Linear and nonlinear surface waves in two-dimensional, hexagonal close-packed granular crystals

Abstract

Granular crystals (GCs) are ordered arrays of spherical particles in contact, which have been shown to exhibit rich nonlinear dynamics stemming from Hertzian contact interactions. They are typically modeled as discrete networks of rigid body spheres, which may undergo translational and rotational motion, with Hertzian interactions modeled as effective nonlinear normal and shear springs. Most of the existing literature on GCs concerns one-dimensional chains of spheres, though a growing body of recent work has explored two-dimensional systems in linear and nonlinear regimes, with multiple packing geometries. In a recent analytical study, Rayleigh-like surface waves were shown to exist in the linear regime for two-dimensional GCs with square packing [Phys. Rev. E 93, 023008 (2016)]. In the present work, we report on surface waves in two-dimensional granular crystals with hexagonal close-packed (HCP) geometry. We demonstrate analytically that, in the linear regime, HCP GCs support Rayleigh-like surface waves analogous to the square-packed case, as well as a leaky surface wave mode with quasi-longitudinal polarization. Extensions of the linear surface modes into the nonlinear regime are demonstrated via numerical examples and compared to results from one-dimensional granular chains. [Work supported by the postdoctoral program at ARL:UT.]