The mixed symmetrical temperature problem for a transversely isotropic elastic layer

Abstract

The problem of the uniform heating of a symmetrical three-layer plate with absolutely rigid outer layers, deformed solely due to thermal expansion, is solved. The generalized plane temperature problem is reduced to determining the stress-strain state, which is symmetrical with respect to two coordinates, of the inner layer (a soft filler) of transversely isotropic material using the equations of the mixed problem of elasticity theory. The layers are in ideal mutual contact. The conditions at the ends of the filler boundary layer (a thin contact layer) are not specified. On the remaining part of the ends of the filler the boundary conditions correspond to a free boundary. The problem has a finite smooth solution. The construct the exact solution a modification of Mathieu's method [1] is proposed, which consists of the fact that, in addition to ordinary Fourier series, solutions in polynomials are used. It is shown that the presence of these solutions in polynomials enables the convergence of the Fourier series to be accelerated considerably.