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

TL;DR

The paper tackles the numerical challenge of solving strain gradient dependent damage models, which typically require high continuity in finite elements. It introduces a discontinuous Galerkin two-field formulation that uses standard C0 displacement elements together with a gradient measure, enforcing interelement continuity weakly via interior-penalty terms. In 1D, with a simple element (piecewise linear displacement and constant gradient measure), the method regularizes the problem and achieves excellent agreement with a high-continuity benchmark. This approach offers a simple, robust finite element framework for gradient-enhanced continuum models with potential extension to higher-order elements and multi-dimensional problems.

Abstract

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 $C^{1}$ continuity of the shape functions. Here, recent development in discontinuous Galerkin methods are explored and exploited for the solution of a prototype nonlinear strain gradient dependent continuum model. A formulation is developed that allows the rigorous solution of a strain gradient damage model using standard $C^{0}$ shape functions. The formulation is tested in one-dimension for the simplest possible finite element formulation: 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.

Figures (7)

• Figure 1: Two element configuration. Displacement degrees of freedom are located at the circular nodes, and $\overline{\epsilon}$ degrees of freedom are located at the squares.
• Figure 2: Linearly tapering bar.
• Figure 3: Load-displacement response for the tapered bar.
• Figure 4: Damage profiles for the tapered bar.
• Figure 5: Bar with narrow central section.