Analytical Solutions of the Quasistatic Thermoelasticity Task with Variable Physical Properties of a Medium

The high-precision approximate analytic solution of the nonlinear quasi-static problem of thermoelasticity for an infinite hollow cylinder with variable along the radial coordinate physical properties is obtained using the orthogonal Bubnov–Galerkin method developed by the construction of systems of coordinate functions exactly satisfying inhomogeneous boundary conditions in any approximation. The mathematical formulation includes non-linear equations for the unknown function of displacement and inhomogeneous boundary conditions. The desired solution is supposed to precisely satisfy the boundary conditions in advance. The exact fulfillment of the boundary conditions is achieved using the coordinate functions of special design. The unknown coefficients are found by constructing the disparity of original differential equation, that should be orthogonal to all the coordinate functions. Hence, the unknown coefficients of solution yields a system of linear algebraic equations, which number is equal to the number of approximations of the solution. It is shown that the solution accuracy increases substantially with increasing the number of approximations. Thus, already in the ninth approximation the disparity of original differential equation is zero almost the entire range of the spatial variable. The maximum disparity in the sixth approximation is $\varepsilon = 5 \cdot 10^{-4}$.