On the relation of Gamma-convergence parameters for pressure-driven quasi-static phase-field fracture

Abstract

The proof of Γ-convergence builds the base of the well-known Ambrosio–Tortorelli functional leading to an energy functional for quasi-static phase-field fracture problems. Three parameters in a monolithic quasi-static phase-field fracture model are very relevant for the quality of the results: the length-scale ϵ, the regularization parameter κ to avoid ill-posedness of the system and the discretization parameter h. The work on hand presents numerical results considering a pressure-driven cavity in 2d with two quantities of interest, the crack opening displacement and the total crack volume. The focus will be to discuss the assumptions of Γ-convergence which demand: h=o(κ)and κ=o(ϵ)and ϵ→0. An error analysis of the chosen quantities of interest allows to identify a proper setting for the three mentioned model parameters.

Keywords

• Γ-convergence
• Phase-field
• Bulk regularization
• Pressure-driven fracture
• Error analysis