his article presents a comprehensive analysis of methods for modeling the mechanical behavior of elastomers and other incompressible polymeric materials using hyperelasticity. The fundamental principles underlying constitutive finite-strain models are reviewed, including the concept of elastic energy density, strain tensor invariants, and the incompressibility condition. The physical and mathematical basis of key hyperelastic models-such as Neo-Hooke, Mooney-Rivlin, Ogden, and Yeo-is detailed, with an emphasis on deriving expressions for the true stress under uniaxial tension. Particular attention is paid to explaining the relationship arising from isotropy and conservation of volume, which is critical for correctly modeling the behavior of elastomers. A practical approach to identifying material parameters based on experimental data is described. Recommendations for conducting mechanical tests (uniaxial, biaxial tension, and shear) are provided, and the least-squares method as a fundamental tool for parametric identification is discussed in detail. The paper also presents directed elastic energy theory (DSET), a modern approach to modeling anisotropic hyperelastic materials. Two model variants are considered: one with additive directional decomposition and one with volumetric separation. It is shown that separating the energy into shape and volume contributions ensures thermodynamic correctness and better applicability to compressible and reinforced materials. DSET naturally accounts for orthotropy, simplifies parameter identification, and effectively describes behavior under large deformations. The paper covers the entire process from theoretical foundations to practical application: from model selection to its implementation in CAE packages such as Abaqus and ANSYS. The importance of model verification on data not used in parameter selection is emphasized to ensure its generalizability. The presented systematic approach can be applied to both standard industrial elastomers and newly synthesized polymers for which reference data is lacking.
HYPERELASTICITY, ELASTOMER, ENERGY DENSITY, DEFORMATION INVARIANTS, LEAST SQUARES METHOD, MOONEY-RIVLIN, OGDEN, INCOMPRESSIBILITY, STRETCH FACTOR, ANISOTROPY
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