We use experiments, numerical simulations, and theoretical analysis to investigate the propagation of highly nonlinear solitary waves in periodic arrangements of dimer (two-mass) and trimer (three-mass) cell structures in one-dimensional granular lattices. To vary the composition of the fundamental periodic units in the granular chains, we utilize beads of different materials (stainless steel, bronze, glass, nylon, polytetrafluoroethylene, and rubber). This selection allows us to tailor the response of the system based on the masses, Poisson ratios, and elastic moduli of the components. For example, we examine dimer configurations with two types of heavy particles, two types of light particles, and alternating light and heavy particles. Employing a model with Hertzian interactions between adjacent beads, we find very good agreement between experiments and numerical simulations. We find equally good agreement between these results and a theoretical analysis of the model in the long-wavelength regime that we derive for heterogeneous environments (dimer chains) and general bead interactions. Our analysis encompasses previously-studied examples as special cases and also provides key insights on the influence of heterogeneous lattices on the properties (width and propagation speed) of the nonlinear wave solutions of this system.

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...

We study the effects of a line of spherical interstitial particles (or intruders) placed between two adjacent uncompressed chains of larger particles in a square packing of spheres, using experiments and numerical simulations. We excite one of the chains of particles adjacent to the intruders with an impact and show how energy is transmitted across the system until equipartition is reached from the excited (or impacted) chain to the absorbing (or adjacent) chain. The coupling of the two chains, although a purely two-dimensional effect, is modeled by a simplified one-and-a-half-dimensional (1.5-D) system in which transverse motions of the particles are neglected.PACS numbers: 46.40.Cd Introduction. Granular crystals are unique nonlinear systems that exhibit interesting properties stemming from the nonlinear contact interactions (Hertzian [1]) between two individual particles. Uniform one-dimensional chains of spheres have been shown to support the propagation of a new type of solitary wave. The width of these waves is independent of its velocity [2][3][4][5]. The degree of nonlinearity can be tuned from highly nonlinear to linear by the addition of a precompression force [3,4,6]. Additionally, it was shown that homogeneous granular media can support families of strongly nonlinear traveling and standing waves [7], whereas heterogeneous media can exhibit resonance [8] and anti-resonance phenomena [9]. Another interesting property of these materials is the reflection of the solitary waves at an interface between two granular crystals [10][11][12], which could be used to develop new impulse trapping granular materials [13][14][15]. Several groups studied the interaction of a solitary wave with defect particles [16][17][18][19]. Twodimensional (2-D) granular crystals have been relatively unexplored, and prior works mainly consisted of numerical studies or experiments visualizing dynamic stress in photo-elastic disks [20][21][22][23]. Solitary waves have been observed in 2-D square packings of spherical particles [24]. Granular crystals have been proposed as new structured materials for the control and redirection of stress waves (see for example [10,13,14,25,28,33]). The experiments reported in this paper provide the first observation of energy equipartition between two adjacent and nonlinearly coupled chains of particles. In particular, we show that when one chain is excited by an impulse while the other is as rest, the energy is redistributed between the two chains within a short spatial distance. A similar equipartition phenomenon was studied numerically in an earlier work [26]. This phenomenon is of interest to create new acoustic wave guides, delay lines and stress mitigating materials. Energy transfer and equipartition phenomena in weakly coupled one-dimensional granular chains were studied [26,27], and in [28] through a macroscopic realization of the Landau-Zener tunneling quantum effect. The energy equipartition principle is well known for elastic waves. Seismic waves for example have well known regimes wh...

We investigate the propagation and scattering of highly nonlinear waves in granular systems composed of spheres in contact arranged in a square packing, and study how the presence of small and light spherical interstitial defects, also referred to as intruders, affects the wave propagation. The effects of a single defect are investigated experimentally and compared to numerical simulations, showing very good quantitative agreement. Transmitted and scattered waves are formed, whose characteristics depend on the material properties of the defect in relation to the properties of the particles in the lattice. Experiments and numerical simulations reveal that stiffer defects are more efficient at redistributing energy outside the impacted chain and soft defects induce a localization of the energy at the defect. Finally, the effects of the presence of two defects, placed diagonally or aligned in the square packing are also investigated, as well as how their interaction depends on their relative positions.