Sharp-Interface Limit of the Cahn-Hilliard-Biot Equations

TL;DR

This paper derives the sharp-interface limit of the Cahn-Hilliard-Biot equations, revealing how phase interfaces evolve with decreasing diffuse-interface width, integrating elasticity, fluid flow, and interface curvature effects.

Contribution

It provides a formal derivation of the sharp-interface limit for the coupled Cahn-Hilliard-Biot equations using asymptotic expansions, highlighting interface conditions and phase evolution.

Findings

Interface conditions include continuous displacement and pore pressure, balanced volumetric fluid content and normal stress.

Normal velocity of the interface relates to the jump in phase-field potential derivative.

Numerical experiments show phase evolution is consistent as diffuse-interface width decreases.

Abstract

In this letter, we derive the sharp-interface limit of the Cahn-Hilliard-Biot equations using formal matched asymptotic expansions. We find that in each sub-domain, the quasi-static Biot equations are obtained with domain-specific material parameters. Moreover, across the interface, material displacement and pore pressure are continuous, while volumetric fluid content and normal stress are balanced. By utilizing the energy of the system, the phase-field potential is shown to be influenced by the curvature, along with contributions from both flow and elasticity at the interface. The normal velocity of the interface is proportional to the jump in normal derivative of the phase-field potential across the interface. Finally, we present a numerical experiment that demonstrates how the location of each phase evolves consistently as the diffuse-interface width parameter becomes smaller; only the width of the diffuse interface changes.