Study of the velocities and modes of propagation of axisymmetric waves along an orthotropic hollow cylinder

Abstract

Elastic waves in a cylindrical waveguide made of isotropic and transversely isotropic (plane of isotropy perpendicular to the axis of the cylinder) materials have been investigated by many authors [i, 5, 6]. If the cylinder material is orthotropic, such as is characteristic of the structures of composites [3], then the problem is complicated considerably because the corresponding boundary-value problem does not permit solutions in cylindrical functions. In this case, it is correct to use the method of power series [5]. The solution algorithm here is similar to the solution in cylindrical functions for the above special cases in the sense that, in numerically analyzing dispersion ratios in both cases, it is necessary to add power series. The main difference between the methods is that the asymptotic behavior of the series is know~n for cylindrical functions with a large argument. This makes it possible to study the asymptotic properties of the frequency spectrum. As regards the general case, analysis shows that the frequency spectrum of a cylindrical layer asymptotically degenerates into the frequency spectrum of a plane layer, and the exact solution in elementary functions of the latter problem (Lamb problem) for an orthotropic layer is well known. Thus, the solution proposed in the present article makes it possible to fully study the exact spectrum of the phase and group velocities of elastic waves for a hollow cylinder made of a cylindrically orthotropic material.