I’ve been experimenting with a gold-braided circular device. The modeling result isn’t too bad (unless someone has a better technique/tutorial of how to do that look–I’m all ears!).
Here’s the file: http://pasteall.org/blend/index.php?id=44282
As you can see in the file, the unwrap is badly distorted from one end to the other. I tried different variations on that unwrap but I get the same result. Why won’t it just lay out flat and undistorted?
You need to add seams to this mesh before unwrapping. the mesh is connected all the way round so it’s impossible to unwrap without distortion unless you use seams. But bare in mind when using seams, you get UV islands, and not a fully connected set of UVs. so you can’t just add seamles textures to your material expecting them to flow right 
This is the best i could get it in my short time available. There is still mild distortion, mainly because the topology of this mesh is a little difficult to work with 
Gold Braid UV ProblemUnwrapped.blend (176 KB)
Some of the islands may be overlapped, as i didn’t take to much care with the unwrapping and packing. this is just an example to show you how i would work with the UVs of a mesh like yours
OK, wow. That’s a hell of a lot of islands, but it looks like it would be fine, thanks.
I put a seam on the bottom of the mesh. I figured I’d treat it like a cylinder–cut it around on one side and it’ll lay flat. I guess with the mesh twisted like this then it’s expecting a bit much from Blender’s UV tools?
How did you break it apart? I don’t see any red edges to indicate you did a “Mark Seam” operation.
I removed the seams after i unwrapped, just to save you the effort of doing it yourself… i should of left them on really to show you the setup 
You could try adding a loop cut around the middle of the ring and then mark it as a seam, then place seams at the top and bottom of the ring, and also on the part where the mesh folds in on the inside of the ring. This will give you less islands, but more probably distortion.
You can check distortion and fix it manually though in the UV panel, in the properties panel you can select an option that will show you either the angle distortion or area distortion. (Under the display section tick the stretch option)
And also, try this - https://blenderartists.org/forum/showthread.php?394506-Addon-Triplanar-UV-mapping
It works best on simple meshes, but may work on yours.
Unwrap with tri-planar add-on from the link (Same as normal uv unwrap - it appears at the bottom of the unwrap menu after installed) then press ctrl+p to pack UVs but disable rotation of islands in the menu that appears in the 3d viewport toolbar after pressing ctrl+p 