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; earthquake1 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 thermomicrostretch 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 breathingtype 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 BoussinesqSomiglianaGalerkin 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 [811] we tackle some questions with regards to the microstretchthermoelastic 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 microstretchthermoelastic 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 microstretchthermoelastic 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 nonhomogeneous 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 seismicsources 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
Let us denote by
As usual, we denote by
In the dynamic theory of the thermoelasticity of microstretch bodies the fundamental system of field equations consists of:
– the equations of motion
– the balance of the equilibrated forces
– the energy equation
For an anisotropic and homogeneous microstretch thermoelastic material, the constitutive equations have the form
where
In the above equations we have used the following notations:
–
–
– L the generalized external body load;
– ϱ is the reference constant mass density;
– J and
–
– η is the entropy per unit mass;
– S is the heat supply per unit mass;
–
The components of the strain tensors
where
The second law of thermodynamics implies that
that is, the conductivity tensor k is positive semidefinite only.
In what follows we need the following regularity assumptions:
(1) the constitutive coefficients are continuously differentiable functions on
(2) ϱ and d are continuous functions on
(3)
We denote by
at regular points of the surface ∂B.
Here,
Along with the system of field equations (1)(4) we consider the following initial conditions:
and the following prescribed boundary conditions:
where
We assume that:
(1)
(2)
(3)
By a solution of the mixed initial boundary value problem of the theory thermoelasticity
of microstretch bodies in the cylinder
3 Main results
Let us consider the functions K and U defined on the interval
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
where we have denoted by
for all
Proof Using the constitutive equations (4) and the symmetry relations (5), we obtain
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
Similarly,
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
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:
for all
The functionPis defined by
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
From this equality it is easy to deduce that
By integrating this relation over the interval
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,
(iii) dis strictly positive or strictly negative;
(iv) the conductivity tensor
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,
Let us denote by
the difference of two solutions, where
Because of linearity, this difference is also a solution of our problem, but it corresponds to null data.
Thus, from (17) we obtain
By using the hypotheses (ii) and (iv) of the theorem, (20) implies that
and
But
Taking into account (22) and (23), (16) reduces to
Since
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
As is well known, the convolution product of the function u and v is defined by the integral
Also, let us consider the functions
For a continuous function h defined on
Using these considerations, we can write the energy equation (3) and the initial condition
in the equivalent form
where
Consider two external data systems
and denote by
Also, we use the following notations:
Lemma 2Assume that the symmetry relations (5) are satisfied. Consider the functions
Then we have
for all
Proof We introduce the notation
From (32) and the constitutive equations (4) we deduce
Using the symmetry relations (5), from (33) we obtain
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
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
where we have used the notations
Proof We use the substitution
It is easy to prove that
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
for all
Then we have
Proof Using the result on
Let us apply (42) to the process
From (30) and (40), we obtain the equality
Similarly,
It is easy to prove the relations
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
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,
Let us denote by
the difference of two solutions, where
If we apply (41) for the difference, we are lead to
From this equality, by using the hypotheses (ii) and (iv) of Theorem 2, we obtain
Then, from (16), (17), and hypothesis (iii) of Theorem 2, we obtain
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 semidefined as results from the ClausiusDuhem 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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