A Stochastic Model for Controlled Discontinuous Crack Growth in Brittle Materials

Abstract

Statistical Fracture Mechanics aims at describing fracture of brittle solids with complex microstructure and pronounced combination of scatter and’ scale effect’ for conventional fracture parameters such as fracture toughness, critical energy release rate, etc. It achieved significant progress in prediction of distributions of critical loads, crack lengths, displacements, etc., for a stressed structural element under a variety of conditions. The present paper addresses another important question: predicting life-time scatter for a stressed brittle structural element. We model an observed mode of slow crack growth in brittle materials, namely a Markovian stochastic pattern of a microscopic random jump, followed by a random waiting time, followed by a random jump, and so on. The waiting times are related, on physical grounds, to random energy barriers at the arrest points, whereas random magnitudes of the jumps are treated within the existing framework of Crack Diffusion Theory. The transition probability density for the resulting random process of crack growth is shown to satisfy a differential equation which admits a solution in quadratures. A simple illustrative example is considered.