Low tension cables subject to torque may form complex three-dimensional (spatial) equilibria. The resulting nonlinear static deformations, which are dominated by cable flexure and torsion, may produce interior loops or kinks that can seriously degrade the performance of the cable. Using Kirchhoffrod assumptions, a theoretical model governing cable flexure and torsion is derived herein and used to analyze (1) globally large equilibrium states, and (2) local equilibrium stability. For the broad class of problems described by pure boundary loading, the equilibrium boundary value problem is integrable and admits closed-form elliptic integral solutions. Attention is focused on the example problem of a cable subject to uni-axial torque and thrust. Closed-form solutions are presented for the complex three-dimensional equilibrium states which, heretofore, were analyzed using purely numerical methods. Moreover, the stability of these equilibrium states is assessed and new and important stability conclusions are drawn.

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A set of three nonlinear partial-differential equations is derived for general finite deformations of a thin membrane. The material that composes the membrane is assumed to be hyperelastic. Its mechanical property is represented by the neo-Hookean strain-energy function. The equations reduce to special cases known in the literature. A fast convergent algorithm is developed. The numerical solutions to the finite-difference approximation of the differential equations are computed iteratively with a trivial initial iterant. As an example, the problem of inflating a rectangular membrane with fixed edges by a uniform pressure applied on one side is presented. The solutions and their convergence are displayed and discussed.

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