TOPOLOGY OF THE FOUR-MODE STRAIN ENERGY OF THERMALLY BUCKLING PLATES

Abstract

Using the Galerkin representation of heated nonlinear plate equations, we derived modal equations for the first four symmetric plate modes of a simply supported and clamped isotropic plate. The modal equations have myriad cubic amplitude terms, yet each and every one of which must account for the energy conservation; hence, the Hamiltonian property is preserved. The Hamiltonian consists of the kinetic energy and the strain (potential) energy of plate bending, membrane stretching, and thermal expansion. Since the strain energy enters into the exponent of the stationary Fokker-Planck distribution for displacement, we investigate the topological structure of strain energy under a uniform temperature plate heating. The strain energy is concave with a single zero minimum for a prebuckled plate; however, it develops a double-well potential as the plate temperature exceeds a certain critical buckling temperature for each modal coordinate. The peaks of the bimodal Fokker-Planck distribution are determined mainly by the double-well potential of the primary plate mode, which is not significantly affected by modal truncations under a uniform temperature plate heated up to five times the critical buckling temperature. Hence, this explains how the single-mode Fokker-Planck formulation was used to validate the response statistics of thermally buckled aluminum and composite plates.