Irreversibility and alternate minimization in phase field fracture: a viscosity approach

TL;DR

This paper analyzes the convergence of an alternate minimization algorithm in phase field fracture models, incorporating viscosity and irreversibility constraints, and studies the limit as viscosity vanishes.

Contribution

It introduces a viscous regularization in an alternating minimization scheme for phase field fracture and proves convergence to viscous and then non-viscous evolutions.

Findings

Convergence of the scheme to a viscous evolution for positive viscosity.

Analysis of the vanishing viscosity limit as viscosity approaches zero.

Abstract

This work is devoted to the analysis of convergence of an alternate (staggered) minimization algorithm in the framework of phase field models of fracture. The energy of the system is characterized by a nonlinear splitting of tensile and compressive strains, featuring non-interpenetration of the fracture lips. The alternating scheme is coupled with an $L^{2}$-penalization in the phase field variable, driven by a viscous parameter~$\delta>0$, and with an irreversibility constraint, forcing the monotonicity of the phase field only w.r.t.~time, but not along the whole iterative minimization. We show first the convergence of such a scheme to a viscous evolution for $\delta>0$ and then consider the vanishing viscosity limit $\delta\to 0$.