A new concept for the representative directions method: Directionalisation of first and second invariant based hyperelastic models

The representative directions method is a continuum mechanical based practical approach to transfer 1D material models to 3D. The selection of the directional energy and the corresponding directional stress is generally based on the standard uniaxial tension (UT) solution of hyperelastic models. However, this approach results in a somewhat different model than hyperelastic models in the context of elasticity and inelasticity. For instance, enrichment of the UT based directional stress with the non-affine stretch does not provide close results unless huge p-root values are considered. Hence, the main objective of this contribution is to determine the directional stresses, which can provide equivalent or close results to first and second invariant based hyperelastic models. Accordingly, the directionalisation concept in the framework of affine representative directions method is introduced. Directional stresses are obtained with a top-down approach from the first and second invariant based hyperelastic models. The standard Mooney-Rivlin model is directionalised to obtain the corresponding 1D energies and stresses using the micro-stretch and the macro-area-stretch. The approach is then utilised to directionalise several hyperelastic-like models as alternatives for statistical-thermomechanics based chain models. Moreover, a new optimisation strategy is proposed to improve the material asymmetry resulting from numerical integration schemes. Optimisation results demonstrate that the material symmetry of the standard Bažant points can be improved. Finally, two nonhomogenous finite-element (FE) simulations demonstrate that the directionalisation approach presented here contributes a good step towards numerically robust inelastic extensions.