New family of solitary waves in granular dimer chains with no precompression

Abstract

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.

Solitary waves propagation described by Korteweg-de Vries equation in the granular chain with initial prestress

“…[48][49][50][51][52][53] In general, the magnitude of n depends upon the contact geometry between the beads and typically varies between n = 5/2 and 3. 28,47,54 If there is great enough prestress, the bead radius will be R ′ = R − δ 0 2 , δ 0 is the initial overlap deformation between two beads.…”

Characterizing traveling-wave collisions in granular chains starting from integrable limits: The case of the Korteweg–de Vries equation and the Toda lattice

“…Geometry and/or material anisotropy between particles composing the systems allows for the observation of interesting dynamic phenomena deriving from the interplay of discreteness and nonlinearity of the problem (i.e. anomalous reflections, breathers, energy trapping and impulse fragmentation) [11,12,19,[24][25][26][27][28][29][30][31][32][33][34][35][36][37][38][39][40]. These findings open up a large parameter space for new materials design with unique properties sharply departing from classical engineering systems.…”

Mechanical wave propagation within nanogold granular crystals

“…when  approaches 0, the lattice becomes a so-called "auxiliary system" 32 . The radii of GNS in homogeneous lattice and heavy GNS in diatomic lattice are both R0 = 2.5 nm.…”

“…It is observed that n is a decreasing function of , but with a slowing-down rate, suggesting that for 15 nanoparticles, contact nonlinearity can be increased by creating mass difference. It is shown in literature that nonlinear resonances in uncompressed diatomic granular crystals result in a significant wave attenuation, which, from a practical point of view, could be employed as passive shock attenuators 32 . A transmission rate  is defined as P max,last imp / vv   to evaluate the attenuation performance of the diatomic GNS granular crystals, where P max,last v is the maximum particle velocity of the last GNS, Different from macroscale diatomic lattices governed by Hertzian contact, governing Eq.…”

“…For example, in compressed diatomic chains of alternating steel and aluminum spherical granules, the existence of discrete breathers was theoretically and experimentally proved, where sustained, exponentially localized oscillations emerge at a frequency inside the forbidden band of the linear Fourier spectrum 31 . For uncompressed diatomic lattices assembled by two different masses, strongly nonlinear resonances and anti-resonances are described by theory and experiments, resulting in 4 an optimal wave attenuation and a new type of solitary wave respectively [32][33][34] .…”