Mapping a texture on curvelinear plane

I am trying to figure out how one would map a square texture such as the one I attached onto a circular plane. In the one image I attached called cobblestone circle 2, how would you map the entire area you see? My thought was this would need unwrapped but then how does one get the texture to follow the shapes?

I have searched the web a bit but I cannot find something this specific. The things I seem to find are related more to curved roads which are similar but one can deal with those as sweeps or splines where UVs can be automatically generated.

I am finding other cases where this challenge exists and I want to figure out the most efficient way to deal with it. I attached a chair arm I applied a checker grid with some UVs I created through 3dsmax quick pelt. How would one get the wood grain to follow the form of the arm side? It seems it is the same challenge as the curvlinear cobble.

Any help would be greatly appreciated.

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In the case of your chair arm, all you need to do is to straighten your UV’s. The wavy parts of your uv shell need to be straight.