Convexity of the Strain-Energy Function in a Two-Scale Model of Ideal Fabrics

A two-scale model is used to generate the macro-scale constitutive response of a sheet of woven fabric from a micro-scale model of interacting yarns regarded as crossed elasticae in contact. The model furnishes a macro-scale strain-energy function for an orthotropic membrane idealized as being weak in shear compared to the extensional resistance of material curves representing the yarns. The operative Legendre–Hadamard inequality for the membrane is derived and shown to be satisfied by a suitably relaxed version of the computed strain-energy function.

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