A discontinuous Galerkin formulation of a model of gradient plasticity at finite strains

The work presented here constitutes an extension to the finite-strain regime of a discontinuous Galerkin based, strain gradient plasticity formulation presented in Djoko et al. [J.K. Djoko, F. Ebobisse, A.T. McBride, B.D. Reddy, A discontinuous Galerkin formulation for classical and gradient plasticity – Part 1: formulation and analysis, Comput. Methods Appl. Mech. Engrg. 196 (2007) 3881–3897; J.K. Djoko, F. Ebobisse, A.T. McBride, B.D. Reddy, A discontinuous Galerkin formulation for classical and gradient plasticity – Part 2: algorithms and numerical analysis, Comput. Methods Appl. Mech. Engrg. 197 (2007) 1–21]. The focus here is on algorithmic and computational aspects of the formulation at finite strains. The adoption of a logarithmic hyperelastic–plastic formulation preserves the essential features of the infinitesimal formulation. This key ingredient allows the predictor–corrector solution algorithms developed for the infinitesimal gradient formulation to be extended readily to the finite-strain regime. The use of low-order elements is essential to contain the computational expense of the formulation but these elements are prone to locking. The method of enhanced assumed strains for geometrically nonlinear problems is utilised to circumvent this limitation. The form of the consistent tangent modulus is derived for the case of gradient plasticity. Two numerical examples are presented to illustrate aspects of the approximation scheme and the algorithm, as well as features of the model of gradient plasticity.