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I have a cylinder of original dimensions r and h, where h is the original height and r is the radius. It is fixed on the surface f the Earth and its z axis is along the axis of the cylinder hanging down into a hole in the earth. Upon stretching along its axis only I have an h' = h + a for the stretched cylinder which will produce an r(z,a) deformation of the radius, that is, a non-constant one along its length but dependent upon the z coordinate of the cross-section and the displacement a of the free cylinder end.

I know this is something to do with the Poisson Ratio, but that suggests radial contraction is uniform which it simply could not be. I know that I can just find the total strain energy of the cylinder if I have r(z,a) by a solid of revolution method but I'm at a loss as to what type of function could describe the varying radius. Any help would be much appreciated, I'm trying to teach myself nonlinear elasticity and it's quite hard!