Three-wave resonance interaction in elastoplastic solid

Resonant interaction of longitudinal and transverse waves in elastoplastic solid is considered. The media behavior is described by cross dependencies between the first invariants of stress and strain tensors and second invariants of stress and strain deviators. The dependence on shear strains is quadratic. The aim of this investigation is to establish the nature of the process of energy exchange between modes in the absence of dissipation. It is assumed that the solid is in the plane strain state. Two cases are considered, with prevailing longitudinal and transverse strains respectively. In both cases the solution of systems of nonlinear partial differential equations are searched in the form of traveling harmonic waves with slowly changing amplitudes. Method of averaging by “fast” variables is used taking into consideration only solutions and items with the order of smallness which is below one. It is obtained that in both cases the form of systems of equations for wave amplitudes is the same. Continual analogues for Manley-Rowe relations are obtained. The solution of the system for a stationary state with boundary conditions for strong input and weak signal waves is presented, as well as the diagram qualitatively illustrating the process of a three-wave interaction in case of decay instability (i.e. frequencies and wave numbers of signal and idler waves are in total equal to the frequency of the input wave) and relation determining the distance of effective energy exchange between the interacting waves. The behavior of magnitudes of wave amplitudes depending on frequency ratios, wave number ratios and media properties (density, shear modulus, limit of intensity of shear strains) has been analysed.