Exact solutions of some mixed problems of uncoupled thermoelasticity for a truncated hollow circular cone with a groove along the generatrix

An elastic body of finite dimensions in the form of a truncated hollow circular cone with a groove along the generatrix is considered. The uncoupled problem of thermoelasticity is formulated for this body for different types of boundary conditions on all the surfaces. These are the conditions for specifying the displacements or sliding clamping on surfaces with fixed angular coordinates and the conditions for specifying the stresses on surfaces with a fixed radial coordinate (shear stresses are assumed to be zero). It is assumed that the temperature is a specified function of all the spherical coordinates. Some auxiliary functions, related to the displacements, are introduced first, and equations for these functions are then derived using Lamé's equations. A finite integral Fourier transformation with respect to one of the angular variables is then employed. After this, by solving certain Sturm-Liouville problems, a new integral transformation is constructed and is applied to the equations with respect to the other angular variable. As a result a one-dimensional system of differential equations is obtained, to solve which an integral Mellin transformation is employed in a special way. Finally, exact solutions of some problems of thermoelasticity are constructed in series for this body.