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

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 ...

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