Nonlinear shear wave resonator consisting of a relaxing material

Abstract

Soft materials such as rubbers, polymers, and tissue exhibit low shear wave speeds, facilitating the generation of shear waves with large acoustic Mach numbers. In addition to finite-amplitude effects that result from cubic nonlinearity, plane shear wave propagation in these materials is subject to frequency-dependent attenuation and dispersion that result from viscoelastic effects. A wave equation for plane shear waves in a relaxing material is obtained from a nonlinear Zener constitutive model that accounts for cubic nonlinearity as well as the attenuation and dispersion associated with relaxation. The wave equation is used to analyze a one-dimensional shear wave resonator comprised of a nonlinear relaxing material that is shaken at one end and free at the other. For excitation of the lowest mode the wave equation is approximated by an augmented Duffing equation, and the resulting frequency-response equation is compared with numerical finite-difference solutions of the original wave equation. Frequency ...