A Γ -convergence result for fluid-filled fracture propagation
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 1003-1023.

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(u →)=∫ Ω e(u →):ℂe(u →)+p∇·u →+∇p,u →dx+ℋ n-1 (J u → ),u →∈GSBD(Ω)∩L 1 (Ω;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.

DOI : 10.1051/m2an/2020016
Classification : 49J45, 74R10, 65N30
Mots-clés : Γ-convergence, phase-field approximation, pressure-driven crack propagation, discontinuous Galerkin method 
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     title = {A $\Gamma$-convergence result for fluid-filled fracture propagation},
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     publisher = {EDP-Sciences},
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Bach, Annika; Sommer, Liesel. A $\Gamma$-convergence result for fluid-filled fracture propagation. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 54 (2020) no. 3, pp. 1003-1023. doi : 10.1051/m2an/2020016. http://www.numdam.org/articles/10.1051/m2an/2020016/

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