Computing exact, elastodynamic linear three-dimensional solutions for plates from classical two-dimensional solutions

Abstract

Given an approximate time-dependent distribution of midplane vertical displacement and three-dimensional transverse shear and normal stresses in a platelike elastic body undergoing flexure – the quantities delivered by the Kirchhoff (classical) theory – we construct exact solutions of the equations of motion of linear three-dimensional elasticity. This is accomplished by (1) solving an auxiliary spatially hyperbolic system of partial differential equations (in which time enters only parametrically) and (2) choosing residual body and surface forces and initial conditions to insure satisfaction of all three-dimensional field equations, boundary, and initial conditions. The residual quantities which, in general, are significant only near the edges of the plate, serve as meaningful physical measures of the errors in classical plate theory. The special difficulties posed by plates with sharp corners are mentioned, but are left for future treatment.