Stresses in equilibrium configurations of inextensible nets with slack

Abstract

The paper deals with nets formed by two families of fibers (cords) which can grow shorter but not longer, in a deformation. The nets are treated as two-dimensional continua in the three-dimensional space. The inextensibility condition places unilateral constraint on the partial derivatives y,1 and y,2 of the deformation of the form | y , 1 ( x ) | ≤ 1 , | y , 2 ( x ) | ≤ 1 , There is no deformation energy, the total energy reduces to the potential energy of the net under external forces. Equilibrium configurations are those of minimum energy. The stresses in equilibrium configurations thus reduce to the reactions to the constraints. Nonzero stresses occur only in tense regions where one or two constraints are satisfied with the equality sign. The paper follows the work of Paroni in treating the stress problem via the dual variational problem in the sense of convex analysis. Unlike in the work of Paroni, where stresses are modeled as finitely additive set functions, here a (perhaps more economic) choice of spaces is made that leads to more accessible stresses represented by (countably additive) measures. The present development is made possible by an observation, of independent value, that the space of measures with divergence measure is the dual of another Banach space, in the present context naturally interpreted as the space of strains. Our measures generalize stress fields represented by ordinary functions to account for stress concentrations along folded lines in tension, frequently occurring in equilibrium configurations of the net.