Interaction between a screw dislocation and an elastic elliptical inhomogeneity by using the angular basis function

Abstract

The behavior of displacement field due to a screw dislocation is similar to the angular basis function (ABF) Arg(z). It is different from the radial basis function (RBF) ln(r) that is used to describe the velocity potential of a sink or source. Nevertheless, the complex-valued fundamental solution ln(z) contains the two parts of RBF ln(r) and ABF Arg(z). In this paper, not only the RBF in the null-field boundary integral equation (BIE) but also the ABF for the screw dislocation are employed to study the interaction between a screw dislocation and an elastic elliptical inhomogeneity. This problem is decomposed into a free field with a screw dislocation and a boundary value problem containing an elliptical inhomogeneity. The boundary value problem is solved by using the RBF and the null-field BIE. Since the geometric shape is an ellipse, the degenerate kernel is expanded to a series form under the elliptical coordinates, while the unknown boundary densities are expanded to eigenfunctions. By combining the degenerate kernel and the null-field BIE, the boundary value problem can be easily solved. The inconsistency between Sendeckyj (In: Simmons JA, et al (eds) Fundamental aspects of dislocation theory. US National Bureau of Standards, Gaithersburg, pp 57–69, 1970) and Gong and Meguid (Int J Eng Sci 32(8):1221–1228, 1994) for the problem was also found by using the present approach. The error in Gong and Meguid (Int J Eng Sci 32(8):1221–1228, 1994) was also printed out. Finally, some examples are demonstrated to verify the validity of the present approach.