespecttotheloadingdirection)ofthe disconti-nuityareconsidered,i.e.0° (brittle-likefracture),54.7° (ductile-like fracture[23]) and45° (worstnumericalcase).Forthe0° case,the crackisforcedtoappearextremelyclosetothelocalizedelements’ side,ensuringagainthatoneofthecreatedsub-volumesdoesnot containanyintegrationpoints. Forthetwo other cases,thecrack positionwithinthe elementsisnot imposedsince thenumberof cut elementsensures the existence of multiple sub-volumesthat willsatisfy theabove mentioned extremecondition. The angleof 45° isthemostcriticalnumerical casein thesense that,for reg-ularmeshing,itguarantees thatoneoutoftwoelementswillnot haveGausspointsononesideofthecrack.The54.7° casedoesnot necessarilyimplytheaforementioned problembutitisaddressed forphysicalreasons.ThefracturedspecimensareshowninFig.8.

Thetwointegrationproceduresaredetailedbelow.

• Volumeaveragingintegration(VAI)technique

The localizationband propagatesthroughoutthe whole width ofthespecimen(withthechosenangle)whenoneelementattains thecriticalcumulatedstrain inoneincrement.Thisincrement de-notedkincLOC issavedforlateruse.

• Sub-divisionintegrationtechnique

Abandisdefinedattheverybeginningoftheanalysiswiththe samepathastheonecreatedconsideringthepreviousintegration technique,resultinginasubdivisionintetrahedra(whosenumber dependsonthewaytheelementiscutintothetwosub-volumes) anda distributionofGauss pointsfromthefirstincrementofthe analysis,although theenricheddofs ofthelocalizedelementsare activatedonlywhentheanalysisreacheskincLOC.

Fig. 9. Tensile specimen reaction force graph (0 ° case).

Fig. 10. Tensile specimen reaction force graph (45 ° case).

Fig. 11. Tensile specimen reaction force graph (54.7 ° case).

Indeed,inordertomakeasuccessfulandpertinentcomparison betweenVAIandthesub-divisionschemesitisimportantto con-trolthelocalizationintermsof(i)timeofoccurrence,(ii)pathand (iii)tetrahedracreated.Itis notedherethat inorderto calculate the sub-volume values in VAI a triangulation technique is used, dividingthevolumeintotetrahedrawhosevolumescan beeasily calculatedandthensummed(thesetetrahedrahavenoother par-ticipationintheintegrationprocedure).Forbothcases,byforcing theband toappearthroughoutthe wholewidthofthe specimen inoneincrement(kincLOC)andonthesamepath,itispossibleto

control(i),(ii)and(iii)inawaythattheyareexactlythesamefor thetwoschemes.Havingisolatedthesethreefactorswecan com-paretheschemessolelyontheirabilitytointegrate thesystemof equilibriumequations.

The numerical results forthe three numericalsimulations are superimposedinFigs.9,10and11forthethreediscontinuity ori-entationsintermsofreactionforcevs.displacement.Accordingto

Figs.9,10and11,VAIandsubdivisiontechniquesgivesimilar re-sponses.Avery slightdivergencecanbe seeninthe casesof45° and54.7°,butthiscanbeattributedtoadifferenceinthelasttime stepoftheanalysis.

It shouldbe notedthat the standard 8Gauss point schemeis notpresentedinanyofthetwocasesbecausewhenitwasusedno analysiswascapabletoconvergeaftertheonsetofthelocalization, soitisomitted.

4.3. Discussion

Itisremindedthatthesubdivisionofelementsconsistsin trian-gulatingthesub-volumesV+_andV−_{,separatedbythediscontinuity}

inseveral tetrahedraandprojecting the state variables knownat the currentintegrationpoints ontothe newly formed integration pointsof the tetrahedra. Theseoperations are costly in terms of computationtimeandpotentialsourcesofnumericalerrors.

The volume averaging integration (VAI) technique does need simply thevalues ofthe sub-volumesV+_{and V}−_{.The cost inthis}

caseisverylowandevennegligible.

ThenumericalexamplestreatedinSections4.1and4.2have ev-idencedthatbothtechniquesleadtothesameresults, demonstrat-ing the interest of using the volume averaging integration tech-nique.

It isnoteworthy thatfor tensionloadingthe softening partof theCZMlawgivestheshapeoftheoverallresponseinthe soften-ing regime. Theapparent similarity inthebrittle andductile-like responses inSection 4.2isduetothefact thatthesamelawwas utilizedforthesakeofsimplicityandmustnotbeconsideredasa generality.

5. Concludingremarks

A volume averaging integration technique or VAIis proposed for the integration of the equilibriumequations when using the XFEMcoupledwithcohesivezonemodel.Twodifferentnumerical tests, i.e.asinglelapjointtestanda tensiletest, havebeen real-izedinordertoverifytheaccuracyoftheproposedscheme.Ithas beenshownthattheproposedVAIschemeleadstothesamelevel ofaccuracyasthe well-knownsubdivisiontechnique,whilebeing muchsimplertoimplementandlesscostlytouse.

Acknowledgement

The authors would like to acknowledge the French Direction Généraledel’Armement(DGA)foritssupport.

DeclarationofCompetingInterests None.

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