Derivation of a variational model for brittle fracture from a random heterogeneous particle chain

TL;DR

This paper derives a variational model for brittle fracture by analyzing the continuum limit of a random particle chain, revealing how microscopic heterogeneities influence macroscopic fracture behavior.

Contribution

It introduces a new formula linking microstructure and boundary conditions to fracture and elastic behavior in brittle materials, using stochastic homogenization techniques.

Findings

Either elastic behavior or crack is energetically favored.

The model explicitly depends on microstructure and boundary conditions.

Mathematical analysis employs $ ext{Gamma}$-convergence and ergodic theorems.

Abstract

A mathematical continuum limit of the interaction energy of a random particle chain is shown to yield new insight into the effect of microscopic heterogeneities on macroscopic fracture laws in brittle materials. We derive a formula which yields that either elastic behaviour or a crack is energetically preferred. The formula explicitly shows the dependence on the boundary condition and the microstructure of the chain. The mathematical analysis is based on a variational convergence $\Gamma$-convergence of convex-concave potentials together with ergodic theorems which are common tools in stochastic homogenization.