On critical loads of compressed elastic rectangular plate with dislocations and disclinations

A problem on critical loads of the compressed rectangular plate containing continuously distributed sources of inherent stress is considered. The task analysis is based on the modification of the Karman nonlinear equations system for large deflections of elastic plates with dislocations and disclinations under different boundary conditions. By the introduction of a replacement for the stress function, the problem reduces to the treatment of two tasks: a linear boundary value problem concerning the stress function caused by internal sources and a system of nonlinear equations concerning the deflection and the stress function caused by external compressive loads, which possesses a trivial solution. The classical critical load is defined as the smallest eigenvalue of the linear boundary value problem obtained by the linearization of the nonlinear equations system relative to the trivial solution. Four types of boundary conditions are treated: all edges are variably restrained; all edges are simply supported; two opposite edges are stress-free, and the other two are either variably restrained or simply supported. Uniformly distributed compressive loads are equal on the opposite edges. It is established that if the measure of inconsistency is odd on one variable and odd or even on another variable, then the stresses caused only by internal sources, do not lead to the loss of the flat equilibrium state and do not affect the critical values of compressive loads.