A discontinuous Galerkin formulation for a strain gradient-dependent continuum model

The numerical solution of strain gradient-dependent continuum problems has been dogged by continuity demands on the basis functions. For most commonly accepted models, solutions using the finite element method demand C1 continuity of the shape functions. Here, recent developments in discontinuous Galerkin methods are explored and exploited for the solution of a prototype non-linear strain gradient-dependent continuum model. A formulation is developed that allows the rigorous solution of a strain gradient damage model using standard C0 shape functions. The formulation is tested in one dimension for the simplest possible finite element formulation: continuous piecewise linear displacement and constant (on elements) internal variable. Numerical results are shown to compare excellently with a benchmark solution. The results are remarkable given the simplicity of the proposed formulation.