On the central role of the invariant I2 in nonlinear elasticity

The necessity of the inclusion of the second invariant of the left Cauchy-Green deformation tensor B, namely I2, in the strain energy function W of rubber-like materials is analysed. Universal relationships that underline such necessity are revisited, and experimental data are examined to establish the trends in the variation of ∂W/∂I2. Corroborated by the established experimental trends, we consider (meso)structural arguments to devise a plausible approach for incorporation of I2 into the W function. On the basis of the molecular theory of rubbers and considering the entanglements as a topological tube constraint, our analysis confers that a first approximation of W(I1,I2) is of the form W(I1,I2)=f(I1)+g(I2). The f(I1) contribution may be that of any classical generalised neo-Hookean model, and the functional form of g(I2) is directly deduced from the tube model of entangled molecules. An additional logarithmic functional form of I2 is also devised based on the rational approximation of the response function β−1. The ensuing additive-type W(I1,I2) models are then compared with experimental datasets. While this additive consideration may not be sufficient to account for all aspects of the mechanics of rubber-like materials, the fitting results demonstrate an eminent improvement in the predictions of the additive-type models compared with generalised neo-Hookean models having the same number of constitutive parameters. These analyses underline the central role of I2 in modelling the finite deformation of rubber-like materials.