Rivlin’s Representation Formula is Ill-Conceived for the Determination of Response Functions via Biaxial Testing

Abstract

The experimental determination of a strain energy function W for a rubber specimen must address departures from an elastic ideal in a rational fashion. Herein, such a rational experimental method is developed for biaxial stretching experiments and applied to rubber data in the literature. It is shown that Rivlin’s representation formula is experimentally ill-conceived because experimental error is magnified to the extent that error obscures trends in the response function plots. Upon developing direct tensor expressions for the response function calculations, we show that Rivlin’s representation formula (or any such constitutive law that has high covariance amongst the response terms) magnifies experimental error greatly. By “high covariance”, we mean the inner product amongst the response terms in the constitutive law is nearly equal to the maximum possible value — i.e., the product of their magnitudes. Moreover, we show that the second partials of W with respect to I₁ and I₂ should approach infinity as the strain decreases. Using an alternate set of invariants with minimal covariance (i.e., a null inner product amongst the response terms), a W for rubber can be determined forthwith.