Path-Following Methods
for Quasi-Static Crack Propagation
in Phase-Field Fracture Models

European Mechanics of Materials Conferences

Flavien Loiseau
flavien.loiseau@ensta.fr

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

Véronique Lazarus
veronique.lazarus@ensta.fr

May 28, 2026
Florence, Italy

Objective

What we want to simulate

Crack propagation in a 3D-printed specimen (Zhai et al., 2025)



How to numerical represent the load ?

Triclot et al. (2024) identified that force loads best reproduce the experimental crack tip fields in FEM simulations of CT specimens!


Outline of the presentation

  1. Why force loads are a problem ?
  2. How to circumvent this issues in :
      a. Linear Elastic Fracture Mechanics (LEFM) ?
      b. Variational phase-field fracture (PFF) ?
  3. Application : LEFM vs PFF models

Loss of force balance

Limitations of incremental load in fracture simulations

Compact Tension specimen with a secondary crack

Force displacement curve

In quasi-static phase-field fracture simulations

  • Under force load, once the peak force is reached, no further configuration verifies the force balance !

Load control

Idea : Control the load during crack propagation

Compact Tension specimen with a secondary crack

Load factor \(\lambda\) to stay at the crack propagation threshold \(G=G_c\)

How to control the load in practice ? / How to follow the equilibrium path ?

  a. in Linear Elastic Fracture Mechanics      b. Variational Phase-Field Fracture models

Path-following in LEFM - Load control

Linearity

At fixed crack length, linear elasticity imposes that

displacement \(\boldsymbol{u}(\boldsymbol{x}) \propto\) load factor \(\lambda\),

which implies that the energy release rate \(G \propto \lambda^2\).


Griffith criterion

Energy minimization in a cracked structure leads to

  • \(G < G_c \implies\) no crack propagation,
  • \(G = G_c \implies\) crack propagation.

\[ \color{grey}{G = \frac{\partial \mathcal{P}}{\partial a}} \]

Idea

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

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


Path-following in LEFM - Algorithm


Algorithm

For a given crack :

  1. Solve the elastic problem for \(\lambda = 1\)
  2. Calculate crack propagation direction \(\varphi\)
    using 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 \(\varphi\)

Towards phase field fracture


Variational approach to fracture (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 \(\mathcal{H}(\Gamma) - \mathcal{H}(\Gamma_0) = \Delta a\).

Variational phase-field fracture


Introducing the phase-field (Francfort & Marigo, 1998)

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: \(\displaystyle \mathcal{H}_\ell (\alpha) - \mathcal{H}_\ell(\alpha_0) = \Delta a\).

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 \(\alpha\),

leading to important changes to classic solvers.

Idea

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

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

Our choice for the constraint

Inspired by Chen & Schreyer (1990)

Control the maximum strain increment
outside the (already) cracked region

Limit strain increment in the new crack portion!

\[ \max_{\boldsymbol{x}\in \Omega} \left( \color{red}{(1-\alpha_0(\boldsymbol{x})) \Delta\boldsymbol{\varepsilon}(\boldsymbol{x})} : \frac{\boldsymbol{\varepsilon}(\boldsymbol{u}_{0}(\boldsymbol{x}))}{\|\boldsymbol{\varepsilon}(\boldsymbol{u}_{0}(\boldsymbol{x}))\|} \right) - \Delta \tau= 0 \]

Adaptation of alternate minization


Algorithm

During an iteration of alternate minimization :

  1. Solve the elastic problem for \(\lambda = 1 \rightarrow \bar{\boldsymbol{u}}(\boldsymbol{x})\)
  2. Solve the path-following constraint \(\rightarrow \lambda, \boldsymbol{u}(\boldsymbol{x})\)
    1. Nested interval algorithm (Lorentz & Badel, 2004)
    2. Rescale the displacement field \(\boldsymbol{u}(\boldsymbol{x}) = \lambda \bar{\boldsymbol{u}}(\boldsymbol{x})\)
  3. Solve the phase problem for \(\boldsymbol{u}(\boldsymbol{x}) \rightarrow \alpha(\boldsymbol{x})\)
  4. Check for convergence

Remark

Minimal changes to the solver!

Results - SENT specimen

Convergence with respect to mesh size




Observations

Snapback is captured
PFF matches LEFM results

Load factor not affected by
  mesh size1


Note
Crack length = \(\mathcal{H}_\ell(\alpha)\) for PFF

Results - SENT specimen

Numerical aspects

Crack growth per load step \(\approx\) constante

Better distribution of the computational cost

Results - CT with secondary crack

Numerical \(\Gamma\)-convergence





Observations

Snapback is captured
PFF matches LEFM results
Force boundary conditions

Constant crack growth
  per load step

Note
Crack length = \(\mathcal{H}_\ell(\alpha)\) for PFF

Thank you
for your attention !


F. Loiseau, V. Lazarus

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

More details on this work
Loiseau, F., & Lazarus, V. (in press).
Path-following methods for phase-field simulation of quasi-static crack propagation
International Journal of Solids and Structures.
https://10.1016/j.ijsolstr.2026.113974

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
Chen, Z., & Schreyer, H. L. (1990). A numerical solution scheme for softening problems involving total strain control. Computers & Structures, 37(6), 1043–1050. https://doi.org/10.1016/0045-7949(90)90016-U
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
Lorentz, E., & Badel, P. (2004). A new path-following constraint for strain-softening finite element simulations. International Journal for Numerical Methods in Engineering, 60(2), 499–526. https://doi.org/10.1002/nme.971
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. (2024). Toughening effects of out-of-crack-path architected zones. International Journal of Fracture. https://doi.org/10.1007/s10704-024-00811-5
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