Fracture of disordered solids in compression as a critical phenomenon: I. Statistical mechanics formalism

TL;DR

This paper develops a statistical mechanics framework for understanding fracture localization in disordered solids under compression, treating it as a critical phenomenon through ensemble averages and crack disorder analysis.

Contribution

It introduces a novel statistical mechanics formalism for fracture in disordered solids, based on ensemble averages and crack state probabilities derived from energy constraints.

Findings

Establishes a thermodynamics-based model for crack distributions.

Defines a partition function and temperature for fracture systems.

Provides a probabilistic description of crack states in disordered materials.

Abstract

This is the first of a series of three articles that treats fracture localization as a critical phenomenon. This first article establishes a statistical mechanics based on ensemble averages when fluctuations through time play no role in defining the ensemble. Ensembles are obtained by dividing a huge rock sample into many mesoscopic volumes. Because rocks are a disordered collection of grains in cohesive contact, we expect that once shear strain is applied and cracks begin to arrive in the system, the mesoscopic volumes will have a wide distribution of different crack states. These mesoscopic volumes are the members of our ensembles. We determine the probability of observing a mesoscopic volume to be in a given crack state by maximizing Shannon's measure of the emergent crack disorder subject to constraints coming from the energy-balance of brittle fracture. The laws of thermodynamics, the partition function, and the quantification of temperature are obtained for such cracking systems.