Some problems in the mathematical theory of thermoelasticity

The mathematical theory of thermoelasticity entered a new phase in 1956 when M.A. Biot discovered the form of the heat conduction equation for an elastic solid - an equation containing a term in the displacement vector, thereby linking this equation to the well known thermoelastic equations of motion. This thesis is concerned with obtaining formal solutions to some problems in thermoelasticity, using the full linked equations and using integral tranforms as the means of solution. In addition two approximated solutions are considered: the classical solution (obtained by omitting the linking term from the heat conduction equation) and the quasi-static solution (obtained by omitting the inertia terms from the equations of motion). These solutions and the 'complete linked' solution coincide in steady-state problems. The first chapter, which contains an introduction to the basic equations and a summary of published work in this field, is followed by the solution of the steady-state problem for the half-space, the thick plate and the elastic layer on a rigid foundation. In each case formal solutions are obtained for arbitrary temperature distributions on the traction free boundaries, and in the special cases considered the thermoelastic analogues of the isochromatic lines of photoelasticity are constructed. The next chapter deals with simple dynamical problems - the sphere, spherical shall and infinite medium with a spherical cavity, subjected to radially symmetrical thermal and elastic disturbances. For the case of the sphere under an exponentially time-dependant surface temperature, it was possible to compare the three types of solution mentioned in the first paragraph above. In the special case considered it was found that the error introduced by neglecting the inertia terms was of the same order as that introduced by neglecting the linking term. The linked quasi-static solution for the cavity problem was then compared with the corresponding classical solution for the case where the boundary is subjected to a sudden rise in temperature. Chapter IV contains the description of a method of obtaining solutions to a class of boundary value problems, by considering modified problems in an infinite space. This method is used in the following chapter to study the thermoelastic effects of longitudinal wave propagation in infinitely long circular cylinders and tubes. The thermoelastic equivalents of Rayleigh surface waves are also considered in this chapter and a brief summary is given of another author's work on plane-wave propagation. The foregoing problems were all simplified by some special feature - steady-state, radial symmetry, wave propagation of a prescribed form. In the last two chapters of the thesis, a beginning is made on the difficult task of solving the linked equations in their most general form. The infinite medium under the action of time dependent heat sources and body forces is treated first, and then a formal solution is obtained for the semi-infinite medium subjected to arbitrary heat sources, body forces, thermal and elastic boundary conditions. In all of these cases the solutions appear in the form of multiple integrals, and in any particular application it would seem to be necessary to evaluate these integrals numerically on a digital computer. In the thesis a very special example has been considered as an application of the solutions for the semi-infinite medium, and here the integrals reduce to a simple form. Within the quasi-static theory, it was found that the linked problem for the infinite medium is identical with a classical problem, in which an extra 'equivalent heat source' is introduced and one of the elastic constants is suitably changed