Necking instabilities in elasto-viscoplastic materials

TL;DR

This paper develops a theoretical framework to analyze necking instabilities in various elasto-viscoplastic materials, deriving criteria for localization onset and validating with models for amorphous and crystalline plastics.

Contribution

It introduces a general eigenvalue-based stability analysis for necking in elasto-viscoplastic materials, connecting physical effects to localization criteria across different material models.

Findings

Derived a general expression for the eigenvalue governing necking stability.

Proposed criteria for the onset of necking and localization.

Validated predictions with models for amorphous and crystalline materials.

Abstract

Necking instabilities, in which tensile (extensional) deformation localizes into a small spatial region, are generic failure modes in elasto-viscoplastic materials. Materials in this very broad class --- including amorphous, crystalline, polycrystalline and other materials --- feature a predominantly elastic response at small stresses, plasticity onset at a rather well-defined yield stress and rate-dependence. Necking instabilities involve a unique coupling between the system's geometry and its constitutive behavior. We consider generic elasto-viscoplastic constitutive relations involving an internal-state field, which represents the structural evolution of the material during plastic deformation, and study necking in the long-wavelength approximation (sometimes termed the lubrication or slender-bar approximation). We derive a general expression for the largest time-dependent eigenvalue in an approximate WKB-like linear stability analysis, highlighting various basic physical effects involved in necking. This expression is then used to propose criteria for the onset of necking and more importantly for the emergence of strong localization. Applications to strain-softening amorphous plasticity, in the framework of the Shear-Transformation-Zone model, and to strain-hardening crystalline/polycrystalline plasticity, in the framework of the Kocks-Mecking model, are presented. These quantitative analyses of widely different material models support the theoretical predictions, most notably for the strong localization during necking.