On gradient field theories: gradient magnetostatics and gradient elasticity

TL;DR

This paper reviews gradient field theories, specifically gradient magnetostatics and gradient elasticity, deriving non-singular expressions for key physical quantities and highlighting their similarities and regularization benefits.

Contribution

It provides a comprehensive comparison of gradient magnetostatics and gradient elasticity, deriving non-singular formulas and emphasizing their regularization advantages.

Findings

Derived non-singular magnetic vector potential and Biot-Savart law.

Presented non-singular dislocation key-formulas.

Highlighted similarities between electric current and dislocation loops.

Abstract

In this work the fundamentals of gradient field theories are presented and reviewed. In particular, the theories of gradient magnetostatics and gradient elasticity are investigated and compared. For gradient magnetostatics, non-singular expressions for the magnetic vector gauge potential, the Biot-Savart law, the Lorentz force and the mutual interaction energy of two electric current loops are derived and discussed. For gradient elasticity, non-singular forms of all dislocation key-formulas (Burgers equation, Mura equation, Peach-Koehler stress equation, Peach-Koehler force equation, and mutual interaction energy of two dislocation loops) are presented. In addition, similarities between an electric current loop and a dislocation loop are pointed out. The obtained fields for both gradient theories are non-singular due to a straightforward and self-consistent regularization.