Conformal Dynamics of Precursors to Fracture

TL;DR

This paper derives an exact conformal map equation to model the evolution of cavities in stressed solids, accurately predicting sharp groove formation and fracture precursors influenced by surface diffusion.

Contribution

It introduces a novel integro-differential equation for cavity boundary evolution, incorporating surface diffusion effects, and demonstrates its application to elliptical and circular cavities under stress.

Findings

Predicts formation of sharp grooves leading to fracture

Accurately models cavity dynamics with surface diffusion

Analyzes stability of cavities under biaxial stress

Abstract

An exact integro-differential equation for the conformal map from the unit circle to the boundary of an evolving cavity in a stressed 2-dimensional solid is derived. This equation provides an accurate description of the dynamics of precursors to fracture when surface diffusion is important. The solution predicts the creation of sharp grooves that eventually lead to material failure via rapid fracture. Solutions of the new equation are demonstrated for the dynamics of an elliptical cavity and the stability of a circular cavity under biaxial stress, including the effects of surface stress.