On the compression of a short cylinder between rough end-blocks
1. In this paper we consider non-uniform elastic compression between parallel endblocks of a cylindrical test specimen with circular cross-section. It is assumed that there is sufficient friction between the end-blocks and the end faces of the specimen to prevent slippage. Thus, points on the end faces of the cylinder undergo only axial displacements. It is the purpose of this paper to determine the relation between the force F acting on the specimen, the compression 2a, and Young's modulus E of the material, assuming Poisson's ratio v to be known. If lateral expansion were not obstructed by the friction at the end surfaces, these quantities would be connected by the relation E = Fh/Aa, where 2h denotes the height of the specimen, b its radius and A its cross-sectional area. If there is friction at the end surfaces, a correction must be applied to this simple formula; the derivation of this correction will now be sketched. The above problem is solved to a good approximation by finding upper and lower bounds for the strain energy of the specimen, using the method of Prager and Synge.1 It is advantageous to resolve the original problem, which we shall denote by P0 , into two simpler problems Pc and P, where Pc is the problem of simple compression and P is that problem which will yield the initial problem P0 when superimposed on Pc. In the natural cylindrical coordinates, the original problem P0 involves the boundary conditions