On gradient field theories: gradient magnetostatics and gradient elasticity

摘要

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