Equilibrium of elastic nets

Abstract

A general equilibrium theory for nets constructed from two families of perfectly flexible elastic fibres is presented. The fibres are assumed to be continuously distributed and to offer negligible resistance to shear distortion. Configurations of nets are shown to be minimizers of the potential energy of deformation only if the associated fibre stretches are points of convexity of the fibre strain energy functions and the stresses in the fibres are tensile. These results are used to construct a relaxed energy density that automatically accounts for wrinkling of the network. Universal solutions are obtained. These are the deformations that can be maintained in every elastic net by the application of edge tractions and lateral pressure alone. A detailed study of the differential geometry of nets is included to aid in their interpretation. The equilibrium theory for half-slack (wrinkled) nets is developed and applied to the solution of some representative examples.