A geometrically regularized gradient-damage model with energetic equivalence

- A geometrically regularized gradient-damage model with energetic equivalence is proposed.
- The damage evolution law emerges from the geometric regularization and energetic equivalence.
- The crack is implicitly represented, and cumbersome crack tracking strategies are not required.
- The length scale can be regarded either as a small numerical parameter or as a material property.
- The localization bandwidth approaches to a finite limit value, not exhibiting spurious damage growth.

Aiming to bridge the gap between damage and fracture mechanics for modeling localized failure in solids, this paper addresses a novel geometrically regularized gradient-damage model with energetic equivalence for cracking evolution. With the free energy potential defined as usual in terms of the local strain and damage fields, the constitutive relations are derived consistently from the standard framework of thermodynamics. Upon the sharp crack topology geometrically regularized by the damage localization band with a length scale, the ensuing energetic equivalence naturally yields the damage evolution law of gradient-type and the associated boundary condition of Neumann-type. Compared to other gradient-damage models, no extra assumptions like the nonlocal energy residual and insulation condition are introduced. Moreover, the damage gradient, physically accounting for microscopic nonlocal interactions, is fully dissipative as expected. In line with the unified phase-field theory recently proposed by the author (Wu, 2017) during failure processes the material behavior is uniquely characterized by two constitutive functions, i.e., the degradation function defining the free energy potential of the bulk and the geometric function regularizing the sharp crack topology. In particular, optimal constitutive functions defining an equivalent cohesive zone model of general softening laws are postulated, with the involved parameters calibrated from standard material properties. The proposed model is numerically implemented into the multi-field finite element method and applied to several benchmark tests of concrete under mode-I and mixed-mode failure. It is found that the incorporated length scale can be regarded either a numerical parameter or a material property. For the former considered in this work, the length scale has negligible, if not no, effects on the global responses, so long as the sharp crack topology and the damage field of high gradients within the localization band are well resolved. Furthermore, the localization bandwidth does not exhibit spurious widening, but rather, it approaches to a finite value proportional to the length scale. Comparison between the numerical and experimental results, regarding the curve of load versus displacement and crack path, confirms validity of the proposed gradient-damage model for characterizing localized failure in quasi-brittle solids.