Interactions of longitudinal and rotation waves
in a nonlinearly elastic cosserat medium
V.I. Erofeyev
Nizhny Novgorod Branch of Mechanical Engineering
Research Institute of Russian Academy of Sciences,
Nizhny Novgorod, Russia
Based on analysis of experimental data, it is shown that high-frequency wave processes in polycrystalline structures, composite materials and polymers are comprehensively described by the equations of the theory of elastic solids with microstructure (Cosserat continuum, Lerux continuum, Eringen micromorphic medium, etc.).
With the help of the Hamilton-Ostrogradsky variational principle, the equations of dynamics, the energy and the wave pulse variation laws are derived for different models of micropolar media. Peculiarities of the wave processes in such media are analyzed. It is demonstrated that the presence of microstructure results, on the one hand, in dispersion of longitudinal, shear and surface Rayleigh waves and, on the other hand, in a possible existence of new types of elastic waves. The shear-rotation waves are, in particular, referred to the latter. Their properties are characterized by two dispersion curves. One dispersion branch is described by the Korteweg-de Vries equation while the Shroedinger equation is used to describe another.
Resonant interactions between the quasiharmonic shear-rotation waves with the longitudinal waves in a nonlinear continuum are studied. Four the qualitatively different instances of resonant triplets are revealed. A breaking instability of a high-frequency longitudinal wave can be observed in the Cosserat medium which leads to the generation of two shear-rotation waves of lower frequencies. The breaking instability of the high-frequency shear-rotation waves can be also observed leading to excitation of a low-frequency wave of the same type and a low-frequency longitudinal wave.