Application of generic path-following methods to phase-field fracture

NewFrac Conference, FEUP Porto

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

May 9, 2024

Fracture in 3D-printed structures

Polymers (Fused Deposit Modelling)

Polycarbonate CT specimen (Zhai, 2023)

Metals (Direct Energy Deposition)

Duplex stainless steel (Roucou et al., 2023)

 Global objective  

Modelling and simulating crack propagation in 3D-printed structures

In this presentation

We will not discuss

  • Anisotropy/Heterogeneity
  • Comparison to experiments

What do we need?

  • Simulation of crack propagation
  • Force boundary conditions1
  • Comparison to LEFM2

Equilibrium path


Snap-back

Displacement loading

Instable propagation

Snap-back

Force loading

Instable propagation + No equilibrium

 During the simulation, we need to adapt the load to follow the equilibrium path!  

Path-following methods

Origins (arc-length methods)

  • Geometrical non-linearities
  • Applications to FEM calculations


Idea

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

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

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

Path-following and phase-field fracture


Systematic literature review1 \(\implies\) approx. 15 applications to phase-field fracture.


Control quantity Energy release/Dissipation Nodal displacement
Origin Gutiérrez (2004) De Borst (1987)
Phase-field May et al. (2016), Singh et al. (2016) Wu (2018)
Pros Problem-independent Model-independent
Cons Model-depend (dissipation), Control switch for elastic phase A-priori choice of control nodes (problem-dependent)


  What can be done?     A method independent with both problem and model.

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).

Problem

We consider a domain \(\Omega\) submitted to a controlled load:

\[ \begin{array}{ll} \forall \boldsymbol{x}\in \partial \Omega_{\boldsymbol{u}}, & \boldsymbol{u}(\boldsymbol{x}) = \lambda \boldsymbol{u}_{\mathrm{ctrl}}(\boldsymbol{x}) \\ \forall \boldsymbol{x}\in \partial \Omega_{\boldsymbol{t}}, & \boldsymbol{t}(\boldsymbol{x}) = \lambda \boldsymbol{t}_{\mathrm{ctrl}}(\boldsymbol{x}) \end{array} \] where \(\lambda\) is the load factor.

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

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

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

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 \(a_1 = \frac{\boldsymbol{\varepsilon}_{n-1} (\boldsymbol{x})}{\| \boldsymbol{\varepsilon}_{n-1}(\boldsymbol{x}) \|} : \boldsymbol{\varepsilon}(\bar{\boldsymbol{u}})\).
  1. Select the load factor using the nested interval algorithm (Lorentz & Badel, 2004).
  1. Calculate the displacement field \(\boldsymbol{u}(\boldsymbol{x}) = \lambda \bar{\boldsymbol{u}}(\boldsymbol{x})\).

Staggered solver with path-following

Summary

While the displacement and crack phase are not converged

  1. Solve the displacement \(\boldsymbol{u}(\boldsymbol{x})\) and load factor \(\lambda\) problem (fixed \(\alpha(\boldsymbol{x})\))
  2. Solve the crack-phase \(\alpha(\boldsymbol{x})\) problem (fixed \(\boldsymbol{u}(\boldsymbol{x})\) and \(\lambda\))

Not too intrusive

  • “Only” need to change the elastic solver
  • Crack phase solver remains unchanged

Other extensions (implemented but not investigated)

  • Adaptative step size \(\Delta \tau\)
  • Both imposed and controlled loads

To application \(\implies\)

SENT specimen: Displacement control

SENT specimen: Max strain increment control

CT specimen with force boundary conditions

Conclusion


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


In terms of robustness, mesh dependency has been observed on the crack path 1.


Yet, it proved to be:

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


Promising results, still room for improvement.

Thank you for your attention !

F. Loiseau, V. Lazarus

flavien.loiseau@ensta-paris.fr

GnuplotProduced by GNUPLOT 5.4 patchlevel 9 0250500750100002.5×10-65×10-67.5×10-6ABBCCDDAForce [N]Displacement [m]CMSI

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
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
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