Wave Propagation in a Strongly Nonlinear Locally Resonant Granular Crystal

TL;DR

This paper investigates wave dynamics in a novel locally resonant granular crystal, revealing complex interactions, decay of primary pulses, emergence of secondary waves, and the formation of weakly nonlocal solitary waves with oscillatory tails.

Contribution

It introduces a new model for wave propagation in resonant granular crystals and uncovers the existence of weakly nonlocal solitary waves with oscillatory tails through analytical and numerical analysis.

Findings

Decay of primary traveling pulses in the system

Emergence and interference of secondary waves

Existence of weakly nonlocal solitary waves with oscillatory tails

Abstract

In this work, we study the wave propagation in a recently proposed acoustic structure, the locally resonant granular crystal. This structure is composed of a one-dimensional granular crystal of hollow spherical particles in contact, containing linear resonators. The relevant model is presented and examined through a combination of analytical approximations (based on ODE and nonlinear map analysis) and of numerical results. The generic dynamics of the system involves a degradation of the well-known traveling pulse of the standard Hertzian chain of elastic beads. Nevertheless, the present system is richer, in that as the primary pulse decays, secondary ones emerge and eventually interfere with it creating modulated wavetrains. Remarkably, upon suitable choices of parameters, this interference "distills" a weakly nonlocal solitary wave (a "nanopteron"). This motivates the consideration of such nonlinear structures through a separate Fourier space technique, whose results suggest the existence of such entities not only with a single-side tail, but also with periodic tails on both ends. These tails are found to oscillate with the intrinsic oscillation frequency of the out-of-phase motion between the outer hollow bead and its internal linear attachment.