Localization Analysis of an Energy-Based Fourth-Order Gradient Plasticity Model

TL;DR

This paper develops analytical and numerical solutions for localized plastic zones in a uniaxial bar using an energy-based fourth-order gradient plasticity model, highlighting the effects of variable cross-section and softening.

Contribution

It introduces an energy-based variational approach for a fourth-order gradient plasticity model with explicit derivation of governing equations and boundary conditions, including multiple example analyses.

Findings

Plastic zone size and distribution depend on yield stress regularity.

Energy balance and plastic elongation are consistent with the model.

Results differ from non-variational gradient formulations.

Abstract

The purpose of this paper is to provide analytical and numerical solutions of the formation and evolution of the localized plastic zone in a uniaxially loaded bar with variable cross-sectional area. An energy-based variational approach is employed and the governing equations with appropriate physical boundary conditions, jump conditions, and regularity conditions at evolving elasto-plastic interface are derived for a fourth-order explicit gradient plasticity model with linear isotropic softening. Four examples that differ by regularity of the yield stress and stress distributions are presented. Results for the load level, size of the plastic zone, distribution of plastic strain and its spatial derivatives, plastic elongation, and energy balance are constructed and compared to another, previously discussed non-variational gradient formulation.