Validating Griffith fracture propagation in the phase-field approach to fracture: The case of Mode III by means of the trousers test

TL;DR

This work addresses Mode III fracture propagation within the Griffith framework by validating a phase-field approach against classical Griffith energy balance predictions. It combines full-field finite-element simulations (to compute the energy release rate $G$ and expose limitations of the Rivlin-Thomas formula $G_{RT}$) with a Neo-Hookean phase-field model that incorporates strength and toughness, enabling simultaneous nucleation and propagation analysis. The key finding is that the phase-field formulation can reproduce Griffith-style propagation in Mode III and exposes significant inaccuracies in $G_{RT}$ for standard trousers-test geometries, where bending and twist of the legs render $G$ different from the RT estimate. The results demonstrate the practical viability of phase-field models for Mode III fracture propagation and advocate their use in validating fracture analyses that historically relied on simplified RT/Greensmith approaches.

Abstract

At present, there is an abundance of results showing that the phase-field approach to fracture in elastic brittle materials -- when properly accounting for material strength -- describes the \emph{nucleation} of fracture from large pre-existing cracks in a manner that is consistent with the Griffith competition between bulk deformation energy and surface fracture energy. By contrast, results that demonstrate the ability of this approach to describe Griffith fracture \emph{propagation} are scarce and restricted to Mode I. Aimed at addressing this lacuna, the main objective of this paper is to show that the phase-field approach to fracture describes Mode III fracture propagation in a manner that is indeed consistent with the Griffith energy competition. This is accomplished via direct comparisons between phase-field predictions for fracture propagation in the so-called \emph{trousers} \emph{test} and the corresponding results that emerge from the Griffith energy competition. The latter are generated from full-field finite-element solutions that -- as an additional critical contribution of this paper -- also serve to bring to light the hitherto unexplored limitations of the classical Rivlin-Thomas-Greensmith formulas that are routinely used to analyze the trousers test.

Figures (9)

• Figure 1: Schematic of the trousers test. The dimensions in the initial configuration are such that $B\ll H<A < L$. The two bottom ends of the trousers legs are held firmly by stiff grips and then pulled apart either by an applied force $P(t)$ or by an applied deformation $l(t)$.
• Figure 2: (a) One of the FE meshes used to compute the energy release rate (\ref{['FE-G']}) for the Neo-Hookean trousers tests. (b,c) Contour plots of the stress components $S_{22}({\bf X},t)$ and $S_{21}({\bf X},t)$ over the deformed configuration of the specimen shown in part (a) at the applied separation $l(t)=106$ mm between the grips. The figures pertain to the specimen with crack length $A=50$ mm.
• Figure 3: Comparison between the energy release rate $G$ computed from full-field FE results (solid line) and the Rivlin-Thomas approximation $G_{RT}$ for the Neo-Hookean trousers test with crack length $A=50$ mm. The results are shown as a function of the separation $l(t)$ between the grips.
• Figure 4: Relative error $(G_{RT}-G)/G$ of the Rivlin-Thomas approximation $G_{RT}$ with respect to the energy release rate $G$ computed from full-field FE results for the Neo-Hookean trousers test with crack length $A=50$ mm. The result is shown as a function of $G$.
• Figure 5: Comparisons between the two different contributions in the energy release rate $G=P(t)\partial l/\partial \mathrm{\Gamma}|_{P}-\partial \mathcal{W}/\partial \mathrm{\Gamma}|_{P}$ computed from full-field FE results (solid lines) and from the Rivlin-Thomas approximations for the Neo-Hookean trousers test with crack length $A=50$ mm. (a) The change in stored elastic energy $-\partial \mathcal{W}/\partial \mathrm{\Gamma}|_{P}$. (b) The change in the work done by the external forces $P(t)\partial l/\partial \mathrm{\Gamma}|_{P}$. The results are shown as a function of the force $P(t)$ at the grips.

Theorems & Definitions (1)

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