Stress of a spatially uniform dislocation density field

TL;DR

This paper investigates the stress effects of uniform dislocation density fields in elastic materials, showing that linear elasticity results do not extend to nonlinear cases, highlighting a fundamental geometric limitation.

Contribution

It proves that the stress vanishing in linear elasticity does not hold in nonlinear elasticity and establishes a geometric constraint on rotation fields.

Findings

Stress vanishes for uniform dislocation density in linear elasticity.

In nonlinear elasticity, the stress may not vanish under the same conditions.

A geometric limitation on rotation fields is demonstrated.

Abstract

It can be shown that the stress produced by a spatially uniform dislocation density field in a body comprising a linear elastic material under no loads vanishes. We prove that the same result does not hold in general in the geometrically nonlinear case. This problem of mechanics establishes the purely geometrical result that the $\curl$ of a sufficiently smooth two-dimensional rotation field cannot be a non-vanishing constant on a domain.