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An assessment of phase field fracture: crack initiation and growth
P. K. Kristensen, C. F. Niordson, E. Martínez-Pañeda
TL;DR
The paper examines whether phase field fracture models reproduce classical fracture-mechanics predictions across crack initiation, stable growth, and size effects. It reports agreement with Griffith-based predictions, while emphasizing dependence on modelling and constitutive choices.
Problem
The paper addresses whether phase field fracture models can predict crack initiation and growth consistently with classical fracture mechanics.
Method
The study implements phase field fracture models and evaluates crack nucleation and growth using fracture-mechanics boundary-value problems.
Results
Phase field models predict crack initiation near G ≈ Gc and stable crack growth in agreement with beam theory and Griffith’s energy balance.
Takeaways & Limitations
Phase field fracture can deliver accurate approximations of classical fracture-mechanics predictions, including for solids containing cracks.
Takeaways & Limitations
The reported findings concern solids containing cracks, and the approximation depends on modelling and constitutive choices.
Abstract
from arXiv · showhide
The phase field paradigm, in combination with a suitable variational structure, has opened a path for using Griffith's energy balance to predict the fracture of solids. These so-called phase field fracture methods have gained significant popularity over the past decade, and are now part of commercial finite element packages and engineering fitness-for-service assessments. Crack paths can be predicted, in arbitrary geometries and dimensions, based on a global energy minimisation - without the need for \textit{ad hoc} criteria. In this work, we review the fundamentals of phase field fracture methods and examine their capabilities in delivering predictions in agreement with the classical fracture mechanics theory pioneered by Griffith. The two most widely used phase field fracture models are implemented in the context of the finite element method, and several paradigmatic boundary value problems are addressed to gain insight into their predictive abilities across all cracking stages; both the initiation of growth and stable crack propagation are investigated. In addition, we examine the effectiveness of phase field models with an internal material length scale in capturing size effects and the transition flaw size concept. Our results show that phase field fracture methods satisfactorily approximate classical fracture mechanics predictions and can also reconcile stress and toughness criteria for fracture. The accuracy of the approximation is however dependent on modelling and constitutive choices; we provide a rationale for these differences and identify suitable approaches for delivering phase field fracture predictions that are in good agreement with well-established fracture mechanics paradigms.
1. Introduction
The paper reviews phase field fracture as a variational framework for predicting crack evolution from Griffith’s energy balance. It assesses agreement with classical fracture mechanics across crack initiation, stable growth, and size effects.
- Phase field fracture provides a mathematical and computational framework for Griffith’s energy balance.
- Crack nucleation, growth, merging, branching, and arrest can be predicted without ad hoc criteria in arbitrary dimensions and geometries.
- The paper reviews phase field fracture fundamentals and examines agreement with classical fracture mechanics theory developed by Griffith and his contemporaries.
- The finite element study investigates crack initiation using a remote K-field, stable growth using a double-cantilever beam, and size effects using a finite plate with an edge crack.
2. A variational framework for Griffith’s energy balance
The variational framework replaces the unknown crack surface with a diffuse phase field while preserving Griffith’s energetic structure. A length scale makes the formulation computationally tractable and introduces finite material strength for nonzero values.
- Griffith’s balance equates the energy reduction from crack growth with the surface energy required to create new free surfaces.
- Direct minimisation is computationally difficult because the crack surface Γ is unknown.
- Phase field regularisation: An auxiliary phase field φ smears a sharp crack interface into a diffuse region and represents intact and fully cracked material with distinct values.
- Phase field regularisation: The phase field regularisation converts crack-surface work into a volume integral, enabling numerical computation while degrading stiffness as φ approaches the cracked phase.
- As ℓ→0+, the regularised functional converges to Griffith’s functional; for ℓ>0+, ℓ becomes a material length governing finite strength.
- Constitutive theory: The AT2 choice uses w(φ)=φ^2, whereas AT1 uses w(φ)=φ and introduces an elastic regime before damage onset.
3. Numerical implementation
The finite element implementation solves a coupled displacement–phase-field problem while enforcing damage irreversibility. It uses history-field methods and a monolithic BFGS strategy to obtain the numerical solution.
- Damage irreversibility: The implementation introduces a history field to ensure damage irreversibility and prevent crack healing.
- Finite element discretisation: The weak formulation is discretised into residuals and consistent tangent stiffness matrices for the displacement and phase-field variables.
- Solution schemes: Monolithic schemes solve displacement and phase-field subsystems simultaneously, whereas staggered schemes solve them sequentially.
- Solution schemes: Although the total potential energy is non-convex jointly in u and φ, fixed-variable subproblems are convex, making staggered schemes robust.
- Solution schemes: The reported implementation uses a BFGS-based monolithic approach to obtain robust and efficient solutions.
4. Results
Across boundary-layer, double-cantilever-beam, and size-effect problems, phase field predictions generally agree with classical fracture mechanics, but accuracy depends on modelling choices.
- Overview: The study examines crack initiation, stable propagation, and size effects using three paradigmatic boundary value problems.The problems include remote-energy-release-rate initiation, double-cantilever-beam propagation, and transition-flaw-size analysis.
- Initiation of crack growth: Mesh-converged initiation predictions approach Gc, with agreement improving when the initial crack is introduced through the phase field.Both AT1 and AT2 attain G/Gc close to unity for phase-field-induced cracks, whereas geometrically defined cracks can yield G/Gc ≈1.3.
- Initiation of crack growth: AT2 predicts higher initiation G values than AT1, while prescribing a geometric crack adds an energy barrier that explains its larger initiation value.The phase field must develop around a geometric crack tip, whereas it is already developed along faces when the crack is phase-field-induced.
- Stable crack growth: Double-cantilever-beam simulations reproduce the analytical stable-cracking shape, with closer quantitative agreement for crack-density-based extension measurements and AT1.All constitutive choices slow crack growth relative to the analytical solution, and results show sensitivity to ℓ/H.
- Size effects and the transition flaw size: An internal length scale produces a transition from Griffith-like toughness-controlled failure for large cracks to strength-driven failure below the transition flaw size.The models reconcile stress and toughness criteria, with agreement slightly better for AT2 in the limiting cases discussed.
5. Discussion
The discussion finds that phase field models generally approximate Griffith-based fracture predictions for crack initiation, stable growth, and size effects, but accuracy depends on modelling choices and crack configuration.
- Crack initiation: Prescribing an initial sharp crack can delay fracture because constraining the phase field near the crack surface requires additional energy around the crack tip.The discussion attributes this delay to the natural boundary condition and the energy needed to build a highly constrained phase-field region.
- Modelling choices: Different crack-density constitutive choices can produce different approximation quality, although the AT1–AT2 initiation difference remains small for phase-field-induced cracks.The discussion compares this dependence with constitutive-law effects in cohesive-zone models.
- Crack initiation: Phase field models predict crack initiation near the Griffith condition G ≈ Gc for phase-field-induced cracks, with AT1 and AT2 giving similar results.AT2 predicts initiation at a slightly larger G, while damage irreversibility has negligible effect under the considered long, sharp-crack conditions.
- Stable crack growth: Stable crack-growth predictions agree satisfactorily with beam theory and fracture-energy balance, although quantitative differences depend on crack-extension measurement and constitutive model.The best result used AT1 and measured crack extension through the crack-density function.
- Size effects: The internal length scale enables phase field models to capture the transition flaw size and the shift from toughness-driven to strength-driven failure.This behavior is obtained for both AT1 and AT2, with slightly better performance reported for AT2.
- Scope and limitations: Phase field models can reconcile toughness and strength without an elastic phase, but their applicability is constrained for pristine samples, blunted notches, and ductile solids.The conclusions discussed for existing long cracks may not apply to crack nucleation from defects, and a Griffith-like balance is questionable for ductile materials because plastic flow produces localized heat.
6. Conclusions
The reviewed phase field models can reproduce key classical fracture-mechanics predictions across crack initiation, stable growth, and size effects, but accuracy depends on modelling choices. Defining initial cracks through the phase field and measuring extension with the crack-density functional generally improves agreement.
- Scope and approach: The study reviews widely used phase field fracture models and evaluates their agreement with classical fracture-mechanics predictions using variational energy minimisation.The analysis addresses crack nucleation and growth through several relevant boundary value problems.
- Crack initiation: Accurate initiation at G = Gc requires prescribing the initial phase-field value; geometrically induced cracks overestimate the critical energy release rate.The mismatch is attributed to the natural boundary condition ∇φ · n = 0 and an energy barrier near the crack tip.
- Stable crack growth: Stable crack-growth predictions agree satisfactorily with beam theory and Griffith’s energy balance, with improved agreement when extension is measured using the crack-density functional.For the AT1 model, defining the crack through the phase field further improves the agreement.
- Size effects: A constant ℓ > 0 captures the vanishing effect of small flaws on fracture strength and reconciles toughness and strength criteria in both AT1 and AT2 models.This size effect is not captured by Griffith’s theory alone.
- Modelling choices: For the conditions examined, AT1 versus AT2 plays a secondary role, while suitable crack definition and extension measurement noticeably improve prediction accuracy.The authors conclude that phase field models can deliver accurate fracture predictions when appropriate modelling choices are made.