Based upon Eringen’s model of non-local anisotropic elasticity, new solutions were derived for the stress fields of screw dislocations in anisotropic materials. In the theory of non-local anisotropic elasticity, the anisotropy was two-fold. The anisotropic material behaviour was not only included in the anisotropy of the elastic stiffness properties, but also in the anisotropy of the non-locality which was expressed by the anisotropy of the length scale parameters; incorporated into the anisotropy of the non-local kernel function. In particular, a new two-dimensional anisotropic kernel which was the Green’s function of a linear differential operator with three length-scale parameters was derived analytically. New solutions for the stresses of straight screw dislocations in anisotropic (monoclinic and hexagonal) materials were found. The stresses did not have singularities, and possessed interesting features of anisotropy. The new mathematical solutions for the stress fields were simpler than those of Eringen and Balta (1978) and were consequently easier to apply. It was assumed that, in the general case, the non-local constitutive moduli of the anti-plane strain problem were given in terms of the local constitutive moduli and a non-local kernel. For the construction of the non-singular stresses it was assumed that the anisotropy of the non-locality was determined by the anisotropy of the elastic moduli c44, c45 and c55. Because the non-locality was predominant in the region near to the dislocation line (dislocation core), the anisotropy of the dislocation core behaviour was characterized for instance, in the case of hexagonal materials, by the elastic constants c44, c55, c11 and c22. However, it seemed that the anisotropy of the anti-plane strain problem, if specified, should be characterized only by the elastic moduli of the anti-plane strain problem, c44 and c55, and not by the elastic constants c11 and c22; that belonged to the plane strain problem in anisotropic elasticity. A generalization of the results was suggested to be the construction of the stress fields using the unspecified non-local kernel. Such solutions were expected to be more complicated (e.g. integral representations) and given in terms of unknown parameters. It was also questionable in this case whether the difference between the anisotropy of the non-locality and the anisotropy of the elastic stiffness properties would give rise to an effect which was measurable. In order to remove the singularities of the strain fields, it was possible to use an inhomogeneous Helmholtz-type equation for the strain in an anisotropic gradient formulation. Finally, the present approach could also be applied to edge dislocations in anisotropic non-local elasticity.

Screw Dislocation in Nonlocal Anisotropic Elasticity. M.Lazar, E.Agiasofitou: International Journal of Engineering Science, 2012, 49[12], 1404-14