|University of Illinois Urbana-Champaign
|Granular media; Acoustic metamaterial
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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 chains 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 primaryAdvisors/Committee Members: Vakakis, Alexander F. (advisor), Vakakis, Alexander F. (Committee Chair), Bergman, Lawrence A. (committee member), McFarland, Donald Michael (committee member), Tawfick, Sameh (committee member).