An uncoupled thermoelasticity problem for a semi-infinite layer with regard of its proper weight

The exact solution of the uncoupled thermoelasticity problem for a semi-infinite elastic layer with regard of its proper weight was constructed. The solving method is differ from the known previous and is based on reducing of Lame equations to two jointly and one separately solvable equation. It allows application of integral transformations directly to the transformed equations of equilibrium and makes it possible to reduce the initial problem to a one-dimensional vector boundary problem. A special technique is given to calculate multiple integrals containing oscillating functions that appear during the inversion of the transformations. The character of the temperature and proper weight influence on the value of normal stresses on the lateral face of the semi-infinite layer, the zone of tensile stresses depending on the shapes of the distributed load section and the temperature and Poisson's ratio is established. The parameters of dimensionless mechanical load and temperature, when the separation of the side wall of the semi-infinite layer can be eliminated, were established. A study of the influence of the layer’s proper weight on the stress emerging on the layer’s edge is conducted. The constructed exact solution can be used as a model for solving a similar class of problems by numerical methods.