Classic hyperelastic models, such as the Neo-Hookean model and Arruda-Boyce eight-chain model, have been widely adopted to describe the mechanical response of rubbers. However, the experimental data has shown that using a single set of parameters these models have difficulty in accurately predicting the measured stress-strain relationship of rubbers under various loading modes. For example, the Arruda-Boyce model fails to describe the stress response in biaxial loading conditions of Treloar's classic experiments. To address this limitation, this work develops a hyperelastic theory incorporation the entanglement effect. At the microscale, the Langevin statistical model is adopted for the entropic part and the tube model is used for the entanglement part. The affine assumption is used to construct the micro-macro mapping. Macroscopically, the Helmholtz free energy of the model consists of both an entropic part and an entanglement part. The entropic part has the same form as the eight-chain model, depending on the first invariant of the Cauchy-Green deformation tensor. In contrast, the entanglement part is a function of the second invariant of the Cauchy-Green deformation tensor. Compared with the eight-chain model and Neo-Hookean model, the developed model with three parameters provides a greatly improved prediction on the experimentally measured stress response of rubbers in uniaxial, pure shear and equibiaxial loading conditions, as well as that of biaxial tension tests with different pre-stretch ratios. The model shows superior prediction ability compared with the classic models, such as the Neo-Hookean model, the eight-chain model, the Yeoh model and the generalized Rivlin model. Finally, the work also compares the free energy density of entanglement part developed in this work and those of the related models in the literature. The constitutive theory developed in this work can accurately predict the large deformation behaviors of rubbers and other related soft materials, which can potentially benefit their engineering applications.
2021, 53(4): 1028-1037.