40—AN ENERGY ANALYSIS OF WOVEN-FABRIC MECHANICS BY MEANS OF OPTIMAL-CONTROL THEORY PART II: PURE-BENDING PROPERTIES

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Abstract
The energy analysis of woven-fabric mechanics developed in Part I of this series is extended by considering the pure-bending behaviour of the plain-weave structure as a generalization of the tensile deformation. The boundary conditions for the plain-weave fabric in bending are evaluated, and the ratio of fabric bending rigidity per thread to yarn bending rigidity is computed for a range of fabric structures of different values of weave crimp and degree of set by introducing inequality constraints on the curvature or control variable, a concept borrowed from optimal-control theory. The theoretical results for fabric bending rigidity are compared with experimental work reported by previous workers, and the theoretical shape of the yarn axis in a bent fabric is computed for different fabric curvatures. A comparison of the theoretical results computed by means of optimal-control theory for plain-weave, plain-knitted, and 1 × 1 rib-knitted structures is presented in terms of the following parameters expressed as dimensionless units: modular yarn length, inter-yarn forces, strain energy, total energy, initial fabric tensile modulus. Poisson's ratio, and yarn-decrimping modulus. The results show a surprising similarity in fabric properties expressed in terms of dimensionless parameters, and the implications of the work reported in this paper for future experimental studies of fabric mechanical properties are discussed.