Mathematical Problems in Engineering

Volume 2010, Article ID 415304, 16 pages

http://dx.doi.org/10.1155/2010/415304

## On the Nonlinear Theory of Micromorphic Thermoelastic Solids

Faculty of Mathematics, University of Iaşi, 700506 Iaşi, Romania

Received 27 September 2010; Accepted 18 November 2010

Academic Editor: J. Rodellar

Copyright © 2010 C. Galeş. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

This work aims to study some dynamical problems in the framework of nonlinear theory of micromorphic thermoelastic solids. First, the continuous dependence of smooth admissible thermodynamic processes upon initial state and supply terms is investigated in the region of state space where the internal energy is a convex function and the elastic material behaves as a definite conductor of heat. Then, a uniqueness result is demonstrated.

#### 1. Introduction

Motivated by experimental studies, various continuous models of deformable bodies have been proposed in literature in order to describe the thermomechanical behavior of media with microstructure. In the micromorphic theory introduced by Eringen and Şuhubi [1] and Eringen [2], a material body is envisioned as a collection of a large number of deformable particles (subcontinua or microcontinua). Each particle possesses finite size and directions representing its microstructure. The microdeformation gives rise to extra degrees of freedom. Thus, the particle has nine independent degrees of freedom describing both rotations and stretches, in addition to the three classical translational degrees of freedom of its center.

Many deformable solids point to the necessity for the incorporation of micromotions into mechanics. Porous solids with deformable grains and pores, composites, polymers with deformable molecules, crystals, solids with microcracks, dislocation and disclinations, and biological tissues (bones and muscles) are just a few examples of deformable solids which require the degrees of freedom given by the micromorphic theory. As a consequence, the micromorphic mechanics is the subject of detailed studies both from theoretical and practical reasons. In the linear context, uniqueness theorems have been proved by Soós [3] and Ieşan [4], variational principles have been established by Maugin [5] and Nappa [6], applications to earthquake problems have been suggested by Teisseyre [7], Dresen et al. [8], and Teisseyre et al. [9], plane harmonic waves have been studied by Eringen [2], reciprocal and existence theorems have been demonstrated by Ieşan [4], and material constants for isotropic materials have been determined by Chen and Lee [10].

On the other hand, the theory has been generalized to mixtures of micromorphic materials by Twiss and Eringen [11, 12], to higher-grade materials by Eringen [13], and to electromagnetic micromorphic thermoelastic solids by Eringen [14]. Moreover, the constitutive theory of micromorphic thermoplasticity has been formulated by Lee and Chen [15], the concept of material forces was extended to micromorphic theory by Lee and Chen [16], the problem of heat flow in a micromorphic continua with microtemperatures has been investigated by Ieşan and Nappa [17].

This paper deals with the nonlinear micromorphic thermoelasticity. The main purpose is to investigate the stability of smooth thermodynamic processes. In this sense, we use the method developed for nonlinear thermoelastic solids which are nonconductor of heat by Dafermos [18] and updated later by Chiriţă [19] to heat-conducting thermoelastic materials. In the general context of heat-conducting nonlinear micromorphic solids, we prove the continuous dependence of smooth admissible thermodynamic processes upon initial state and supply terms. We present also a uniqueness theorem. Both results are obtained in the region of state space where the internal energy is a convex function and the elastic material behaves as a definite conductor of heat.

We recall that the Dafermos method has been utilized recently to prove continuous dependence results for nonconductor-of-heat mixtures [20–22] and for materials with voids [23].

The paper is organized as follows. In the next section, we recall the basic equations of the nonlinear theory of micromorphic thermoelasticity. Then, in Section 3, we use the consequences of the second law of thermodynamics to prove a uniqueness theorem and the continuous dependence of smooth thermodynamic processes upon initial state and body loads.

#### 2. Basic Formulation

We consider a micromorphic continuum, and we assume that at time the body occupies the region of the Euclidean three-dimensional space and is bounded by a piecewise smooth surface . Following [2], a point in the body is characterized by its rectangular coordinates , , , in a fixed system of rectangular Cartesian axes , , and a deformable vector . One writes . Deformation carries to in the spatial configuration at time , where , , and are the rectangular coordinates with respect to a new coordinate Cartesian system , , and is the vector attached to . The motion can be expressed as where, and henceforth, the summation convention on repeated indices is understood and , .

The inverse motion can be written as where

In what follows denotes the material time rate of . Moreover, we use the notations and .

The balance laws of micromorphic continua can be expressed as follows [1, 2]:(i)the conservation of mass: where and are the mass densities at time and at present time, respectively, , and are the components of velocity vector;(ii)the conservation of micromorphic inertia: where is the microgyration tensor defined by and are the components of the inertia stress tensor at time and are the components of the microinertia tensor at time . Clearly, and are symmetric and positive definite;(iii)the balance of momentum:(iv)the balance of moment of momentum:(v)the balance of energy: where , , , , , are the stress tensor, moment stress tensor, microstress average, spin inertia, internal energy, and heat input, respectively, and , , are the body force, external body moment, and heat source, respectively. The tensor is symmetric, that is, . The spin inertia may be expressed as

The second law of thermodynamics is written as where is the entropy density and is the absolute temperature.

The above formulation is described in detail in the book by Eringen [2]. Since we deal with micromorphic solids, we reformulate the basic equations in Lagrangean description. Thus, introducing the Piola-Kirchhoff tensors and making the notation then, with the help of the relations (2.4), (2.6), and (2.10) and the identity we obtain the equations of motion the energy equation and the inequality Introducing Helmholtz's free energy as then, with the help of (2.15), the energy equation can be expressed as Moreover, from (2.16), (2.17), and (2.18), we deduce

The response of a micromorphic thermoelastic solid is characterized by the following constitutive equations: where and , ,, , , and are sufficient smooth functions.

We assume that there is no kinematical constraint. Then, it follows from the inequality (2.20) that The inequality (2.20) reduces to

#### 3. Uniqueness and Continuous Dependence

In this section, we establish a uniqueness result and the continuous dependence of smooth thermodynamic processes upon initial state and supply terms.

We assume that is a bounded region and that is sufficiently regular to assure the common laws of transformation of surface integrals. We will employ the following notations: are the components of the unit outward normal vector to the surface ; denotes the time interval , where may be infinity; the symbol denotes a norm, either in Euclidean vector space or in a tensor space, while denotes the -norm.

*Definition 3.1. *By a thermodynamic process on one means an ordered array of functions which satisfy the equations of motion (2.15), the energy equation (2.19), and the constitutive relations (2.21).

*Definition 3.2. *A thermodynamic process will be called admissible if it also satisfies the Clausius-Duhem inequality (2.17). From the previous section it follows that the Clausius-Duhem inequality (2.17) holds for all admissible processes in if and only if (2.23), (2.24), and (2.25) are satisfied.

For admissible thermodynamic processes, one may write the energy equation in reduced form

*Definition 3.3. *One will say that is an admissible state corresponding to the load if is an admissible thermodynamic process. The admissible state is smooth if , , and are functions of class .

Let and be two smooth admissible states on corresponding to the loads and , respectively. We define the function on by where

On account of (2.23) and (3.3) it is easy to see that is of quadratic order in where , , and . The evolution in time of this function is described by the following.

Theorem 3.4. *If and are two smooth admissible states on corresponding to the loads and in , then
**
where
*

*Proof. *From (3.2) we obtain
Using the balance laws (2.15) and (2.19), we may write (3.9) in the form
where

It follows from (3.3) that
With the help of (3.12) and (3.3), we find that
On the other hand, using (3.1), we have
Collecting (3.10), (3.13), and (3.14) and using (3.6), (3.7), and (3.8) and the divergence theorem we conclude that (3.5) holds. The proof is complete.

From (2.21), it follows that where and is given by In (3.15), is a function of order , being defined by where , , , and .

Let us introduce the following definition for a definite heat conductor material (see Chiriţă [19]).

*Definition 3.5. *One says that the admissible state resides in the region where the material is a definite heat conductor if
for any nonzero , where is given by

We introduce the following notation:

Theorem 3.6. *Let be a smooth admissible state residing in the region where the material is a definite conductor of heat. Then there exist the positive constants , , and with the following property: if is any smooth admissible process defined on , such that
**
on , then
*

*Proof. *In view of (3.8), (3.15), and (3.21), it follows that
where

Using the arithmetic-geometric inequality
Schwarz's inequality, and (3.17), we deduce
where , () and , () are arbitrary nonzero constants and

In view of (3.20), we conclude that there exists a positive constant such that
Collecting (3.25), (3.28), and (3.30), we conclude that there exists a positive constant with the property that whenever (3.23) holds, we have
where
Now, taking the constants , such that and setting
from (3.22) and (3.31), we obtain (3.24), and the proof is complete.

Following [18], we introduce the following.

*Definition 3.7. *One says that the admissible state resides in the convexity region of the internal energy if the following two conditions are satisfied:(i)for each , there exists a positive constant *μ* such that
for all , , and (ii)and

Our study on stability and uniqueness is based on the following Gronwall-type inequality [18].

Lemma 3.8. *Assume that the nonnegative functions and satisfy the inequality
**
with , , , and being nonnegative constants. Then
**
where .*

Now, we are ready to state the following stability result.

Theorem 3.9. *Let be a smooth admissible state on corresponding to the loading and residing in the region where the internal energy is a convex function and the material is a definite conductor of heat. Then there exist the positive constants , , , and with the following property: if is any smooth admissible state on corresponding to the loading , such that
**
on , and
**
then for any , one has
**
where
*

*Proof. *From (3.6) and (3.39), we have . In view of (3.5), (3.7), (3.24), and Schwarz inequality, it follows that there exist the positive constants , , and such that whenever (3.23) holds, we have
where is defined in (3.22) and
Let us fix and integrate (3.42) over , with . Then, we have
Here we used the inequalities and , .

On the other hand, in view of (2.23) and (3.3), we obtain
It follows from (3.2), (3.34), (3.35), and (3.45) that there exist the positive constants and such that whenever
we have
Setting in (3.38), from (3.44) and (3.47), we obtain
Using the estimate
and the notations
then (3.48) implies that
An application of Lemma 3.8 completes the proof.

A direct consequence of the above theorem is the following uniqueness result.

Theorem 3.10. *Let and be as in Theorem 3.9. Assume that the corresponding body loads coincide on and and originate from the same state, namely,
**
Then, and coincide on .*

#### Acknowledgment

This work was supported by CNCSIS-UEFISCSU, project PN II-RU TE code 184, no. 86/30.07.2010.

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