Experimental Characterization of Cohesive Zone Models for Thin Adhesive Layers Loaded in Mode I, Mode II, and Mixed-Mode I/II by the use of a Direct Method

an author's

https://oatao.univ-toulouse.fr/20854

https://doi.org/10.1016/j.ijsolstr.2018.09.005

Lélias, Guillaume and Paroissien, Eric and Lachaud, Frédéric and Morlier, Joseph Experimental Characterization of

Cohesive Zone Models for Thin Adhesive Layers Loaded in Mode I, Mode II, and Mixed-Mode I/II by the use of a

Direct Method. (2019) International Journal of Solids and Structures, 158. 90-115. ISSN 0020-7683

Experimental

characterization

of

cohesive

zone

models

for

thin

adhesive

layers

loaded

in

mode

I,

mode

II,

and

mixed-mode

I/II

by

the

use

of

a

direct

method

G. Lélias

a,b

_,

_E.

_Paroissien

a

_,

_F.

_Lachaud

a,∗

_,

_J.

_Morlier

a

a Institut Clément Ader (ICA), Université de Toulouse, UPS, INSA, ISAE-SUPAERO, MINES-ALBI, CNRS, 3 rue Caroline Aigle, Toulouse 31400, France b Sogeti High Tech, Aeropark, 3 Chemin Laporte, Toulouse 31300, France

Keywords:

Adhesively bonded joint Cohesive zone model Macro-element Mode I Mode II Mixed-mode I/II

Singular value decomposition Digital image correlation technique

a

b

s

t

r

a

c

t

Thedemandfordesigninglightweightstructureswithoutanylossofstrengthorstiffnesshasconducted manyengineersandresearcherstoseekforalternativejoiningmethods.Inthiscontext,adhesivebonding mayappearasanattractivejoiningmethod.Howevertheinterestofadhesivebondingremainswhilethe structuralintegrityofthejointisensured.Accordingtorecentliteraturethecohesivezonemodel(CZM) appearsasasuitableapproachabletopredictbothstaticandfatiguestrengthofadhesivelybondedjoints. Thisapproachofthefractureprocessofadhesivelayersisbasedonthemodelingoftheadhesive me-chanicalbehaviorthroughasetofadhesivecohesivepropertiesineithermodeI,modeIIormixed-mode I/II.ThestrengthpredictionofadhesivelybondedjointsisthenhighlydependentontheCZMparameters. Themethodsusedtoexperimentallycharacterizethemarethusessential.Anewmethodology,termed directmethod,ispresentedandtested.Itisbasedonthemeasurementofdisplacementfieldofbonded adherendsatthecracktipofclassicalspecimensallowingfortheloadingoftheadhesivelayerinpure modeI,puremodeIIandmixed-modeI/II.Thetestedadhesiveisamethacrylate–basedtwo-component adhesivepastefoundunderthereference SAF30MIBmanufacturedbyAECPolymers(ARKEMAgroup). Theadherendsareinaluminium6060.Itisshownthatitispossibletocharacterizethecohesive prop-ertiesofthe adhesivelayer usingthe directmethod.The numericaltests involvebothadherendsand adhesivenonlinearities. Nevertheless,thepresented experimentalimplementationpassesbythe devel-opmentofadedicateddatapre-processingtointerprettheexperimentalmeasurements,highlightingthe significanceofthechoiceofthemeasurementmeanslinkedtothedesignofspecimen.

1. Introduction

Intheframeofstructuraldesign,thechoiceofjoining technolo-giesisdecisivesincetheyguaranteetheintegrity ofthe manufac-turedsystem.Themechanicalfastening,suchasrivetingor screw-ing, appears the reliable solution for the designers. Nevertheless aloneorincombinationwiththemechanicalfastening, the adhe-sive bonding joiningtechnology may offersignificantly improved mechanicalperformance in termsof stiffness, staticstrength and fatigue strength (Hart-Smith, 1980; Kelly, 2006). The use of this higherlevel ofmechanical performance allows forthe design of lighterjoints.Inotherwords,theadhesivebondingoffersthe pos-sibilitytoreduce thestructuralmasswhile ensuringthe mechan-ical strength.The optimizationof the strength-to-mass ratiois a

∗ _{Corresponding author.}

E-mail address: [email protected] (E. Paroissien).

challengeforseveralindustrialsectors,suchasaerospace, automo-tive,railornavaltransportindustries.But,thereductionof struc-turalmassmakes senseonlyifthestructuralintegrity isensured. As result to take benefit from the adhesive bonding in view of mass reduction, it is required to be able to predict the strength ofbondedjoints. Thestrength predictionconsistsin the compar-isonof computedstrength criteriatodesign allowablevalue. The strength criteria could be based on theoretical, empirical, semi-empirical investigations and possibly including in-service feed-back.Thestressanalysisallowsforthecomputationofinputdata, mandatorytotheassessmentofstrengthcriteria.Theexperimental characterizationallowsthen forthedefinitionofdesignallowable valueaswellasofmechanicalbehaviortobeusedasinputdataof themechanicalanalysis.Ashighlightedin(Jumelandal.,2013),the strength ofa same joiningsystemat macroscaledependson the experimentaltestspecimenandprocedureused,whichcontributes inrestrictedreliabilityorinextensiveandexpensiveexperimental

Nomencaltureandunits BBe bonded-beamselement

CZM cohesivezonemodel

DIC digitalimagecorrelation

DoE designofexperiments

DCB doublecantileverbeam

ENF endnotchedflexure

FE finiteelement

ME macro-element

MMB mixedmodebending

OSRA optimalsubrankapproximation

SLJ single-lapjoint

SVD singularvaluedecomposition

Aj extensionalstiffness(N)ofadherendj

Bj extensionalandbendingcouplingstiffness(N mm)

ofadherendj

D_j bendingstiffness(Nmm2 )ofadherendj E adherendYoung’smodulus(MPa)

GI strain energyreleaserate(energyper unitof area:

mJorN/mm)inpeel

GII strain energyreleaserate(energyper unitof area:

mJorN/mm)inshear

GIc criticalstrainenergyreleaserate(energyperunitof

area:mJorN/mm)inpeel

GIe adhesive elastic strain energy stored (energy per

unitofarea:mJorN/mm)inpeel

GIIc criticalstrainenergyreleaserate(energyperunitof

area:mJorN/mm)inshear

GIIe adhesive elastic strain energy stored (energy per

unitofarea:mJorN/mm)inshear

H magnitudeofapplieddisplacement(mm)

J J-integralparameter

KBBe elementarystiffnessmatrix ofabonded-beam

ele-ment

L length(mm)ofbondedoverlap

M_j bendingmoment(Nmm)inadherendjaroundthe

zdirection

Nj normalforce(N)inadherendjinthexdirection

P magnitudeofappliedforce(N)

S adhesivepeelstress(MPa)

Smax maximaladhesivepeelstress(MPa)

T adhesiveshearstress(MPa)

Tmax maximaladhesiveshearstress(MPa)

Vj shearforce(N)inadherendjintheydirection

a cracklength(mm)

b width(mm)oftheadherends

d damageparameter

e thickness(mm)oftheadhesivelayer

h_j halfthickness(mm)ofadherendj

kI adhesiveelasticstiffness(MPa/mm)inpeel

kII adhesiveelasticstiffness(MPa/mm)inshear

n powerusdintheadhesivemateriallaw

n_ME numberofmacro-elements

t adherendthickness(mm)

uj displacement(mm)ofadherendjinthexdirection

vj displacement(mm)ofadherendjintheydirection



overlaplength(mm)ofamacro-element



j characteristicparameterofadherendjinN2 mm2

α

angle(rad)usedforthedefinitiontheload

applica-tioninMCBtest

β

mixed-modeparameter

δ

t numericaltimestep(s)

δ

u displacementjump(mm)oftheinterface alongthe

x-axis

δ

ue displacementjump(mm)oftheinterface alongthe

x-axisatinitiation

δ

uf displacementjump(mm)oftheinterface alongthe

x-axisatpropagation

δ

v displacementjump(mm)oftheinterface alongthe

y-axis

δ

ve displacementjump(mm)oftheinterface alongthe

x-axisatinitiation

δ

vf displacementjump(mm)oftheinterface alongthe

x-axisatpropagation

λ

normofdisplacementjump(mm)oftheinterface

λ

e norm of displacement jump (mm) ofthe interface

atinitiation

λ

f norm of displacement jump (mm) ofthe interface

atpropagation

ν

adherendPoisson’sratio

θ

j bendingangle(rad)oftheadherendjaroundthez

direction

testcampaign. Accordingto (Liandal.,2006; Khoramishadetal., 2010,2011;DaSilvaandCampilho,2012),thecohesivezone mod-eling – denoted CZM – appears as one of the most suitable ap-proachable tomodel both thestatic andthefatigue behavior of adhesive joints.According to Khoramishadet al.(2010), the CZM have the advantage of: (i) considering finite strains and stresses at the adhesive crack tip, (ii) indicating both damage initiation and propagation as direct outputs of the model, (iii) advancing thecrack tip assoon asthe local energyrelease ratereachesits criticalvalue with noneed of complexmoving mesh techniques. Based on Continuum Damage Mechanics and Fracture Mechan-ics, the CZM enablesa diagnostic of the currentstate of the ad-hesive interface damage along the overlap. The damage, associ-atedtomicro-cracksand/orvoidscoalescence,resultsina progres-sive degradation of the material stiffness before failure. An ide-alization of a CZM bilinear stress-strain relationship or CZM bi-lineartraction separationlawis presentedinFig. 1.The CZM bi-linear traction separation law is a well-established interface be-havior that first assumes a linearly dependency relationship be-tween the interface separation (deformation) and the resulting traction(stress). Onceaprescribed value ofseparationisreached by the adhesive, the damage initiation is described in the form of a linearly decreasing resulting traction. Finally, the propaga-tion of the damage is described by voluntarily fixing the result-ingtractiontozero,hencemodelingthecreationoftwo traction-free surfaces (i.e.: physical cracking). Both damage initiation and damage propagationphases are addressed inthe model with no need ofassuming any initial crackin the material (Valoroso and Champaney,2004; De Mouraet al., 2009;Campilho etal., 2013). The strength prediction of adhesively bonded joints is then highlydependent on the CZM parameters. The methods used to experimentally characterize them are thus essential. As a result, numbersofauthorshaveaddressedthiscriticalpointoverthepast fewyears(AndersonandStigh,2004;Alfredssonetal.,2003,2004; Leffleretal., 2007;Högberg,2006; HögbergandStigh,2006; Cui etal.,2014;Azarietal.,2009;Gowrishankaretal.,2012;Wuetal., 2013).Mostofthesemethodsmakeuseoftheconceptofthe ener-geticalbalanceassociatedtothecomputationofthepath indepen-dent J-integral (Rice, 1968) along a closed contour of specifically designedjointspecimens,knownastheinversemethod(Anderson andStigh,2004;Alfredsson etal.,2003,2004;Leffleretal.,2007, Högberg,2006; HögbergandStigh,2006).The inversemethod is basedontheenergeticalbalanceassociatedwiththecomputation

Fig. 26. (a) Experimental adhesive traction separation law in pure mode II. (b) Comparison between experimental results and numerical predictions in terms of load versus displacement curves.

numberofexperimentsNofafullfactorialDoEisthen:

N=p



_k  i=1 ni



(28)

Here is considered a full factorialDoE of fivefactors with re-spectively3× 3× 3× 3× 2levels,sothatthelinearTaguchi’sgraph of effects and interactions can be represented in the form of

Fig.15.

In Fig. 15, the main effects and interactions are represented, termed respectively E(i) and I(ij), of factors i,j=A, B, C,D andE

onto the objective function that is r2_{. Each experiment is}

repli-cated15timestocapturethe impactofthemeasurement disper-sion,sothatthetotalnumberofexperimentsis(3× 3× 3× 3× 2)× 15=2430. The different factorlevels are givenin Table3. where

SNRreferstothesimulatedSignal-to-Noiseratio,x₌ytothe spa-tialresolutionofeachdisplacementfieldinstantaneousimage,tto thenumberofinstantaneousimagestakenduringtheexperiment

Table 4

Definition of aluminum bulk material law identified. σ= true stress. ε= true strain. Elastic Plastic 1 Plastic 2

Model (EPB) σ( ε) = E ε σ( ε) = σ1 + E T,1 ( ε−ε1 ) σ( ε) = σ2 + E T,2 ( ε−ε2 ) Parameters E = 660 0 0MPa ET,1 = 8800 MPa σ1 = 200 . 31 MPa ε1 = 0 . 003035 ET,2 = 250 MPa σ2 = 232 . 84 MPa ε2 = 0 . 04 Validity 0 ≤ε≤ε1 ε1 ≤ε ≤ε2 ε2 ≤ε Table 5

Controlled geometries of the ENF, DCB, MMB and SLJ joint specimens.

a L l t e b ENF 29.82 mm 71.43 mm N.A. 3.96 mm 0.230 mm 22.0 mm DCB 30.69 mm 70.0 mm N.A. 3.96 mm 0.180 mm 22.0 mm MMB 30.21 mm 70.89 mm N.A. 3.96 mm 0.180 mm 22.0 mm SLJ N.A. 51.4 mm 29.35 mm 3.96 mm 0.120 mm 22.0 mm Table 6

CZM properties in pure mode I.

Elastic Plastic Softening Model (CZM) S(ε) = kSε1 ln(eS)( 1 − exp ( ln(eS) ε1 ε)) S ( ε) = S 1 + k S,1 ( ε−ε1 ) S(ε) = S 2 ε3−ε ε3−ε2 Parameters kS = 250 MPa eS = 0 . 030 ε1 = 0 . 15 kS,1 = 2 . 5 MPa S1 = 10 . 37 MPa ε1 = 0 . 15 S2 = 10 . 99 MPa ε2 = 0 . 4 ε3 = 0 . 75 Validity 0 ≤ε≤ε1 ε1 ≤ε ≤ε2 ε2 ≤ε ≤ε3 Table 7

CZM properties in pure mode II.

Elastic Plastic Softening Model (CZM) T(γ) = kTγ1 ln(eT)( 1 − exp ( ln(eT) γ1 γ)) T ( γ) = T 1 + k T,1 ( γ−γ1 ) T(γ) = T 2 γ33−γ3 γ33−γ23 Parameters kT = 110 MPa eT = 0 . 075 γ1 = 0 . 2 kT,1 = 2 . 5 MPa T1 = 7 . 85 MPa γ1 = 0 . 2 T2 = 10 . 48 MPa γ2 = 1 . 25 γ3 = 1 . 675 Validity 0 ≤γ ≤γ1 γ1 ≤γ≤γ2 γ2 ≤γ≤γ3

relationships.Themechanicalstiffnessofthetensiletestmachine is characterized so that the resulting displacement measured by thebuild-in machinedisplacementcelliscorrectedtofitthetrue displacement of the adhesive test specimens. Four specimens of eachconfiguration(e.g.ENF,DCB,MMBandSLJ)aretested.TheSLJ specimensaretestedforrelevanceassessmentpurposesonly. Cor-relationsbetweenexperimentalandnumericalforceversus result-ingdisplacementcurvesareusedtoassesstheabilityofthedirect approachtocharacterizetheCZMproperties.Aparticular empha-sisisgiventotheabilityofthesuggestedapproachtoprovideboth theexperimentalstiffnessandthemaximumloadbearing capabil-ityof each adhesive specimen.All the numerical tests presented inthispaperarebasedonthesimplifiedstressanalysesusingME, alreadypresentedindetailsin(Léliasetal.,2015).

4.2.2. Testresultsandmodellingunderpuremodes

The constitutive traction separation law of the adhesive layer obtainedin the caseof pure mode I and pure mode II loadings ispresented in Figs. 25-(a)and 26-(a)respectively. The CZM pa-rameters forpure mode I andpure mode II l are then provided inTables6and7respectively.Oneach puremode,themodel ob-tainediscomposedby3parts:anelasticpart,aplasticpartanda softeningpart.TheidentificationofCZM parametersisperformed onthe envelope curves builtfrom thecyclic response, usingthe elastic stiffness of adherends. The hypothesis underlying is that theYoung’s modulus does not varysignificantly during the plas-ticphase.Evenifthishypothesis doesnot holdatlarge strain of adherends,itallowsdrasticsimplificationoftheidentification pro-cess.Itmeansthattheadherendstiffnessesusedforthe identifica-tioninEqs.(18),(21),(23)and(24)correspondtotheinitialelastic

values.A numericaltest is then performedassuming a nonlinear behavior forboth the adherends (following the trilinear approxi-mationinTable4,seeAppendixC)andtheadhesivelayer.The nu-mericaltestpredictions,intermsofload/displacementcurve,are then compared to the experimental test resultson DCB andENF specimens in Figs. 25-(b) and 26-(b) respectively. A good agree-mentisshown.

4.2.3. Testresultsandmodellingundermixed-modeI/II

As previously, the constitutive relationships of the adhesive layerwhenfacingmixed-modeI/IIsolicitationsareinvestigated us-ingthedirectmethod.Nevertheless,theexploitationoftestresults fails, due to the limited axial displacements of both upper and loweradherendsnearbytheadhesive cracktip. Itresultsinbadly conditionedmeasures oftheadherendsaxial displacements,from which the differentiationwithrespect to x isinsufficiently accu-rate.Analternativecharacterizationmethodisthendevelopedfor determiningtheeffectivemixed-modeI/IIpropertiesofthetested MMB specimens. It is suggested to use an inverse characteriza-tionmethodbasedonnumericaltests.TheCZMpropertiesinpure mode I and pure mode II are considered as well as the nonlin-earadheredbehavior.Bothinitiationandpropagationmixed-mode criteriaare assumedasfollowing power lawenergetical relation-ships(seeEq.(7)).Theideaisthentoadjustthevalueofthevalue oftheexponentn₌minordernumericalpredictionfitthe exper-imental tests. In this study, the identification was made accord-ing totestswithtwo differentvalue ofmixed-moderatio,by the useoftwodifferentleverarmdenotedc.Similarlytotheprevious pure modetests, theexperimentaltest resultsandnumericaltest predictionsforbothmixed-moderatiosare comparedintermsof

load/displacementcurve inFig.27.Thepresentedcomparisonis forn=m=1.Goodagreementisshown.

In order to assess the relevance of the measured constitutive stress-strain relationshipsoftheadhesive layer subjectedto pure mode I,puremode II andmixed-modeI/II adhesiveloadings, ex-perimental test results and numerical test predictions are com-paredon theconfiguration oftheSLJ configuration.Asshown, in

Fig. 28, goodagreement is shown in termsof both stiffness and maximumloadbearingcapabilityoftheSLJjointspecimen. Never-theless,thesingle-lapjointconfigurationisknowntobesubmitted significantlymoreinmodeIthaninmodeIIduetotheeccentricity oftheloadpathgeneratingsecondary bendingmomentandlarge peel stresses at both overlap ends. To validate the behavior law undermixed-mode,other experimentaltestsshouldbeconducted basedonvariousloadingandgeometricalconfigurations.

5. Conclusion

In thispaper, adirect method fortheassessment ofthe CZM parameters ofa thinadhesivelayer ispresentedandthen imple-mented. This method is based on the measurement of the dis-placement field of adherends at the crack tip of classical adhe-sively bondedspecimens(i.e.:ENF,DCB,MMB),allowing forpure mode I, pure mode II and mixed-mode I/II loadings. Neverthe-less, the identificationpresentedremains dependentof the mod-ellingframework.The experimentalimplementationmakesuseof amethacrylate-basedtwo-componentadhesivepaste,foundunder the commercialreference SAF30MIB manufactured by AEC Poly-mers / Bostik(ARKEMA Group). The adherendsare made in alu-minum alloy(6060series). Theadhesive constitutivestress-strain relationshipsare derived fromthemonitoringof theevolution at crack tip of both the relative displacement of interfaces andthe displacement field ofthe adherend,using theDIC technique. The main difficulty encountered within the experimental implemen-tation concerns the experimental measurements. Indeed, a dedi-cateddatapreprocessing(seeAppendixB)isdevelopedtobestfit theexperimental datacomingfromtheDICtechnique.The useof experimental measurement providing a higherresolution such as Speckleinterferometrycouldbemoresuitable.

In pure mode Iand pure mode II, it isshown that the adhe-sivelayerexperiences threedistinctphases. Thefirstone,the lin-ear elastic phase, appears asextremely limited compared to the whole deforming capability of the adhesive layer. A mathemati-calmodelisthenprovidedforeachmode.Undermixed-mode,the datapreprocessingfailsininterpretingtheexperimental measure-ments,sothatadedicatedmethodissuggested.Themathematical models arethenimplemented toperformnumericaltestsusinga simplifiedstressanalysisbasedonME.Intermsofglobalbehavior, the predictions of numericaltests are ina close agreement with the resultsof experimental tests, up to thefinal failure of speci-mens.Besides,itisindicated thattheidentificationofCZM prop-erties presented inthis paperinvolves specimens, the adherends of which experienced plasticization. Evenif the simplified stress analysis basedon ME allowing fornumerical testssupports both nonlinearadhesiveandadherendmaterialbehavior,theembedded levelofcomplexityintheexperimental testprocedureappears as elevated.Theimplementationofthedirectmethodshouldbethen tested through various combinations of adherends and adhesive materials andvarious geometries, some of which should prevent theadherendstoplasticize,toassessthereliability ofthe experi-mentalprocedure.Theeffectoftheadhesivethicknessonthe ma-teriallawcouldbeinvestigatedusingmeasurement meanswitha betterperformance.

Acknowledgment

The authors gratefullyacknowledge Mr Christian Bret, CEO of AECPolymers,forthesupplyingofmaterials.

AppendixA

Consideringthe local equilibriumequationsEq. (8),the adhe-sivestressesare replacedby theirexpressions asfunctionsof ad-herenddisplacementsEq.(9).InconjunctionEq.(10),itresultsin asystemoftwelvelinearfirst-orderordinarydifferential:

⎧

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎨

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎪

⎩

du1 dx = D1 1N1 + B1 1M1 du2 dx = D2 2N1 + B2 2M2 dv1 dx =

θ

1 dv2 dx =

θ

2 dθ1 dx = B1 1N1 + A1 1M1 dθ2 dx = B2 2N2 + A2 2M2 dN1 dx =bkIIu1 +bkIIh1

θ

1 − bkIIu2 +bkIIh2

θ

2 dN2 dx =−bkIIu1 − bkIIh1

θ

1 +bkIIu2 − bkIIh2

θ

2 dV1 dx =bkI

v

1 − bkI

v

2 dV2 dx =−bkI

v

1 +bkI

v

2 dM1 dx =bh1 kIIu1 +bkIIh1 h1

θ

1 − bh1 kIIu2 +bkIIh1 h2

θ

2 − V1 dM2 dx =bh2 kIIu1 +bkIIh2 h2

θ

1 − bh2 kIIu2 +bkIIh2 h2

θ

2 − V2 (A1)

Thissystemcanbewrittenas dX

dx=AX whereAis12× 12

ma-trix with real constant components and the unknown vector X

suchthatt_X=(u

1 u2 v1 v2

θ

1

θ

2 N1 N2 V1 V2 M1 M2 ).Butthe el-ementarystiffnessmatrixcorrespondstotherelationshipbetween thevector of nodalforces andthevector of nodaldisplacements (Paroissien,2006a;Paroissienetal.,2006b,2007,2013;Léliasetal., 2015),suchas:

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

−N1

(

0

)

−N2

(

0

)

N₁

(

)

N2

(

)

−V1

(

0

)

−V2

(

0

)

V1

(

)

V₂

(

)

−M1

(

0

)

−M2

(

0

)

M1

(

)

M2

(

)

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

=KBBe

⎛

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎜

⎝

u1

(

0

)

u2

(

0

)

u₁

(

)

u2

(

)

v

1

(

0

)

v

2

(

0

)

v

1

(

)

v

2

(

)

θ

1

(

0

)

θ

2

(

0

)

θ

1

(

)

θ

2

(

)

⎞

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎟

⎠

(A2)

ThefundamentalmatrixofA,termed



A,iscomputedatx=0

andx₌



;usingtheSCILAB software,theassociatedcommand is “expm”:



A

(

x=0

)

=expm

(

A.0

)



A

(

x=

)

=expm

(

A.

)

(A3)

Fromtheseboth12∗12matrices,twomatricesM’andN’are ex-tracted.M’(N’)iscomposedofthelinesrelatedtothenodal dis-placements(forces). Foreach, afirstblock ofsixlinesandtwelve

rowscomes from



A(x=0)andthesecond blockofsixlinesand

twelverowscomefrom



A(x=



),suchthat:

⎧

⎪

⎪

⎨

⎪

⎪

⎩

M=



U

(

0,

)

=



[



A

(

x=0

)

]i=1 ,2 ,3 ,4 ,5 ,6 ; j=1:12 [



A

(

x=

)

]i=1 ,2 ,3 ,4 ,5 ,6 ; j=1:12



N=



F

(

0,

)

=



[



A

(

x=0

)

]i=7 ,8 ,9 ,10 ,11 ,12 ; j=1:12 [



A

(

x=

)

]i=7 ,8 ,9 ,10 ,11 ,12 ; j=1:12



(A4)

wherei(j)indicatestheline(row)number.

As KBBeisdefinedaccordingto([u1 (0)u2 (0)u1 (



)u2 (



)v1 (0)

v₂ (0)v₁ (



)v₂ (



)

θ

₁ (0)

θ

₂ (0)

θ

₁ (



)

θ

₂ (



)]),asimple rearrange-mentoftheorderoflinesofM’isperformedtoproducethe ma-trixM. Similarly, the matrix N’ is submitted to the same opera-tion. In a same way, the terms related to nodal forces at x₌0

aremultiplied by −1 to followthe arrangement([−N1(0) −N2(0)

N1(



)N2(



)−V1(0)−V2 (0)V1 (



)V2 (



)−M1 (0)−M2 (0)M1 (



)

M2(



)]).ItleadstothedefinitionofthematrixN.Theelementary stiffnessmatrixKBBe isequaltothe productofN andtheinverse

ofM(Paroissienetal.,2013;Léliasetal.,2015):KBBe=N.M−1 .

Evenifit isnot the topicofthispaper,it isobviousthat this previousapproachcanbeeasilyusedtodevelopME,under differ-entlocalequilibriumequations(e.g.Hart-Smith(Hart-Smith1973; LuoandTong2009;Paroissienetal.,2018)orunderdifferent con-stitutiveequations(e.g.Tsaietal.,1998)and/orincludingdifferent numberlayers of adhesives and adherends (e.g. double lap joint configuration).It is indicated that the resolution usingthe expo-nentialmatrixwasalreadybeenusedinpreviousworks(Gustafson etal.,2006;GustafsonandWaas,2007;Gustafson,2018;Gustafson andWaas,2009;StapletonandWaas,2007;Stapletonetal.,2010; Stapleton,2012;Stapletonetal., 2012;Stapletonetal., 2017).The useoftheresolutionschemeusingtheexponentialmatrixis suit-ableinthecaseofnonlinearanalysissinceameshisrequired.Itis suitableinthe caseofnon-homogeneouselastic adhesive proper-tiestoo(Paroissienetal.,2018).Besides,itisusefulforthe

formu-lationofnewmacro-elementsundervarioussimplifiedhypotheses (Paroissienetal.,2018).

AppendixB

The data pre-processingalgorithm used to reduce experimen-tal noises from the measured adherend-to-adherend displace-ment fieldsthen lieson the digitalmapping ofthe adherend-to-adherendaxial(transverse)displacementfieldsasasetof2D ma-trices. First, the evolution of the axial (transverse) displacement field of each adherend is mapped as 3D tensors resuming both thedistributions oftheadherendaxial(transverse) displacements nearby the adhesive crack tip as well as their respective evolu-tions. Then, the constructed 3D tensors of dimensionsx,y and t

are rearranged in the form ofsimpler 2D matrices so that their newdimensionsarerespectivelyyandx∗t(seeFig.B1).The con-structed 2D matricesarethen filteredusingthe rank-R reduction approximationbased ontheSVDoftherawexperimentalresults, so thatR ischosen to capture95%of theoriginal dataenergyin the sense of the Frobenius norm (see Fig. B2). The evolution of each adherend axial and transverse displacement fields are then reconstructedfromtheirrespectivedecompositionsandrearranged intheformof3Dtensors,sothat thedisplacementsoftheupper (lower) neutral fiberare finally extractedfrom the reconstructed axial and transverse displacement fields and formatted into the relevant beamor platetheory (see Fig. B3). Finally, the differen-tiationoftheadherendscross-sectionrotationisensuredbyfitting apolynomialseriessothattheverticaldeviationwith experimen-taldataisminimizedinthesenseoftheleastsquaresmethodby usingtheMoore–Penrosepseudoinversetechnique.

Besides, afull factorialdesign of experimentsis performedin ordertoassessthealgorithmicparametersontheaccuracyofthe measure.Itconsistsinthefollowing:(i)varyonefactoratatime, (ii) performexperimentsforalllevelsandcombinationoflevelsfor allfactors,(iii)henceperformalargenumberofexperiments,(iv)

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