Prevent texture deformation when mesh deforms I have a plane, which has a texture UV mapped to it.  Let's say this plane is square and the texture is square.

I am trying to make a mouth (basically an opening in the plane which will have teeth, tongue, etc. that goes "down into" the plane) in this plane, but not deform the image on the plane.  Now this mouth will change shape, particularly when speaking (which I will do using shapekeys), and I don't know how to prevent the texture from getting deformed when the mouth is getting deformed due to speaking.

Now, I have found a modifier called UV Project, however, that does not work in Cycles, which is what I'm using.

So the main question is: How can I prevent a texture from deforming due to the mesh moving?

Edit  Added a .blend file, it should have the texture packed into it (a 1000x1000 pixel png).  I have made some vertex groups on the "Face" object, with the teeth.  Shapekeys and inside mouth color not finalized yet, so they are not in this .blend.
So I took Todd McIntosh's advice on messing with the mesh to try to prevent it from deforming, and I have (IMO) good results:



This image has the top and bottom of the mouth not closed at all, but the sides "fully" (only partially due to how my texture will work) closed using shapekeys.

Basically loop cut and slide a ring around the edge, then from the loop extrude and scale in.  Make sure to only scale in on the X and Y axes, or whatever two axes you want to prevent the surface from bowing in.

I have the shapekeys set up so that the sides of the mouth can't come more than halfway to the center, but the top and bottom can go all the way.  There is a little bit of intersection if you have the top mostly closed, and the sides all the way in, but for my purposes, I can work around that.

And here is a blend file with some shapekeys and vertex groups if you want to test.