When invariants matter: The role of I1 and I2 in neural network models of incompressible hyperelasticity

Machine learning-based approaches enable flexible and precise material models. For the formulation of these models, the usage of invariants of deformation tensors is attractive, since this can a priori guarantee objectivity and material symmetry. In this work, we consider incompressible, isotropic hyperelasticity, where two invariants I ¯ 1 and I ¯ 2 are required for depicting a deformation state. First, we aim at enhancing the understanding of deformation invariants. We provide an explicit representation of the set of invariants that are actually admissible, i.e. for which ( I ¯ 1 , I ¯ 2 ) ∈ R 2 a physical deformation state does indeed exist. Furthermore, we prove that uniaxial and equi-biaxial deformation states correspond to the boundary of the set of admissible invariants. Second, we study how the experimentally-observed constitutive behaviour of different materials can be captured by means of neural network models of incompressible hyperelasticity, depending on whether both I ¯ 1 and I ¯ 2 or solely one of the invariants, i.e. either only I ¯ 1 or only I ¯ 2 , are taken into account. To this end, we investigate three different experimental data sets from the literature. In particular, we demonstrate that considering only one invariant – either I ¯ 1 or I ¯ 2 – can allow for good agreement with experiments in case of small deformations. In contrast, it is necessary to consider both invariants for precise models at large strains, for instance when rubbery polymers are deformed. Moreover, we show that multiaxial experiments are strictly required for the parameterisation of models considering I ¯ 2 . Otherwise, if only data from uniaxial deformation is available, significantly overly stiff responses could be predicted for general deformation states. On the contrary, I ¯ 1 -only models can make qualitatively correct predictions for multiaxial loadings even if parameterised only from uniaxial data, whereas I ¯ 2 -only models are completely incapable in even qualitatively capturing experimental stress data at large deformations. • Importance of the deformation invariants I1 and I2 as input in neural network models • Explicit representation of the set of admissible isochoric invariants • Performance comparison between I1-only, I2-only and I1 and I2-models • Parameterisation with limited data: The role of I2