Strongly nonlinear dynamics and acoustics of coupled granular sonic vacua: theoretical and experimental studies
Passive control of energy flow by means of trapping, redirection and scattering has been the subject of considerable attention in recent years. The use of ordered granular media is a promising candidate for such designs. Ordered granular media are ordered aggregates, where neighboring particles (or granules) interact elastically in a highly nonlinear fashion. These highly discontinuous media have strongly nonlinear behavior owing to Hertzian interactions between granules in compression, and separations – collisions of the same granules in the absence of compression. Depending on the ratio of static to dynamic interparticle deformations and displacements, the nonlinear dynamics of ordered granular media is highly adaptive, ranging from being essentially nonlinear (even non-linearizable) and non-smooth (in the absence of static pre-compression), to being weekly nonlinear and smooth or even linear (with high static pre-compression). Moreover, such interplay between strongly nonlinear and weakly nonlinear dynamics may occur in different phases of the same dynamic response of a granular medium subject to a given external excitation. In this dissertation, we aim to analyze the strongly nonlinear dynamics of coupled ordered granular media and investigate interesting response regimes such as, passive wave redirection / redistribution and targeted energy transfer (TET). These studies are performed using numerical computations, analytical calculations, and experimental tests. In particular, we consider weakly coupled granular chains with or without on-site potentials, as well as two-dimensional granular networks with regularly placed intruders that act as effective coupling elements. Unlike previous studies of weakly coupled oscillatory chains, the dynamical systems considered herein incorporate both non-smooth effects due to possible separations between interacting neighboring beads (granules), as well as strongly nonlinear inter-particle Hertzian interactions. We show that these systems exhibit very rich and complex dynamics that, however, can be completely captured by our analytical approximations. For the case of weakly interacting granular networks, three independent mechanisms of efficient transport of energy from one chain to another are found. The first mechanism is a simple exchange of energy between the weakly interacting granular chains providing equi-partition of Nesterenko solitary waves through the chains. The second mechanism is a complete and recurrent exchange of energy (beating phenomenon) between the propagating breathers through the weakly coupled granular chains laying on a strong elastic foundation. The last mechanism is the most intriguing one and demonstrates targeted (irreversible) energy transfer between coupled granular chains due to appropriate stratification of their elastic foundations, in a macroscopic analogue of the well-known Landau-Zener Quantum effect in space. The aforementioned mechanisms of energy transfer and redirection in highly nonlinear granular chains are conceptually new and were presented for the first time. Analytical and computational studies of these nonlinear energy transfer mechanisms are addressed in the present work, and their potential for future predictive designs of highly discontinuous and adaptive granular acoustic metamaterials for shock wave redirection and control are discussed. Then we focus on another class of granular acoustic metamaterials, namely, one-dimensional single or coupled granular chains embedded in elastic matrix, and present experimental and theoretical studies on pulse transmission and non-linear energy exchange in these systems. Three different matrices are considered in the experiments: Poly-di-methyl-siloxane (PDMS), polyurethane and geopolymer. Specifically, we examine two rows of granular chains embedded in elastic matrix and show that when an impulse is applied to one of the chains, the resulting pulse gets partially transferred to a neighboring chain and energy gets distributed among the entire embedded granular network. Based on our experimental measurements we validate a theoretical model and then use it for predictive design. Then, we experimentally study and verify the existence of acoustic pass- and stop-bands in harmonically excited embedded granular chains, and prove the existence of traveling breathers in these systems. We report a very rich structure of nonlinear acoustic phenomena in these highly discontinuous and strongly nonlinear granular metamaterials, and prove conclusively that traveling breathers are realized robustly in granular chains embedded in three widely different types of matrices, over wide frequency and energy ranges. In addition to experimentally confirming prior theoretical predictions regarding the existence of breathers in these media, we provide a new avenue for exploring the highly complex dynamics and acoustics of granular metamaterials for a variety of practical applications. Finally, we study the propagatory and oscillatory dynamics of two-dimensional coupled granular networks with discontinuous lateral boundary conditions. We numerically show that depending on the strength of coupling between chains, different dynamical response regimes can be realized, namely, propagating breathers leading to pulse equi-partition in both chains, or strong scattering of propagating pulses leading to spatial energy localization in the network. These regimes are caused by the strong on-site potentials of the two chains provided by the lateral discontinuous boundary conditions. Hence, we provide a new paradigm of designing the nonlinear dynamics of ordered granular metamaterials by appropriate forming of their boundaries.
Granular Media, Solitary Waves, Standing Breathers, Traveling Breathers, Wave Equipartition, Targeted Energy Transfer, Landau–Zener Tunneling, Acoustic Metamaterial, Acoustic Stop- And Pass-Bands
University of Illinois - Urbana-Champaign
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University of Illinois at Urbana-Champaign