Nonlinear transformation of acoustic waves in microinhomogeneous media with relaxation

A nonlinear equation of state and the corresponding wave equation are derived in the framework of a rheological model suggested for elastic media containing defects with relaxation. The defects in the model are considered as highly compliant visco-elastic inclusions that exhibit a nonlinear stress-strain dependence. For real solids, these inclusions may model, for example, cracks, intergrain contacts and other similar defects. The proposed model thus can be applied to a wide class of microinhomogeneous materials (Earth rocks, engineering materials, damaged metals, etc.), which are also called mesoscopic solids. The derived equations consistently comprise the following material properties connected to the presence of the defects: (i) the microstructure-induced absorption (including near-constant Q-factor), (ii) the complementary dispersion of sound velocity, (iii) increased magnitude of the mesoscopic elastic nonlinearity and (iv) frequency dependence of the nonlinearity. To illustrate the aforementioned manifestations of the material microstructure, a few basic nonlinear effects (second and difference-frequency harmonics generation, and self-demodulation of high-frequency pulses) are analyzed in the framework of the derived equations. Main distinctions of the effects compared to the case of classical lattice (atomic) nonlinearity of homogeneous solids are pointed out.