|Abstract:||Ordered arrays of granular particles (beads) have attracted considerable attention in recent years due to their rich dynamical behaviors and interesting properties. Depending on the ratio of static to dynamic deformations between particles the dynamics of granular media is highly tunable ranging from being strongly nonlinear and non-smooth in the absence of static pre-compression, to reducing to weakly nonlinear and smooth for large static pre-compression. The nonlinearity in uncompressed granular media arises from two sources: First, nonlinear Hertzian interactions, which can be modeled mathematically, between beads in contact, and second, bead separations in the absence of compressive forces between them leading to collisions between adjacent beads. When no applied pre-compression exists there is complete absence of linear acoustics in ordered granular media, which results in zero speed of sound as defined in the sense of linear acoustics through the classical wave equation; thus, these media have been characterized as “sonic vacua”. However, various nonlinear waves can still propagate in these media with energy tunable properties.
The first part of this dissertation aims to study the frequency responses of a single homogenous granular chain. We consider a one-dimensional uncompressed granular chain composed of a finite number of identical spherical elastic beads with Hertzian interactions. The chain is harmonically excited by an amplitude- and frequency-dependent boundary drive at its left end and has a fixed boundary at its right end. We computationally and experimentally detect time-periodic, strongly nonlinear resonances whereby the particles (beads) of the granular chain respond at integer multiples of the excitation period, and which correspond to local peaks of the maximum transmitted force at the chain’s right, fixed end. In between these resonances we detect local minima of the maximum transmitted forces corresponding to anti-resonances, where chimera states (i.e., coexistence of different stationary and nonstationary waveforms) are noted, in the steady-state dynamics. Furthermore, we construct a mathematical model which can completely capture the rich and complex dynamics of the system.
The second part of the study is primarily concerned with the propagatory dynamics of geometrically coupled ordered granular media. In particular, we focus on primary pulse transmission in a two-dimensional granular network composed of two ordered chains that are nonlinearly coupled through Hertzian interactions. Impulsive excitation is applied to one of the chains (denoted as “excited chain”), and the resulting transmitted primary pulses in both chains are considered, especially in the non-directly excited chain (denoted as “absorbing chain”). A new type of mixed nonlinear solitary pulses – shear waves is predicted for this system, leading to primary pulse equi-partition between chains, indicating strong energy exchange between two chains through the geometric coupling. Then, an analytical reduced model for primary pulse transmission is derived to study the strongly nonlinear acoustics in the small-amplitude approximation. In contrast to the full equations of motion the simplified model is re-scalable with energy and parameter-free, and is asymptotically solved by extending the one-dimensional nonlinear mapping technique. The nonlinear maps, which are derived for this two-dimensional system and governing the amplitudes of the mixed-type waves, accurately capture the primary pulse propagation in this system and predict the first occurrence of energy or pulse equi-partition in the network. Moreover, to confirm the theoretical results we experimentally test a series of two-dimensional granular networks, and prove the occurrence of strong energy exchanges leading to eventual pulse equi-partition between the excited and absorbing chains, provided that the number of beads is sufficiently large.
Then we analyze the dynamics of a granular network composed of two semi-infinite, ordered homogeneous granular chains mounted on linear elastic foundations and coupled by weak linear stiffnesses under periodic excitation. We first review the acoustic filtering properties of linear and nonlinear semi-infinite periodic media containing two attenuation zones (AZs) and one propagation zone (PZ) in the frequency domain. In both linear and nonlinear systems, under suddenly applied, high-frequency harmonic excitations, “dynamic overshoot” phenomena are realized whereby coherent traveling responses are propagating to the far fields of these media despite the fact that the high frequencies of the suddenly applied excitations lie well within the stop bands of these systems. For the case of the linear system we show that the transient dynamic overshoot can be approximately expressed in terms of the Green’s function at its free end. A different type of propagating wave in the form of a “pure” traveling breather, i.e., of a single propagating oscillatory wavepacket with a localized envelope, is realized in the transient responses of a nonlinear granular network. The pure breather is asymptotically studied by a complexification/averaging technique, showing nearly complete but reversible energy exchanges between the excited and absorbing chains as the breather propagates to the far field. We analytically prove that the reason for this dynamic overshoot phenomenon in both linear and nonlinear networks is the high rate of application of the high-frequency harmonic excitation, which, in essence, amounts approximately to an impulsive excitation of the periodic medium. Verification of the analytical approximations with direct numerical simulations is performed.
We further study passive pulse redirection and nonlinear targeted energy transfer in the aforementioned weakly coupled granular network. Periodic excitation in the form of repetitive half-sine pulses is applied to the excited chain. The frequency of excitation is within the pass band of the granular system. At the steady state nearly complete but reversible energy exchanges between the two chains are noted. We show that passive pulse redirection and targeted energy transfer from the excited to the absorbing chain can be achieved by macro-scale realization of the spatial analog of the Landau-Zener quantum tunneling effect. This is realized by finite stratification of the elastic foundation of the excited chain, and depends on the system parameters (e.g., the percentage of stratification) and on the parameters of the periodic excitation. We detect the existence of two distinct nonlinear phenomena in the periodically forced network; namely, (i) energy localization in the absorbing chain due to sustained 1:1 resonance capture leading to irreversible pulse redirection from the excited chain, and (ii) continuous energy exchanges in the form of nonlinear beats between the two chains in the absence of resonance capture. Our results demonstrate that steady state passive pulse redirection in these networks can be robustly achieved under periodic excitation.
The final part of present work is concerned with propagating breathers in granular networks under impulsive excitation. We apply a complexification-averaging methodology leading to smooth slow flow reduced models of the dynamics to reveal the nature of 1:1 resonance at fundamental steady-state responses of the system. The primary aim of this analytical study is to provide a predictive way to excite the system at its resonance conditions. In addition to the fundamental resonance we numerically verify the occurrences of subharmonic steady-state responses in such granular networks. We experimentally detect the propagating breathers in a single chain mounted on elastic foundations. Our experimental measurements show good correspondence with the computational results which validate our previous theoretical predications. The results of this work contribute to the design of practical nonlinear acoustic metamaterials and provide a new avenue for understanding ofthe complex nonlinear dynamics of granular media.