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On gradient field theories: gradient magnetostatics and gradient elasticity

机译：关于梯度场理论：梯度静磁学和梯度弹性

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摘要

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 gradientmagnetostatics, 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.
机译：在这项工作中，提出并回顾了梯度场理论的基础。特别是对梯度静磁学和梯度弹性理论进行了研究和比较。对于梯度静磁，推导并讨论了磁矢量规范电势的非奇异表达式，Biot-Savart定律，洛伦兹力和两个电流环的相互作用能量。对于梯度弹性，提出了所有位错关键公式（Burgers方程，Mura方程，Peach-Koehler应力方程，Peach-Koehler力方程以及两个位错环的互作用能）的非奇异形式。另外，指出了电流回路和位错回路之间的相似性。由于直接和自洽的正则化，两种梯度理论的获得字段都不是奇异的。

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