Influence of the mesh on the crack path in phase-field fracture simulations

GAMM PF 25 and Materials/Microstructure modelling

F. Loiseau
flavien.loiseau@ensta-paris.fr

IMSIA, ENSTA

E. Zembra
edgar.zembra@polytechnique.edu

PMC, Ecole Polytechnique

H. Henry
herve.henry@polytechnique.edu

PMC, Ecole Polytechnique

V. Lazarus
veronique.lazarus@ensta-paris.fr

IMSIA, ENSTA

February 12, 2025

Crack propagation in 3D-printed structures

Duplex stainless steel by DED (Roucou et al., 2023)

Polycarbonate CT specimen by FDM (Zhai, 2023)

Global objective

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

The problem we want to solve

Fracture mechanics

We consider

  • a domain \(\Omega\) with a crack \(\Gamma_0\),
  • an elastic material (\(E\), \(\nu\)),
  • a force and/or displacement load,

and we want to determine

  • the crack path,
  • the evolution of the displacement field.

To solve this problem, we want to employ numerical methods.

Linear Elastic Fracture Mechanics (LEFM)

State
The state of a domain \(\Omega\) is described by:

  • the displacement field \(\boldsymbol{u}(\boldsymbol{x})\),
  • the crack length \(a\).

Griffith criterion (Griffith, 1920)
The crack propagates when: \[ \begin{array}{ccc} G & = & G_c \\ % \downarrow & \quad & \downarrow \\ \text{elastic energy} & & \text{fracture energy} \\ \text{release} & & \text{required} \\ \end{array} \]

Variational approach to fracture (Francfort & Marigo, 1998)
The state minimizes the potential energy \(\mathcal{P}\), \[ (\boldsymbol{u}, a) = \arg\min_{\substack{\boldsymbol{u}' \in \mathcal{U} \\ a' \in \mathcal{A}}} \mathcal{P}(\boldsymbol{u}, a), \quad \mathcal{P}(\boldsymbol{u}, a) = \underset{\text{elastic}}{\mathcal{E} (\boldsymbol{u}', a')} + \underset{\text{dissipation}}{\mathcal{D} (a')} - \underset{\text{external work}}{\mathcal{W}_{\mathrm{ext}} (\boldsymbol{u}')}. \] with \(\quad \displaystyle \mathcal{D} (a) = \int_{\Gamma(a)} G_c \, \mathrm{d}S\).

Variational Phase-Field Model for fracture (VPFM)

State
The state of a domain \(\Omega\) is described by:

  • the displacement field \(\boldsymbol{u}(\boldsymbol{x})\),
  • the crack phase field \(\alpha(\boldsymbol{x})\),
    • \(\alpha=0 \rightarrow \text{unbroken}\),
    • \(\alpha=1 \rightarrow \text{broken}\),
    • \(\alpha(t + \Delta t) \geq \alpha(t)\).


Variational phase field model (Bourdin et al., 2000; Francfort & Marigo, 1998)
The state minimizes the regularized potential energy \(\mathcal{P}\), \[ (\boldsymbol{u}, \alpha) = \arg\min_{\substack{\boldsymbol{u}' \in \mathcal{U} \\ \alpha' \in \mathcal{A}}} \mathcal{P}(\boldsymbol{u}, \alpha), \quad \mathcal{P}(\boldsymbol{u}, \alpha) = \underset{\text{elastic}}{\mathcal{E} (\boldsymbol{u}', \alpha')} + \underset{\text{dissipation}}{\mathcal{D} (\alpha')} - \underset{\text{external work}}{\mathcal{W}_{\mathrm{ext}} (\boldsymbol{u}')}. \]

\(\Gamma\)-convergence of VPFM towards LEFM

We consider the classic dissipation functional

\[ \mathcal{D}(\alpha) = \frac{G_0}{c_w} \int_{\Omega} \frac{w(\alpha)}{\ell} + \ell |\nabla \alpha|^2 \,\mathrm{d}x. \]

With continous field \(\alpha\) (Braides, 1998; Giacomini, 2005), \[ \mathcal{D}(\alpha) \underset{\ell \to 0}{\to} \int_{\Gamma} G_0 \, \mathrm{d}S. \]

However, with a discrete field \(\alpha\) (Negri, 1999, 2003), \[ \mathcal{D}(\alpha) \underset{\ell \to 0}{\to} \int_{\Gamma} G_0 \, \varphi (\theta) \, \mathrm{d}S. \]

Observation

The discretization induces artificial anisotropy: \(G_0 \, \phi (\theta) = G_c(\theta)\).

Illustration of mesh-induced anisotropy

Negri (1999), Negri (2003)



Polar plot of the mesh-induced anisotropy function \(\phi(\boldsymbol{n})\) for different triangulation (based on the calculations of Negri (2003)).

Numeric analysis :
Mesh influence on crack path


Objective
Preliminary analysis of the mesh influence on the crack path in variational phase-field models


Outline

  • Presentation of the benchmark proposed by H. Henry
  • Comparison of crack path on structured and unstructured meshes for:
    • Instable propagation case
    • Stable propagation case

Benchmark: Eccentric Pure Shear (H. Henry)


\[H = 1 \qquad W = 8 H \qquad a_0 = H / 2 \qquad e = H / 4\]

Other parameters are given at the end of the presentation (see appendix in Section 5.1)

Benchmark: Expected results


Two types of meshes


Structured mesh

Unstructured mesh

Remarks

  • Meshes have been coarsened for illustration purposes (\(h = H /12\)).
  • Initial crack is one-element wide and \(\alpha = 1\) on it (see preprint Loiseau & Lazarus (2025)).

Unstable propagation1 – Structured mesh

Unstable propagation1 – Unstructured mesh

Unstable propagation1 – Struct vs Unstruct

Stable crack propagation

Change of load

Stable crack propagation

Conclusion

Discussion of the results


Unstable crack propagation1

  • Unstructured mesh : large bias which decreases with mesh refinement.
  • Structured mesh : mesh only adds noise to the path.


Stable crack propagation

  • Unstructured mesh : no convergence.
  • Structured mesh : convergence but high noise.

Crack increments are more sensitive to the (local) discretization-induced anisotropy.

Summary

In this work, we:

  • Propose a benchmark to assess the influence of the mesh on crack path.
  • Encourage to extend mesh-sensitivity study to its structure (not only mesh size).
  • Show that structured meshes severely bias the crack path (stable propagation).

Thank you for your attention !

F. Loiseau, E. Zembra, H. Henry, V. Lazarus

flavien.loiseau@ensta-paris.fr

Presentation




The work was supported by Agence de l’Innovation de Défense – AID – via Centre Interdisciplinaire d’Etudes pour la Défense et la Sécurité – CIEDS – (projects 2022 - FracAddi).

Appendices

Appendix : Parameters for the simulations

  • 2D : plane stress.
  • Phase-field model: AT1.
  • Elastic parameters: \(\lambda = \mu = 1 Pa\).
  • Fracture parameters: \(G_c = 1\).
  • Regularization length: \(\ell = 0.0625 = H/16\).
  • Mesh sizes: \(h \in {H/64, H/128, H/256}\).
  • Load \(u_x = 0\) and \(u_y(x,t) = \pm (t + \tan(\pi/180) * (H - x))\)
    • Unstable : \(\theta = 0\)
    • Stable : \(\theta = \pi/180\)

References

References

Bourdin, B., Francfort, G. A., & Marigo, J.-J. (2000). Numerical experiments in revisited brittle fracture. Journal of the Mechanics and Physics of Solids, 48(4), 797–826. https://doi.org/10.1016/S0022-5096(99)00028-9
Braides, A. (1998). Approximation of Free-Discontinuity Problems (Vol. 1694). Springer. https://doi.org/10.1007/BFb0097344
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
Giacomini, A. (2005). Ambrosio-Tortorelli approximation of quasi-static evolution of brittle fractures. Calculus of Variations and Partial Differential Equations, 22(2), 129–172. https://doi.org/10.1007/s00526-004-0269-6
Griffith, A. A. (1920). The phenomena of rupture and flow in solids. Philosophical Transactions of the Royal Society of London. Series A, Containing Papers of a Mathematical or Physical Character, 221(582-593), 163–198. https://doi.org/10.1098/rsta.1921.0006
Loiseau, F., & Lazarus, V. (2025). How to introduce an initial crack in phase field simulations to accurately predict the linear elastic fracture propagation threshold? arXiv. https://doi.org/10.48550/arXiv.2502.03900
Negri, M. (1999). The anisotropy introduced by the mesh in the finite element approximation of the mumford-shah functional. Numerical Functional Analysis and Optimization, 20(9-10), 957–982. https://doi.org/10.1080/01630569908816934
Negri, M. (2003). A finite element approximation of the Griffith’s model in fracture mechanics. Numerische Mathematik, 95(4), 653–687. https://doi.org/10.1007/s00211-003-0456-y
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
Zhai, X. (2023). Crack propagation in elastic media with anisotropic fracture toughness: Experiments and phase-field modeling [PhD thesis].