Statistical Mechanics of Cracks: Thermodynamic Limit, Fluctuations, Breakdown, and Asymptotics of Elastic Theory

TL;DR

This paper models brittle fracture using elastic theory, revealing a first-order transition to fracture, analyzing crack fluctuations, and deriving asymptotic behaviors of elastic coefficients at finite temperatures.

Contribution

It introduces a thermodynamic framework for crack formation, proves shape independence of energy release, and calculates fracture lifetime and elastic coefficient divergences.

Findings

Fracture transition resembles a first-order liquid-gas transition.

Energy release for cracks is shape-independent under certain boundary conditions.

High-order elastic coefficients diverge asymptotically, determined by linear theory.

Abstract

We study a class of models for brittle fracture: elastic theory models which allow for cracks but not for plastic flow. We show that these models exhibit, at all finite temperatures, a transition to fracture under applied load similar to that at a first order liquid-gas transition. We study this transition at low temperature for small tension. We discuss the appropriate thermodynamic limit of these theories: a large class of boundary conditions is identified for which the energy release for a crack becomes independent of the macroscopic shape of the material. Using the complex variable method in a two-dimensional elastic theory, we prove that the energy release in an isotropically stretched material due to the creation of an arbitrary curvy cut is the same to cubic order as the energy release for the straight cut with the same end points. We find the normal modes and the energy spectrum for crack shape fluctuations and for crack surface phonons, under a uniform isotropic tension. For small uniform isotropic tension in two dimensions we calculate the essential singularity associated with fracturing the material in a saddle point approximation including quadratic fluctuations. This singularity determines the lifetime of the material (half-life for fracture), and also determines the asymptotic divergence of the high-order corrections to the zero temperature elastic coefficients. We calculate the asymptotic ratio of the high-order elastic coefficients of the inverse bulk modulus and argue that the result is unchanged by nonlinearities --- the ratio of the high-order nonlinear terms are determined solely by the linear theory.