Equilibrium of Tchebychev nets
type: Item
instance of: scholarly article
title: Equilibrium of Tchebychev nets
zbMATH Open document ID: 0556.73046
DOI: 10.1007/BF00250867
author: A. C. Pipkin
published in: Archive for Rational Mechanics and Analysis
publication date: 1984-01-01T00:00:00Z
review text: A network is a continuum where every line \((x=ct\) or \(y=ct)\), is regarded as a fiber. Cloth is deformed by changing the angle between the threads of the warp and woof. The deformation due to fiberstretching is negligible by comparison to the finite equations for reinforced networks and discussed are the solutions and the singularities of these equations. The author investigates the general solution of the tangential equilibrium equations and points out that this general solution is equally well applicable to all developable surfaces. \textit{R. S. Rivlin} [Arch. Ration. Mech. Anal. 2, 447-476 (1959; Zbl 0082.386)], has studied the mechanical theory for networks that are not initially planar by way of general tensor notation. The author gives vector forms of the equilibrium equations for reinforced networks and discusses these equations. He discusses the singularities that can arise because the equations are hyperbolic. He remarks that \textit{J. J. Stoker} [Differential Geometry (1969; Zbl 0182.546); pp. 198-202] proved that although the geometric problem of clothing a surface is interesting, the solution can not have any physical meaning because equilibrium equations require the fibers to lie along geodesics. This would be true for a single fiber isolated from all the others, but the author remarks that in general the tension has a jump if the fiber is not geodesic. Giving an interesting example the author points out a particular way of clothing a paraboloid. This is remarquable and makes clear that this work is very interesting.
Mathematics Subject Classification ID: 74B99
Mathematics Subject Classification ID: 74C99
Mathematics Subject Classification ID: 74D99
Mathematics Subject Classification ID: 53A60
Mathematics Subject Classification ID: 53B50
Mathematics Subject Classification ID: 74K05
Mathematics Subject Classification ID: 35L67
zbMATH DE Number: 3887194
zbMATH Keywords: Cloth
zbMATH Keywords: finite equations for reinforced networks
zbMATH Keywords: general solution
zbMATH Keywords: tangential equilibrium equations
zbMATH Keywords: vector forms of the equilibrium equations
zbMATH Keywords: singularities
zbMATH Keywords: equations are hyperbolic
zbMATH Keywords: clothing a paraboloid
MaRDI profile type: Publication
cites work: Q5850220
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full work available at URL: https://doi.org/10.1007/bf00250867
OpenAlex ID: W4253371450
description: scientific article; zbMATH DE number 3887194