First contact breaking distributions in strained disordered crystals

TL;DR

This paper derives exact probability distributions for the strain at which the first stress drop occurs in disordered crystals, linking contact breaking events to convex polytope volumes and validating results with simulations.

Contribution

It introduces a novel theoretical framework connecting first contact breaking strains to convex polytope volumes, validated by numerical simulations.

Findings

Exact distribution of first stress drop strains derived.

Polytope volume computation matches simulation data.

Uncorrelated contact breaking assumption reproduces distributions.

Abstract

We derive exact probability distributions for the strain ($\epsilon$) at which the first stress drop event occurs in uniformly strained disordered crystals, with quenched disorder introduced through polydispersity in particle sizes. We characterize these first stress drop events numerically as well as theoretically, and identify them with the first contact breaking event in the system. Our theoretical results are corroborated with numerical simulations of quasistatic volumetric strain applied to disordered near-crystalline configurations of athermal soft particles. We develop a general technique to determine the $distribution$ of strains at which the first stress drop events occur, through an exact mapping between the cumulative distribution of first contact breaking events and the volume of a convex polytope whose dimension is determined by the number of defects $N_d$ in the system. An exact numerical computation of this polytope volume for systems with small numbers of defects displays a remarkable match with the distribution of strains generated through direct numerical simulations. Finally, we derive the distribution of strains at which the first stress drop occurs, assuming that individual contact breaking events are uncorrelated, which accurately reproduces distributions obtained from direct numerical simulations.