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.