Contribution of Gaussian Curvature to Strain Energy of Plates

Abstract

It is shown that when a plate with a smooth boundary, on every point of which either the displacement or slope is zero, there is, in general, a contribution to the flexural strain energy from the Gaussian curvature term. When the plate boundary is approximated by a polygon with a large, but finite, number of sides, it is shown that the contribution of the Gaussian curvature, converted to a line integral around the boundary, is zero both along the sides and on corners where the displacement gradient is continuous. The singuiar behavior is shown to be the result of discontinuities of the Gaussian curvature term as a functional of the boundary contour.