On Slow Processes in Piezothermoelastic Plates

We consider a plate infinite in extent and bounded by two parallel planes. The plate is filled by a heat-conducting piezoelectric material with the symmetry of a hexagonal crystal class C6ν = 6mm, so that e.g. ferroelectric ceramics are included. We assume that the panel is subject to a thermic exposure on the upper face which varies very slowly with time. On the lower face the displacement is prescribed, as when e.g. the panel is welded above a fixed flat body. We study processes which are homogeneous on each plane parallel to the boundary planes, i.e. they depend only on the thickness coordinate, and moreover they vary very slowly with time. Hence they form a family of equilibrium configurations, for the thermopiezoelastic panel, indexed by a parameter τ, which is a function of time with everywhere small time-derivative; they can be identified with quasi-static processes of the body, induced in particular by a given thermic exposure on the upper bounding face. Thus we formulate some appropriate boundary-value problems involving the equilibrium field equations. Such problems are completely solved by means of their explicit solutions. In particular, we show that whatever temperature is given at the upper face, by the thermic exposure, the temperature at the lower face can be controlled by the difference of electric potential between the two bounding planes.