Effect of intrinsic rotations, microstructural expansion and contractions in initial boundary value problem of thermoelastic bodies
© Marin et al.; licensee Springer. 2014
Received: 26 February 2014
Accepted: 12 May 2014
Published: 22 May 2014
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.
Keywordsthermoelastic microstretch seismic waves earthquake
The theory of micromorphic elastic solids was first elaborated by Eringen (see, for instance, ). Then Eringen has generalized  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  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  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  Iesan and Pompei have presented a solution of Boussinesq-Somigliana-Galerkin type for the boundary value problem in this context.
In the paper , 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  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 , 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  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 .
The study  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  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.
2 Basic equations
Let the body occupy, at time , 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 . We refer the motion of the body to a fixed system of rectangular Cartesian axes , and adopt the Cartesian tensor notation. Points in B are denoted by and 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 . All Latin subscripts are understood to range over the integers , while the Greek indices have the range . 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 the components of the displacement vector and by 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 of the body in its reference state.
As usual, we denote by the components of the stress tensor and by the components of the couple stress tensor over B. Also, we denote by the components of the microstress vector.
the equations of motion(1)
the balance of the equilibrated forces(2)
the energy equation(3)
the components of body force;
the components of body couple;
L the generalized external body load;
ϱ is the reference constant mass density;
J and are the coefficients of microinertia;
are the components of the internal hypertraction vector;
η is the entropy per unit mass;
S is the heat supply per unit mass;
are the components of heat flux vector.
where is the alternating symbol.
that is, the conductivity tensor k is positive semi-definite only.
the constitutive coefficients are continuously differentiable functions on ;
ϱ and d are continuous functions on ;
, and L are continuous functions on , where .
at regular points of the surface ∂B.
Here, are the components of the outward unit normal of the surface ∂B.
, , , , , , and are continuous function on ;
, , and are continuous function on their domains;
, , and 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 we mean an ordered array which satisfies (1), (2), and (3) for all , the initial conditions (7) and the boundary conditions (8).
3 Main results
In the next theorem we prove a relationship between the functions K and U.
for all .
Here we have suppressed, for convenience, the dependence of the functions on argument x, because there is no likely confusion.
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).
for all .
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.
the symmetry relations (5) are satisfied;
ϱ, and J are strictly positive;
d is strictly positive or strictly negative;
the conductivity tensor 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.
Because of linearity, this difference is also a solution of our problem, but it corresponds to null data.
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 and continuous with respect to time.
for all .
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.
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  in the classical isothermal case.
for all .
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 , but using some strong hypotheses on the thermoelastic coefficients.
If the conductivity tensor is assumed be positive definite, then the result of Theorem 4 can be used to obtain the uniqueness result established in .
which proves Theorem 4.
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.
We express our gratitude to the referees for their valuable criticisms of the manuscript and for helpful suggestions.
- Eringen AC: Theory of micromorphic materials with memory. Int. J. Eng. Sci. 1972, 10: 623-641. 10.1016/0020-7225(72)90089-4View Article
- Eringen AC: Theory of thermo-microstretch elastic solids. Int. J. Eng. Sci. 1990, 28: 1291-1301. 10.1016/0020-7225(90)90076-UMathSciNetView Article
- Ciarletta M: On the bending of microstretch elastic plates. Int. J. Eng. Sci. 1995, 37: 1309-1318.MathSciNetView Article
- Ciarletta M, Svanadze M, Buonanno L: Plane waves and vibrations in the theory of micropolar thermoelasticity for materials with voids. Eur. J. Mech. A, Solids 2009, 28: 897-903. 10.1016/j.euromechsol.2009.03.008MathSciNetView Article
- Iesan D, Pompei A: Equilibrium theory of microstretch elastic solids. Int. J. Eng. Sci. 1995, 33: 399-410. 10.1016/0020-7225(94)00067-TMathSciNetView Article
- Agarwal RP, O’Regan D, Lashmikantham V, Leela S: A generalized upper and lower solution method for singular initial value problems. Comput. Math. Appl. 2004, 47: 739-750. 10.1016/S0898-1221(04)90061-2MathSciNetView Article
- Agarwal RP, O’Regan D: Boundary value problems on the half line in the theory of colloids. Math. Probl. Eng. 2002, 8: 143-150. 10.1080/10241230212905MathSciNetView Article
- Marin M: A partition of energy in thermoelasticity of microstretch bodies. Nonlinear Anal., Real World Appl. 2010, 11(4):2436-2447. 10.1016/j.nonrwa.2009.07.014MathSciNetView Article
- Marin M: Lagrange identity for microstretch thermoelastic materials. J. Math. Anal. Appl. 2010, 361(1):275-286.View Article
- Marin M: Some estimates on vibrations in thermoelasticity of dipolar bodies. J. Vib. Control 2010, 16(1):33-47. 10.1177/1077546309103419MathSciNetView Article
- Marin M: An evolutionary equation in thermoelasticity of dipolar bodies. J. Math. Phys. 1999, 40(3):1391-1399. 10.1063/1.532809MathSciNetView Article
- Marin M, Agarwal RP, Mahmoud SR: Modeling a microstretch thermoelastic body with two temperatures. Abstr. Appl. Anal. 2013., 2013: Article ID 583464 10.1155/2013/583464
- Abd-Alla AM, Mahmoud SR, Abo-Dahab SM, Helmy MI: Propagation of S-wave in a non-homogeneous anisotropic incompressible and initially stressed medium under influence of gravity field. Appl. Math. Comput. 2011, 217(9):4321-4332. 10.1016/j.amc.2010.10.029MathSciNetView Article
- Abd-alla AM, Abbas IA: Magnetoelastic longitudinal wave propagation in a transversely isotropic circular cylinder. Appl. Math. Comput. 2002, 127(2-3):347-360. 10.1016/S0096-3003(01)00012-1MathSciNetView Article
- Othman MIA: Effect of rotation on plane waves in generalized thermoelasticity with two relaxation times. Int. J. Solids Struct. 2004, 41: 2939-2956. 10.1016/j.ijsolstr.2004.01.009View Article
- Othman MIA, Singh B: The effect of rotation on generalized micropolar thermoelasticity for a half-space under five theories. Int. J. Solids Struct. 2007, 44(9):2748-2762. 10.1016/j.ijsolstr.2006.08.016View Article
- Boschi E, Mainardi F: Body loadings in thermo-microstretch elastic solids. Geophys. J. R. Astron. Soc. 1973, 34: 313-320. 10.1111/j.1365-246X.1973.tb02398.xView Article
- Knops RJ, Payne LE: On uniqueness and continuous dependence in dynamical problems of thermoelasticity. Int. J. Solids Struct. 1970, 6: 1173-1184. 10.1016/0020-7683(70)90054-5View Article
- Reiss R: Minimum principles for linear elastodynamics. J. Elast. 1978, 8(1):35-45. 10.1007/BF00044509MathSciNetView Article
This article is published under license to BioMed Central Ltd. 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.