Primary wave transmission in the hexagonally packed, damped granular crystal with a spatially varying cross section

Abstract

Present study concerns the dynamics of a primary pulse propagating through the uncompressed, 2D hexagonally packed granular crystal with spatially varying cross-section and given to onsite perturbation. In this study, the system under consideration is referred to as a fundamental model. We demonstrate that application of the uniform, soliton like loading on the narrow end of granular crystal leads to the formation of a spatially localized, traveling, primary pulse. At the first stage of propagation, the shape of a spatial distribution of a primary pulse is nearly straight, consequently preserving the main features of incoming pulse (i.e. preserving the waveform and the local speed of Nesterenko soliton). However, as it propagates, the shape of a spatial distribution slowly deviates from a straight line. Primary pulse becomes distributed along the curve after sufficient number of layers, rather than follows a straight line. In current work we primarily focus on the first stage of propagation of the primary pulse. We show that a spatial evolution of the strongly localized primary pulse can be efficiently described by a reduced order model comprising the perturbed, purely nonlinear (Hertz interaction law) chain of effective particles with the linearly increasing masses and stiffness coefficients. Using the recently developed analytical procedure of nonlinear maps and a subsequent homogenization, we derive a closed form, analytical approximation depicting the evolution of the traveling primary pulse. It is worthwhile noting that the devised analytical approximation accounts for the presence of an onsite perturbation (e.g. dissipation) imposed on the full model. Results of the numerical simulations of the reduced order model as well as these of the analytical approximation are found to be in a spectacular agreement with the results of numerical simulations obtained for the full 2D model.