A generation of second harmonic of shear wave in elasto-plastic media

A.M. Doronin, V.I. Erofeev

Abstract

Elasto-plastic media, whose behavior is described by cross dependencies between first invariants of stress and strain tensors and second invariants of stress and strain deviators, is considered. The dependency has quadratic behavior in terms of shear strains. The goal of this article is to determine the possibility of generation of second harmonic of shear wave, whose appearance is not predicted by classical theory of elasticity, and the character of the process of energy transfer from first harmonic to second. In case of prevailing shear strains with only shear wave existing, the search of solutions of nonlinear equation, determining the dynamic behavior of the media, in form of travelling quasiharmonic waves with slowly changing amplitudes is performed. During the examination of the system of shortened equations for complex amplitudes, depending on time and travel distance, the unsymmetric character of harmonic interaction process is shown. It is determined that in stationary state even the presence of small plastic strains leads to appearance of doubled frequency (second harmonic) in the spectrum of shear wave, propagating in the material, at that second harmonic affects first only if there is a first harmonic’s signal. The distance, at which the harmonic’s amplitudes become equal, is obtained. The qualitative character of process of energy transfer is illustrated. The character of influence of media constants (density, shear modulus, limit intensity of shear strains) and wave parameters (frequency, wave number, first harmonic’s initial amplitude) on a distance, corresponding to a state of equality of amplitudes of harmonics, is established

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