Inversion symmetry in separable line, area and volume elastic energies

Abstract This work introduces a constructive framework for formulating isotropic compressible hyperelastic constitutive laws based on a geometrically transparent generalization of the Valanis–Landel (VL) hypothesis. The strain energy is decomposed into independent line, area, and volume contributions, each governed by a scalar function of the corresponding deformation measure. Physical admissibility is ensured by imposing inversion symmetry on these functions, which automatically enforces a stress-free reference state. The resulting formulation provides a systematic procedure for generating physically coherent models without ad hoc assumptions and naturally reduces to classical incompressible VL-type models as a special case. A central result of the theory is the emergence, both in linear and finite elasticity, of three fundamental Poisson’s ratios–0, 1/3, and 1/2–associated with pure line, area, and volume deformation mechanisms, respectively. In the general case, the macroscopic linear elastic moduli are explicitly recovered as weighted combinations of these mechanisms. The Hencky strain energy is identified as the simplest symmetric member of this class. The framework thus offers a unified, symmetry-based toolbox for the design and analysis of compressible materials, with direct applicability to the microstructurally informed engineering of elastomers, polymers, and metamaterials.