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

TL;DR

This paper demonstrates that the Griffith phase-field fracture theory with material strength can be reformulated as a variational problem involving the minimization of two separate functionals, offering a new perspective on fracture modeling.

Contribution

It introduces a variational formulation of the Griffith phase-field fracture theory with material strength, connecting PDE solutions to functional minimization.

Findings

Recasts PDE-based fracture theory as a variational problem.

Shows the solution pair minimizes two distinct functionals.

Discusses advantages of variational approach for fracture analysis.

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.