A theoretical analysis for the normal impact fracture in thin brittle plates based on the principle of minimum external work

Abstract

This paper presents a theoretical methodology for predicting the fracture pattern of a thin brittle plate subjected to normal impact loads. The approach is based on the principle of minimum external work, which is obtained directly from the principle of least action. Global scaling laws of the number of radial cracks, the maximum radial crack speed, and the critical impact velocity when radial cracks appear are established. It is found that the fracture pattern is controlled by two dimensionless parameters: one is the ratio of the striking velocity to the longitudinal wave speed, and the other is the ratio of the surface energy to Young's modulus (bridging the material length to the macroscopic geometrical size). The physical meaning of this material length is further investigated by assuming that a body's internal energy should simultaneously consider the surface and the volumetric parts. On the one hand, this length represents the competition of the internal energy between the surface and the volumetric parts. On the other hand, this material length is consistent with the distance of neighbor atoms in equilibrium.