The Strain-Energy Function for Elastomers

It has been shown that the strain-energy function for an incompressible isotropic elastic material must be expressible as a function of two strain invariants I1, I2 which are elementary symmetric functions of the squares of the principal extension ratios λ1, λ2, λ3. Valanis and Landel have proposed a strain-energy function of the form f(λ1)+f(λ2)+f(λ3). It is shown how such a strain-energy function can be expressed explicitly as a function of I1 and I2. Also, necessary and sufficient conditions are derived for any function of I1 and I2 to be expressible in the form proposed by Valinis and Landel. It is shown how the results of problems which have been solved using the general form of the strain-energy function can be conveniently specialized to the case when the Valanis-Landel strain-energy function is valid.