The Surface Fitness of Textile Fabrics

Skewing of the warp and weft yarns of fabrics, fitted to any surface, is described as a function of the pre-deformed position of a fabric on the assumptions that (1) yarns are inextensible, that (2) the size of a yarn is ignored and that (3) the configuration of yarns on the surface agrees with geodesic curves. The differential equation satisfying such a conditions is derived thus: ∂2θ/∂x ∂y+K/√EG-F2=0 where x and y are curvilinear coordinates along the warp and weft yarns; θ (x, y) is the crossing angle between warp and weft yarns at (x, Y); F, G and F are the fundamental differential forms of the first kind of surface; and K is the Gaussian curvature of a surface. When the surface is a sphere R in radius, the solution is obtained under the following two types, (1) Type 1: assuming xy/R2 to be t, and boundary conditions, t=0, θ=π/2, θ=-1 θ≈π/2-t+t3/18-7t5/5×5!………… (2) Type 2 : 1et x-y/R be t, and boundary conditions t=0, θ=π/2, θ=0 θ≈π/2-1/2t23/6! t6-189/10! t10………… This coincides with the results by Mack and Taylor.