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

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 \((\textbf{u},v)\) $(\mathbf{u},v)$ defined by the PDEs for the displacement field \(\textbf{u}\) $\mathbf{u}$ and the phase field v is shown to correspond to the fields that minimize separately two different functionals, much like the solution pair \((\textbf{u},v)\) $(\mathbf{u},v)$ 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.