Analytical solutions of the quasistatic thermoelasticity task with variable physical properties of a medium. (Russian. English summary) Zbl 1413.74043

Summary: A 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 constructing the disparity of the original differential equation, that should be orthogonal to all the coordinate functions. Hence, the unknown coefficients of the 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 number of approximations. Thus, already in the ninth approximation the disparity of the original differential equation is zero in almost the entire range of the spatial variable. The maximum disparity in the sixth approximation is \(\varepsilon = 5 \cdot 10^{-4}\).


74F05 Thermal effects in solid mechanics
74B20 Nonlinear elasticity
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