Cauchy problems for linear thermoelastic systems of type III in one space variable

The goal of the paper is to analyse properties of solutions for linear thermoelastic systems of type III in one space variable. Our approach does not use energy methods, it bases on a special diagonalization procedure which is different in different parts of the phase space. This procedure allows to derive explicit representations of solutions. These representations help to prove results for well‐posedness of the Cauchy problem, L^P–L^q decay estimates on the conjugate line and results for propagation of singularities. Copyright © 2005 John Wiley & Sons, Ltd.     

Eigenvalue approach to study the effect of rotation and relaxation time in generalized magneto-thermo-viscoelastic medium in one dimension

The fundamental equations of the problems of generalized thermoelasticity with one relaxation parameter including heat sources in infinite rotating magneto-thermo-viscoelastic media have been derived in the form of a vector matrix differential equation in the Laplace transform domain for a one dimensional problem. These equations have been solved by the eigenvalue approach to determine deformations, stress, and temperature. The results have been compared to those available in the existing literature. The graphs have been drawn to show the effect of rotation in the medium.     

EFFECTIVE SPECIFIC HEATS OF MULTI-PHASE THERMOELASTIC COMPOSITES

This paper studies the effective properties of multi-phase thermoelastic composites. Based on the Helmholtz free energy and the Gibbs free energy of individual phases, the effective elastic tensor, thermal-expansion tensor, and specific heats of the multi-phase composites are derived by means of the volume average of free-energies of these phases. Particular emphasis is placed on the derivation of new analytical expressions of effective specific heats at constant-strain and constant-stress situations, in which a modified Eshelby’s micromechanics theory is developed and the interaction between inclusions is considered. As an illustrative example, the analytical expression of the effective specific heat for a three-phase thermoelastic composite is presented.     

Reflection of plane waves in generalized thermoelasticity of type III with nonlocal effect

The generalized thermoelasticity theory based upon the Green and Naghdi model III of thermoelasticity as well as the Eringen's nonlocal elasticity model is used to study the propagation of harmonic plane waves in a nonlocal thermoelastic medium. We found two sets of coupled longitudinal waves, which are dispersive in nature and experience attenuation. In addition to the coupled waves, there also exists one independent vertically shear-type wave, which is dispersive but experiences no attenuation. All these waves are found to be influenced by the elastic nonlocality parameter. Furthermore, the shear-type wave is found to face a critical frequency, while the coupled longitudinal waves may face critical frequencies conditionally. The problem of reflection of the thermoelastic waves at the stress-free insulated and isothermal boundary of a homogeneous, isotropic nonlocal thermoelastic half-space has also been investigated. The formulae for various reflection coefficients and their respective energy ratios are determined in various cases. For a particular material, the effects of the angular frequency and the elastic nonlocal parameter have been shown on phase speeds and the attenuation coefficients of the propagating waves. The effect of the elastic nonlocality on the reflection coefficients and the energy ratios has been observed and depicted graphically. Finally, analysis of the various results has been interpreted.     

Spectral analysis and stability of thermoelastic Bresse system with second sound and boundary viscoelastic damping

In this paper, we consider the energy decay rate of a thermoelastic Bresse system with variable coefficients. Assume that the thermo-propagation in the system satisfies the Cattaneo's law, which can eliminate the paradox of infinite speed of thermal propagation in the assumption of the Fourier's law in the classical theory of thermoelasticity. Meanwhile, we also discuss the effect of a boundary viscoelastic damping on the stability of this system. By a detailed spectral analysis, we obtain the expressions of the spectrum and deduce some spectral properties of the system. Then based on the distribution of the spectrum, we prove that the energy of the system with a boundary viscoelastic damping decays exponentially. However, it no longer decays exponentially if there is no boundary damping. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonaxisymmetric thermal stress state of laminated rotational bodies of orthotropic materials under nonisothermic loading

A procedure for numerical investigation of nonaxisymmetric temperature fields and the elastic stress-strain state of laminated rotational bodies of cylindrically and rectilinearly orthotropic materials under nonisothermal loading is proposed. The deformation of orthotropic materials is described by the equations of anisotropic elasticity theory. The equations of state are written in the form of Hookes law for homogeneous materials, with additional terms which take into account the thermal deformation, changes in the mechanical properties of materials in the circumferential direction, and their dependence on temperature. A semianalytic finite-element method in combination with the method of successive approximations is used. An algorithm for numerical solution of the corresponding nonlinear boundary problem is elaborated, which is realized as a package of applied FORTRAN programs. Some numerical results are presented.Translated from Mekhanika Kompozitnykh Materialov, Vol. 40, No. 6, pp. 731–752, November–December, 2004.     

The null controllability of thermoelastic plates and singularity of the associated minimal energy function

We consider the null controllability problem for thermoelastic plates, defined on a two dimensional domain Ω, and subject to hinged, clamped or free boundary conditions. The uncontrolled partial differential equation system generates an analytic semigroup on the space of finite energy. Consequently, the concept of null controllability is indeed appropriate for consideration here. It is shown that all finite energy states can be driven to zero by means of just one L2((0,T)×Ω) control be it either mechanical or thermal. The singularity, as T↓0, of the associated minimal energy function is the main object studied in the paper. Singularity and blow-up rates for minimal energy function are not only of interest in their own right but are also of critical importance in Stochastic PDEs. In this paper, we establish the optimal blow-up rate for this function. It is shown that the rate of singularity is the same as for finite-dimensional truncations of the model. In view of sharp estimates available in the finite dimensional setting [Math. Control Signals Systems 9 (1997) 327], the singularity rates provided in this paper are optimal.     

Axisymmetric free vibrations of infinite thermoelastic plate

The propagation of axisymmetric free vibrations in an infinite homogeneous isotropic micropolar thermoelastic plate without energy dissipation subjected to stress free and rigidly fixed boundary conditions is investigated. The secular equations for homogeneous isotropic micropolar thermoelastic plate without energy dissipation in closed form for symmetric and skew symmetric wave modes of propagation are derived. The different regions of secular equations are obtained. At short wavelength limits, the secular equations for symmetric and skew symmetric modes of wave propagation in a stress free insulated and isothermal plate reduce to Rayleigh surface wave frequency equation. The results for thermoelastic, micropolar elastic and elastic materials are obtained as particular cases from the derived secular equations. The amplitudes of displacement components, microrotation and temperature distribution are also computed during the symmetric and skew symmetric motion of the plate. The dispersion curves for symmetric and skew symmetric modes and amplitudes of displacement components, microrotation and temperature distribution in case of fundamental symmetric and skew symmetric modes are presented graphically. The analytical and numerical results are found to be in close agreement.     

Axisymmetric Thermoelastic Interactions without Energy Dissipation in an Unbounded Body with Cylindrical Cavity

The linear theory of thermoelasticity without energy dissipation is employed to study thermoelastic interactions in a homogeneous and isotropic unbounded body containing a cylindrical cavity. The interactions are supposed to be due to a constant step in radial stress or temperature applied to the boundary of the acvity, which is maintained at a constant temperature or zero radial stress (as the case may be). By using the Laplace transform technique, it is found that the interactions consist of two coupled waves both of which propagate with a finite speed but with no attenuation. The discontinuities that occurs at the wavefronts are computed. Numerical results applicable to a copper-like material are presented.