Torsional Waves in Composite Co-axial Cylinders with Imperfect Interfaces

We consider the propagation of time-harmonic torsional waves in composite bi-material elastic cylinders. The interface between the core and the surrounding cladding is assumed to be imperfect: the tractions are continuous across the interface but the displacement jump is proportional to the stress acting on the interface. For torsional waves, this requires the introduction of a single constant of proportionality, F . We derive a frequency equation for the cylinder. The analysis recovers the known dispersion curves for a bimaterial rod with a perfect (welded) interface (F = 0), and has the correct limiting behavior for large F . We show that the modes, at any given frequency, are orthogonal, and outline how the problem of reflection of a torsional mode by a planar defect (such as a circumferential crack) can be treated.