A Γ-convergence result for fluid-filled fracture propagation

¹ Zentrum Mathematik, M7, TU München, Boltzmannstr. 3, 85748 Garching, Germany
² Angewandte Mathematik, WWU Münster, Einsteinstr. 62, 48149 Münster, Germany

^* Corresponding author: annika.bach@ma.tum.de

Received: 5 February 2019
Accepted: 3 March 2020

Abstract

In this paper we provide a rigorous asymptotic analysis of a phase-field model used to simulate pressure-driven fracture propagation in poro-elastic media. More precisely, assuming a given pressure p ∈ W ^1,∞ (Ω) we show that functionals of the form $$E\left(\stackrel{}{u}\right)={\int }_{\mathrm{\Omega }}\mathrm{}e\left(\stackrel{}{u}\right):\mathbb{C}e\left(\stackrel{}{u}\right)+p\nabla \cdot \stackrel{}{u}+\langle \nabla p,\stackrel{}{u}\rangle \mathrm{d}x+{\mathcal{H}}^{n-1}\left(J_{\stackrel{}{u}}\right), \stackrel{}{u}\in \mathrm{G}\mathrm{SBD}\left(\mathrm{\Omega }\right)\cap L^1(\mathrm{\Omega };{\mathbb{R}}^n)$$can be approximated in terms of Γ-convergence by a sequence of phase-field functionals, which are suitable for numerical simulations. The Γ-convergence result is complemented by a numerical example where the phase-field model is implemented using a Discontinuous Galerkin Discretization.