Graded damage in quasi-brittle solids

A novel approach to damage modeling for quasi-brittle solids is presented relying upon a differential inclusion that is closely related to the one of implicit gradient models. The proposed formulation naturally fits in the so-called nonlocal standard approach, whereby the framework of generalized standard materials is extended to include gradients of internal variables to account for the physics of the fracture phenomenon in a regularized sense, that is, via extended constitutive equations in which a length scale parameter brings to the macro level information about material microstructure. This concept is fully embodied into the present approach to quasi-brittle fracture, whereby progressive damage occurs in layers of finite thickness where the gradient of damage is bounded and a fully damaged region is understood as a fracture with no ambiguity. Key to the effective implementation of the model are the choice of two constitutive functions and the implicit tracking of regions in a state of progressive damage via Lagrange multipliers acting on internal constraints. The ideas are developed for a general Cauchy continuum and representative numerical simulations are included that demonstrate the model capabilities.