Energetic Exhaustiveness: tracing energy derivatives from nonlinear homogeneous deformations

The mechanical characterization of soft materials is traditionally performed by fitting stress components or esultants obtained from standard states of deformation, such as uniaxial tension/compression, biaxial tests, torsion, or shear. Once a constitutive framework is assumed, the identification process focuses on determining the corresponding material parameters.
An alternative and more direct approach consists in experimentally tracking the two- or three-dimensional behavior of the derivatives of the energy function, for incompressible and compressible materials, respectively. Depending on the choice of deformation invariants [1] and on the adopted experimental protocol, homothetic tests can be classified into three families [2]: energetically exhaustive, partially exhaustive, and unexhaustive tests. A test is defined as energetically exhaustive when its matrix formulation yields a determined system of equations for the unknown derivatives of the energy function.

Within this framework, the unequal-biaxial test is shown to possess a remarkable characterization capability, as it alone allows for the direct identification of the functional form of the energy derivatives. The theoretical basis of this property is discussed, and dedicated experimental design tools are proposed. Experimental results obtained from unequal-biaxial tests on three different rubber-like materials are presented and analyzed.



These experiments reveal two unexpected findings. First, two commonly adopted empirical inequalities fail to describe the behavior of two of the tested materials, despite the fact that the classical Baker–Ericksen inequalities are satisfied. Second, alternative formulations of the Baker–Ericksen inequalities [3] are proposed, together with a new hierarchical empirical inequality, providing fresh insights into the constitutive behavior of hyperelastic isotropic materials.

References
[1] Criscione, J. C., Humphrey, J. D., Douglas, A. S., Hunter, W. C, “An invariant basis for natural strain
which yields orthogonal stress response terms in isotropic hyperelasticity”, J. Mech. Phys. Solids,
48(12), 2445-2465 (2000).
[2] Falope, F. O., Lanzoni, L., Tarantino, A. M., “Energetic exhaustiveness for the direct character-
ization of energy forms of hyperelastic isotropic materials”, J. Mech. Phys. Solids, 193, 105885
(2024).
[3] Baker, M., Ericksen, J., “Inequalities restricting the form of the stress-deformation relations for
isotropic elastic solids and Reiner-Rivlin fluids”,J. Wash. Acad. Sci., 44(2), 33–35 (1954).