Eur. Phys. J. Plus (2020) 135: 7
https://doi.org/10.1140/epjp/s13360-019-00014-1

Review

New results for the effects of higher-order moduli and material properties on strain waves propagating through elastic circular cylindrical rods

¹ Department of Basic Science, Faculty of Engineering at October 6 University, Cairo, Egypt
² Department of Mathematics, Faculty of Science, Cairo University, Giza, 12613, Egypt

^* e-mail: mtantawymath@gmail.com

Received: 4 May 2019
Accepted: 26 September 2019
Published online: 7 January 2020

Abstract

The equation that describes the propagation of macrostrain waves is studied using a unified approach quoted earlier by one of the authors. New exact traveling wave solutions, mainly solitary, soliton, periodic and elliptic waves are obtained. It is believed that the family of obtained solutions covers all possible explicit solutions that could be found by all the other methods. Here, attention is focused on the case when the speed of the moving frame is equal to the wave speed. It is found that the nonlinearity produced by higher order of elasticity moduli acts as an impeding factor to the wave intensity. While the double dispersion relative to properties of the material leads to the existence of critical values for some nonlinearity and double dispersion parameters for which the wave amplitude reduces to zero. Such critical values do not seem to have been reported earlier. Finally, the stability of solution is analyzed and stability criteria are obtained.