A finite strain non-parametric hyperelastic extension of the classical phenomenological theory for orthotropic compressible composites

The phenomenological linear theory of orthotropic compressible materials is employed systematically in the engineering and scientific analysis of a large scope of bulk and composite materials. However, at finite elastic strains, multiscale and fiber-based parametric models are typically employed, which material parameters are fitted to macroscopic experimental data using optimization procedures. Phenomenological extensions of the linear theory capable of effectively modelling such a large scope of materials are not available. What we present in this work is a simple extension of the linear theory to finite strains such that at every deformation level, the infinitesimal theory is fully recovered. The model is based in non-parametric spline complementary energies employing an energy decomposition compatible with the classical infinitesimal expression at all strain levels. No material parameter is explicitly involved because the spline-based stress energies are numerically computed (not fitted) directly from experimental data. We show the applicability of the model to capture (1) the behavior of orthotropic bimodulus materials consistent with hyperelasticity (a novel formulation also presented herein), (2) orthotropic auxetic foams, and (3) composites at finite strains.