Thermoelastic Displacement Potentials

Abstract

As in the two-dimensional case (Chapter 14), three-dimensional problems of thermoelasticity are conveniently treated by finding a particular solution — i.e. a solution which satisfies the field equations without regard to bound ary conditions — and completing the general solution by superposition of an appropriate representation for the general isothermal problem, such as the Papkovich-Neuber solution.

In this section, we shall show that a particular solution can always be obtained in the form of a strain potential — i.e. by writing

$$2\mu u = \nabla \phi$$

((22.1))

The thermoelastic stress-strain relations (14.3, 14.4) can be solved to give

$$\sigma _{xx} = \lambda e + 2\mu e_{xx} - (3\lambda + 2\mu )\alpha T$$

((22.2))

$$\sigma _{xy} = 2\mu e_{xy} $$

((22.3))

etc.