Following the equilibrium path during crack propagation under quasi-static loads :
LEFM and phase-field approaches

CFRAC 2025, Porto, Portugal

F. Loiseau
flavien.loiseau@ensta.fr

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

V. Lazarus
veronique.lazarus@ensta.fr

June 5, 2025

Context : Fracture in 3D-printed structures

Polymers (Fused Deposit Modelling)

Polycarbonate CT specimen (Zhai et al., 2025)

Metals (Direct Energy Deposition)

Duplex stainless steel (Roucou et al., 2023)

Global objective in my postdoc

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

Objective in this presentation


Difficulties

  • Modelling (fracture anisotropy, boundary conditions , etc.)
  • Numerical bias (proper crack initialization, mesh influence on crack path)


Previous work on boundary conditions (Triclot et al., 2023)

showed that stress intensity factors in elastic FEM simulation are best represented when using force boundary condition in CT specimens (contact between the hole and the pin).


How to simulate crack propagation with force boundary conditions ?

  1. Why is it diffult to impose force boundary conditions ?
  2. How to tackle this problem in LEFM ? and in phase-field fracture ?

In practice, the methods presented here can find many more applications.

Incremental force load in fracture mechanics

No sufficient reaction to counteract the imposed load,
regardless of crack length.

Application of path-following to Griffith theory

Linear Elastic Fracture Mechanics


At fixed crack length, linear elasticity imposes that displacement \(\boldsymbol{u}(\boldsymbol{x}) \propto\) load factor \(\lambda\).

Moreover, the energy release rate \(G \propto \lambda^2\).


💡 Idea

Rather than incrementing the load factor, control it using the Griffith criterion:

\[\lambda^2 \overline{G} = G_c \implies \lambda = \sqrt{G_c / \bar{G}}\]

Load control based on crack length

Linear Elastic Fracture Mechanics


Algorithm

For a given crack :

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

Example: SENT specimen (problem)

Linear Elastic Fracture Mechanics

Example: SENT specimen (results)

Linear Elastic Fracture Mechanics

Ensuring \(G = G_c\) \(\implies\) the load factor \(\lambda\searrow\)

Snapback in the force-displacement curve.

Formalization in variational approach to fracture

Framework proposed by Francfort & Marigo (1998)

The energy minimization principle of Griffith can be written as

\[ (\boldsymbol{u}, \Gamma) = \arg\min_{\substack{\boldsymbol{u}' \in \mathcal{U}_{\Gamma} \\ \Gamma' \supseteq \Gamma_0}} \underbrace{\mathcal{P}(\boldsymbol{u}', \Gamma')}_{\substack{\text{Potential} \\ \text{energy}}} + \underbrace{G_c \mathcal{H}(\Gamma')}_{\substack{\text{Fracture} \\ \text{dissipation}}}, \]

where \(\displaystyle \mathcal{P}(\boldsymbol{u}', \Gamma') = \mathcal{E}(\boldsymbol{u}', \Gamma') - \mathcal{W}_{\mathrm{ext}}(\boldsymbol{u}')\), and \(\mathcal{H}\) is the Hausdorff measure.


Adding the path-following constraint

With the path-following constraint, the problem writes

\[ (\overline{\boldsymbol{u}}, \Gamma) = \arg\min_{\substack{\overline{\boldsymbol{u}}' \in \overline{\mathcal{U}}_{\Gamma} \\ \Gamma' \supseteq \Gamma_0}} {\color{red}\lambda^2} \mathcal{P}(\overline{\boldsymbol{u}}', \Gamma') + G_c \mathcal{H}(\Gamma'), \] where \(\lambda\) is such that \(\mathcal{H}(\Gamma) - \mathcal{H}(\Gamma_0) = \Delta S\).

Variational phase-field approach to fracture

Francfort & Marigo (1998)

Key idea

Using a continuous phase field \(\alpha\) to represent the discontinuity

\[ \mathcal{H}_\ell(\alpha) = \frac{1}{c_w} \int_{\Omega} \frac{w(\alpha)}{\ell} + \ell \| \nabla \alpha \|^2 \,\mathrm{d}x\underset{\ell \to 0}{\to} \mathcal{H}(\Gamma). \]

Updating the path-following constraint

The minimization problem with the path-following constraint becomes

\[ (\overline{\boldsymbol{u}}, \alpha) = \arg\min_{\substack{\overline{\boldsymbol{u}}' \in \overline{\mathcal{U}} \\ \alpha' \geq \alpha0}} \lambda^2 \mathcal{P}(\overline{\boldsymbol{u}}', \alpha') + G_c \mathcal{H}_\ell(\alpha) . \] where \(\lambda\) is such that: \(\displaystyle \mathcal{H}_\ell (\alpha) - \mathcal{H}_\ell(\alpha_0) = \Delta S\).

In practice, it is not that easy

This approach has been proposed (in another format) by Singh et al. (2016).


❗ Drawbacks

The path-following constraint

  • does not explicitely depend on \(\lambda\),
  • further constrains the (non-linear) minimization with respect to crack phase \(\alpha\).

💡 Idea

Indirectly control/limit the crack growth using a path-following constraint

\[ f(\lambda, \boldsymbol{u}, ...) - \Delta \tau = 0, \] where, ideally, \(f(\lambda, \boldsymbol{u}, ...) \propto \mathcal{H}(\Gamma)\).


Lots of possibilities for \(f\)

We aim at finding a constraint that is:

  1. independent with model, geometry and BC,
  2. easy to implement in a staggered solver.

Finding a constraint to limit crack growth

Studies

  • Systematic literature review on path-following and phase-field fracture \(\approx\) 20 articles.
  • Implementation and comparison of different path-following constraints.

Based on the ideas of Gutiérrez (2004), and similar to Fayezioghani et al. (2019) and Zambrano et al. (2023), we obtained the following constraint.

At fixed displacement \(\boldsymbol{u}\), the growth of the crack is governed by the PDE in \(\alpha\) \[ \tag{1st order optimality} G_c \frac{\partial \mathcal{H}_\ell}{\partial \alpha} = - \frac{\partial \mathcal{P}}{\partial \alpha}, \quad \text{ with } \mathcal{P} = \int_\Omega p \,\mathrm{d}x \] allowing us to introduce the control by maximum crack driving force \[ \int_\Omega h(t) - h(t_0) \,\mathrm{d}x= \Delta \tau, \quad h(t) = \max_{t' \leq t} \frac{\partial p}{\partial \alpha},\] … but the implementation is not easy (not detailed here).

Application to the SENT problem




Application with force boundary conditions

TDCB specimen with soft region adapted from Triclot et al. (2024)

Enable to simulate crack propagation with force boundary conditions.

Conclusion


How to simulate crack propagation with force boundary conditions ?

  • In LEFM, use load factor to verify \(G=G_c\) at each crack increment.
  • In regularized fracture, use a path-following constraint to limit the crack growth.
    • Check the review of Rastiello et al. (2022) for constraints.


Main result

It provides a framework to study quasi-static crack propagation under any parameterized load.
The extension to non-linear loads has been done (e.g., moving load \(\boldsymbol{u}_\mathrm{imp} = \overline{\boldsymbol{u}}_\mathrm{imp}(x-\lambda)\)).


Important underlying result?

Path-following ensures that the energy is minimized for a crack increment (not the whole crack!).

Thank you for your attention !

F. Loiseau, V. Lazarus

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


Based on this work
Loiseau, F., & Lazarus, V. (in press).
How to introduce an initial crack in phase field simulations
to accurately predict the linear elastic fracture propagation threshold?
JTCAM. https://hal.science/hal-04931758v1

References

Amestoy, M., & Leblond, J. B. (1992). Crack paths in plane situationsII. Detailed form of the expansion of the stress intensity factors. International Journal of Solids and Structures, 29(4), 465–501. https://doi.org/10.1016/0020-7683(92)90210-K
Fayezioghani, A., Vandoren, B., & Sluys, L. J. (2019). A posteriori performance-based comparison of three new path-following constraints for damage analysis of quasi-brittle materials. Computer Methods in Applied Mechanics and Engineering, 346, 746–768. https://doi.org/10.1016/j.cma.2018.09.014
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
Gutiérrez, M. A. (2004). Energy release control for numerical simulations of failure in quasi-brittle solids. Communications in Numerical Methods in Engineering, 20(1), 19–29. https://doi.org/10.1002/cnm.649
Rastiello, G., Oliveira, H. L., & Millard, A. (2022). Path-following methods for unstable structural responses induced by strain softening: A critical review. Comptes Rendus. Mécanique, 350(G2), 205–236. https://doi.org/10.5802/crmeca.112
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
Singh, N., Verhoosel, C. V., Borst, R. de, & Brummelen, E. H. van. (2016). A fracture-controlled path-following technique for phase-field modeling of brittle fracture. Finite Elements in Analysis and Design, 113, 14–29. https://doi.org/10.1016/j.finel.2015.12.005
Triclot, J., Corre, T., Gravouil, A., & Lazarus, V. (2023). Key role of boundary conditions for the 2D modeling of crack propagation in linear elastic Compact Tension tests. Engineering Fracture Mechanics, 277, 109012. https://doi.org/10.1016/j.engfracmech.2022.109012
Triclot, J., Corre, T., Gravouil, A., & Lazarus, V. (2024). Toughening effects of out-of-crack-path architected zones. International Journal of Fracture. https://doi.org/10.1007/s10704-024-00811-5
Zambrano, J., Toro, S., Sánchez, P. J., Duda, F. P., Méndez, C. G., & Huespe, A. E. (2023). An arc-length control technique for solving quasi-static fracture problems with phase field models and a staggered scheme. Computational Mechanics. https://doi.org/10.1007/s00466-023-02388-7
Zhai, X., Corre, T., & Lazarus, V. (2025). A FDM-based experimental benchmark for evaluating quasistatic crack propagation in anisotropic linear elastic materials. Engineering Fracture Mechanics, 324, 111175. https://doi.org/10.1016/j.engfracmech.2025.111175