ENOC 2011, 24-29 July 2011, Rome, Italy Primary Wave Transmission in Systems of Elastic Rods with Granular Interfaces Y. Starosvetsky, A.F. Vakakis Department of Mechanical Science and Engineering, University of Illinois at Urbana-Champaign, Illinois, USA Summary. We study analytically and numerically primary pulse transmission in a periodic system of identical linearly elastic non- dispersive rods connected by identical homogeneous granular interfaces. The main challenge in studying pulse transmission in this strongly nonlinear periodic medium is to analyze the ‘basic problem’, namely, the dynamical response of a single intermediate granular layer, confined from both ends by barely touching linear elastic rods, subject to impulsive excitation of the left rod. The analysis of the basic problem is carried out under two basic assumptions: (i) of sufficiently small duration of the shock excitation applied to the first layer of the system, and (ii) of sufficiently small mass of each bead in the granular interface compared to the mass of each rod (in fact, this defines the small parameter of the problem). Both assumptions are reasonable from the point of view of practical applications. In the analysis we focus only in primary pulse propagation, by neglecting secondary pulse reflections caused by wave scattering at each granular interface and considering only the transmission of the primary pulse across the interface to the neighboring elastic rod. The influence of the number of beads of the granular interface on the wave transmission is studied, and shown that at granular interfaces with relatively low number of beads fast time scale oscillations are excited, whose amplitudes increase with increasing number of beads. For larger number of beads primary pulse transmission is by means of solitary wave trains resulting from the dispersion of the shock pulse; in that case fast oscillations result due to interference phenomena that originate from the scattering of the main pulse at the boundary of the interface. Considering the periodic system of rods we demonstrate significant reduction of the primary pulse when transmitted through a sequence of granular interfaces, a result that indicates the efficacy of applying granular interfaces for shock mitigation in layered elastic media. Introduction Wave propagation in linear periodic or layered structures was intensively studied in prior works. Thus Mead and co- workers [1-4] analyzed wave propagation in mono-coupled and multi-coupled structures. In these works they defined propagation and attenuation zones (PZs and AZs) in the frequency domain where traveling and standing waves exist, respectively. The notions of PZs and AZs were extended to nonlinear repetitive systems [5-7]. The principal aim of this work will be to study the dynamical response of a periodic system composed of identical non-dispersive linearly elastic rods connected by granular interfaces. To simplify this complex problem we consider only primary pulse transmission at each granular interface, by focusing on the transmission of the primary pulse through the periodic medium and omitting secondary waves formed by delayed reflections of scattered waves at the boundaries of the rods. Our analysis will extend the methodology developed in [8] where primary stress transmission in a periodic system of rods coupled by strongly nonlinear stiffnesses was examined. Key in our study will be the analysis of primary pulse transmission across a one-dimensional granular interface between two linearly elastic non-dispersive rods; this we refer to as the ‘basic problem’ of primary pulse transmission. Then, we will apply the results of the analysis of the basic problem to the periodic set of rods with granular interfaces. We will show that depending on the dimensionality of the granular interface qualitatively different dynamics occur, with significant implications regarding the character of primary stress transmission through the periodic medium. Results The basic problem for studying primary pulse transmission across a granular interface between two elastic rods is presented in Figure 1. It consists of array of non-dispersive linearly elastic rods connected by an uncompressed, one dimensional homogeneous granular layer. One-dimensional (longitudinal) elastic wave propagation is considered in the rods and no dissipation effects are taken into account (that is, perfectly elastic collisions between beads are considered). Assuming that the system is initially at rest, a shock pulse () i Ft of finite duration is considered to be applied at time 0 t  on the left end of rod i . Figure 1. The basic problem of wave transmission across a granular interface between two rods Rod i Rod i+1 () i Ft Rigid disks 1 2 n (, ) i utL 1 ( ,0)  i u t 1 () vt () n v t x x L L , , mEA Granular interface