A review on material models for isotropic hyperelasticity

Dozens of hyperelastic models have been formulated and have been extremely handy in understanding the complex mechanical behavior of materials that exhibit hyperelastic behavior (characterized by large nonlinear elastic deformations that are completely recoverable) such as elastomers, polymers, and even biological tissues. These models are indispensable in the design of complex engineering components such as engine mounts and structural bearings in the automotive and aerospace industries and vibration isolators and shock absorbers in mechanical systems. Particularly, the problem of vibration control in mechanical system dynamics is extremely important and, therefore, knowledge of accurate hyperelastic models facilitates optimum designs and the development of three-dimensional finite element system dynamics for studying the large and nonlinear deformation behavior. This review work intends to enhance the knowledge of 15 of the most commonly used hyperelastic models and consequently help design engineers and scientists make informed decisions on the right ones to use. For each of the models, expressions for the strain-energy function and the Cauchy stress for both arbitrary loading assuming compressibility and each of the three loading modes (uniaxial tension, equibiaxial tension, and pure shear) assuming incompressibility are provided. Furthermore, the stress–strain or stress–stretch plots of the model's predictions in each of the loading modes are compared with that of the classical experimental data of Treloar and the coefficient of determination is utilized as a measure of the model's predictive ability. Lastly, a ranking scheme is proposed based on the model's ability to predict each of the loading modes with minimum deviations and the overall coefficient of determination.