Interacting damage models mapped onto Ising and percolation models

TL;DR

This paper introduces a class of damage models on lattices with isotropic interactions, demonstrating their isomorphism to Ising and percolation models, and explores their statistical mechanics properties and universality classes.

Contribution

It establishes a theoretical mapping between damage models with quenched disorder and well-known statistical physics models like Ising and percolation, including analysis of interactions and entropy.

Findings

Damage models with global load sharing are isomorphic to percolation.

Damage models with local load sharing are isomorphic to the Ising model.

The probability distribution over damage configurations maximizes Shannon's entropy under energetic constraints.

Abstract

We introduce a class of damage models on regular lattices with isotropic interactions, as e.g. quasistatic fiber bundles. The system starts intact with a surface-energy threshold required to break any cell sampled from an uncorrelated quenched-disorder distribution. The evolution of this heterogeneous system is ruled by Griffith's principle which states that a cell breaks when the release in elastic energy in the system exceeds the surface-energy barrier necessary to break the cell. By direct integration over all possible realizations of the quenched disorder, we obtain the probability distribution of each damage configuration at any level of the imposed external deformation. We demonstrate an isomorphism between the distributions so obtained and standard generalized Ising models, in which the coupling constants and effective temperature in the Ising model are functions of the nature of the quenched-disorder distribution and the extent of accumulated damage. In particular, we show that damage models with global load sharing are isomorphic to standard percolation theory, that damage models with local load sharing rule are isomorphic to the standard Ising model, and draw consequences thereof for the universality class and behavior of the autocorrelation length of the breakdown transitions corresponding to these models. We also treat damage models having more general power-law interactions, and classify the breakdown process as a function of the power-law interaction exponent. Last, we also show that the probability distribution over configurations is a maximum of Shannon's entropy under some specific constraints related to the energetic balance of the fracture process, which firmly relates this type of quenched-disorder based damage model to standard statistical mechanics.