A brittle material with tunable elasticity: Crêpe paper

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A brittle material with tunable elasticity: Crêpe paper

Nicolas Vandenberghe^∗,Emmanuel Villermaux

AixMarseilleUniversité,CNRS,CentraleMarseille,IRPHE,Marseille,France

a rt i c l e i n f o a b s t ra c t

Articlehistory:

Received18January2019 Accepted22February2019 Availableonline12April2019

Keywords:

Patterns Fracture

Westudythemechanicalresponse,andtearingfeaturesofcrêpepaper,atwo-dimensional, veryanisotropicmaterial,withonedirectionmuchlessstiffthantheotherone.Depending onhowthesoftdirectionhasbeenpre-stretchedornot,theapparent Youngmodulusof thematerialcanbevariedoverabroadrange,whileitsfractureenergyremainsunaltered.

Theclassicaltearingconcertinaproblemshowsthatamacroscopicmeasurement(theshape ofthetearedregion)provides adirectaccessto the fracturepropertiesof thematerial (effectiveYoung’s modulus,and fracture energy). Theoverall discussionis conductedin theframeofGriﬃth’stheoryoffracture.

©²⁰¹⁹Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccess articleundertheCCBY-NC-NDlicense (http://creativecommons.org/licenses/by-nc-nd/4.0/).

1. Introduction

When strongly stressed,a sample ofa solid material breaks intopieces by the formationandextension of cracks. As multiplecracksdevelopandextend,thestressfieldismodified,and,inreturn,thepathsfollowedbythecracksarealtered [1–3].In other words, cracks mutuallyinteractand thisinteraction ismediated by the stress field in the sample, which mayalsoincorporateacomponentofnoise,eitherthermal,orintrinsic(see[4] andreferencestherein).Thepredictionof themotionofasinglecracktipinageneralstress fieldremainsachallenging problem[5,6] andnosimplelawoffering a

“particle-like”descriptionofthemotionandinteractionofcracktipsiscurrentlyavailable.However,simplifiedapproaches based on geometry have been used successfullyto address crack paths in thin films [7,8]. These models are based on Griffith’stheoryoffracture[9]i.e.totalenergy(elasticandfracture)minimisation.Inthesimplestlimitofthe“inextensible fabricmodel”[7,10],thepatterncanusuallybedeterminedbyapurelygeometricapproachinavarietyofcases,including thepeelingofathinstripfromanadheringsubstrateorthetearingthroughathinfilmbyabluntobject.Evencomplicated crackpathssuchasoscillatingorspirallingcrackscanbeunderstoodfromthissimplemodel.Thisapproach,however,does not addressthevariationofthepatternwiththeelasticpropertiesofthematerial.Suchdependencecanbecapturedbya strictapplicationoftotalenergyminimisation[8].Inthisframework,anevaluationoftheelasticenergystoredinasample, generallyneglectingthedetailofthestressfieldnearthecracktip,isneeded.

We willbe discussingthe physicsofpatterns,a topicfamiliarto PierreCoullet,towhom we dedicatethispaper,and moreprecisely theselectionofapatternbyapairofexpandingcracks,highlightingtheroleofthematerialpropertiesof the sample. Thepattern ofinterestinvolves two cracks thatare formed simultaneouslywhen a bluntobject tearsa thin ﬁlmofamaterial.Thisconﬁgurationisoftencalledtheconcertinatearing,becausethematerialthatisleftbetweenthetwo cracksbuckles andfoldsasdoesaconcertina[11,12].Werelyonasimpleexperimentthatcanbeconductedbyhandand

*Correspondingauthor.

E-mailaddress:nicolas.vandenberghe@univ-amu.fr(N. Vandenberghe).

https://doi.org/10.1016/j.crme.2019.03.013

1631-0721/©²⁰¹⁹Âcadémie^des^sciences.^Published^byÊlsevier^Masson^SAS.^Thisîsânôpenâccessârticleûnder^the^CC^BY-NC-ND^license (http://creativecommons.org/licenses/by-nc-nd/4.0/).

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Fig. 1.3Dstructureofcrepepaperrevealingthestructuremadeofridgesandtroughs.(a)Aviewofthesurface,(b–c)measurementoftheheightproﬁle.

Notethatthezdirectionisnottoscale.(d–e)Acutthroughanunstretched(d),andstretched(e)sheetofcrêpepaper,alongwiththedeﬁnitionsofthe variablesenteringouranalysis.

useacommonlyavailablematerial(crêpepaper).Thematerialpropertiesandinparticularitsstiffnesscanbetunedatwill by initiallystrainingthe samples,whiletheir fracture energyisconstant,independent oftheinitial strain.We show that thepatternchangesdramaticallyasthestiffnessisvariedoverabroadrange.Weproposeageometricalmodeltoestimate thetotalenergyofthecrackingmaterial,whichcomparesfavourablywiththeobservations.

2. Materialproperties 2.1. Materialcharacterization

Weconductexperimentsonthinsheetsofcrêpepaper(Cansonbrand,typicaldensity32g·^m⁻²^,^thicknessⁱⁿ^the^range from60to 100μ^m).^Crêpe^paperîsâ^materialûsed^for^the^creationôfârtwork.Ît^consistsôf^thin^paper^that îs^foldedⁱⁿ one direction,thus creatingtroughsandbumps. Thepaperis coatedwitha glue-like substancethat holdsthe patternof folds. Avisualisation ofits3D structurewasperformedwitha 3D-Microscope(Fig. 1)revealingacorrugationoftheplate withatypicalpeak-to-peakamplitudeof80μ^mândâ^wavelengthôf¹⁸⁰μ^m.

Amechanicaltensiletesthasbeenperformedonatestingmachineonthinstripsofcrêpepaper(typicalwidth10mm andlength100mm). Duetothepreferreddirectionofthefolds,when pulledupon,theresponse ofthesamplepresents a strong anisotropy.When stretched inthe direction parallel to the directionof troughand ridges, the sample presents a classical linear elasticresponse with astretching modulus Eh=^{65 N/mm.} ^When ^stretched ⁱⁿ ^the^direction ^normal^to the folds, the sample is much softer and it presents a strong nonlinear response as shown in Fig. 2. The response is characterizedbydifferentfeatures.Firstifoneconsidersthestress–strainrelationitself(i.e.theupperenvelopeofthecurve in Fig.2 omitting the presence ofunloading cycles), we note that it is similar to the curve obtained inother materials formedbynetworksofelasticcomponentssuch asrubber[13] orfoldedpaper[14].Theloadingcurve presentsamarked softening atintermediate strains,while athigherstrainsthematerial getsstiffer.Anobservationofthematerialafterthe pullingtest(Fig.1d,e)revealsthat themorphology ofthestretched samplechanges:theamplitude ofthecorrugation has beenreduced,indicatingplasticityinthemechanicalresponseofthesample.Thenwealsonotethattheunloading–loading cycles performedduring the testing ofthe sample reveal an hystereticbehaviour characteristic ofplasticity. Finally,it is worthnotingthatthemaximalstrainattainedduringtesting(afterwhichthesamplebreaks)iscompatiblewiththeproﬁle measurement:theunfoldedlengthistypically

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Fig. 2.Measuredloadingcurveforastripofcrêpepaper(width10mm,length100mm)loadedinthedirectionperpendiculartoitsfolds(forceapplied alongthexdirection).Unloadingperformedatregularintervalduringloadingrevealsaplasticbehaviour,evenatmoderatestrain.Thegraycurvescorre- spondtounloadingphases.Thesteepdashedlineshowstheelasticbehaviourwhenstretchinginthedirectionparalleltothefolds(obtainedinadistinct test).Theinsetshowsthelocalslopesoftheloadingcurve(disks)andoftheunloadingcurves(diamonds)obtainedatdifferentstrains.Theapparent moduliarenormalisedbythestretchingmodulusEh=^{65 N/mm.}

lun=

lap

0

1+w(x)²dx (1)

where lap is the apparent length (in the x direction) of the profile w(x). With, for instance, w(x)=â^sin(2π^x/λ), the maximalstrainl_un/l_ap−^{1 is}approximately(πâ/λ)²,yieldingapproximately0.5witha=⁸⁰μ^mândλ=¹⁸⁰μ^m^(Fig.^2).

Theresponseofthematerialinthedirectionperpendiculartothefoldsisthuscharacterisedbyanapparentstretching modulus thatdependsonthestrain. Twodifferentmodulicanbedistinguished:amodulus E_l obtainedby measuringthe slopeduringtheloadingphaseandanunloadingmodulus Eun obtainedbymeasuringtheslopeduringtheunloadingone.

ThesetwoquantitiesareplottedintheinsetofFig.2.

2.2. Discretemodel

The behaviour observed in Fig. 2 can be qualitatively reproduced using a simple model of an elastoplastic ridge (a discrete version of the creased sheet model [15]).We build a model upon an element consistingof two rigid plates of length merging at a ridge; it will be further convenient to assimilate the ridge to a hinge characterized by a spring constantkandayieldmomentmy abovewhichthereststateofthehingeismodiﬁed.

Whenextendedtoanangleθ,themomentoftheridgelinearizedaroundθ0 ism=^k(θ−θ0)(wherekistypicallyrelated to the sheetbending stiffness throughk≈^Eh³/12) in theelasticdomain (i.e. m<my). Intheplastic domain,the ridge momentism=^myand,iftheangleisextendedabovem_y/k,therestangleθ0 increasesuptoθ0=θ−^my/k.Theforceis

f= ^m

cosθ ⁽²⁾

andthus astheangle θ increases, the stiffness increasesbya purely geometricaleffect.Ina typical pullingtest,starting from thereststate x=^x0,forwhich f =^0, ^as^x=^x0(1+)is increased, themoment increases, ﬁrstreaching the yield moment. At this point, the strain–load curve presents a slope change.As the strain further increases, the moment is constant,m=^my,buttherestangleofthespringincreases.Thepullingforce duringtheplasticphaseofthetest isthen

f=(my/)[¹−(x/)²]⁻¹^/²^and^the^associated^stiffness∂f/∂intheloadingphasewhenthestresshasreached 1 is

kp=^m^y ²

_x₀

2 1+1

[¹−(x0/)²(1+1)²]³^/² ⁽³⁾

The stiffnessincreaseswith ₁ (thatis,increasing θ)andthisstress hardeningispurelygeometrical,asseenfromEq.(2).

Thestiffnessinthisphasedoesnotdependontheelasticconstantk.Whenthestrainhasreached 1,therestangleofthe hingeisθ1=^arcsin[(x0/)(1+1)]−^my/k.

Ifstartingfrom 1 thestructureisunloaded(i.e. xislowered),thestructurereactselasticallybecausethemomentatthe hingedecreasesbelowm_y.Theforceisnow f=(k/){ârcsin[(^x0/)(1+)]−θ1}[¹−(x/)²]⁻¹^/² ând^thus^the^stiffnessîs

k_u=^k

(my/k)(x0/)²(1+1)+(x0/)[1−(x0/)²(1+1)²]¹^/²

[¹−(x₀/)²(1+1)²]³^/² ⁽⁴⁾

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Fig. 3.(a)Mechanicalresponseofasingleelastoplastichingewithx0/=0.67andmy=0.1.Theresponseexhibitsthesofteningcharacteristicofthe plasticityand thegeometricalstiffeningat highstrains,whenthehinge ﬂattens.(b)Whenpulledupon,anassemblyofhingesdisplays abehaviour qualitativelysimilartothatofcrepepaper.Thecurveisobtainedforasequenceof20elementsformedof30hingeswhoselengthsarenormallydistributed (withamean1andastandarddeviation0.25).

Fig. 4.Atoolisusedtotearasheetofcrepepaper,startingfromasmallnotch.Asthetoolispulled,twocrackopens,andthepatterncanbeparameterised byitshalf-width(x).Thesheetofcrepepapercanbeinitiallystretchedinthedirectionnormaltox(thefoldofthepaperareinthe xdirection)by adjustingL.Theinitialstretchingis0=(L−^L0)/L0.

Thusthissingle elastoplasticridgepartlyreproducessome thebehaviour ofcrepepaper;geometryinducesastiffening of the structure; andunloading during the pullingtest impliesan elastic response that resultsin unloading cycleswith an apparentstiffnessthatincreaseswiththestrain.(SeeFig.3.)

However,thissimplemodeldoesnotreproducethehystereticbehaviour observedduringtheunloadingloop.Whenthe wedgeisunloadedandthenloadedagain,theforce–displacementcurvefollowsthesamepath.

Themodelcanbe furtherextendedbyconsidering anassemblyofhingesinparallelandinsequence,withdistributed segmentlength.Thoughthemodelpresentsamoreconstrainedgeometricalorganisationthananactualcreasedsheet,it sharessome ofits features:thegeometryofasinglehingemimicsthegeometric nonlinearityoftheelasticresponseofa fold,andtheplasticityintroducedatthehingeissimilartotheplasticityobservedonthecrêpepapersheet.Eventhougha quantitativecomparisonisoutofthescopeofthepresentpaper,itisworthnotingthatpullinganassemblyofhingeswith distributedlengths(allotherparametersbeinguniform)leadstoabehaviourqualitativelysimilartothatofcrêpepaper.In particular,itexhibitsthestresssofteningassociatedwithplasticity,thestresshardeningathighstrainsassociatedwiththe flattening ofthehinges.Italsoexhibits thestiffeningoftheunloading curvesasthestrain increases, whichisassociated withthehingesdeformingplasticallyintoflatterconfigurations.

3. Tearingbehaviour 3.1. Theconcertinatearing

Weare nowinterestedinthe signatureofthestress–strainbehaviourpresentedabove oncrackpatterns.Ofparticular interest istherole ofthe changeofthe localstretching moduluson thepattern. Theexperiment isconductedonsheets ofcrêpepaper(170mm×¹⁷⁰^mm)âttached^toâ^rigid^substrate^by â^rigid^tape^(Fig. ^4).Ânînitial^cutôfâbout²^mmîs performedwitha sharpbladeonthesheet. Atoolofsmallsize(afew millimetrewideatthecontactwiththepaper)is thenintroduced inthenotchandispulledtowardsx>0.The setupallowsforanadjustmentoftheinitial tensioninthe xdirection.Theinitialstretching 0=(L−^L0)/L0 canbeadjustedbetween0upto0.4.Thewidthofasampleis150mm beforeapplicationofthestretching.

Asthetoolispulled,ittearsthesheetofcrepepaper.Twocracksextend,leavingadivergingpatternofcracks(Fig.1).In thedomainaroundthecracktipsandthetool,thethinpaperisstronglyfolded(thusthenameConcertinatearing[11,12]).

AsseeninFig.5,thehalf-widthofthecrackpatterncanbeapproximatedbytheshape

(x)∼^{a x}³^/⁴ ⁽⁵⁾

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lefttoright:0=⁰.053,0.21,0.26,0.40.Thereddashedlineindicatesashape(x)∼^x³^/⁴^.^The^otherexperimentalconditionsarekeptidentical.

Fig. 6.(a)Sketchofcrackextension.ThetoolislocalisedinA.Thegeometryisusedtocomputethechangeinstrainenergyasthecracktipismovedfrom CtoC’.(b)Sketchofcrackextensionasthetooladvances.AsthetooladvancesfromAtoA’,thecracktipmovesfromCtoC’.

Thisshapeissuggestedbythetheoreticalargumentsdevelopedinthenextsection,whichalsoprovidethemissingdimen- sionallengtha⁴inEq. (5).Remarkably,thereisastronginﬂuenceoftheinitialstretching:pre-stretchingthematerialleads tonarrowerpatterns.

4. Geometryofthecrackpattern

Modelling isbasedonGriﬃth’slinearelasticfracturemechanicsandfollowstheanalysisofrefs. [7,8].Suchanapproach requires the calculationof thevariation ofelastic energystoredin thematerial during loading andwe willassume that this energy is storedin stretching only (i.e. the bending energy associated with the folding is neglected).Moreover, in Griﬃth’s approach,thechangeof elasticenergythat occursduring the extensionofa crackisassociated withunloading.

Forthisreason,weconsiderthattherelevantelasticenergywritesUs∼

Eunh²dS whereEunhcharacterisesthesample unloadingatagivenpre-stretchand isthestrain.

Theﬁrsttaskisthecomputationofthestrainassociatedwithagivengeometry.ConsiderthecrackgeometryofFig.6a.

The tearing force is applied to point A towards x>0 atthe apex of the pattern. The segment ACof initial length is stretched toalength/cosα andthusthestrainis =¹/cosα₋1.Theanalysisproceedsintwostages:ﬁrstweconsider that theangleθ isknownandwecomputethepositionofthecracktip (parameterisedby theangle α) thatensuresthat thetotalenergyisminimal.Thenanoptimisationamongallthecrackpath(i.e.thedifferentvaluesofθ)isusedtoﬁndthe actualshapeofthepattern.

AsthecracktipsadvancesfromCtoC’,theelasticenergyisreduced, whereasthecrackadvanceresultsinan increase offractureenergy.Parameterisingthecrackadvancebythe(small)angleβ=α₋α,wecancomputethelengthsandu usingthefollowingrelations:

tanα−tan(α−β)=−

tanθ , and u=−

sinθ ⁽⁶⁾

yielding(forβ small)

=[1+βtanα₊βtan(θ−α)] and u= β

cosαcos(θ−α) ⁽⁷⁾

Thevariationoftotalenergy(sumofelasticandfractureenergy)isδU∼Êunh²l²−Êunh²l²+huwiththegeometrical parameterβ canbecomputedusingEq. (7).ImposingδU=^{0 at}^firstôrderâsβ changesyieldsanequationfor α

cosα−⁸ ccosα

2 −θ

sin³α

2 =⁰ ⁽⁸⁾

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α³₌ ^c

cosθ ⁽¹⁰⁾

Equation(8) oritssimpliﬁedform(10) setsthepositionofthecracktipwhenevertheangleθ isknown.

Wenowconsidertheenergyinvolvedasthetooldrivesthecrack.Asthetooladvancesbyadistanced(seeFig.6b),the geometryofthepatternisgivenby

ucosθ+tanα₌d+tanα and u=−

sinθ ⁽¹¹⁾

Combiningthetwoequations,andwriting=(1+λ),for α small,oneobtains

λ≈^d ^tanθ

1+²

3

_c

cosθ 1/3

tanθ ₋1

(12) and

u= λ sinθ = ^d

cosθ

1+² 3

_c

cosθ 1/3

tanθ ₋1

(13) ThetotalenergyvariationwritesδU∼Eunhl²²−Eunhl²²+huwith

= ¹

cosα⁻¹^≈ α²

2 ≈¹ 2

_c

cosθ 2/3

(14) Withthedifferentgeometricquantities,thevariationofenergywrites

δU

Eunh ≈²^/³ 4

_c

cosθ 4/3

−²^/³ 4

_c

cosθ 4/3

+ λ c sinθ

≈²^/³¹ 6

_c

cosθ 4/3

λ+ c sinθλ

Tocomputetheoptimalconﬁguration,weconsiderthevalueofθ thatensuresthat∂(δU)/∂disminimalwhenδ→0.The coeﬃcientthatshouldbeminimizedis

C_d= 1

6

_c

cosθ 4/3

+ ^c sinθ

tanθ

1−²

3

_c

cosθ 1/3

tanθ

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≈^c −¹

2θ _c

4/3

+θ²

c

2−¹ 9

_c

5/3

(16) MinimizingC_d withrespecttoθ yieldsatleadingorder

θ=¹ 2

_c

1/3

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Thisequationcanthenbeusedtocompute theshapeofthecrackasitextends:withd/dx=^tanθ≈θ≈¹/2(c/)¹^/³,we obtainfortheshape[8]

(x)≈c¹^/⁴x³^/⁴ (18)

wherec=h/Eunh=/Eunisanintrinsicfeatureofthematerial.