On the central role of the invariant I2 in nonlinear elasticity

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