Question

A stent is a small, tubular device implanted into arteries orsome other hollow organ in order to provide structural supportagainst collapse. Some modern stents can even provide gradual drugdelivery. Clinical failures are some of the most serious reasonsfor continual R&D into stent improvement. Models are anindispensable part of further research as experimental testing isoften expensive and time-consuming. For example, a stent with a10-year life expectancy corresponds to approximately 400 millioncyclic loadings from heartbeats!

A common design for a stent is a hollow woven mesh of helicalwire, as shown in the above figure. Much like Chinese finger traps,a change in length is accompanied by a corresponding change inradius of the stent. The goal of this problem is to examine howthis deformations are coupled.

(a) Let’s start out by looking at a single helix of wire.Convince yourself that we can parametrize the curve of the wireby:

x(θ) =rcosθ

y(θ) = rsinθ

z(θ) = (lθ/2π)sinα

where r is the radius and α is the ‘lead angle’ of the helix -the angle the wire makes with the xy-plane as it crosses it, and lis the length of wire in one turn of the helix. Determine thislength of wire l in one turn of the helix by using l =∫ ds withappropriate bounds, where ds =√(dx²+dy²+dz² ). Express l in terms of rand α.