From the onset of damage to rupture: construction of responses with damage localization for a general class of gradient damage models

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A gradient-enhanced damage model coupled to plasticity—multi-surface formulation and algorithmic concepts (2018, background): “…approximations of free discontinuity problems (Ambrosio and Tortorelli, 1990), have received much attention—see, for example, the gradient damage functional studied in Pham and Marigo (2013) and their role in general variational approaches to fracture discussed in Bourdin et al. (2008).”
A numerical assessment of phase-field models for brittle and cohesive fracture: Γ-Convergence and stress oscillations (2015, background): “…g(1) = 0, since ford = 1 the damaged partψd has to vanish; • g′(0) , 0, since the damage has to be initiated at the onset; • g′(1) = 0, since the energy must converge to a finite value for the fully broken state, seeBraides(1998); Pham et al.(2011); Pham and Marigo(2013) for a general discussion.”
Phase‐field modeling and simulation of fracture in brittle materials with strongly anisotropic surface energy (2015, background): “/ consistent with the variational theory of brittle fracture; see, for example, [43, 44] for a discussion.”