Self-similarity of crack propagation in inhomogeneous materials

A non-equilibrium statistical method is used to study the origin of self-similarity of crack propagation. Here, crack propagation is regarded as a stochastic process due to the inhomogeneity of the material's microstructure. The crack propagation rate is expressed by a deterministic term combined with a nonlinear stochastic term. From the rate expression, the statistical evolution equation of microcracks is established. Then, from its solution, the origin of self-similarity of crack propagation is discussed and the necessary condition for self-similar behaviour of crack propagation is given. The probability distribution function of crack density, which has a self-similar property, is found to be of the form of the Weibull function modified by a confluent hypergeometric function.