Asymptotic Analysis of Elliptic Membrane Shells in Thermoelastodynamics

In this paper we consider a family of three-dimensional problems in thermoelasticity for elliptic membrane shells and study the asymptotic behaviour of the solution when the thickness tends to zero. We fully characterize with strong convergence results the limit as the unique solution of a two-dimensional problem, where the reference domain is the common middle surface of the family of three-dimensional shells. The problems are dynamic and the constitutive thermoelastic law is given by the Duhamel-Neumann relation.


Introduction
In the last decades, asymptotic methods have been used to derive and justify simplified models for three-dimensional solid mechanics problems for beams, plates and shells. The  1 foundation for these methods was established by Lions in [15] and some of the first applications were to plate bending models in [10,11]. Many other models for plates have been justified by using asymptotic methods and a comprehensive review concerning plate models can be found in [8].
Since then, its application has been extended over the years to many other problems like beam bending, rod stretching and elastic shells. For example, Bernoulli-Navier model was justified in [1] and the Saint-Venant, Timoshenko and Vlassov models of elastic beams were justified in [23], while a model for Kelvin-Voigt viscoelastic beams was justified in [22] and models for piezoelectric beams were obtained in [24], all of them by using the asymptotic expansion method followed by rigorous convergence results. The asymptotic modelling of rods in linearized thermoelasticity was also studied in [23].
Regarding elastic shells, a complete theory can be found in [9], where models for elliptic membranes, generalized membranes and flexural shells are presented. In there, the reader can find a full description of the asymptotic procedure that leads to the corresponding sets of two-dimensional equations. More recently, in a series of papers we studied the asymptotic analysis of viscoelastic shells [4][5][6][7] and contact problems for elastic shells [3,[19][20][21]. For the dynamic case, the authors in [13,14] use the asymptotic analysis to derive two-dimensional sets of equations for elastic membranes and flexural shells, though no strong convergence results are provided. Dynamic problems for shells is a topic which is attracting a considerable effort in modeling, analysis and numerical approximation, due to the abundance of real world applications, see for example [18] and references therein.
The aim of the present paper is to provide the first results of the asymptotic analysis devoted to thermoelastic shells in a dynamic regime. Here we briefly describe the formal asymptotic analysis and the limit two-dimensional problem and we focus in the case of elliptic membrane shells, for which we provide a rigorous convergence result. We also discuss the existence and uniqueness of solution for both the three-dimensional problem and the corresponding two-dimensional limit problem.
The structure of the paper is the following: in Sect. 2 we shall describe the variational and mechanical formulations of the problem in cartesian coordinates in a general domain, and present a result of existence and uniqueness of solution for that problem. In Sect. 3 we consider the particular case when the deformable body is, in fact, a shell and reformulate the variational formulation in curvilinear coordinates. Then we give the scaled formulation. To do that, we will use a projection map into a reference domain and we will introduce the scaled unknowns and forces as well as the assumptions on coefficients. We also devote this section to recall and derive results that will be needed later. In Sect. 4 we briefly describe the formal asymptotic analysis which leads to the formulation of limit two-dimensional problems. Then, in Sect. 5 we discuss the existence and uniqueness of solution for the twodimensional limit problem and then we focus on the elliptic membrane case, for which we provide a rigorous convergence result. Finally, in Sect. 6 we show that the solution to the re-scaled version of this problem, with true physical meaning, also converges. The paper ends with Sect. 7, devoted to the conclusions and future work.

A Three-Dimensional Dynamic Problem for Thermoelastic Bodies
LetΩ ε be a three-dimensional bounded domain and assume thatΩ ε is the reference configuration of a deformable body made of an elastic material, which is homogeneous and isotropic, with Lamé coefficientsλ ε ≥ 0,μ ε > 0. LetΓ ε = ∂Ω ε denote the boundary of the body, which is divided into two disjoint partsΓ ε N andΓ ε 0 , where the measure of the latter is strictly positive. Letx ε = (x ε i ) be a generic point ofΩ ε . Notice that at first glance, the notation for sets, variables and functions seems unnecessarily complex. Indeed, the ε andˆmarks are only meaningful in the context of the shells setting, to be detailed in the forthcoming sections. But, given that there we are going to recall results and arguments developed in this current section, we decided to keep here this notation, in favor of future coherence.
We suppose that the material has a thermal dilatation coefficientα ε T , a thermal conductivity coefficientk ε , a specific heat coefficientβ ε and a specific mass densityρ ε . The constitutive equation relating the stress tensor componentsσ ε ij to the linearized strain tensorê ε ij (û ε ) components, and the temperatureθ ε is given by the linearized Duhamel-Neumann law (see, for example [23] and references therein): denotes the deformation operator. Here δ ij represents the Kronecker's symbol and∂ i the partial derivative with respect tox ε i . Notice that here and below, and for the sake of a clearer exposition, we shall omit the explicit dependence of the various functions on space and time variables, as long as there is no ambiguity. We assume that the body is subjected to a boundary condition of place; in particular, the displacements field vanishes onΓ ε 0 . The body is under the effect of a heat sourceq ε and applied volume forces of densitŷ f ε = (f i,ε ) acting inΩ ε , and tractions of densityĥ ε = (ĥ i,ε ) acting uponΓ ε N . Then, the set of equations describing the mechanical behaviour of a regular threedimensional deformable solid in thermoelasticity are the following: Problem 1 Find the displacements fieldû ε = (û ε i ) and the temperature fieldθ ε verifyinĝ

Remark 1
We only consider homogeneous boundary and initial conditions for the sake of simplicity. Besides, our main interest is in the asymptotic analysis which follow in the sections below, and nonhomogenous initial conditions do not introduce major additional difficulties in that sense. Now, to derive the variational formulation of the problem, we first define the space of admissible displacements We also define the space of admissible temperatures which is a Hilbert space equipped with the norm Besides, as long as there is no room for confusion, we shall avoid specifying the domain in the subindices for the norms' notation. Further, for the sake of simplicity in the formulations to come, we define the following operators: -The bilinear, continuous and coercive forms whereÂ ij kl,ε =λ ε δ ij δ kl +μ ε (δ ik δ jl + δ il δ jk ) denotes the elasticity fourth-order tensor. -The continuous form -The functionalĴ ε (·) is defined a.e. in (0, T ) as where we use the notation for a duality pair ·, · in V (Ω ε ) × V (Ω ε ), and similarly, Then, it is straightforward to obtain the following variational formulation: β ε θε ,φ ε + a S,ε (θ ε ,φ ε ) + c ε (φ ε ,u ε ) = Q ε (t),φ ε ∀φ ε ∈ S(Ω ε ), a.e. in (0, T ), (3) withu ε (·, 0) =û ε (·, 0) = 0 andθ ε (·, 0) = 0.
In favour of simplicity, we are going to assume that the different parameters of the problem (thermal conductivity, thermal dilatation, specific heat coefficient, mass density, Lamé coefficients) are constants.

Theorem 1 Let us assume that
Then, there exists a unique pair (û ε (x, t),θ ε (x, t)) solution to Problem 2 such that Remark 2 The regularity results in (4c) and (5b) imply that the duality products involvinĝ u ε andθ ε in (2) and (3) are actually the usual inner products in L 2 (Ω ε ).

A Three-Dimensional Dynamic Problem for Thermoelastic Shells
In this section we consider the particular case when the deformable body is, in fact, a shell. We first provide key notations and some preliminary results in a summarised form. The interested reader can consult [9] and [20] for a more detailed exposition.
Let ω be a bounded domain of R 2 , with a Lipschitz-continuous boundary γ = ∂ω. Let y = (y α ) be a generic point of its closureω and let ∂ α denote the partial derivative with respect to y α . Above and in what follows, Greek indices take their values in the set {1, 2}, whereas Latin indices do it in the set {1, 2, 3}. We will use summation convention on repeated indices. Let θ ∈ C 2 (ω; R 3 ) be an injective mapping such that the two vectors a α (y) := ∂ α θ (y) are linearly independent. These vectors form the covariant basis of the tangent plane to the surface S := θ (ω) at the point θ (y). We also consider the two vectors a α (y) of the same tangent plane defined by the relations a α (y) · a β (y) = δ α β , that constitute its contravariant basis. We define a 3 (y) = a 3 (y) := a 1 (y) ∧ a 2 (y) |a 1 (y) ∧ a 2 (y)| , the unit normal vector to S at the point θ (y), where ∧ denotes vector product in R 3 . We can define the first fundamental form, given as metric tensor, in covariant or contravariant components, respectively, by a αβ := a α · a β , a αβ := a α · a β . The second fundamental form, given as curvature tensor, in covariant or mixed components, respectively, is given by b αβ := a 3 · ∂ β a α , b β α := a βσ · b σ α , and the Christoffel symbols of the surface S as Γ σ αβ := a σ · ∂ β a α . The area element along S is √ ady where a := det(a αβ ). We define the three-dimensional domain Ω ε := ω × (−ε, ε) and its boundary Γ ε = ∂Ω ε . We also define the following parts of the boundary, Γ ε N := ω × {ε} (it could also be the lower face or the union of both), we cast this setting into the more general three dimensional framework of the preceding section, as a particular case. Further, in [9, Th. 3.1-1] it is shown that if the injective mapping θ :ω → R 3 is smooth enough, the mapping Θ :Ω ε → R 3 is also injective for ε > 0 small enough and the vectors are linearly independent. Therefore, the three vectors g ε i (x ε ) form the covariant basis at the point Θ(x ε ) and g i,ε (x ε ) defined by the relations g i,ε · g ε j = δ i j form the contravariant basis at the point Θ(x ε ). The covariant and contravariant components of the metric tensor are defined, respectively, as g ε ij := g ε i · g ε j , g ij,ε := g i,ε · g j,ε , and Christoffel symbols as Γ p,ε . We now define the corresponding contravariant components in curvilinear coordinates for the applied forces densities: and the covariant components in curvilinear coordinates for the displacements field: Remark 3 Notice that forces above depend also on the time variable t ∈ [0, T ], but we decided to keep it implicit for the sake of readiness, since the subject of the change of variable is the spatial component. The same comment applies in a number of situations below.

Both are real Hilbert spaces with the induced inner product of
The corresponding norm is denoted by · 1,Ω ε in both cases, since no confusion is possible. With these definitions it is straightforward to derive from the Problem 2 the following variational problem (see [9] for the case in linear elasticity and use similar arguments): A ij kl,ε := λg ij,ε g kl,ε + μ(g ik,ε g jl,ε + g il,ε g jk,ε ), represent the contravariant components of the three-dimensional elasticity tensor, and the functions e ε i||j (v ε ) = e ε j ||i (v ε ) ∈ L 2 (Ω ε ) that represent the covariant components of the linearized change of metric tensor, or strain tensor, are defined by 3 , where ∂ ε i denotes partial derivative with respect to x ε i . Note that the following simplifications are verified, as a consequence of the definition of Θ in (6). The definitions of the fourth order tensor (7) imply that (see [9, Theorem 1.8-1]) for ε > 0 small enough, there exists a constant C e > 0, independent of ε, such that, for all x ε ∈Ω ε and all t = (t ij ) ∈ S 3 (vector space of 3 × 3 real symmetric matrices).

Remark 4
We recall that the vector field u ε = (u ε i ) : Ω ε × [0, T ] → R 3 solution of Problem 3 has to be interpreted conveniently. The functions u ε i :Ω ε × [0, T ] → R 3 are the covariant, time dependent, components of the "true" displacements field For convenience, we consider a reference domain independent of the small parameter ε. Hence, let us define the three-dimensional domain Ω := ω × (−1, 1) and its boundary Γ = ∂Ω. We also define the following parts of the boundary, be a generic point inΩ and we consider the notation ∂ i for the partial derivative with respect to x i . We define the projection map π ε :Ω →Ω ε , such that For the sake of simplicity, from now on, we are going to assume that the different parameters of the problem (thermal conductivity, thermal dilatation, specific heat coefficient, mass density, Lamé coefficients) are all independent of ε. Also, let the functions, Γ p,ε ij , g ε , A ij kl,ε be associated with the functions Γ p ij (ε), g(ε), A ij kl (ε), defined by sym , which we also denote as (e i||j (ε; v)), defined by Note that with these definitions it is verified that Remark 5 The functions Γ p ij (ε), g(ε), A ij kl (ε) converge in C 0 (Ω) when ε tends to zero.

Remark 6
When we consider ε = 0 the functions will be defined with respect to y ∈ω. Notice the singularities in (11) and (12) for that case. We shall distinguish the three-dimensional Christoffel symbols from the two-dimensional ones associated to S by using Γ σ αβ (ε) and Γ σ αβ , respectively.
See for example [18]. Further, as commented earlier, we usually omit the explicit time dependence for the sake of a shorter notation.
We now present some additional results which will be used in the next section. In [9, Theorem 3.4-1], we find the following useful result:
We provide here, as a standalone theorem, a result which can be found inside the proof of [9,. for all v ∈ X(Ω). As consequence there exists a constant c 2 > 0 such that

Formal Asymptotic Analysis
In this section we briefly describe the formal procedure to identify possible two-dimensional limit problems, depending on the geometry of the middle surface, the set where the boundary conditions are given, the order of the applied forces (the procedure is described in detail in [9] for elastic shells in the static case). We consider scaled applied forces and heat source of the form where m is an integer number that will show the order of the volume, heat source and surface forces, respectively. We substitute in (20) to obtain the following problem:
Assume that θ ∈ C 3 (ω; R 3 ) and that the scaled unknowns u(ε), ϑ(ε) admit asymptotic expansions of the form The assumption (25) implies an asymptotic expansion of the scaled linear strain as follows Besides, the functions e i||j (ε; v) admit the following expansion, Upon substitution on (23) and (24), we proceed to characterize the terms involved in the asymptotic expansions by considering different values for m and grouping terms of the same order. In this way, taking in (23) the order m = −2 and particular cases of test functions, we reason that f −2 = h −1 = 0, which leads to ∂ 3 u 0 = 0. From (24), we reason that q −2 = 0 and find that ∂ 3 ϑ 0 = 0. Thus the zeroth order terms of both unknowns would be independent of the transversal variable x 3 . Particularly, u 0 can be identified with a function ξ 0 ∈ V (ω), and ϑ 0 can be identified with a function ζ 0 ∈ S(ω) where Taking m = −1, and using particular cases of test functions, we reason that f −1 = h 0 = 0 and we find that denote the covariant components of the linearized change of metric tensor associated with a displacement field η i a i of the surface S. From (24) we reason that q −1 = 0 and find that Having these results in mind, for m = 0, developing A ij kl (0) and taking v = η ∈ V (ω) and ϕ ∈ S(ω) leads to the following two-dimensional problem, to which we shall refer as thermoelastic membrane problem: 1 (·, +1), and Q 0 := 1 −1 q 0 dx 3 . Also, a αβσ τ denotes the contravariant components of the fourth order twodimensional elasticity tensor, defined as follows: a αβ a σ τ + 2μ(a ασ a βτ + a ατ a βσ ).
The problem above will be analyzed in more detail in the following section. There, we shall study the existence and uniqueness of solution under additional hypotheses of geometric nature and a more suitable set of functional spaces, and provide a rigorous convergence result. To that end, the following ellipticity result for the elasticity tensor will be used. There exists a constant c e > 0 independent of the variables and ε, such that α,β |t αβ | 2 ≤ c e a αβσ τ (y)t σ τ t αβ , for all y ∈ω and all t = (t αβ ) ∈ S 2 (vector space of 2 × 2 real symmetric matrices).

Elliptic Membrane Case. Convergence
In what follows, we assume that the family of three-dimensional linearly thermoelastic shells consist of elliptic membrane shells, that is, the middle surface of the shell S is uniformly elliptic and the boundary condition of place is considered on the whole lateral face of the shell, that is, γ 0 = γ . Further, from the formal asymptotic analysis made in the preceding section, we assume the hypotheses which led to Problem 6, namely, Since there is no possible ambiguity, in what follows we drop the superindices indicating the order of the different functions. We also recall that for elliptic membranes it is verified the following two-dimensional Korn's type inequality (see, for example, [ where V M (ω) := H 1 0 (ω) × H 1 0 (ω) × L 2 (ω), is the right space for the well-posedness of Problem 6. In this section and in the sequel, C represents a positive generic constant whose specific value may change from line to line, independent of ε and the unknowns. Besides, for the sake of simplicity, we assume that all the parameters involved are constant. Also, the notationv stands for the average on x 3 , i.e., v := 1 To favour a clearer exposition, let us reformulate Problem 6: Problem 7 Find a pair t → (ξ (y, t), ζ(y, t) withξ (·, 0) = ξ (·, 0) = 0 and ζ(·, 0) = 0.
Above, we have used +1) and Q := 1 −1 qdx 3 . The following shows that there is a unique solution for this problem.

Theorem 4
Let ω be a domain in R 2 , let θ ∈ C 2 (ω; R 3 ) be an injective mapping such that the two vectors a α = ∂ α θ are linearly independent at all points ofω. Let f i and q ∈ H 1 (0, T ; L 2 (Ω)), h i ∈ H 2 (0, T ; L 2 (Γ N )). Then the Problem 7, has a unique solution (ξ , ζ ) such that Like in Theorem 1, we can cast this problem into the setting of problems solved in [12] or [16, p. 359], for example. Details can be found in [2]. Now, we present here the main result of this paper, namely that the scaled threedimensional unknowns (u(ε), ϑ(ε)) converge, as ε tends to zero, towards a limit (u, ϑ) independent of the transversal variable, and that this limit can be identified with the solution (ξ , ζ ) of the Problem 7, posed over the two-dimensional set ω.
In what follows, and for the sake of simplicity, we assume that for each ε > 0 the initial condition for the scaled linear strain is this is, the domain is on its natural state with no strains on it at the beginning of the period of observation.  Proof We follow the structure of the proof given in [9, for the case of elastic elliptic membrane shells. Hence, we shall reference some steps which apply in the same manner and omit some details. Also, for the sake of readability we may use the shorter notations e i||j (ε) := e i||j (ε; u(ε)). In addition to that, all references to (23) or (24) have to be considered as for m = 0 and drop the superindices. The proof is divided into several parts, numbered from (i) to (v).
(i) A priori boundedness and extraction of weak convergent sequences. For ε > 0 sufficiently small, there exist bounded sequences, also indexed by ε, and weak limits as specified below: Moreover, ϑ, u α = 0 on Γ 0 .
For the proof of this step we take v =u(ε) in (23) (see Remark 8) and ϕ = ϑ(ε) in (24) and sum both expressions to find We now introduce the following norms: which is equivalent to the usual norm | · | 0,Ω because of the ellipticity of (g αβ (ε)) and the regularity of Θ. Also, which is a norm in V (Ω) because of the Korn inequality (see [9,) and the ellipticity of A ij kl (ε). Finally, which is a seminorm in S(Ω). Because of the uniform ellipticity of the tensors and matrices involved, and the properties of g(ε), we are going to be able to use constants independent of ε in the estimates below. Indeed, going back to (33), we obtain Integrating in [0, t] with respect to the time variable, using the equivalences mentioned above, together with the uniformity with respect to ε of the constants involved in those equivalences, integrating by parts the term with the tractions h i , using Theorem 3 and Young's inequality, we find that there exist a constant C > 0 independent of ε such that Hence, by using Gronwall's inequality and the three-dimensional Korn's inequality that can be found in [9,, all the assertions of (i) follow.
(ii) The limits of the scaled unknowns, u i , ϑ found in Step (i) are independent of x 3 .
The part corresponding to u i is analogous to the Step (ii) in [9, Theorem 4.4-1], so we omit it. Regarding ϑ , its independence on x 3 is a consequence of the boundedness of {ε −1 ∂ 3 ϑ(ε)}.
(iii) The limits e i||j found in (i) are independent of the variable x 3 . Moreover, they are related with the limits u := (u i ) and ϑ by Indeed, first considering v = u(ε) in (10) and η = u in (26) (par abus de langage, since u is independent of x 3 , but actually u ∈ [H 1 (Ω)] 2 × L 2 (Ω)), taking into account Step (i) and the convergences Γ σ αβ (ε) → Γ σ αβ and Γ 3 αβ (ε) → b αβ in C 0 (Ω) given by (16)-(18), we have that in L 2 (Ω) a.e. in (0, T ). Moreover, e α||β are independent of x 3 , as a straightforward consequence of the independence on x 3 of u i (Step (ii)). In addition, let v ∈ V (Ω). As a consequence of the definition of the scaled strains in (10)-(12), we find (23) we can take as test function εv ∈ V (Ω). Then, taking into account (8), we have Passing to the limit as ε → 0, decomposing A ij kl (ε) into the components with different asymptotic behaviour (see (13)- (14)), the properties of g(ε) (see (19)) and the convergences in Step (i), we obtain the following equality: By taking particular test functions and using Theorem 2, we deduce (34). Then, we go back to (36) and use again Theorem 2 to deduce (35). The independence of e 3||3 on x 3 is a consequence of this relation, as well.
(v) The weak convergences are, in fact, strong.
Besides, we use Lebesgue Theorem where needed, as well. Thus, the limit of the terms with traction (47)  We can undo the integration by parts, then reason like in (46). Finally, the strong convergences e i||j (ε) → e i||j in L ∞ (0, T ; L 2 (Ω)) also imply the strong convergences for u i (ε), by following arguments not depending on the particular set of equations, but on arguments of differential geometry and functional analysis which do not differ from those used in [9,. Therefore, we just omit them and refer the interested reader to the book.