An improved Rivlin model of hyperelastic rubber materials

The phenomenological Rivlin constitutive model of incompressible rubber materials is investigated in this paper. The advantages and disadvantages of several models, including the neo-Hookean formulation, the Mooney formulation, the three-order Rivlin formulation and the Ogden formulation, are analyzed. Subsequently, an improved Rivlin formulation is proposed as W=B₀₀(A₃-3)+∑(i+j=1)(N) B_ij(A₁-3)ⁱ (A₂-3)^j, which has two significant advantages: (1) in the case of N=1 and small strain, the new function is the same as the strain-energy of the linear elastic incompressible material, and depicts approximately linear shear deformation and exactly linear tension and compression deformation. (2) when N≥2, in the condition of small strain, using the new formulation, the strain-energy and shear stress in simple shear experiment are polynomial with γ_xy as geometric proportion. While using Rivlin formulation, the strain-energy and shear stress τ_xy under small deformation are polynomial with (γ_xy)² as geometric proportion. Besides, when the deformation is small, Rivlin formulation with N=1 (Mooney) provides constant shear module and non-linear relationship in tension and compression. The improved form, compared with the same order original Rivlin formulation, shows significant advantages in prediction of test data in single test, and has a good performance in the mutual prediction between different tests.