A numerical assessment of phase-field models for brittle and cohesive fracture: Γ-Convergence and stress oscillations

Abstract

Article available under the terms of the CC-BY-NC-ND licence (https://creativecommons.org/licenses/by-nc-nd/4.0/) eprints@whiterose.ac.uk https://eprints.whiterose.ac.uk/ Reuse Unless indicated otherwise, fulltext items are protected by copyright with all rights reserved. The copyright exception in section 29 of the Copyright, Designs and Patents Act 1988 allows the making of a single copy solely for the purpose of non-commercial research or private study within the limits of fair dealing. The publisher or other rights-holder may allow further reproduction and re-use of this version -refer to the White Rose Research Online record for this item. Where records identify the publisher as the copyright holder, users can verify any specific terms of use on the publisher's website. This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Takedown M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPT• A numerical demonstration that in phase-field models for brittle fracture the smeared crack length does not necessarily converge to the discrete crack length upon mesh refinement.• A demonstration that the numerical results of boundary value problems that use the phase-field model for brittle fracture are very sensitive to how the boundary conditions are applied.• A proof that the phase-field model for cohesive fracture does not satisfy a two-dimensional patch test, even when the interpolation orders of the displacement field, the phase field and the crack-opening field are balanced. *Highlights (for review)M A N U S C R I P T A C C E P T E D ACCEPTED MANUSCRIPTA numerical assessment of phase-field models for brittle and cohesive fracture: Γ-convergence and stress oscillations AbstractRecently, phase-field approaches have gained popularity as a versatile tool for simulating fracture in a smeared manner. In this paper we give a numerical assessment of two types of phase-field models. For the case of brittle fracture we focus on the question whether the functional that describes the smeared crack surface approaches the functional for the discrete crack in the limiting case that the internal length scale parameter vanishes. By a one-dimensional example we will show that Γ-convergence is not necessarily attained numerically. Next, we turn attention to cohesive fracture. The necessity to have the crack opening explicitly available as input for the cohesive traction-relative displacement relation requires the independent interpolation of this quantity. The resulting three-field problem can be solved accurately on structured meshes when using a balanced interpolation of the field variables: displacements, phase field, and crac...

A convergence study of phase-field models for brittle fracture

“…A natural requirement is that the functional that describes the smeared crack surface, converges to the original functional that describes the discrete crack surface. While proofs for this so-called Γ-convergence have been given (Bellettini and Coscia, 1994;Chambolle, 2004), doubt has arisen whether Γ-convergence can actually be achieved in numerical computations where both the internal length scale and the spacing of the discretisation are small, but finite (Vignollet et al, 2014;May et al, 2015).…”

“…This correction has been applied in the numerical studies of Borden et al (2014) and May et al (2015).…”

“…Likely, the criterion of Equation (22) is an upper bound, and in analyses in which the phase-field is coupled to the mechanical field, smaller values for h can be expected to be necessary. A study in which a coupling with the mechanical field was maintained, can be found in May et al (2015), and Equation (22) was respected as a minimum requirement. The simple, one-dimensional boundary value problem of Figure 2 was considered.…”

“…Π ℓ,h converges to Π for ℓ → 0 under the condition that h ≪ ℓ, where h is the mesh spacing. Doubt has been cast on whether Γ-convergence can be achieved in actual computations, since, using the phase-field model for brittle fracture as developed by Miehe et al (2010a,b), Vignollet et al (2014) and May et al (2015) have shown by numerical analyses of some simple boundary value problems that there exists a ratio ℓ/h for which the difference |Γ ℓ − Γ| attains a minimum. Moreover, at this minimum the error…”

A Phase Field Model for Rate-Dependent Ductile Fracture

“…In addition, the finite element implementation becomes straightforward. In the phase field fracture model, the main idea is to evaluate a sharp discontinuity Γ by a smeared surface Γ l [6]. For this end, an auxiliary field, called order phase field variable, is introduced to represent the crack.…”