On the directional approach in constitutive modelling: A general thermomechanical framework and exact solutions for Mooney–Rivlin type elasticity in each direction

In order to represent process-induced anisotropies in continuum mechanics or to transfer one-dimensional material models to three spatial dimensions the directional approach is a helpful technique. Since the essential equations are defined in the orientation space it is also denoted as microsphere approach. In the current article, the relation for the directional stress tensor of the second Piola–Kirchhoff type is motivated using the volumetric/isochoric split of the deformation gradient and the Clausius–Duhem inequality. Owing to inherent nonlinearities, numerical discretisation techniques are usually applied to calculate the total stress by averaging the directional stress tensors over the unit sphere. In order to investigate the accuracy of such simulations, the availability of exact solutions in closed form is essential. To this end, the tension/compression behaviour which belongs to a certain direction in the orientation space is modelled by an elasticity relation of the Mooney Rivlin type. The exact solutions are calculated, visualized and discussed for uniaxial tension and compression as well as for equibiaxial tension.