Wave curves for the Riemann problem of plane waves in isotropic elastic solids
type: Item
instance of: scholarly article
title: Wave curves for the Riemann problem of plane waves in isotropic elastic solids
zbMATH Open document ID: 0637.73020
DOI: 10.1016/0020-7225(87)90014-0
author: Zhijing Tang
author: Ting, T. C. T.
published in: International Journal of Engineering Science
publication date: 1987-01-01T00:00:00Z
review text: (From authors' summary.) A modified Riemann problem in which the initial and boundary conditions are constants is considered for plane waves in a half-space occupied by an elastic solid. The governing quasilinear differential equations form a system of hyperbolic conservation laws which possesses three wave speeds \(c_ 1\geq c_ 2\geq c_ 3\). The system is genuinely nonlinear with respect to \(c_ 1\) and \(c_ 3\) and linearly degenerate with respect to \(c_ 2\). Thus it is sufficient to study a two-wave-speed system with \(c_ 1\) and \(c_ 3\). Wave curves for simple waves and shock waves are used to construct the solution. Second- order hyperelastic materials which contain four material constants are considered and the solution in the form of wave curves is obtained of all possible combinations of initial and boundary conditions. The two-wave-speed system has one umbilic point at which \(c_ 1=c_ 3\) and hence the system is not totally hyperbolic (or not strictly hyperbolic). Several interesting and unexpected results are obtained due to the existence of the umbilic point. In one example, we find that a shock wave satisfies the Lax stability condition for a \(V_ 1\) shock as well as a \(V_ 3\) shock. In another, a shock wave which involves only one stress component does not satisfy the Lax stability condition for either a \(V_ 1\) shock or a \(V_ 3\) shock. However, it satisfies the Lax condition if we consider it under the context of a one-wave-speed system.
Mathematics Subject Classification ID: 74J10
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 76L05
Mathematics Subject Classification ID: 35L65
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 4039403
zbMATH Keywords: modified Riemann problem
zbMATH Keywords: initial and boundary conditions
zbMATH Keywords: plane waves
zbMATH Keywords: quasilinear differential equations
zbMATH Keywords: system of hyperbolic conservation laws
zbMATH Keywords: Wave curves
zbMATH Keywords: simple waves
zbMATH Keywords: Second-order hyperelastic materials
zbMATH Keywords: two-wave-speed system
zbMATH Keywords: umbilic point
zbMATH Keywords: shock wave
reviewed by: P. Laasonen
MaRDI profile type: Publication
full work available at URL: 0020-7225(87)90014-0
OpenAlex ID: W2052312859
cites work: Q5801144
cites work: Hyperbolic systems of conservation laws II
cites work: Q3213564
cites work: Q4084659
cites work: Q4124409
cites work: Q3320768
cites work: Finite amplitude waves in incompressible perfectly elastic materials
cites work: ONE-DIMENSIONAL NON-LINEAR WAVE PROPAGATION IN INCOMPRESSIBLE ELASTIC MATERIALS
cites work: THE PROPAGATION AND INTERACTION OF ONE-DIMENSIONAL NON-LINEAR WAVES IN AN INCOMPRESSIBLE ISOTROPIC ELASTIC HALF-SPACE
cites work: FINITE SIMPLE WAVES IN A COMPRESSIBLE TRANSVERSELY ISOTROPIC ELASTIC SOLID
cites work: SIMPLE WAVES IN CONSTRAINED ELASTIC MATERIALS
cites work: The Riemann problem for general systems of conservation laws
cites work: On the solution of the Riemann problem with general step data for an extended class of hyperbolic systems
cites work: Plane waves in simple elastic solids and discontinuous dependence of solution on boundary conditions
cites work: Plane Waves Due to Combined Compressive and Shear Stresses in a Half Space
cites work: The partial differential equation ut + uux = μxx
cites work: Q3999241
cites work: Solution of the Riemann problem for a prototype 2\(\times 2\) system of non-strictly hyperbolic conservation laws
cites work: The classification of 2 × 2 systems of non-strictly hyperbolic conservation laws, with application to oil recovery
cites work: On wave propagation problems in which 𝑐_{𝑓}=𝑐_{𝑠}=𝑐₂ occurs
cites work: Q5646485
cites work: Q4023641
cites work: Q3327849
cites work: Non-linear wave propagation with applications to physics and magnetohydrodynamics
cites work: Solution of the Riemann problem for a class of hyperbolic systems of conservation laws by the viscosity method
description: scientific article; zbMATH DE number 4039403