A Method for Gradient Enhancement of Continuum Damage Models
A method for the regularization of continuum damage material models based on gradient-type enhancement of the free-energy functional is presented. Direct introduction of the gradient of the damage variable would require C1 interpolation of the displacements, which is a complicated task to achieve with quadrilateral elements. Therefore a new variable ﬁeld is introduced, which makes the model non-local in nature, while preserving C0 interpolation order of the variables at the same time. The strategy is formulated as a pure minimization problem, therefore the LBB-condition does not apply in this case. However, we still take the interpolation of the displacement ﬁeld one order higher than the interpolation of the ﬁeld of additional (non-local) variables. That leads to increased accuracy and removes the post-processing step necessary to obtain consistent results in the case of equal interpolation order. Several numerical examples which show the performance of the proposed gradient enhancement are presented. The pathological mesh dependence of the damage model is efﬁciently removed, together with the difﬁculties of numerical calculations in the softening range. Calculations predict a development of the damage variable which is mesh-objective for ﬁxed internal material length.