A probabilistic model of brittle crack formation

Abstract

Probability of a brittle crack formation in an elastic solid with fluctuating strength is considered. A set Ω of all possible crack trajectories reflecting the fluctuation of the strength field is introduced. The probability P(X) that crack penetration depth exceeds X is expressed as a functional integral over Ω of a conditional probability of the same event taking place along a particular path. Various techniques are considered to evaluate the integral. Under rather nonrestrictive assumptions we reduce the integral to solving a diffusion-type equation. A new characteristic of fracture process, ‘‘crack diffusion coefficient,’’ is introduced. An illustrative example is then considered where the integration is reduced to solving an ordinary differential equation. The effect of the crack diffusion coefficient and of the magnitude of strength fluctuations (ratio of minimal and mean values of the strength field) on probability density of crack penetration depth is presented. Practical implications of the proposed model are discussed.