Spatial behaviour of thermoelasticity with microtempera- tures and microconcentrations

We consider a thermoelastic material with microtemperatures and microconcentrations. The mathematical model is represented by a system of partial differential equations with the coupling of the displacement, temperature, chemical potential, microconcentrations and microtemperatures fields. The processes of heat and mass diffusion play an important role in many engineering applications, such as satellite problems, manufacturing of integrated circuits or oil extractions. We study the spatial behaviour in a prismatic cylinder occupied by an anisotropic and inhomogeneous material. We impose final prescribed data that are proportional, but not identical, to their initial values. Moreover, we have zero body forces and zero lateral boundary conditions. The spatial behaviour is analysed in terms of some cross-sectional integrals of the solution that depend on the axial variable.


Introduction
The processes of heat and mass diffusion play an important role in many engineering applications, such as satellite problems, manufacturing of integrated circuits or oil extractions [1]. In order to show that the microelements have different microtemperatures, R. Grot introduced the concept of microtemperatures [4]. In order to show that the microelements have different concentrations, M. Aouadi, M. Ciarletta and V. Tibullo introduced the concept of microconcentrations [1].
We consider a thermoelastic material with microtemperatures and microconcentrations, as in [1], [2]. The mathematical model is represented by a system of partial differential equations with the coupling of the displacement, temperature, chemical potential, microconcentrations and microtemperatures fields. In the anisotropic case, well-posedness was shown in [1] by means of the semigroup theory of linear operators. Moreover, the asymptotic behaviour of the solution was discussed. In the isotropic case, the problem was analysed from a numerical point of view in [2] by means of the finite element method and the implicit Euler scheme.
The concept of microtemperatures was also used to model thermal effects at the microlevel in dipolar materials, see for example [7] and [8]. Other types of mathematical models for materials with microstructures were studied in [5] and [6].
We study the spatial behaviour in a prismatic cylinder occupied by an anisotropic and inhomogeneous material, following the approach from [3]. We impose final prescribed data that are proportional, but not identical, to their initial values. Moreover, we have zero body forces and zero lateral boundary conditions. The spatial behaviour is analysed in terms of some cross-sectional integrals of the solution that depend on the axial variable.

Preliminaries
We consider a thermoelastic material with microtemperatures and microconcentrations in the three-dimensional euclidean space and we use a fixed system of rectangular axes Ox i , i = 1, 2, 3. The Latin subscripts take the values 1, 2, 3, Greek subscripts take the values 1, 2 and repeated indices indicate the use of the Einstein summation convention. Differentiation is represented by a superposed dot in the case of material time derivatives and by a comma followed by a subscript in the case of derivatives with respect to a spatial coordinate.
First, we consider a domain Ω. Then, we restrict our considerations to a cylinder that is filled by an anisotropic and inhomogeneous thermoelastic material with microtemperatures and microconcentrations.
Below, we use the following notations: u i is the displacement, t i j is the stress tensor, ρ is the density in the reference configuration, f i is the body force per unit mass, s is the heat source per unit mass, T 0 is the absolute temperature in the reference configuration, S is the microentropy, q k is the heat flux vector, η k is the flux vector of mass diffusion, ε i is the first moment of energy vector, q i j is the first heat flux moment tensor, ς i is the microheat flux average, µ i is the first moment of the heat source vector, Ω i is the first moment of mass diffusion, η i j is the first mass diffusion flux moment tensor, σ i is the micromass diffusion flux average, T is the absolute temperature, T i is the microtemperature vector, C i is the microconcentration vector, P is the particle chemical potential, C is the concentration, θ = T − T 0 .
The equations of motion are [1] ρü i = t ki,k + ρ f i , ρT 0Ṡ = q i,i + ρs, The constitutive equations of the linear theory of centrosymmetric materials are [1] t i j = α i jkl e kl + γ i j θ + β i j P, ρS = −γ i j e i j + cθ + κP, where and the following symmetries hold true Furthermore, we have [1] cθ 2 + 2κθP + mP 2 > 0, Moreover, we have [1] q i = k i j θ , j + K i j T j , with the following symmetries for the coefficients Moreover, we denote bȳ and we haveK (s) ≥ 0.
Furthermore, we assume that ρ and the constitutive coefficients are continuous and bounded functions onΩ and

Spatial behaviour
We introduce the following notations where inΛ 1 (t) the integrand from relation (13) is evaluated over ∂Ω.
Lemma 3.2 (Properties) We obtain In the sequel, we assume that the region Ω ⊂ R 3 is a prismatic cylinder. Moreover, we consider that its bounded uniform cross-section D ⊂ R 2 has piecewise continuously differentiable boundary ∂D. The region Ω is assumed to be filled with an anisotropic and inhomogeneous thermoelastic material with microtemperatures and microconcentrations. The base of the cylinder contains the origin of the Cartesian coordinate system. The axis of the cylinder is parallel to the positive x 3 -axis. We use the following notation Furthermore, D(x 3 ) is used to show that the respective quantities are considered over the cross-section whose distance from the origin is x 3 . The lateral surface of the cylinder is denoted by Π, with Π = ∂D × [0, L], where L is the length of the cylinder. The constitutive coefficients are given functions that depend on the spatial variable x. The supply terms are assumed to be zero.