Determination of the steady-state thermal stresses in an elastic space through the use of Vihak’s functions

The efficiency in constructing analytical solutions to the spatial elasticity and ther­mo­elas­ticity problems in terms of stresses relies on clarifying the problem concerned with the overdetermined system of governing equations. The latter problem is closely connected to the classical Southwell paradox which implies the selection of a sufficient number of independent equations for the determination of the six stress tensor components out of the system of three equilibrium and six Beltrami–Michell equations. In this paper, the direct integration method is employed along with Vihak’s functions for solving a spatial thermoelasticity problem in an unbounded solid subject to an arbitrary local distribution of the temperature field. It is shown that three Vihak’s functions for this problem can be expressed through a single function, for which an integral equation is derived. In the mapping domain of the triple Fourier integral transform, an explicit expression is obtained for the introduced function through the temperature field. It is shown that without loss of generality, there are seventeen different ways to select three governing equations out of six compatibility equations in terms of stresses which agrees well with conclusions drawn by other means. Cite as: M. Y. Yuzvyak, Y. V . Tokovyy, “Determination of the steady-state thermal stresses in an elastic space through the use of Vihak’s functions,” Prykl. Probl. Mekh. Mat., Issue 22, 37–45 (2024) (in Ukrainian), https://doi.org/10.15407/apmm2024.22.37-45