What happens to matter when there's distortion in space This is all hypothetical, but if you had control of space (to do distortions to space as very heavy objects do), what happens to the matter in the space? To have a concrete example, let's say that you have a rectangular volume of space, in it there's air and a solid beam going lengthwise. Now you do different distortions to the space in the middle of this rectangle, leaving the two lengthwise corners fixed to the "original" space. Some examples of the distortions are that you compress/expand the space, or twist it, or curve it. What happens to the bar and air in it? Now suppose you leave the space distorted, and you push a bar from one end of the rectangle to the other, what happens to the bar?
 A: Your question is slightly unclear, but I think you are asking about the distortion or warping that is described by general relativity. So, to carry out the distortion you have in mind, I could, for example, bring along a neutron star and put it near to the region you have in mind.
Suppose you have some wooden things, some iron bars, and some pins fixed in a rubber lattice with a very very low Young's modulus. What will happen is, mainly, tidal forces or stress. The pins will follow geodesics in spacetime, which means they will move nearer to one another in some areas, and further from one another in other areas, as time goes on. Imagine them falling down towards the neutron star on radial trajectories, for example. The rubber will weakly restrict this motion.
Meanwhile the iron bars will also be squeezed or stretched, but not so much because their internal electromagnetic and other forces will tend to keep them close to fixed proper length. But all the distance measurements can only be done in a relative way. So if the pins move relative to the bars, for example so that a given bar no longer reaches between a given pair of pins, then you might ask: did the pins get further apart or did the bar shrink? The answer is that the pins got further apart. This is an example of the strong equivalence principle, which says that each iron bar and each pair of pins is essentially unaffected in their own rest frame, apart from the tidal stress. That stress has a much bigger effect on freely floating or loosely joined things than it does on rigid materials such as iron.
A: An excellent analysis by @AndrewSteane. But I'll try to answer your question through another viewpoint. Suppose instead of using mass as your space-time warper, we use energy as our source ( kind of like how wormholes are created not by huge masses, but rather using lots and lots of energy). This prevents the gravitational collapsing of matter (so the bar won't break), and will lead to interesting results. If you distort your rectangular volume in some manner using energy, let's say you twisted the volume like how you wring a wet towel, then neither the air nor the bar will develop stresses within them. This is because the very manifold of space is twisted, not the matter contained within it. 
To see why, imagine this - suppose you drew a good sized circle on a flat piece of paper. Then you held the paper with your hands and crumpled it to the size of a small ball. What's interesting is that although you crumpled the paper (just like you warped 3D space), you didn't physically alter the circle you drew in any manner (much like the air and bar don't strain). The circle is still a circle! It's just on a very weirdly curved space now. Only by reopening the paper and erasing a part of the circle do you alter the object contained within the 2D space. 
So nothing actually happens to the air and wooden bar in your towel-wringed curved space. Sure you warped time along with warping space, but that is another issue in itself. The wood may age slower or quicker based on how you warp space, but the air won't be compressed at some places or the bar won't break. 
Note: Now we certainly don't know how to use energy to achieve warping of space-time, so this is purely hypothetical (if we did know, we'd be all be going to work via wormholes!)
Hope this helps and possibly answers your question!
