Onset of fracture in random heterogeneous particle chains

TL;DR

This paper introduces a new method to determine the onset of fracture in heterogeneous particle chains, linking discrete models to continuum limits and simplifying threshold calculations for practical applications.

Contribution

It proposes a novel notion of fracture in discrete systems and demonstrates that its continuum limit matches the $ ext{Gamma}$-convergence results, simplifying threshold computation.

Findings

Discrete fracture notion aligns with continuum $ ext{Gamma}$-limit thresholds

New method simplifies fracture threshold calculations

Potential for practical application in heterogeneous systems

Abstract

In mechanical systems it is of interest to know the onset of fracture in dependence of the boundary conditions. Here we study a one-dimensional model which allows for an underlying heterogeneous structure in the discrete setting. Such models have recently been studied in the passage to the continuum by means of variational convergence ($\Gamma$-convergence). The $\Gamma$-limit results determine thresholds of the boundary condition, which mark a transition from purely elastic behaviour to the occurrence of a crack. In this article we provide a notion of fracture in the discrete setting and show that its continuum limit yields the same threshold as that obtained from the $\Gamma$-limit. Since the calculation of the fracture threshold is much easier with the new method, we see a good chance that this new approach will turn out useful in applications.