RAMIFICATION OF EQUILIBRIA OF A COMPRESSED ELASTIC ORTHOTROPIC PLATE WITH INTERNAL STRESSES

Abstract

The problem of loss of stability and post-critical behavior of a compressed orthotropic rectangular plate on a nonlinearly elastic base containing continuously distributed fields of dislocations and disclinations and under the influence of a small normal load is considered. The compressive force components are evenly distributed along the edges and act parallel to the main directions of elasticity. The problem is formulated as an analogue of a system of nonlinear Karman equations for an orthotropic plate containing a function called the incompatibility measure, which is expressed in terms of the densities of edge dislocations and wedge disclinations. The system of equations takes into account the small transverse pressure and the reaction of the elastic base in the form of a polynomial of the third degree of deflection. The following boundary conditions are considered: the edges of the plate are freely pinched or movably pivotally supported; two opposite edges of the plate are freely pinched or movably pivotally supported, and the other two are free from loads. The stress function is sought in the form of two components: the stress function caused by the presence of internal sources, determined from the linear boundary value problem, and the stress function caused by the external impact of compressive loads and a nonlinear elastic base, which is determined from the nonlinear boundary value problem. The nonlinear boundary value problem is investigated by the Lyapunov – Schmidt method. To solve the linearized equation from which the critical value of the compressive load is determined, the variational method is used in combination with the difference method. A system of branching equations of the Lyapunov – Schmidt method is constructed, which is investigated by numerical methods. The post-critical behavior of the plate is investigated and asymptotic formulas for new equilibria in the vicinity of the critical load are derived. For various values of compressive loads, the orthotropy parameters of the plate and the intensity parameter of internal stresses, the relations between the values of the base parameters are established, at which its bearing capacity is preserved in the vicinity of the classical value of the critical load.