The fundamentals of non-singular dislocations in the theory of gradient elasticity: dislocation loops and straight dislocations

TL;DR

This paper develops a comprehensive theory of non-singular dislocations within gradient elasticity, providing regularized formulas for dislocation fields and extending classical dislocation theory to non-singular, analytical solutions.

Contribution

It introduces a general framework for non-singular dislocation fields using gradient elasticity of Helmholtz and bi-Helmholtz types, including new regularized formulas for dislocation loops and straight dislocations.

Findings

Derived non-singular dislocation fields using Green's functions.

Presented modified Mura, Peach-Koehler, and Burgers formulas.

Obtained exact analytical solutions for dislocation problems.

Abstract

The fundamental problem of non-singular dislocations in the framework of the theory of gradient elasticity is presented in this work. Gradient elasticity of Helmholtz type and bi-Helmholtz type are used. A general theory of non-singular dislocations is developed for linearly elastic, infinitely extended, homogeneous, and isotropic media. Dislocation loops and straight dislocations are investigated. Using the theory of gradient elasticity, the non-singular fields which are produced by arbitrary dislocation loops are given. `Modified' Mura, Peach-Koehler, and Burgers formulae are presented in the framework of gradient elasticity theory. These formulae are given in terms of an elementary function, which regularizes the classical expressions, obtained from the Green tensor of the Helmholtz-Navier equation and bi-Helmholtz-Navier equation. Using the mathematical method of Green's functions and the Fourier transform, exact, analytical, and non-singular solutions were found. The obtained dislocation fields are non-singular due to the regularization of the classical singular fields.