Discussion: “Axial Loading of Bonded Rubber Blocks” (Horton, J. M., Tupholme, G. E., and Gover, M. J. C., 2002, ASME J. Appl. Mech., 69, pp. 836–843)

Abstract

The paper by Horton, Tupholme, and Gover (1) analyzes the deformation behavior of rubber using the isotropic, infinitesimal strain theory of elasticty for an incompressible material. Accordingly, the Poisson’s ratio ν=1∕2 and Young’s modulus is three times the shear modulus (E=3μ). In this setting, the stresses σij determine the (infinitesimal) strains ϵij, but the strains only determine the stresses up to a hydrostatic pressure. Hence, the strains determine the shear stresses and also determine the normal stress differences. The strains follow from the displacements ui in the usual fashion ϵij=(ui,j+uj,i)∕2. In their analysis of a rectangular block with Cartesian coordinates (x,y,z) and associated displacement components (u,v,w) the authors in their analytic development arrive at a displacement field (Eqs. (14), (12), (24))u=−xdwdz,v=0,w=3F4EA{z−2sinhαz2cosh[α2(h−z)]αcoshαh2},where α=43∕b and F, E, A, b, h are constants with units of force, stress, area, length, length, respectively. On the basis of the stress equation of equilibrium associated with the x direction, and in conjunction with the stress-strain-displacement relations reviewed above, they also obtain [Eq. (17)]σzz=E3[4dwdz−12(b24−x2)d3wdz3]−FA.These fields are central to the ensuing development, in particular to the calculation of percentage errors associated with previous treatments (e.g., Table 1).