Effect of Heat on Deformations in Material with a Defect

Abstract

A system of thermoelasticity equations is considered. Boundary transmission conditions are specified by the differences in temperature, heat fluxes, deformations, and their first derivatives on the boundary. The stationary case is studied. The boundary (crack) is represented by the interval \([ - 1;1]\) of the \(O{{x}_{1}}\) axis. The given problem is investigated, its solution is found, and the well-posedness of its formulation is proved. The results of previous works are generalized. The subject of greatest interest is the asymptotic behavior, as \({{x}_{1}} \to \pm 1,\;{{x}_{2}} \to 0\), of the displacements \(u({{x}_{1}},{{x}_{2}}),\)\(v({{x}_{1}},{{x}_{2}})\) of a point \(({{x}_{1}},{{x}_{2}})\) under material deformations and the asymptotic behavior of their derivatives. Here, the functions \(u({{x}_{1}},{{x}_{2}}),\)\(v({{x}_{1}},{{x}_{2}})\) are assumed to depend on the material temperature \(T({{x}_{1}},{{x}_{2}})\) at the point \(({{x}_{1}},{{x}_{2}})\).