Constructing Exact Dynamic Elasticity Solutions for Axisymmetrically Deformed Plates From Classical Plate Theory Solutions

Abstract

This paper addresses the question of how to assess the errors made when the exact three-dimensional linear elasticity solution for the axisymmetric dynamic deformation of an elastic plate is approximated by a solution inferred from the classical plate theory of Kirchhoff. Following the strategy used by Ladeve`ze and Simmonds for beams, the exact solution of a “nearby” three-dimensional problem, which differs from the original problem by the addition of incremental, computable body forces, face shears, and initial conditions—error increments, for short—is expressed in terms of the solution of a wave equation in which distance normal to the plate’s midplane plays the role of a time-like variable while the physical time itself enters only as a parameter. The error increments which, ultimately, can be computed in terms of the solution delivered by plate theory, can be regarded as an “engineering norm” because with them an engineer can decide if such a shift in the external data lies within acceptable bounds.