An elliptic approximation for phase separation in a fractured material

TL;DR

This paper introduces an elliptic approximation method for modeling phase separation and fracture in elastic materials, using Gamma-convergence to connect a modified Cahn-Hilliard energy with fracture and elastic energies.

Contribution

It presents a novel elliptic approximation framework that captures phase separation and fracture phenomena through Gamma-convergence from a combined Cahn-Hilliard and Ambrosio-Tortorelli model.

Findings

Gamma-convergence of the approximation to the sharp energy

Effective modeling of phase separation in fractured materials

Heuristic approximation of interfacial energy in cracked regions

Abstract

We consider a free-boundary and free-discontinuity energy connecting phase separation and fracture in an elastic material. The energy excludes the contribution of phase boundaries in the cracked region, providing a heuristic approximation of the interfacial energy in the current material configuration. Our primary result shows that the sharp energy may be recovered via Gamma-convergence from a modified Cahn-Hilliard energy coupled with an Ambrosio-Tortorelli-type approximation of the (linear) elastic and fracture energy.