Highly nonlinear solitary waves in periodic dimer granular chains

Abstract

We investigate the propagation of highly nonlinear solitary waves in heterogeneous, periodic granular media using experiments, numerical simulations, and theoretical analysis. We examine periodic arrangements of particles in experiments in which stiffer and heavier beads ͑stainless steel͒ are alternated with softer and lighter ones ͑polytetrafluoroethylene beads͒. We find good agreement between experiments and numerics in a model with Hertzian interactions between adjacent beads, which in turn agrees very well with a theoretical analysis of the model in the long-wavelength regime that we derive for heterogeneous environments and general bead interactions. Our analysis encompasses previously studied examples as special cases and also provides key insights into the influence of the dimer lattice on the properties ͑width and propagation speed͒ of the highly nonlinear wave solutions. Over the past several years, the highly nonlinear dynamic response of granular materials has drawn increased attention from the scientific community ͓1-16͔. The corresponding theory developed for uniform lattice systems ͓1͔ supports the formation of a novel type of wave in materials, setting a paradigm for the design and creation of systems with unprecedented properties. A simple setup for the study of highly nonlinear dynamics in solids is provided by one-dimensional ͑1D͒ granular media consisting of chains of interacting spherical particles that deform elastically when they collide. The broad interest in such systems has arisen because they possess qualitatively different features from weakly nonlinear systems. For example, their solitary-wave solutions have a finite support that is independent of their amplitude ͓1͔, providing perhaps the most experimentally tractable application of the notion of "compactons" ͓17͔. There have also been several recent studies on the effects of defects ͑inhomo-geneities, particles with different masses, etc.͒ in such systems, allowing the observation of interesting physical responses such as fragmentation, anomalous reflections, and energy trapping ͓4-6,8-12,15͔. Moreover, chains of granular media have been shown to be highly tunable ͓1-3͔ and have the potential to be used in many engineering applicationsincluding shock and energy absorbing layers ͓9-12͔, sound focusing devices ͑tunable acoustic lenses and delay lines͒, sound absorption layers, and sound scramblers ͓13,14͔.Chains of granular media provide an ideal setting for investigating the interplay between nonlinearity and periodicity. The study of nonlinear oscillator chains has a timehonored history, originating with the Fermi-Pasta-Ulam problem ͓18-20͔. Its applications arise in numerous areas of physics, including coupled waveguide arrays and photorefractive crystals in nonlinear optics ͓21,22͔, Bose-Einstein condensates in optical lattices in atomic physics ͓23͔, and DNA double-strand dynamics in biophysics ͓24͔. A particular theme that often arises in this context is that of "heterogeneous" versus "uniform" lattices. Here, we focus on the pr...

Highly nonlinear solitary waves in chains of ellipsoidal particles

“…The particles were cut along their major diameter direction and carved to accommodate wires. The final assembly of the sensor particles was achieved following a procedure similar to the one described in [15,16] for spherical particles. All the sensors were precalibrated to obtain the relation between voltage output from the oscilloscope and the corresponding force on the particle assuming conservation of linear momentum.…”

“…One-dimensional chains of spherical particles have been extensively studied in the literature and are known to present highly nonlinear dynamic properties [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19]. When the chains are uncompressed, they support the formation and propagation of compact solitary waves that have been predicted theoretically [1], and observed numerically [1,6] and experimentally [4,5].…”

“…These systems derive their unique characteristics from a double nonlinearity in the particles interaction: a powerlaw-type contact potential in compression and a zero tensile strength. One of the fundamental characteristics of such systems is their high degree of tunability in terms of control over the traveling pulse width, speed, and the number of separated pulses that can be generated in the chain [2,3,9,[13][14][15][16]. The tunability of these systems is evident when the addition of static precompression on the chain of particles enables the system to change from the highly nonlinear, to the weakly nonlinear, to the linear dynamic regime [2,3,5,13].…”

“…Following previous investigations [9,13,15,16], we modeled a chain composed of N uniform ellipsoidal particles as a onedimensional system, where the ellipsoidal particles are treated as point masses connected by nonlinear springs according to the elliptical contact law in Eq. (1).…”

“…Changing the type and duration of the initial excitation applied to the system, it is possible to generate a single or train of pulses with variable amplitudes [2,14]. Changing the particle diameters, the particle material properties, and/or their periodicity, it is also possible to change the solitary waves' amplitude and traveling speed [9,13,15,16]. This ability to control the wave properties in such chains has been proposed for a variety of practical engineering applications; for example, it could be employed for shock absorbing materials [7,12,19,21], vibration damping [20], sound scramblers [9,10], and sound focusing [22].…”