A continuum model of skeletal muscle tissue with loss of activation

0.6 MB·

by  Giulia Giantesio

A CONTINUUM MODEL OF SKELETAL MUSCLE TISSUE WITH LOSS OF
ACTIVATION
GIULIAGIANTESIOANDALESSANDROMUSESTI
7
1
0
2
n ABSTRACT. Wepresentacontinuummodelforthemechanicalbehavioroftheskeletal
a muscletissuewhenitsfunctionalityisreducedduetoaging. Thelossofabilityofacti-
J vatingistypicalofthegeriatricsyndromecalledsarcopenia.Thematerialisdescribedby
ahyperelastic,polyconvex,transverseisotropicstrainenergyfunction.Thethreematerial
6
parametersappearingintheenergyarefittedbyleastsquareoptimizationonexperimen-
2
taldata,whileincompressibilityisassumedthroughaLagrangemultiplierrepresentingthe

hydrostaticpressure.Theactivationofthemusclefibers,whichisrelatedtothecontraction
]
O ofthesarcomere,ismodeledbythesocalledactivestrainapproach. Thelossofperfor-
manceofaneldermuscleisthenobtainedbyloweringofsomepercentagetheactivepart
T
ofthestress.Themodelisimplementednumericallyandtheobtainedresultsarediscussed
o. andgraphicallyrepresented.
i
b
- 1. INTRODUCTION
q
[ Skeletalmuscletissueisoneofthemaincomponentsofthehumanbody,beingabout

40%ofitstotalmass. Itsprincipalroleistheproductionofforce,whichsupportsthebody
1
andbecomesmovementbyactingonbones. Themechanismbywhichamuscleproduces
v
3 forceiscalledactivation.
2 Skeletalmuscletissueisahighlyorderedhierarchicalstructure. Thecellsofthetissue
8 arethemuscularfibers,havingalengthuptoseveralcentimeters;theyareorganizedinfas-
7
cicles,whereeveryfiberismultiplyconnectedtonerveaxons,whichdrivetheactivation
0
ofthetissue. Connectivetissue, whichisessentiallyisotropic, fillsthespacesamongthe
.
1
fibers. Everyfibercontainsaconcatenationofmillionsofsarcomeres,whicharethefun-
0
damentalunitofthemuscle. Withalengthofsomemicrometers,asarcomereiscomposed
7
1 bychainsofproteins,mainlyactinandmyosin,whichcanslideoneachother. Thissliding
: movementproducesthecontractionofthesarcomereand,ultimately,thecontractionofthe
v
wholemuscleandtheproductionofforceandmovement.
i
X The aim of this Chapter is to propose a mathematical model of skeletal muscle tissue
r withareducedactivation,whichistypicalofageriatricsyndromenamedsarcopenia[16].
a
About thirty years ago, the term sarcopenia (from Greek sarx or flesh and penia or loss)
hasbeenintroducedinordertodescribetheage-relateddecreaseofmusclemassandper-
formance. Sarcopeniahassincethenbeendefinedasthelossofskeletalmusclemassand
strength that occurs with advancing age, which in turn affects balance, gait and overall
abilitytoperformeventhesimpletasksofdailylivingsuchasrisingfromachairorclimb-
ingsteps. Accordingto[6],sarcopeniaaffectsmorethan50millionspeopletodayandit
willaffectmorethan200millionsinthenext40years. Thereisstillnogenerallyaccepted
test for its diagnosis and many efforts are made nowadays by the medical community to
betterunderstandthissyndrome. Thereforeitisdesirabletobuildamathematicalmodel
of muscle tissue affected by sarcopenia. However, to the best of our knowledge, in the
biomathematicalliteraturethetopicoflossofactivationhasneverbeenaddressed.
1
2 GIULIAGIANTESIOANDALESSANDROMUSESTI
InordertousethevaluabletoolsofContinuumMechanics,duringthelastdecadesthe
skeletalmuscletissuehasbeenoftenmodeledasacontinuummaterial[7,11,4,5],which
isusuallyassumedtobetransverselyisotropicandincompressible.Theformerassumption
ismotivatedbythealignmentofthemuscularfibers,whilethelatterisensuredbythehigh
water content of the tissue (about 75% of the total volume). Moreover, in view of some
experimentaltests,thematerialisassumedtobenonlinearandviscoelastic. Focusingour
attentiononlyonthesteadypropertiesofthetissue,hereweneglecttheviscouseffectsand
wesetintheframeworkofhyperelasticity.
In the model that we propose, there are three constitutive prescriptions: one for the
hyperelastic energy when the tissue is not active (passive energy), one for the activation
andoneforthelossofperformance. Asfarasthepassivepartisconcerned,weassumean
exponentialstressresponseofthematerial,whichiscustomaryinbiologicaltissues. The
particular form that we choose, being a slight simplification of the one proposed in [7],
hastheadvantageofbeingpolyconvexandcoercive,givingmathematicalsoundnesstothe
modelandstabilitytothenumericalsimulations.
Arecentandverypromisingwaytodescribetheactivationistheactivestrainapproach,
wheretheextraenergyproducedbytheactivationmechanismisencodedinamultiplicative
decompositionofthedeformationgradientinanelasticandanactivepart(seeSection2.2).
Unliketheclassicalactivestressapproach,inwhichtheactivepartofthestressismodeled
in a pure phenomenological way and a new term has to be added to the passive energy,
theactivestrainmethoddoesnotchangetheformoftheelasticenergy,keepinginpartic-
ularallitsmathematicalproperties. Moreover,atleastinthecaseofskeletalmuscles,the
active strain approach seems to be more adherent to the physiology of the tissue, in the
sensethatatthemolecularleveltheproductionofforceisactuallygivenbyadeformation
ofthematerial, thankstothecontractionofthesarcomeres. Theactivepartofthedefor-
mationgradientisamathematicalrepresentationofsuchacontraction. Themultiplicative
decomposition of the deformation gradient has been applied to an active striated muscle
in[18,12]. However,thisdecompositioninvolvesonlyapartofthewholeelasticenergy,
which is written as the sum of two terms for the case of a fiber-reinforced material. As
farasweknow,theactivestrainapproachhasneverbeenpreviouslyappliedtothewhole
elastic energy of a skeletal muscle tissue. As a drawback, the active strain approach can
beasourceofsometechnicaldifficulties;forinstance,inourcasefittingthemodelonthe
experimentaldataisnotsosimple,seeeq.(13).
Furthermore, we consider the loss of performance, which is one of the novelties of
our model. Unfortunately, there are no experimental data on the elastic properties of a
sarcopenic muscle tissue, at least to our knowledge; hence we adopt the naive strategy
ofreducingtheactivepartofthestress(whichisthedifferencebetweenthestressofthe
material with and without activation) by a given percentage, represented by the damage
parameterd (seeSection3.2). Inthisway,thereisasingleparameterinthemodelwhich
conciselyaccountsforanyeffectofthedisease.
Theproposedmodelcanbenumericallyimplementedusingfiniteelementmethods. In
Section 4 we present some results obtained using FEniCS, an open source collection of
Python libraries. Actually, we consider a cylindrical geometry with radial symmetry, so
thatthenumericaldomainistwo-dimensionalandthecomputationalcostisreduced. As
farastheboundaryconditionsareconcerned,weprescribethedisplacementonthebases
ofthecylinderandletthelateralsurfacetraction-free. Suchsimulationsshowthattheex-
perimental results of [10] on the passive and active stress-strain healthy curves, obtained
in vivo from a tetanized tibialis anterior of a rat, can be well reproduced by our model.
ACONTINUUMMODELOFSKELETALMUSCLETISSUEWITHLOSSOFACTIVATION 3
Further, the behavior of the tissue when d increases is analyzed. An ongoing task is to
performafiniteelementimplementationofthemodelwhengenericloadsareapplied,and
toconsiderarealisticthree-dimensionalmusclemesh. Wearenowdevelopingatrulyhy-
perelasticmodel,wheretheexpressionofthestresstakesintoaccountalsothedependence
of the activation on the deformation gradient. Actually, in this chapter the stress is com-
putedasthederivativeofthehyperelasticenergykeepingtheactivepartofthedeformation
gradientfixed.
Inthefuture, itwillbeveryinterestingtofindsomeconnectionsbetweenthedamage
percentage (the parameter d) and other physiological quantities, such as the mass of the
musculartissueortheneuronalactivity. Anotherimportanttopicwillbetheapplicationof
somehomogenizationtechniquesinordertodeduceanimprovedconstitutiveequationfor
theskeletalmusclestartingfromitsmicrostructure.
2. CONSTITUTIVEMODEL
Skeletalmuscletissueischaracterizedbydenselypackedmusclefibers, whicharear-
ranged in fascicles. Filling the spaces between the fibers and fascicles, connective tissue
surrounds the muscle and it is responsible of the elastic recoil of the muscle to elonga-
tion. Besidesalargeamountofwater,thefibersthemselvescontaintitin,actinandmyosin
filaments. The latter two sliding elements form the actual contractile component of the
muscle,whichiscalledsarcomere. Sincethefiberslocallyfollowapredominantunidirec-
tional alignment, transverse isotropy with respect to that main direction can be assumed.
We hence begin by modelling the skeletal muscle tissue as a transversely isotropic non-
linearhyperelasticmaterialwithprincipaldirectionm,whichfollowsthealignmentofthe
musclefibers.
2.1. Passive model. Let F denote the deformation gradient tensor, C=FTF the right
Cauchy-GreentensorandM=m mthesocalledstructuraltensor. IfΩdenotestheref-
⊗
erenceconfigurationoccupiedbythemuscle,wedescribeitspassivebehaviorbychoosing
ahyperelasticstrainenergyfunction
(cid:90)
W(C)dV,
Ω
wherethestrainenergydensityisoftheform
µ (cid:26)1 (cid:104) (cid:105) (cid:27)
(1) W(C)= eα(Ip−1) 1 +Kp 1 ,
4 α − −
with
w w
Ip= 0tr(C)+(1 w0)tr(CM), Kp= 0tr(C−1)+(1 w0)tr(C−1M).
3 − 3 −
Here µ is an elastic parameter and α and w are positive dimensionless material param-
0
eters. The generalized invariants I and K are given by a weighted combination of the
p p
isotropicandanisotropiccomponents;inparticular,w measurestheratioofisotropictis-
0
sue constituents and 1 w0 that of muscle fibers. Moreover, the term tr(CM) represents
−
the squared stretch in the direction of the muscle fiber and is thus associated with lon-
gitudinal fiber properties, while the term tr(C−1M) describes the change of the squared
cross-sectional area of a surface element which is normal to the direction m in the refer-
ence configuration and thus relates to the transverse behavior of the material [20, 9] (see
Fig.2).
4 GIULIAGIANTESIOANDALESSANDROMUSESTI
µ [kPa] α [-] w [-]
0
0.1599 19.35 0.7335
TABLE1. Materialparametersofthepassivemodel.
Oneofthemathematicalfeaturesoftheenergydensity(1)isthatitispolyconvexand
coercive [8, 20], hence the equilibrium problem with mixed boundary conditions is well
posed.
We remark that C is the identity tensor I in the reference configuration, so that Ip =
K =1,i.e. wehavetheenergy-andstress-freestateofthepassivemuscletissue(see[8]).
p
Thehighcontentofwaterisresponsibleofthenearlyincompressiblebehaviorwhichis
experimentallyreportedformusclefibers,sothatwecanassume
(2) detC=1.
Asiscustomaryinhyperelasticity,thefirstPiola-Kirchhoffstresstensor,knownasnom-
inalstresstensor,canbedirectlycomputedbydifferentiatingthestrainenergyfunction:
∂W ∂W
(3)P = pF−T =2F pF−T =
∂F − ∂C −
µ (cid:110) (cid:104)w (cid:105) (cid:104)w (cid:105) (cid:111)
= F eα(Ip−1) 0I+(1 w0)M C−1 0I+(1 w0)M C−1 pF−T,
2 3 − − 3 − −
where p is a Lagrange multiplier associated with the hydrostatic pressure which results
fromtheincompressibilityconstraint(2).
Thematerialparametersofthemodelcanbeobtainedfromrealdata. Moreprecisely,
concerning the elastic parameter µ, we use the value given in [7], while the other two
parametershavebeenobtainedbyleastsquaresoptimizationusingtheexperimentaldata
byHawkinsandBey[10]aboutthestretchresponseofatetanizedtibialisanteriorofarat
(seeFig.1). InTable2.1wefurnishthevaluesoftheparameters.
60
Hawkins and Bey
a]50 Passive model
P
k
[

1140
P

s
s
e30
r
t
s

l
a20
n
i
m
o10
N
0
1.0 1.1 1.2 1.3 1.4
Muscle stretch
FIGURE 1. Comparison of the passive model in uniaxial tension with
theexperimentaldataofarattibialisanteriormusclereportedin[10].
ACONTINUUMMODELOFSKELETALMUSCLETISSUEWITHLOSSOFACTIVATION 5
2.0
Longitudinal uniaxial
Transverse uniaxial
] 1.5
a Transverse biaxial
P
k
[ 1.0
s
s
e
r
t 0.5
s

l
a
n
i 0.0
m
o
N
0.5
1.0
0.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4
Muscle stretch
FIGURE2. Transverselyisotropicbehaviorofthemodel.
Weremarkthatthestrainenergyfunction(1)isaslightsimplificationoftheonepro-
posedbyEhret,Bo¨landItskovin[7]:
µ (cid:26)1 (cid:104) (cid:105) 1 (cid:104) (cid:105)(cid:27)
(4) WEBI(C)= eα(Ip−1) 1 + eβ(Kp−1) 1 ,
4 α − β −
whereα=19.69,β =1.190,w =0.7388.Actually,oursimplificationconsistsinlineariz-
0
ingthetermrelatedtoK , whichdescribesthetransversebehavior. Thisismotivatedby
p
the fact that the parameter β is much smaller than α. In Fig. 3 we can see the compari-
sonbetweenthenominalstressinthedirectionofthestretchofthetwomodelswhenthe
musclefibersareelongatedintheirdirection.
5
Proposed
a] Ehret et al.
P4
k
[ 0.00
1 0.02
1
P3 0.04
s
s 0.06
e 0.08
r
st2 0.10
l 0.12
a
n 0.14
i
m 0.16
1 0.70 0.75 0.80 0.85 0.90 0.95 1.00
o
N
0
0.7 0.8 0.9 1.0 1.1 1.2 1.3
Muscle stretch
FIGURE 3. Comparison between the passive stress here proposed and
theonestudiedin[7]duringuniaxialtensionalongthefibers.
6 GIULIAGIANTESIOANDALESSANDROMUSESTI
2.2. Activemodel. Oneofthemainfeaturesoftheskeletalmuscletissueisitsabilityof
beingvoluntarilyactivated.Skeletalmusclesareactivatedthroughelectricalimpulsesfrom
motor nerves; the activation triggers a chemical reaction between the actin and myosin
filaments which produces a sliding of the molecular chains, causing a contraction of the
musclefibers.
During the last decades, many authors tried to mathematically model the process of
activation,mainlywithtwodifferentapproaches(forareviewsee[2]). Themostfamous
approachfollowedintheliteratureiscalledactivestressanditconsistsinaddinganextra
term to the stress, which accounts for the contribution given by the activation (see for
example[15,3,11]). However,thisisanadhocmethod,usuallynotrelatedtothesliding
movementofthefilamentsinthesarcomeres,whichisthemainmechanismofcontraction
atthemesoscale.
Morerecently,theactivestrainapproachwasproposedbyTaberandPerucchio[21]in
ordertodescribetheactivationofthecardiactissue,followingprevioustheoriesofgrowth
and morphogenesis, as well as several models of plasticity. The method for soft living
tissuesisexplainedin[17]. Differentlyfromtheactivestressapproach,thismethoddoes
notchangetheformofthestrainenergyfunction;rather,itassumesthatonlyapartofthe
deformation gradient, obtained by a multiplicative decomposition, is responsible for the
storeofelasticenergy. Thismethodisrelatedtothebiologicalmeaningofactivationand
canbereasonablyadoptedalsoinourcase. Tothebestofourknowledge,theactivestrain
approachhasneverbeenfollowedfortheskeletalmuscletissueinliterature.
We begin by rewriting the deformation gradient as F=FeFa, where Fe is the elastic
partandFa describestheactivecontribution(seeFig.4). TheactivestrainFa representsa
F
F
a
F =FF 1
e −a
FIGURE4. Pictorialviewoftheactivestrainapproach.
changeofthereferencevolumeelementsduetothecontractionofthesarcomeres,sothat
itdoesnotcontributetotheelasticenergy. Areferencevolumeelement, distortedbyFa,
needsafurtherdeformationFe tomatchtheactualvolumeelement,whichaccommodates
both the external forces and the active contraction. Notice that neither Fa nor Fe need
to be the gradients of some displacement, that is, it is not necessary that they fulfill the
compatibilityconditioncurlFa=0orcurlFe=0.
The volume elements are modified by the internal active forces without changing the
elastic energy, hence the strain energy function of the activated material has to be com-
puted using Ce =FTeFe and taking into account Fe =FF−a1. If Fa =gradχa for some
ACONTINUUMMODELOFSKELETALMUSCLETISSUEWITHLOSSOFACTIVATION 7
displacementχ ,thenfromFig.4byachangeofvariablesitiseasytoseethat
a
(cid:90) (cid:90)
W(Ce)dV(cid:98)= W(F−aTCF−a1)(detFa)dV.
χa(Ω) Ω
Theright-handsideofthepreviousequationiswelldefinedalsowhenFa doesnotcome
fromaglobaldisplacement,anditdescribesthestrainenergyoftheactivebody. Wethen
obtainthemodifiedhyperelasticenergydensity
W(cid:98)(C)=(detFa)W(Ce)=(detFa)W(F−aTCF−a1).
WenowhavetomodeltheactivepartFa. Sincetheactivationofthemuscleconsistsin
acontractionalongthefibers,wechoose
(5) Fa=I γm m,
− ⊗
where0 γ <1isadimensionlessparameterrepresentingtherelativecontractionofac-
≤
tivatedfibers(γ =0meaningnoactivation). Thenthemodifiedstrainenergydensitybe-
comes
µ (cid:26)1 (cid:104) (cid:105) (cid:27)
(6) W(cid:98)(C)=(1 γ)W(Ce)=(1 γ) eα(Ie−1) 1 +Ke 1 ,
− − 4 α − −
w w
Ie= 0tr(Ce)+(1 w0)tr(CeM), Ke= 0tr(Ce−1)+(1 w0)tr(Ce−1M).
3 − 3 −
ThecorrespondingfirstPiola-Kirchhoffstresstensorisgivenby
∂W
(P(cid:98)7)= detFa∂FeF−a1−p(cid:98)F−T =
µ (cid:110) (cid:104)w (cid:105) (cid:104)w (cid:105) (cid:111)
= 2(1−γ)Fe eα(Ie−1) 30I+(1−w0)M −C−e1 30I+(1−w0)M C−e1 F−a1
p(cid:98)F−T,
−
where p(cid:98)accounts for the incompressibility constraint detC=1. Notice that, since the
activation(5)doesnotpreservevolumeandthematerialhastobegloballyincompressible,
onehasthatdetCe=1,sothatthematerialiselasticallycompressible. Asfarasthestrain
(cid:54)
energydensityisconcerned, afactor(1 γ)appearsin(6)whichkeepsintoaccountthe
−
compressibilityofFa. Itwouldbeinterestingtostudyalsootherkindsofpassiveenergies,
involving the quantity detC, in order to better describe the elastic compressibility of the
material.
InFig.5werepresent,forseveralvaluesoftheparameterγ,thestress-straincurvefor
a uniaxial tension along the fibers. If the muscle is activated (γ >0), then (the absolute
valueof)thestressincreaseswithγ andthevalueofthestretchsuchthatthestressiszero
becomeslessthanone.
3. MODELLINGTHEACTIVATIONONEXPERIMENTALDATA
The activation parameter γ, which was assumed constant in the previous section, in
fact usually depends on the deformation gradient. In typical experiments on a tetanized
skeletalmuscleitisapparentthatthecontractionofthefibersduetoactivationvarieswith
theirstretch,reachingamaximumvalueandthendecreasing. Fig.6showsthequalitative
relationbetweentheelongationandthedevelopedstress. Thissectionwillbedevotedto
takingintoaccountthisphenomenon. Specifically,theexpressionofγ willbedetermined
matchinganexperiment-basedrelationbetweenstressandstrainwithourmodel(7).
8 GIULIAGIANTESIOANDALESSANDROMUSESTI
100
a] 80
P
K
[ 60

1
1
P
s 40
s
e
r
t 20
s

l
a
n 0
i
m
o 20
N
40
0.6 0.8 1.0 1.2 1.4
Muscle stretch
FIGURE5. Stress-stretchcurvesinuniaxialtensionforseveralvaluesof
γ from0to0.4.
Inordertofindtherelationbetweenstressandstrain,theexperimentsinvivoareusu-
allyperformedintwosteps. First,thestress-straincurveisobtainedwithoutanyactivation
(passive curve). Second, by an electrical stimulus the muscle is isometrically kept in a
tetanizedstateandthetotalstress-straincurveisplotted. Thelastcurve,whichisqualita-
tivelyrepresentedinFig.6,dependsonthereciprocalpositionofactinandmyosinchains.
Bytakingthedifferenceofthetwocurvesonecanobtaintheactivecurve,describingthe
amount of stress due to activation. It is useful to find a mathematical expression of such
acurve,inordertotakeintoaccounttheexperimentalbehavioroftheactivecontraction.
Thisissuehasalreadybeenaddressedinseveralpapers,seee.g. [7,11,22,23,13,3,4].
Denoting with λ the ratio between the current length of the muscle and its original
length,weassumetheactivecurvetobeoftheform
 (cid:34) (cid:35)
(λ2 λ2 )2
Poptexp k − opt ifλ >λmin,
(8) Pact(λ)= − λ−λmin
0 otherwise,
where λ is the minimum stretch value after which the activation starts (i.e. the lower
min
boundforthestretchatwhichthemyofilamentsbegintooverlap)andkismerelyafitting
parameter. Thecoordinates(λ ,P )identifythepositionofthemaximumofthecurve.
opt opt
As it is explained in [7], the value of P takes into account some information at the
opt
mesoscalelevel,suchasthenumberofactivatedmotorunitsandtheinterstimulusinterval;
accordingtotheliterature[7,11],itissetatP =73.52kPa. Thenumericalvaluesofthe
opt
other three parameters, deduced through least squares optimization on the data reported
in [10], are given in Table 2. The expression (8) has the advantage of describing the
asymmetrybetweentheascendinganddescendingbranchesoftheactivecurveobtainedin
[10].Indeed,eveniftheasymmetryisnotsoevidentintheircurve,duetothefactthatthere
areonlyfewdataonthedescendingbranch,itisatypicalfeatureofseveralexperimentally
measured sarcomere length-force relation. Moreover, as one can easily see in Fig. 7, the
convexbehaviorofthedatanearbyλ iswellfitted.
min
ACONTINUUMMODELOFSKELETALMUSCLETISSUEWITHLOSSOFACTIVATION 9
total force
active force
passive force
n
o
nsi
e
d t
e
p
o
el
v
e
d
Lmin L0 Lopt sarcomere length (L)
FIGURE6. Length-tensionrelationshipofasarcomere. Herewedenote
byL thelengthatwhichthesarcomereproducesthemaximumforce
opt
inisometricexperiments,byL therestlengthandbyL theminimal
0 min
lengthofthesarcomere(fullyactivated).
TABLE2. Materialparametersoftheactivemodel.
λ [-] λ [-] k[-] P [kPa]
min opt opt
0.6243 1.1704 0.4342 73.52
80
Hawkins and Bey
70 approximation
60
]
a50
P
k
[40
)
(
act30
P
20
10
0
min 1 opt 1.5 2 2.5
FIGURE 7. Plotoftheactivecurve(8)withtheparametersreportedin
Table 2 together with the representation of the experimental data given
in[10].
10 GIULIAGIANTESIOANDALESSANDROMUSESTI
3.1. The activation parameter γ as a function of the elongation. Now our aim is to
obtainP (λ)givenin(8)fromthemodeldescribedinSection2.2. Inordertoreachour
act
purpose,wehavetomodeltheactivationparameterγ asafunctionofthestretch.
AsintheexperimentsofHawkinsandBey[10],letusconsiderauniaxialsimpletension
along the fibers. For simplicity, we assume that the fibers follow the direction m=e1.
Since the skeletal muscle tissue is modeled as an incompressible transversely isotropic
material,thegeneralformofthedeformationgradientFisgivenby
 
λ 0 0
F=0 √1λ 0 .
0 0 1
√λ
ThenusingthenotationintroducedinSection2.2,onehas
 
λ2 0 0
(1 γ)2
Ce = −0 1 0,
λ
0 0 1
λ
w (cid:20) λ2 2(cid:21) λ2
I = 0 + +(1 w ) ,
e 3 (1 γ)2 λ − 0 (1 γ)2
− −
w (cid:20)(1 γ)2 (cid:21) (1 γ)2
K = 0 − +2λ +(1 w ) − .
e 3 λ2 − 0 λ2
In this case, it is convenient to look at the strain energy as a function of the stretch λ
andtheactivationparameterγ:
µ (cid:26)1 (cid:104) (cid:105) (cid:27)
(9) W(cid:98)(λ,γ)=(1 γ)W(λ,γ)=(1 γ) eα(Ie−1) 1 +Ke 1 .
− − 4 α − −
Thenthenominalstressalongthefiberdirectionisgivenby
∂W(cid:98) µ (cid:104) (cid:105)
(10) Ptot(λ,γ):= ∂λ =(1−γ)4 Ie(cid:48)eα(Ie−1)+Ke(cid:48) ,
where
(cid:20) (cid:21)
∂I w λ 1 λ
I = e =2 0 +2(1 w ) ,
e(cid:48) ∂λ 3 (1 γ)2 −λ2 − 0 (1 γ)2
− −
∂K w (cid:20) (1 γ)2 (cid:21) (1 γ)2
K = e =2 0 − +1 2(1 w ) − .
e(cid:48) ∂λ 3 − λ3 − − 0 λ3
Wecangetthepassivestressbysettingγ =0:
(cid:26)(cid:20)(cid:18) (cid:19) (cid:21)
Ppas(λ):=Ptot(λ,0)= µ2 1−23w0 λ−w30λ12 eα[(1−23w0)λ2+w30λ2−1]
(cid:18) (cid:19) (cid:27)
2 1 w
(11) 1 w + 0 .
− −3 0 λ3 3
We remark that the values of P and P can also be obtained by computing the first
tot pas
componentofthestressgivenby(7)and(3)afterfindingthehydrostaticpressurefromthe
conditionsP(cid:98)22=P(cid:98)33=P22=P33=0(traction-freelateralsurface).
Ouraimistofindthevalueofγ suchthat
(12) P (λ,γ)=P (λ)+P (λ),
tot act pas