Line-integral solution for the stress and displacement fields of an arbitrary dislocation segment in isotropic bi-materials in 3D space

Abstract

The solutions for the stress and displacement fields due to an arbitrary dislocation segment in an isotropic bi-material medium consisting of joined three-dimensional (3D) half spaces are derived and expressed in terms of line integrals, integrands of which are given in an exact analytical form that, in turn, can also be integrated to yield analytical expressions for the stress–displacement field. The solution is constructed by employing a general solution derived by Walpole [Int. J. Eng. Sci. 34 (1996) p.629] for any elastic singularity in joined isotropic half space, and combining it with Mura's integral formula for the displacement gradient of an arbitrary dislocation segment in homogeneous medium. The resulting new solution provides a framework for deriving analytical expressions for stress and displacement fields of dislocation curves of arbitrary shapes and orientations. The benefit of the method developed, as compared with other methods found in the literature, is that the new solution presented is naturally divided into two components, a homogenous component representing the field of a dislocation in an infinitely homogenous medium, and an image component. This makes it easy and straightforward to modify existing dislocation dynamics codes that already include the homogenous part. To illustrate the accuracy of the method, the stress field expressions of an edge dislocation with Burgers vector perpendicular to the bi-material interface are derived as a degenerate case of the general result. It is shown that our solution is identical to that found in the literature for this case.

Keywords:

• dislocation
• image stress
• interface

Acknowledgement

This work was supported by the Division of Materials Science and Engineering, Office of Basic Energy Science at the U.S. Department of Energy under grant number DE-FG02-07ER4635.