The Hanging Rope of Minimum Elongation for a Nonlinear Stress-Strain Relation

We consider the problem of determining the shape that minimizes the elongation of a rope that hangs vertically under its own weight and an applied force, subject to either a constraint of fixed total mass or fixed total volume. The constitutive function for the rope is given by a nonlinear stress-strain relation and the mass-density function of the rope can be variable. For the case of fixed total mass we show that the problem can be explicitly solved in terms of the mass density function, applied force, and constitutive function. In the special case where the mass-density function is constant, we show that the optimal cross-sectional area of the rope is as that for a linear stress-strain relation (Hooke's Law). For the total fixed volume problem, we use the implicit function theorem to show the existence of a branch of solutions depending on the parameter representing the acceleration of gravity. This local branch of solutions is extended globally using degree theoretic techniques.