Phase field fracture: Towards a generic method to follow the equilibrium path of the structure

Congrès des Jeunes Chercheurs en Mécanique

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

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

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

August 28, 2024

Fracture in 3D-printed structures

Polycarbonate CT specimen by FDM (Zhai, 2023)

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

Global objective

Modelling and simulating crack propagation in 3D-printed structures

Fracture mechanics problem

We consider

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

and we want to determine

  • the crack-path,
  • the force-displacement curve,
  • the evolution of the displacement field.

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

Phase-field fracture

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

  • a displacement field \(\boldsymbol{u}(\boldsymbol{x})\),
  • a crack phase field \(\alpha(\boldsymbol{x})\).

The state fields are governed by an energy minization problem, \[ (\boldsymbol{u}, \alpha) = \arg\min_{\substack{\boldsymbol{u}^* \in \mathcal{U} \\ \alpha^* \in \mathcal{A}}} \ \underset{\text{elastic}}{\mathcal{E} (\boldsymbol{u}^*, \alpha^*)} + \underset{\text{dissipation}}{\mathcal{D} (\alpha^*)} - \underset{\text{external work}}{\mathcal{W}_{\mathrm{ext}} (\boldsymbol{u}^*)} \]

Issues due to fracture (displacement loading)


Issue due to fracture (force loading)


Path-following methods 1

Principle

A load factor \(\lambda\) with a control equation,

\[ f(\lambda, \boldsymbol{u}, ...) - \Delta \tau= 0, \]

where \(\Delta \tau\): step size of control quantity \(f\).


From geometrical non-linearities

Wempner (1971), Riks (1972), Riks (1979),
Crisfield (1981), Ramm (1981).

Applications to phase-field fracture

Energy release/Dissipation
Gutiérrez (2004)

Application to phase-field fracture by

Pros

  • Problem-independent

Cons

  • Model-depend (dissipation)
  • Control switch for elastic phase

Nodal displacement
De Borst (1987)

Applied to phase-field by


Pros

  • Model-independent

Cons

  • A-priori choice of control nodes
    • Problem-depend

Can we introduce path-following in simulations of phase-field fracture?

With some constraints:

  • Simplicity (minimize intrusivity, no switch).
  • Flexibility (no model dependency, no problem dependency).
  • Robustness (no mesh dependency, no divergence).

Control equation

Control by Maximum Strain Increment (Chen & Schreyer, 1990)

The idea is to limit the maximum strain variation in the domain using the equation \[ \max_{\boldsymbol{x}\in \Omega} \left( \frac{\boldsymbol{\varepsilon}(\boldsymbol{u}_{n-1})}{\| \boldsymbol{\varepsilon}(\boldsymbol{u}_{n-1}) \|} : {\color{\red}\Delta \boldsymbol{\varepsilon}(\boldsymbol{u})} \right) - \Delta \tau= 0, \]

where \(\Delta \tau\) is the step size.

Solving the problem (alternate minimization)

Applications

SENT specimen: Presentation

SENT specimen: Displacement control

SENT specimen: Path-following control

CT specimen with force boundary conditions

Conclusion

We applied a path-following method based on control by maximum strain increment.

It proved to be:

  • simple: small changes to the elastic solver, none to the crack-phase solver.
  • flexible: no problem nor model dependencies.

Yet, we observed some divergence of the resolution in some specific cases.

Thank you for your attention !

F. Loiseau, V. Lazarus

flavien.loiseau@ensta-paris.fr

References

References

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
Crisfield, M. A. (1981). A fast incremental/iterative solution procedure that handles “snap-through.” Computers & Structures, 13(1), 55–62. https://doi.org/10.1016/0045-7949(81)90108-5
De Borst, R. (1987). Computation of post-bifurcation and post-failure behavior of strain-softening solids. Computers & Structures, 25(2), 211–224. https://doi.org/10.1016/0045-7949(87)90144-1
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
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
May, S., Vignollet, J., & Borst, R. de. (2016). A new arc-length control method based on the rates of the internal and the dissipated energy. Engineering Computations, 33(1), 100–115. https://doi.org/10.1108/EC-02-2015-0044
Ramm, E. (1981). Strategies for Tracing the Nonlinear Response Near Limit Points. In W. Wunderlich, E. Stein, & K.-J. Bathe (Eds.), Nonlinear Finite Element Analysis in Structural Mechanics (pp. 63–89). Springer. https://doi.org/10.1007/978-3-642-81589-8_5
Riks, E. (1972). The Application of Newton’s Method to the Problem of Elastic Stability. Journal of Applied Mechanics, 39(4), 1060–1065. https://doi.org/10.1115/1.3422829
Riks, E. (1979). An incremental approach to the solution of snapping and buckling problems. International Journal of Solids and Structures, 15(7), 529–551. https://doi.org/10.1016/0020-7683(79)90081-7
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
Wempner, G. A. (1971). Discrete approximations related to nonlinear theories of solids. International Journal of Solids and Structures, 7(11), 1581–1599. https://doi.org/10.1016/0020-7683(71)90038-2
Wu, J.-Y. (2018). Robust numerical implementation of non-standard phase-field damage models for failure in solids. Computer Methods in Applied Mechanics and Engineering, 340, 767–797. https://doi.org/10.1016/j.cma.2018.06.007
Zhai, X. (2023). Crack propagation in elastic media with anisotropic fracture toughness: Experiments and phase-field modeling [PhD thesis].

Appendices

Systematic literature review

Query

Search on Scopus using the following query in April, 2024.

("load control" OR "path following" OR "path-following" OR "arc length" OR "arc-length" OR "continuation")
AND ("phase-field")
AND ("fracture")

Results

  • 15 articles
  • 2 filtered out
    • 1 written in chinese
    • 1 unrelated
  • 2 added articles (not in the search, found by previous researches)

Notes

  • Main contributors at TU Delft
  • Few load control methods applied to PF
  • Various solving algorithm (staggered/monolithic)
  • Small/No comparison to LEFM
  • No comparison to expe

Phase-field fracture

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

  • a displacement field \(\boldsymbol{u}(\boldsymbol{x})\)
  • a crack phase field \(\alpha(\boldsymbol{x})\)

The state of the domain verifies \[ (\boldsymbol{u}, \alpha) = \arg\min_{\substack{\boldsymbol{u}^* \in \mathcal{U} \\ \alpha^* \in \mathcal{A}}} \mathcal{E}_{\mathrm{tot}} = \arg\min_{\substack{\boldsymbol{u}^* \in \mathcal{U} \\ \alpha^* \in \mathcal{A}}} \mathcal{E} (\boldsymbol{u}^*, \alpha^*) + \mathcal{D} (\alpha^*) - \mathcal{W}_{\mathrm{ext}} (\boldsymbol{u}^*) \]

where the energy functionals are,

\[ \begin{split} \mathcal{E} (\boldsymbol{u}, \alpha) &= \int_{\Omega} \frac{1}{2} a(\alpha) \boldsymbol{\varepsilon}(\boldsymbol{u}) : \mathbf{E}: \boldsymbol{\varepsilon}(\boldsymbol{u}) \,\mathrm{d}x, \\ \mathcal{D} (\alpha) &= \int_{\Omega} \frac{G_c}{c_w} \left( \frac{w(\alpha)}{\ell} + \ell \nabla \alpha \cdot \nabla \alpha \right) \,\mathrm{d}x, \\ \mathcal{W} (\boldsymbol{u}) &= \int_{\partial \Omega} \lambda \boldsymbol{t}_{\mathrm{ctrl}}\cdot \boldsymbol{u}^* \,\mathrm{d}x. \end{split} \]

Staggered scheme (alternate minimization)


The minimization problem can be solved by splitting the problem into two subproblems:

  • minimization with respect to the displacement field \(\boldsymbol{u}(\boldsymbol{x})\) \[ \boldsymbol{u}(\boldsymbol{x}) = \arg\min_{\boldsymbol{u}^* \in \mathcal{U}} \mathcal{E}_{\mathrm{tot}} (\boldsymbol{u}^*, \alpha), \]
  • minimization of the functional with respect to the crack phase field \(\alpha(\boldsymbol{x})\) \[ \alpha(\boldsymbol{x}) =\arg\min_{\alpha^* \in \mathcal{A}} \mathcal{E}_{\mathrm{tot}} (\boldsymbol{u}, \alpha^*). \]

We need to introduce the solving of the control equation.

Solving procedure (displacement + load factor)

  1. Solve the displacement field \(\bar{\boldsymbol{u}}\) associated with a unitary load (\(\lambda = 1\))

\[ \begin{split} \forall \boldsymbol{v}\in \mathcal{U}_0, \quad & \int_{\Omega} a(\alpha) \boldsymbol{\varepsilon}(\bar{\boldsymbol{u}}) : \mathbf{E}: \boldsymbol{\varepsilon}(\boldsymbol{v}) \,\mathrm{d}x + \int_{\partial \Omega} \boldsymbol{t}_{\mathrm{ctrl}}\cdot \boldsymbol{v}\,\mathrm{d}x = 0, \end{split} \]

  1. Solve the control equation for each \(\boldsymbol{x}\in \Omega\) with \[ \frac{\boldsymbol{\varepsilon}(\boldsymbol{u}_{n-1})}{\| \boldsymbol{\varepsilon}(\boldsymbol{u}_{n-1}) \|} : \overbrace{({\color{\red}\lambda(\boldsymbol{x})} \boldsymbol{\varepsilon}(\boldsymbol{u}_{n-1}) - \boldsymbol{\varepsilon}(\boldsymbol{u}_{n-1}))}^{\Delta \boldsymbol{\varepsilon}(\boldsymbol{u})} - \Delta \tau= 0 \iff \lambda(\boldsymbol{x}) = \frac{\Delta \tau- a_0}{a_1}, \] where \(a_0 = \frac{\boldsymbol{\varepsilon}_{n-1} (\boldsymbol{x})}{\| \boldsymbol{\varepsilon}_{n-1}(\boldsymbol{x}) \|} : \boldsymbol{\varepsilon}_{n-1}(\boldsymbol{x})\), and \(a_1 = \frac{\boldsymbol{\varepsilon}_{n-1} (\boldsymbol{x})}{\| \boldsymbol{\varepsilon}_{n-1}(\boldsymbol{x}) \|} : \boldsymbol{\varepsilon}(\bar{\boldsymbol{u}})\).

  2. Select the load factor using the nested interval algorithm (Lorentz & Badel, 2004).

  3. Calculate the displacement field \(\boldsymbol{u}(\boldsymbol{x}) = \lambda \bar{\boldsymbol{u}}(\boldsymbol{x})\).