Existence and nonexistence for the full thermomechanical Souza-Auricchio model of shape memory

17 Pages·2011·0.59 MB·English

Article
MathematicsandMechanicsofSolids
16(4):349–365
Existence and non-existence for the ©TheAuthor(s)2011
Reprintsandpermission:
full thermomechanical sagepub.co.uk/journalsPermissions.nav
DOI:10.1177/1081286510386935
Souza–Auricchio model of shape mms.sagepub.com
memory wires
Pavel Krejcˇí
Matematický ústav AVCˇR, Praha, Czech Republic
Ulisse Stefanelli
IMATI – CNR, Pavia, Italy
Received 15June2010;accepted14September2010
Abstract
Weprovideanexistencetheoryforthefullthermomechanicalquasi-staticevolutionofashapememorywiredescribed
bytheSouza–Auricchioconstitutivemodel.Theanalysisrequiressomemildrestrictiononthechoiceofthethermome-
chanical coupling term in the expression on the free energy of the material. This restriction slightly deviates from the
originalSouza–Auricchiomodelingframe,whilestillbeingcompatiblewithrealsituations.Weadditionallyshowthat,by
imposingnosuchrestriction,theoriginalSouza–Auricchiomodelisill-posed.
Keywords
shapememoryalloys,thermomechanics,existenceresult,blowupinfinitetime
1. Introduction
Shape memory materials are metallic alloys (but also polymers and ceramics) exhibiting an amazing ther-
momechanical behavior: severely deformed specimens regain their original shape after a thermal cycle. This
is the so-called shape memory effect. Moreover, at some higher temperature regimes, the same materials are
superelastic,namely,theyrecoverlargedeformationsduringmechanicalloading–unloadingcycles[1].
These features are not present (at least to a comparable extent) in materials traditionally used in engi-
neering and are at the basis of a variety of innovative applications to aeronautical, structural, and earthquake
technologies [2, 3]. The most successful application field of shape memory alloys (SMAs) is probably that
ofbiomedicaldevices.Nowadays,SMAsaresuccessfullyusedinorthodontics(archwires),orthopedics(bone
anchors,intramedullaryfixations,bonestaples),medicalinstrumentsandminimalinvasivesurgery/technology
(endoguidewires, grippers, cutters, vena cava filters), and drug delivery systems. The most emerging biomed-
ical applications of SMAs are stents for intra-vascular or extra-vascular scaffolding (cardiovascular stenting,
bronchialbiliary,aorticaneurysm,carotidstenosis).
Correspondingauthor:
UlisseStefanelli,IMATI–CNR,viaFerrata1,I-27100Pavia,Italy
Email:[email protected]
350 MathematicsandMechanicsofSolids16(4)
The relevance of these applications recently trimmed an intense and always increasing interest in the
description of the thermomechanical behavior of SMAs and a whole menagerie of models has been pro-
posedbyaddressingdifferentalloys(NiTi,CuAlNi,Ni MnGa,andmanyothers)atdifferentscales(atomistic,
2
microscopic with micro-structures, mesoscopic with volume fractions, macroscopic) and emphasizing differ-
ent principles (minimization of stored energy vs. maximization of dissipation, phenomenology vs. rational
crystallography and thermodynamics) and different structures (single crystals vs. polycrystalline aggregates,
possibly including intragranular interaction). Correspondingly, the engineering literature on SMA modeling
is vast. By limiting ourselves to macroscopic-phenomenological models we can refer, without claim of com-
pleteness, to a variety of references [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16]. The referred models have
of course ambitions for different ranges of applicability (from laboratory single-crystal experiments to com-
mercially exploitable tools) and different abilities to fit particular experiments and to explain microstructures,
stress/strainrelations,orhysteresis.Themutualrelationsbetweenmodelsatdifferentscalesaretoalargeextent
stilltobeinvestigated[17].
Weshallfocushereonaphenomenological,internal-variable-typemodelmodelforpolycrystallinemateri-
als,whichiscapableofdescribingboththeshapememoryandthesuperelasticeffect.Themodelwasoriginally
advancedinthesmall-strainregimebySouzaetal.[18]andthencombinedwithfiniteelementsbyAuricchio
andPetrini[19,20,21].WehencerefertoitastheSouza–Auricchiomodelwhatfollows.
The interest in this model is motivated by its robustness with respect to parameters and discretizations
despite its simplicity: in the three-dimensional situation, the constitutive behavior of the specimen is deter-
minedbytheknowledgeofjusteightmaterialparameters,see(2.4)(notethatlinearizedthermoplasticitywith
linearhardeningalreadyrequiresfivematerialparameters).Theseparametersaredirectlyavailableastheycan
be easilyfittedfrom experimentaldata (see Section2 below). The Souza–Auricchiomodel has been analyzed
from the viewpoint of the existence and approximation of solutions of the three-dimensional quasi-static evo-
lution problem [22]. Later, convergence rates for space-time discretization of the problem were obtained [23,
24]. Extensions of the original model to the case of permanent inelastic effects have been provided [25, 27]
and further improvements in the direction of the description of more realistic non-symmetric behaviors and
transformation-dependentmaterialparametershavealsobeenreported[28].
All the above-mentioned contributions on the Souza–Auricchio model focus on the isothermal situation,
namely the temperature of the specimen is assumed to be fixed, uniform, and known. In particular, the cited
literature is basically concerned with the description of the super-elastic effect only, for no shape memory
behavioroccurswithouttemperaturechanges.
Some first results in the direction of including temperature changes in the Souza–Auricchio model have
beenobtainedby Mielkeetal.[29,30].Inthesepapers,thetemperatureofthebodyisassumedtobechanging
in time, being however given a priori. This simplification is justified if the body is relatively thin in at least
one direction and the mechanical evolution is so slow that the produced heat may be assumed to be (almost)
instantaneouslydissipated(almost)intheenvironment.However,simplecyclicloading–unloadingexperiments
onwiresrevealthattheheatproductionduetothedissipativephasetransformationisnotatallnegligible[31]
and the temperature rise of the specimen with respect to the surrounding environment can indeed be relevant.
This applicative observation and the interest in a complete material theory are the leading motivations for
ouranalysisinthefullthermomechanicallycoupledevolutionofSMAspecimensundertheSouza–Auricchio
model.
The main issue of this paper is hence that of addressing a full thermomechanical quasi-static evolution
problem, letting indeed the temperature of the body be one of the unknowns of the system. Our main result
is an existence theory for the resulting system of nonlinear PDEs derived from the conservation of energy
and momentum (quasi-static) and the constitutive equations of the material. The analysis deeply relies on the
specific form of the free energy for the Souza–Auricchio model and is obtained by space-discretization and
passagetothelimit.StatementsandproofsaregiveninSections3–5.
Acrucialby-productofourinvestigationistheconstructiveproofofthefactthattheoriginalformulationof
theSouza–Auricchiomodelnecessarilyrequiressome(minor)modificationinordertocomplywiththermody-
namics,thismodificationbeingindeedcompatiblewithrealexperimentaldata.Inparticular,weshowthatthe
modelisill-posedifthedependenceofthelatentheatontemperatureisnotsmoothenough,orifthehardening
KrejcˇíandStefanelli 351
constant is too small, or if the dissipation is too large. In these cases, we explicitly construct solutions which
failtoexistforalltimes.DetailsinthisdirectionwillbegiveninSection4.1.
Before closing this introduction let us mention that, besides the Souza–Auricchio framework, at least two
othermodelsforSMAshaveprovedtopermitsomeanalyticdiscussioninthefullycoupledthermomechanical
situation. These are the Frémond [6] and the Falk et al. models [5, 32]. With respect to these models, the
Souza–Auricchiomodelprovidessomeclearadvantagesasitdirectlyfeaturesarate-independentevolutionof
internalvariableswithrespecttostressandtemperatureanditiscapableofreproducingthecrucialphenomenon
of martensitic reorientation in the three-dimensional situation. It is beyond our purposes even to attempt a
completeliteraturereviewonmathematicalresultsforSMA-relatedsystemsofPDEs.Thereaderis,however,
referredtovariouscontributions[33–43]andtherelatedreferencesforacomprehensivecollectionofresults.
2. The model
We shall start by briefly recalling the Souza–Auricchio model together with some notation. The interested
readerisreferredtotheoriginalpapers[18,19,22]forsomeextradetail.
The SMA body is modeled within the frame of generalized standard materials (see Maugin [44]) and the
small-strain approximation by additively decomposing the linearized deformation ε ε(u) (ε (u))
ij
= = =
(ui,j uj,i)/2 (where u : " R3 is the displacement from a fixed reference configuration " R3) into an
+ → ⊂
elasticpartεel R3×3 andaninelastic(ortransformation)partεtr R3×3 as
∈ ∈
ε εel εtr. (2.1)
= +
At the microscopic level the super-elastic effect is interpreted as the result of a structural phase transition
betweendifferentconfigurationsofthemateriallattice,namelythenon-orientedphase(austeniteandtwinned
martensite) and the oriented phase (detwinned martensite). In particular, the tensorial internal variable εtr is
assumedtobedescriptiveofthemechanicaleffectofthedetwinningobservedinthematerial.
Thefreeenergyofthebodyisprescribedas
1 1
ψ(θ,ε,εtr) = cθ(1−logθ)+ 2(ε−εtr) : C(ε−εtr)+ 2εtr : Hεtr +f(θ)|εtr|+IεL(εtr). (2.2)
Thefirsttermintheexpressionofψ isthepurelycaloricpartandcstandsforthespecificheatofthematerial.
Thesecondandthirdtermsareclassicalinlinearizedelastoplasticitywithlinearhardening.Inparticular,Cis
theelasticitytensor(isotropyassumed)
E E
C : (I4 I2 I2) I2 I2
= 1 ν + ⊗ + 1 2ν ⊗
+ −
where E is the Young modulus, ν is the Poisson ratio, and I4 and I2 denote the identity 4- and 2-tensor,
respectively.Instead,H : EhI4 whereEh > 0isthehardeningtensor.
=
The last two terms in the expression of ψ are the distinguishing traits of the Souza–Auricchio model. The
function f is convex, non-negative, non-decreasing, and suitably smooth. Note that the original choice of the
Souza–Auricchiomodelforthefunctionf is
f (θ) b(θ θ ) (2.3)
SA M +
= −
where θ is the critical temperature for the austenite–martensite transition at zero stress and b > 0. Note,
M
however,thatthechoicef f isnotadmissiblehereforthesakeofprovingtheglobalexistenceofsolutions
SA
=
aswillbedetailedinSection4.Finally,thevalueε > 0representsamaximalamountoftransformationstrain
L
obtainable by martensitic reorientation in a uniaxial test and I is the indicator function of the closed ball
εL
centeredat0withradiusε .Namely,I (εtr) 0ifandandif εtr ε andI (εtr) elsewhere.
L εL = | |≤ L εL =∞
352 MathematicsandMechanicsofSolids16(4)
Figure1. Parameterfittingontwothermaltestsatdifferenttensionlevels.
The evolution of the material will be prescribed by means of the specification of the rate-independent
dissipationrelatedtothephasetransformationviathecorrespondingpseudo-potentialofdissipation
ϕ(εtr) R εtr
˙ = |˙ |
whereR > 0istheso-calledtransformationradius.
Notethatthematerialisassumedtohaveconstantmassdensity,topresentthesamespecificheatandelastic
behavior in all phases, and to show no thermal dilation. These are, of course, crude simplifications which
are, however, motivated by simplicity and partly justified by the good quantitative agreement of the model to
experimental data. In particular, Figure 1 illustrates (thin lines) the outcome of two thermo-cycle experiments
with different (fixed) tensions for a commercial NiTi wire (straight annealed NiTi FWM #1 superelastic wire
withdiameter 0.1mmandlength 50mm[45]).
= =
These experimental data were obtained by the SMA Group at the Institute of Physics of the Academy of
Sciences of the Czech Republic in Prague within an international action on SMA modeling validation [46].
Bold lines in Figure 1 are obtained by partly fitting the experimental data for the Souza–Auricchio model
(henceforf f )bytheSMAgroupinPavia.Forthesakeoflaterconsiderationletusreportherethevalues
SA
=
ofthematerialparameters:
c 5.2MPa/K, E 53.6GPa, ν 0.33, E 750MPa, ε 0.058,
h L
= = = = =
b 5.6MPa/K, θ 235K, R 90MPa. (2.4)
M
= = =
AllthesevaluesarefittedfromtheexperimentsinFigure1exceptforthespecificheatcwhichistakeninstead
fromRoubícˇek[17].
Some validation of the Souza–Auricchio model is illustrated in Figure 2 where the fitted parameters are
usedinapuretensiontest.Inparticular,experimental(thin)andnumericalresults(boldline)forcompleteand
incompletetensioncyclesarecompared.
Given the expression of the free energy (2.2), the corresponding entropy s and internal energy e are given
bytheformulas
∂ψ
s clogθ f (θ) εtr , (2.5)
)
=− ∂θ = − | |
1 E
e = ψ +θs = cθ + 2(ε−εtr) : C(ε−εtr)+ 2h|εtr|2+(f(θ)−θf)(θ))|εtr|+IεL(εtr). (2.6)
KrejcˇíandStefanelli 353
Figure2. Comparisonbetweenexperimentalresultsandnumericalpredictionsinacyclictensiontest.Materialparametersarefitted
fromthethermo-thermo-cyclingexperimentillustratedinFigure1.
From the expression of the entropy (2.5) it is already clear that some restriction on f has to be required for
the model to comply with thermodynamics. Indeed, as the temperature–entropy relation needs to be strictly
increasing,bycomputing
∂s c
f (θ) εtr > 0, (2.7)
))
∂θ = θ − | |
wefindthatthevalueoff (θ)cannotbetoolarge(and,inparticular,cannotbeaDiracdeltaasforthechoice
))
f ).
SA
Forthestressσ andtransformationstrainεtr weprescribetheconstitutiveequations
∂ψ
σ , (2.8)
= ∂ε
∂ϕ ∂ψ
0 , (2.9)
∈ ∂εtr + ∂εtr
˙
which,owingtothespecificchoiceforthefreeenergyin(2.2),readas
σ C(ε εtr) Cεel, (2.10)
= − =
σ R∂ εtr f(θ)∂ εtr E εtr ∂I (εtr). (2.11)
∈ |˙ |+ | |+ h + εL
The process is governed by the mechanical equilibrium equation (quasi-static) and by the energy balance,
thatis,
divσ 0, (2.12)
− =
e divq σ : ε r (x,t) (2.13)
t t 0
+ = +
in" (0,T),whereqistheheatfluxvectorwhichweassumetoobeyFourier’slawq κ θ withaconstant
× =− ∇
heat conductivity κ, and r (x,t) 0 are given heat sources (e.g., the Joule heating). From the construction
0
≥
it is clear that every regular solution of the system with positive temperature θ satisfies the Clausius–Duhem
inequality
q r (x,t)
0
s div 0. (2.14)
t
+ θ − θ ≥
! "
354 MathematicsandMechanicsofSolids16(4)
Indeed,forθ> 0thelatterisequivalenttotheinequality
q θ
θs σ : ε ψ ·∇ 0.
t t t
− + − − θ ≥
Bythechainrule,wehaveψ(θ,ε,εtr) θs σ : ε εtr : ∂ ψ,andhence,byvirtueof(2.9),theleft-hand
t t t εtr
=− + +˙
sideof(2)equalsR εtr κ θ 2/θ 0.
|˙ |+ |∇ | ≥
2.1. One-dimensional model
Letusnowcommentontheone-dimensionalmodelweshallbeconsideringinthefollowing.Weshallassume
thattheprocesstakesplaceinadomain" R3 oftheform" (0,)) ω,where)> 0isfixedandω R2
⊂ = × ⊂
isaverysmallboundedconnecteddomain(i.e.thecrosssectionofawire).Weassumethatallstatevariables
depend only on x [0,)] and t [0,T], where T > 0 is some final time of the process and that we are
∈ ∈
concernedwithanuniaxialtractiontest,namelytheonlynon-zerocomponentofthestressis
σ : σ . (2.15)
11
=
Moreover, we assume that the body is so thin (or so long) that the displacement out of the x-direction is
negligible. This amounts to say that u u 0. Hence, owing to (2.15), the only non-zero components
2 3
= =
ofthedisplacementsare
ε : ε , εtr : εtr.
= 11 = 11
Correspondingly,thefreeenergyofthemediumcanberewrittenas
E E
ψ(θ,ε,εtr) cθ(1 logθ) (ε εtr)2 h(εtr)2 f(θ) εtr I (εtr).
= − + 2 − + 2 + | |+ εL
Hence,thebalanceequations(2.12)–(2.13)andtheconstitutiverelation(2.11)read
σ 0, (2.16)
x
=
e κθ σε r(θ,x,t), (2.17)
t xx t
− = +
σ R∂ εtr f(θ)∂ εtr E εtr ∂I (εtr) (2.18)
∈ |˙ |+ | |+ h + εL
in (0,)) (0,T). We offer two motivations for the θ-dependence of the heat source r in the energy balance
×
(2.17).Atfirst,givenanexternaltemperaturedistributionθ (x,t),theRobinboundaryconditionalongthewire
,
leadsafterone-dimensionalreductiontotheformula
r(θ,x,t) r (x,t) h (θ θ (x,t)) h (θ4 θ4(x,t))
= 0 − 0 − , − 1 − ,
with r 0, θ (x,t) θ > 0, and with constants h ,h 0. Secondly, the θ-dependence in r allows the
0 , 0 1
≥ ≥ ∗ ≥
possibilityofprescribingtheinequalityr(θ,x,t)H(θ θ) 0,whereH istheHeavisidefunction.Thisinturn
∗− ≥
ensuresthevalidityofamaximumprinciplegivingthepositivityofthetemperature.
Wecomplementthesystem(2.16)–(2.18)withtheboundaryconditions
u(0,t) 0, σ(),t) τ(t), (2.19)
= =
θ (0,t) θ (),t) 0, (2.20)
x x
= =
which in particular entail that the wire is thermally insulated from the actuators, it is fixed at x 0, and a
=
knowntractionτ : [0,T] Risappliedatx ).
→ =
KrejcˇíandStefanelli 355
3. The constitutive relation
Before formulating our existence result we shall devote a preliminary discussion to the specific form of the
material constitutive relation (2.18). Our aim here is to advance an equivalent formulation of the functional
relation (θ,σ) εtr prescribed by (2.18) in terms of (a function of) elementary hysteresis operators [35, 47,
-→
48]. In particular, the functions f f(θ) and σ are supposed to be known functions of time throughout this
=
section. Indeed, for given functions f,σ W1,1(0,T), f(t) 0 for all t [0,T], we consider the differential
∈ ≥ ∈
inclusionfortheunknownfunctionη,
σ(t) R∂ η(t) f(t)∂ η(t) η(t) ∂I (η(t)), (3.1)
[ 1,1]
∈ |˙ |+ | |+ + −
withagiveninitialconditionη(0) η [ 1,1].Notallinitialconditionsareadmissible.Weeasilycheckthe
0
= ∈ −
implications
η 1 σ(0) f(0) 1 R,
0
= ⇒ − − ≥ −
η 1 σ(0) f(0) 1 R,
0
 =− ⇒ + + ≤
η 0 σ(0) [ f(0) R,f(0) R], (3.2)
 0
 η0 = (0,1) ⇒ σ(0) ∈ f−(0) η−0 [ R,+R],
∈ ⇒ − − ∈ −
η ( 1,0) σ(0) f(0) η [ R,R].
0 0
∈ − ⇒ + − ∈ −


We first prove that thereexists at most one solution to (3.1) for any given admissible initial condition. Let

η ,η be two solutions and let m(t) ∂ η(t) sign η(t), i 1,2 be arbitrary selections. We use f(t) 0
1 2 i i i
∈ |˙ | = ˙ = ≥
anddeducethat
R(m (t) m (t))(η (t) η (t)) (η (t) η (t))2 0 almosteverywhere,
1 2 1 2 1 2
− − + − ≤
whencetheimplicationη (t) >η (t) η (t) < η (t)holdsforalmosteveryt [0,T].Interchangingtheroles
1 2 1 2
⇒˙ ˙ ∈
ofη andη ,weobtain
1 2
(η (t) η (t))(η (t) η (t)) 0 almosteverywhere,
1 2 1 2
˙ −˙ − ≤
henceη η .
1 2
≡
We now construct an explicit solution to (3.1) in terms of the so-called play operator p with threshold
R
R > 0.Recallthatforeverygivenfunctionv W1,1(0,T)andeveryz [ R,R]thereexistsauniquesolution
0
∈ ∈ −
ξ W1,1(0,T)totheproblem
∈
v(t) ξ(t) R t [0,T],
| − |≤ ∀ ∈
v(0) ξ(0) z , (3.3)
0
 − =
ξ(t)(v(t) ξ(t) y) 0 a.e.in(0,T), y [ R,R].
 ˙ − − ≥ ∀ ∈ −
Theoperatorp : [ R,R] W1,1(0,T) W1,1(0,T)isdefinedasthesolutionmappingξ p [z ,v]of(3.3).
R R 0
− × → =
It is Lipschitz continuous and admits a Lipschitz continuous extension to p : [ R,R] C[0,T] C[0,T].
R
− × →
Thefollowingscalingproperty
pcR[cz0,cv] cpR[z0,v] c R (3.4)
= ∀ ∈
isanimmediateconsequenceofthedefinition.
WenowstateaclassicalcomparisonresultbyKrasnosel’skiiandPokrovskii[49].
Lemma3.1Let v (t) v (t) for all t [0,T], let z z , and let ξ p [z ,v] for i 1,2. Then
1 2 01 02 i R 0i i
≤ ∈ ≥ = =
ξ (t) ξ (t)forallt [0,T].
1 2
≤ ∈
Proof.Fromtheinequality
(ξ (t) ξ (t))(v (t) v (t) ξ (t) ξ (t)) 0 almosteverywhere
˙1 − ˙2 1 − 2 − 1 + 2 ≥
356 MathematicsandMechanicsofSolids16(4)
weinfertheimplication
ξ (t) >ξ (t) ξ (t) ξ (t),
1 2 ⇒ ˙1 ≤ ˙2
whichyieldsinturnthat
(ξ (t) ξ (t))H(ξ (t) ξ (t)) 0 almosteverywhere,
˙1 − ˙2 1 − 2 ≤
whereH istheHeavisidefunction.Hence,
(ξ (t) ξ (t)) (ξ (0) ξ (0)) (v (0) v (0) z z ) 0
1 2 + 1 2 + 1 2 01 02 +
− ≤ − = − − + =
foralltandtheassertionfollows.
Wearenowreadytostateandprovethemainresultofthissection.
Proposition3.2Let f,σ W1,1(0,T), f(t) 0 for t [0,T], and η [ 1,1] be given satisfying the
0
∈ ≥ ∈ ∈ −
compatibilityconditions(3.2).Then,(3.1)admitsauniquesolutionηgivenbytheformula
η Q(p [z ,σ f]) Q( p [z ,σ f]), (3.5)
R 0 R 0
= − − − − + +
whereQ : R [0,1]istheprojectiondefinedasQ(y) max 0,min y,1 fory R,andthevaluesz0 are
→ = { { }} ∈ ±
chosenasz σ(0) f(0) η ,z σ(0) f(0) η .
0 0 0 0
− = − − + = + −
The relation between the initial condition η , on the one hand, and z , on the other, is not one-to-one for
0 0
±
otherchoicesofz arepossibleforthesameη .
0 0
±
Proof. We already know that (3.1) admits at most one solution for a given initial condition. A straightforward
computation shows that p [z ,σ f](0) p [z ,σ f](0) η , hence η(0) η . Moreover, by Lemma
R 0 R 0 0 0
+ + = − − = =
3.1wehave
p [z ,σ f](t) p [z ,σ f](t)
R 0 R 0
+ + ≥ − −
for all t [0,T], since f(t) 0. Hence η Q(p [z ,σ f]), and η Q( p [z ,σ f]). It remains
+ R 0 − R 0
∈ ≥ = − − = − + +
to check that the function η defined by (3.5) satisfies the inclusion (3.1) almost everywhere. We define the set
M (0,T)asthesetofallLebesguepointsofallfunctionsσ,f,ξ p [z ,σ f],ξ p [z ,σ f],η,
R 0 R 0
⊂ − = − − + = + +
andconsideranyt M.Wedistinguishagainthefivecasesasabove.
∈
η(t) 1:Thenη(t) 0,ξ (t) 1,henceσ(t) f(t) 1 R,and(3.1)follows;
= ˙ = − ≥ − ≥ −
η(t) 1:Thenη(t) 0,ξ (t) 1,henceσ(t) f(t) 1 R,and(3.1)followsagain;
=− ˙ = + ≤ − + ≤ − +
η(t) 0: Then ξ (t) 0 ξ (t), hence f(t) R σ(t) f(t) R, and (3.1) follows provided η(t) 0.
= − ≤ ≤ + − − ≤ ≤ + ˙ =
Thecaseη(t) 0canonlyoccuriff(t) 0andσ(t) sign(η(t))R,hence(3.1)holdsagain;
˙ 1= = = ˙
η(t) (0,1):Thenη(t) ξ (t),henceη(t)(σ(t) f(t) η(t) y) 0forally [ R,R],whichisequivalent
∈ = − ˙ − − − ≥ ∈ −
to(3.1);
η(t) ( 1,0):Thenη(t) ξ (t),henceη(t)(σ(t) f(t) η(t) y) 0forally [ R,R],whichisequivalent
∈ − = + ˙ + − − ≥ ∈ −
to(3.1).
TheproofofProposition3.2iscomplete.
4. Main existence result
We shall now exploit the results of Section 3 in order to provide a useful reformulation of the system (2.16)–
(2.18). Equation (2.16) with boundary conditions (2.19) has a unique solution σ(x,t) τ(t) for all (x,t)
= ∈
KrejcˇíandStefanelli 357
(0,)) (0,T). We prescribe initial conditions θ (x) > 0 for θ and εtr(x) [ ε ,ε ] for εtr, satisfying the
× 0 0 ∈ − L L
compatibilityconditionsanalogousto(3.2),thatis,
εtr(x) ε τ(0) f(θ (x)) E ε R,
0 = L ⇒ − 0 − h L ≥ −
εtr(x) ε τ(0) f(θ (x)) E ε R,
 0 =− L ⇒ + 0 + h L ≤
εtr(x) 0 τ(0) [ f(θ (x)) R,f(θ (x)) R], (4.1)
 ε00tr(x) =∈ (0,εL) ⇒⇒ τ(0)−∈ f−(θ0(x0))−E−hε0tr(x)0∈ [−+R,R],
εtr(x) ( ε ,0) τ(0) f(θ (x)) E εtr(x) [ R,R].
0 ∈ − L ⇒ + 0 − h 0 ∈ −




Wenowdividetheconstitutiverelation(2.18)byE ε andobtain
h L
τ(t) R εtr f(θ(x,t)) εtr εtr εtr
∂ ˙ ∂ ∂I .
[ 1,1]
EhεL ∈ EhεL εL + EhεL εL + εL + − εL
’ ’ ’ ’ ! "
’ ’ ’ ’
ByProposition3.2andidentity(3.4)we’thu’shave ’ ’
1
εtr(x,t) ε Q p [z (x),τ f(θ(x, ))](t)
L R 0
= EhεL − − ·
! 1 "
ε Q p [z (x),τ f(θ(x, ))](t) , (4.2)
L R 0
− − EhεL + + ·
! "
with
z (x) τ(0) f(θ (x)) E εtr(x), z (x) τ(0) f(θ (x)) E εtr(x). (4.3)
0− = − 0 − h 0 0+ = + 0 − h 0
Theenergybalance(2.17)canbewrittenintheform
E
cθ (f(θ) θf (θ)) εtr h(εtr)2 κθ τ(t)εtr r(θ,x,t) (4.4)
+ − ) | |+ 2 t − xx = t +
! "
or,alternatively,
c θf (θ) εtr θ κθ θf (θ) εtr R εtr r(θ,x,t). (4.5)
− )) | | t − xx = ) | |t + | t |+
Weshallstartbycolle(ctingourassum)ptionsondatainthefollowinghypothesis.
Hypothesis4.1Thereexistconstantsθ > 0,r > 0,C > 0andanincreasingfunctionC : (0, ) (0, )
∗ f
∗ ∞ → ∞
withthefollowingproperties.
(i) Thedatahavetheregularityεtr C[0,)],θ W1,2(0,)),τ W1, (0,T),θ θ ,εtr(x) [ ε ,ε ]for
0 ∈ 0 ∈ ∈ ∞ 0 ≥ ∗ 0 ∈ − L L
allx [0,)],andthecompatibilitycondition(4.1)holds.
∈
(ii) f : (0, ) [0, )isanon-decreasingconvexfunctionofclassC1,1 suchthatf(θ) 0for0 <θ<θ ,
∞ → ∞ = ∗
f (θ) C forallθ> 0,and
) f
≤
f (θ)
)
c : inf c θf (θ)ε (θf (θ) R) > 0, (4.6)
0 )) L )
= θ>0 − + Eh −
! "
c : inf c θf (θ)ε > 0. (4.7)
1 )) L
= θ>0 −
! "
(iii) r : (0, ) (0,)) (0,T) Risameasurablefunctionsuchthat
∞ × × →
r(θ,x,t) r , r(θ,x,t) r almosteverywhere,
∗ t ∗
• ≤ | |≤
r(θ,x,t)H(θ θ) 0almosteverywhere,
• ∗− ≥
r(θ ,x,t) r(θ ,x,t) C(max θ ,θ ) θ θ almosteverywhereforallθ ,θ > 0,
1 2 1 2 1 2 1 2
• | − |≤ { } | − |
wherewerecallthatH istheHeavisidefunction.
358 MathematicsandMechanicsofSolids16(4)
Theultimatemotivationforintroducingtherestrictions(4.6)–(4.7)isthatofpreservingtheparabolicityof
the problem. Restriction (4.7) is nothing but the former (2.7) and it ensures that the coefficient of θ in the
t
energybalance(4.5)ispositive.Ontheotherhand,(4.6)entailsthattheoverallthermomechanicalevolutionof
thebodyissuchthatenergygetsdissipatedwithtime.
Anexampleforafunctionf fulfillingtheassumptionsis
0 for θ (0,(1 δ)θ ],
M
∈ −
b
f)(θ)  (θ (1 δ)θM) for θ ((1 δ)θM,(1 δ)θM), (4.8)
=  2δθM − − ∈ − +

b for θ (1 δ)θ
M
≥ +



forδ (0,1).Notethatf coincideswiththeoriginalfunctionf from(2.3)oftheSouza–Auricchiomodelout
SA
∈
of the temperature interval centered in θ with radius ρ δθ . This radius cannot be taken arbitrarily small
M M
=
without violating (4.6)–(4.7). Indeed, by exploiting the actual material parameters from (2.4), the restrictions
(4.6)–(4.7)entailρ 7.5K.
≥
Ourmainexistenceresultreadsasfollows.
Theorem4.2Let Hypothesis 4.1 hold. Then system (4.4), (2.20), and (4.2) with z given by (4.3) admits a
0
solutionθ C([0,)] [0,T])suchthatθ,θ L2((0,)) (0,T)),andθ(x,t) > 0for±all(x,t) [0,)] [0,T].
t xx
∈ × ∈ × ∈ ×
Ifmoreovernocompressiontakesplace,thatisεtr(x) 0forallx [0,)]andτ(t) 0forallt [0,T],then
0 ≥ ∈ ≥ ∈
condition(4.6)canbereplacedby
c : inf c θf (θ)ε f (θ)ε > 0. (4.9)
0 )) L ) L
= θ>0 − −
! "
TheproofofthisresultisdetailedinSection5.
4.1. Counterexamples to global existence
Conditions (4.6), (4.7), and (4.9) are probably not optimal. The following examples show, however, that if
the material parameters are completely arbitrary, and, in particular, these conditions are not fulfilled, a global
solutiontothefullthermodynamicsystemmayfailtoexist.
Intheexamplesbelow,weconsiderthefunctionf givenby(4.8),andspacehomogeneousheatsourcer
=
r(t) 0andinitialconditionsθ andεtr.Thephysicallyrelevantsolutionsθ andεtr arehomogeneousinspace
≥ 0 0
as well. Namely the problem reduces to a (nonlinear) system of ordinary differential equations. We construct
solutions to system (4.2) and (4.4) which exists only for (small) finite time. No general uniqueness proof is
available, hence the existence of a global pathological solution cannot be rigorously excluded. Nevertheless,
in the class of piecewise monotone functions of time, the play operator can be represented by a Lipschitz
continuous superposition (Nemytskii) operator, and uniqueness follows from the general theory of ordinary
differentialequations.
Example4.3We shall start by directly showing the the Souza–Auricchio model may not admit global weak
solutions. This follows from the fact that the choice f f violates (4.7). As f is not smooth, strong
SA SA
=
solvability is clearly out of range and we resort to checking that physically reasonable weak solutions cannot
existgloballyin(0,T).Tothisaim,letusfixθ <θ andr(t) 1andassumethatθ L1(0,T)solvesweakly
0 M
= ∈
(4.5)and(4.2)alongwiththechoicesεtr(x,t) εtr(x,0) ε andτ(t) E ε f(θ(t)),namely
L h L
= = = +
(cθ(t) bθ H(θ(t) θ )ε ) 1 inthedistributionalsense.
M M L t
− − =
Wehavethat
cθ(t) bθ H(θ(t) θ )ε Cθ t almosteverywhere.
M M L 0
− − = +