General theory of the solutions of the equations of motion of an elastic medium of different moduli
type: Item
instance of: scholarly article
title: General theory of the solutions of the equations of motion of an elastic medium of different moduli
zbMATH Open document ID: 0608.73034
DOI: 10.1016/0021-8928(85)90031-0
author: V. P. Maslov
author: P. P. Mosolov
published in: Journal of Applied Mathematics and Mechanics
publication date: 1985-01-01T00:00:00Z
review text: Consider the one dimensional partial differential equation \(u_{tt}=(u_ x-a| u_ x|)_ x\) with \(a=const.\), which describes the longitudinal motion of an elastic medium. Depending upon the value of a, the medium will be of different modulii, granular or elasto-plastic and will be applicable to geophysical and two-phases systems. The general solution is attempted through integral transforms with piecewise smooth solutions, leading to the Hugoniot conditions. The study of the local structure is then carried out. For the case \(0<a<1\), shocks, simple discontinuities, signotons, semi-signotons and other solutions are classified. For \(a=1\), an additional type, a discontinuity in the continuity of displacement, occurs. The evolution of the bifurcation of these features is next considered. Reflections of a compressional wave from a free edge or a rigid wall and the particle motion in an elastic granular medium with body forces are considered in detail in which substantial nonlinear effects occur. Comments are also made on the general Cauchy and associated boundary/initial value problems.
Mathematics Subject Classification ID: 74H45
Mathematics Subject Classification ID: 35L60
Mathematics Subject Classification ID: 74S30
zbMATH DE Number: 3984687
zbMATH Keywords: longitudinal motion
zbMATH Keywords: different modulii, granular
zbMATH Keywords: elasto-plastic
zbMATH Keywords: piecewise smooth solutions
zbMATH Keywords: Hugoniot conditions
zbMATH Keywords: shocks
zbMATH Keywords: simple discontinuities
zbMATH Keywords: signotons
zbMATH Keywords: semi-signotons
zbMATH Keywords: discontinuity in the continuity of displacement
zbMATH Keywords: bifurcation
zbMATH Keywords: Reflections
zbMATH Keywords: compressional wave
zbMATH Keywords: free edge
zbMATH Keywords: rigid wall
zbMATH Keywords: particle motion
zbMATH Keywords: elastic granular medium
reviewed by: E. S. R. Gopal
MaRDI profile type: Publication
full work available at URL: 0021-8928(85)90031-0
OpenAlex ID: W2079353007
cites work: Q5566685
cites work: Formation of singularities in one-dimensional nonlinear wave propagation
cites work: Solutions in the large for nonlinear hyperbolic systems of equations
cites work: Q5630591
cites work: Q3846827
cites work: Q3844581
cites work: Q3658762
description: scientific article; zbMATH DE number 3984687