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Effect of intrinsic rotations, microstructural expansion and contractions in initial boundary value problem of thermoelastic bodies

Marin I Marin1*, Ravi P Agarwal2 and Ibrahim A Abbas34

Author Affiliations

1 Department of Mathematics, Transilvania University of Brasov, Brasov, Romania

2 Department of Mathematics, Texas A&M University-Kingsville, Kingsville, USA

3 Department of Mathematics, Faculty of Science and Arts-Khulais, King Abdulaziz University, Jeddah, Saudi Arabia

4 Department of Mathematics, Faculty of Science, Sohag University, Sohag, Egypt

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Boundary Value Problems 2014, 2014:129  doi:10.1186/1687-2770-2014-129

The electronic version of this article is the complete one and can be found online at: http://www.boundaryvalueproblems.com/content/2014/1/129


Received:26 February 2014
Accepted:12 May 2014
Published:22 May 2014

© 2014 Marin et al.; licensee Springer.

This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly credited.

Abstract

This study is dedicated to some basic theorems in the thermoelastodynamics of microstretch bodies. Our intention is to show that the presence of the microstretch does not affect the main characteristics of the mixed initial boundary value problem for thermoelastic bodies. The result regarding the uniqueness theorem is derived with no definiteness assumptions on the elastic coefficients and in the absence of the restriction that the conductivity tensor is positive definite. In the last part of the paper we establish a basic relation which leads to the reciprocal theorem and to another uniqueness result.

MSC: 35M30, 35Q74, 74A15, 74A60, 74M25.

Keywords:
thermoelastic; microstretch; seismic waves; earthquake

1 Introduction

The theory of micromorphic elastic solids was first elaborated by Eringen (see, for instance, [1]). Then Eringen has generalized [2] this theory in order to cover the theory of thermo-microstretch elastic materials. In short, this is a theory of thermoelasticity with microstructure that includes intrinsic rotations and microstructural expansion and contractions.

The micromorphic theory was introduced to describe adequately the behavior of materials such a liquid crystal, fluid suspensions, polycrystalline aggregates, and granular media. For this it is necessary to introduce into the continuum theory some terms reflecting the microstructure of the materials. In the context of this theory, each material point has three deformable directors.

A continuum body is a microstretch continuum if the directors are constrained to have only breathing-type microdeformations. All points of a microstretch continuum can stretch and contract independently of their translations and rotations.

This theory is expected to find applications in the treatment of composites materials reinforced with chopped fibers. Also, this theory can be useful in applications which deal with porous materials as geological materials, solid packed granular materials, and many others.

On the other hand, materials which operate at elevated temperatures will invariably be subjected to heat flow at some time during normal use. Such heat flow will involve a nonlinear temperature distribution, which will inevitable give rise to thermal stresses. For these reasons, the development, design, and selection of materials for high temperature applications require a great deal of care. The role of the pertinent material properties and other variables which can affect the magnitude of thermal stress must be considered.

The theory of microstretch elastic bodies is generalized from the micropolar theory introduced by Cosserat. There are many papers which are concerned with this theory. For instance, Ciarletta in [3] has used the basic results deduced by Eringen in order to investigate the isothermal bending of microstretch elastic plates. Ciarletta et al. dedicated the paper [4] to the study of some basic properties of wave numbers of the longitudinal and transverse plane harmonic waves, in the context of thermoelasticity for materials with voids.

In the paper [5] Iesan and Pompei have presented a solution of Boussinesq-Somigliana-Galerkin type for the boundary value problem in this context.

In the paper [6], Agarwal et al. presented new existence results for initial value problems. The nonlinearity may be singular in its dependent variable and is allowed to change sign. Also, the paper of Agarwal and O’Regan [7] presents existence result for some boundary value problems definite on infinite intervals, which, in particular, includes a problem which arises in the theory of colloids. In the studies [8-11] we tackle some questions with regards to the microstretch-thermoelastic materials. Thus, in the paper [9], we use the Lagrange identity to prove the uniqueness theorem and some continuous dependence theorems without recourse to any energy conservation law, or to any boundedness assumptions on the thermoelastic coefficients. Also, the paper [8] is concerned with microstretch-thermoelastic materials. In this context, for the mixed initial boundary value problem, we prove that the Cesaro means of the kinetic and strain energies of a solution with finite energy become asymptotically equal as time tends to infinity. A model of microstretch-thermoelastic body with two temperatures is presented in the paper [12].

The study [13] is dedicated to the study of propagation of shear waves in a non-homogeneous anisotropic incompressible gravity field, and an initially stressed medium is studied.

The paper [14] investigates the longitudinal wave propagation in a perfectly conducting elastic circular cylinder in the presence of an axial initial magnetic field. Other results regarding generalized thermoelasticity can be found in the papers [15,16].

The present paper must be considered as a first step toward a better understanding of microstretch and thermal stress in the study of the above enumerated materials.

The reciprocity and representation relations that appear in our study constitute powerful theoretical tools in the assessment of the theory of seismic-sources mechanism, in the studies connected with seismic wave propagation.

Also, we think that this paper is a good help to understanding the application of the microstretch mechanism to earthquake problems.

There are many results regarding the mechanism of earthquakes, as, for instance, in the papers [17] and [18] and in the references therein.

2 Basic equations

For convenience the notations and terminology chosen are almost identical to those of our studies [8,9]. The present paper is concerned with an anisotropic and homogeneous material.

Let the body occupy, at time <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M1','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M1">View MathML</a>, a properly regular region B of the three-dimensional Euclidian space, bounded by the piece-wise smooth surface ∂B; and we denote the closure of B by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M2">View MathML</a>. We refer the motion of the body to a fixed system of rectangular Cartesian axes <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M3','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M3">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M4','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M4">View MathML</a> and adopt the Cartesian tensor notation. Points in B are denoted by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M5">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M6','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M6">View MathML</a> is the temporal variable. Throughout this work the Einstein summation convention over repeated indices is used. The subscript j after a comma indicates partial differentiation with respect to the spatial argument <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M5','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M5">View MathML</a>. All Latin subscripts are understood to range over the integers <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M8','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M8">View MathML</a>, while the Greek indices have the range <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M9','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M9">View MathML</a>. A superposed dot denotes the derivatives with respect to the time variable t. Also, the spatial argument and the time argument of a function will be omitted when there is no likely confusion.

Let us denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M10">View MathML</a> the components of the displacement vector and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M11">View MathML</a> the components of the microrotation vector. Also, we denote by ω a scalar function that characterizes the microstretch and by θ the temperature measured from the constant absolute temperature <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M12','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M12">View MathML</a> of the body in its reference state.

As usual, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M13','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M13">View MathML</a> the components of the stress tensor and by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M14','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M14">View MathML</a> the components of the couple stress tensor over B. Also, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M15">View MathML</a> the components of the microstress vector.

In the dynamic theory of the thermoelasticity of microstretch bodies the fundamental system of field equations consists of:

– the equations of motion

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M16','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M16">View MathML</a>

(1)

– the balance of the equilibrated forces

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M17','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M17">View MathML</a>

(2)

– the energy equation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M18','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M18">View MathML</a>

(3)

For an anisotropic and homogeneous microstretch thermoelastic material, the constitutive equations have the form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M19','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M19">View MathML</a>

(4)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M20','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M20">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M21','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M21">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M22','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M22">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M23','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M23">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M24','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M24">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M25','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M25">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M26','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M26">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M27','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M27">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M28','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M28">View MathML</a>, d, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M29">View MathML</a> are the characteristic constitutive coefficients, and they obey the following symmetry relations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M30','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M30">View MathML</a>

(5)

In the above equations we have used the following notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M31">View MathML</a> the components of body force;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M32">View MathML</a> the components of body couple;

L the generalized external body load;

ϱ is the reference constant mass density;

J and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M33','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M33">View MathML</a> are the coefficients of microinertia;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M15','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M15">View MathML</a> are the components of the internal hypertraction vector;

η is the entropy per unit mass;

S is the heat supply per unit mass;

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M35','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M35">View MathML</a> are the components of heat flux vector.

The components of the strain tensors <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M36','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M36">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M37','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M37">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M38','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M38">View MathML</a> are defined by means of the geometric equations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M39','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M39">View MathML</a>

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M40','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M40">View MathML</a> is the alternating symbol.

The second law of thermodynamics implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M41','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M41">View MathML</a>

(6)

that is, the conductivity tensor k is positive semi-definite only.

In what follows we need the following regularity assumptions:

(1) the constitutive coefficients are continuously differentiable functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M2">View MathML</a>;

(2) ϱ and d are continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M2">View MathML</a>;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M31','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M31">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M32','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M32">View MathML</a> and L are continuous functions on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M46','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M46">View MathML</a>, where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M47','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M47">View MathML</a>.

We denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M48','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M48">View MathML</a> the components of surface traction, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M49','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M49">View MathML</a> the components of surface couple, λ the microsurface traction and q the heat flux. These quantities are defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M50','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M50">View MathML</a>

at regular points of the surface ∂B.

Here, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M51','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M51">View MathML</a> are the components of the outward unit normal of the surface ∂B.

Along with the system of field equations (1)-(4) we consider the following initial conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M52','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M52">View MathML</a>

(7)

and the following prescribed boundary conditions:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M53','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M53">View MathML</a>

(8)

where <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M54','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M54">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M55','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M55">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M56','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M56">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M57','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M57">View MathML</a> with respective complements <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M58','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M58">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M59','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M59">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M60','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M60">View MathML</a>, and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M61','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M61">View MathML</a> are subsets of the surface ∂B such that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M62','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M62">View MathML</a>

We assume that:

(1) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M63','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M63">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M64','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M64">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M65','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M65">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M66','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M66">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M67','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M67">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M68','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M68">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M69','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M69">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M70','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M70">View MathML</a> are continuous function on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M2','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M2">View MathML</a>;

(2) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M72','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M72">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M73','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M73">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M74','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M74">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M75','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M75">View MathML</a> are continuous function on their domains;

(3) <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M76','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M76">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M77','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M77">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M78','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M78">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M79','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M79">View MathML</a> are piecewise regular functions in their domains and continuous with regards to the time variable t.

By a solution of the mixed initial boundary value problem of the theory thermoelasticity of microstretch bodies in the cylinder <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M80','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M80">View MathML</a> we mean an ordered array <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M81','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M81">View MathML</a> which satisfies (1), (2), and (3) for all <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M82','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M82">View MathML</a>, the initial conditions (7) and the boundary conditions (8).

3 Main results

Let us consider the functions K and U defined on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M83','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M83">View MathML</a> by the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M84','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M84">View MathML</a>

(9)

In the next theorem we prove a relationship between the functions K and U.

Theorem 1Assume that the symmetry relations (5) are satisfied. Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M85','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M85">View MathML</a>

(10)

where we have denoted by<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M86','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M86">View MathML</a>the following expression:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M87','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M87">View MathML</a>

(11)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M88','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M88">View MathML</a>.

Proof Using the constitutive equations (4) and the symmetry relations (5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M89','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M89">View MathML</a>

(12)

Here we have suppressed, for convenience, the dependence of the functions on argument x, because there is no likely confusion.

By using the equations of motion (1), the balance of equilibrated forces (2), the energy equation (3) and the geometric equations, we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M90','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M90">View MathML</a>

(13)

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M91','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M91">View MathML</a>

(14)

Now, we integrate, over B, both sides of the equality (12) and then use (5), (13), (14), and the divergence theorem, and it follows that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M92','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M92">View MathML</a>

(15)

Finally, we integrate the equality (15) from 0 to t and arrive at the desired result (10) so that the theorem is demonstrated. □

In the next lemma we prove another relation between the functions U and K defined by (9).

Lemma 1Assume that the symmetry relations (5) are satisfied. Then we have the following relations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M93','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M93">View MathML</a>

(16)

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M94','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M94">View MathML</a>

(17)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M95','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M95">View MathML</a>.

The functionPis defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M96','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M96">View MathML</a>

(18)

Proof Taking into account the equations of motion (1), the balance of equilibrated forces (2), the energy equation (3), and the constitutive equations (4), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M97','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M97">View MathML</a>

From this equality it is easy to deduce that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M98','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M98">View MathML</a>

By integrating this relation over the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M99','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M99">View MathML</a> we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M100','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M100">View MathML</a>

(19)

If we add (10) and (19), term by term, (16) follows.

If we subtract (10) from (19), term by term, (17) follows and this concludes Lemma 1. □

The uniqueness result from the next theorem is based on the results from Theorem 1 and Lemma 1.

Theorem 2Assume that:

(i) the symmetry relations (5) are satisfied;

(ii) ϱ, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M101','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M101">View MathML</a>andJare strictly positive;

(iii) dis strictly positive or strictly negative;

(iv) the conductivity tensor<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M29">View MathML</a>is positive semi-definite.

Then the mixed initial boundary value problem of thermoelasticity of microstretch materials consisting of (1)-(3), the initial condition (7), and the boundary condition (8) has at most one solution.

Proof Suppose, to the contrary, that our mixed problem has two solutions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M103','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M103">View MathML</a>

Let us denote by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M104','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M104">View MathML</a>

the difference of two solutions, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M105','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M105">View MathML</a>

Because of linearity, this difference is also a solution of our problem, but it corresponds to null data.

Thus, from (17) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M106','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M106">View MathML</a>

(20)

By using the hypotheses (ii) and (iv) of the theorem, (20) implies that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M107','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M107">View MathML</a>

(21)

and

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M108','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M108">View MathML</a>

(22)

But <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M10','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M10">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M11','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M11">View MathML</a>, and ω vanish initially, so that from (21) we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M111','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M111">View MathML</a>

(23)

Taking into account (22) and (23), (16) reduces to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M112','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M112">View MathML</a>

Since <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M113','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M113">View MathML</a> or <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M114','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M114">View MathML</a>, this relation yields

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M115','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M115">View MathML</a>

(24)

From (23) and (24) we deduce that the difference of the two solutions is null, i.e., we have the uniqueness of solution and Theorem 2 is demonstrated. □

Consider two scalar functions u and v which are defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M116">View MathML</a> and continuous with respect to time.

As is well known, the convolution product of the function u and v is defined by the integral

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M117','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M117">View MathML</a>

Also, let us consider the functions <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M118','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M118">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M119','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M119">View MathML</a> defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M120','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M120">View MathML</a>

(25)

For a continuous function h defined on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M116','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M116">View MathML</a>, we denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M122','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M122">View MathML</a> the convolution product <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M123','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M123">View MathML</a>, i.e.

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M124','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M124">View MathML</a>

(26)

Using these considerations, we can write the energy equation (3) and the initial condition

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M125','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M125">View MathML</a>

in the equivalent form

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M126','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M126">View MathML</a>

(27)

where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M127','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M127">View MathML</a>

(28)

Consider two external data systems <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M128">View MathML</a>, which act on the thermoelastic-microstretch material, defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M129','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M129">View MathML</a>

and denote by <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M130">View MathML</a> a solution of the mixed problem which corresponds to <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M128">View MathML</a>

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M132','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M132">View MathML</a>

Also, we use the following notations:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M133','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M133">View MathML</a>

(29)

Lemma 2Assume that the symmetry relations (5) are satisfied. Consider the functions<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M134','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M134">View MathML</a>defined by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M135','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M135">View MathML</a>

(30)

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M136','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M136">View MathML</a>

(31)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M137','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M137">View MathML</a>.

Proof We introduce the notation

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M138','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M138">View MathML</a>

(32)

From (32) and the constitutive equations (4) we deduce

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M139','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M139">View MathML</a>

(33)

Using the symmetry relations (5), from (33) we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M140','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M140">View MathML</a>

(34)

On the other hand, using the equations of motion (1), the balance of equilibrated forces (2), and the energy equation (3) in (27) and (32), we are lead to the following expression of <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M141','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M141">View MathML</a>:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M142','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M142">View MathML</a>

(35)

Now, we integrate (35) over B, then we use the symmetry relations (34) and the divergence theorem so that we obtain the desired result (31) and Lemma 2 is proved. □

Based on the result of Lemma 2 we can prove the reciprocal result from the next theorem.

Theorem 3Assume that the symmetry relations (5) are satisfied. Let<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M130','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M130">View MathML</a>be a solution corresponding to the external data<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M128','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M128">View MathML</a>, <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M145','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M145">View MathML</a>. Then the following reciprocal relation is valid:

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M146','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M146">View MathML</a>

(36)

where we have used the notations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M147','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M147">View MathML</a>

(37)

Proof We use the substitution <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M148','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M148">View MathML</a> and <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M149','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M149">View MathML</a> in (31) and integrate the resulting relation on the interval <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M150','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M150">View MathML</a> such that, by using the symmetry relations (5), we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M151','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M151">View MathML</a>

(38)

It is easy to prove that

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M152','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M152">View MathML</a>

(39)

Taking the convolution of (38) with g and using (39), we obtain the reciprocal relation (36) and the proof of Theorem 3 is complete. □

Remark In the case of null boundary data, from (36) we deduce that the operator of the thermoelastodynamics of microstretch bodies is symmetric with regard to the convolution.

Based on this symmetry, we can obtain some variational theorems of Gurtin type in classical thermoelasticity.

Also, based on the symmetry relations (31) we can obtain a minimum principle similar to those obtained by Reiss [19] in the classical isothermal case.

Theorem 4Assume that the symmetry relations (5) are satisfied. Let us consider the function

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M153','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M153">View MathML</a>

(40)

for all<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M154','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M154">View MathML</a>.

Then we have

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M155','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M155">View MathML</a>

(41)

Proof Using the result on <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M156','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M156">View MathML</a> from Lemma 2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M157','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M157">View MathML</a>

(42)

Let us apply (42) to the process

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M158','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M158">View MathML</a>

From (30) and (40), we obtain the equality

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M159','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M159">View MathML</a>

(43)

Similarly,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M160','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M160">View MathML</a>

(44)

It is easy to prove the relations

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M161','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M161">View MathML</a>

(45)

Using the symmetry relations (5) and (45), from (42), (43), and (44) we obtain the desired result (41) and the proof of Theorem 4 is complete. □

A similar reciprocal result has been obtained in [11], but using some strong hypotheses on the thermoelastic coefficients.

If the conductivity tensor <a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M29','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M29">View MathML</a> is assumed be positive definite, then the result of Theorem 4 can be used to obtain the uniqueness result established in [18].

Now, our intention is to give another proof of Theorem 2 by using the result of Theorem 4. Let us consider that the mixed problem formulated above has two solutions,

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M163','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M163">View MathML</a>

Let us denote by

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M164','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M164">View MathML</a>

the difference of two solutions, where

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M165','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M165">View MathML</a>

If we apply (41) for the difference, we are lead to

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M166','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M166">View MathML</a>

From this equality, by using the hypotheses (ii) and (iv) of Theorem 2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M167','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M167">View MathML</a>

Then, from (16), (17), and hypothesis (iii) of Theorem 2, we obtain

<a onClick="popup('http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M168','MathML',630,470);return false;" target="_blank" href="http://www.boundaryvalueproblems.com/content/2014/1/129/mathml/M168">View MathML</a>

which proves Theorem 4.

Remark Using a similar procedure as in [18] and [9], we can use (16), (17), and (41) to obtain some continuous dependence results.

4 Concluding remarks

The intrinsic rotations, microstructural expansion, and contractions do not affect the existence nor the uniqueness and continuous dependence of the solution of the mixed initial boundary value problem for thermoelastic bodies.

Also, it is not necessary to constrain the conductivity tensor to be positive definite to obtain the basic results of the theory of thermoelasticity of microstretch materials. It is sufficient for this tensor to be positive semi-defined as results from the Clausius-Duhem inequality.

Competing interests

The authors declare that they have no competing interests.

Authors’ contributions

MIM proposed main results of the paper and verified all calculations and demonstrations. RPA proposed the method of demonstration of results, without using a sophisticated mathematical apparatus. Also, he controlled the final shape of the paper. IAA performed all calculations and demonstrations and took into account the suggestions given by MIM. All authors read and approved the final manuscript.

Acknowledgements

We express our gratitude to the referees for their valuable criticisms of the manuscript and for helpful suggestions.

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