Models of microscopic fracture
Two kinds of models have traditionally been studied: ones based on looking at arrays of atoms, and ones based on continuum descriptions of materials. At the atomic level, a simple model suggested fairly recently is that atoms are connected by bonds with a random distribution of strengths, and that cracks follow paths that minimize the total strength of bonds to be broken. It is not clear why in a crystal bonds should be of different strengths, and there is some evidence that this model yields incorrect predictions for the statistical properties of actual cracks. A slightly better model, related to the one in the main text, is that the bonds between atoms are identical, and act like springs which break when they are stretched too far. In recent years, computer simulations with millions of atoms have been carried out—usually with realistic but complicated interatomic force laws—and some randomness has been observed, but its origins have not been isolated. A set of nonlinear partial differential equations known as the Lamé equations are commonly used as a continuum description of elastic materials. Various instabilities have been found in these equations, but the equations are based on small deformations, and presumably cannot be relied upon to provide information about fracture.