ВЫПУЧИВАНИЕ И ПОСЛЕКРИТИЧЕСКОЕ ПОВЕДЕНИЕ СЖАТОЙ ПРЯМОУГОЛЬНОЙ ПЛАСТИНЫ С ВНУТРЕННИМИ НАПРЯЖЕНИЯМИ, ЛЕЖАЩЕЙ НА НЕЛИНЕЙНО-УПРУГОМ ОСНОВАНИИ

Abstract

The problem of the effect of initial imperfections in the form of small transverse loads on the loss of stability and the post-critical behavior of a compressed elastic rectangular plate lying on a non-linearly elastic foundation is considered. The plate contains in a flat state continuously distributed edge dislocations and wedge disclinations or other sources of internal stresses. The research is conducted on the basis of a modified system of non-linear Karman equations for an elastic plate with dislocations and disclinations which additionally takes into account the reaction of the base in the form of a second or third degree polynomial in deflection. Two cases of boundary conditions are considered: free pinching and movable hinged support of the edges. The problem is reduced to solving a non-linear operator equation which is investigated by the Lyapunov-Schmidt method. The linearized equation is a multiparameter boundary value problem for eigenvalues which is solved by a finite-difference method. The coefficients of the system of ramification equations are calculated numerically. The post-buckling behavior of the plate is investigated and asymptotic formulas are derived for new equilibria in the neighborhood of critical loads. For different values of the parameters of compressive loads and the parameter of internal stresses, relations have been established between the values of the parameters of the base, at which its bearing capacity is preserved in the neighborhood of the classical value of the critical load.