# Analysis of staggered evolutions for nonlinear energies in phase field fracture

## TL;DR

This paper investigates the behavior of staggered minimization schemes for nonlinear phase field energies in fracture mechanics, revealing how discrete evolutions converge to continuous models and exhibit complex behaviors at discontinuities.

## Contribution

It provides a novel analysis of the convergence of staggered schemes to balanced viscosity evolutions in phase field fracture models, including discontinuity behavior.

## Key findings

- Discrete evolutions converge to balanced viscosity solutions.
- Discontinuous times can exhibit alternate behaviors.
- Energy balance analysis reveals complex evolution dynamics.

## Abstract

We consider a class of separately convex phase field energies employed in fracture mechanics, featuring non-interpenetration and a general softening behavior. We analyze the time-discrete evolutions generated by a staggered minimization scheme, where fracture irreversibility is modeled by a monotonicity constraint on the phase field variable. After recasting the staggered scheme by means of gradient flows, we characterize the time-continuous limits of the discrete solutions in terms of balanced viscosity evolutions, parametrized by their arc-length with respect to the L2-norm (for the phase field) and the H1-norm (for the displacement field). By a careful study of the energy balance we deduce that time-continuous evolutions may still exhibit an alternate behavior in discontinuity times.