Discontinuous Dependency of Plate Strain Energy on Boundary Contour

Abstract

This note shows that the strain energy of a plate suffers a finite jump when the boundary contour of the plate is subject to a change, however small, from a smooth curve to a piecewise linear curve, if either the displacement or the slope, both of which are prescribed, is homogeneous at every point along the plate boundary. The proof is based on an observation that the plate strain energy depends linearly on Poisson's ratio when the lateral load, the boundary displacement and slope are specified. It also makes use of a previous result that the Gaussian curvature term in plate strain energy is a discontinuous functional of the boundary contour, and extends the theorem to that of the plate strain energy itself. This note suggests that this singular behavior may result from a discontinuous dependency of the solution to plate equation on small changes of the domain of existence.