Necessary condition for the existence of solution of plane problem of elasticity in a rectangular domain under external loadings (Q2754723)

The authors consider the solution of the general problem of elastostatics in a rectangle. The main attention is paid to the selection of elementary solutions of equilibrium equations. It is proved that a solution exists when external loadings on the boundary satisfy certain necessary conditions. The authors find elementary solutions caused by non-equilibrated loadings; they are represented by bilinear functions for normal stresses and by quadratic functions for tangential stresses. The derived approach enables one to examine in detail some well-known usual techniques of constructing elementary solutions of elastostatics problem, and to point out shortcomings of these techniques.