Two-dimensional finite difference analysis of shape instabilities in plates

Abstract

A two-dimensional finite difference analysis is applied to surface diffusion-controlled instabilities of plates. Plates can evolve into “cylinders,” or if the plates have longitudinal internal boundaries, they may split into two segments. The evolution process of plates containing internal boundaries into equilibrium shapes depends on both the initial plate aspect ratio (plate width to thickness) and the ratio of the internal boundary energy to the plate-matrix interface energy. When the internal boundary energy is relatively low or the initial plate aspect ratio is relatively small, the transverse equilibrium cross-sectional area shape is composed of two circular segments, with an appropriate dihedral angle dictated by the ratio of the interface energy terms. As either the internal boundary energy or the initial aspect ratio increases, plate splitting, rather than cylinderization, becomes the dominant instability mode. The results of this work are compared to a recent theory of Courtney and Malzahn Kampe (CMK) on shape instability diagrams.^[1] The complicated interactive effects between cylinderization and boundary splitting were not considered in the analytical CMK approach; thus, when they are minimal, the results of this finite difference calculation are in reasonable accord with the CMK results, as far as predicting instability times are concerned. However, when the interaction is significant, cylinderization and/or splitting times are markedly changed. The present accurate calculations allow refinement of the CMK plate instability diagrams.