Nonlinear response of a relaxing shear wave resonator to elliptical driving motion

Abstract

Soft elastic media such as rubber and soft tissue possess small shear moduli, facilitating the generation of large shear deformations. These materials may also exhibit viscoelasticity such as stress relaxation over the frequency range of interest. An augmented form of the Duffing equation was recently developed to model the response near the lowest resonance of a shear wave resonator formed with a nonlinear relaxing material that is shaken at one end and free at the other [J. Acoust. Soc. Am. 143, 1035 (2018)]. The augmented Duffing model was found to accurately describe the response of the resonator when the driving motion is linearly polarized. Here the model is extended to account for elliptical driving motion at frequencies near the lowest resonance. The augmented Duffing model in this case consists of two coupled ordinary differential equations for the two displacement components. Amplitude-dependent phenomena such as amplification of the minor displacement component and an induced phase shift between the displacement components are predicted. Approximate analytical solutions of the coupled equations are obtained for weakly nonlinear motions and for driving motions that are close to either circular or linear polarization.