Uniform asymptotic analysis for waves in an incompressible elastic rod I. Disturbances superimposed on an initially stress-free state

Abstract

This paper studies the propagation of disturbances in an initially stress-free elastic circular rod. Starting from the Mindlin-Hermann equations, we derive a fourth-order partial differential equation as the governing equation for small axial-radial deformations in a rod composed of an incompressible material. Then, we consider an initial-value problem with an initial singularity in the shear strain. Using the technique of Fourier transforms, we manage to express the physical quantities in terms of integrals. The classical method of stationary phase is first used to obtain some important information. However, for material points in a neighbourhood behind the shear-wave front, the phase function of the integrals has a stationary point which approaches positive infinity. Consequently, the classical method of stationary phase fails completely. Here, instead, we use a new method developed by us (Dai & Wong 1994 Wave Motion 19, 293-308) to handle this case. An asymptotic expansion for the shear strain, which is uniformly valid in a neighbourhood behind the shear-wave front, is derived. This uniform asymptotic expansion reveals that for the shear strain there is a transition from O(1) disturbance to O(t -1/4 ) disturbance as the distance to the shear-wave front increases. We also find that the shear strain has three other jumps in terms of asymptotic orders. The first jump is from a larger O(t -2/3 ) disturbance to a smaller O(t -1 ) disturbance behind and ahead of the bar-wave front. The second jump is from a larger O(1) disturbance to a smaller O(t -1/2 ) disturbance behind and ahead of the bar-wave front. This also implies that in an asymptotic sense the initial singularity in the shear strain will be preserved and propagates with the shear-wave speed as time progresses. The third jump is from a smaller O(t -1 ) disturbance to a larger O(t -1/3 ) disturbance behind and ahead of a third wave front.