On the Second Order Approximation in the Phenomenological

Abstract

From the phenomenological theory of finite deformation, a stress-strain relation was obtained with a second order approximation to the stored energy function w with higher degree accuracy than that shown by L.R.G. Treloar or R.S. Rivlin.In the phenomenological theory of large elastic deformation, the stored energy function w can be written in terms of three strain invariantes I1, I2 and I3, and one way of expressing this is in terms of power series:(1)where I1=α12+α22+α32, I2=α12α22+α22α32+α12α32, I3=α12α22α32 and αi, (i=1, 2, 3) are the principal extension ratios. If the principal extension ratios α1, α2 and α3 are small and assumed to be nearly in unity, the second order approximation to the expression (1) for w will be given as follows, in the case of incompressible materials(2)because the squared term (I>1-3)2 is equivalent to the term (I2-3) through the principal extension ratios, from the view-point of the second order approximation. From this type of the stored energy function, the following stress-strain relation has been obtained(3)for the case of simple extension or uniaxial compression defind by the extension ratio or the compression ratio α, where A≡2(c100-6c200), B≡2(4c200+c010), C≡4c200. Upon comparing the resulting stress-strain relation with the experimental stress-strain behavior, it seems that this relation is valid for rubber vulcanizates, and even for larger deformations to which the assumption that deformations are small (α«2) is inapplicable, and that this relation almost coincides with the observed stress-strain relation in the extension range wider than that predicted by Mooney-Rivlin equation.