Mathematical Problems in Engineering 
Volume 2006 (2006), Article ID 76041, 11 pages
doi:10.1155/MPE/2006/76041

Multiwave nonlinear couplings in elastic structures

D. A. Kovriguine,1 G. A. Maugin,2 and A. I. Potapov1

 1Mechanical Engineering Institute, Russian Academy of Sciences, 85 Belinsky Street, Nizhny Novgorod 603024, Russia
2Laboratoire de Modélisation en Mécanique (UMR 7607 CNRS), Université Pierre et Marie Curie, 4 Place Jussieu, Paris Cedex 75252, France
Received 27 December 2004; Revised 13 April 2005; Accepted 4 May 2005

Abstract

This short contribution considers the essentials of nonlinear wave properties in typical mechanical systems such as an infinite straight bar, a circular ring, and a flat plate. It is found that nonlinear resonance is experienced in all the systems exhibiting continuous and discrete spectra, respectively. Multiwave interactions and the stability of coupled modes with respect to small perturbations are discussed. The emphasis is placed on mechanical phenomena, for example, stress amplification, although some analogies with some nonlinear optical systems are also obvious. The nonlinear resonance coupling in a plate within the Kirchhoff-Love approximation is selected as a two-dimensional example exhibiting a rich range of resonant wave phenomena. This is originally examined by use of Whitham's averaged Lagrangian method. In particular, the existence of three basic resonant triads between longitudinal, shear, and bending modes is shown. Some of these necessarily enter cascade wave processes related to the instability of some mode components of the triad under small perturbations.