On ideal dynamic climbing ropes

TL;DR

This paper models the properties of an ideal dynamic climbing rope that minimizes peak force during falls, considering factors like a carabiner and nonlinear elasticity, and explores shape memory materials as potential solutions.

Contribution

It introduces a mathematical framework for designing optimal dynamic ropes using nonlinear elasticity and analyzes shape memory materials as ideal rope candidates.

Findings

Optimal ropes minimize peak force during falls.

Shape memory materials exhibit tension-strain features of ideal ropes.

Hysteresis in shape memory materials absorbs fall energy effectively.

Abstract

We consider the rope climber fall problem in two different settings. The simplest formulation of the problem is when the climber falls from a given altitude and is attached to one end of the rope while the other end of the rope is attached to the rock at a given height. The problem is then finding the properties of the rope for which the peak force felt by the climber during the fall is minimal. The second problem of our consideration is again minimizing the same quantity in the presence of a carabiner. We will call such ropes \textit{mathematically ideal.} Given the height of the carabiner, the initial height and the mass of the climber, the length of the unstretched rope, and the distance between the belayer and the carabineer, we find the optimal (in the sense of minimized the peak force to a given elongation) dynamic rope in the framework of nonlinear elasticity. Wires of shape memory materials have some of the desired features of the tension-strain relation of a mathematically ideal dynamic rope, namely a plateau in the tension over a range of strains. With a suitable hysteresis loop, they also absorb essentially all the energy from the fall, thus making them an ideal rope in this sense too.