Stretched fabric cylinder between two hoops - what is the shape?

[sperryrand]

Hi Everyone,

Imagine a tube of fabric stretched between two hoops, like the one's seen here..
http://www.stretchshapes.net/blog/wp-content/uploads/2012/08/A1.png
That is, a cylinder made of an elastic material, under tension.

Is there a name for this shape?
Is there a generic equation describing it?
So far, searching the web I've come up with nothing, but I assume I'm using the wrong vocabulary.

Any tips appreciated!
MM

 

[Danger]

I severely doubt that this would be considered an official term, but in my circles the shape would be called a "spool". In the case of multiples as in the picture, they would be "stacked spools".

 

[A.T.]

If done with a soap film in zero gravity you get a minimal surface called Catenoid:
http://en.wikipedia.org/wiki/Catenoid

A fabric however can have nonisotropic stresses, so its not exactly a minimal surface anymore.

 

[sperryrand]

Thanks. Catenoid has put me in the right direction.

 

[sperryrand]

http://www.membranes24.com/theory-form-finding.html?showall=1

 

[haruspex]

Seems to me there is a crucial difference between soap films and elastic materials. The tension in a soap film is constant, so it adopts a minimal surface area to minimise the surface energy. An elastic material has a modulus, the tension increasing with the stretch.
In the illustration in the OP, the difference is clearest in the left-hand image. The spool shaft is almost straight over the middle third, only flaring at the ends. A soap film would become much narrower in the middle.

 

[A.T.]

Seems to me there is a crucial difference between soap films and elastic materials. The tension in a soap film is constant, so it adopts a minimal surface area to minimise the surface energy. An elastic material has a modulus, the tension increasing with the stretch.
In the illustration in the OP, the difference is clearest in the left-hand image. The spool shaft is almost straight over the middle third, only flaring at the ends. A soap film would become much narrower in the middle.

Yes, that's what I meant by "nonisotropic stresses". You can stretch the fabric in one direction more than in the other.

So only in special cases (I assume when you make sure the strains are uniform and isotropic) those fabrics will be an exact minimal surface. If minimizing the amount of fabric used was the sole goal, this would be the optimal configuration. But obviously they are other criteria for designing declarations than that.

 

[haruspex]

Yes, that's what I meant by "nonisotropic stresses". You can stretch the fabric in one direction more than in the other.
So only in special cases (I assume when you make sure the strains are uniform and isotropic) those fabrics will be an exact minimal surface.

Not just isotropic, but isotopic. My point is that if you stretch a bit of soap bubble the tension stays constant, but if you stretch an elastic fabric it will increase. So the cases in which the fabric will assume a minimal surface are trivial.