Stability of a nonlinear elastic plate under uniaxial loading with respect to finite perturbations

Abstract

The stability of a plate made of nonlinear elastic material with respect to finite perturbations is considered. The analysis of the main process of plate deformation is reduced to the solution of the nonlinear boundary value problem with respect to finite perturbations. Solutions for perturbations of displacements are chosen in the form of eigenfunction series, that are solutions of the corresponding linearized problems and satisfy the geometric boundary conditions. After applying the principle of possible displacements, the question on the stability of the ground state of the nonlinear problem is reduced to the study of the stability of zero solution of an infinite system of ordinary differential equations with constant coefficients, the number of terms in which is specified by the elastic potential. For the obtained system of equations under certain restrictions on initial perturbations, a Lyapunov function is constructed. The dimension of the strange attractor of the dynamical system is found which allows to limit the number of terms in the Bubnov-Galerkin series.