Yarn cross-section and path length

[ldanielrosa]

TL;DR

How much does yarn 'squish'?

I'm trying to find the "ideal" cross-section for woven yarns. As seen in the first image, a circular monofilament (assuming inviolable cross-section but infinite flexibility) will have a limit of 2d thickness, and repeat period of 2d√3.

This allows me to calculate how long the yarn path length is, whether I want to know the weight or how much yarn to supply for a given amount of product.

However when the yarn is untwisted multifilament, the cross section will take a different shape. I'm having trouble determining exactly what is the optimum cross section.

I do not know whether the shape will be governed by circular arcs, parabolic arches, sinusoidal form, or something else entirely. I am assuming that the football/ellipsoid will be symmetrical across a line drawn between corners because this will equalize strain, but what I can intelligently guess ends there.

I want to know not only because I want to be able to calculate the path length for the reasons mentioned above (materials required and weight), but also because I want to know the thickness and pore size. My application is on a tubular braider (16 carrier), and soon on a 24 carrier braider, and 17 and 33 carrier flat braiders as well.

Can someone please direct me to an article I can read that will explain this?

 

[Lnewqban]

Could this article be helpful?
https://www.mdpi.com/2073-4360/15/3/637

 

[Rive]

Smells like 'spherical cows' for me... A practical approach might be more fruitful.

I would try to look for old books/literature from textile industry, about the properties of finished products and weaving methods/materials. I think it won't be too hard, just different: likely needs to be done offline (old style papers & libraries thing).

 

[Baluncore]

The path length for a flat braid?
The golden braid.