Closed Solutions of Boundary-Value Problems of Coupled Thermoelasticity

Coupled equations of thermoelasticity take into account the effect of nonuniform heating on the medium deformation and that of the dilatation rate on the temperature distribution. As a rule, the coupling coefficients are small and it is assumed, sometimes without proper justification, that the effect of the dilatation rate on the heat conduction process can be neglected. The aim of the present paper is to construct analytical solutions of some model boundary-value problems for a thermoelastic bounded body and to determine the body characteristic dimensions and the medium thermomechanical moduli for which it is necessary to take into account that the temperature and displacement fields are coupled. We consider some models constructed on the basis of the Fourier heat conduction law and the generalized Cattaneo-Jeffreys law in which the heat flux inertia is taken into account. The solution is constructed as an expansion in a biorthogonal system of eigenfunctions of the nonself-adjoint operator pencil generated by the coupled equations of motion and heat conduction. For the model problem, we choose a special class of boundary conditions that allows us to exactly determine the pencil eigenvalues.