Calculus of Variations and Geometric Measure Theory

A. Bach - L. Sommer

A $\Gamma$-convergence result for fluid-filled fracture propagation

created by bach on 29 May 2019
modified on 07 Oct 2020


Published Paper

Inserted: 29 may 2019
Last Updated: 7 oct 2020

Journal: ESAIM Math. Model. Numer. Anal.
Volume: 54
Year: 2020


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 $p\in W^{1,\infty}(\Omega)$ we show that functionals of the form \[E(u)=\int_\Omega e(u)\colon\mathbb{C}e(u)+p\nabla\cdot u+\langle\nabla p,u\rangle\,\mathrm{dx}+\mathcal{H}^{n-1}(J_{u}),\quad u\in GSBD(\Omega)\cap L^1(\Omega;\mathbb{R}^n)\] can be approximated in terms of $\Gamma$-convergence by a sequence of phase-field functionals, which are suitable for numerical simulations. The $\Gamma$-convergence result is complemented by a numerical example where the phase-field model is implemented using a Discontinuous Galerkin Discretization.