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Numerical analysis of shear stiffness of an entangled
cross-linked fibrous material
Fadhel Chatti, Christophe Bouvet, Guilhem Michon, Dominique Poquillon
To cite this version:
Fadhel Chatti, Christophe Bouvet, Guilhem Michon, Dominique Poquillon. Numerical analysis of
shear stiffness of an entangled cross-linked fibrous material. International Journal of Solids and
Struc-tures, Elsevier, 2020, 184, pp.221-232. �10.1016/j.ijsolstr.2018.12.001�. �hal-02430768�
an author's
https://oatao.univ-toulouse.fr/21848
https://doi.org/10.1016/j.ijsolstr.2018.12.001
Chatti, Fadhel and Bouvet, Christophe and Michon, Guilhem and Poquillon, Dominique Numerical analysis of shear
stiffness of an entangled cross-linked fibrous material. (2020) International Journal of Solids and Structures, 184.
221-232. ISSN 0020-7683
Numerical
analysis
of
shear
stiffness
of
an
entangled
cross-linked
fibrous
material
Fadhel
Chatti
a,
Christophe
Bouvet
a,
Guilhem
Michon
a,
Dominique
Poquillon
b,∗ a Institut Clément Ader, CNRS UMR 5312, Université de Toulouse, ISAE-SUPAERO, 3 Rue Caroline Aigle, Toulouse 31400, France b CIRIMAT, Université de Toulouse, INP-ENSIACET, 4, allée Emile Monso, BP 44362, Toulouse Cedex 4, Toulouse 31432, FranceKeywords:
Entangled cross-linked fibres Finite element model Shear stiffness
a
b
s
t
r
a
c
t
Theobjectiveofthispaperistounderstandand studythe effectofmorphologicalparametersonthe shearstiffnessofanentangledcross-linkedfibrousmaterialmadewithcarbonfibreswheresomeofthe contactsare bonded by theepoxyresin. Thiscurrentwork presents a3Dfiniteelement modelusing ABAQUS/Standardinordertocharacterize themechanicalbehaviourofdifferentcarbonfibrenetworks rigidifiedbyepoxycross-links.Numericalsimulationsareachievedonarepresentativevolumeelement (RVE)withtheorientationdistributionofthefibresbasedonatestedsample.Sincenotallthestrands areperfectlyseparated,anequivalentdiameteroffibreisdeterminedtoobtaintherigidity experimen-tallymeasuredinshear.Then,aninvestigationoftheinfluenceofmorphologicaldescriptors,suchasthe distancebetweencross-links,distributionfibreorientationsand junctionproperties,iscarriedout.For theentangledcross-linkedfibrousmaterialwithasmallfibrevolumefraction,therelationshipbetween theshearstiffnessandthefibrevolumefractionisalinearfunctionwhereastherelationbetweenthe shearstiffnessandthedistancebetweenjunctionsisapowerlawwithexponentof−3/2.Theshear stiff-nessdependsslightlyonthetwistingjointstiffness,anditsrelationshipwiththetensionjointstiffnessis alogarithmicfunction.TheeffectoffibrestiffnessisalsoinvestigatedbytakingYoung’smodulusvalues correspondingtothoseofglassfibres,inoxfibresoraramidfibres.Alinearfunctionisobtainedbetween theshearstiffnessandtheYoung’smodulus.Theseresultsareconsistentwiththeanalyticalmodelsin theliteratureforacross-linkedfibrousmaterial.
1. Introduction
Compositesareincreasinglyusedinindustrialapplicationsand especiallyintheaerospacefieldduetotheirattractivestiffnessto weight ratios (Castanié et al., 2008; Belouettar etal., 2009). The sandwichstructureoccupiesanimportantplaceinthe manufactur-ing ofcomposite parts.Its composition of tworigid skinsspaced by a light core ofgreat thickness andlow stiffness offers a high stiffness for bending applications. Numerous core materials with differentconfigurationshavebeendesignedandstudiedinthe lit-erature(Mezeixetal.,2009).Thehoneycombisstillthemostused corematerialforthesandwichstructurebecauseitoffersthebest stiffness to weight ratio. However, it presents many drawbacks, suchasthedifficultyimplementingitincomplexstructures,closed porosity anddifficult qualitycontrol processes. Thelow vibration dampingisstillthemostimportantproblemwiththehoneycomb
∗ Corresponding author.
E-mail addresses: fadhel.chatti@isae.fr (F. Chatti), christophe.bouvet@isae.fr (C. Bouvet), guilhem.michon@isae.fr (G. Michon), dominique.poquillon@ensiacet.fr (D. Poquillon).
structure,andmanymethodshavebeenpresentedtoenhancethe dampingofthesandwichstructure(Caietal.,2004;Fotsingetal., 2013;Li,2006).
Based on the study of Poquillon et al. (2005),
Mezeix et al. (2009),(2015a) developed a new entangled
cross-linked material that can offer good vibration damping (Piolletetal., 2016) duetothe energydissipatedthrough friction betweenfibres.The honeycombcannot yetbe substitutedby this newmaterial in theaerospace field because its shearstiffness is very low.The numerical modelling of theentangled cross-linked material seems necessary tounderstand and studyits behaviour. Whereas the honeycomb has been significantly studied in the literature(SilvaandGibson,1997; KallmesandCorte,1960), only a few investigations have been conducted to study such fibrous materials.
The creation of models aimed at enhancing the understand-ing ofthe mechanical behaviour of fibre assemblies startedwith analytical models (Gibson and Ashby, 1997; vanWyk, 1946). Al-though the previous analytical models were confined to rather simpleuniformarrays,theyprovidedataofgreatinterest.In1952, Cox(Cox,1952)proposed ananalytical modelinwhichthefibres
were not supposed to interactwith each other,extend fromone edgeofthenetworktotheother andtransmit onlyanaxialload. Hismodelderived theelastic coefficientsasa function of the fi-breorientation distribution. According to Cox’s studies and fora three-dimensionalisotropicassembly,theresultsare:
Theelasticmodulus: E = k
6 (1)
Theshearmodulus: G = k
15 (2)
where k =
ρ
sρ
fE f
and
ρ
s,ρ
f andEf denotethefibrous materialdensity, thedensity ofthefibreandtheelasticmodulusoffibre,respectively.The pre-dictedstiffnessvaluescouldbeviewedasanupperlimitthat can-notbereachedinrealfibrenetworks.Van denAkker (1962)assumed that the non-bonded parts of
the fibres can also support shear and bending, apart from axial strain.Heconsideredthebondstoberigidandtorotatetoan ex-tent equal to the mean of the rotations of the two cross-linked fibres. He concluded that two groups of stress cause rupture of fibre-to-fibre bonds: those associated with tension in the fibres andthose associatedwith torqueon the bonds dueto the shear forceinthefibresegments.
In 1960, Kallmesand Corte(1960) presenteda model dealing withthemorphology offibre networks. Theirmodelwasapplied tothe studyof paper. The authorsassumed that the fibreswere independentlydepositedofeachotherandthefibrenetworkhada uniformorientationdistribution.Theystatedthatthegeometric ar-rangementofthefibreswasaneffectofthepapermakingprocess andthecauseofthepaper’sproperties.Therelationshipsbetween thepaperpropertiesandthedifferentmorphologicalpropertiesof theassembly, suchastheaveragelengthofthestraightsegments betweencontacts,thenumberoffibre crossings,thefibrevolume fractionand the elastic properties of the fibres, were presented. Amongits results,the followingequation established theaverage numberoffibrescrossingsinasquareofsidelength,L:
n c=
n fl f 2 c 2 L 2π
(3)wherenfisthenumberoffibres,lfistheaveragelengthbetween contactsand cdenotes theaverage curl index, which is the dis-tancebetweenthefibreendpointsdividedbythefibrelength.
AnotherequationfromKallmesandCorteprovidestheaverage lengthlsoffreesegmentwithoutcontactonafibre:
l s=
n fl fc 2n c
(4) Kallmes et al. (1963) proposed an investigation based on the calculationsofVandenAkker(1962)thatemphasizedtheeffectof thefreeandbondedpartsofthefibresegments.
Eqs.(3)and(4)havebeenpresentedfor3D fibre networksby KomoriandMakishima (1977).Their theoriesusedcurved beams andconsidered theinfluenceofcurliness,fibresliding andcrimp. Althoughtheir modelqualitativelypresentsthebehaviouroffibre assembly,itsapplicability islimitedbecauseitassumesaffine de-formationoftheinter-fibrecontactpoints.
The fibre network made by inducing cross-links at the fibre contact can be considered to be a particular kind of foam-like material.Thus, the Ashbyand Gibson’s methods, concerned with the modelling ofopen cellular solids (Gibson and Ashby, 1997b; Ashbyetal., 2000),canbe usedforthefibre cross-linked assem-bly.The authors model is basedon beam deflections.Ashby and
Gibsonproposedanexpressioncalculatingthecompression modu-lusofopencellfoam:
E = 3
π
E f 4LD
4 (5)whereEfistheYoung’smodulusoffibre,Listhedistancebetween jointsandDisthefibrediameter.
Heyden (2000) developed a finite element model for the ge-ometrical linear deformation of cellulose fibre assemblies with an isotropicorientation distribution.The fibresweremodelled by beamelementswithaconstantcurvatureandintroducedina pe-riodicrepresentativevolume. Thefibre contactswere represented byasetofspringsthatallowedstick-slip.Theywerebondedinthe sensethat oncecreated,thespringscannotbe broken.Themodel providedsimulationresultsthatwereingoodagreementwith ex-perimentresults.However,itwastestedonlyforlowfibrevolume fraction,f=3%.
Delince and Delannay (2004) developed an analytical model
that was based on a periodic network architecture to deter-mine the elastic properties of transverse isotropic steel fibre ar-rays. The Young’s moduli calculated from their model were be-low experimental values. They assumed that this difference was due to the negligence of triangulation effects in the model. Delannay(2005)proposedanewmodelthataccountsforthe neg-ative Poisson ratio behaviour and the random triangulation. His model predicted the shear stiffness, G13, properly. However, no
such agreement was verified for the predictions of the in-plane modulus, E1, and the out-of-plane modulus, E3, which were
no-tably lower thanthe measured values.Evenifthemodeldid not provide a full quantitative agreement with the experiment re-sults,it can offer indications foroptimizing thedesign of the fi-breorientationdistributionthat cangivethemostsuitable elastic anisotropyfortheaimedapplication.Itfollowsthatthepredictions ofthenewmodelforthein-planeshearmodulus,G13,appearedto agreewiththevaluemeasuredforthereferencematerial.
A simple analytical model was developed by Markaki and Clyne(2005)topredictthemechanicalbehaviour ofnon-periodic bondedfibreassemblieswhensubjectedtoeithermagneticor me-chanical loads. This model assumed that the bending of the in-dividual fibres dominated the deformation. None of the changes in morphology duringcompression wastakeninto account. Only theinitial geometryofthematerial wasconsidered.It can there-fore be applied only to cases in which the fibre segments be-tween thebondsare slender(L/D≥ 3).An expressionwasderived for the Young’s modulus, andit is similar to that of the Gibson andAshby’smodel(GibsonandAshby,1997b;Ashbyetal.,2000) thatisbasedonaregularandorthogonalsetoffibres,exceptthat thefibre segmentaspectratio,L/D,andthe fibrevolume fraction,
f,canbe independentlyspecified.Comparedwithexperiment,the resultsprovidedby thismodelwere limitedduetothedifficulties inreliablyestimatingL/D.
Mezeixetal.(2015b)developedananalyticalmodelto investi-gatethebehaviourofentangledcross-linkedfibresunderatensile andcompressiveload.LiketheMarkakiandClyne(2005),theform oftheirpredictionwasbasedonthefibresbending.However,the cross-links (epoxy joints) were modelledby twisting springs that wereappliedattheextremityofthebeam.Thismodelconsidered the additional rigidity in tension due to the quasi-vertical fibres thatmadethestiffnessintensionhigherthantheonein compres-sion. The quasi-vertical fibres were considered to buckle quickly duringthecompressiontest.
Recently, a 3D periodic beam model was proposed by Ma etal.(2018) toinvestigatethe elasto-plastic characteristicsof porous metal fibre sintered sheets (MFSSs).The establishment of the finite element model was carried out progressively with the
2Dcoordinatesconsideredfirst.Bypilingupthefibresonebyone until Nfibres were layered inthe model,the growthinthe out-of-planedimension wasconsidered.At thefibreintersections, ad-ditional beamelementswere inserted tomodel the cross-linkers whosedensityhadasignificantrole inboththestrength andthe stiffness of the stochastic fibre network. The simulation results showedgoodagreementwiththedimensionalanalysisand exper-imentaldata(Jinetal.,2013).
In the literature, the previous analytical models, investigating thestiffnessofafibrousmaterialwithpermanentjunctions,donot considerthevariationsinthemorphology oftheassemblyduring mechanicalloading.In addition,thecompressionstiffnessofa fi-brous material wasmuch more studied than the shear stiffness, whichcanbemoreimportantwhenthefibrousmaterialisusedas acorematerialinasandwichstructure,such asfortheentangled cross-linkedfibrousmaterialofthepresentstudy.
Theobjectiveofthisworkistoinvestigatetheshearstiffnessof an entangledcross-linkedfibrous materialandthe dependenceof the shearmodulus onthe morphologicalparameters,such asthe distancebetweenthejunctions,thefibrevolumefraction,thejoint stiffnessandthedistributionofthefibreorientations.
2. Simulationmethods
An entangled cross-linked material can be made by different typesoffibres,suchascarbon,glassoraramid(Mezeixetal.,2009, 2015a), that were cross-linked by a thermo-hardening resin. Car-bonfibreswere mainlyusedto manufacturetheentangled cross-linked material in order to obtain a higher mechanical perfor-mance, thoughits costis higher.Filamentcarbonis characterised by a diameterof7μm,an elasticmodulus of240GPaanda bulk density of 1770kg/m3. Epoxy was chosen as the resin for
cross-linking because it is widely used in aerospace applications. The manufacturingprocessofthesematerialswascarriedoutmanually atthe laboratory scaleand hasbeendetailed inprevious studies (Mezeixetal., 2009,2015a).Theyarnswere cuttoa fixedlength (12mm),introduced ina 64lparallelepipedboxinwhicha com-pressed airflow (6 bars)wasapplied toseparate the fibres. Sep-arationandentanglementwerecarriedout together.Thematerial thus obtainedwasexposed to an epoxy sprayto obtain attached epoxy droplets insome ofcontacts betweenfibres.Agivenmass ofthismaterialwasthenplacedintoamouldforthelaststep of theprocess:thepolymerizationoftheepoxyresin(8hintoa tem-peraturechamberat70°C).Thesampleswerethendemoulded.An image of themanufactured material’s microstructure isshownin Fig.1.
The numerical modelling wasbased on experimental tests in which a sample of entangled cross-linked material, with a size of 20× 40× 60 mm3, wassolicited in shear (Piollet etal., 2016).
The whole sample contained more than three million carbon fi-bres fora fibre volume fraction of8.5%. Inthis case, the numer-ical modellingofan entiresamplewouldbe complicatedand in-curahighcomputationalcost. Therefore,a representativevolume element (RVE)waschosen forthemodel.Its side dimension was 1mm,whichisthemostappropriatechoiceto guaranteethe sta-bilityofthemorphologicalparameters(distancebetweencontacts, distribution of fibre orientations) ineach draw witha minimum of computational cost (Chatti et al., 2018a). The architecture of the RVEwasgeneratedfroman in-housepre-processingprogram coded inFORTRANandpresented inprevious work (Chatti etal., 2018a,2018b).
Thisnumerical modellingconsidered theinitial distributionof fibre orientations that were close to the architecture of the real sample before loading. Indeed, when the fibres were packed in the mould, the entangled fibres lost their initial isotropic distri-bution,soatransversallyisotropicmaterialwasobtainedafterthe
Fig. 1. SEM observation of entangled cross-linked carbon fibres with a volume frac- tion, f , of 8.5%; The Blue circles indicate some of epoxy cross-links.
polymerisationstep.Then,anumericaltechniquewasdevelopedto identifythefibredirectiondistributionafterthemouldingprocess. Fig. 2 shows the distribution of fibre orientations (green curve) that was used to represent the architecture of the real sample, and it is compared to the theoretical isotropic fibre distribution (blackcurveinFig.2).Inthenumericalanisotropicfibre distribu-tion,therearemorefibreswithdirectionscharacterizedbyan an-gle
θ
closeto90°.Thisanglecorrespondstotheperpendicular di-rectiontotheverticalaxisanditdescribesthequasi-horizontality offibres afterbeing packagedin themould. Moredetails ofthis techniquecanbefoundinthepreviouspaper(Chattietal.,2018b). ThefiniteelementsolverinABAQUS/Standardwasusedinthe actualworktoaccuratelydeterminetheshearstress.Ituses tech-nologies that are ideal for the determination of highly accurate stress solutions.Eachfibre wasmodelledby a seriesof3D beam elements(ABAQUSelementtypeB31)dependingonboththe num-berofintersectionsbetweenthefibresandontheir orientationin theRVEthatdeterminedtheirlengthinthecomputation.This ele-menttypeisbasedontheTimoshenkobeamtheorythatallowsfor sheardeformation.Thestrainsconsideredhereareofsmall ampli-tude,lessthan 1%.Thedominantmode ofdeformationofthe as-semblyisthebendingoftheelasticcarbonfibres.Forthisreason, it isthe tensile/compressive behaviour ofindividual fibresin the fibredirectionthatmustbe takenintoaccount. Moreover,evenif thetransversestiffnessofthefibreisdifferentofthelongitudinal one, this transverse behaviour does not affect the bending stiff-ness of fibre. Then, based on previous work (Piollet et al., 2016; Chattietal., 2018a), thebehaviour ofthe fibreshasbeen simpli-fied and each fibre is assumed to have an isotropic linear elas-ticbehaviour.Springs were introduced in themodel through the userelementtomodeltheepoxycross-link.Inthepreviouspaper (Chattietal.,2018a),afiniteelementmodeloftwofibresthatwere bondedbyepoxyjunctionwasusedtodeterminethestiffnessthat shouldbeintroducedinthespringandcandescribethebehaviour oftheepoxyjunction.Thematerialpropertiesoftheepoxy cross-linkswereassumedtobeanisotropiclinearmaterialwithYoung’s modulusE=5GPaandPoissonratioν
=0.3. Attheendofthe nu-mericalinvestigation,twostiffnessesweredetermined.Thetension stiffness,Kte,wasequalto 0.4N/mm,whilethe twistingstiffness,Ktw,wasequalto3.10−4N.mm(Chatti etal.,2018a). Each simula-tionused fourparallel processors and12 GBofmemory intotal, andit lasted a few minutes. A few thousand carbon fibreswere
Fig. 2. Different fibre orientation distributions. The solid black curve (sinus) represents the theoretical distribution obtained in the case of a perfectly random orientation of the fibres in 3D, i.e. an isotropic distribution of the fibre assembly. The solid green curve represents the anisotropic distribution that is closer to the experimental sample ( Chatti et al., 2018 b). The angle θis the angle between the fibre direction and the vertical axis, z .
Fig. 3. Three-dimensional view of the RVE with the size of 1 × 1 × 1 mm 3 . Kine-
matic coupling, of the nodes of the upper and bottom face with two reference nodes, were used to apply the shear load.
generatedinsidethecubicRVEwithasizeof1× 1× 1mm3in
or-dertoreachatargetedinitialvolumefractionof8.5%.
In the experimental tests, 3mm-thick aluminium plates were glued on each side of the entangled cross-linked sample (Piolletetal.,2016) whiletheother faceswere free.Toapply the boundaryconditions that respected thegeneral conditionsofthe experimental test, the nodes for the upper and bottom faces of the RVEwere constrained to the upper referencenode and bot-tomreferencenode,respectively,asillustratedinFig.3.Thesetwo referencenodesgoverned themotion ofthe nodesforthe upper andbottomfaces.Inordertoapplytheshearload,theupperone moved to the left while no motion was allowed for the bottom
one. The shear strain applied to the upper reference node was equalto1% sothattheexpressionoftheshearstrainwas
γ
=u/L. Alltheothernodeswerefreetomove.The ABAQUS/Standard (User Documentation) cannot manage theinteraction betweenpairsoffibresandtheinteraction ofone fibre with itself. Indeed,this commercialsoftware basingon im-plicit schema of integration cannot detect the surface ofcontact betweenthe intermediate parts of beamelements;it just allows the interaction betweentheextremities ofbeamelements. How-ever, theother software“ABAQUS/Explicit” (User Documentation) ismoreappropriate forthemanagement ofdifferentcontactsbut we did not use it in thisactual work because the objective was toobtain fasterresultsforthe linearpartofthe behaviourofthe entangledcross-linked fibre material,which corresponds tosmall deformations andduringwhich the morphology of the assembly changeslittleasdoesthenumberofcontacts(Piolletetal.,2016; Gibson andAshby,1997b;Markaki andClyne,2005; Chatti etal., 2018a). So, ABAQUS/Standard is the best candidate in this case, whentherigidityofentangledcross-linkedmaterialismainlydue tothefibrenetworkandtotheepoxybonds(Mezeixetal.,2009). Thetotaltimeperiodusedintheloadingstepwasequalto1s. Itis afictivetimethat doesnotimpactthe elasticlinear calcula-tion.The initialtime incrementchosen was0.1s.The time incre-mentallowedwasbetween1μsand0.1s.
3. Resultsanddiscussion 3.1. Sheartest
First,the1× 1× 1mm3cubicRVE,thatcontained2430carbon
fibres witha diameter of7μm andYoung’s modulus of240GPa, was solicited in shear. The initial volume fraction was 8.5%. The numerical fibre diameter corresponded to the real carbon fibre, which wasequalto 7μm.Thus, a numerical caseofperfectyarn separation was treated. The anisotropic fibre orientation (Fig. 2) wasgenerated in thisRVE following the previous data. This mix ofmorphologicalparametersprovidedaround25,000contacts be-tweenthefibresandmorethan60,000beamelements. The tech-nique used to block the contacts (thus positioning the springs)
Fig. 4. The view in front of the RVE geometry before and after shear loading where γ= 1%.
Fig. 5. Comparison between the experimental curve ( Piollet et al., 2016 ) in black and numerical simulation in green for f = 8.5% and a fibre network with fibre orientation representative of the experimental data. This consisted of perfectly separated strands with a fibre diameter of 7 μm on which two out of three contacts were glued. The black dashed black line corresponds to the value of the linear rigidity of the network identified in ( Piollet et al., 2016 ), where G = 6 MPa.
consistsinblocking agivenproportionofthe contacts.Thisis il-lustrated inFig. 9.In the previously describedconfiguration,two thirdsofthecontactswerebondedwithcross-linksintheshapeof springsinordertogetan averagedistancebetweenthejunctions equalsto122μm.Thisvalueissimilartotheexperimentalvalueof 120+140−70 μmestimatedexperimentallybyMezeixetal.(2015a).
Fig. 5 shows the curve of the shear stress versus the shear straininthecaseofaperfectfibreseparation.Thecurve iselastic linearinview ofthefiniteelementhypothesis.Itisshownwitha greenlineandissteeperthantheexperimentalcurve(Piolletetal., 2016).Theshearmodulusforthiscasewasequalto12MPa,which isabouttwotimesthevalueobtainedduringexperimentaltests.
This difference can be explained by the yarns that were not well separatedduring the manufacturing of theentangled cross-linked samples (Fig. 6). The entangled cross-linked material was heterogeneousandallyarnswerenotperfectlyseparatedafterthe
entanglementstep.ThemodellingoftheyarnsintheRVEwas dif-ficultduetothelackofthenecessarydataaboutthesizeandthe positionoftheseyarnsinthesample.Duetothesmallsizeofthe fibre,itwasnotpossibleuntilnowtogetstatisticaldataforaRVE throughthe3Dtomographywork.Then,thequantity andthesize ofthe remaining yarnswere not known enough toget statistical dataformodellingtheRVE.Furthermore,thedevelopednumerical approachallowedtheuseofonlybeamswithahomogeneous di-ameter.Itwasthencomplicatedtomodelthenon-separatedyarns andtheseremainingyarnsarebeamsofvariablediameters(twoto tenfibrescanbejoinedtogether)andofvariablestiffness.The as-sumptionmadeisthefollowing:thisdistributioncanbereplaced byhavinga beamwitha diametergreaterthan theoneofa sin-glefibre,whosecross-sectionandrigidityareequaltothatoftwo adjacent fibres.This equivalence is explained in Fig. 7. A homo-geneousfibre withadiameterDh of7
√
modu-Fig. 6. The SEM observation of an entangled cross-linked material made with car- bon fibres of Ø 7 μm; some of the yarns still existed and were heterogeneously dispersed.
lusofEh=120GPa, whichisthe halfofthat fora realcarbon fi-bre,wasusedtotarget abendingstiffnessequaltothatofoneof yarnthatiscomposedoftwofibres.Thisassumptionwas success-fullytestedandusedtomodelthehysteresisloop intheprevious work(Chattiet al., 2018). As thedeformation ofthe material re-sultsmainlyfrom thebending ofthebeams (Markaki andClyne, 2005;van Wyk,1946), thedecrease ofthe axial stiffnessof seg-mentsby half,hasnot an importanteffectonthe generalresults becausethe stretching can be neglected against the bending de-formation(GibsonandAshby,1997;MarkakiandClyne,2005;van Wyk,1946).
In this new case, 1230 carbon fibres were generated in the RVEtoobtaina fibrevolume fractionequalto8.5%.The
distribu-tion offibre orientationsis thesame than inthe previous calcu-lations and induced anisotropy,as illustrated in Fig. 2.The aver-agedistancebetweenthejunctionswasaround160μm.This aver-agedistance wasobtainedby blocking two out ofthree contacts (Fig.9-b).Around6000among9000contactswereblockedbythe springs. This value of160μm is comparableto the experimental value of120+140−70 μmreportedbyMezeixetal.(2009)butslightly larger.
Fig.8showscurve ofthe shearstress versusshear strain that allowed the evaluationofthe shearmodulus in thecaseof non-separatedfibres.Thecomparisonbetweenthenumericalcurveand experimental data indicates a good agreement in terms of shear stiffness. Numerically, the shear modulus value is of the same order of magnitude as the experimental one. It is equal around 7MPa,whereas the experimental value isabout6MPa (Silva and Gibson,1997).
In order to check the repeatability of the results of the nu-mericalsimulation,threedifferentgeometriesweregenerated. In-deed,inthepreviousworks(Chattietal.,2018a,2018b),wehave checked the volume chosen forthe simulation (a cubic millime-tre), was a RVE in the morphological sense (Kanit et al., 2003). Itwasthen importanttoverifythat wehada representative ele-mentaryvolumeformechanics.Table1presentstheparametersof eachdrawandthevaluesoftheshearmoduluscalculatedineach case.Numericalvaluesareveryclosewithasmalldispersion that can beexplainedby the dispersionbetweenthedraws that were generated.Inthissmalldispersion,thenumericalresultsshowthat theshearstiffnessincreasedlinearlywiththefibrevolumefraction and it increased non-linearlywith the average distance between junctions.Theseobservationsagreewiththepreviousstudiesof fi-brousmaterials.
Inthe followingsections,some morphologicalparameters will bemodifiedinthegeometryoftheRVEmadewithhomogeneous fibresin orderto studythe evolution ofthe shearmodulus. Five parameters will be involved: the fibre volume fraction, the ini-tialdistributionoffibreorientations,theaveragedistancebetween cross-links, thejointtensionstiffnessandthe jointtwisting stiff-ness.
Fig. 7. Technique used to consider the non-separated fibres. A big fibre with a diameter D h = 7
√
2 μm is equivalent to two non-separated fibres with a diameter 7 μm in terms of bending stiffness.
Fig. 8. Comparison between experimental data and numerical results for a fibre volume fraction f = 8.5%. The blue curve corresponds to the simulation using a fibre diameter equal to D h = D
√
2 = 7 √ 2 μm and a Young’s modulus E h = E/2 = 120 GPa. The green curve corresponds to the simulation using a fibre diameter equal to D = 7 μm and a
Young’s modulus E = 240 GPa ( Fig. 7 ). Table 1
Details of the three different simulations carried out with three different random draws of the position of the fibres. Number of beam elements Fibre Volume fraction (%) Number of cross-links Average distance between non-bonded contacts (μm) Average distance between cross-links (μm) Shear modulus G (MPa) RVE 1 21646 8.58 6115 330 165 7.2 RVE 2 21543 8.45 5952 336 168 6.8 RVE 3 21608 8.51 6082 334 167 7
3.2. Effectofmorphologicalparametersontheshearmodulus 3.2.1. Effectofthedistancebetweencross-linksontheshearmodulus fordifferentdistributionsoffibreorientations
Thedistancebetweenthecross-linksisconsideredtobeoneof theprincipalparametersrelatedtothestiffnessofafibrous mate-rial.Itdependsonthequantityofpulverizedresinandonthefibre volumefraction.
Inthisparagraph,theinfluenceofthedistancebetween cross-linksfordifferentdistributions offibre orientationswillbe evalu-ated.Thevaluesofothermorphologicalparameterswerefixed.The initialfibrevolumefractionwasequalto8.5%,thetensionstiffness,
Kte,wasequal to 0.4N/mmwhile the twisting stiffness, Ktw, was equalto3.10−4N.mm.
In thiswork, fourinitial distributions ofthefibre orientations werestudied,asillustratedinFig.10-b.Theredcolourcurve rep-resentsthedistributionobtainedinthecaseofarandom orienta-tion of the fibresin 3D, i.e.an isotropicdistribution of the fibre assembly. The three other distributions were anisotropic charac-terised bymorefibreswithhorizontaldirections (theshear load-ingdirectionintheFig.3).InFig.10-a,fourcurvesareplottedand each curve wasformed byfour pointscorrespondingto theratio ofthebondedcontact.Themethodthatwasusedtoperformeach ratio ispresented inthe Fig.9 (a, b,c andd), whichcorrespond tothefourratiosoftheboundedcontact,namely1/1,2/3,1/2and 1/3,respectively.
In Fig. 10, four curves present the shear modulus as a func-tion of distance between the cross-links, Dc. When the last one
increased, the shear modulus decreased for thefour cases. More precisely,forthefourassembliesoffibrestestedwithfour differ-ent fibre orientation distributions, the results between the shear stiffness G and the distance between junctions D seem to fol-low a power law G=a.Db with an exponent b=−3/2. This
ex-ponent is slightly lower than the one deducted by Markaki and Clyne(2005) (−2). This may be due to our boundary conditions at the level of the junctions which are more flexible than the infinite rigid cross-links in the analytical model of Markaki and Clyne(2005). The shear stiffness increasedin the same way, re-gardless of the distribution of the fibre orientations (Fig. 10-a). Thus, the effect of the distance between the cross-links did not change if the distribution of fibre orientations was isotropic or anisotropic.
Theshearmodulusclearlydependedonthetwomorphological parameters, namelythe distribution oforientation fibresand the distancebetweencross-links,asillustratedinFig.10-a.Forthe dis-tributionofexperimentalsamples(thegreencurve),thenumerical simulationresultprovidedashearmodulusequalto16MPawhen eachcontactbetweenthefibreswasbonded.
The highshear modulus value wasobtained when the distri-butionwasisotropicandtheaveragedistancebetweencross-links wasaround110μm,whichcorrespondstothecaseinwhichallthe contactsbetweenthefibreswere gluedby epoxyjunctions.It can reachmorethan50MPa.Thisshearmodulusvalue isstill smaller than for a Nomex honeycomb, which is equal to about 120MPa (Zenkert,1997).However,itisclosetotheshearmodulusoffoam PMI110,whichisequaltoabout50MPa.
Fig. 9. The way used to generate springs linking the contacts of one fibre, which is coloured in green, to the other fibres. These are illustrations in the case where (a) all contacts were blocked, (b) two out of three contacts were blocked, (c) one out of two contacts was blocked and (d) one out of three contacts was blocked.
The manufacturing of an entangled cross-linkedfibrous mate-rial inwhich all contactswould be bonded by epoxy cross-links could increase the stiffness of the material but probably reduce the vibration damping,which is the most important property of thecross-linked material.Acompromise betweenenergy dissipa-tionthroughdryfrictionandshearstiffnessshouldbeconsidered.
3.2.2. Effectofthedistancebetweencross-linksandfibrevolume fractionontheshearmodulus
Inthissection,theinfluenceofthedistancebetweenthe cross-linksontheshearstiffnesswasevaluatedfordifferentfibrevolume fractions.Maintainingthesame sizeofthe RVEas1× 1× 1mm3
andthe same distribution of fibre orientations (Fig. 2), the vari-ation of the fibre volume fractionwas followed necessarily by a variationinthenumberofcontactsandasubsequentvariationin thedistancebetweenjunctions.Asintheprevioussection,four ra-tiosofbondedcontactwerestudied:1/1,2/3,1/2,and1/3.The ef-fectofthedistancebetweenthecross-linksontheshearmodulus fordifferentfibrevolumefractionsisshowninFig.11-a.Three dif-ferentinitialfibre volumefractionswere tested.The smallerfibre volumefraction,whichwastested, wasequalto6.5%.As checked in(Chatti etal., 2018a), a cubicvolume withthesize of1mm is a morphological RVE fora fibre volume fraction equal to 6%. As long as the fibre volume fraction is bigger than 6%, the conclu-sion is the same. For the three fibre volume fractions, f, tested: 6.5%,8.5%and10.5%,asseen intheprevious paragraph,the rela-tionshipbetweentheshearmodulusandthedistancebetweenthe cross-linksisagainapowerlawfunction withan exponentvalue of−3/2(Fig.11-a).Intheconsideredrange,thedependenceofthe shearstiffness, asa function of the distance betweencross-links distance,doesnotseemtodependonthefibrevolumefraction.
Thefibrevolumefractionisamongtheprinciplemorphological parametersinafibrousmaterial.Itcangreatlyinfluencethe char-acteristicsofanentangledcross-linkedmaterial.Itdoesnot influ-enceonlythemechanicalpropertiesbutalsothermicandacoustic properties.Inthiswork,theinfluenceofthefibrevolumefraction on the shearstiffness wasstudied. Concerningthe fibrous mate-rial, the relationship betweenthe stiffnessand the fibre volume fractionisconsidered asone ofthemostimportantpropertiesin practicalapplications.Thefibrevolumefractioncanbeadjustedby changingthenumberoffibresintroducedintheboxoftheRVEfor FEsimulations.
The technique to generate the cross-links illustrated in Fig. 9 does not allow the fixation of the average distance be-tween cross-links. This value is computed in each case and de-pendsonfibrevolumefraction,onfibrediameterandonthe pro-portion of. So, three average distances between cross-links were takenfromthecurveplottedinFig.11-a(Dc=150μm,Dc=205μm,
Dc=260μm). For each average distance betweenthe cross-links, the effect of the fibre volume fraction on the shear modulus is shownasafunctionoffibrevolumefraction(Fig.11-b).Thefinite elementsimulationresultsindicatethattheshearmodulusisa lin-ear function of the fibre volume fraction ofthe entangled cross-linked fibrous material.This resultisconsistent withEq.(2) that providestheshearmodulusproposedbyCox(1952).Itagreesalso withthestiffnessstudyofMarkakiandClyne(2005).Theslopesof thethreecurvesinFig.11-baredifferentbecausetheydependon theaveragedistancebetweencross-links.Theslopeofeach curve increasedwiththedecreaseoftheaveragedistancebetween cross-links.Thus, thefibrevolume fractionimpactedtheshearstiffness moreforthesmallaveragedistancebetweencross-links(Table2). The increase of the fibre volume fraction slightly impacted the
Fig. 10. (a) Influence of the average distance between cross-links on shear modulus for (b) different distributions of fibre orientations and different proportion of bonded contacts (one-third, one-half, two-thirds and one).
Fig. 11. Influence of fibre volume fraction and distance between cross-links on the shear stiffness.
Table 2
Values of slope for different average distance between cross-links:
Dc = 150 μm, D c = 205 μm and D c = 260 μm.
Average distance between cross-links D c (μm) Value of slope(MPa/%)
150 0.77
205 0.75
260 0.45
shear modulus. Nevertheless, it should be chosen carefully so it doesnotimpactthegoodratioweight/stiffnessofmaterial.
3.2.3. Effectofnatureoffibreontheshearmodulusfordifferent distributionsoffibreorientations
Anentangledcross-likedmaterialcan bemanufacturedby dif-ferent types of fibres, such as carbon, stainless steel, aramid or
Table 3
Young’s modulus of different fibre nature: car- bon, Inox, glass and aramid.
Nature Young’s modulus E (GPa) Carbon 240
Stainless steel 180 Glass 73 Aramid 36
glass.Theactual numericalmodel canprovide agood ideaabout theeffectofthefibre type bycomparingfourdifferentmaterials. The study of the effect of the fibre type is always done for the no-perfectseparatedfibrescase(Dh=
√
2D). TheYoung’smodulus foreach kindoffibre ispresented inTable 3.Torespect the ap-proachofhomogenization,halfoftheYoung’smoduluswas
intro-Fig. 12. The effect of Young’s modulus corresponding to different types of fibre (carbon, Inox, glass and aramid) on the shear stiffness of the RVE for a fibre volume fraction of 8.5%; three distributions of fibre orientations were studied (one isotropic and two anisotropic).
Table 4
Values of the slope for three different distributions of fibre orientations presented in Fig. 12 -b.
Distribution of fibre orientations Value of slope (%)
Anisotropic
distribution Green curve ( Fig. 12 -b) 1.3 Purple curve
( Fig. 12 -b)
4.2 Isotropic distribution 12
ducednumericallyinthemodelforeachcase.Thevolumefraction wasequal to8.5% foreach caseandthesame jointstiffnesswas used, namely 0.4N/mm for the spring tension stiffness, Kte, and 3.10−4N.mm for the joint twisting stiffness, Ktw. 2/3 of contacts betweenfibres were bonded by 6000 springs. Thus, the average distancebetween the cross-links wasaround 160μm forall RVE madewiththethreedifferenttypesoffibre.
Theeffectofthetypeoffibreontheshearmodulusforan en-tangled cross-linked fibrous material can be evaluated from the relationship betweenthe shear modulus and the Young’s modu-lus of the fibre. The influence of the type of fibre on the shear moduluswastestedfordifferentdistributionsoffibreorientations (Fig. 12-b). In each one, the shear stiffness increased with the increase of the fibre Young’s modulus. The relationship between the shear stiffness and the Young’s modulus was a linear func-tion,which agrees withthe literature studying the stiffness ofa fibrousmaterial(Cox,1952;GibsonandAshby,1997b;Markakiand Clyne,2005).
The slope of the curve corresponding to the isotropic distri-bution of fibre orientations is the biggest, with a value of 12%. The value of the slope decreased when there were more fibres withhorizontaldirection(anisotropicdistribution),asillustratedin Table4.The intrinsicstiffnessofthefibresseemsto havea more pronouncedeffectontheshearmodulus foran isotropic distribu-tionoffibre orientations.As theanisotropyofthedistribution in-creases(increaseoftheproportionoffibrebetween45° and135°), theshearmodulusdecreasesbuttheeffectoftheintrinsicstiffness ofthematerialalsoseemstodecrease.
3.2.4. Effectofjointtensionstiffnessontheshearmodulus
Inthissection,the influenceofthespringtensionstiffnesson theshearmodulus fordifferentdistances betweenthe cross-links was studied (Fig. 13). The other morphological parameters were fixed. The initial fibre volume fraction was equal to 8.5%, two-thirdsofthecontactswerebondedbyepoxyjunctionsandthe dis-tributionoffibreorientationswasasshowninFig.2(greencurve). Thetwistingstiffnesswasequalto3.10−4N.mm.
For all tested geometries, the shear stiffness of the fibrous material increased with the value of the joint tension stiffness (Fig. 13). The shear stiffness of material can be significantly in-creasedbyusinga higherrigidity resin.Therelationshipbetween theshearmodulusandthejointtensionstiffnessiswellfittedby logarithmicfunction,asillustrated inFig13.However,thechange oftheresinalsocausedamodificationintheviscosity,andthe av-eragedistancebetweenthejunctionsintherealmaterialandthese effectsonthemanufacturingprocess,cannotbemodelledherebut mustbekeptinmind.
Theeffectofthespringstiffnessfordifferentdistancesbetween the junctions is shown in Fig. 13. Firstly, asexpected, when the averagedistancebetweenjunctions, Dc,isshorter,then theshear stiffnessofthematerialishigher.Secondly,themorethedistance betweenjunctions decreases, the more thematerial stiffness can increase when the joint tension stiffness increases, as shown in Fig. 13 (curves I and II). The displacement of the fibres in this case was mainly due to the bending displacement of the beam. Whenthe distancebetweenthe junctionsdecreased,thestiffness wasmainly due to the springstiffness, which indicated that the springhadagreaterinfluence.
3.2.5. Effectofjointtwistingstiffnessontheshearmodulus
In Fig. 14, the twisting stiffness of the joint was varied to studytheinfluenceofthisparameterontheshearbehaviour.The other morphologicalparameters were fixed. The initial fibre vol-ume fraction was equal to 8.5%, the distribution of fibre orien-tations is as presented in Fig. 2 and the tension stiffness was equalto0.4N/mm.Threedifferentvaluesforthedistancebetween thecross-linkswerestudied.Theshearmodulusincreasedslightly with theincrease of the springtwisting stiffness, andit became stable starting with Ktw equal to 0.0015N.mm. There was not a clear effect of the twisting stiffness on the shear modulus for
Fig. 13. Influence of joint tension stiffness on shear modulus for different distances between cross-links. All tested RVE had the same configuration and a fibre volume fraction equal to 8.5%. The values of D c = 110 μm, D c = 165 μm and D c = 310 μm correspond to the bonding of all contacts, 2/3 of contacts and 1/3 of contacts, respectively.
Fig. 14. Influence of joint twisting stiffness on shear modulus for different distances between cross-links. The same configuration is used for all tested RVE with a fibre volume fraction equal to 8.5%. The D c = 110 μm corresponds to all bonded contacts, D c = 165 μm to 2/3 of the bonded contacts and D c = 310 μm to 1/3 of the bonded contacts.
different distances between the cross-links. Even if the twisting stiffness, Ktw,wasincreased significantly (log scaleof the curve), there wasnot a clear change on the shear modulus. It was ne-glectedcomparedtotheinfluenceofthetensionstiffness.This re-sultshowsthatthejunctionswere moresolicitedintensionthan intwisting.
4. Conclusions
Themain objectiveofthecurrentworkwastostudythe stiff-ness in shear of the entangled cross-linked fibrous material. The numerical model focused on the influence of the morphological parameters on the shear behaviour ofthe entangled cross-linked material. Thisinvestigation wasan interesting tool tounderstand the behaviourof entangledcross-linked materialsin orderto op-timizeit andenhance their mechanicalproperties.The work rep-resentsarobustbasefordevelopingamanufacturing processthat
can offer an entangled cross-linked material with the best ofits stiffnessrigidity.
Afinite elementmodel wasdevelopedto predict thestiffness ofanentangledcross-linkedmaterialinshear.Itused3Dbeam el-ementstosimulateallmodesoffibredeformation.Thecomparison betweenthenumericalresultsandtheexperimentaldatashowed agoodagreementintermsofshearmoduluswhenanassumption concerning non-separatedyarns wasconsidered. Then, a numeri-calinvestigationwascarriedouttostudytheeffectof morpholog-icalparametersontheshearstiffness.Theobtainedresultsshowed thattheperfectseparationofyarnsduringmanufacturingprocess canimprovethemechanicalpropertiesofthefibrousmaterial.
The relationship between the shear modulus and the average distancebetween cross-links wasfitted by a power lawfunction withan exponent of−3/2. The increase of the numberof epoxy cross-links,whichcausesthedecreaseoftheaveragedistance be-tween junctions, can allow the increase of the rigidity of our fi-brous material. Thiscan be done by spraying moreresin, but in
this case, the density of the sample will increase and the ratio stiffness/masswillthen decrease.Processimprovements,in order tohaveamoreefficientbondingofthejunctions, aretobe stud-ied.The question is: how tohave the finestdroplets condensing onall the contactpointswhile minimizing the presence ofresin elsewhere?Thecreationofagreaternumberofjunctionsrequires asprayingofveryfinedropsofresinofgoodfluidity.An improve-ment in the actual manufacturing process, andin particular, the processofentanglementandsprayingoftheresin,couldthus gen-eratesignificantgainsintheactualrigidityofourentangled cross-linkedmaterial.
Therelationshipbetweentheshearmodulusandthefibre vol-umefractionortheYoung’s modulusisalinearfunction.This re-sultisingoodagreementwiththeformerstudies.Itconfirmsthat these two parameters have a first-rate effect on the rigidity of thesetypesoffibrous material.Duetotheir highspecificrigidity, highmoduluscarbonfibresaretheidealcandidatesforthis appli-cation.The increaseofthetensionjointstiffnesscanincreasethe shearmodulus,butthetwistingjointstiffnessdidnothavean im-portanteffectontheshearstiffnessoftheentangledcross-linked fibrous material.The increase ofthe rigidity ofthe junctionscan be obtained experimentally through the modification of the me-chanical propertiesof theresin and / orthe increase of thesize ofthesprayedresindroplets.Thelatterisquitedelicatetocontrol duringthephase of spray.Surface treatments ofthe fibrescould modifythe surfaceenergyofthefibres andthusmodify thesize andmorphologyofthejunctions.
Itisalsoimportanttorememberthatinatargetedapplication ofsandwichpanel corematerials, the propertiesof thecomplete sandwichmustbestudiedandmodelledforbothstaticand vibra-toryloads,forwhichthedampingpotentialoftheinterlocking ma-terialsisveryinteresting.
Themanagementofmorphologicalparameterscanimprovethe shearstiffnessofentangledcross-linkedfibrousmaterialsuchthat a good value of the shear modulus can be reached that is com-parable to that of Nomex honeycombs and PMIfoam. Neverthe-less,thesemorphologicalparametersshouldbecarefullychosento enhance the mechanical propertiesof the entangled cross-linked materialwithoutimpactingits goodvibrationdampingproperties (Piollet et al., 2016; Chatti et al., 2018). This improvement will maketheentangled cross-linkedfibrous materialrelevant foruse instructuralapplicationsasacorematerialforvibrationdamping.
Acknowledgements
Thanks are due to CALMIP under the reference of program P1026 for the support and the computing power. Financial sup-portforthisworkwasobtainedthankstothejointprojectMAFIVA fundedbytheMidi-PyrénéesregionandbyISAE-SUPAERO.
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