A fast and robust numerical treatment of a gradient-enhanced model for brittle damage

Damage processes are modeled by a softening behavior in a stress/strain diagram. This reveals that the stiffness loses its ellipticity and the energy is thus not coercive. A numerical implementation of such ill-posed problems yields results that are strongly dependent on the chosen spatial discretization. Consequently, regularization strategies have to be employed that render the problem well-posed. A prominent method for regularization is a gradient enhancement of the free energy. This, however, results in field equations that have to be solved in parallel to the Euler-Lagrange equation for the displacement field. An usual finite element treatment thus deals with an increased number of nodal unknowns, which remarkably increases numerical costs. We present a gradient-enhanced material model for brittle damage using Hamilton’s principle for nonconservative continua. We propose an improved algorithm, which is based on a combination of the finite element and strategies from meshless methods, for a fast update of the field function. This treatment keeps the numerical effort limited and close to purely elastic problems. Several boundary value problems prove the mesh-independence of the results.