Our paper is concerned with some basic theorems for nonsimple thermoelastic materials. By using the Lagrange identity, we 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. Moreover, we avoid the use of positive definiteness assumptions on the thermoelastic coefficients.
Keywords:Lagrange identity; nonsimple materials; uniqueness; continuous dependence
Even classical elasticity does not consider the inner structure, the material response of materials to stimuli depends in a relevant way on its internal structure. Thus, it has been needed to develop some new mathematical models for continuum materials where this kind of effects was taken into account. Some of them are nonsimple elastic solids. It is known that from a mathematical point of view, these materials are characterized by the inclusion of higher-order gradients of displacement in the basic postulates.
The theory of nonsimple elastic materials was first proposed by Toupin in his famous article . Also, among the first studies devoted to this material, we must mention those belonging to Green and Rivlin  and Mindlin .
The interest to introduce high-order derivatives consists in the fact that the possible configurations of the materials are clarified more and more finely by the values of the successive higher gradients.
As it is known, the constitutive equations of nonsimple elastic solids are known to contain first- and second-order gradients, both contributing to dissipation. It is then interesting to understand the relevance of the two different dissipation mechanisms which can appear in the theory. In fact, the simultaneous presence of both mechanisms can be analyzed as well, with inessential changes in the proofs. In that situation, the behavior turns out to be the same as if only the higher-order dissipation appears in the equations.
In the last decade many studies have been devoted to nonsimple materials. We remember only three of them, differing in issues addressed, though in essence they are dedicated to nonsimple materials. So, in the paper of Pata and Quintanilla , the theory is linearized, and a uniqueness result is presented.
Also, the study  of Martinez and Quintanilla is devoted to study the incremental problem in the thermoelastic theory of nonsimple elastic materials.
A linearized theory of thermoelasticity for nonsimple materials is derived within the framework of extended thermodynamics in the paper of Ciarletta . The theory is linearized, and a uniqueness result is presented. A Galerkin-type solution of the field equations and fundamental solutions for steady vibrations are also studied.
Previous papers on the uniqueness and continuous dependence in elasticity or thermoelasticity were based almost exclusively on the assumptions that the elasticity tensor or thermoelastic coefficients are positive definite (see, for instance, the paper ).
In other papers, authors recourse the energy conservation law in order to derive the uniqueness or continuous dependence of solutions. For instance, a uniqueness result was indicated in paper  of Green and Lindsay by supplementing the restrictions arising from thermodynamics with certain definiteness assumptions.
We want to outline that there are many papers which employ the various refinements of the Lagrange identity, of which we remember only a few, namely papers [9,10] and . Also, a lot of papers are dedicated to Cesaro means, as [12-14] and  for instance.
The objective of our study is to examine by a new approach the mixed initial-boundary value problem in the context of thermoelasticity of nonsimple materials. The approach is developed on the basis of Lagrange identity and its consequences. Therefore, we establish the uniqueness and continuous dependence of solutions with respect to body forces, body couple, generalized external body load and heat supply. We also deduce the continuous dependence of solutions of our problem with respect to initial data and, finally, with respect to thermoelastic coefficients. The results are obtained for bounded regions of the Euclidian three-dimensional space. We point out, again, that the results are obtained without recourse to the energy conservation law or to any boundedness assumptions on the thermoelastic coefficients. Also, we avoid the use of definiteness assumptions on the thermoelastic coefficients.
2 Basic equations
We assume that a bounded region B of the three-dimensional Euclidian space is occupied by a nonsimple elastic body, referred to the reference configuration and a fixed system of rectangular Cartesian axes. Let denote the closure of B and call ∂B the boundary of the domain B. We consider ∂B to be a piecewise smooth surface and designate by the components of the outward unit normal to the surface ∂B. Letters in boldface stand for vector fields. We use the notation to designate the components of the vector v in the underlying rectangular Cartesian coordinates frame. Superposed dots stand for the material time derivative. We employ the usual summation and differentiation conventions: the subscripts are understood to range over integer . 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 the 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 , . Let us denote by the components of the displacement vector and by θ the temperature measured from the constant absolute temperature of the body in its reference state.
As usual, we denote by the components of the stress tensor and by the components of the hyperstress tensor over B.
We here will use the theory and the notation in the way developed by Iesan in his book, which tackles also nonsimple materials .
The equations of motion from thermoelasticity of nonsimple materials are as follows (see also ):
The equation of energy is given by
For an anisotropic and homogeneous nonsimple thermoelastic material, the constitutive equations have the form
The kinematic characteristics of the body (components of the strain tensors) and are defined by means of the geometric equations
In the above equations, we have used the following notations:
– the components of body force;
– ϱ is the reference constant mass density;
– and are the components of the stress;
– S is the entropy per unit mass;
– r is the heat supply per unit mass;
– are the components of heat flux vector.
Also, the coefficients , , , , , , a and are the characteristic constitutive constants of the material and they satisfy the following symmetry relations:
Also, the second law of thermodynamics implies that
We denote by the components of surface traction and by q the heat flux. These quantities are defined by
at regular points of the surface ∂B. Here, 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:
and the following prescribed boundary conditions:
where is some instant that may be infinite.
Also, and with respective complements and are subsets of the surface ∂B such that
Assume that , , σ, , , and are prescribed smooth functions in their domains.
To avoid repeating the regularity assumptions, we assume from the beginning that:
(i) all constitutive coefficients are continuously differentiable functions on ;
(ii) ϱ is continuous on ;
(iii) and r are continuous functions on ;
(iv) , and σ are continuous functions on ;
(v) and are continuous functions on and , respectively;
(vi) and are piecewise regular functions on and , respectively, and continuous in time.
Taking into account the constitutive equations (3), from (1) and (2) we obtain the following system of equations:
By a solution of the mixed initial boundary value problem of the theory of thermoelasticity of nonsimple bodies in the cylinder , we mean an ordered array which satisfies the system of equations (8) for all , the boundary conditions (7) and the initial conditions (6).
3 Main result
Let us consider and , two functions assumed to be twice continuously differentiable with respect to the time variable t. By direct calculations, it is easy to deduce that
For the sake of simplicity, the spatial argument and the time argument of the functions and are omitted because there is no likelihood of confusion.
In the above equality, we substitute the functions and by the functions and , respectively, which also are assumed to be twice continuously differentiable with respect to the time variable, and then we obtain the following well known Lagrange identity:
Let us denote by ( ) two solutions of the mixed initial boundary value problem defined by (8), (6) and (7) which correspond to the same boundary data and same initial data, but to different body forces and heat supplies,
We introduce the following notations:
The constitutive equations become
So, we deduce that the differences satisfy the following equations and conditions:
– the equations of motion
– the equations of energy
– the initial conditions
– the boundary conditions
We are now in a position to prove the first basic result.
Theorem 1For the differences of two solutions of the mixed initial boundary value problem (8), (6) and (7), the Lagrange identity becomes
Proof Because of the linearity of the problem defined by (8), (6) and (7), we deduce that the differences represent the solution of a mixed initial boundary value problem analogous to (8), (6) and (7), namely the problem consisting of equations (12) and (13) with loads , respectively , the initial conditions (14) and boundary conditions (15). By setting
then the identity (9), after some straightforward calculation, becomes
where we have used the fact that the initial and boundary data are null.
We shall eliminate the inertial terms on the right-hand side of the relation (17) by means of the equations of motion for the differences .
So, in view of equation (12), we have
After we use the symmetry relations (5), this equality takes on the form
We integrate by parts equality (18), and after using boundary conditions (15), we get the equality
Now we integrate equation (13) on the interval and take into account the zero initial data in (14) so that we obtain the relation
After we multiply in equality (20) by and use a similar result obtained for , multiplied by , we find
If we introduce (21) into (19), we obtain
Based on the symmetry of the tensor , we get
On the other hand, integrating by parts, we obtain
From (23) and (24) we deduce
We now substitute (25) in (22), and so we are led to equality (16). With this, the proof of Theorem 1 is completed. □
Remark It is important to note that the identity (16) is just like in the classical thermoelasticity (see ).
The identity (16) constitutes the basis on which we shall prove the uniqueness and the continuous dependence results.
We proceed first to obtain the uniqueness of the solution of the mixed initial boundary value problem defined by (8), (6) and (7).
Theorem 2Assume that the conductivity tensor is positive definite in the sense there exists a positive constant such that
Also, we suppose that the symmetry relations (5) are satisfied. If is not empty or onB, then the mixed initial boundary value problem in thermoelastodynamics of nonsimple materials has at most one solution.
Proof Suppose, by contrary, that our mixed problem defined by (8), (6) and (7) has two solutions ( ) that correspond to the same initial and boundary data, to the same body force and the same heat supply.
If we denote by
then we shall prove that
It is clear that the differences from (26) also represent a solution of our problem but with null body force and null heat supply. If we write the identity (16) for this particular case, we have
Now, we integrate this equality on the interval , and obtain
Taking into account the properties of ϱ and , the above identity proves that
If is not empty, considering the boundary conditions (7), then from (28) we deduce that (27) holds. If , from the equation of energy (written for the differences), we get . However, χ vanishes initially, such that (27) again holds true.
If is infinite, then the proof of Theorem 2 is complete. If is finite, then we set
and repeat the above procedure on the interval such that we extend the conclusion (27) on , and so on.
Finally, we obtain (27) on and this concludes the proof of Theorem 2. □
We are ready to state and prove the continuous dependence theorem with regard to body force and heat supply on the compact subintervals of the interval for the solution of the mixed initial boundary value problem defined by the system of equations (8), the initial conditions (6) and the boundary conditions (7).
Theorem 3Suppose the same conditions as in Theorem 2. Let ( ) be two solutions of our mixed problem which correspond to the same initial and boundary data but to different body force and different heat supply, ( ), where
Moreover, we suppose that there exists such that
Then we have the following estimate:
where and are the differences defined in (26) and .
Proof We will use the identity (16). On the right-hand side of this identity, we employ the Schwarz inequality for each integral.
For instance, we have
where, at last, we use the substitution .
We proceed analogously with other integrals in the identity (16). Finally, we integrate the resulting inequality over , and we obtain the inequality (30) and the proof of Theorem 3 is complete. □
In the following theorem, we use the estimate (30) in order to deduce a continuous dependence result upon initial data.
Theorem 4Assume that the symmetry relations (5) are satisfied. Consider
two solutions of the mixed initial boundary value problem defined by (8), (6) and (7) which correspond to the same body force and heat supply and to the same boundary data, but to different initial data
Here the perturbations obey the following restrictions:
where we used the notation
Using perturbation andχ, we define the functions and by
If the functions satisfy the conditions (29), then we have the following estimate:
Proof Integrating by parts in (31), we deduce
It is easy to prove that the difference functions satisfy the equations of motion and the equation of energy as in (8), but with null loads
Also, by direct calculations, we deduce that the difference functions satisfy the initial conditions in the form
Then, a straightforward calculation proves that the functions defined in (31) satisfy the equations of motion and the equation of energy as in (8), but with the following body force and heat supply:
By using these specifications, the estimate (32) follows from (30) and Theorem 4 is concluded. □
Finally, we obtain a continuous dependence result of the solution to problems (8), (6) and (7) upon the thermoelastic coefficients, again as a consequence of Theorem 3.
Theorem 5Assume that the symmetry relations (5) are satisfied and consider
two solutions of the mixed initial boundary value problem defined by (8), (6) and (7) which correspond to the same body force and heat supply and to the same boundary and initial data, but to different thermoelastic coefficients
Suppose that the perturbations satisfy the conditions (29). Then any solution of the initial boundary value problem defined by (8), (6) and (7) that satisfies the condition
depends continuously on the thermoelastic coefficients on the interval in
Proof A straightforward calculation proves that the perturbations of two solutions verify the equations of motion and the equation of energy with the following body force load and heat supply:
Thus the problem is analogous to the problem from Theorem 4. Therefore, according to the estimates (32) and (30), we obtain the desired result. □
4 Concluding remarks
The uniqueness theorem and the continuous dependence theorems were proved without recourse to any conservation laws or to any boundedness assumptions on the thermoelastic coefficients. In various papers, the existence of the solution to the mixed initial boundary value problem defined by (8), (6) and (7) is obtained by assuming some strong restrictions. For instance, in the paper  the existence of the solution is obtained under assumption that the internal energy density is positive definite.
The authors declare that they have no competing interests.
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. SRM performed all calculations and demonstrations and took into account the suggestions given by MIM.
We express our gratitude to the referees for their valuable criticisms of the manuscript and for helpful suggestions.
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