On the spatial formulation of discontinuous Galerkin methods for finite elastoplasticity

In this paper, we present a consistent spatial formulation for discontinuous Galerkin (DG) methods applied to solid mechanics problems with finite deformation. This spatial formulation provides a general, accurate, and efficient DG finite element computational framework for modeling nonlinear solid mechanics problems. To obtain a consistent formulation, we employ the Incomplete Interior Penalty Galerkin (IIPG) method. Another requirement for achieving a fast convergence rate for Newton’s iterations is the consistent formulation of material integrators. We show that material integrators that are well developed and tested in continuous Galerkin (CG) methods can be fully exploited for DG methods by additionally performing stress returning on element interfaces. Finally, for problems with pressure or follower loading, stiffness contributed from loaded surfaces must also be consistently incorporated. In this work, we propose the Truesdell objective stress rate for both hypoelastoplastic and hyperelastoplastic problems. Two formulations based on the co-rotational and multiplicative decomposition-based frameworks are implemented for hypoelastoplasticity and hyperelastoplasticity, respectively. Two new terminologies, the so-called standard surface geometry stiffness and the penalty surface geometry stiffness, are introduced and derived through consistently linearizing the virtual work contributed from interior surface integrals. The performance of our DG formulation has been demonstrated through solving a cantilever beam problem undergoing large rotations, as well as a bipolar void coalescence problem where the voids grow up to several hundred times of their original volumes. Fast convergence rates for Newton’s iterations have been achieved in our IIPG implementation.