A phase-field model for fractures in nearly incompressible solids

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.

Numerical Studies of Different Mixed Phase-Field Fracture Models for Simulating Crack Propagation in Punctured EPDM Strips

“…Phase-field fracture modeling in mixed form can be used to simulate cracks in nearly incompressible materials [14]. The work on hand focuses on the crack paths in punctured EPDM strips where the underlying phase-field fracture model differs in terms of three different energy functional definitions (Ambrosio-Tortorelli functionals AT2 and AT1 and the model of Wu [20]) and two different stress splitting approach (Miehe et al [15] and Amor et al [4]).…”

“…For the spatial discretization, we employ a Galerkin finite element method in each incremental step, where the domain Ω is partitioned into quadrilaterals. Stable Taylor-Hood elements with biquadratic shape functions (Q 2 ) for the displacement field u and bilinear shape functions (Q 1 ) for the pressure variable p and the phase-field variable ϕ are used [10,14]. The implementation is embedded in the finite element library deal.II [6].…”

“…The commonly used quasi-static phase-field fracture model in its classical displacement-based formulation fails due to locking effects if the considered solid is nearly incompressible. To allow simulating crack growth in rubber-like materials like EPDM rubber, this classical phase-field fracture model is extended to a mixed form of the solid-displacement equation resulting in two unknowns: a displacement field and a hydro-static pressure variable [14]. To fulfill an inf-sup condition, Taylor-Hood elements are employed for the displacement-pressure system.…”

Crack path comparisons of a mixed phase‐field fracture model and experiments in punctured EPDM strips

“…For the spatial discretization, we employ a Galerkin finite element method in each incremental step, where the domain Ω is partitioned into quadrilaterals. To fulfill a discrete inf-sup condition, stable Taylor-Hood elements with biquadratic shape functions (Q 2 ) for the displacement field u and bilinear shape functions (Q 1 ) for the pressure variable p and the phase-field variable ϕ are used as in [4]. The open-source code to derive the proposed problem formulation is available at https://github.com/tjhei/cracks which is embedded in deal.II [12].…”

A comparative review of peridynamics and phase-field models for engineering fracture mechanics

“…On the other hand, accessing raw experimental data is a different challenge [370], and one of the Sandia Fracture Challenges could be used to validate peridynamics and phase-field models against the same experimental data. [208,209] and PD [103,375,376]. Note that modeling of hyper elastic material behavior is challenging for any numerical method since the constitutive material law must reflect material behaviors such as a neo-Hookean [377] or Mooney-Rivlin [378] solids.…”