Dynamic fiber inclusions with elliptical and arbitrary cross-sections and related retarded potentials in a quasi-plane piezoelectric medium

Abstract

A piezoelectric medium of transversely isotropic symmetry with continuous fiber inclusion parallel to the axis of symmetry is considered. The problem is equivalent to a two-dimensional ‘ quasi-plane’ piezoelectric medium containing a 2D inclusion. The inclusion is assumed to undergo a spatially uniform δ( t)-type time domain transformation. The continuous fiber has elliptical, circular and arbitrary cross-sections. The solutions of the inclusion problem is expressed by scalar potentials. In the time domain two of these functions correspond to the retarded potential integrals of the inclusion. Their frequency domain representation which we shall call the ‘ dynamic potentials of the inclusion’ are also considered. Integral formulae are derived for continuous fiber inclusions with elliptical cross-sections. Known closed-form solutions are reproduced for circular fibers. For fibers with arbitrary cross-sections a numerical method based on Gauss quadrature is applied. High accuracy and efficiency of the numerical method is confirmed. Characteristic superposition and runtime effects for the inclusions are found.