Modelling of crack propagation
in strongly anisotropic media
using phase-field fracture and LEFM

12th European Solid Mechanics Conference, Lyon, France

F. Loiseau
flavien.loiseau@ensta.fr

IMSIA, CNRS, EDF, ENSTA Paris,
Institut Polytechnique de Paris,
91120 Palaiseau, France

X. Zhai
xinyuan.zhai@ensta.fr
V. Lazarus
veronique.lazarus@ensta.fr

July 7, 2025

Context : Fracture in 3D-printed structures

Global objective in my postdoc

Modelling and simulating quasi-static crack propagation in 3D-printed structures

Metals (Direct Energy Deposition)

Duplex stainless steel (Roucou et al., 2023)

Polymers (Fused Deposit Modelling)

Polycarbonate CT specimen (Zhai et al., 2025)

Crack propagation in anisotropic media

Objective in this presentation

Model and simulate crack propagation in strongly anisotropic material using

  1. Linear Elastic Fracture Mechanics
  1. Variational phase-field models

Reference problem: Compact Tension (CT) test

Expected output of the model
Force-displacement curve and the crack path.

Load factor \(\lambda\)
Dimensionless parameter controlling the load amplitude.

Assumptions
2D problem (plane strain)
Isotropic elasticity (\(E = 1.9\) GPa, \(\nu = 0.34\))

Reference data (Corre & Lazarus, 2021; Zhai et al., 2025)
Experiments on 3D-printed polycarbonate specimens.

Linear Elastic Fracture Mechanics

Modelling crack propagation (1/3)

Straight crack, isotropic fracture toughness

Crack propagation threshold

Griffith (1921) defined the energy release rate \[ G = - \frac{\partial \mathcal{P}}{\partial a}. \] Griffith (1921) criterion states that the crack propagates when it reaches a critical value \[G = G_c.\] Having \(G = \lambda^2 \bar{G}\), the propagation occurs for the critical load factor \[\lambda_c = \sqrt{G_c/ \bar{G}}.\]

Modelling crack propagation (2/3)

Kinked crack, isotropic fracture toughness

Crack kink direction

According to the Maximum Energy Release Rate (MERR) criterion, the kink angle is \[\varphi = \arg\max_{\varphi'} G^*(\varphi'),\] where \(G^*(\varphi)\) is the energy release rate accross a kink of angle \(\varphi\) (Amestoy & Leblond, 1992).

Combined propagation and kink criteria

Kink angle \(\varphi\) and propagation threshold \(\lambda_c\) states that the kink angle \[ \varphi = \arg\min_{\varphi'} \lambda_c (\varphi'). \]

Modelling crack propagation (3/3)

Kinked crack, anisotropic fracture toughness

To account for anisotropy fracture toughness, the critical load factor recasts as \[ \lambda_c (\varphi) = \sqrt{\frac{\color{red}{G_c(\varphi)}}{\bar{G}^*(\varphi)}}, \] and the MERR criterion remains \[ \varphi = \arg\min_{\varphi'} \lambda_c (\varphi). \] We choose a strongly anisotropic toughness \[ G_c (\varphi) = G_0 \left( 1 + \frac{\Delta G}{G_0} | \sin 2(\varphi - \theta_0) | \right). \]

Solution to the crack propagation problem

Starting from an initial crack, the following procedure is repeated iteratively until structural failure:

  1. Solve the elastic problem for \(\lambda = 1\)
  2. Calculate crack propagation direction \(\varphi\) (GMERR criterion)
    1. Compute the Stress Intensity Factors (\(I\)-integral)
    2. Apply Amestoy & Leblond (1992) formula to get \(\bar{G}^*(\varphi)\)
    3. Compute the kink angle \(\displaystyle \varphi = \arg\min_{\varphi'} \lambda_c (\varphi')\)
  3. Compute the critical load factor \(\displaystyle \lambda_c(\varphi) = \sqrt{G_c(\varphi) / \bar{G}^*(\varphi)}\)
  4. Extend the crack in the mesh by length \(\Delta a\) with a kink angle \(\varphi\)

Results with LEFM : Crack path

Experiments

Simulations

Results with LEFM : Force-displacement curve

Experiments

Simulations

Observations

  • Orders of magnitude and tendancies are accurately captured.
  • More accurate results needs to account material property scatter.
    • See the work of Zhai et al. (2025) for more experimental results.

Variational phase-field model
for strongly anisotropic fracture

I promised a bit too much … so we will focus on one specific model.

Presentation of the model

From the works of Focardi (2001) and Gerasimov & De Lorenzis (2022).

Variational Approach to Fracture (anisotropic toughness)

Similarly to Francfort & Marigo (1998), the problem recasts as an energy minimization problem \[ (\boldsymbol{u}, \Gamma) = \arg\min_{\boldsymbol{u}', \Gamma'} \mathcal{E}(\boldsymbol{u}', \Gamma'), \quad \mathcal{E}(\boldsymbol{u}, \Gamma) = \mathcal{P}(\boldsymbol{u}, \Gamma) + G_0 \int_{\Gamma} {\color{red}\phi (\boldsymbol{n})} \mathrm{d}S. \]

The function \({\color{red}\phi} : \mathbb{R}^d \to [0, +\infty[\) is a norm carrying the fracture anisotropy.


Regularization by Focardi (2001)

\[ (\boldsymbol{u}, \alpha) = \arg\min_{\boldsymbol{u}', \alpha'} \mathcal{E}(\boldsymbol{u}', \alpha'), \quad \mathcal{E}(\boldsymbol{u}, \Gamma) = \mathcal{P}(\boldsymbol{u}, \alpha) + G_0 \int_{\Omega} \frac{w(\alpha)}{\ell q} + \frac{\ell^{p-1}}{p} {\color{red}\phi^p(\nabla \alpha)} \,\mathrm{d}\boldsymbol{x}. \] where \(\alpha\) is the crack phase field (\(0=\)sane, \(1=\)cracked), \(\ell\) denotes the regularization length.

Gerasimov & De Lorenzis (2022) implemented and investigated this model in anti-plane shear.

To improve numerical convergence

OLD \[ G_c (\varphi) = G_0 \left( 1 + \frac{\Delta G}{G_0} {\color{red}| \sin 2(\varphi - \theta_0) |} \right) \]

NEW \[ G_c (\varphi) = G_0 \left( 1 + \frac{\Delta G}{G_0} {\color{red}\sin^2 2(\varphi - \theta_0)} \right) \]

Results : Crack propagation

Observations

  • Crack arrests and jumps inconsistent with LEFM (and experiments).
  • Bifurcations are induced by misalignement between crack path and FEM mesh.
  • It is a purely numerical bias, which is reinforced by strong fracture anisotropy.

Conclusion

Linear Elastic Fracture Mechanics

  • Framework for anisotropic crack propagation using the GMERR.
  • Result close to experiments to experiments in terms of
    • Crack path, Force-displacement (up to material scatter)

Variational phase-field fracture models

  • Few quantitative study for strong anisotropy in the literature.
  • Investigated a model based on the work of Focardi (2001).
  • While crack path is OK, significant issues remains :
    • Mesh bias (smooth displacement jump?, isotropic mesh?)
    • Stability of the solutions
    • Robustness of the numerical solver (line search)

Thank you for your attention !

F. Loiseau, V. Lazarus

flavien.loiseau@ensta.fr
Find this presentation at https://floiseau.github.io.


Recent publication
Loiseau, F., & Lazarus, V. (2025).
How to introduce an initial crack in phase field simulations
to accurately predict the linear elastic fracture propagation threshold?
JTCAM. https://doi.org/10.46298/jtcam.15198

References

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Corre, T., & Lazarus, V. (2021). Kinked crack paths in polycarbonate samples printed by fused deposition modelling using criss-cross patterns. International Journal of Fracture, 230(1), 19–31. https://doi.org/10.1007/s10704-021-00518-x
Focardi, M. (2001). On the variational approximation of free-discontinuity problems in the vectorial case. Mathematical Models and Methods in Applied Sciences, 11(04), 663–684. https://doi.org/10.1142/S0218202501001045
Francfort, G. A., & Marigo, J.-J. (1998). Revisiting brittle fracture as an energy minimization problem. Journal of the Mechanics and Physics of Solids, 46(8), 1319–1342. https://doi.org/10.1016/S0022-5096(98)00034-9
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Roucou, D., Corre, T., Rolland, G., & Lazarus, V. (2023). Effect of the deposition direction on fracture propagation in a Duplex Stainless Steel manufactured by Directed Energy Deposition. Materials Science and Engineering: A, 878, 145176. https://doi.org/10.1016/j.msea.2023.145176
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