A cube of linear elastic material is subjected to a vertical compressive stress sigma1 in the 1-direction, but it is now constrained ( e = 0) in both the 2- and 3- directions.
a) Find expressions for the induced transverse stresses, sigma2 and sigma3 in terms of sigma1. Hence derive an expression for the 'effective modulus' (sigma1/sigma2) in this case.
b) Sketch the variation of effective modulus with poison’s ratio and comment on the limiting values when v = 0 (foam) and v=0.5 (rubber).
c) Explain why the rubber soles of running shoes are designed with some combination of air or gel pockets, partially foamed rubber and a tread.