Buckling of the Nonuniformly Compressed Plate with Dislocations and Disclinations

Abstract

The buckling problem of the elastic rectangular plate, which is compressed by the nonuniformly distributed along the edges external loads, is considered. This plate contains continuously distributed sources of internal stresses and is found under the action of a small normal load. The components of the external loads are the given continuous functions and act in direction, which is parallel to the coordinate axes. The coordinate system has its beginning in the center of the plate. The study is based on a modified system of elastic plates’ nonlinear von Karaan equations, which take into account influence of dislocations and disclinations or other sources of intrinsic stresses. The boundary conditions correspond to the free clamped plates’ edges or their movable hinge support. The considered problem is reduced to a sequence of three linear boundary value problems for determining the forms of the stress functions, which correspond to internal sources and two components of compressive forces, and a system of nonlinear equations with a trivial solution. Numerical parameters characterizing the intensity of stresses caused by internal sources and components of external loads are introduced. The critical buckling load is defined as the solution of the linearized nonlinear problem on a trivial solution. The stability loss problem is reduced to a parametric eigenvalue boundary value problem. The problem of eigenvalues is solved by the variational method. To study the post-critical behavior of a compressed plate, the Lyapunov–Schmidt method is used in combination with numerical methods for calculating the coefficients of the system of the branching equations. The cases of equilibrium branching of a compressed plate with one and two eigenforms are considered. Asymptotic formulas of new equilibria in the vicinity of the bifurcation point are constructed in the case when a small normal load acts and without it as well. It is obtained that the presence of a small normal load does not reduce the bearing capacity of the plate in the case of even forms of the incompatibility function and even forms of distribution along the edges of the compressive forces.