Addendum to "On Extensional Oscillations and Waves in Elastic Rods"

Abstract

In a previous paper on infinitesimal waves in an infinite elastic Cosserat rod [1], we proposed values for various material parameters, hereafter referred to as KB2. These parameters were determined by matching the response of the Cosserat rod with that of a three-dimensional elastic cylinder. Subsequently, we have studied extensional vibrations in rods of finite length [2]. For a small range of intermediate frequencies, it was found that a slightly different set of parameters yielded more accurate results. This latter set referred to as KB 1 is the same as KB2, except that (1;2 of KB 1 is computed with .8 being the first nonzero root, rather than the second, of (1 2v)J1(.8a) = (1 v).8aJo(.8a). Here, a is the radius of the rod, v is Poisson's ratio, and In is the Bessel function of order n. The phase and group speeds of the Pochhammer-Chree solution are commonly plotted only for the first two or three modes. Figures 1 and 2 depict these speeds for the first nine modes. We compare the dispersion curves (Figs. 3-8) and group speeds (Figs. 9-11) arising from the Cosserat theory using KB 1 and KB2, with those from the Pochhammer-Chree solution. In these figures, results for the Cosserat theory using material parameters GN, given by Green and Naghdi, and reported in [1], are also included. The nondimensional wave numbers k and k are defined by k =kaj21f and k =kajd, where k is the wave number, and d is the first nonzero root of J1 (x) = 0 (see Onoe et al. [3]). Both the phase speed C and group speed Cg are nondimensionalized by the classical bar wave speed Co and are denoted by a superimposed hat. All plots have been generated for v = 0.29. We discuss some of the salient features of these results. First, we note that all the qualitative features of the Pochhammer-Chree solution are captured by the Cosserat solution; the only exception is that the Cosserat solution seems to smear out the rapid variation of Cg for the two higher modes in a small range of low k. The low-frequency limit of Pochhammer-Chree solution for the first mode is C = 1. The higher two modes have no cutoff frequency. In the high-frequency limit, the phase speed of the first mode approaches the Rayleigh wave speed CR, whereas the second mode tends to the shear wave speed cs.