Nonlinear Waves in Rods and Beams of Power-Law Materials

Abstract

Some novel traveling waves and special solutions to the 1D nonlinear dynamic equations of rod and beam of power-law materials are found in closed forms. The traveling solutions represent waves of high elevation that propagates without change of forms in time. These waves resemble the usual kink waves except that they do not possess bounded elevations. The special solutions satisfying certain boundary and initial conditions are presented to demonstrate the nonlinear behavior of the materials. This note demonstrates the apparent distinctions between linear elastic and nonlinear plastic waves.

Stability Properties of 1-Dimensional Hamiltonian Lattices with Non-analytic Potentials

“…In this paper we focus on 1D Hamiltonian N particle systems, whose potential is a nonanalytic function of the position coordinates. Such systems are important for applications involving "graphenetype" materials [Cadelano et al, 2009;Lu & Huang, 2009;Colombo & Giordano, 2011;Hazim et al, 2015;Wei et al, 2017a], and micro-electrical-mechanical systems (MEMS) [Esposito et al, 2010;Younis, 2013;Khan et al, 2017] obeying Hollomon's power-law and exhibiting "work-hardening" properties [Wei & Liu, 2012;Wei et al, 2017b]. As in earlier studies Antonopoulos et al, 2006;Bountis & Skokos, 2012], we concentrate here on the (local and global) stability properties of certain socalled simple periodic orbits (SPOs), which represent continuation of linear normal modes of the system and are characterized by the return of all the variables to their initial state after only one maximum and one minimum in their oscillations.…”

“…In these systems, the nonlinear differential equations and the associated initial/boundary value problems arise through the so-called Hollomon's power-law and are governed by nonlinear springmass equations of the form mẍ = −Kx + x|x| p−2 , 1 ≤ p < 2, for a single oscillator in the absence of external load. While for linear elastic materials, the principal operator is the bi-Laplacian, for Hollomon's power-law materials, it is a bi-p-Laplacian [Wei & Liu, 2012;Wei et al, 2017b]. Here we plan to generalize these models by considering an array of N such coupled oscillators described by the Hamiltonian…”