How to introduce an initial crack in phase field simulations to accurately predict the linear elastic fracture propagation threshold?

TL;DR

This study addresses how the method chosen to initialize a crack affects the propagation threshold in variational phase field fracture simulations, benchmarking against a sharp Griffith model via a quasi-static, path-following approach. It formalizes the phase-field regularization with an AT1-type energy and demonstrates that initial cracks must be represented at least one element wide and fully damaged along the crack boundary to prevent artificial energy penalties and overestimation of the critical load. Through a SENT benchmark, eight initialization techniques are compared, revealing that infinitely thin (T0) cracks induce spurious peaks while one-element-wide (T1) implementations—whether GEO-T1-WHL or PHA-T1—yield unbiased predictions with comparable computational cost. The work provides practical guidance for reliable phase field simulations, outlines extensions to curved cracks and history-variable models, and reinforces the importance of preserving $Γ$-convergence fidelity in crack evolution predictions.

Abstract

Variational phase field fracture models are now widely used to simulate crack propagation in structures. A critical aspect of these simulations is the correct determination of the propagation threshold of pre-existing cracks, as it highly relies on how the initial cracks are implemented. While prior studies briefly discuss initial crack implementation techniques, we present here a systematic investigation. Various techniques to introduce initial cracks in phase field fracture simulations are tested, from the crack explicit meshing to the replacement by a fully damaged phase field, including different variants for the boundary conditions. Our focus here is on phase field models aiming to approximate, in the $Γ$-convergence limit, Griffith quasi-static propagation in the framework of Linear Elastic Fracture Mechanics. Therefore, a sharp crack model from classic linear elastic fracture mechanics based on Griffith criterion is the reference in this work. To assess the different techniques to introduce initial cracks, we rely on path-following methods to compute the sharp crack and the phase field smeared crack solutions. The underlying idea is that path-following ensures staying at equilibrium at each instant so that any difference between phase field and sharp crack models can be attributed to numerical artifacts. Thus, by comparing the results from both models, we can provide practical recommendations for reliably incorporating initial cracks in phase field fracture simulations. The comparison shows that an improper initial crack implementation often requires the smeared crack to transition to a one-element-wide phase band to adequately represent a displacement jump along a crack. This transition increases the energy required to propagate the crack, leading to a significant overshoot in the force-displacement response. The take-home message is that to predict the propagation threshold accurately and avoid artificial toughening; the crack must be initialized either setting the phase field to its damage state over a one-element-wide band or meshing the crack explicitly as a one-element-wide slit and imposing the fully cracked state on the crack surface.

Figures (7)

• Figure 1: Illustration of the different crack initialization techniques. The green line on the GEO-T0 illustrations corresponds to the line where the node are duplicated. The color corresponds to the phase field with blue being uncracked ($\alpha=0$) and red being fully cracked ($\alpha=1$). The abbreviation BC corresponds to boundary condition.
• Figure 2: Illustration of the definition of phase field initial crack (PHA). The red color corresponds to the crack ($\alpha$) whereas the blue color corresponds to the uncracked parts ($\alpha=0$).
• Figure 3: Geometry of the SENT specimen and boundary conditions.
• Figure 4: Comparison of the phase field fracture solutions with the different crack initialization techniques for an infinitely thin initial crack (T0). The top part shows the force-displacement curves along with the reference solution from the sharp crack model. The lower part shows the crack phase field after crack propagation with blue being uncracked ($\alpha=0$) and red being fully cracked ($\alpha=1$).
• Figure 5: Comparison of the phase field fracture solutions with the different crack initialization techniques for a one-element-wide initial crack (T1). The top part shows the force-displacement curves along with the reference solution from the sharp crack model. The lower part shows the crack phase field after crack propagation with blue being uncracked ($\alpha=0$) and red being fully cracked ($\alpha=1$).