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A variational formulation of Griffith phase-field fracture with material strength

C. J. Larsen, J. E. Dolbow, O. Lopez-Pamies

Abstract

In this expository Note, it is shown that the Griffith phase-field theory of fracture accounting for material strength originally introduced by Kumar, Francfort, and Lopez-Pamies (J Mech Phys Solids 112, 523--551, 2018) in the form of PDEs can be recast as a variational theory. In particular, the solution pair $(\textbf{u},v)$ defined by the PDEs for the displacement field $\textbf{u}$ and the phase field $v$ is shown to correspond to the fields that minimize separately two different functionals, much like the solution pair $(\textbf{u},v)$ defined by the original phase-field theory of fracture without material strength implemented in terms of alternating minimization. The merits of formulating a complete theory of fracture nucleation and propagation via such a variational approach -- in terms of the minimization of two different functionals -- are discussed.

A variational formulation of Griffith phase-field fracture with material strength

Abstract

In this expository Note, it is shown that the Griffith phase-field theory of fracture accounting for material strength originally introduced by Kumar, Francfort, and Lopez-Pamies (J Mech Phys Solids 112, 523--551, 2018) in the form of PDEs can be recast as a variational theory. In particular, the solution pair defined by the PDEs for the displacement field and the phase field is shown to correspond to the fields that minimize separately two different functionals, much like the solution pair defined by the original phase-field theory of fracture without material strength implemented in terms of alternating minimization. The merits of formulating a complete theory of fracture nucleation and propagation via such a variational approach -- in terms of the minimization of two different functionals -- are discussed.
Paper Structure (14 sections, 42 equations)

This paper contains 14 sections, 42 equations.

Table of Contents

  1. Introduction
  2. The problem
  3. The phase-field fracture theory of Kumar, Francfort and Lopez-Pamies (2018)
  4. The driving force $c_{\emph{e}}({\bf X},t)$ and coefficient $\delta^{\varepsilon}$.
  5. The variational formulation of the phase-field fracture theory (\ref{['BVP-u-theory']})-(\ref{['BVP-v-theory']})
  6. The deformation energy functional $\mathcal{E}^{\varepsilon}_{d}({\bf u}_k;v_k)$
  7. The fracture functional $\mathcal{E}^{\varepsilon}_{f}(v_k;{\bf u}_k)$
  8. The variational principle
  9. Some remarks on the competitions set up by the variational principle
  10. Uniform fields.
  11. Non-uniform fields due to the presence of large cracks.
  12. Arbitrary non-uniform fields.
  13. A comment on the variational structure of the original formulation of Griffith phase-field fracture without material strength
  14. Final comments

Theorems & Definitions (10)

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