Modeling of brittle fracture based on the concept of crack trajectory ensemble

Abstract

The objective of this paper is to present a comprehensive review of an approach which stands aside from the mainstream of statistical modeling of fracture. The approach is essentially based on the concept of an ensemble of macroscopically identical fracture specimens and on averaging over it. Equivalently, an ensemble Ω of virtual crack trajectories is associated with a single specimen; the averaging is then expressed in the form of functional integration over Ω. The approach combines the concepts of weakest link theories with fracture mechanics formalism and models crack propagation through a brittle microheterogeneous solid. The statistics of microheterogeneity, e.g. the population of pre-existing defects, is reflected in a random field of specific fracture energy γ and in the statistical features of Ω. The fracture parameters employed in the approach are: parameters of the pointwise distribution of the γ-field; its correlation distance; and the characteristics of roughness of the fracture surfaces, including their fractal dimension. The probability of crack formation between any two points in a two-dimensional solid (referred to as “crack propagator”) is introduced as the main building block of the approach. It is expressed as a functional integral (over the set Ω) of the probability of crack formation along a particular path. The probability distributions of critical loads, critical crack lengths, G 1c, crack arrest locations, etc., are derived in terms of crack propagator. The dependence of the distributions on the statistical characteristics of the material as well as on the roughness of the crack trajectories is analyzed by both analytical and numerical means.