Analytical Solution of Stationary Coupled Thermoelasticity Problem for Inhomogeneous Structures

A mathematical statement for the coupled stationary thermoelasticity is given on the basis of a variational approach and the contact boundary problem is formulated to consider inhomogeneous materials. The structure of general representation of the solution from the set of the auxiliary potentials is established. The potentials are analyzed depending on the parameters of the model, taking into account the restrictions associated with additional requirements for the positive definiteness of the potential energy density for the coupled problem in the one-dimensional case. The novelty of this work lies in the fact that it attempts to take into account the effects of higher order coupling between the gradients of the temperature fields and the gradients of the deformation fields. From a mathematical point of view, this leads to a change in the roots of the characteristic equation and affects the structure of the solution. Contact boundary value problems are formulated for modeling inhomogeneous materials and a solution for a layered structure is constructed. The analysis of the influence of the model parameters on the structure of the solution is given. The features of the distribution of mechanical and thermal fields in the region of phase contact with a change in the parameters, which are characteristic only for gradient theories of coupled thermoelasticity and stationary thermal conductivity, are discussed. It is shown, for example, that taking into account the additional parameter of connectivity of gradient fields of deformations and temperatures predicts the appearance of rapidly changing temperature fields and significant localization of heat fluxes in the vicinity of phase contact in inhomogeneous materials.

AVERAGED CONTROLLABILITY OF THERMOELASTICITY EQUATIONS. AVERAGE STATE OF A RECTANGULAR PLATE

The concept of averaged controllability has been introduced relatively recently aiming to analyse the controllability of systems or processes containing some important parameters that may affect the controllability in usual sense. The averaged controllability of various specific and abstract equations has been studied so far. Relatively little attention has been paid to averaged controllability of coupled systems. The averaged state of a thermoelastic rectangular plate is studied in this paper using the well-known Green's function approach. The aim of the paper is to provide a theoretical background for further exact and approximate controllability analysis of fully coupled thermoelasticity equations which will appear elsewhere.

Trefftz Method of Solving a 1D Coupled Thermoelasticity Problem for One- and Two-Layered Media

This paper discusses a 1D one-dimensional mathematical model for the thermoelasticity problem in a two-layer plate. Basic equations in dimensionless form contain both temperature and displacement. General solutions of homogeneous equations (displacement and temperature equations) are assumed to be a linear combination of Trefftz functions. Particular solutions of these equations are then expressed with appropriately constructed sums of derivatives of general solutions. Next, the inverse operators to those appearing in homogeneous equations are defined and applied to the right-hand sides of inhomogeneous equations. Thus, two systems of functions are obtained, satisfying strictly a fully coupled system of equations. To determine the unknown coefficients of these linear combinations, a functional is constructed that describes the error of meeting the initial and boundary conditions by approximate solutions. The minimization of the functional leads to an approximate solution to the problem under consideration. The solutions for one layer and for a two-layer plate are graphically presented and analyzed, illustrating the possible application of the method. Our results show that increasing the number of Trefftz functions leads to the reduction of differences between successive approximations.

Interface Models in Coupled Thermoelasticity

This work proposes new interface conditions between the layers of a three-dimensional composite structure in the framework of coupled thermoelasticity. More precisely, the mechanical behavior of two linear isotropic thermoelastic solids, bonded together by a thin layer, constituted of a linear isotropic thermoelastic material, is studied by means of an asymptotic analysis. After defining a small parameter ε, which tends to zero, associated with the thickness and constitutive coefficients of the intermediate layer, two different limit models and their associated limit problems, the so-called soft and hard thermoelastic interface models, are characterized. The asymptotic expansion method is reviewed by taking into account the effect of higher-order terms and defining a generalized thermoelastic interface law which comprises the above aforementioned models, as presented previously. A numerical example is presented to show the efficiency of the proposed methodology, based on a finite element approach developed previously.

Plane thermoelastic harmonic waves in hemitropic micropolar media

В работе рассматривается решение задачи о распространении плоской термоупругой гармонической волны в гемитропной микрополярной среде. Приводятся два варианта динамических уравнений гемитропного микрополярного континуума. Определены пространственные поляризации волн перемещений и микровращений относительно волнового вектора плоской волны. Обсуждается качественный характер возможных волновых решений уравнений связанной термоупругости. Отдельно рассматривается случай атермической волны. Вычисление волновых чисел приводится к исследованию одного кубического уравнения с вещественными коэффициентами. The paper is devoted to the problem of a plane thermoelastic harmonic wave propagation in hemitropic micropolar media. Two versions of the dynamic equations of the hemitropic micropolar continuum are presented. The spatial polarizations of the displacements and microrotations waves relative to the wave vector of a plane wave are determined. The characterictic features of possible wave solutions of the coupled thermoelasticity problems are discussed. The case of athermal waves is separately considered. Computation of wave numbers is reduced to the analysis of a cubic equation with real coefficients.