Using cylindrical coordinates, we wish to determine and compare the uniaxial stress strain response of both an unconfined and confined isotropic homogeneous elastic cylindrical sample as shown. For both cases we will assume that the shear stresses and strains will vanish. For the unconfined sample, the stresses σ_r = σ_θ = 0, while for the confined sample, the strains e_r = e_θ= 0. Using Hooke’s law in cylindrical coordinates (A.8), show that the uniaxial response for the unconfined case is given by the expected relation σ_z = Ee_z, while the corresponding relation for the confined situation is σ_z = E^∗e_z, whereE^∗ = [(1 – v)E/(1 + v)(1 – 2v)]. Finally make a plot of E^∗/E versus n over the range 0≤v≤0.5 and discuss the results.

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0 = 0 = 0 % % (Unconfined) % % (Confined) e = = e = 0