Strongly nonlinear acoustics of one-dimensional granular sonic vacua

This research is concerned with the nonlinear dynamics and acoustics of one dimensional ordered granular media. In particular the considered granular systems are composed of discrete elastic spherical beads that mutually interact through strongly nonlinear Hertzian force interaction law. In this work we primarily consider the granular chains in the limit of zero pre-compression, thus leading to complete absence of linear acoustics and zero speed of sound (as defined in the classical sense), hence their characterization as 'sonic vacua'. Furthermore, we sparingly incorporate a dissipative mechanism between interacting neighboring beads.;The first part of the study is primarily focused on the oscillatory dynamics of finite dimensional homogeneous granular chains. We initiate the study by investigating the existence of nonlinear normal modes (NNMs) in these systems with fixed boundary conditions. The realized modes which have energy dependent frequency, when represented on frequency-energy plots (FEP) divide these plots into two mutually exclusive regions separated by the out of phase NNM which corresponds to the highest possible oscillation frequency. All the NNMs realized are situated in the region below the out-of-phase NNM, which is denoted as propagation band; the complementary region is then the attenuation band. When the chain is harmonically base excited, frequencies in the propagation band are spatially extended whereas those in the attenuation band are spatially localized wherein the beads experience constant compression.;The second part of the study is concerned with periodic ordered diatomic (dimer) granular chains consisting of spherical beads of two types. Initially we consider the most simple dimer chain wherein each bead of type 1 is preceded and followed by a bead of type 2; such chains are denoted as 1: 1 dimers and the dynamics of such chains is governed by a single parameter (&egr;) scaling the mass of the two types of beads. Due to the periodic variation of the masses, a propagating pulse loses energy in the form of radiating waves in its trail and thus the pulse attenuates in dimer chains with an arbitrary value of mass ratio. Interestingly, at certain discrete values of mass ratios, the energy leakage from the propagating pulse ceases and the pulse propagates without attenuation, thus such pulses are called solitary waves. At the particular mass ratios where solitary waves are realized, these waves form an important energy transfer mechanism and any arbitrary pulse eventually disintegrates into a train of solitary waves. This is the first exposition of the realization of solitary waves in 1: 1 dimer granular chains. Moreover, we show that these chains support a countable infinity of solitary waves parameterized by energy. The contrasting (but more intuitive) effect of substantial energy radiation from the propagating pulse is also observed at a discrete set of mass ratios. This phenomenon is designated as resonance. A complete analytical formulation for these two phenomena is provided.;The spatial periodicity of traveling waves radiated in the trail of the propagating pulse (at arbitrary mass ratio) is found to depend only on the specific mass ratio of the dimer. The effect of the mass ratio on the realization of traveling waves, and in turn their significance to the resonance and pulse attenuation is studied by considering reduced order dimer chains composed of finite number of beads with periodic boundary conditions. Interesting bifurcations of the traveling waves have been discovered and correlation between the bifurcations and resonances is noted.;We further study the dynamics of a general class of 1:N (N ≥ 2) dimer chains. The dynamics of these chains is governed by two non-dimensional parameters, the mass ratio (&egr;) and the stiffness ratio (alpha) scaling the respective properties of the two types of beads. We report on a countable infinity of traveling solitary waves and resonances and prove numerically and asymptotically their existence in the 1:2 dimer chain. These solitary waves studied in homogeneous and 1: 1 dimer chains possess symmetric velocity waveforms, In contrast, the traveling solitary waves velocity waveforms of the 1: 2 dimers of the heavy beads are symmetric, whereas those of the light beads are non-symmetric. Interestingly, we show that no such solitary waves or resonances can be realized in general 1: N granular dimers with N > 2.;The final part of this study is concerned with the nonlinear dynamics of granular containers. The primary aim of this study is to apply the binary collision approximation (BCA) for the analytic estimation of the amplitude of the scattered pulses (solitary waves) in the granular container. In addition, we provide a numerical study showing the qualitatively different dynamics of these containers depending on the frequency and amplitude of an applied harmonic excitation.