Energetic exhaustiveness for the direct characterization of energy forms of hyperelastic isotropic materials

It is common practice to characterize the constitutive law of a material indirectly. This takes place by fitting a specific stress component, which is given as a combination of response functions or derivatives of the energy function of the material. Yet, it is possible to characterize each energy derivative of the material directly. Not only that but, through a few well-designed tests, getting a set of well-distributed data that defines the evolution of the energy derivatives in the invariant space is attainable, but not for all tests. Here, each test is portrayed as an equilibrium path on the surfaces (or volumes) of the derivative of the energy function. In the framework of the homothetic tests of hyperelastic isotropic materials, we propose the definition of energetic exhaustiveness. This definition relates to the capability of a test, via its analytic formulation according to a proper set of deformation invariants, to directly provide a closed-form solution for the derivatives of the energy function. In reaching this definition and retracing the Baker-Ericksen and the empirical inequalities, an alternative form of Baker-Ericksen inequalities is presented. We demonstrate that the unequal-biaxial test alone is energetically exhaustive and that it can provide (the same and more) information on the energy compared to the uniaxial, equi-biaxial, and pure shear tests. Unequal-biaxial experiments on three rubbers are presented. The outcomes of experiments contradict the empirical inequalities and seem to suggest new hierarchical empirical inequalities. Compact and nearly exact solutions are provided to perform and design tests at a constant magnitude of distortion, thus reaching a direct and comprehensive representation of the energy.