Crosslinking most likely occurs in an equilibrated polymer without any solvent. The free energy expression derived from the
Neo-Hookean model of rubber elasticity is in terms of free energy change due to
deformation per unit volume of the sample. The strand concentration, v, is the number of strands over the volume which does not depend on the overall size and shape of the elastomer. Beta relates the end-to-end distance of polymer strands across crosslinks over polymers that obey random walk statistics. \Delta f_d = \frac{\Delta F_d}{V} = \frac{K_BT\nu_{el}\beta\lambda_1p^2 + \lambda_2p + 2\lambda_3p^2 - 3}{2} v_{el} = \frac{n_{el}}{V} , \beta = 1 In the specific case of shear deformation, the elastomer besides abiding to the simplest model of rubber elasticity is also incompressible. For pure shear we relate the shear strain, to the extension ratios lambdas. Pure shear is a two-dimensional stress state making lambda equal to 1, reducing the energy strain function above to: \Delta f_{d}= \frac{k_{B}T\nu_{s}\beta\gamma^2}{2} To get
shear stress, then the energy strain function is differentiated with respect to shear strain to get the shear modulus, G, times the shear strain: \sigma_{12} = \frac{d(\Delta f_{d})}{d\gamma} = G\gamma Shear stress is then proportional to the shear strain even at large strains. Notice how a low shear modulus correlates to a low deformation strain energy density and vice versa. Shearing deformation in elastomers, require less energy to change shape than volume. \Delta f_d = W = \frac{G(\lambda_{1p}^2+\lambda_{2p}^2+\lambda_{3p}^2-3)}{2} ==See also==