Bifurcations in von Karman problem for rectangular, thin, elastic plate resting on elastic foundation of Winkler type

Abstract

The von Kármán model for thin, elastic, rectangular plate resting on a linear elastic foundation of Winkler type is studied. The plate is simply supported along all four edges and is subjected to a compressive loading in one from two directions. The critical values of the loading parameter and buckling modes are found on the basis of investigation of linearized problem. In this research, authors have developed their approach from their prior papers. The aim of this approach is to examine the corresponding nonlinear mathematical model using the methods of functional analysis and bifurcation theory. This model is reduced to operator equation with Fredholm-type operator of index 0, which is dependent on parameters and defined in corresponding Sobolev spaces. The Crandall-Rabinovitz bifurcation theorem (gradient case) is used to prove the bifurcation theorems and to examine the postcritical behavior of the plate.