Qualitative analysis of response caused by growing plane waves by underdetermined system theory

Abstract

A qualitative analysis of the mechanical response of rate-dependent media caused by one-dimensional plane smooth- and continuous-wave fronts with the growing peaks of strain, particle velocity, and stress is performed by an underdetermined system of nonlinear partial differential equations. The response found by the analysis reveals that strain, particle velocity, and stress profiles in the smooth-wave front are not similar and that the front is composed of five partial waves having different properties. The property is represented by the set of strain rate, acceleration, and stress rate as in a previous study. The front of the wave front is necessarily a contraction wave in which strain, particle velocity, and stress increase with time. The second partial wave is another contraction wave. We call the wave a vice-contraction wave. The rear is assumed to be a rarefaction wave where they all decrease with time. Between these two partial waves there are two remaining partial waves. We call these waves mesorarefaction waves I and II. Wave II is a wave in which particle velocity and stress increase, notwithstanding the decrease in strain with time. It is followed by wave I in which the increase in stress with time occurs in spite of the decrease in strain and particle velocity. The continuous-wave front, which has discontinuous-movement velocities at the continuous, but nonsmooth, positions in the profiles of strain, particle velocity, and stress, is composed of five independent waves. These waves are a contraction wave, a vice-contraction wave, evolutional rarefaction waves II and I, and a rarefaction wave which possess the same properties as the corresponding partial waves in the smooth-wave front mentioned above. Both in the smooth-growing-wave front and in the continuous one the peak precedence is in the order of the strain, particle velocity, and stress peaks. The stress-strain path and stress-particle velocity path at a position in a rate-dependent medium which is passed by the continuous-wave front are schematically shown.