A single Gauss point continuum finite element formulation for gradient-extended damage at large deformations

Abstract

Some years ago, a family of large deformation continuum finite elements based on reduced integration was investigated by Reese (2005), Schwarze and Reese (2011) and Frischkorn and Reese (2015). Many structural components with different kinds of elastic and inelastic material behavior were considered and these elements showed accurate results while being more efficient than similar three-dimensional formulations based on full integration. The objective of the present contribution is to extend the analysis to non-local damage and fracture. To this end, the incorporation of the gradient-extended damage plasticity model for large deformations of Brepols et al. (2020) into the framework of reduced integration-based continuum elements is presented. For the sake of brevity, the present work is restricted to solids with only one integration (Gauss) point in the center of the element. The weak form of the formulation, which is based on a two-field variational functional closely related to the enhanced-assumed strain (EAS) method is extended by the weak form of the micromorphic balance equation. The steps required in order to transform the extended formulation into a stable, robust and efficient single Gauss point concept are described in detail. Due to the analogy to fully-coupled thermomechanical problems, the derivation of a novel micromorphic hourglass stabilization is based on earlier contributions of the research group. Therein, the Taylor series expansion of all constitutively dependent quantities plays a crucial role. Representative numerical examples of quasi-brittle and ductile fracture reveal the accuracy and efficiency of the proposed approach. Besides the ability to deliver mesh-independent results, the framework is especially suitable for constrained situations in which conventional low-order finite elements suffer from well-known locking phenomena.