Theoretical and computational aspects of a thermodynamically consistent framework for geometrically linear gradient damage Original Research Article

This paper presents the theory and the numerics of an isotropic gradient damage formulation within a thermodynamical background. The main motivation is provided by localization computations whereby classical local continuum formulations fail to produce physically meaningful and numerically converging results. We propose a formulation in terms of the Helmholtz free energy incorporating the gradient of the damage field, a dissipation potential and the postulate of maximum dissipation. As a result, the driving force conjugated to damage evolution is identified as the quasi-nonlocal energy release rate, which essentially incorporates the divergence of a vectorial damage flux besides the strictly local energy release rate. On the numerical side, besides balance of linear momentum, the algorithmic consistency condition must be solved in weak form. Thereby, the crucial issue is the selection of active constraints which is solved by an active set search algorithm borrowed from convex nonlinear programming. In the examples, we compare the behavior in local damage with the performance of the gradient formulation.