Dynamic Green's functions for multiple elliptical inclusions with imperfect interfaces

Abstract

The problem of an unbounded elastic solid with multiple elliptical inclusions subjected to a time-harmonic anti-plane concentrated force is semi-analytically solved by using the collocation multipole method. The displacement of matrix and inclusion are represented by angular and radial Mathieu functions. The imperfect interface between the matrix and the inclusion is characterized as a linear spring model with vanishing thickness. It is the derivative that the imperfect condition is involved. The addition theorem of Mathieu function is frequently used to solve multiply-connected domain problems in the traditional multipole method. An alternate here is a direct computation. The associated normal derivative with respect to a non-local elliptical coordinate system is developed by means of directional derivative. Besides simple computation, no truncation error is caused. The displacement field is determined by using the imperfect interface conditions through collocating points along the interface. Several numerical experiments are done to investigate the effects of the driving frequency of the concentrated force, imperfect interface and the convexity of elliptical inclusions on the dynamic Green's functions.