Plane nonlinear shear waves in relaxing media

This dissertation investigates plane nonlinear shear wave motion in a material that possesses a single relaxation mechanism. Derivations that employ three common yet separate descriptions of viscoelasticity are followed to arrive at the same mathematical model for a nonlinear and relaxing medium. The model is then used to obtain a single wave equation for linearly polarized particle motion, and two coupled wave equations for elliptically polarized particle motion. The resulting wave equations account for cubic nonlinearity and viscoelasticity in the form of a single relaxation mechanism. Progressive wave solutions are obtained from a Burgers-type evolution equation, thereby illustrating competition between nonlinearity and viscoelasticity in the wave motion. Energy loss due to relaxation is found insufficient to prevent the occurrence of multivalued solutions of the evolution equation when the wave amplitude is sufficiently large, thus indicating that the model is lacking essential physics in those cases. Multivalued solutions are prevented by employing weak shock theory in the obtained analytical solution for a step shock, or by introducing shear viscosity when solving the evolution equation numerically. A coordinate transformation is employed that allows simulation of waveform evolution up to and beyond the point of waveform overturning, which permits determination of the parameter space in which multivalued solutions exist. The analysis is also applied to nonlinear shear waves in media characterized by attenuation that is proportional to frequency raised to some power. Finally, standing nonlinear shear waves are investigated by developing an augmented Duffing equation that describes the nonlinear response of a shear wave resonator near its lowest resonance. Both linearly and elliptically polarized motions are described with analytical implicit solutions of the augmented Duffing equation.