# A phase-field model for fractures in incompressible solids

## TL;DR

This paper introduces a phase-field model for simulating fractures in incompressible solids, addressing volume-locking issues by using a mixed displacement-pressure formulation and stable finite element discretization.

## Contribution

It develops a novel phase-field fracture model for incompressible materials employing a mixed formulation with four variables and demonstrates its effectiveness through numerical studies.

## Key findings

- Stable finite element choices influence results
- Mesh refinement improves accuracy
- Approaching incompressibility affects fracture simulation

## Abstract

Within this work, we develop a phase-field description for simulating fractures in incompressible materials. Standard formulations are subject to volume-locking when the solid is (nearly) incompressible. We propose an approach that builds on a mixed form of the displacement equation with two unknowns: a displacement field and a hydro-static pressure variable. Corresponding function spaces have to be chosen properly. On the discrete level, stable Taylor-Hood elements are employed for the displacement-pressure system. Two additional variables describe the phase-field solution and the crack irreversibility constraint. Therefore, the final system contains four variables: displacements, pressure, phase-field, and a Lagrange multiplier. The resulting discrete system is nonlinear and solved monolithically with a Newton-type method. Our proposed model is demonstrated by means of several numerical studies based on two numerical tests. First, different finite element choices are compared in order to investigate the influence of higher-order elements in the proposed settings. Further, numerical results including spatial mesh refinement studies and variations in Poisson's ratio approaching the incompressible limit, are presented.