A Finite Strain Plastic-damage Model for High Velocity Impacts using Combined Viscosity and Gradient Localization Limiters: Part II - Numerical Aspects and Simulations

In this companion article, we present within the finite element context the numerical algorithms for the integration of the thermodynamically consistent formulation of geometrically nonlinear gradient-enhanced viscoinelasticity derived in the first part of the article. The proposed unified integration algorithms are extensions of the classical rate-independent return mapping algorithms to the rate-dependent problems. An operator split structure is used consisting of a trial state followed by the return map by imposing the generalized viscoplastic and visco-damage consistency conditions simultaneously. Furthermore, a trivially incrementally objective integration scheme is established for the rate constitutive relations. The proposed finite deformation scheme is based on hypoelastic stress-strain representations and the proposed elastic predictor and coupled viscoplastic-viscodamage corrector algorithm allows for the total uncoupling of geometrical and material nonlinearities. A simple and direct computational algorithm is also used for calculation of the higher-order gradients. This algorithm can be implemented in the existing finite element codes without numerous modifications as compared to the current numerical approaches for integrating gradient-dependent models. The nonlinear algebraic system of equations is solved by consistent linearization and the Newton-Raphson iteration. The proposed model is implemented in the explicit finite element code ABAQUS via the user subroutine VUMAT. Model capabilities are preliminarily illustrated for the dynamic localization of inelastic flow in adiabatic shear bands and the perforation of a 12 mm thick Weldox 460E steel plates by deformable blunt projectiles at various impact speeds. The simulated shear band results well illustrated the potential of the proposed model in dealing with the well-known mesh sensitivity problem. Consequently, the introduced implicit and explicit length-scale measures are able to predict size effects in localization failures. Moreover, good agreement is obtained between the numerical simulations and experimental results of the perforation problem.