sigma2 = confining stress (applied at the sample-sleeve contact),
sigma1 = axial stress (or load),
nu = Poisson's ratio.
To demonstrate the above formula you just take the equations that give epsilon1 (axial deformation) and epsilon2 as functions of sigma1, sigma2 and sigma3 (here equal to sigma2) and impose that epsilon2 = 0 (non-deformable sleeve).