On gradient field theories: gradient magnetostatics and gradient elasticity

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.

arXiv comment40 pages, to appear in: Philosophical Magazine

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• Distributional and regularized radiation fields of non-uniformly moving straight dislocations, and elastodynamic Tamm problem
• Atomistically enabled nonsingular anisotropic elastic representation of near-core dislocation stress fields in $\alpha$-iron
• Green functions and propagation in the Bopp-Podolsky electrodynamics
• Electronic structure study of screw dislocation core energetics in Aluminum and core energetics informed forces in a dislocation aggregate
• A non-singular continuum theory of point defects using gradient elasticity of bi-Helmholtz type
• On the gradient of the Green tensor in two-dimensional elastodynamic problems, and related integrals: Distributional approach and regularization, with application to nonuniformly moving sources
• Second gradient electromagnetostatics: electric point charge, electrostatic and magnetostatic dipoles
• On gradient enriched elasticity theories: A reply to "Comment on 'On non-singular crack fields in Helmholtz type enriched elasticity theories' " and important theoretical aspects
• Dislocations and cracks in generalized continua