Granular Matter (2014) 16:141–150 DOI 10.1007/s10035-013-0475-z ORIGINAL PAPER Plane wave propagation in 2D and 3D monodisperse periodic granular media Mohith Manjunath · Amnaya P. Awasthi · Philippe H. Geubelle Received: 15 October 2013 / Published online: 22 January 2014 © Springer-Verlag Berlin Heidelberg 2014 Abstract Plane wave propagation in periodic ordered gran- ular media comprising of elastic spherical particles is inves- tigated. The spheres are under zero precompression and are assumed to interact via the Hertzian contact potential. Vari- ous two- and three-dimensional granular structures such as hexagonal packing (2D and 3D), face-centered cubic and body-centered cubic packings are considered in the present study, with the plane impact either normal or oblique to the granular system. For the normal impact case, 1D chains equivalent to the 2D and 3D structures are obtained. A uni- versal relation between the wavefront speed and the force amplitude is derived, valid for all the granular structures stud- ied. In the angular impact case, the shear component of the amplitude of the particle velocity is found to initially decay exponentially and further in a series of linear regimes. By employing simpler models, semi-analytical predictions are obtained for the decay of shearing effect. Keywords Granular media · Plane wave propagation · Hertzian contact · Solitary wave 1 Introduction Due to the unique features associated with the nonlinear con- tact between particles, wave propagation in granular media comprising of spherical particles has been a subject of several studies over the past few decades [1]. The nonlinearity arises from the Hertzian law which provides the force of interac- tion between two spherical particles under compression [2]. Nesterenko demonstrated that the Hertzian interaction gives M. Manjunath · A. P. Awasthi · P. H. Geubelle (B ) Department of Aerospace Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA e-mail: geubelle@illinois.edu rise to solitary waves in 1D homogeneous granular chains [3]. These solitary waves are characterized by the nonlinear dependence of the wave speed on its amplitude, which has also been demonstrated in some experiments [4, 5]. Further- more, owing to the interplay between discreteness and non- linearity, granular media offers wide flexibility in directing the force transmission response suitable for different appli- cations. Some of the potential applications include granular protectors [6], acoustic lens [7] and shock mitigation and absorption [8]. Most of the studies related to force transmission response of granular media have focused on 1D granular chains. Some studies are centered on heterogeneous chains such as deco- rated, tapered or stepped chains [9–11], dimer chains [12], chains with different geometries or mass defect [13, 14] and chains with randomness or disorder [15–17]. In 2D granu- lar media, a few experimental investigations using dynamic photoelasticity have been performed on arrays of disks arranged in body-centered cubic, hexagonal or random pack- ing [18, 19]. A key aspect of these studies was to predict the angular dependence of the load transfer between any two grains in contact. In the recent years, extensive experimental and numerical investigations have been conducted on square packing of spheres with intruders at interstitials [20–23], and have reported a wide range of different wavefronts depend- ing on the ratio of the material and geometric properties of the two types of spheres. In 2D granular media, most studies so far have focused on wave propagation associated with a point impact. These point impact studies address some fundamental aspects such as shape of the wavefront and decay of the force amplitude due to dimensionality [22, 24]. The present study focuses on the other ‘extreme’ case in which the periodic granular medium is subjected to a plane impact, considering uniform perpendicular or oblique impact. 123