The aim of this paper is to model the unique dynamic deformation properties of textile materials. Fabric deformation is modeled as a nonlinear dynamic system so that a fabric can be completely specified under general boundary conditions. Fabric deformation or fabric drape is dynamically analogous to waves traveling in a fluid. A localized twodimensional deformation evolves through the fabric to form a three-dimensional drape or fold configuration. The nonlinear differential equations arising from the analysis of fabric deformation belong to the Klein-Gordon family of equations, and become the sine-Gordon equation in three dimensions. The sine-Gordon equation has its origins in the study of Bäcklund Transformations is used to transform a trivial solution into a series of solitary waves within the fabric. These analytical expressions describing the curvature parameters of a surface represent actual solutions of fabric dynamic systems.

Aims to analyse unique deformation properties of textile materials in terms of basic mechanical properties. Models fabric deformation as a nonlinear dynamical system so that a fabric can be completely specified in terms of its mechanical behaviour under general boundary conditions. Fabric deformation is dynamically analogous to waves travelling in a fluid. A localized two‐dimensional deformation evolves through the fabric to form a three‐dimensional drape or fold configuration. The nonlinear differential equations arising in the analysis of fabric deformation belong to the Klein‐Gordon family of equations which becomes the sine‐Gordon equation in three dimensions. The sine‐Gordon equation has its origins in the study of Bäcklund Transformations in differential geometry. Describes fabric deformation as a series of transformations of surfaces, defined in terms of curvature parameters using Gaussian representation of surfaces. By considering a deformed fabric as a two‐dimensional surface, algebraically constructs analytical solutions of fabric deformation by solving the sine‐Gordon Equation. The theory of Bäcklund Transformations is used to transform a trivial solution into a series of solitary wave solutions. These analytical expressions describing the curvature parameters of a surface represent actual solutions of fabric dynamical systems.

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The buckling behaviour of engineering materials has been researched extensively since the 1890s and more recently, thin shell theory has generalised the analysis to include complicated boundary conditions. However, the approximations and assumptions which form the basis of engineering models make them inappropriate for textile materials. Very small stresses on textile materials cause extremely large strains so that the deformations are highly nonlinear. In this paper, we develop a nonlinear mathematical method. In the final section, the nonlinear differential equations used are generalised into a nonlinear evolution equation which is completely integrable and thus solved analytically obtaining dynamical solution for three‐dimensional fabric drape. These analytical solutions are applicable under all conditions and are not subject to computational difficulties associated with finding numerical solutions for highly nonlinear problems. The use of this analytical approach to fabric mechanics and dynamics provides us with a very powerful tool to formulate and solve many long‐standing problems in fabric and clothing technology.

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 ...