Quasi-Static Brittle Fracture in Inhomogeneous Media and Iterated Conformal Maps: Modes I, II and III

TL;DR

This paper extends the iterated conformal maps method to model quasi-static brittle fracture in inhomogeneous media across all fracture modes, revealing differences in crack geometry and stress distribution.

Contribution

It generalizes previous mode III theory to include modes I and II, involving bi-Laplace equations and multiple analytic functions, and considers material randomness.

Findings

Mode I and II produce different crack geometries than Mode III.

The method accounts for material inhomogeneity and randomness in fracture events.

Distinct stress distributions are identified for different fracture modes.

Abstract

The method of iterated conformal maps is developed for quasi-static fracture of brittle materials, for all modes of fracture. Previous theory, that was relevant for mode III only, is extended here to mode I and II. The latter require solution of the bi-Laplace rather than the Laplace equation. For all cases we can consider quenched randomness in the brittle material itself, as well as randomness in the succession of fracture events. While mode III calls for the advance (in time) of one analytic function, mode I and II call for the advance of two analytic functions. This fundamental difference creates different stress distribution around the cracks. As a result the geometric characteristics of the cracks differ, putting mode III in a different class compared to modes I and II.