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.

Boudinage and folding as an energy instability in ductile deformation

“…Hence, both theories are related through their perspective. The latter approach offers the great opportunity to explicitly calculate the energetics of (micro‐)physical processes and their feedbacks, which is relevant across many scientific disciplines, e.g., manufacturing processes of metals and polymers [e.g., Chrysochoos et al , ; Horstemeyer et al , ; Rittel , ; Steif , ; Yalcinkaya and Lancioni , ; Zheng et al , ]. A general way of approaching the theory of localization is to start from the class of viscoplastic materials and to obtain the solid mechanical type of instability as a limiting case of the mathematical system of equations.…”

Strain Gradient Crystal Plasticity: Thermodynamics and Implementation

“…Depending on the plastic slip potential for the energetic hardening, the models are distinguished as convex and nonconvex in nature. The convex type of model, which is used in this chapter, is essentially used for the size effect predictions, while the non-convex type of models could be employed for the simulation of intrinsic microstructure evolution or macroscopic localization and necking (see, e.g., Yalcinkaya et al , 2012Yalçinkaya 2013;Yalcinkaya and Lancioni 2014). The purpose of this section is to study the thermodynamics and the derivation of the convex-type strain gradient crystal plasticity model and to address the ways to incorporate different physical phenomena into the developed framework.…”

“…In addition to their success in predicting size effects, strain gradient crystal plasticity models have been improved further to simulate the intragranular microstructure evolution, intergranular grain boundary behavior, and macroscopic strain localization leading to necking in metallic materials (see, e.g., also Yalcinkaya et al , 2012Özdemir and Yalçinkaya 2014;Yalcinkaya and Lancioni 2014;Lancioni et al 2015a). A thermodynamically consistent incorporation of a proper plastic potentials results in different responses in terms of both strain distribution and the global constitutive response.…”

Strain Gradient Crystal Plasticity: Intragranular Microstructure Formation

“…The free energy is assumed to depend only on the elastic strain and the dislocation density. In comparison to recent works of the authors (see, e.g., Yalcinkaya et al 2011, Yalçinkaya et al 2012, Yalçinkaya 2013, Yalcinkaya and Lancioni 2014, Lancioni et al 2015a where the framework is developed and used for modeling the inhomogeneous deformation field (microstructure) formation and evolution, the current chapter does not include the energetic hardening term in the bulk material, which could be convex or non-convex in nature. The details of such modeling approaches are discussed in one of the chapters here.…”