On Some Models in Linear Thermo-Elasticity with Rational Material Laws
Santwana Mukhopadhyay, Rainer Picard, Sascha Trostorff, Marcus Waurick
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Technische Universität Dresden
Herausgeber: Der Rektor
On Some Models in LinearThermo-Elasticity with Rational MaterialLaws.
S. Mukhopadhyay, R. Picard, S. Trostorff, M. Waurick
Institut für AnalysisMATH-AN-04-2014 n Some Models in LinearThermo-Elasticity with Rational MaterialLaws.
S. Mukhopadhyay ∗ , R. Picard, S. Trostorff, M. Waurick † October 1, 2018
Abstract:
We shall consider some common models in linear thermo-elasticity within a com-mon structural framework. Due to the flexibility of the structural perspective we will obtainwell-posedness results for a large class of generalized models allowing for more general materialproperties such as anisotropies, inhomogeneities, etc.
The coupled dynamical thermoelasticity (CTE) theory was developed by Biot [1] to eliminatethe drawback of uncoupled theory of thermoelasticity that the elastic changes in a materialhave no effects on temperature. Like other classical thermodynamical theories of continua,this theory is developed on the basis of firm grounds of irreversible thermodynamics by em-ploying Fourier’s law and has been used to study the coupling effects of elastic and thermalfields over the years. However, this theory suffers from the paradox of infinite heat propagationspeed and predicts unsatisfactory descriptions of a solid’s response to some situations, like fasttransient loading at low temperature, etc. Generalized thermoelasticity theories are thereforedeveloped in last few decades with the aim to eliminate this drawback. Extended thermoelas-ticity (ETE) theory was introduced by Lord and Shulman [2] by employing a modified Fourierlaw proposed by Catteneo [3] and Vernotte [4, 5] that includes one thermal relaxation timeparameter. Temperature-rate dependent thermoelasticity (TRDTE) theory by Green andLindsay [6] and thermoelasticity theories of type I, II and III by Green and Naghdi [7, 8, 9]are also advocated in this context. Later on, Chandrasekharaiah [10] modified the govern-ing equations of thermoelasticity on the basis of a so-called dual phase-lag heat conductionequation due to Tzou [11, 12] and proposed two different models of thermoelasticity, namelydual phase-lag model- I (DPL-I) and dual phase-lag model- II (DPL-II). The dual phase lagheat conduction law is supposed to be the macroscopic formulation of the microscopic effectsin heat transport processes. A possible application of this generalized heat conduction law ∗ S. Mukhopadhyay:
Department of Mathematical Sciences, Indian Institute of Technology (BHU), Varanasi-221005, India: Corresponding author: E-mail: [email protected] † R. Picard , S. Trostorff, M. Waurick : Institute for Analysis, Faculty of Mathematics and Sciences, TU-Dresden, Germany. − s). When the response time becomes shorter, thenon-equilibrium thermodynamic transition and the microscopic effects in the energy exchangeduring heat transport procedure become pronounced (Tzou [13]). The formulation thereforebecomes microscopic in nature. The dual phase-lag heat conduction law incorporates thismicroscopic effects in heat transport process by introducing two macroscopic lagging (or de-layed) responses as possible outcomes. A detailed history about the development of somewell- established non-Fourier heat conduction models and their importance are available inthe references [14, 10, 15, 16, 17, 18, 19, 20, 21, 22, 23].Recently, a structural formulation for linear material laws in classical mathematical physicswas reported by Picard [24]. Here, a class of evolutionary problems is considered to cover anumber of initial boundary value problems of classical mathematical physics and the solutiontheory is established. The well-posedness of classical thermoelasticity and Lord Shulmantheory was shown to be covered by this model. The main objective of this present work isto show that the aforementioned models of generalized thermoelasticity can be treated withinthe common structural framework of evolutionary equations. Due to the flexibility of thestructural perspective we will obtain well-posedness results for a large class of generalizedmodels allowing for more general material properties such as anisotropies, inhomogeneities,etc. The solution strategy is not based on constructions involving fundamental solutions(semi-group theory), which will allow for even more general materials resulting for example inchanges of type (e.g. from parabolic to hyperbolic) or for suitable non-local material propertiesinvolving e.g. spatial integral operators. It should be noted that evolutionary equations inthe form just discussed have also been studied with regards to homogenization theory, seee.g. [25, 26, 27]. Hence, the general perspective on thermo-elasticity to be presented may alsoshed some new light on the theory of homogenization of such models.The article is structured as follows. We begin to introduce the framework of evolutionaryequations and recall the general well-posedness result. We will focus on so-called rationalmaterial laws defined as functions of the time-derivative ∂ , which is established as a normaloperator in a suitable exponentially weighted L -space. In the proceeding sections we willshow, how the different models of generalized elasticity can be incorporated into this frameworkand we will derive assumptions on the material coefficients yielding the well-posedness of thecorresponding evolutionary equations. The family of Hilbert spaces ( H ̺, ( R , H )) ̺ ∈ R , H complex Hilbert space, with H ̺, ( R , H ) := L ( R , µ ̺ , H ) , where the measure µ ̺ is defined by µ ̺ ( S ) := R S exp ( − ̺t ) d t , S ⊆ R a Borelset, ̺ ∈ R , provides the desired Hilbert space setting for evolutionary problems (cf. [28, 29]).The sign of ̺ is associated with the direction of causality, where the positive sign is linked toforward causality. Since we have a preference for forward causality, we shall usually assume4hat ̺ ∈ ]0 , ∞ [ . By construction of these spaces, we can establish exp ( − ̺ m ) : H ̺, ( R , H ) → H , ( R , H ) (= L ( R , H )) ϕ exp ( − ̺ m ) ϕ where (exp ( − ̺ m ) ϕ ) ( t ) := exp ( − ̺t ) ϕ ( t ) , t ∈ R , as a unitary mapping. We use m as anotation for the multiplication-by-argument operator corresponding to the time parameter.In this Hilbert space setting the time-derivative operation, defined as the closure of ˚ C ∞ ( R , H ) ⊆ H ̺, ( R , H ) → H ̺, ( R , H ) ϕ ˙ ϕ, where by ˚ C ∞ ( R , H ) we denote the space of arbitrary differentiable functions from R to H having compact support, generates a normal operator ∂ ,̺ with Re ∂ ,̺ = ̺, Im ∂ ,̺ = 1i ( ∂ ,̺ − ̺ ) . The skew-selfadjoint operator i Im ∂ ,̺ is unitarily equivalent to the differentiation operator ∂ , in L ( R , H ) = H ( R , H ) with domain H ( R , H ) - the space of weakly differentiablefunctions in L ( R , H ) - via i Im ∂ ,̺ = (exp ( − ̺ m )) − ∂ , exp ( − ̺ m ) and has the Fourier-Laplace transformation as its spectral representation, which is the unitarytransformation L ̺ := F exp ( − ̺ m ) : H ̺, ( R , H ) → L ( R , H ) , where F : L ( R , H ) → L ( R , H ) is the Fourier transformation given as the unitary extensionof ˚ C ∞ ( R , H ) ∋ ϕ s √ π Z R exp( − i s · t ) ϕ ( t ) d t . Indeed, this follows from the well-known fact that F is unitary in L ( R , H ) and a spectralrepresentation for ∂ , in L ( R , H ) . In particular, we have Im ∂ ,̺ = L ∗ ̺ m L ̺ Recall that for normal operators N in a Hilbert space H Re N := 12 ( N + N ∗ ) , Im N := 12i ( N − N ∗ ) and N = Re N + i Im N. It is D ( N ) = D ( Re N ) ∩ D ( Im N ) . ∂ ,̺ = L ∗ ̺ (i m + ̺ ) L ̺ . It is crucial to note that for ̺ = 0 we have that ∂ ,̺ has a bounded inverse. Indeed, for ̺ > we find from Re ∂ ,̺ = ̺ that (cid:13)(cid:13)(cid:13) ∂ − ,̺ (cid:13)(cid:13)(cid:13) ̺, = 1 ̺ , (1)where k · k ̺, , ̺ ∈ ]0 , ∞ [ denotes the operator norm on H ̺, ( R , H ) . For continuous functions ϕ with compact support we find (cid:16) ∂ − ,̺ ϕ (cid:17) ( t ) = Z t −∞ ϕ ( s ) d s, t ∈ R , ̺ ∈ ]0 , ∞ [ , (2)which shows the causality of ∂ − ,̺ for ̺ > . Since it is usually clear from the context which ̺ has been chosen, we shall, as it is customary, drop the index ̺ from the notation for the timederivative and simply use ∂ instead of ∂ ,̺ .We are now able to define operator-valued functions of ∂ via the induced function calculusof ∂ − as M (cid:0) ∂ − (cid:1) := L ∗ ̺ M (cid:16) (i m + ̺ ) − (cid:17) L ̺ . Here, we require that z M ( z ) is a bounded, analytic function defined on B C (cid:16) ̺ , ̺ (cid:17) forsome ̺ ∈ ]0 , ∞ [ attaining values in L ( H ) , the space of bounded linear operators on H . Then,for ̺ > ̺ the operator M (cid:16) (i m + ̺ ) − (cid:17) defined as (cid:16) M (cid:16) (i m + ̺ ) − (cid:17) f (cid:17) ( t ) := M (cid:16) (i t + ̺ ) − (cid:17) f ( t ) ( t ∈ R , f ∈ L ( R , H )) is bounded and linear, and hence M ( ∂ − ) ∈ L ( H ̺, ( R , H )) . Moreover, due to the analyticityof M we obtain that M ( ∂ − ) becomes causal (see [24, Theorem 2.10]).We recall from [24] (and the concluding chapter of [28]) that the common form of standardinitial boundary value problems of mathematical physics is given by (cid:0) ∂ M (cid:0) ∂ − (cid:1) + A (cid:1) U = F, (3)where A is the canonical skew-selfadjoint extension to H ̺, ( R , H ) of a skew-selfadjoint oper-ator in H . We recall the well-posedness result for this class of problems. Theorem 1.1 ([24, Solution Theory]) . Let A : D ( A ) ⊆ H → H be a skew-selfadjoint operatorand M : B (cid:16) ̺ , ̺ (cid:17) → L ( H ) an analytic and bounded mapping, where ̺ ∈ ]0 , ∞ [ . Assumethat there is c ∈ ]0 , ∞ [ such that for all z ∈ B (cid:16) ̺ , ̺ (cid:17) the estimate Re z − M ( z ) = 12 (cid:16) z − M ( z ) + (cid:0) z − (cid:1) ∗ M ( z ) ∗ (cid:17) ≥ c (4) If ̺ < the operator ∂ ,̺ is also boundedly invertible and its inverse is given by (cid:0) ∂ − ,̺ ϕ (cid:1) ( t ) = − ∞ Z t ϕ ( s ) ds ( t ∈ R ) for all ϕ ∈ ˚ C ∞ ( R , H ) . Thus, ̺ < corresponds to the backward causal (or anticausal) case. olds. For ̺ > ̺ we denote the canonical extension of A to H ̺, ( R , H ) again by A . Then theevolutionary problem (cid:0) ∂ M (cid:0) ∂ − (cid:1) + A (cid:1) U = F is well-posed in the sense that (cid:0) ∂ M (cid:0) ∂ − (cid:1) + A (cid:1) has a bounded inverse on H ̺, ( R , H ) . More-over, the inverse is causal. For the models under consideration it suffices to consider M as a rational, bounded-operator-valued function, which, possibly by eliminating removable singularities, is analytic at (in [24]these material laws are called − analytic). This means in particular that M can be factorizedin the form M ( z ) = s Y k =0 Q k ( z ) − P k ( z ) (5)where P k , Q k are polynomials. In this case, condition (4) simplifies to ̺M (0) + Re M ′ (0) ≥ c (6)for some c > and all sufficiently large ̺ > . Indeed, the only difference between theexpression in (6) and (4) are terms multiplied by a multiple of | z | = (cid:12)(cid:12)(cid:12) it + ̺ (cid:12)(cid:12)(cid:12) , which are eventuallysmall, if ̺ > is chosen sufficiently large. A finer classification of these models can beobtained by looking at the (unbounded) linear operator A and the “zero patterns” of M (0) and Re M ′ (0) . We start with the classical equations of irreversible thermo-elasticity in an elastic body Ω ⊆ R due to Biot [1]. Before we can formulate these equations properly, we need to define the spatialdifferential operators involved. Definition 1.2.
We define the operator ˚grad as the closure of grad | ˚ C ∞ (Ω) : ˚ C ∞ (Ω) ⊆ L (Ω) → L (Ω) φ ( ∂ φ, ∂ φ, ∂ φ ) , where we recall that ˚ C ∞ (Ω) denotes the space of smooth functions with compact support in Ω . Likewise we define ˚div as the closure of div | ˚ C ∞ (Ω) : ˚ C ∞ (Ω) ⊆ L (Ω) → L (Ω)( φ , φ , φ ) X i =1 ∂ i φ i . The form U = M (cid:0) ∂ − (cid:1) V may be interpreted as coming from solving an integro-differential equation of theform ∂ N Q (cid:0) ∂ − (cid:1) U = ∂ N P (cid:0) ∂ − (cid:1) V, where N ∈ N is the degree of the operator polynomial Q . ˚grad ⊆ − (cid:16) ˚div (cid:17) ∗ =: grad and, similarly, ˚div ⊆ − (cid:16) ˚grad (cid:17) ∗ =: div . Moreover, we define the operator sym : L (Ω) × → L (Ω) × Φ (cid:18) x (cid:16) Φ( x ) + Φ( x ) ⊤ (cid:17)(cid:19) , which clearly is the orthogonal projector onto the closed subspace L (Ω) × := n Φ ∈ L (Ω) × | Φ( x ) = Φ( x ) ⊤ ( x ∈ Ω a.e. ) o of L (Ω) × . Similar to the definition above, we define the operator ˚Grad as the closure of
Grad | ˚ C ∞ (Ω) : ˚ C ∞ (Ω) ⊆ L (Ω) → L (Ω) × ( φ , φ , φ ) (cid:18)
12 ( ∂ j φ i + ∂ i φ j ) (cid:19) i,j ∈{ , , } and ˚Div as the closure of Div | sym[˚ C ∞ (Ω) × ] : sym[ ˚ C ∞ (Ω) × ] ⊆ L (Ω) × → L (Ω) ( φ ij ) i,j ∈{ , , } X j =1 ∂ j φ ij i ∈{ , , } . By integration by parts we again obtain ˚Grad ⊆ − (cid:16) ˚Div (cid:17) ∗ =: Grad as well as ˚Div ⊆− (cid:16) ˚Grad (cid:17) ∗ =: Div .We are now able to formulate the equations of thermo-elasticity. Let u ∈ H ̺, ( R , L (Ω) ) denote the displacement-field of the elastic body Ω and σ ∈ H ̺, ( R , L (Ω) × ) the stress.Then u and σ satisfy the balance of momentum equation ̺ ∂ u − Div σ = f, (7)where ̺ ∈ L ∞ (Ω) denotes the mass-density of Ω and f ∈ H ̺, ( R , L (Ω) ) is an externalforcing term. Furthermore, let η ∈ H ̺, ( R , L (Ω)) denote the entropy and q ∈ H ̺, ( R , L (Ω) ) the heat flux. Then, these quantities satisfy the conservation law ̺ ∂ η + div( T − q ) = T − h, (8)where T denotes the reference temperature and h ∈ H ̺, ( R , L (Ω)) is a heating source term.The equations are completed by the following relations σ = Cε − Γ θ, (9) ̺ η = Γ ∗ ε + νθ, (10) q = − κ grad θ. (11) For simplicity we have set the reference temperature T in the introduction (and also later on) to T = 1 . Inequation (8) we let T ∈ ]0 , ∞ [ be arbitrary to keep the formulation more comparable with the classicallyproposed models. ε = Grad u is the strain, θ ∈ H ̺, ( R , L (Ω)) denotes the temperature, C ∈ L ( L (Ω) × ) is the elasticity tensor, ν ∈ L ∞ (Ω) stands for the specific heat, κ ∈ L ∞ (Ω) denotes the thermalconductivity and Γ ∈ L ( L (Ω) , L (Ω) × ) is the thermo-elasticity tensor that results fromthe Duhamel-Neumann law linking the stress to strain and temperature. Assuming that C isinvertible, we may rewrite (9) as ε = C − σ + C − Γ θ. (12)Consequently, (10) can be written as ̺ η = Γ ∗ C − σ + (cid:0) Γ ∗ C − Γ + ν (cid:1) θ (13)and, hence, with v := ∂ u, σ, θ and q as our basic unknowns, (7),(11),(12) and (13) can becombined to the following equations on H ̺, ( R , H ) , where H := L (Ω) ⊕ L (Ω) × ⊕ L (Ω) ⊕ L (Ω) : ∂ ̺ C − C − Γ 00 Γ ∗ C − Γ ∗ C − Γ + ν
00 0 0 0 + κ − + − Div 0 0 − Grad 0 0 00 0 0 div0 0 grad 0 vσθq = f h . This systems is, at least formally, of the form (3), where M ( ∂ − ) = M + ∂ − M with M = ̺ C − C − Γ 00 Γ ∗ C − Γ ∗ C − Γ + ν
00 0 0 0 , M = κ − and A = − Div 0 0 − Grad 0 0 00 0 0 div0 0 grad 0 . (14)To make A become skew-selfadjoint, we need to impose boundary conditions on our unknowns.For instance, one could require homogeneous Dirichlet-conditions for v and θ, which can beformulated by v ∈ D ( ˚Grad) and θ ∈ D ( ˚grad) . Then, A becomes A = − Div 0 0 − ˚Grad 0 0 00 0 0 div0 0 ˚grad 0 , (15)which clearly is skew-selfadjoint. Of course, other boundary conditions can be imposed making A skew-selfadjoint, see e.g. [30]. 9s we shall see, the Lord-Shulman model [2], the two dual-phase lag models [11, 12] andthe three Green-Naghdy models [7-9] are based on the same relations (9), (10), differencesonly appearing in the modification of Fourier’s law (11). In the case of the Green-Lindsaymodel [6], although of the same formal shape, the meaning of the temperature θ is replacedby the differential expression (1 + n ∂ ) applied to temperature. Therefore, in order to avoidconfusion, we shall use in this case Θ := θ + n ∂ θ instead of re-dedicating the symbol θ , where n is the thermal relaxation time, a characteristicof this model. In this section we will show that the models of thermo-elasticity mentioned in the introductioncan be written as evolutionary problems in the sense of Section 1.1 and thus, their well-posedness can be shown with the help of Theorem 1.1. In fact, we will show that a generalizedmodel of the basic Green-Lindsay type allows to recover all other models as special cases. Wewill begin to formulate this abstract model and prove its well-posedness. In the subsequentsubsection, we will show how the classical models can be recovered from the abstract one andwhich conditions yield their well-posedness.
We consider the following material law M ( ∂ − ) = M + ∂ − M ( ∂ − ) , where M = ̺ C − C − Γ 00 Γ ∗ C − ν + Γ ∗ C − Γ + ζ ∗ a ζ ζ ∗ a a ζ a , (16) M ( ∂ − ) = a ( ∂ − ) 00 0 0 a ( ∂ − ) . (17)Here a ∈ L ( L (Ω) ) is a selfadjoint operator, ζ ∈ L ( L (Ω) , L (Ω) ) and a : B (0 , r ) → L ( L (Ω)) and a : B (0 , r ) → L ( L (Ω) ) are rational functions for some r > . We recallfrom the previous section that ̺ , ν ∈ L ∞ (Ω) denote the mass density and the specific heat,respectively, which will be assumed to be real and strictly positive, i.e. ̺ ( x ) , ν ( x ) ≥ c forsome c > and almost every x ∈ Ω . Moreover, the elasticity tensor C ∈ L ( L (Ω) × ) isassumed to be selfadjoint and strictly positive definite. Theorem 2.1.
Let M and M ( ∂ − ) be as in (16) and (17) , respectively. We assume that ̺ , ν ∈ L ∞ (Ω) are real-valued and strictly positive and C ∈ L ( L (Ω) × ) is selfadjoint and trictly positive definite. Moreover, we assume that a is strictly positive definite on its rangeand Re a (0) is strictly positive definite on the kernel of a . Then the evolutionary problem (cid:0) ∂ M + M ( ∂ − ) + A (cid:1) vσ Θ q = f h (18) is well-posed in the sense of Theorem 1.1, where A is given by (15) and Θ = (1 + n ∂ ) θ .Proof. According to Theorem 1.1, we need to verify condition (4) for M ( ∂ − ) = M + ∂ − M ( ∂ − ) . Or, equivalently, by the structural properties assumed for M , we need toverify that there exists ̺ > such that for all ̺ > ̺ we have that ̺M + Re M (0) ≥ c for some c > . Indeed, the latter equation is precisely the reformulation of (6) for theparticular M under consideration. For ̺ > , we compute ̺M + Re M (0)= ̺ ̺ C − C − Γ 00 Γ ∗ C − ν + Γ ∗ C − Γ + ζ ∗ a ζ ζ ∗ a a ζ a + Re a (0) 00 0 0 Re a (0) = ∗ ζ ∗ × ̺ ̺ C − ν
00 0 0 a + Re a (0) + ζ ∗ Re a (0) ζ − ζ ∗ Re a (0)0 0 − Re a (0) ζ Re a (0) ζ . We read off that the latter is strictly positive definite if and only if the operator ̺ ̺ C − ν
00 0 0 a + Re a (0) + ζ ∗ Re a (0) ζ − ζ ∗ Re a (0)0 0 − Re a (0) ζ Re a (0) is strictly positive definite. As, by assumption, the operators ̺ and C − are positive definiteanyway, we only have to study the positive definiteness of the operator ̺ (cid:18) ν a (cid:19) + (cid:18) Re a (0) + ζ ∗ Re a (0) ζ − ζ ∗ Re a (0) − Re a (0) ζ Re a (0) (cid:19) . (19) Or any other skew-selfadjoint restriction of (14). L (Ω) ⊕ L (Ω) = R (cid:18)(cid:18) ν a (cid:19)(cid:19) ⊕ N (cid:18)(cid:18) ν a (cid:19)(cid:19) , which can be done, since both a and ν are strictly positive definite on the respective ranges,we realize that (19) is strictly positive definite on H ̺, (cid:18) R , R (cid:18)(cid:18) ν a (cid:19)(cid:19)(cid:19) with positive def-initeness constant arbitrarily large, depending on the choice of ̺ > . By Euklid’s inequality( ab ≤ ε a + εb , a, b ∈ R , ε > ), the assertion follows, if we show that the operator in (19) isstrictly positive definite on the nullspace of a . However, by assumption, Re a (0) is strictlypositive on N ( a ) . This yields the assertion. Remark . We write down Equation (18) line by line. It is ∂ ̺ v − Div σ = f,∂ C − σ + ∂ C − ΓΘ − ˚Grad v = 0 ,∂ Γ ∗ C − σ + ∂ (cid:0) ν + Γ ∗ C − Γ + ζ ∗ a ζ (cid:1) Θ + ∂ ζ ∗ a q + a ( ∂ − )Θ + div q = h,∂ a ζ Θ + ∂ a q + a ( ∂ − ) q + ˚gradΘ = 0 . Defining u := ∂ − v, ε := Grad u and η := ̺ − (cid:0) Γ ∗ ε + ( ν + ζ ∗ a ζ ) Θ + ζ ∗ a q + ∂ − a ( ∂ − )Θ (cid:1) we get from the second line σ = Cε − ΓΘ . Moreover, the fourth line reads as ∂ a q + a ( ∂ − ) q = − ∂ a ζ Θ − ˚gradΘ and the first line is ∂ ̺ u − Div σ = f. Finally, the third line reads as ∂ ̺ η + div q = ∂ (cid:0) Γ ∗ ε + ( ν + ζ ∗ a ζ ) Θ + ζ ∗ a q + ∂ − a ( ∂ − )Θ (cid:1) + a ( ∂ − )Θ + div q = ∂ Γ ∗ C − σ + ∂ (cid:0) Γ ∗ C − ΓΘ + ν + ζ ∗ a ζ (cid:1) Θ + ∂ ζ ∗ a q + a ( ∂ − )Θ + div q = h, where we have used σ = Cε − ΓΘ . Summarizing, our material relations are ∂ ̺ u − Div σ = f, (20) ∂ ̺ η + div q = h, (21) σ = Cε − ΓΘ , (22) ̺ η = Γ ∗ ε + ( ν + ζ ∗ a ζ ) Θ + ζ ∗ a q + ∂ − a ( ∂ − )Θ , (23) ∂ a q + a ( ∂ − ) q = − ∂ a ζ Θ − ˚gradΘ , (24) Θ = (1 + n ∂ ) θ. (25)We note that for n = a = ζ = a ( ∂ − ) = 0 and a ( ∂ − ) = κ − we recover the equations ofirreversible thermo-elasticity (compare Subsection 1.2).12e will now discuss several models of thermo-elasticity and we will show that they all arecovered by the model proposed above. Due to the importance of M (0) = M , M ′ (0) = M (0) in the discussion of well-posedness, compare (6), we are first lead to distinguish two classes ofmodels. • Generic models.These models are characterized by M (0) = Re M (0) being strictly positive definite. Forthese (6) is always satisfied. Moreover, M (0) + ε is then also strictly positive definitefor any sufficiently small selfadjoint operator ε . • Degenerate models.These models fail to have the remarkable stability with regards to perturbations of thegeneric models. They are characterized by M (0) = Re M (0) having a non-trivial nullspace. In these cases (6) can be ensured for example by assuming that M (0) = Re M (0) is strictly positive definite on its own range M (0) [ H ] , i.e. h x | M (0) x i H ≥ c > for all x ∈ M (0) [ H ] , and Re M ′ (0) being strictly positive definite on the null space [ { } ] M (0) , i.e. (cid:10) x | Re M ′ (0) x (cid:11) H ≥ c > for all x ∈ [ { } ] M (0) . In contrast to the model for irreversible thermo-elasticity (compare Subsection 1.2), Lord andShulman ([2]) proposed to replace Fourier’s law (11) by the so-called Cattaneo modificationof Fourier’s law (see [3]), which reads as ∂ a q + q = − κ ˚grad θ, where a ∈ L ∞ (Ω) is assumed to be real-valued and strictly positive definite. This results ina system of the form (18), where n = ζ = a ( ∂ − ) = 0 and a ( ∂ − ) = κ − . (26)In consequence, we obtain the well-posedness for this model by Theorem 2.1: Corollary 2.3.
Let M , M ( ∂ − ) and A be given by (16) , (17) , and a skew-selfadjoint restric-tion of (14) , respectively. Assume that a , ̺ , ν ∈ L ∞ (Ω) , κ ∈ L ( L (Ω) ) , C ∈ L ( L sym (Ω) × ) are selfadjoint and strictly positive definite as well as (26) . Then (18) is well-posed in thesense of Theorem 1.1.Proof. It suffices to observe that the kernel of a is trivial. For ease of formulation, note that we identified a , ̺ and ν with the induced multiplication operators on L . In this way, selfadjointness is just the same as to say the respective L ∞ -functions assume only realvalues and, thus, strict positivity coincides with strict positivity of the respective functions. emark . If we mark possible non-zero entries in the operator matrix M (0) by a star, wehave the zero-pattern M (0) = ⋆ ⋆ ⋆ ⋆ ⋆
00 0 0 ⋆ . The zero-pattern of M is M ( ∂ − ) = M (0) = ⋆ . Hence, we observe that there are no higher order terms in the material law operator in caseof Lord-Shulman model.
In the models proposed by Green and Naghdy (see [7, 8, 9]) a modified heat flux of the form q = − ∂ − ( k ∗ + k∂ ) ˚grad θ = − ( ∂ − k ∗ + k ) ˚grad θ. is assumed by considering k, k ∗ ∈ R as thermal conductivity and conductivity rate, respec-tively. Depending on k ∗ and k , we distinguish between three types of this model. If k = 0 and k ∗ = 0 , we speak about the Green-Naghdy Model of Type II. In this case, the above heatflux satisfies ∂ ( k ∗ ) − q = − ˚grad θ. This system is covered by the abstract one if n = ζ = a ( ∂ − ) = a ( ∂ − ) = 0 and a = ( k ∗ ) − . (27)Hence, M (0) has the zero-pattern M (0) = ⋆ ⋆ ⋆ ⋆ ⋆
00 0 0 ⋆ , while M ( ∂ − ) = . The corresponding well-posedness result in a generalized fashion reads as:14 orollary 2.5.
Let M , M ( ∂ − ) and A be given by (16) , (17) , and a skew-selfadjoint restric-tion of (14) , respectively. Assume that k ∗ , ̺ , ν ∈ L ∞ (Ω) , C ∈ L ( L sym (Ω) × ) are selfadjointand strictly positive definite as well as (27) . Then (18) is well-posed in the sense of Theorem1.1.Proof. Again, the assertion follows when applying Theorem 2.1 while observing that N ( a ) = { } . As indicated earlier in Section 1.2, here the material relations (9)-(11) are modified to σ = Cε − Γ ( θ + n ∂ θ ) ,̺ η = dθ + h∂ θ + Γ ∗ ε − b ∗ ˚grad θ,q = − b∂ θ − κ ˚grad θ. Here b, d are material parameters and h, n ( = 0) are the thermal relaxation times. Now,letting Θ := θ + n ∂ θ we get θ = (1 + n ∂ ) − Θ= ∂ − (cid:0) ∂ − + n (cid:1) − Θ and the material relations above turn into ε = C − σ + C − ΓΘ ,q = − (cid:0) ∂ − + n (cid:1) − (cid:16) b Θ + κ∂ − ˚gradΘ (cid:17) , which yields, ∂ n κ − q + κ − q = − ∂ κ − b Θ − ˚gradΘ . Moreover, we have, using the Neumann series, ̺ η = dθ + h∂ θ + Γ ∗ ε − b ∗ ˚grad θ = ( d + h∂ ) ∂ − (cid:0) ∂ − + n (cid:1) − Θ + Γ ∗ ε + b ∗ κ − b (cid:0) ∂ − + n (cid:1) − Θ + b ∗ κ − q = Γ ∗ ε + (cid:0) ∂ − d + h (cid:1) n − ∞ X j =0 (cid:0) − ∂ − n − (cid:1) j Θ + b ∗ κ − bn − ∞ X j =0 ( − ∂ − n − ) j Θ + b ∗ κ − q = Γ ∗ ε + (cid:0) hn − + b ∗ κ − bn − (cid:1) Θ + b ∗ κ − q + ∂ − (cid:0) dn − − (cid:0) h + b ∗ κ − b (cid:1) n − (cid:1) ∞ X j =0 (cid:0) − ∂ − n − (cid:1) j Θ= Γ ∗ ε + (cid:0) hn − + b ∗ κ − bn − (cid:1) Θ + b ∗ κ − q + ∂ − (cid:0) d − (cid:0) h + b ∗ κ − b (cid:1) n − (cid:1) (cid:0) n + ∂ − (cid:1) − Θ . a = n κ − , a ( ∂ − ) = κ − , ζ = bn − , ν = hn − and (28) a ( ∂ − ) = (cid:0) d − (cid:0) h + b ∗ κ − b (cid:1) n − (cid:1) (cid:0) n + ∂ − (cid:1) − In this case the operator matrix M (0) has the zero-pattern M (0) = ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ ⋆ . The zero-pattern of M (0) is now M (0) = ⋆
00 0 0 ⋆ . Also, we note that there are higher order terms in the material law operator. The correspond-ing well-posedness result is therefore noted in the following corollary.
Corollary 2.6.
Let M , M ( ∂ − ) and A be given by (16) , (17) , and a skew-selfadjoint re-striction of (14) , respectively. Let n > and assume that h, ̺ ∈ L ∞ (Ω) , κ ∈ L ( L (Ω) ) , C ∈ L ( L sym (Ω) × ) are selfadjoint and strictly positive definite, b ∈ L ( L (Ω) , L (Ω) ) as wellas (28) . Then (18) is well-posed in the sense of Theorem 1.1.Proof. Again, note that the kernel of a is trivial. Apply Theorem 2.1. In case of the model DPL-II, apart from (9), (10) we have here the modified Fourier law as (cid:18) n ∂ + 12 n ∂ (cid:19) q = − κ (1 + n ∂ ) ˚grad θ. where n , n ∈ R \ { } are called phase-lags. Assuming that κ is invertible, we can write thelatter relation as16 ˚grad θ = (1 + n ∂ ) − (cid:18) n ∂ + 12 n ∂ (cid:19) κ − q = (cid:18) ∂ − + n + 12 n ∂ (cid:19) (cid:0) ∂ − + n (cid:1) − κ − q = 12 n ∂ (cid:0) ∂ − + n (cid:1) − κ − q + (cid:0) n + ∂ − (cid:1) (cid:0) ∂ − + n (cid:1) − κ − q = 12 n n − ∂ ∞ X j =0 (cid:0) − ∂ − n − (cid:1) j κ − q + (cid:0) n + ∂ − (cid:1) (cid:0) ∂ − + n (cid:1) − κ − q = 12 n n − ∂ κ − q − n n − ∞ X j =0 (cid:0) − ∂ − n − (cid:1) j κ − q + (cid:0) n + ∂ − (cid:1) (cid:0) ∂ − + n (cid:1) − κ − q = 12 n n − ∂ κ − q + (cid:18)(cid:0) n + ∂ − (cid:1) − n n − (cid:19) (cid:0) ∂ − + n (cid:1) − κ − q. Thus, this corresponds to the abstract situation when n = ζ = a ( ∂ − ) = 0 and (29) a = 12 n n − κ − , a ( ∂ − ) = (cid:18)(cid:0) n + ∂ − (cid:1) − n n − (cid:19) (cid:0) ∂ − + n (cid:1) − κ − . Therefore, the zero-pattern of M (0) is M (0) = ⋆ ⋆ ⋆ ⋆ ⋆
00 0 0 ⋆ . and the zero-pattern of M (0) is M = Re M = ⋆ . It is seen that this is similar to the case of the Lord-Shulman model. Thus, the well-posednessconditions are similar to the ones in Corollary 2.3 using (29) instead of (26). However, thereare (different) higher order terms in the material law.
Recall that in the Green-Naghdy model (see Subsection 2.2.2), Fourier’s law is replaced by q = − ( ∂ − k ∗ + k ) ˚grad θ.
17n the Green-Naghdy model of type I, it is assumed that k ∗ = 0 , k > . Thus, the aboverelation becomes q = − k ˚grad θ, which is the classical Fourier law and so we have M (0) = ⋆ ⋆ ⋆ ⋆ ⋆
00 0 0 0 , and M ( ∂ − ) = M (0) = ⋆ , with no higher order terms. This is the classical model of thermoelasticity discussed in theintroduction, compare e.g. [31, 32, 33]. In case of the Green-Naghdy model of type III, wehave that k, k ∗ > . This yields, that the modified Fourier law becomes (cid:0) ∂ − k ∗ + k (cid:1) − q = − ˚grad θ, and hence, we are in the situation of Subsection 2.1 with n = ζ = a = a ( ∂ − ) = 0 and (30) a ( ∂ − ) = (cid:0) ∂ − k ∗ + k (cid:1) − . Thus, the zero-patterns of M (0) and M (0) look the same as above with the difference thathigher order terms appear (i.e. M ( ∂ − ) = M (0) ). The well-posedness result reads as follows. Corollary 2.7.
Let M , M ( ∂ − ) and A be given by (16) , (17) , and a skew-selfadjoint restric-tion of (14) , respectively. Assume that ̺ , ν ∈ L ∞ (Ω) , k ∈ L ( L (Ω) ) , C ∈ L ( L sym (Ω) × ) are selfadjoint and strictly positive definite, k ∗ ∈ L (cid:0) L (Ω) (cid:1) as well as (30) . Then (18) iswell-posed in the sense of Theorem 1.1.Proof. By the strict positive definiteness of k , it follows that Re a (0) = Re k − is strictlypositive on L (Ω) = N (0) = N ( a ) . Now, apply Theorem 2.1 to obtain the required result. We conclude our considerations by the study of the DPL-I model. Here again, we assume (9)and (10) to hold, while Fourier’s law (11) is replaced by (1 + n ∂ ) q = − κ (1 + n ∂ ) ˚grad θ, with two phase-lags n , n ∈ R \ { } . The latter gives − ˚grad θ = (1 + n ∂ ) − (1 + n ∂ ) κ − q = (cid:0) ∂ − + n (cid:1) − (cid:0) ∂ − + n (cid:1) κ − q, n = ζ = a = a ( ∂ − ) = 0 and (31) a ( ∂ − ) = (cid:0) ∂ − + n (cid:1) − (cid:0) ∂ − + n (cid:1) κ − . Therefore, the zero-pattern of M (0) is M (0) = ⋆ ⋆ ⋆ ⋆ ⋆
00 0 0 0 and the zero-pattern of M (0) is M (0) = ⋆ . Similarly to Corollary 2.7, using (31) instead of (30) and imposing n · n > , we get thecorresponding well-posedness result also for this type of equation.
4. Conclusion
Various models of thermoelasticity are writen as evolutionary problems and their well posed-ness results are shown. We formulate an abstract model with rational material laws which is ofbasic Green-Lindsay type model and we prove its well-posedness. All other models are shownto be recovered from this abstract one and we find the conditions which yield well posedness.
Anowledgement:
One of the authors (SM) thankfully acknowledges the extended facili-ties provided by the Institute of Analysis, Technical University- Dresden, Germany during theperiod when the present work was carried out.
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