טכניון מכון טכנולוגי לישראל
הטכניון מכון טכנולוגי לישראל - בית הספר ללימודי מוסמכים  
M.Sc Thesis
M.Sc StudentShiffer Adi
SubjectTheoretical study and characterization of the response
regimes of highly heterogeneous granular crystals
DepartmentDepartment of Mechanical Engineering
Supervisor Professor Yuli Starosvetsky
Full Thesis textFull thesis text - English Version


Abstract

Granular crystals also known as nonlinear acoustic meta-materials are the assembly of discrete solid granular elements of various shapes arranged in a regular lattice structure. These elements are initially in contact and interact one with another through a highly nonlinear, repulsive contact force (Hertzian contact law). Recently these materials have attracted substantial attention for their unique dynamical properties of a significant practical importance as well as for their relatively simple fabrication. Recent studies of the dynamics of di-atomic (dimer) 1D granular crystals (chains composed of only two types of elements with different properties such as different masses, shapes, radii etc.)  have shown that the motion of these waves can be efficiently controlled ranging from the regimes of strong attenuation of propagating disturbances and up to the formation of an all new type of stationary pulses and nonlinear normal dynamical regimes. This spectacular adaptivity of granular materials is of major importance in various engineering applications dealing with shock and vibration isolation in micro and macro scales.  In the present work we studied the dynamic response of granular tri-atomic chains (trimers) subjected to impulsive loading and/or given a certain initial condition of initial displacements and velocities.

In the first part of the present study we consider the impulsive response of the perfectly aligned, uncompressed, tri-atomic granular lattice. It is demonstrated that under particular choice of the system parameters - impulsively loaded, tri-atomic granular lattice can support formation of highly localized, weakly attenuated pulses. These pulses are manifested by the completely non-symmetric wave profiles and can be attributed to the special family of solitary like waves forming in the non-homogenous, periodic tri-atomic granular lattice in the state of acoustic vacuum. We derive a simplified reduced order model predicting the special regions in the space of the system parameters corresponding to the formation of the weakly attenuated pulses by using recently developed analytical procedures based on the singular, multi-scale perturbation analysis. The predictions of the asymptotical model are found to be in a good agreement with the results of numerical simulations of the full tri-atomic granular lattice.

The second part of the study primarily concerned with the nonlinear normal modes and traveling waves of the one-dimensional periodic granular trimer chain in the state of acoustic vacuum. In this part of the study two different classes of periodic solutions namely the traveling and standing waves were demonstrated. The objective of this part of the study is the numerical analysis of the bifurcation structure of these solutions with emphasis on the dynamics of traveling waves. In fact, understanding of the bifurcation structure of the traveling wave solutions emerging in the unit-cell granular trimer is rather important and can shed light on the more complex nonlinear wave phenomena emerging in semi-infinite trimer chains.