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We derive the variational formulation of a gradient damage model by applyingthe energetic formulation of rate-independent processes and obtain aregularized formulation of fracture. The model exhibits different behavior attraction and compression and has a state-dependent dissipation potential whichinduces a path-independent work. We will show how such formulation provides thenatural framework for setting up a consistent numerical scheme with theunderlying variational structure and for the derivation of additional necessaryconditions of global optimality in the form of a two-sided energeticinequality. These conditions will form our criteria for making a better choiceof the starting guess in the application of the alternating minimization schemeto describe crack propagation as quasistatic evolution of global minimizers ofthe underlying incremental functional. We will apply the procedure for two- andthree-dimensional benchmark problems and we will compare the results with thesolution of the weak form of the Euler-Lagrange equations. We will observe thatby including the two-sided energetic inequality in our solution method, wedescribe, for some of the benchmark problems, an equilibrium path when thedamage starts to manifest, which is different from the one obtained by solvingsimply the stationarity conditions of the underlying functional.

A variational asymmetric phase-field model of quasi-brittle fracture: Energetic solutions and their computation