Generalized Incremental Moduli of Hyperelastic Materials

The field of hyperelasticity, though well-established for isotropic materials, remains an area of ongoing research, particularly regarding the evolution of mechanical properties as a function of strain state. This article presents a generalized approach to calculating incremental moduli, which describe material stiffness under any given stress–strain condition, extending traditional incremental moduli definitions used for uniaxial, volumetric, and shear stress situations. In detail, given a hyperelastic potential and the stress corresponding to a generic strain state, the approach allows the identification of generalized incremental Young's moduli, Poisson's ratios, shear moduli, and bulk modulus corresponding to such a generic strain state. The approach reveals the progressive stiffening, softening, and anisotropization of the material under deformation, which contrasts with the behavior of materials with fixed stiffness properties, such as linear elastic ones. The method is applied to a compressible neo-Hookean material to evaluate its mechanical response under uniaxial, volumetric, and deviatoric stress–strain states. The results show that under uniaxial loading or deviatoric loading, the generalized incremental moduli exhibit directional sensitivity, demonstrating the material's increasing anisotropic behavior as strain progresses. In contrast, isotropic trends are maintained under volumetric deformations. This study provides valuable insights into the evolution of material properties under whatever loading condition and offers a more general framework for understanding and characterizing the behavior of hyperelastic materials.