Imagine one requests a weaver to put two or three 3k weaves in the warp for every one 1k weave in the weft, such that the fabric is mostly unidirectional. Is there a general technical name for this kind of -shall we say- asymmetrical fabric?
How would the weft stay in place? It would be unstable. The only way would be if it was stitched diagonally.
The solution is to lay unidirectional at 90, then lay a much lighter unidirectional at 0.
If you mean something like this , it is a 4-harness satin weave . The top pattern is a 12k/7x7 and the bottom is a 3k/12x12 .
No, he means a weave which is 90% warp, 10% weft.
If you flip a satin weave cloth over, you’ll see the opposite. One side the warp goes over 3 under 1, the other side the warp goes over 1 under 3.
Not necessarily 90/10, but some kind of unequal weave. I cannot see why a plain weave with say 3k tow in the warp and 1k in the weft would be unstable, but then again, I am just starting to learn about weaves.
I know I could use a combination of unidirectional and a weave, but I want an unbalanced fabric and want to know the name of such. Surely there must be some industry nomenclature for a weave that is significantly heavier in one direction.
3x3k = 9k warp
1x1k = 1k weft
That’s 90/10 
The reason it wouldn’t work is because 1k is 3 times lighter and therefore 3 times less dense than 3k. If you had a square cutout, it would have 100 3k warp tows and 300 1k weft tows. overall the warp and weft would be of equal density, in which case its no different. A pointless exercise.
And if you somehow forced the weft tows to stay 2 tows apart from each other (the only way to do it is to stitch them in place diagonally), you’ll have big gaps where there should be fabric. If you don’t stitch them together, the fabric would distort horribly as the weft tries to fill those gaps.
Specialty fabrics can be woven. They just cost hundreds of dollars per yard. The only way it would ever get woven is if a very large company like Boeing requests it. The rest of the composites industry would just get the leftovers. It’s much cheaper to vary the thread count over spooling up 1k carbon. 1K is more costly to produce than 3k or 12k. The demand for 1k is also lower.
Btw, the yarn used or the thread count does not dictate the weave. The over under nature of the yarns dictates the weave. There are heaps of woven fabrics that are not balanced (varying yarn and threat count) that all have the save weave. When looking at a fabric you need to look at the weave (plain, twill, harness satins, leno, etc), thread count, and yarn used. The variations end up with style #s. Each style has a designated weave, fiber, thread count, and yarn bundle.
There are some fun glass fabrics like Style #1576 - 12H Satin, 120 x 24 thread count, ECG 150 1/2 x ECG 150 1/0, 600 x 100 breaking strength. There are some carbons like 262 - Plain weave, 12x8, 3K x 3K. Some hyprid carbons use carbon in warp and glass in fill. I can’t remember any of their style #s.
Your desired fabric has no Style # that I’m aware off. It would be much cheaper for you to use available formats than pay through the nose and every other bodily orifice for something special. To get the attention of a weaver would cost you $$$$$$$$$$$. Nobody, at least that I’m aware of, has found your style in great enough demand to justify the expense and setup.
Hmmm… I hate to be sceptical, particularly when I am entirely new to carbon fiber and because I greatly appreciate the responses of others, but a few things in the last replies just do not make sense. Carboncactus, you seem to be intimating that making an unbalanced fabric is impossible. I watched a few videos recently posted in response to one of my questions, and after doing so, and noting the sophistication of the looms, I cannot believe it is impossible. Maybe my example does not work as I just conjured it out of thin air, but surely a “mostly” unidirectional fabric is possible.
So far as demand goes, again, I cannot believe that the entire CF industry survives on just uni and balanced weaves. I am a novice and even I can think of several advantages and applications for an unbalanced fabric. Lastly, in today’s economic climate, I cannot believe that every company that has looms to weave carbon fiber is so busy that all could command hundreds of dollars a yard. Mind you, it might take some shopping around to find the right company to do a role or two.
You can go to Hexcel’s website and download a style guide that shows you all the styles. While your at it you should give them a call and see if they will shut down their looms, set one up for your fabric, and weave a couple hundred yards. 
It my help you to know that Wovens are often avoided in high performance structures. Many of them are made with multiple layers UD because they perform better.
I’m just thankful that prices are as low as they are. It wasn’t that long ago when the average person couldn’t even get it in any form. What you could get was crazy expensive.
The example you gave with 3x3k warp and 1x1k weft to form a fabric that is 9 times stronger at the warp than it is at the weft is impractical, if not impossible. Without stitching it would fall apart and the stitching needed to hold the fabric together would have to be close together and make it the whole thing heavier.
There wouldnt be a need for a mostly unidirectional fabric, you just lay up 5 layers of uni at 90 and one at 0, or whatever.
The only fabric could be “unbalanced” would be a hybrid. Like this one:
That has carbon at the warp, glass at the weft, so It would be more rigid and have more tensile strength one way, more flexible the other, but not much. These sorts of fabrics are mostly used for cosmetic applications. It would also be heavier than modulating a layup schedule out of unidirectional carbons.
An unbalanced woven is possible and quite common. If you look at a hot-melt carbon uni-directional it is a woven with carbon in the warp and a polyester thread in the weft that is then melted slightly to stabilize the fabric. I have seen carbon uni with light weight glass in the weft as well but they still use a polyester thread woven in which is melted to stabilize everything. So to answer the original question, yes it is possible but I am not aware of any standard nomenclature and it will be a special setup for the weaver which means a minimum order quantity will apply and the price will also be high.
I also agree that unless you are requiring hundreds of pounds of custom material or you can afford to have a small quantity of custom fabric produced I would stick with standard products. You can also contact some weavers and see if they have any products that may fit your needs as they may manufacture them but not advertise them since they are niche products.