Modeling a Microstretch Thermoelastic Body with Two Temperatures

Abstract

We consider a theory of thermoelasticity constructed by taking into account the heat conduction in deformable bodies which depends on two temperatures. The first one is the conductive temperature, the second is the thermodynamic temperature, and the difference between them is proportional to the heat supply.

1. Introduction

In the classical uncoupled theory of thermoelasticity the equation of the heat conduction does not contain any elastic term. On the other hand, the heat equation is of parabolic type, which leads to infinite speeds of propagation for heat waves.

In view of eliminating these two phenomena which are not compatible with physical observations, some researchers proposed different generalizations of the classical theory. We restrict our attention to two such extensions. According to the model proposed by Lord and Shulman [1], the classical Fourier's law of heat conduction is replaced by a wave type heat equation. This new equation ensures finite speeds of propagation for the heat and elastic waves.

It is important to remark that in this model the equation of motion and constitutive equations remain the same as those for the coupled and uncoupled theories.

Another generalization is known as the theory of temperature-rate-dependent thermoelasticity or the thermoelasticity with two relaxation times. This theory contains two constants that act as relaxation times and modify the heat equation and, also, the equation of motion and constitutive equations. This theory was first proposed by Green and Lindsay [2] and has aroused much interest in recent years. Unlike the coupled thermoelasticity theory, this theory includes temperature rate among the constitutive variables and consequently predicts a finite speed for the propagation of thermal signals. Since thermal signals propagating with finite speeds have actually been observed in solids, the theory of temperature-rate-dependent thermoelasticity is more general and physically more realistic than the coupled theory.

In [3], Chen et al. gave a theory of thermodynamics of nonsimple elastic materials with two temperatures For the linearized form, Iesan establish in [4] some general theorems.

For time-dependent problems, in particular for wave propagation, the conductive temperature is different from thermodynamic temperature, regardless of presence of a heat supply.

The two temperatures have representation in the form a travelling wave plus a response which occurs instantaneously through the body.

First studies dedicated to the theory of microstretch elastic bodies were published by Eringen [5, 6]. This theory is a generalization of the micropolar theory and a special case of the micromorphic theory. In the context of this theory each material point is endowed with three deformable directors. A body is a microstretch continuum if the directors are constrained to have only breathing-type microdeformations. Also, the material points of a microstretch solid can stretch and contract independently of their translations and rotations.

The purpose of this theory is to eliminate discrepancies between the classical elasticity and experiments, since the classical elasticity failed to present acceptable results when the effects of material microstructure were known to contribute significantly to the body's overall deformations, for example, in the case of granular bodies with large molecules (e.g., polymers), graphite, or human bones (see [6]).

These cases are becoming increasingly important in the design and manufacture of modern day advanced materials, as small-scale effects become paramount in the prediction of the overall mechanical behaviour of these materials.

Other intended applications of this theory are to composite materials reinforced with chopped fibers and various porous materials.

In [7, 8], we find some basic results regarding thermoelastic microstretch bodies.

In [9], the governing equations are modified in the context of Lord and Shulman's theory of generalized thermoelasticity to include the two temperatures.

2. Basic Equations

Let us summarize the basic equations of the theory of thermoelasticity of microstretch bodies with two temperatures. Let 𝐵 be a bounded regular region of three-dimensional Euclidian space 𝑅³ occupied by a microstretch elastic body, referred to the reference configuration (at time 𝑡=0). Let 𝐵 denote the closure of 𝐵 and call 𝜕𝐵 the boundary of the domain 𝐵. We consider 𝜕𝐵 a piecewise smooth surface designated by 𝑛_𝑖 the components of the outward unit normal to the surface 𝜕𝐵. Letters in boldface stand for vector fields. We use the notation 𝑣_𝑖 to designate the components of the vector 𝐯 in the underlying rectangular Cartesian coordinates frame. A superposed dot stands for the material time derivative. We will employ the usual summation and differentiation conventions: the subscripts are understood to range over integer (1,2,3). Summation over repeated subscripts is implied and subscripts preceded by a comma denote partial differentiation with respect to the corresponding Cartesian coordinate.

The spatial argument and time argument of a function will be omitted when there is no likelihood of confusion. We refer the motion of the body to a fixed system of rectangular Cartesian axes 𝑂𝑥_𝑖,𝑖=1,2,3, and to the reference configuration.

The behaviour of the thermoelastic microstretch body is characterized by the following kinematic variables: 𝑢_𝑖=𝑢_𝑖(𝑥,𝑡),𝜑_𝑖=𝜑_𝑖(),(),()𝑥,𝑡𝜓=𝜓𝑥,𝑡𝑥,𝑡∈𝐵×0,𝑡₀, (1) where 𝑢_𝑖 are the components of the displacement field, 𝜑_𝑖 are the components of the microrotation field, and 𝜓 is the microstretch function.

The the components of the strain tensors 𝜀_𝑖𝑗, 𝑘_𝑖𝑗, and 𝛾_𝑖 are defined by means of the geometric equations: 𝜀_𝑖𝑗=𝑢_𝑗,𝑖+𝜀_𝑖𝑗𝑘𝜑_𝑘,𝜂_𝑖𝑗=𝜑_𝑗,𝑖,𝛾_𝑖=𝜓_,𝑖, (2) where 𝜀_𝑖𝑗𝑘 is the alternating symbol.

As usual, we denote by 𝑡_𝑖𝑗 the components of the stress tensor and by 𝑚_𝑖𝑗 the components of the couple stress tensor over 𝐵. Also, we denote by 𝜆_𝑖 the components of the internal hypertraction vector and by 𝑞_𝑖 the components of the heat flux vector.

Now, we define the surface traction 𝑡_𝑖, the surface couple 𝑚_𝑖, the microstretch traction 𝑝, and the heat flux 𝑞 at regular points of the surface 𝜕𝐵 by means of the components of the outward unit normal to the surface 𝜕𝐵 and components of stress tensors: 𝑡_𝑖=𝑡_𝑗𝑖𝑛_𝑗,𝑚_𝑖=𝑚_𝑗𝑖𝑛_𝑗,𝑝=𝜆_𝑖𝑛_𝑖,𝑞=𝑞_𝑖𝑛_𝑖. (3)

If we refer the motion relative to a stress-free and undistorted reference state, then the basic equations for the theory of thermoelasticity of microstretch bodies with two temperatures are(i)the equations of motion: 𝑡_{𝑖𝑗,𝑗}+𝐹_𝑖=𝜚𝑢̈_𝑖,𝑚_{𝑖𝑗,𝑗}+𝜀_𝑖𝑗𝑘𝑡_𝑗𝑘+𝐺_𝑖=𝐼_𝑖𝑗𝜑̈_𝑗, (4)(ii)the balance of the equilibrated forces: 𝜆_𝑖,𝑖+𝐿=𝐽𝜓̈, (5)(iii)the energy equation: 𝜑₀𝜓̇=−𝑞_𝑖,𝑖+𝑟. (6)

For an isotropic and homogeneous microstretch elastic material, the constitutive equations have the following form: 𝑡_𝑖𝑗=𝜆𝜀_𝑘𝑘+𝜆₀𝜓−𝛽𝜃𝛿_𝑖𝑗+𝜇+𝜈𝜀_𝑖𝑗+𝜇𝜀_𝑗𝑖,𝑚_𝑖𝑗=𝛼𝜂_𝑘𝑘𝛿_𝑖𝑗+𝛽𝜂_𝑗𝑖+𝛾𝜂_𝑖𝑗+𝑏₀𝜀_𝑘𝑗𝑖𝛾_𝑘,𝜆_𝑖=𝑎₀𝛾_𝑖+𝜀_𝑖𝑗𝑘𝜂_𝑗𝑘,𝑞_𝑖=−𝑘𝜑_,𝑖,𝜂=𝛽𝜀_𝑟𝑟+𝑐𝜑₀𝜃, (7) where 𝛿_𝑖𝑗 is the Kronecker symbol.

We define the thermodynamic temperature 𝜃 measured from a constant reference temperature 𝜑₀ by means of formula 𝜃=𝜑−𝑎𝜑_,𝑘𝑘 (8) in which 𝑎 is the two-temperature parameter, 𝑎>0.

In the above equations we have used the following notations:(i)𝐹_𝑖 are the components of body force;(ii)𝐺_𝑖 are the components of body couple;(iii)𝐿 is the generalized external body load;(iv)𝜚 is the reference mass density;(v)𝐼_𝑖𝑗=𝐼_𝑗𝑖 and 𝐽 are the coefficients of inertia;(vi)𝜂 is the entropy per unit volume measured from the entropy of the reference state;(vii)𝑟 is heat supply per unit mass;(viii)𝜑 is the conductive temperature measured from a constant reference temperature 𝜑₀.

Also, the constants 𝜆, 𝜆₀, 𝜇, 𝛽, 𝜈, 𝛼, 𝛾, 𝑎₀, 𝑏₀, 𝑘, 𝑎 and 𝑐, from the above relations, are the characteristic coefficients of the material and full characterize the mechanical properties of the body.

As usual in mechanics of continua, we assume that the internal energy density is a positive definite quadratic form and this hypothesis implies (see Eringen [6]) 𝜆3𝜆+2𝜇+𝜈>²⁰𝜆₁𝛼,2𝜇+𝜈>0,𝜈>0,3𝛼+𝛽+𝛾>0,𝛾+𝛽>0,𝛾−𝛽>0,₀>0,𝜆₁>0,𝑐>0,𝑘>0,𝑎≥0. (9) To the system of field equations (4)–(7) we adjoin the following initial conditions: 𝑢_𝑖(𝑥,0)=𝑎⁰_𝑖(𝑥),𝑢̇_𝑖(𝑥,0)=𝑎¹_𝑖(𝑥),𝜑_𝑖(𝑥,0)=𝑏⁰_𝑖(𝑥),𝜑̇_𝑖(𝑥,0)=𝑏¹_𝑖(𝑥),𝜓(𝑥,0)=𝜓⁰(𝑥),𝜓̇(𝑥,0)=𝜓¹(𝑥),𝜂(𝑥,0)=𝜂⁰(𝑥),𝑥=𝑥₁,𝑥₂,𝑥₃∈𝐵. (10) and the following prescribed boundary conditions: 𝑢_𝑖=𝑢_𝑖on𝜕𝐵₁×0,𝑡₀,𝑡_𝑖=𝑡_𝑖̃on𝜕𝐵^𝑐₁×0,𝑡₀,𝜑_𝑖=𝜑_𝑖on𝜕𝐵₂×0,𝑡₀,𝑚_𝑖=𝑚_𝑖on𝜕𝐵^𝑐₂×0,𝑡₀,𝜓=𝜓on𝜕𝐵₃×0,𝑡₀,𝑝=𝑝on𝜕𝐵^𝑐₃×0,𝑡₀,𝜑=𝜑on𝜕𝐵₄×0,𝑡₀,𝑞=𝑞on𝜕𝐵^𝑐₄×0,𝑡₀, (11) where 𝑎⁰_𝑖, 𝑎¹_𝑖, 𝑏⁰_𝑖, 𝑏¹_𝑖, 𝜓⁰, 𝜓¹, 𝜂⁰, 𝑢_𝑖, 𝑡_𝑖̃, 𝜑_𝑖, 𝑚_𝑖, 𝜓, 𝜑, and 𝑞 are prescribed functions in their domains of definition. 𝑡₀ is some instant that may be infinite.

Also, 𝜕𝐵₁, 𝜕𝐵₂, 𝜕𝐵₃, and 𝜕𝐵₄ with respective complements 𝜕𝐵^𝑐₁, 𝜕𝐵^𝑐₂, 𝜕𝐵^𝑐₃, and 𝜕𝐵^𝑐₄ are subsets of the surface 𝜕𝐵 such that 𝜕𝐵₁∩𝜕𝐵^𝑐₁=𝜕𝐵₂∩𝜕𝐵^𝑐₂=𝜕𝐵₃∩𝜕𝐵^𝑐₃=𝜕𝐵₄∩𝜕𝐵^𝑐₄=∅,𝜕𝐵₁∪𝜕𝐵^𝑐₁=𝜕𝐵₂∪𝜕𝐵^𝑐₂=𝜕𝐵₃∪𝜕𝐵^𝑐₃=𝜕𝐵₄∪𝜕𝐵^𝑐₄=𝜕𝐵. (12)

For a function 𝑓 defined on the cylinder 𝐵×[0,𝑡₀), we will use the notations 𝑓∈𝐶^𝑀,𝑁 which means that the derivative, 𝜕^𝑚𝜕𝑥_𝑖𝜕𝑥_𝑗⋯𝜕𝑥_𝑘𝜕^𝑛𝑓𝜕𝑡^𝑛, (13) exists and is continuous on 𝐵×[0,𝑡₀) for 𝑚=0,1,2,…,𝑀, 𝑛=0,1,2,…,𝑁, and 𝑚+𝑛≤max{𝑀,𝑁}.

We say that the ordered array, 𝑆=𝑢_𝑖,𝜑_𝑖,𝜓,𝑞_𝑖,𝜀_𝑖𝑗,𝜂_𝑖𝑗,𝛾_𝑖,𝜂,𝑡_𝑖𝑗,𝑚_𝑖𝑗,𝜆_𝑖,𝜑, (14)

is an admissible state if 𝑢_𝑖∈𝐶^1,2,𝜑_𝑖∈𝐶^1,2,𝜓∈𝐶^1,2,𝜀_𝑖𝑗∈𝐶^0,0,𝜂_𝑖𝑗∈𝐶^0,0,𝛾_𝑖∈𝐶^0,0,𝜂∈𝐶^0,1,𝑡_𝑖𝑗∈𝐶^1,0,𝑚_𝑖𝑗∈𝐶^1,0,𝜆_𝑖∈𝐶^1,0,𝜑∈𝐶^2,0. (15)

It easy to verify that the set of all admissible states is a linear space with regard to the addition of two admissible states and multiplication of a admissible state by a scalar 𝑆₁+𝑆₂=𝑢¹_𝑖+𝑢²_𝑖,𝜑¹_𝑖+𝜑²_𝑖,𝜓¹+𝜓²,𝑞¹_𝑖+𝑞²_𝑖,𝜀¹_𝑖𝑗+𝜀²_𝑖𝑗,𝜂¹_𝑖𝑗+𝜂²_𝑖𝑗,𝛾¹_𝑖+𝛾²_𝑖,𝜂¹+𝜂²,𝑡¹_𝑖𝑗+𝑡²_𝑖𝑗,𝑚¹_𝑖𝑗+𝑚²_𝑖𝑗,𝜆¹_𝑖+𝜆²_𝑖,𝜑¹+𝜑²,𝜆𝑆=𝜆𝑢_𝑖,𝜆𝜑_𝑖,𝜆𝜓,𝜆𝑞_𝑖,𝜆𝜀_𝑖𝑗,𝜆𝜂_𝑖𝑗,𝜆𝛾_𝑖,𝜆𝜂,𝜆𝑡_𝑖𝑗,𝜆𝑚_𝑖𝑗,𝜆𝜆_𝑖,𝜆𝜑. (16)

By a solution of the mixed initial boundary value problem of the theory of thermoelasticity of microstretch bodies with two temperatures in the cylinder Ω₀=𝐵×[0,𝑡₀) we mean an admissible state which satisfies (4)–(7), the initial conditions (10) and the boundary conditions (11) for all (𝑥,𝑡)∈Ω₀.

3. Main Results

Let us denote by 𝜀 the specific internal energy. Using a usual procedure, we can write 𝜀̇=𝑡_𝑖𝑗𝜀̇_𝑖𝑗+𝑚_𝑖𝑗𝜂̇_𝑖𝑗+𝜆_𝑖𝛾̇_𝑖+𝜃𝜂̇. (17)

Theorem 1. For an admissible state one has the following equality: 12𝑑𝑑𝑡_𝐵2𝜀+𝜚𝑢̇_𝑖𝑢̇_𝑖+𝐼_𝑖𝑗𝜑̇_𝑖𝜑̇_𝑗+𝐽𝜓̇²=𝑑𝑉_𝐵𝐹_𝑖𝑢̇_𝑖+𝐺_𝑖𝜑̇_𝑖̇1+𝐽𝜓+𝜑₀−𝑘𝑟𝜃𝑑𝑉𝜑_0𝐵𝜑_,𝑘𝜑_,𝑘+𝑎𝜑_,𝑖𝑖𝜑_,𝑗𝑗+𝑑𝑉_𝜕𝐵𝑡_𝑖𝑢̇_𝑖+𝑚_𝑖𝜑̇_𝑖̇𝑞+𝑝𝜓−𝜑₀𝜑𝑑𝐴. (18)

Proof. Taking into account, first the geometric equations (2) and then (4)–(6), we can write 𝑡_𝑖𝑗𝜀̇_𝑖𝑗+𝑚_𝑖𝑗𝜂̇_𝑖𝑗+𝜆_𝑖𝛾̇_𝑖+𝜃𝜂̇=𝑡_𝑖𝑗𝑢̇_𝑖,𝑗+𝜀_𝑗𝑖𝑘𝜑̇_𝑘+𝑚_𝑖𝑗𝜑̇_𝑗,𝑖+𝜆_𝑖𝜓̇_,𝑖−1𝜑₀𝑞_𝑖,𝑖𝜑−𝑎𝜑_,𝑘𝑘+1𝜑₀=𝑡𝑟𝜃_𝑖𝑗𝑢̇_𝑖,−𝑡_{𝑖𝑗,𝑗}𝑢̇_𝑖+𝜀_𝑗𝑖𝑘𝑡_𝑖𝑗𝜑̇_𝑘+𝑚_𝑖𝑗𝜑̇_𝑗,𝑖−𝑚_{𝑖𝑗,𝑖}𝜑̇_𝑗+𝜆_𝑖𝜓̇_,𝑖−𝜆_𝑖,𝑖̇1𝜓+𝜑₀1𝑟𝜃−𝜑₀𝑞_𝑖𝜑_,𝑖+1𝜑₀𝑞_𝑖𝜑_,𝑖=𝑡_𝑖𝑗𝑢̇_𝑖,𝑗+𝐹_𝑖−𝜚𝑢̈_𝑖+𝐹_𝑖𝑢̇_𝑖−𝜚𝑢̇_𝑖𝑢̈_𝑖+𝜀_𝑗𝑖𝑘𝑡_𝑖𝑗𝜑̇_𝑘+𝑚_𝑖𝑗𝜑̇_𝑗,𝑖+𝜀_𝑖𝑗𝑘𝑡_𝑗𝑘𝜑̇_𝑖+𝐺_𝑖𝜑̇_𝑖−𝐼_𝑖𝑗𝜑̈_𝑗𝜑̇_𝑖+𝜆_𝑖𝜓̇_,𝑖̇𝜓̇𝜓̈+1+𝐿𝜓−𝐽𝜑₀1𝑟𝜃−𝜑₀𝑞_𝑖𝜑_,𝑖+1𝜑₀𝑞_𝑖𝜑_,𝑖+𝑎𝜑₀𝑞_𝑖,𝑖𝜑_,𝑘𝑘=𝑡_𝑖𝑗𝑢̇_𝑖,𝑗+𝑚_𝑖𝑗𝜑̇_𝑗,𝑖+𝜆_𝑖𝜓̇_,𝑖−1𝜑₀𝑞_𝑖𝜑_,𝑖+𝐹_𝑖𝑢̇_𝑖+𝐺_𝑖𝜑̇_𝑖̇1+𝐿𝜓+𝜑₀𝑢̇𝑟𝜃−𝜚_𝑖𝑢̈_𝑖−𝐼_𝑖𝑗𝜑̇_𝑖𝜑̈_𝑗𝜓̇̈𝑘−𝐽𝜓−𝜑₀𝜑_,𝑖𝜑_,𝑖−𝑎𝑘𝜑₀𝜑_,𝑖𝑖𝜑_,𝑘𝑘. (19) Using (17) and (19), we can write 𝜀̇+𝜚𝑢̇_𝑖𝑢̈_𝑖+𝐼_𝑖𝑗𝜑̇_𝑖𝜑̈_𝑗+𝐽𝜓̇𝜓̈=𝐹_𝑖𝑢̇_𝑖+𝐺_𝑖𝜑̇_𝑖̇1+𝐿𝜓+𝜑₀+𝑡𝑟𝜃_𝑖𝑗𝑢̇_𝑖,𝑗+𝑚_𝑖𝑗𝜑̇_𝑗,𝑖+𝜆_𝑖𝜓̇_,𝑖−1𝜑₀𝑞_𝑖𝜑_,𝑖−𝑘𝜑₀𝜑_,𝑖𝜑_,𝑖+𝑎𝜑_,𝑖𝑖𝜑_,𝑘𝑘. (20) Now, we integrate equality (20) over 𝐵 and by using the divergence theorem, we obtain 12𝑑𝑑𝑡_𝐵2𝜀+𝜚𝑢̇_𝑖𝑢̇_𝑖+𝐼_𝑖𝑗𝜑̇_𝑖𝜑̇_𝑗+𝐽𝜓̇²=𝑑𝑉_𝐵𝐹_𝑖𝑢̇_𝑖+𝐺_𝑖𝜑̇_𝑖̇1+𝐿𝜓+𝜑₀−𝑘𝑟𝜃𝑑𝑉𝜑_0𝐵𝜑_,𝑖𝜑_,𝑖+𝑎𝜑_,𝑖𝑖𝜑_,𝑘𝑘+𝑑𝑉_𝜕𝐵𝑡_𝑖𝑗𝑛_𝑗𝑢̇_𝑖+𝑚_𝑖𝑗𝑛_𝑗𝜑̇_𝑖+𝜆_𝑖𝑛_𝑖̇1𝜓−𝜑₀𝑞_𝑖𝑛_𝑖𝜑𝑑𝐴. (21) On the last integral in the right-side of equality (21) we take into account the boundary condition (11) such that we are led to _𝜕𝐵𝑡_𝑖𝑗𝑛_𝑗𝑢̇_𝑖+𝑚_𝑖𝑗𝑛_𝑗𝜑̇_𝑖+𝜆_𝑖𝑛_𝑖̇1𝜓−𝜑₀𝑞_𝑖𝑛_𝑖𝜑=𝑑𝐴_𝜕𝐵𝑡_𝑖𝑢̇_𝑖+𝑚_𝑖𝜑̇_𝑖̇1+𝑝𝜓−𝜑₀𝑞𝜑𝑑𝐴. (22)
Finally, from (21) and (22) we deduce the desired equality (18) and the proof of theorem is complete.

Based on equality (18) we can prove the uniqueness result of the solution in the following theorem.

Theorem 2. The mixed problem of the thermoelasticity of microstretch materials with two temperatures has at most one solution.

Proof. Suppose, by contrast, that our problem admits two solutions: 𝑢_𝑖^(𝛼),𝜑_𝑖^(𝛼),𝜓_𝑖^(𝛼),𝜑^(𝛼), (23) where 𝛼=1,2 and denote by 𝑢_𝑖, 𝜑_𝑖, 𝜓_𝑖, and 𝜑 the difference of two solutions; that is 𝑢_𝑖=𝑢_𝑖⁽¹⁾−𝑢_𝑖⁽²⁾,𝜑_𝑖=𝜑_𝑖⁽¹⁾−𝜑_𝑖⁽²⁾,𝜓=𝜓⁽¹⁾−𝜓⁽²⁾,𝜑=𝜑⁽¹⁾−𝜑⁽²⁾. (24)
Of course, because of the linearity of the problem, the differences (24) satisfy the equations and conditions of the problem, but in their homogeneous form. If we write relation (18) for the differences (24) and take into account the hypothesis (9) we obtain 12𝑑𝑑𝑡_𝐵2𝜀+𝜚𝑢̇_𝑖𝑢̇_𝑖+𝐼_𝑖𝑗𝜑̇_𝑖𝜑̇_𝑗+𝐽𝜓̇²𝑘𝑑𝑉=−𝜑_0𝐵𝜑_,𝑘𝜑_,𝑘+𝑎𝜑_,𝑖𝑖𝜑_,𝑗𝑗𝑑𝑉, (25) where 𝜀 is the specific internal energy corresponding to the differences (24).
Clarly, from (25) taking into account that 𝑘>0, 𝜑₀>0, and 𝑎>0 we deduce 𝑑𝑑𝑡_𝐵2𝜀+𝜚𝑢̇_𝑖𝑢̇_𝑖+𝐼_𝑖𝑗𝜑̇_𝑖𝜑̇_𝑗+𝐽𝜓̇²𝑑𝑉≤0 (26) and this inequality assures that the function, _𝐵2𝜀+𝜚𝑢̇_𝑖𝑢̇_𝑖+𝐼_𝑖𝑗𝜑̇_𝑖𝜑̇_𝑗+𝐽𝜓̇²𝑑𝑉, (27) is decreasing.
According to the homogeneous initial conditions for differences (24), this integral is null at time 𝑡=0. On the other hand, this integral cannot be negative, such that we deduce that the integral vanishes for all time 𝑡>0.
Thus we obtain 𝑢_𝑖=0,𝜑_𝑖=0,𝜓=0,𝜑=0,𝜃=𝜑−𝑎𝜑_,𝑘𝑘=0. (28) It is easy to see that _𝐵𝜑𝜑−𝑎𝜑_,𝑘𝑘=𝑑𝑉_𝐵𝜑²+𝑎𝜑_,𝑖𝜑_,𝑖𝑎𝑑𝑉+𝑘_𝜕𝐵𝑞𝜑𝑑𝐴. (29)
Using (28) and (29) and taking into account the fact that 𝑞 and 𝜑 satisfy the homogeneous boundary conditions, we obtain 𝜑=0 and Theorem 2 is concluded.

This theorem generalizes Iesan's uniqueness result from the classical thermoelasticity with two temperatures.

Now, we give an alternative form of our mixed problem, by using the convolution of two functions.

If 𝛼 and 𝛽 are two functions defined on 𝐵×[0,∞), assumed be continuous on [0,∞), with respect to 𝑡, for each 𝑥∈𝐵, then we define the convolution 𝛼∗𝛽 by 𝛼∗𝛽(𝑥,𝑡)=^𝑡₀𝛼(𝑥,𝑡−𝜏)𝛽(𝑥,𝜏)𝑑𝜏 (30) which has the known properties: =∗,𝛼∗𝛽=𝛽∗𝛼,𝛼∗𝛽∗𝛾=𝛼∗𝛽∗𝛾=𝛼∗𝛽∗𝛾,𝛼∗𝛽+𝛾𝛼+𝛽𝛼+𝛾𝛼∗𝛽=0⟹𝛼=0or𝛽=0. (31) Let us consider the functions ℎ and 𝑔 defined on [0,∞) by ℎ(𝑡)=1,𝑔(𝑡)=𝑡, (32) and the functions 𝑓_𝑖, 𝑔_𝑖, 𝑙, and 𝑊 defined on 𝐵×[0,∞) by 𝑓_𝑖=ℎ∗𝐹_𝑖+𝜚𝑡𝑎⁰_𝑖+𝑎¹_𝑖,𝑔_𝑖=ℎ∗𝐺_𝑖+𝐼_𝑖𝑗𝑡𝑏⁰_𝑗+𝑏¹_𝑗,𝑙=ℎ∗𝐿+𝐽𝑡𝜓⁰+𝜓¹,𝑊=ℎ∗𝑟+𝜑₀𝜂⁰. (33) Then (4)–(6) received the form ℎ∗𝑡_{𝑖𝑗,𝑗}+𝑓_𝑖=𝜚𝑢_𝑖,ℎ∗𝑚_{𝑖𝑗,𝑗}+𝜀_𝑖𝑗𝑘ℎ∗𝑡_𝑗𝑘+𝑔_𝑖=𝐼_𝑖𝑗𝜑_𝑗,ℎ∗𝜆_𝑖,𝑖𝜑+𝑙=𝐽𝜓,₀𝜂=−𝑙∗𝑞_𝑖,𝑖+𝑊. (34)(35)

In this way, we obtain the following result.

Theorem 3. An admissible state 𝑆=𝑢_𝑖,𝜑_𝑖,𝜓,𝑞_𝑖,𝜀_𝑖𝑗,𝜂_𝑖𝑗,𝛾_𝑖,𝜂,𝑡_𝑖𝑗,𝑚_𝑖𝑗,𝜆_𝑖,𝜑 (36) is a solution of the mixed problem of thermoelasticity of microstretch bodies with two temperatures if and only if it satisfies (34), (35), (7), and (8) and the boundary conditions (11).

We must outline that in this form of the mixed problem, the initial conditions are included in the field of equations.

In the following, we propose to find a result of Betti's type regarding our mixed problem.

Assume that the thermoelastic body with two temperatures is subjected to two systems of thermoelastic loadings: 𝐿^(𝛼)=𝐹_𝑖^(𝛼),𝐺_𝑖^(𝛼),𝐿^(𝛼),𝑟^(𝛼),𝑢_𝑖^(𝛼),𝜑_𝑖^(𝛼),𝜓^(𝛼),𝜑^(𝛼),𝑡̃_𝑖^(𝛼),𝑚_𝑖^(𝛼)𝑝^(𝛼),𝑞^(𝛼),𝑎_𝑖^0(𝛼),𝑎_𝑖^1(𝛼),𝑏_𝑖^0(𝛼),𝑏_𝑖^1(𝛼),𝜓^0(𝛼),𝜓^1(𝛼),𝜂₀^(𝛼), (37) where 𝛼=1,2.

These loadings correspond to the thermoelastic configurations: 𝐶^(𝛼)=𝑢_𝑖^(𝛼),𝜑_𝑖^(𝛼),𝜓^(𝛼),𝜑^(𝛼). (38)

Theorem 4. Suppose that the thermoelastic body with two temperatures is subjected to two systems of loadings 𝐿^(𝛼), 𝛼=1,2. Then between the corresponding thermoelastic configurations 𝐶^(𝛼), 𝛼=1,2, there is the following reciprocity relation: _𝐵𝑓_𝑖⁽¹⁾∗𝑢_𝑖⁽²⁾+𝑔_𝑖⁽¹⁾∗𝜑_𝑖⁽²⁾+𝑙⁽¹⁾∗𝜓⁽²⁾−1𝜑₀𝑔∗𝑊⁽¹⁾∗𝜃⁽²⁾+𝑑𝑉_𝜕𝐵𝑔∗𝑡_𝑖⁽¹⁾∗𝑢_𝑖⁽²⁾+𝑚_𝑖⁽¹⁾∗𝜑_𝑖⁽²⁾+𝑝⁽¹⁾∗𝜓⁽²⁾+1𝜑₀ℎ∗𝑞⁽¹⁾∗𝜑⁽²⁾=𝑑𝐴_𝐵𝑓_𝑖⁽²⁾∗𝑢_𝑖⁽¹⁾+𝑔_𝑖⁽²⁾∗𝜑_𝑖⁽¹⁾+𝑙⁽²⁾∗𝜓⁽¹⁾−1𝜑₀𝑔∗𝑊⁽²⁾∗𝜃⁽¹⁾+𝑑𝑉_𝜕𝐵𝑔∗𝑡_𝑖⁽²⁾∗𝑢_𝑖⁽¹⁾+𝑚_𝑖⁽²⁾∗𝜑_𝑖⁽¹⁾+𝑝⁽²⁾∗𝜓⁽¹⁾+1𝜑₀ℎ∗𝑞⁽²⁾∗𝜑⁽¹⁾𝑑𝐴, (39) where 𝑓_𝑖^(𝛼), 𝑔_𝑖^(𝛼), 𝑙^(𝛼), and 𝑊^(𝛼) are given by relations of the form (33).

Proof. Taking into account the properties of the convolution, from the constitutive equations (7) and relation (8), we get 𝑡⁽¹⁾_𝑖𝑗+𝛽𝜃⁽¹⁾𝛿_𝑖𝑗∗𝑒⁽²⁾_𝑖𝑗=𝑡⁽²⁾_𝑖𝑗+𝛽𝜃⁽²⁾𝛿_𝑖𝑗∗𝑒⁽¹⁾_𝑖𝑗,𝑚⁽¹⁾_𝑖𝑗∗𝜂⁽²⁾_𝑖𝑗=𝑚⁽²⁾_𝑖𝑗∗𝜂⁽¹⁾_𝑖𝑗,𝜆_𝑖⁽¹⁾∗𝛾_𝑖⁽²⁾=𝜆_𝑖⁽²⁾∗𝛾_𝑖⁽¹⁾,𝛽𝜀⁽¹⁾_𝑘𝑘−𝜂⁽¹⁾∗𝜃⁽²⁾=𝛽𝜀⁽²⁾_𝑘𝑘−𝜂⁽²⁾∗𝜃⁽¹⁾. (40) Adding these relations we deduce that 𝑡⁽¹⁾_𝑖𝑗∗𝑒⁽²⁾_𝑖𝑗+𝑚⁽¹⁾_𝑖𝑗∗𝜂⁽²⁾_𝑖𝑗+𝜆_𝑖⁽¹⁾∗𝛾_𝑖⁽²⁾−𝜂⁽¹⁾∗𝜃⁽²⁾=𝑡⁽²⁾_𝑖𝑗∗𝑒⁽¹⁾_𝑖𝑗+𝑚⁽²⁾_𝑖𝑗∗𝜂⁽¹⁾_𝑖𝑗+𝜆_𝑖⁽²⁾∗𝛾_𝑖⁽¹⁾−𝜂⁽²⁾∗𝜃⁽¹⁾. (41) Now, we introduce the following notation: 𝐿_𝛼𝛽=_𝐵𝑔∗𝑡^(𝛼)_𝑖𝑗∗𝑒^(𝛽)_𝑖𝑗+𝑚^(𝛼)_𝑖𝑗∗𝜂^(𝛽)_𝑖𝑗+𝜆_𝑖^(𝛼)∗𝛾_𝑖^(𝛽)−𝜂^(𝛼)∗𝜃^(𝛽)𝑑𝑉. (42) Thus, from (41) it is easy to deduce that 𝐿₂₁=𝐿₁₂. (43)
Taking into account the constitutive equations (7), relation (8), the geometric equations (2), and (34) and (35), we obtain 𝑔∗𝑡^(𝛼)_𝑖𝑗∗𝑒^(𝛽)_𝑖𝑗+𝑚^(𝛼)_𝑖𝑗∗𝜂^(𝛽)_𝑖𝑗+𝜆_𝑖^(𝛼)∗𝛾_𝑖^(𝛽)−𝜂^(𝛼)∗𝜃^(𝛽)=𝑔∗𝑡^(𝛼)_𝑖𝑗∗𝑢^(𝛽)_𝑖,𝑗+𝜀_𝑗𝑖𝑘𝑔∗𝑡^(𝛼)_𝑖𝑗∗𝜑_𝑘^(𝛽)+𝑔∗𝑚^(𝛼)_𝑖𝑗∗𝜑^(𝛽)_𝑗,𝑖+𝑔∗𝜆_𝑖^(𝛼)∗𝜓^(𝛽)_,𝑖+1𝜑₀𝑔∗ℎ∗𝑞^(𝛼)_𝑖,𝑖∗𝜑^(𝛽)−𝑎𝜑^(𝛽)_,𝑘𝑘−1𝜑₀𝑔∗𝑊^(𝛼)∗𝜃^(𝛽)=𝑔∗𝑡^(𝛼)_𝑖𝑗∗𝑢_𝑖^(𝛽)_,𝑗+𝑔∗𝑚^(𝛼)_𝑖𝑗∗𝜑_𝑖^(𝛽)_,𝑗+𝑔∗𝜆_𝑖^(𝛼)∗𝜓^(𝛽)_,𝑖−𝑔∗𝑡^(𝛼)_{𝑖𝑗,𝑗}∗𝑢_𝑖^(𝛽)−𝑔∗𝑚^(𝛼)_{𝑖𝑗,𝑗}∗𝜑_𝑖^(𝛽)−𝑔∗𝜆^(𝛼)_𝑖,𝑖∗𝜓^(𝛽)+1𝜑₀𝑔∗ℎ∗𝑞_𝑖^(𝛼)∗𝜑^(𝛽)_,𝑖−1𝜑₀𝑔∗ℎ∗𝑞_𝑖^(𝛼)∗𝜑^(𝛽)_,𝑖−𝑎𝜑₀𝑔∗ℎ∗𝑞^(𝛼)_𝑖,𝑖∗𝜑^(𝛽)_,𝑘𝑘−1𝜑₀𝑔∗𝑊^(𝛼)∗𝜃^(𝛽)=𝑔∗𝑡^(𝛼)_𝑖𝑗∗𝑢_𝑖^(𝛽)_,𝑗+𝑔∗𝑚^(𝛼)_𝑖𝑗∗𝜑_𝑖^(𝛽)_,𝑗+𝑔∗𝜆_𝑖^(𝛼)∗𝜓^(𝛽)_,𝑖+𝑓_𝑖^(𝛼)∗𝑢_𝑖^(𝛽)+𝑔_𝑖^(𝛼)∗𝜑_𝑖^(𝛽)+𝑙^(𝛼)∗𝜓^(𝛽)−1𝜑₀𝑔∗𝑊^(𝛼)∗𝜃^(𝛽)−𝜚𝑢_𝑖^(𝛼)∗𝑢_𝑖^(𝛽)−𝐼_𝑖𝑗𝜑_𝑖^(𝛼)∗𝜑_𝑗^(𝛽)−𝐽𝜓^(𝛼)∗𝜓^(𝛽)+𝑘𝜑₀𝑔∗ℎ∗𝜑^(𝛼)_,𝑖∗𝜑^(𝛽)_,𝑖+𝑘𝑎𝜑₀𝑔∗ℎ∗𝜑^(𝛼)_,𝑖𝑖∗𝜑^(𝛽)_,𝑘𝑘. (44) If we introduce this result in (42) we are led to 𝐿_𝛼𝛽=_𝐵𝑓_𝑖^(𝛼)∗𝑢_𝑖^(𝛽)+𝑔_𝑖^(𝛼)∗𝜑_𝑖^(𝛽)+𝑙^(𝛼)∗𝜓^(𝛽)−1𝜑₀𝑔∗𝑊^(𝛼)∗𝜃^(𝛽)+𝑑𝑉_𝜕𝐵𝑔∗𝑡_𝑖^(𝛼)∗𝑢_𝑖^(𝛽)+𝑚_𝑖^(𝛼)∗𝜑_𝑖^(𝛽)+𝑝^(𝛼)∗𝜓^(𝛽)+1𝜑₀ℎ∗𝑞^(𝛼)∗𝜑^(𝛽)−𝑑𝐴_𝐵𝜚𝑢_𝑖^(𝛼)∗𝑢_𝑖^(𝛽)+𝐼_𝑖𝑗𝜑_𝑖^(𝛼)∗𝜑_𝑗^(𝛽)−𝐽𝜓^(𝛼)∗𝜓^(𝛽)−𝑘𝜑₀𝑔∗ℎ∗𝜑^(𝛼)_,𝑖∗𝜑^(𝛽)_,𝑖−𝑘𝑎𝜑₀𝑔∗ℎ∗𝜑^(𝛼)_,𝑖𝑖∗𝜑^(𝛽)_,𝑘𝑘𝑑𝑉. (45)
Using form (45) of the functional 𝐿_𝛼𝛽, taking into account the symmetry relation (43), we obtain the desired result (39) that concludes the proof of Theorem 4.

Let us make some considerations on the uncoupled problem of the thermoelasticity with two temperatures. So, instead of relation (7)5 we assume that 𝑐𝜂=𝜑₀𝜃. (46) In this way, we can obtain the relation between the thermoelastic configurations 𝐶⁽¹⁾ and 𝐶⁽²⁾ corresponding to the coupled problem, say 𝑃⁽¹⁾, and the uncoupled problem 𝑃⁽²⁾, respectively.

Suppose that the above systems of thermoelastic loadings 𝐿^(𝛼) correspond to the problems 𝑃^(𝛼), 𝛼=1,2.

Using the same procedure as in the proof of Theorem 4, we obtain the following result.

Theorem 5. Assume that the microstretch thermoelastic body with two temperatures is subjected to two thermoelastic bodies 𝐿^(𝛼), 𝛼=1,2; then between the thermoelastic configurations 𝐶^(𝛼), 𝛼=1,2, which correspond to the problems 𝑃^(𝛼), 𝛼=1,2, there is the following reciprocity relation: _𝐵𝑓_𝑖⁽¹⁾∗𝑢_𝑖⁽²⁾+𝑔_𝑖⁽¹⁾∗𝜑_𝑖⁽²⁾+𝑙⁽¹⁾∗𝜓⁽²⁾−1𝜑₀𝑔∗𝑊⁽¹⁾∗𝜃⁽²⁾+𝑑𝑉_𝜕𝐵𝑔∗𝑡_𝑖⁽¹⁾∗𝑢_𝑖⁽²⁾+𝑚_𝑖⁽¹⁾∗𝜑_𝑖⁽²⁾+𝑝⁽¹⁾∗𝜓⁽²⁾+1𝜑₀𝑔∗𝑞⁽¹⁾∗𝜑⁽²⁾=𝑑𝐴_𝐵𝑓_𝑖⁽²⁾∗𝑢_𝑖⁽¹⁾+𝑔_𝑖⁽²⁾∗𝜑_𝑖⁽¹⁾+𝑙⁽²⁾∗𝜓⁽¹⁾−1𝜑₀𝑔∗𝑊⁽²⁾∗𝜃⁽¹⁾+𝑑𝑉_𝜕𝐵𝑔∗𝑡_𝑖⁽²⁾∗𝑢_𝑖⁽¹⁾+𝑚_𝑖⁽²⁾∗𝜑_𝑖⁽¹⁾+𝑝⁽²⁾∗𝜓⁽¹⁾+1𝜑₀𝑔∗𝑞⁽²⁾∗𝜑⁽¹⁾−𝑑𝐴_𝐵𝑔∗𝜀⁽²⁾_𝑘𝑘∗𝜃⁽¹⁾𝑑𝑉. (47)

As an immediately consequence of Theorem 5, we indicate the following particular application. In fact, using the reciprocity relation (47) the problem of coupled thermoelasticity will be reduced to an associated problem of uncoupled thermoelasticity and to an integral equation.

Indeed, we take 𝑟⁽²⁾=𝛿(𝑥−𝜉)𝛿(𝑡),𝐹_𝑖^(𝛼)=𝐺_𝑖^(𝛼)=𝐿^(𝛼)𝑢=0,_𝑖(𝛼)=𝜑_𝑖^(𝛼)=𝜓^(𝛼)=𝜑^(𝛼)𝑡̃=0,𝛼=1,2,_𝑖(𝛼)=𝑚_𝑖^(𝛼)=𝑝^(𝛼)=𝑞^(𝛼)𝑎=0,_𝑖0(𝛼)=𝑎_𝑖^1(𝛼)=𝑏_𝑖^0(𝛼)=𝑏_𝑖^1(𝛼)=𝜓^0(𝛼)=𝜓^1(𝛼)=𝜂₀^(𝛼)=0, (48) where 𝛿 is the Dirac's distribution.

Using the properties of convolution and taking into account definition (33) of the functions 𝑓_𝑖, 𝑔_𝑖, 𝑙 and 𝑊, the reciprocity relation (47) permits to reduce the problem 𝑃⁽¹⁾ of coupled thermoelasticity to solving the problem 𝑃⁽²⁾ of uncoupled thermoelasticity and to find the solution of the following integral equation: 𝜃⁽¹⁾(𝜉,𝑡)=_𝐵𝑟⁽¹⁾∗𝜃⁽²⁾𝑑𝑉−𝜑₀𝛽_𝐵𝑒̇_𝑘𝑘𝑟⁽²⁾∗𝜃⁽¹⁾𝑑𝑉. (49)

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The authors express gratitude to the referees for their criticism of the paper and for helpful suggestions. This paper was funded by the Deanship of Scientific Research (DSR), King Abdulaziz University, Jeddah. The authors, therefore, acknowledge with thanks the DSR technical and financial support.