The α-Mooney–Rivlin model of hyperelasticity

Abstract We introduce the α-Mooney–Rivlin model, a three-parameter generalization of the classical Mooney–Rivlin law based on a new pair of α-dependent invariants for incompressible hyperelasticity. The model preserves the transparent two-term structure of the classical formulation while replacing the quadratic measures of mean elongation and area stretch by nonlinear α-dependent counterparts. Within this framework, the exponent α controls the degree of constitutive nonlinearity, and hence the tendency of the response towards strain-softening or strain-stiffening, whereas the parameters C and D weight the respective contributions of mean elongation and area stretch. The resulting constitutive law is compact, mechanically interpretable and consistent with weakly nonlinear elasticity, for which three parameters are the minimal requirement. We analyse the model with particular emphasis on constitutive inequalities and polyconvexity, and identify the material parameters through simultaneous fits to Treloar's experimental data for simple extension, pure shear and biaxial loading. The results show that the proposed framework provides a robust and structurally consistent description of the experimental response, improving substantially on the classical Mooney–Rivlin model over small-to-moderate finite deformations.