Underdetermined system theory applied to qualitative analysis of response caused by attenuating plane waves

Abstract

Underdetermined system theory applied to quantitative analysis of responses caused by unsteady smoothplane waves Qualitative analysis of response caused by growing plane waves by underdetermined system theory Qualitative analysis of degradation processes of attenuating plane waves by underdetermined system theory A qualitative analysis of the mechanical response of rate-dependent media caused by a onedimensional plane smooth wave front and by a continuous wave front attenuating in the media is performed by an underdetermined system of nonlinear partial differential equations. The analysis reveals that smooth strain, particle velocity, and stress profiles, which the smooth wave front has, are not similar and that the wave front is composed of some partial waves having different properties. The property is represented by a set of strain rate, acceleration, and stress rate. The wave front derived here from the analysis is composed offour different partial waves. The front of the wave front is necessarily a contraction wave in which strain, particle velocity, and stress increase with time, while the rear is a rarefaction wave where they all decrease with time. Between these two wave fronts there are two remaining wave fronts. We call these wave fronts mesocontraction waves I and II. Wave front I is a wave in which stress decreases notwithstanding the increase in strain and particle velocity with time, which is followed by the other, i.e., wave front II, where with time, particle velocity, and stress decrease in spite of the increase in strain. The continuous wave front having continuous and nonsmooth profiles of strain, particle velocity, and stress canalso be composed offour waves. These waves possess the same property as the corresponding waves in the smooth wave front mentioned above. The velocities at three boundaries that the waves have are discontinuous. Therefore, these four wave fronts are independent waves, just as a shock wave and a rarefraction wa~e. Specifically, the front wave, i.e., a contraction wave front is being outrun by a second wave front, the second one is being outrun by a third wave front, and the third is being outrun by a fourth wave front, i.e., a rarefaction wave. We call the second wave front degenerate contraction wave I. We also call the third one degenerate contraction wave II. The stress-strain path and the stress-particle velocity path at a position in a rate-dependent medium which is passed by the continuous wave front are schematically shown. 3857

Generalized smooth and weak-discontinuous unsteady waves

“…It was recently clarified that the effects of the strain rate and strain acceleration induce smooth or weak-discontinuous unsteady wave fronts in those materials and they are responsible for anomalous stress-particle velocity and stress-strain paths. [5][6][7][8][9][10][11][12][13][14][15] In addition, Rankine-Hugoniot jump conditions 16,17 can only be applied insufficiently to some unsteady wave fronts that induce nonzero strain accelerations. 13,14 Use of these jump conditions in translating particle velocity into stress is one of the factors that causes the precursor decay anomaly in single-crystal lithium fluoride.…”

“…where b ii 0 and c iiϩ1 0; iϭ1,2,...,n. This generalization of the previous plane wave theory [5][6][7][8][9][10][11][12][13][14][15] using the conservation equations of mass and momentum also considers that B, C, and d depend on ũ, x, and t, respectively. The fundamental theorem demonstrates the theoretical existence of a smooth wave and a weak-discontinuous wave in a generalized wave.…”

Unsteady plane smooth wave fronts of combined growing-attenuating type

“…If such unsteady wave fronts could be formalized, then we could specify the response caused by the wave fronts. However, only our previous qualitative [3][4][5][6][7] and quantitative 8 work is applicable to the characterization of the wave fronts. For this purpose, the wave fronts were classified into the attenuating type, [3][4][5]7,8 and the growing type.…”

“…However, only our previous qualitative [3][4][5][6][7] and quantitative 8 work is applicable to the characterization of the wave fronts. For this purpose, the wave fronts were classified into the attenuating type, [3][4][5]7,8 and the growing type. 6,8 The peaks of the strain, particle velocity, and stress waves in the former type of wave front are all attenuating, while those in the latter type are all growing.…”

Numerical Analysis of Dynamic Mechanical Properties for Rock Sample under Strong Impact Loading

“…In order to inquiry the blast theory and fragment distribution of impacted rock, it is very important to research the mechanical properties and failure characteristics of rock under impact loading. Some scholars have achieved a series of research findings in the fields of shock characteristic and theoretical model [1][2][3][4][5][6][7][8][9][10][11][12], at the same times, the impacted rock is recovered for deep analysis. At present, there are two methods for impacted rock recover, the first is soft recover method, a certain mechanical parameter is tested in this proceed and the impacted rock accumulate in light-gas gun recover device, the second is hard recover method, the rock sample is implanted in the matched rigid die for the integrity of impacted rock.…”