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 superimposedon a pre-stressed incompressible hyperelastic thin rod. Startingfrom the incremental equations given by Haughton and Ogden (1979,J. Mech. Phys. Solids 27, 179-212 and 489-512), we derive athree parameter-dependent one-dimensional rod equation as thegoverning equation. In particular, it is found that one parameterplays a crucial role. Depending on whether it is larger or smallerthan or equal to a critical value, the shear-wave velocity islarger or smaller than or equal to the bar-wave velocity. Inthe case that these two velocities are equal, there exist travelling-wavesolutions of arbitrary form. This implies that for this particularcase the initial disturbance would propagate along the rod withoutdistortion. To see the influence of the pre-stress in detail,we further consider an initial-value problem with an initialsingularity in the shear strain. The solutions are expressedin terms of integrals through the method of Fourier transform.We then conduct an asymptotic analysis for the solutions. Fora material point in a neighbourhood behind the shear-wave front,the phase function of these integrals has a stationary pointat infinity. Here, we use a technique of uniform asymtotic expansionto handle this case. An asymptotic expansion, correct up toorder O(t^-1), for the shear strain, which is uniformly validin a neighbourhood behind the shear-wave front, is derived.For material points in other spatial domains, the method ofstationary point is applicable, and asymptotic expansions (correctup to order O(t^-1)) are obtained. A novelty is that we are ableto deduce precise qualitative information about the waves inthe far field from our analytic results. Wave profiles for twoconcrete examples are also provided.