Closed-Form Analytical Solutions for Stability Bounds of the Incompressible Mooney–Rivlin Hyperelastic Model Under Standard Homogeneous Loadings

Understanding and predicting the stability of hyperelastic materials under diverse loading conditions is crucial for designing reliable engineering applications and accurately modeling rubberlike materials. The stability bounds of the incompressible Mooney–Rivlin (MR) hyperelastic model, which is widely used for modeling the mechanical behavior of elastic materials, are determined in this study through closed-form analytical solutions. Stability analysis is considered critical for ensuring the robustness of the model under various loading conditions. Explicit analytical expressions for the stability limits are derived for uniaxial, equibiaxial, and planar loading modes by solving quartic polynomials. A dimensionless parameter for the ratio of the material parameters is introduced, and stability maps are provided along with calculations of the stress values at the stability boundaries. The incorporation of stability constraints into parameter fitting is facilitated by the results, ensuring stable model behavior within the desired deformation range. The reliability of the MR model in practical applications is enhanced, and the development of more accurate and stable hyperelastic material models is supported by this work. The derived stability bounds reveal that for the MR model to remain stable under [Formula: see text] engineering strain in the three loading modes, the parameter [Formula: see text] must satisfy either [Formula: see text] or [Formula: see text]. These criteria highlight the relationship between stability and material parameters, providing practical guidelines for parameter selection.