Integral formulae in the coupled problem of the thermoelasticity of an inhomogeneous body. Application in the mechanics of composite materials

Gorbachev, V.I.

Автор: Gorbachev V.I.
Журнал: Journal of Applied Mathematics and Mechanics (English translation of Prikladnaya Matematika i Mekhanika)
Том: 78
Номер: 2
Год издания: 2014
Издательство: Pergamon Press Ltd.
Местоположение издательства: United Kingdom
Первая страница: 192
Последняя страница: 208
DOI: 10.1016/j.jappmathmech.2014.07.013
Аннотация: A coupled unsteady problem of thermoelasticity for an inhomogeneous body, described by a system of four second-order partial differential equations with coefficients that vary depending on the coordinates, is considered, and the same problem for a homogeneous body of the same shape (the concomitant problem) is examined together with this original problem. Integral formulae are obtained that allow one to express the displacements and temperature in the original problem in terms of the displacements and temperature in the concomitant problem. Integral formulae are used to represent the solution of the original problem in the form of series over all possible derivatives of the solution of the concomitant problem. A system of recurrence problems is written for the coefficients of these series. Expressions are found for the coefficients of the concomitant problem (effective coefficients) and special boundary value problems are formulated, from the solution of which specific expressions are found for the effective thermoelasticity coefficients. A theorem concerning the fact that the effective coefficients satisfy the physicomechanical constraints imposed on the thermoelastic constants of real bodies is proved. The case of a layer that is inhomogeneous in its thickness is considered and explicit analytical expressions for all the thermoelasticity coefficients are obtained for it. The case when the thermoelasticity coefficients depend periodically on the coordinates is examined in detail. The fundamentals of the thermoelasticity of homogeneous isotropic solids and the solutions of many classical problems are described in Refs.1, 2, 3 and 4 Thermoelasticity problems have been considered for solids with continuous inhomogeneity and also for piecewise-homogeneous solids.5 An asymptotic averaging method was proposed by Bakhvalov6, 7 and 8 for equations with periodic coefficients which, in certain cases, enable exact solutions to be obtained that are also useful in the case when there is a non-periodic dependence of the coefficients on the coordinates.9 and 10 Averaging of the coupled unsteady thermoelasticity problem has been carried out for regular composite materials. Integral representations were used for the first time15, 16 and 17 to average static and dynamic problems of inhomogeneous elasticity.
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Integral formulae in the coupled problem of the thermoelasticity of an inhomogeneous body. Application in the mechanics of composite materialsстатья

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