ASYMPTOTICS OF CRITICAL LOADS OF A COMPRESSED NARROW ELASTIC PLATE WITH INTERNAL STRESSES

Abstract

The problem of the asymptotic solution of a modified system of nonlinear Karman equilibrium equations for a longitudinally compressed elongated elastic rectangular plate with internal stresses lying on an elastic base is considered. Internal stresses can be caused by continuously distributed edge dislocations and wedge disclinations, or other sources. The compressive pressure is applied parallel to the long sides of the plate to the two short edges. The boundary conditions are considered: the long edges of the plate are free from loads, and the short edges are freely pinched or movably hinged. A small parameter is introduced, equal to the ratio of the short side of the plate to the long side. The solution of the system – the compressive load, the deflection function, and the stress function – is sought in the form of series expansions over a small parameter. The system of Karman equations with dimensionless variables is reduced to an infinite system of boundary value problems for ordinary differential equations with respect to the coefficients of asymptotic expansions for the critical load, deflection, and stress function. In this case, to meet the boundary conditions, the boundary layer functions are additionally introduced, which are concentrated near the fixed edges and disappear when moving away from them. Boundary value problems for determining the functions of the boundary layer are constructed. It is shown that the main terms of the small parameter expansions for the critical load and deflection are determined from the equilibrium equation of a compressed beam on an elastic base with the boundary conditions of free pinching or movable hinge support of the ends. In this case, the main term of the expansion into a series of the stress function has a fourth order of smallness in the parameter of the relative width of the plate.