Here the possibility of developing a better three dimensional model for rubber when Hencky strain is written as an explicit function of the Cauchy stress is examined. Based on the experimental evidence, vulcanized unfilled rubber is assumed to be isotropic, incompressible, and deforming from a stress-free reference configuration. Vulcanized unfilled rubber undergoes strain-induced crystallization when uniaxial stretch exceeds 2. Therefore, in this study, a non-dissipative model is sought for rubber only till the von Mises stress is below a critical value. Using the biaxial response reported in Kawabata et al. (1981), the material functions are arrived adopting Rivlin and Saunders methodology. The material constants in the determined material functions are found using biaxial tests for four other experimental studies on rubber reported in the literature. The ability of the arrived constitutive model to describe the uniaxial and equal biaxial response of the same rubber is evaluated. The proposed model is benchmarked with established models in the classical framework. Among the models studied the proposed model performed as well as the other models for Kawabata et al. data set and marginally better than the other models for the remaining four experimental data sets. However, while the proposed model performance does not critically depend on the experiments used to deduce the material parameters, for the other models the material parameters need to be found from fixed biaxial experiments for better performance. © 2020 Elsevier Ltd