Underdetermined system theory applied to quantitative analysis of responses caused by unsteady smooth-plane waves

The mechanical responses of rate-dependent media caused by unsteady smooth-plane waves are quantitatively analyzed by an underdetermined system of equations without using any constitutive relation of the media; that is, by using the particle velocity field expressed by an algebraic equation that is derived from the mass conservation equation, and the stress field expressed by an algebraic equation that is derived from the momentum conservation equation. First of all, this approach for analyzing unsteady wave motion is justified by the verification of various inferences such as the existence of the nonindependent elementary waves by Sano [J. Appl. Phys. 65, 3857(1989)] and the degradation process by Sano [J. Appl. Phys. 67, 4072(1990)]. Second, the situation under which a spike arises in particle velocity–time and stress–time profiles, and the reason for the arising are clarified. Third, the influence of the spike on stress-particle velocity and stress–strain paths is examined. The spike induced in the profiles by a growing wave greatly influences the paths near the impacted surface. Finally, calculated particle velocity–time profiles are compared with experimental data.