Fracture of heterogeneous materials with continuous distributions of local breaking strengths

We develop a recursion-relation approach for the failure probability of heterogeneous networks with continuously distributed bond strengths and with local stress enhancements after bond failure. Current computational methods to solve these models scale as 2ⁿ where n is sample size, while our method provides a rapidly converging sequence of approximations which scale algebraically with the sample size. The method is applied to systems with uniform and Weibull distributions of local bond-failure thresholds. We find that the characteristic feature which occurs when there is a continuous distribution of local breaking strengths is that, as a function of sample size, the failure probability Fₙ({σ}) shows a minimum at nc, which deepens and moves to higher n as the external stress {σ} is reduced. At large n, there is a stable ``weak-link'' scaling form for the failure probability, in agreement with work by Harlow and Phoenix. For sufficiently large n{}nc, the failure probability is of double-exponential form, and the size effect is logarithmic. For small nnc, however, the low-stress tail of the failure probability appears to be of Weibull form with n-dependent parameters, and the size effect can be algebraic.