To circumvent a mesh dependency of damage models, non-local approaches such as phase-field and gradient-extended damage models have shown a good capability and attracted a lot of attention for modeling fracture. These models can predict crack nucleation, kinking, and branching. The gradient-extended formulation proposed by [1,2], which includes a micromorphic degree of freedom for damage, is connected to a phase-field damage model presented in [3]; by connecting fracture parameters in brittle fracture. The latter is followed by comparing the thermodynamic consistency of these models. Despite having similarities in the formulation, gradient-extended models differ from the standard phase-field ones by having a damage threshold. Besides that, the local iteration exists in the gradient-extended damage models. By employing the cohesive phase-field model or the Angiotensin type 1 (AT1), a damage threshold appears in the formulation; by having a linear term for damage in the crack density function, see [4,5,12]. A comparison between these models is made, by taking several numerical examples and comparing their responses in a quasi-static case. Moreover, the feasibility of different responses is addressed when one uses a standard Newton-Raphson solver or the arc-length one for solving a boundary value problem.
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