39—AN ENERGY ANALYSIS OF WOVEN-FABRIC MECHANICS BY MEANS OF OPTIMAL-CONTROL THEORY PART I: TENSILE PROPERTIES

Abstract

The general energy analysis of fabric mechanics by means of optimal control presented in a previous publication is applied to the woven-fabric structure for deformations whereby the yarns remain in the same plane. The three-dimensional equations of the general theory are reduced to two-dimensional form, the boundary conditions are evaluated for the plain-weave fabric in biaxial tension, and an important mechanism of woven-fabric extension, which was not included in the general analysis, namely, the possibility of yarn extension, is introduced into the theory. The fabric load–extension curves and yarn-decrimping curves for the plain-weave construction are computed for a realistic range of input parameters: the yarn extension rigidity, weave crimp, degree of set, lateral yarn compression, and ratio of the curvilinear length of yarn in the warp direction to that in the weft direction of the fabric. The computed results are discussed in terms of the following dimensionless parameters: the applied tension per thread and the interaction force on the crossing thread (divided by the yarn bending rigidity), the relative fabric extension (the fabric extension divided by the yarn crimp), the initial relative fabric tensile rigidity, the Poisson's ratio of the fabric, and the initial tensile rigidity of the crimped yarn unravelled from the fabric. The tensile properties of plain-weave fabrics in both the grey state and the finished state are illustrated by reference to the computed results, which are also employed to explain the behaviour of yarns during the extension of woven fabrics.