Closed-form solutions of dynamic coupled thermoelasticity problems for a cylinder and a sphere

The closed-form solutions of the linear coupled thermoelasticity problem for a finite cylinder and a sphere are obtained. The solutions are constructed as an expansion in a series of eigenfunctions of the differential operators generated by the initial-boundary value problems under study. Special boundary conditions and symmetries are formulated, which allows us to solve the thermoelasticity problem of bodies of canonic shape without resorting to numerical methods and to estimate the influence of the coupling between the thermal and mechanical fields on their time distribution depending on the size of the examined body. The obtained solutions of the coupled problems and the solutions of the corresponding thermal conductivity problems are compared. It is shown that for micron-scale bodies the amplitude of thermal waves caused by the coupling between the thermal and mechanical fields increases significantly compared to the amplitude of analogous waves in macrobodies and constitutes a few per cent of the initial value of thermal action.