Modeling the Evolution of Deformed Fabric Surfaces as a Nonlinear Dynamical System

Abstract

Fabric drape is modeled as a nonlinear dynamical system. A localized two-dimensional deformation (e.g., folding or buckling), considered as the initial state, evolves, yam by yam, through the fabric, which bifurcates into complex wave configurations to form a three-dimensional fabric surface. Differential equations that arise in modeling fabric deformation such as buckling, folding, and drape can be generalized to the sine-Gordon equation. Tchebyshev nets are used to propose a generalized mathematical model for fabric deformation. The sine-Gordon equation is the compatibility condition that yields the net coordinates over the deformed fabric surface. Complex solutions of the sine-Gordon equation are constructed and plotted in three dimensions.Nonlinear partial differential equations to mechanicallv model fabric deformation arise in studies of fabric d2$ 1 Y d2$ bending [5] -= --s cos $, buckling [3, 81 -as2 B d. ? P a'$ P = --sin $, and woven mechanics [6] -= -sin $ B a. ? B Q B --cos $. These differential equations all belong to the Klein-Gordon family of nonlinear evolution equations, which can be simplified to the sine-Gordon SG equation:where f(s) is some function of s. This nonlinear partial differential equation first arose in the 1880s in differential geometry as a generalization of pseudospherical surfaces (surfaces of constant negative Gaussian curvature) [I]. Forty years later, Klein (1927) and Gordon (1926) derived a relativistic equation for a charged particle in an electromagnetic field, using the recently discovered ideas of quantum theory. Their Klein-Gordon equation reduces to what is now known as the sine-Gordon equation. In this paper, we will show how this equation (and thus all our fabric modeling equations) are invariant under the special Lorentz transformation, enabling the construction of an infinite number of analytical solutions of fabric drape.The problem of mapping an initially plane, orthogonal network of inextensible fibers onto a given curved surface is known to mathematicians as "clothing the surface'' [7]. Tchebyshev first suggested a continuum model for cloth in which the fibers or yams are treated as continuously distributed and inextensible [l 13. The theory presented by Tchebyshev was embraced by differential geometers and has since become known as Tchebyshev nets.A Tchebyshev net can be physically constructed with a ioosely woven fabric made up of inextensible yams in the warp and weft (orthogonal) directions with fixed yam contact points that can be rotated in shear. When this fabric net is draped over a surface, the warp and weft yarn axes become the coordinate lines on the surface. Such a coordinate can always be constructed locally, starting from any two intersecting curves. In fact, any surface can be locally covered by a Tchebyshev net.When the Gauss equations for any given surface are written in terms of the coordinates of a Tchebyshev net as parameters, they reduce to the equation [6] ---K sin II, , a2$ a.xay where x and y are the spatial coordinates, $(x, y ) is the ...