Softening Gradient Plasticity: Analytical Study of Localization under Nonuniform Stress

TL;DR

This paper provides analytical solutions for localization in gradient plasticity models under nonuniform stress, revealing how internal length scales influence strain localization and load-displacement behavior.

Contribution

It offers explicit analytical solutions for one-dimensional localization problems with nonuniform stress, enhancing understanding of softening gradient plasticity models.

Findings

Global load-displacement diagrams can show hardening despite local softening.

Internal length scales affect the shape of strain localization profiles.

Analytical solutions clarify the interplay between material and geometric effects.

Abstract

Localization of plastic strain induced by softening can be objectively described by a regularized plasticity model that postulates a dependence of the current yield stress on a nonlocal softening variable defined by a differential (gradient) expression. This paper presents analytical solutions of the one-dimensional localization problem under certain special nonuniform stress distributions. The one-dimensional problem can be interpreted as describing either a tensile bar with variable cross section, or a beam subjected to a nonuniform bending moment. Explicit as well as implicit gradient formulations are considered. The evolution of the plastic strain profile and the shape of the load-displacement diagram are investigated. It is shown that even if the local constitutive law exhibits softening right from the onset of yielding, the global load-displacement diagram has a hardening part. The interplay between the internal length scales characterizing the material and the geometry is discussed.