טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentLisyansky Alexander
SubjectEffective Particles Approximation of a Primary Pulse
Transmission in Hexagonally Packed, Damped
Granular Crystal with a Spatially
Varing Cross Section: Numerica
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Yuli Starosvetsky
Full Thesis textFull thesis text - English Version


Abstract

Granular crystals are the assembly of discrete solid elements of various shapes arranged in a regular lattice structure. These elements interact one with another through a highly nonlinear, repulsive contact force. One dimensional granular chains support the formation of propagating, highly localized, stress waves. Recent studies have shown that the motion of these waves can be controlled by introducing periodic and local in-homogeneities in the granular structure. Thus granular materials carry a potential to redirect, localize and mitigate the propagating shock waves. Recently, there is a growing interest to the dynamical response of higher dimensional granular crystals. One possibility to characterize the wave propagation in the 2D granular setups is by introducing effective granular scalar model. The current work concerns the dynamics of a primary pulse propagating through uncompressed, 2D hexagonal granular crystal of spheres with spatially varying cross section and given to onsite perturbation, referred to as the fundamental model. We demonstrate that application of shock like loading on the narrow end of the setup leads to the formation of a spatially localized, traveling, primary pulse. At the first stage of propagation the spatial shape of the pulse is nearly straight. However, after sufficient number of layers, the pulse becomes distributed along a curve. We focus on the first stage of propagation, in which the initial excitation energy is spread over large number of particles. We show that a spatial evolution of the localized primary pulse can be described by a reduced order model comprising perturbed, purely nonlinear (Hertz law) chain of effective particles with an increasing masses and stiffness coefficients. Using the recently developed procedure of nonlinear maps and a subsequent homogenization, we derive an analytical approximation depicting the evolution of the traveling primary pulse. Numerical simulations results 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 of the full 2D model.

Theoretical results derived for the fundamental model are further applied on a different dynamical setup concerning the primary pulse transmission in the hexagonally packed, granular crystal given to a radial, shock like loading. Here we show that despite the obvious differences between the two physical models, the analytical approach devised for the fundamental one provides a fairly acceptable approximation also for the problem of internal, radial excitation of the perturbed, hexagonally packed, granular crystal.