Dislocation Loops In Transversely Isotropic Materials

Motivated by both the three-dimensional (3D) discrete dislocation dynamics (DD) simulation and the dislocation theory for seismology, In this paper, the statics for an arbitrary3D dislocation loop of the Burgers type in transversely isotropic media is systematically established for the first time, including displacements, strains, stresses, the self-energy and the interaction energy. For a full space, or half space, or bimaterial, or single-layered half space composed of non-degenerate or degenerate transversely isotropic purely elastic, or piezoelectric, or piezomagnetic, or multiferroic (i.e. magneto-electro-elastic) materials, based upon its static Green’s function expressed newly in terms of second derivatives of certain potential functions, we have derived for the first time the line-integral representations for the displacements, stresses, and interaction energy due to arbitrary3D dislocation loops of the Burgers type in it. These simple and elegant line-integral expressions are the direct and natural extension of the well-known Burgers’ formula for displacements, the Peach-Koehler’s formula for stresses, and the Blin’s formula for the interaction energy to more general and complex cases in the presence of transverse isotropy, multi-field coupling, or material interface/surface. Moreover, for the important case of an arbitrary3D polygonal dislocation loop, the induced displacements, strains, and stresses are obtained directly from the above line-integral representations and expressed analytically in terms of elementary functions.Based upon the line-integral representations for the stresses and the self-energy due to an arbitrary3D dislocation loop of the Burgers type in a transversely isotropic full space, a nonsingular continuum theory of dislocations in transversely isotropic media is proposed, which automatically satisfies the self-consistency between the self-stress and the self-energy.The concept of quasi solid angle is introduced to describe the discontinuities over the dislocation surface. From the line-integral expressions of the quasi solid angle, we have derived an analytical expression for the quasi solid angle or the classical solid angle subtended by an arbitrary3D polygon. This analytical expression is simple and elegant, and is very suitable for numerical computations.Several numerical examples are provided to fully verify the correctness of the formulae shown in this paper, and to investigate in detail the effects of material transverse isotropy, material interface/surface, and multi-field coupling on the static response of dislocation loops.