Highly Nonlinear Wave Propagation in Elastic Woodpile Periodic Structures

Abstract

In the present work, we experimentally implement, numerically compute with, and theoretically analyze a configuration in the form of a single column woodpile periodic structure. Our main finding is that a Hertzian, locally resonant, woodpile lattice offers a test bed for the formation of genuinely traveling waves composed of a strongly localized solitary wave on top of a small amplitude oscillatory tail. This type of wave, called a nanopteron, is not only motivated theoretically and numerically, but is also visualized experimentally by means of a laser Doppler vibrometer. This system can also be useful for manipulating stress waves at will, for example, to achieve strong attenuation and modulation of high-amplitude impacts without relying on damping in the system.

Traveling waves and their tails in locally resonant granular systems

“…This condition is necessary because the above-mentioned experimental observations [62] suggest that ξ R ( ) and ξ S ( ) sometimes bear non-decaying (yet bounded) oscillating tails as ξ | | → ∞. This implies that R and S may be non-integrable and thus their Fourier transforms may not be possible to define.…”

“…For the nth bead = … n N ( 1, 2, , ), we attach a local resonator, effectively coupling it to another kind of bead (the 'MiM' [59] or the 'MwM' [60] discussed above), whose displacement from the original position is denoted by V n . Notice that in the case of the woodpile configuration, this does not constitute a separate mass but rather reflects the internal vibrational modes of the woodpile rods [62]. Here we use K 1 to stand for the spring constant; R for the radius of the bead, ρ for the density, E for the elastic modulus, μ for the Poisson's ratio of the bead so that the mass πρ = m R (or more generally, the effective mass of the locally resonant mode).…”

“…A remarkable find of these locally resonant, highly nonlinear chains was found to be the propagation of traveling waves with an apparently non-decaying tail [62]. It was indeed found that the tail behind the wave may persist in small amplitude oscillations (in the experiment they were found to potentially be three orders of magnitude smaller than the core of the traveling wave) which, however, do not decay and which bear a clear frequency/ wavenumber, namely that of the out-of-phase vibration between the principal and the resonating mass in such a system.…”

“…Admittedly, both of these realizations were chiefly linear (in the presence of externally imposed pre-compression of the chain) and aiming to illustrate the remarkable meta-material type properties that these systems possess. Yet, while a MwM nonlinear prototype was also demonstrated in [61], it was a different type of experiment that very recently realized highly nonlinear propagation in a locally resonant granular system [62]. In particular, the experiment of [62] built a so-called woodpile configuration consisting of orthogonally stacked rods (every second rod is aligned, in this alternating 0-90 degree configuration) and demonstrated that the internal vibrations of the rods can play the role of the local resonator within the granular chain.…”

“…Yet, while a MwM nonlinear prototype was also demonstrated in [61], it was a different type of experiment that very recently realized highly nonlinear propagation in a locally resonant granular system [62]. In particular, the experiment of [62] built a so-called woodpile configuration consisting of orthogonally stacked rods (every second rod is aligned, in this alternating 0-90 degree configuration) and demonstrated that the internal vibrations of the rods can play the role of the local resonator within the granular chain. It was also shown that depending on the properties of the system (i.e., the length of the rods), one can controllably incorporate one or more such resonators, yet here our focus will be in the case of a single such resonator.…”