An arterial constitutive model accounting for collagen content and cross-linking

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Journal of the Mechanics and Physics of Solids

journalhomepage:www.elsevier.com/locate/jmps

An arterial constitutive model accounting for collagen content and cross-linking

Gerhard A. Holzapfel

^a,b,∗

, Ray W. Ogden

^c

aInstitute of Biomechanics, Graz University of Technology, Stremayrgasse 16-II, Graz 8010, Austria

bNorwegian University of Science and Technology (NTNU), Department of Structural Engineering, Trondheim 7491, Norway

cSchool of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8SQ, Scotland, UK

a r t i c l e i n f o

Article history:

Received 22 June 2019 Revised 8 August 2019 Accepted 8 August 2019 Available online 9 August 2019 Keywords:

Artery elasticity Collagen fibers Collagen cross-links Fibrous tissue

a b s t r a c t

Itisapparentfromtheliteraturethatthedensityofcross-linksincollagenoustissuehas astiffeningeffectonthemechanicalresponseofthetissue.Thispaperrepresentsanini- tialattempt tocharacterize thiseffectonthe elasticresponse,specificallyinrespect of arterialtissue.Twoapproachesarepresented.First,asimplephenomenologicalcontinuum modelwithacross-link-dependentstiffnessisconsidered,andtheinfluenceofthecross- linkdensityontheresponseinuniaxialtensionisillustrated.Inthesecondapproach,a 3Dmodelisdevelopedthataccountsfortherelativeorientationandstiffnessof(twofam- iliesof)collagenfibersandcross-linksandtheircouplingusinganinvariant-basedstrain- energyfunction.Thisisalsoillustratedforuniaxialtension,andtheinfluenceofdifferent cross-linkarrangementsandmaterialparametersisdetailed.Specializationofthemodel forplanestrainisthenused toshowtheeffectofthecross-linkorientation (relativeto thefibers)andcross-linkdensityontheshearstressversustheamountofsheardeforma- tionresponse.Theelasticitytensorforthe general(3D)caseisprovidedwithaviewto subsequentfiniteelementimplementation.

© 2019TheAuthors.PublishedbyElsevierLtd.

ThisisanopenaccessarticleundertheCCBYlicense.

(http://creativecommons.org/licenses/by/4.0/)

1. Introduction

Collagen is the mostimportant structuralprotein in the body and is ableto bear significant mechanical loadwithin fibrous tissues (Fratzl, 2008). In such tissues collagen forms a network together with cross-links which, from the solid mechanicspoint ofview,contribute to thetransmissionof forcesbetweenthefibers ofthenetwork inboth healthyand agedtissues(Andriotisetal.,2018).Collagen isa hierarchicalmaterial (Fratzl,2008; FratzlandWeinkamer,2007), which iscomposed ofa tropocollagentriple helix atthe nanoscale, typically about300nm long. Thistropocollagenis held to- getherbyintramolecularbonds.Aggregatesoftropocollagenmoleculesconnectedtogetherbycross-linksformcollagenfib- rilswhichthemselvesgrouptogethertoformcollagenfibers.Enzymaticcross-links,whichconnecttropocollagenmolecules attheirendsandprovidestabilityofthestructure,contributetothemechanicalresistanceofafibrilundertension.Onthe otherhandnon-enzymatic cross-links whichcanattach atanypoint alongthelength ofatropocollagenmolecule canbe

∗ Corresponding author.

E-mail address: [email protected] (G.A. Holzapfel).

https://doi.org/10.1016/j.jmps.2019.103682

0022-5096/© 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY license.

( http://creativecommons.org/licenses/by/4.0/ )

detrimental to normalfibril function. The bonds mayoccur as bivalent and/or trivalentcross-links (Eekhoff et al., 2018).

Undersuitableloadsbondscanbebroken,andslidingbetweenthefilamentsmayoccur.

Clearly,cross-linkingofcollagenfibershasasignificanteffectontheresponseofthetissueswithinwhichthefibersare embedded(Buehler,2008;Eekhoff etal.,2018;Yoshidaetal.,2014);theinfluenceofcross-linksonthefracturemechanicsof collagenfibrilsisdocumentedinSvenssonetal.(2013).Thereisalotofevidence,however,thatindicatesthatthenumber ofcross-linksincreaseswithage,whichisanimportantfactorintheage-relatedstiffeningofarterialwalls(Barodkaetal., 2011;Cantinietal.,2001).However,thereisalackofquantitative dataconcerningage-relatedchangesofcross-linkingin biologicaltissues(HayashiandHirayama,2017).OnthebasisofbovinetailtendonsthestudyofWillettetal.(2010)found thatagedtissuescontainmorematurecovalentcross-links.AccordingtothereviewarticleofTsamisetal.(2013)thereare twodifferentmechanismsresponsiblefortheincreaseofcross-links:(i)increaseoftheamountofcross-linkingaminoacids withincollagen,and(ii)accumulatingadvancedglycationend-products(AGEs)whichformprotein-proteincross-linksalong thecollagen molecules(HayashiandHirayama,2017; WagenseilandMecham, 2012) – for a review oncollagenglycation as a potential driver ofconnective tissue disease see Snedeker andGautieri (2014). Forbackground on the chemistry of cross-linkingofcollagenandelastinwerefertothestudyofEyreetal.(1984).

The effect of the mechanicalproperties of AGEs wasalso investigated by Svensson et al. (2018) for tendons. Inhibi- tion oftheformationofAGE-induced cross-linksreducesthe stiffnessoflargearteriesinrats; seeGreenwald (2007)and referencestherein. Thestudyof Uzel andBuehler(2011)developed asimple molecular modelofthe cross-linkstructure of type Icollagenandshowed that thepresence ofthe cross-links resultedin strengthening ofthecollagen structure at large deformations.The studyofYang etal. (2012) investigatedthe influenceof differentcross-links onthe stress relax- ation behavior of collagenfibrils.Experimental data were analyzed usinga two-term Pronyseries, whichsuggested that fastrelaxationisrelatedtothe relativeslidingofcollagenmicrofibrilsandthat theslowrelaxationprocessresulted from the collagenmoleculesforwhich thereis alarger numberofcross-links. The paperofDavidenko etal.(2015) examined howdifferentlevelsofcross-linkingofcollagenous-basedscaffoldseffecttheirmechanicalproperties.ThestudyofKwansa et al.(2016) used molecular-dynamics simulations based on collagentype Imicrofibril units of both uncross-linked and cross-linkedfibrils.Inparticular,inKwansaetal.(2016)uniaxialtensiontestsweresimulatedtoexaminetheeffectofthe cross-linkingontheelastic moduli,thusshowingthatthedifferentcross-linktypesledtonoalterationsinthelow-strain moduliwhilethefinitestrainelasticmoduluswassignificantlyincreased.

Onecontinuum-based3Dmodeltakingaccountofcross-linkingwasproposedbySáezetal.(2014).Thecross-linkswere accountedforbyaparameterwhichprovideda weightingbetweentheisotropicandtheanisotropicresponse.Themodel wasusedtofitdatafromuniaxialtestsonpigcarotidarteriesforwhichthecross-linkingwasunknown,inwhichcasethe relevanceofamodelaccountingforcross-linkingisunclear.Another3Dcontinuum arterialconstitutivemodelconsidering collagencontent andcross-linking wasproposed by Tianetal. (2016).Thatapproach isbased on theeight-chain model, whichwasdevelopedtocharacterizerubber-likematerialsbutisunsuitableforfibroustissues.Basedonadiscretenetwork ofcross-linkedbiopolymerfibers(Žagaretal.,2015)usesacomputationalapproachtodeterminethestiffeningeffectsofthe cross-links,whileinLinandGu(2015)asimilarcomputationalmodelwasusedtodeterminetheeffectofcross-linkdensity andstiffnessinacollagengel.Ontheotherhand,onthebasisofBuehler(2006),Buehler(2008)proposeda1Dnanoscale modelwhichconsiderstheeffectofdifferentcross-linkdensitiesonthemechanicalresponseofcollagenfibrils.Moredetails oftheeffectofthecross-linkstructureonthemechanicalpropertiesofcollagenfibrilshavebeenconsideredbyDepalleetal.

(2015)withparticularreferencetoenzymaticcross-links.ThereviewarticleofEekhoff etal.(2018)describesthemechanical effectsofcollagencross-linkingspecificallyfortendons,whilethepaperofYoshidaetal.(2014)documentsthemechanical propertiesofmousecervicaltissuewithrespecttocollagencross-links.ThestudyofChenetal.(2017)considersacollagen networkforarticularcartilagebasedon aspring-nodemodelofcross-linkedcollagen,andtheauthorsstudiedchangesof thecross-linkstiffnessanddensityonthemechanicalresponse.

The numberofcross-links hasa significanteffect onthe measuresof theelastic modulus.Thissuggeststhat material parametersinaconstitutivemodelshouldbedependentontheproportionandarrangementofcross-linkswithinthecolla- genstructure.Twokeyingredientsarethecollagenfibercontent,asmeasuredby,e.g.,thevolumefractionsofcollagenand cross-links,andtherelativearrangement.

Thepurposeofthispaperistocharacterizetheeffectofcross-linksonthebasisofaphenomenologicalcontinuummodel thattakesaccountofinformationaboutcross-linksatthemicro-structurelevel.Althoughtherearesomeapproachesdocu- mentedintheliterature,asmentionedabove,thereisnotyetafully3Dmodelavailablethatdescribes,e.g.,theanisotropic responseofarterialwallsthattakesproperaccountofcollagencross-linking.

In the present study we first consider a continuum approach that involves the cross-link density and a cross-link- dependentstiffness,whichisarathersimplephenomenologicalapproach.Forthismodelwespecializetouniaxialextension andexaminetheeffectsofvaryingcross-linkdensitiesonthemechanicalresponseofthematerial.Second,weconsiderex- plicitly the relative orientation of the collagenfibers and the cross-links and their interactions usingan invariant-based energyfunctionthatincorporatescontributionsfromthematrixmaterial,collagenfibers,thecross-linksandtheirinterac- tions withthefibers.Forthesecond approachwe alsoillustrate uniaxialextension andanalyzetheinfluenceofdifferent cross-linkarrangementsandmaterialparameters.Finally,weconsideramodelwithtwo familiesoffibersarrangedin3D, with the fibers alignedwithin each family, and withtwo sets of aligned cross-links connecting the fibers in each fam- ily.Thisgeneralformulationissuitable forfiniteelementimplementation,andtowards thisaim weprovidetheelasticity tensorassociatedwiththemodelinanappendix.

We alsoconsider a planar specializationof the model andillustrate it by application to a simple shear deformation, showingthe effect of the cross-linkorientation (relativeto the fibers) andcross-link densityon the shear stress versus amountof shear deformation. The readerwho requires additional informationon the subjectof nonlinear elasticityand solidmechanics isreferred tothemonographsofHolzapfel (2000)andOgden(1997),while adetailedexplanationofthe underlyingconstitutivetheoryforstronglyanisotropicsolidscanbefoundinSpencer(1984).

2. Modelstructure

Supposethecollagenfibersareembeddedwithinanisotropicmatrixwiththevolumefraction^sothat(¹−)^isthe volume fractionof thematrix. Letthe elastic propertiesof the matrixand fibersbe describedin termsof strain-energy functionsiso andf,respectively.Weconsider iso todepend ontheisotropicinvariant I₁=trC,whereC=F^TFisthe rightCauchy–Greentensor, Fisthedeformation gradient,andf,theenergyassociatedwiththefiberinthedirection M inthereferenceconfiguration,todependonI4=M·CMandalsoonameasureofthedensityofcross-linksperunitlength ofthefiberinthedirection M.

Weconsiderthematerialtobeincompressible(J≡detF=1)withthetotalelasticenergyofthiscompositeas

(

^I1,I4,

ρ )

=

(

¹−

)

iso

(

^I1

)

+

f

(

^I4,

ρ )

, (1) where

ρ

^{canbe thoughtofasthenumberof}cross-links perunit length,subsequentlyreferredtoasthedensityofcross- linkswithdimension1/(length).Asanexampleisocantakeonaneo-Hookeanform,whileforfwecouldhavek(

ρ

)(^I4− 1)²/2,astandard reinforcingmodel.Hereink(

ρ

^)isthecross-link-dependentfiberstiffness.Notethatthederivativekshould bepositivetoreflecttheincreasingstiffnesswithincreasingdensityofcross-links.Hence,from(1),theCauchystresstensor

σ

^{canbecalculatedas}

σ

=F

∂

∂

^{F −pI=}

⁽

¹⁻

⁾ μ

^b+2

^k

( ρ )(

^I4−1

)

^mm−pI, (2) wherepistheLagrangemultiplierassociatedwiththeincompressibilityconstraint, bdenotestheleftCauchy–Greentensor, Iistheidentitytensor,

μ

^{istheshearmodulusofthe}neo-Hookeanmatrixandm=FM.Theanisotropictermisonlyactive ifI₄>1,soundercompressioninthefiberdirection,thematrixbearsthestress.

Becauseoftheexperimentaldataoffibroustissueitisusefultorepresentfasanexponential,inthiscasegivenby

f=k₁

( ρ )

2k₂

{

^exp[k2

(

^I4−1

)

^2]−1

}

^{, (3)}

wherek₁>0isaparameterwiththedimensionofstress,whilek₂>0isadimensionlessparameter.TheCauchy stressof thefibersisthendenotedby

σ

f,i.e.

σ

f=2

^k1

( ρ )(

^I4−1

)

^exp[k2

(

^I4−1

)

^2]mm, (4) whiletheCauchystressforthematrix

σ

isois

σ

iso=

(

¹−

) μ

^b. (5)

Letusnow considera stripoftissue intheaxial/circumferentialplane withtwo familiesoffibers whichare arranged symmetricallywithrespect tothe axes,as indicated inFig.1, where

α

^{isthe anglebetweenthe axial directionandthe}

directionofeachfamilyoffibers. Thedirectionofthesecond fiberfamilyisdenoted by M withm=FM.According to Fig.1thematrixformsof Mand Mare

[M]=[cos

α

^,sin

α

^{,0]T, [M]=[cos}

α

^,−sin

α

^{,0]T, (6)}

wherewehaveassumedthatthecollagenfibershavenoout-of-planecomponent.Apushforwardgives

[m]=[

λ

^1cos

α

^,

λ

^2sin

α

^{,0]T, [m]=[}

λ

^1cos

α

^,−

λ

^2sin

α

^{,0]T, (7)}

where

λ

1and

λ

2aretheprincipalstretchesalongthedirections1(axialdirection)and2(circumferentialdirection),respec- tively,while,fromtheincompressibilitycondition,

λ

3=

λ

⁻1¹

λ

⁻2¹.TheinvariantI₄ isthen

I4=M·CM=

λ

²1cos²

α

+

λ

²2sin²

α

, (8)

andbysymmetrywehaveM·CM=I₄.

TheCauchystresstensor

σ

=

σ

iso+

σ

f−pIthenbecomes

σ

⁼

(

¹−

) μ

^b+2

^k1

( ρ )(

^I4−1

)

^exp[k2

(

^I4−1

)

^2]

(

^mm+mm

)

−pI, (9) whichisdiagonalwithrespecttothechosenaxes(noshearstress),andhenceitscomponentsare

σ

¹¹⁼

(

¹⁻

) μλ

²1+4

^k1

( ρ )(

^I4−1

)

^exp[k2

(

^I4−1

)

^2]

λ

²1cos²

α

^{−p, (10)}

σ

22=

(

¹−

) μλ

²2+4

^k1

( ρ )(

^I4−1

)

^exp[k2

(

^I4−1

)

^2]

λ

²2sin²

α

−p, (11)

σ

³³⁼

(

¹−

) μλ

²3−p. (12)

Fig. 1. Sketch of a rectangular tissue strip reinforced by two in-plane families of aligned fibers symmetric with respect to its edges, with fiber angle α.

We assume that thestrip is underplane stress conditionswithloads parallel tothe circumferential andaxial directions.

Thenthestress

σ

33 iszerowhichallowsforptobeeliminatedfrom(10)and(11)togive

σ

¹¹⁼

(

¹−

) μ ( λ

²1−

λ

²3

)

+4

^k1

( ρ )(

^I4−1

)

^exp[k2

(

^I4−1

)

^2]

λ

²1cos²

α

^{, (13)}

σ

²²⁼

(

¹−

) μ ( λ

²2−

λ

²3

)

+4

^k1

( ρ )(

^I4−1

)

^exp[k2

(

^I4−1

)

^2]

λ

²2sin²

α

^{, (14)}

whereI₄isgivenby(8)₂,and

λ

3bytheincompressibilitycondition.Thus,

σ

11and

σ

22aregivenintermsof

λ

1and

λ

2.By considering uniaxialstress with

σ

22=0, Eq.(14)caninprinciple besolved for

λ

2 asafunction of

λ

1,andthen

σ

11 isa functionof

λ

1alone.

Wenowfocusonaspecialcase,namely

α

=0.Hence,thetwofamiliesoffiberscoincideandthedirectionofthecollagen fibersistheaxialdirection.Forthiscasewechoose

λ

=

λ

1,sobysymmetry

λ

2=

λ

3=

λ

^−1/2,andaccordingto(8)₂wehave I₄=

λ

^2.Weusethedimensionlessquantities

σ

¯11=

σ

11/

μ

^andk¯1=k₁/

μ

,andobtain

σ

¯11=

(

¹−

)( λ

²−

λ

⁻¹

)

+4

^k¯1

( ρ )( λ

²−1

) λ

^2exp[k2

( λ

²−1

)

^{2] (15)}

from(13),whichisanexplicitexpressionfor

σ

¯11 intermsof

λ

^,where,k¯1(

ρ

)^andk2 needtobespecified.

The functional dependence of k¯₁ on

ρ

^{can be modeled by anysuitable function butfor simplicityof} illustration we consider the quadratic equation k¯₁=k¯₀+a¯

ρ

², where k¯₀≥0 is the value ofk¯₁ at

ρ

=0 and a¯>0 is a parameter with dimension(length)².

Fig.2 illustrates theinfluence ofthedensityof cross-links onthe uniaxialresponse inthe directionofthefibers. For particular valuesofthe parameters we haveused =0.1,k¯₀=0.1,a¯=0.25andk₂=0.3.The fourcurves inthe figure correspondto thefourvaluesof

ρ

^{,i.e.0,1,2,and3.Noteinparticularthecase}

ρ

=0forwhich therearenocross-links andthetensionissupportedbythefibersandthematrixwithoutcross-links,asreflectedinthelowervalueofthetension.

As

ρ

^{increasestheresponsebecomesstiffer.}

3. Amodelwithalignedcollagenfibersconnectedbyseparatelyalignedcross-links

We consider collagenfibers to be in the directionofthe unit vector E₁ andlet E_R be the radial unit vector normal to thatdirection. Inaddition weconsider twosymmetricallydisposed familiesofcross-linksin thedirectionsofthe unit vectorsL⁺andL⁻ (Lstandsforlink),whicharedefinedby

L^±=±cos

α

0E₁+sin

α

0E_R, (16)

where

α

0 definestheirorientationrelativetothecollagenfiberdirection;seeFig.3.

Fig. 2. Plots of the dimensionless Cauchy stress ¯σ11versus the stretch λfor four different values of density of cross-links ρ, including ρ= 0 (no cross-links).

Fig. 3. (a) Aligned collagen fibers in the direction E 1with two families of aligned interconnecting cross-links with directions L ⁺and L ⁻making an angle α0with E 1. (b) Focus on a pair of cross-links in (a), indicating their rotational symmetry about E 1with the radial vector E R.

A uniaxial deformation with stretch

λ

^{is applied along the collagen fibers so that by symmetry and}considering the materialtobeincompressiblethedeformationgradienthastheform

F=

λ

^E1E1+

λ

^−1/2ERER. (17)

We define e=FE₁=

λ

^E1 ande_r=FE_R=

λ

^−1/2ER asthe push-forwards of E₁ and E_R under thedeformation. The corre- spondingpush-forwardsofL⁺andL⁻arethen

l^±=FL^±=±

λ

^cos

α

0E₁+

λ

^−1/2sin

α

0E_R. (18) ForthisspecialdeformationtheisotropicinvariantisgivenbyI₁=trC=

λ

²+2

λ

⁻¹,whilethesquaresofthestretches inthedirections E₁,L⁺andL⁻are

I₄=e·e=CE₁·E₁=

λ

², I=l^±·l^±=

λ

^2cos2

α

0+

λ

^−1sin2

α

0, (19) whereinthe invariants I₄ andI are defined. For the cross-links to be extended, i.e.when

λ

>1,

α

0 has to be restricted accordingto

cos²

α

^0>

λ

²⁺¹

λ

^{+1, (20)}

whichissatisfiedforall

λ

>1ifcos

α

0>1/√

3.Theinvariant I₄ isthesquareofthestretchinthecollagenfibersandIis thesquare ofthestretchineach ofthecross-linkdirections.Wealso definethecouplingbetweenthecollagenfiberand cross-linkdirectionsbythequantitiesI⁺₈ andI⁻₈,whicharegivenby

I₈^±=l^±·e=±

λ

^2cos

α

^{0. (21)}

Thesearenotthemselvesinvariants(theirsignchangesunderreversalofeither eor l^±),but(^I+8)²=(^I8⁻)^{2 isinvariant.The} valuesoftheinvariantsI₁,I₄,andIandthequantitiesI₈^±inthereferenceconfigurationare3,1,1and ±cos

α

0,respectively.

Wenowconsiderastrain-energyfunction^{,whichisafunctionofI}1,I₄,I,I⁺₈ andI₈⁻.Specificallyweconsider^tohave theform

=

(

¹−

−

)

iso

(

^I1

)

+

f

(

^I4

)

+

^[

c

(

^I

)

+

fc

(

^I8⁺

)

+

fc

(

^I−8

)

^], (22) where^{and denotethevolumefractionsofthecollagenfibersandthe}cross-links,respectively.

The functionsiso,f andc are theenergies storedin thematrix material,the collagenfibersandthe cross-links, respectively, while thetwo fc-termsrepresentthe interaction energies betweenthe collagenfibers andthe cross-links.

Notingthat_fc(^I₈⁻)=−_fc(^I+₈),theCauchystresstensor

σ

^{canbewrittenintheform}

σ

^=−pI+2

(

¹⁻

⁻

)

iso

(

^I1

)

^b+2

f

(

^I4

)

^ee

+

{

²

c

(

^l+l⁺+l⁻l⁻

)

⁺

fc

(

^I8⁺

)

^[el⁺+l⁺e−

(

^el⁻+l⁻e

)

^]

}

^{, (23)}

wherewehaveusedtheabbreviations

iso

(

^I1

)

=

∂

iso

∂

^I1 ,

f

(

^I4

)

=

∂

f

∂

^I4,

c

(

^I

)

=

∂

c

∂

^{I ,}

^fc

⁽

^I8+

⁾

⁼

∂

fc

∂

^I+8

. (24)

Now,forthesubsequentcomponentformsweneed

el⁺+l⁺e−

(

^el⁻+l⁻e

)

=4

λ

^2cos

α

^0E1E1. (25) Hence,thestresscomponentsare

σ

11=−p+2

(

¹−

−

)

iso

λ

²+2

f

λ

²+4

c

λ

^2cos2

α

0+4

fc

λ

^2cos

α

0, (26)

0=

σ

^rr=−p+2

(

¹−

−

)

iso

λ

⁻¹+4

c

λ

^−1sin2

α

0, (27)

wheretheargumentoffc isI₈⁺.Byeliminating theLagrangemultiplier pby subtractionof(27)from(26)weobtainthe uniaxialstress

σ

=

σ

11 as

σ

⁼²

(

¹−

−

)

iso

( λ

²⁻

λ

⁻¹

)

+2

f

λ

²⁺⁴

c

( λ

^2cos2

α

⁰⁻

λ

^−1sin2

α

⁰

)

+4

fc

λ

^2cos

α

^{0. (28)}

Nowletusconsidersomespecificenergyfunctions.Forthematrixmaterialweusetheisotropicneo-Hookeanmaterial

iso=1

2

μ (

^I1−3

)

, (29)

wheretheconstant

μ

^{isapositiveparameter,andforthecollagenfibersweusethestandard}exponentialform,i.e.

f= k₁

2k2

{

^exp[k2

(

^I4−1

)

^2]−1

}

^{, (30)}

wherek₁>0isastress-likeconstantandk₂>0isadimensionlessconstant.

Thereis verylittleifanyinformationavailable aboutthemechanicalpropertiesofcross-links.Therefore,forsimplicity ofillustration,wemaketheassumptionthatc hasthequadraticreinforcingform

c=1

2

ν (

^I−1

)

², (31)

where

ν

^{isapositiveparameterwiththedimensionofstressthatmeasuresthestrengthofthe}cross-links,andisreferred to asthecross-linkparameter. Notethat the constantk₁ in(30)is differentfromthat in(3),andwe nowwrite itask₀ temporarily. Then, by comparingthe quadratic approximationof (3)with(30) and(31) weobtain

ν

+k0=k1(

ρ

),which relatesthecross-linkstiffness

ν

^{tothecross-linkdensityofthefirstmodel.Similarlyto(31),for}fcwetaketheform

fc=1

2

κ (

^I₈⁺−cos

α

⁰

)

²=1

2

κ (

^I₈⁻+cos

α

⁰

)

², (32)

where

κ

^{is also a positive} stress-like parameter. It measures the strength ofthe interaction between the fibers andthe cross-links.Hence,byusing(24)andaccordingto(28),theCauchystress

σ

^hastheform

σ

=

(

¹−

−

) μ ( λ

²−

λ

⁻¹

)

+2

^k1

( λ

²−1

) λ

^2exp[k2

( λ

²−1

)

^2]

+4

ν ( λ

^2cos2

α

0+

λ

^−1sin2

α

0−1

)( λ

^2cos2

α

0−

λ

^−1sin2

α

0

)

⁺⁴

κ ( λ

²⁻¹

) λ

^2cos2

α

0. (33) InFig.4weplotthedimensionlessstress

σ

¯=

σ

/

μ

^{againstthestretch}

λ

^forarepresentativeselectionoftheparameters involved in(33).Fig. 4(a)showshow the response dependsonthe orientation

α

0 of the cross-links fora fixed value of

ν

¯=

ν

/

μ

,whileFig.4(b)illustrates thedependenceon

ν

¯ ^{forafixed valueof}

α

0,ineachcaseforfixed valuesoftheother parameters,asspecifiedinthecaptionofFig.4.Itisclearthatthecross-linksstiffentheresponse.InFig.4(a)theresponse becomes stiffer asthe cross-links become more aligned withthe fibers, much stiffer than in the absenceof cross-links, whileFig.4(b)showsthatanincreaseinthedensityofthecross-linkslikewisestiffenstheresponse.

Fig. 4. Plots of the dimensionless Cauchy stress ¯σ= σ/μversus the stretch λ: (a) for three values of the cross-link angle α0( π/16, π/6, π/4) compared with the plot for the case of no cross-links. On the basis of (33) the following parameters were used = 0 . 1 , = 0 . 15 , k ¯1= k 1/μ^{= 1 , k} 2= 0 . 3 , ¯ν⁼ ν^/μ^{= 5 , ¯}κ⁼ κ^/μ^{= 1} ; (b) for four values of the dimensionless cross-link parameter ¯ν (10.0, 5.0, 1.0, 0). On the basis of (33) the following parameters were used = 0 . 2 , = 0 . 2 , k ¯1= 1 , k 2= 0 . 16 , α0= π/ 6 , ¯κ= 1 .

4. Formulationforageneralfiberdirection

Intheprevioussectionweconsideredaspecialdeformationwiththefiberdirectionsalignedwiththeaxis E₁ofexten- sion.Inthepresentsection wegeneralize thisforanarbitraryfiberdirectionandthecorresponding cross-links.Consider theaxis E₁ andtheassociated rectangularCartesian axes E₂ and E₃, whichare depictedinFig. 5. Toarrangethe initial generalgeometrywerotatethe systembymeans oftherotation tensor Qsuch thatthe unitbasis vectors E_i,i=1,2,3, become

e_i=QE_i, i=1,2,3, (34)

where

Q=e1E1+e2E2+e3E3, (35)

Fig. 5. Rectangular Cartesian axes E 1, E 2, E 3transformed into orthonormal axes e 1, e 2, e 3, where the transformation is a function of the two spherical polar angles θ^and φ^{. E} 1represents the fiber direction and, according to (39) , L ^±₀ are the directions of representative cross-links, which are rotationally symmetric with respect to ±E 1. E Rrepresents an arbitrary unit vector normal to E 1.

andhencewithrespecttosphericalpolarcoordinates

θ

^and

φ

^showninFig.5

e1=sin

θ

^cos

φ

^E1+sin

θ

^sin

φ

^E2+cos

θ

^E3, (36)

e2=cos

θ

^cos

φ

^E1+cos

θ

^sin

φ

^E2−sin

θ

^E3, (37)

e₃=−sin

φ

^E1+cos

φ

^E2, (38)

with e₁nowidentifiedasthecollagenfiberdirection.

Herethedirectionsoftwosymmetricallydisposedfamiliesofcross-linksaredenotedbytheunitvectorsL⁺₀ andL⁻₀,as distinctfromthenotation L^± usedin(16),sothat

L^±₀ =±cos

α

0E1+sin

α

0ER, (39)

where E_R,whichisanarbitraryvectororthogonalto E₁,canbewrittenas

ER=cos

φ

^0E2+sin

φ

^{0E3, (40)}

with

φ

0 arbitrary.Thenwedefine L^± accordingto

L^±=QL^±₀ =±cos

α

^0e1+sin

α

^{0er, er=QER=cos}

φ

^0e2+sin

φ

^{0e3. (41)}

Nowonapplicationofadeformationgradient FtheinvariantI₄ associatedwiththefiberdirectionisgivenby

I4=

(

^Fe1

)

·

(

^Fe1

)

=

(

^Ce1

)

·e1. (42)

Itfollowsfrom(41)₁ that

FL^±=±cos

α

^0Fe1+sin

α

^{0Fer. (43)}

Hence,theinvariantsI^±,andthequantitiesI^±₈ describingthecouplingbetweenthecollagenfiberandcross-linkdirections are

I^±=

(

^FL±

)

·

(

^FL±

)

=c²₀I₄±2s₀c₀

(

^Ce1

)

·er+s²₀

(

^Cer

)

·er, (44)

I^±₈ =

(

^Fe1

)

^·

(

^FL±

)

^=±c0I₄+s₀

(

^Ce1

)

^{·er, (45)}

where forconciseness we havewritten s₀=sin

α

0 andc₀=cos

α

0.Note that, ingeneral, I⁺=I⁻ and I₈⁺=−I₈⁻, which is unlikethecaseoftheuniaxialtensionconsideredinSection3.

NextwenotethederivativesoftheinvariantsI₄,I^± andthequantitiesI₈^±withrespecttotherightCauchy–Greentensor C,i.e.

∂

^I4

∂

^{C =e1e1, (46)}

∂

^I±

∂

^{C =c20e1e1±s0c0}

⁽

^e1er+ere1

⁾

^{+s20erer, (47)}

∂

^I8^±

∂

^{C =±c0e1e1+}

1

2s0

(

^e1er+ere1

)

. (48)

Nowletusconsiderthestrain-energyfunction(^I1,I₄,I⁺,I⁻,I⁺₈,I⁻₈)^sothat

σ

=−pI+2

ψ

1b+2

ψ

4Fe₁Fe₁+2

ψ

I⁺[c²₀Fe₁Fe₁+s₀c₀

(

^Fe1Fer+FerFe₁

)

+s²₀FerFer] +2

ψ

I⁻[c²₀Fe1Fe1−s0c0

(

^Fe1Fer+FerFe1

)

+s²₀FerFer]

+

ψ

8⁺[2c0Fe1Fe1+s0

(

^Fe1Fer+FerFe1

)

^]+

ψ

8⁻[−2c0Fe1Fe1+s0

(

^Fe1Fer+FerFe1

)

^], (49) where we have used the abbreviations

ψ

1=

∂

/

∂

^I1,

ψ

4=

∂

/

∂

^I4,

ψ

I^±=

∂

/

∂

^{I± and}

ψ

8^±=

∂

/

∂

^I±8. This is the most generalCauchystressexpressionforparallelcollagenfiberswithcross-linksofthetypeindicated.Fortherelatedelasticity tensorinthematerialdescriptionseetheAppendix.

Torecovertheuniaxialcase(Section3)fromthegeneralequationsinthissectionwehavee₁=E₁ foruniaxialtension, andconsequently

Fe₁=

λ

^e1, Fer=

λ

^−1/2er, FL^±=±c₀

λ

^e1+s₀

λ

^−1/2er. (50) Then,withI₄=

λ

^{2weobtainfrom(44)and(45)}

I≡I^±=c²₀

λ

²+s²₀

λ

⁻¹, I₈=I⁺₈ =c₀

λ

², I⁻₈ =−I₈⁺. (51) Then

ψ

I=

ψ

I⁺=

ψ

I⁻,

ψ

8⁺=−

ψ

8⁻=

ψ

8, (52) and(49)specializesto

σ

^=−pI+2

ψ

1

( λ

^2e1e₁+

λ

^−1erer

)

⁺²

ψ

4

λ

^2e1e₁+4

ψ

I

(

^c20

λ

^2e1e₁+s²₀

λ

^−1erer

)

⁺⁴

ψ

8c₀

λ

^2e1e₁. (53) Therelatedcomponentsare

σ

⁼

σ

^11=−p+2

ψ

¹

λ

²⁺²

ψ

⁴

λ

²⁺⁴

ψ

^Ic20

λ

²⁺⁴

ψ

^8c0

λ

^{2, (54)}

0=

σ

^rr=−p+2

ψ

¹

λ

⁻¹⁺⁴

ψ

^Is20

λ

^{−1. (55)}

ByeliminatingtheLagrangemultiplierpweobtain

σ

⁼²

ψ

¹

( λ

²⁻

λ

⁻¹

)

+2

ψ

⁴

λ

²⁺⁴

ψ

^I

(

^c20

λ

^2−s20

λ

⁻¹

)

+4

ψ

^8c0

λ

^{2. (56)}

Nowletususethespecificstrain-energyfunctions(29)–(32),i.e.

^{= 1}₂

μ (

^I1−3

)

⁺₂^k_k¹

2

{

^exp[k2

(

^I4−1

)

^2]−1

}

⁺¹₂

ν (

^I−1

)

²⁺¹₂

κ (

^I8−c0

)

^{2, (57)}

whichgiveswith(56)thesameexpressionfor

σ

^{asin(33)exceptthatherethevolumefractionsare}incorporatedintothe materialconstants

μ

^,k1,

ν

^and

κ

^.

4.1. Planarformulation

Nextweconsiderthesituationinwhichthefibersandcross-linksarerestrictedtothe(E₁, E₂)planeandwedefinethe fiberdirection e₁ anditsnormal eras

e1=cos

α

^E1+sin

α

^{E2, er=−sin}

α

^E1+cos

α

^{E2, (58)}

where

α

^{istheanglebetweenthefiberdirectionandthe E}1 axis(seeFig.6).

Withrespectto e₁ and erthecross-linkdirections L^± aredefinedby

L^±=±c0e1+s0er, (59)

which hasthe same form as(41)₁. The invariant I₄=(^Ce1)·e₁, as in(42)₂, butwith e₁ now defined by (58). We also have

FL^±=±c0Fe₁+s₀Fer, (60)

whichis the sameexpression as(43) andthe invariantsI^± andthe quantities I₈^± are againgiven by (44)and (45). The Cauchystresstensor

σ

^{hasthesameform(49)asin3Dbutnowrestrictedto2D.}