Equilibrium of Tchebychev nets
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.
finite equations for reinforced networks
vector forms of the equilibrium equations
scientific article; zbMATH DE number 3887194
scientific article