Formulation of continuous/discontinuous Galerkin methods for strain gradient-dependent damage

Abstract

Continuum damage models are widely used to represent the development of microscopic defects that coalesce into a macroscopic crack. The microscopic defects cause a progressive weakening or softening of the material (damage). Strain gradient-dependent terms have been included in some damage theories to regularize them, and thereby avoid a pathological mesh-dependence in the solution. A strain gradient-dependent damage model is considered here for the simulation of this feature in quasi-brittle materials. In the model considered, the damage parameter depends upon a regularized equivalent strain. The regularization is introduced through a dependency on the Laplacian of an equivalent strain measure. The introduction of the Laplacian of the strain leads to numerical difficulties as the governing differential equations are fourth-order, and additional boundary conditions must be specified. The application of such a model in a standard finite element framework requires $C^1$ continuity of the shape functions. Here, a continuous/discontinuous mixed Galerkin method is presented which avoids the need for high-order continuity. The formulation allows the use of $C^0$ or $C^{-1}$ interpolations for the regularized strain field and a $C^0$ interpolation of the displacement field. Numerical examples are presented to validate the formulation in one and two dimensions. Several interpolations are tested extensively in one dimension in order to provide guidance for the most appropriate formulations in two dimensions. The formulation is applied to crack propagation in a three-point bending test, with the computed result being independent of the discretization.