Non-singular straight dislocations in anisotropic crystals

Abstract

A non-singular dislocation theory of straight dislocations in anisotropic crystals is derived using simplified anisotropic incompatible first strain gradient elasticity theory. Based on the non-singular theory of dislocations for anisotropic crystals, all dislocation key-formulas of straight dislocations are derived in generalized plane strain, for the first time. In this model, the singularity of the dislocation fields at the dislocation core is regularized owing to the nonlocal nature of strain gradient elasticity. The non-singular dislocation fields of straight dislocations are obtained in terms of two-dimensional anisotropic Green functions of simplified anisotropic strain gradient elasticity. All necessary Green functions, including the two-dimensional Green tensor of the twofold anisotropic Helmholtz-Navier operator and the two-dimensional \(\varvec{F}\)-tensor of generalized plane strain, are derived as sum of the classical part and a gradient part in terms of Meijer G-functions. Among others, we calculate the regularization of the Barnett solution for the elastic distortion of straight dislocations in an anisotropic crystal. In the framework of simplified anisotropic first strain gradient elasticity, the necessary material parameters are computed for cubic materials including aluminum (Al), copper (Cu), iron (Fe) and tungsten (W) using a second nearest-neighbour modified embedded-atom-method interatomic potential. The elastic distortion and stress fields of screw and edge dislocations of \(\frac{1}{2} \langle 111\rangle\) Burgers vector in bcc iron and bcc tungsten and screw and edge dislocations of \(\frac{1}{2} \langle 110\rangle\) Burgers vector in fcc copper and fcc aluminum have been computed and presented in contour plots.