Fracture of disordered solids in compression as a critical phenomenon: III. Analysis of the localization transition

TL;DR

This paper analyzes the critical localization transition in disordered solids under compression, revealing how crack bands form and how the transition depends on material properties and stress regimes, with implications for understanding failure.

Contribution

It introduces an analytical approach to study the localization transition, showing its continuous nature and dependence on material parameters, extending previous models of fracture in disordered solids.

Findings

Localization occurs in the softening regime after peak stress for large bulk modulus.

Localization occurs in the hardening regime before peak stress for small bulk modulus.

The crack correlation length diverges as the transition is approached, indicating critical behavior.

Abstract

The properties of the Hamiltonian developed in Paper II are studied showing that at a particular strain level a ``localization'' phase transition occurs characterized by the emergence of conjugate bands of coherently oriented cracks. The functional integration that yields the partition function is then performed analytically using an approximation that employs only a subset of states in the functional neighborhood surrounding the most probable states. Such integration establishes the free energy of the system, and upon taking the derivatives of the free energy, the localization transition is shown to be continuous and to be distinct from peak stress. When the bulk modulus of the grain material is large, localization always occurs in the softening regime following peak stress, while for sufficiently small bulk moduli and at sufficiently low confining pressure, the localization occurs in the hardening regime prior to peak stress. In the approach to localization, the stress-strain relation for the whole rock remains analytic, as is observed both in experimental data and in simpler models. The correlation function of the crack fields is also obtained. It has a correlation length characterizing the aspect ratio of the crack clusters that diverges as (\xi \sim (\ep_{c}-\ep)^{-2}) at localization.