A well-defined local continuum damage model with softening via convexification and entropic regularization

We introduce a local continuum damage model at small strains which is able to represent material softening behavior and leads to well-posed mechanical boundary value problems at the same time. To be more precise, upon an implicit Euler discretization in time, the condensed incremental potentials are uniformly strongly convex, permitting standard optimization algorithms with convergence guarantees to be used. As no gradients or non-local averaging is required, the innovative material model may be integrated into material routines of standard finite-element codes. An isotropic damage model with rate limitation serves as the starting point. Then, the construction of the material model proceeds in two steps. The material model is convexified first based on an explicit expression of the convex envelope of the condensed incremental potentials, leading to the emergence of damage distributions encoded via probability measures. The resulting model is convex, but not strictly convex. Therefore, we augment the model by an entropic regularization term which is deliberately chosen to produce the desired characteristics. We discuss the mechanical and computational aspects of the model, i.e., thermomechanical consistency, how to efficiently solve the discrete evolution problem for the internal variables and the consistent algorithmic tangent. Last but not least, we present computational examples of the model both at material-point level and in a heterogeneous environment which demonstrate the significantly improved numerical stability due to the uniformly strongly convex condensed incremental energies.