In the present paper we report the existence of a new family of solitary waves in general one-dimensional dimer chains with elastic interactions between beads obeying a strongly nonlinear Hertzian force law. These dimers consist of pairs of "heavy" and "light" beads with no precompression. The solitary waves reported herein can be considered as analogous to the solitary waves in general homogeneous granular chains studied by Nesterenko, in the sense that they do not involve separations between beads, but rather satisfy special symmetries or, equivalently antiresonances in their intrinsic dynamics. We conjecture that these solitary waves are the direct products of a countable infinity of antiresonances in the dimer. An interesting finding is that the solitary waves in the dimer propagate faster than solitary waves in the corresponding homogeneous granular chain obtained in the limit of no mass mismatch between beads (i.e., composed of only heavy beads). This finding, which might seem counterintuitive, indicates that under certain conditions nonlinear antiresonances can increase the speed of disturbance transmission in periodic granular media, through the generation of different ways for transferring energy to the far field of these media. From a practical point of view, this result can have interesting implications in applications where granular media are employed as shock transmitters or attenuators.

We study standing waves (nonlinear normal modes-NNMs) and band zones in finite granular chains composed of spherical granular beads in Hertzian contact, with fixed boundary conditions. Although these are homogeneous dynamical systems in the notation of Rosenberg (Adv. Appl. Mech. 9:155-242, 1966), we show that the discontinuous nature of the dynamics leads to interesting effects such as separation between beads, NNMs that appear as traveling waves (these are characterized as pseudo-waves), and localization phenomena. In the limit of infinite extent, we study band zones, i.e., pass and stop bands in the frequency-energy plane of these dynamical systems, and classify the essentially nonlinear responses that occur in these bands. Moreover, we show how the topologies of these bands significantly affect the forced dynamics of these granular media subject to narrowband excitations. This work provides a classification of the coherent (regular) intrinsic dynamics of one-dimensional homogeneous granular chains with no pre-compression, and provides a rigorous theoretical foundation for further systematic study of the dynamics of granular systems, e.g., the effects of disorders or clearances, discrete breathers, nonlinear localized modes, and high-frequency scattering by local disorders. Moreover, it contributes toward the design of granular media as shock protectors, and in the passive mitigation of transmission of unwanted disturbances.

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Recent numerical studies on an infinite number of identical spherical beads in Hertzian contact showed the presence of frequency bands [Jayaprakash, Starosvetsky, Vakakis, Peeters, and Kerschen, Nonlinear Dyn. 63, 359 (2011)]. These bands, denoted here as propagation and attenuation bands (PBs and ABs), are typically present in linear or weakly nonlinear periodic media; however, their counterparts are not intuitive in essentially nonlinear periodic media where there is a complete lack of classical linear acoustics, i.e., in "sonic vacua." Here, we study the effects of PBs and ABs on the forced dynamics of ordered, uncompressed granular systems. Through numerical and experimental techniques, we find that the dynamics of these systems depends critically on the frequency and amplitude of the applied harmonic excitation. For fixed forcing amplitude, at lower frequencies, the oscillations are large in amplitude and governed by strongly nonlinear and nonsmooth dynamics, indicating PB behavior. At higher frequencies the dynamics is weakly nonlinear and smooth, in the form of compressed low-amplitude oscillations, indicating AB behavior. At the boundary between the PB and the AB large-amplitude oscillations due to resonance occur, giving rise to collisions between beads and chaotic dynamics; this renders the forced dynamics sensitive to initial and forcing conditions, and hence unpredictable. Finally, we study asymptotically the near field standing wave dynamics occurring for high frequencies, well inside the AB.

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