Constructing solutions for two-dimensional quasi-static problems of thermomechanics in terms of stresses for bodies with plane-parallel boundaries

A methodology to construct solutions for two-dimensional quasi-static thermomechanical problems for bodies with plane-parallel boundaries (2D-QS thermomechanical problems) is proposed.  This approach begins with selecting equations for the plane quasi-static thermoelasticity problem in terms of stresses.  The methodology approximates the distribution of non-zero stress tensor components through the body's thickness using cubic polynomials, with coefficients expressed in terms of integral characteristics of the stress tensor components over the thickness variable and their specified boundary values on the body’s lower and upper surfaces.  Consequently, the original two-dimensional boundary problem is simplified to a one-dimensional boundary problem for the integral characteristics.  For an infinite layer, solutions are found using the Fourier transform along the longitudinal coordinate, while for a strip plate, a finite integral transformation is applied along the transverse coordinate. General solutions for 2D-QS thermomechanical problems are formulated for the selected bodies under non-stationary volume forces and temperature fields.  The resulting expressions for the stress tensor components are presented as convolutions of functions representing the boundary values on the bases (and end cross-sections for strip-plates) and functions describing homogeneous solutions to the one-dimensional boundary problems for the integral characteristics of the stress tensor components.