Oscillatory Impact of an Inhomogeneous Viscoelastic Rod

Abstract

A stress is applied to the end of a semi-infinite, inhomogeneous, “standard” viscoelastic rod. The rod is initially unstressed and at rest. The applied stress is uniformly distributed over the end, and it is oscillatory in time. The propagation of small-amplitude, one-dimensional, longitudinal waves is studied. A formal asymptotic expansion of the solution is obtained. A partial justification of the method for homogeneous rods is given. The leading term in the expansion represents a modulated, oscillating, progressive wave that propagates with variable velocity. The modulation depends on the elastic and viscous moduli. The velocity depends only on the elastic moduli. The Maxwell rod is studied as a limiting case of the standard rod. Application of the method to finite rods is discussed.