Uniform asymptotic analysis for waves in an incompressible elastic rod II.Disturbances superimposed on a pre-stressed state

This paper studies the propagation of disturbances superimposed on a pre-st ressed incompressible hyperelastic thin rod. Starting from the incremental equations given by Haughton and Ogden (1979, J. Mech. Phys. Solids 27, 179- 212 and 489-512), we derive a three parameter-dependent one-dimensional rod equation as the governing equation. In particular, it is found that one pa rameter plays a crucial role. Depending on whether it is larger or smaller than or equal to a critical value, the shear-wave velocity is larger or sma ller than or equal to the bar-wave velocity. In the case that these two vel ocities are equal, there exist travelling-wave solutions of arbitrary form. This implies that for this particular case the initial disturbance would p ropagate along the rod without distortion. To see the influence of the pre- stress in detail, we further consider an initial-value problem with an init ial singularity in the shear strain. The solutions are expressed in terms o f integrals through the method of Fourier transform. We then conduct an asy mptotic analysis for the solutions. For a material point in a neighbourhood behind the shear-wave front, the phase function of these integrals has a s tationary point at infinity. Here, we use a technique of uniform asymptotic expansion to handle this case. An asymptotic expansion, correct up to orde r O(t(-1)), for the shear strain, which is uniformly valid in a neighbourho od behind the shear-wave front, is derived. For material points in other sp atial domains, the method of stationary point is applicable, and asymptotic expansions (correct up to order O(t(-1))) are obtained. A novelty is that we are able to deduce precise qualitative information about the waves in th e far field from our analytic results. Wave profiles for two concrete examp les are also provided.