Energy-based Modeling of Localization and Necking in Plasticity

Abstract

In this paper two different non-local plasticity models are presented and compared to describe the necking and fracture through a non-convex energy, where fracture is regarded as the extreme localization of the plastic strain. The difference between the models arises from the evolution of plastic deformation. The first (rate-dependent) approach, proposed in Yalcinkaya et al. (2011) follows the principle of virtual work to get balance equations and the dissipation inequality, in order to obtain the plastic evolution equation. The free-energy is given by the sum of a non-convex plastic term, and two quadratic terms with respect to the elastic deformation and the plastic deformation gradient. In the second (rate-independent) model, developed in Del Piero et al. (2013a), the plastic evolution is determined by incremental minimization of an energy functional which is equal to the free-energy of the previous model. The numerical example considers a convex-concave plastic energy to address the response of a tensile steel bar, where plastic strains localize intrinsically up to fracture. The numerical results exhibit good agreement between the two models. The solutions provided by the rate-dependent model approach those of the rate-independent model, as the imposed deformation rate reduces.