The long-standing challenge of fulfilling all physical requirements for hyperelastic constitutive models at the same time, which has been heavily disputed in recent decades, could be considered "the main open problem of the theory of material behavior"[3]. This applies in particular to the modelling of hyperelastic materials based on neural networks (NN), especially for the compressible case.

Therefore, a hyperelastic constitutive model based on physics-augmented neural networks (PANNs) is presented, which fulfils all common physical requirements by construction, and is particularly applicable for compressible material behavior. Combining the theory of hyperelasticity established in the last decades with the latest advances in machine learning, the model expresses the hyperelastic potential as an input convex neural network (ICNN). This potential satisfies conditions such as compatibility with the balance of angular momentum, objectivity, material symmetry, polyconvexity, and thermodynamic consistency [1,2]. In order to ensure that the model produces physically sensible results, analytical growth terms and normalisation terms are used. These terms, developed for both isotropic and transversely isotropic materials, guarantee that the undeformed deformation state is stress-free and has zero energy in an exact manner [1]. The non-negativity of the hyperelastic potential is verified numerically by sampling the space of admissible deformation states.

Finally, the applicability of the model is demonstrated through various examples, such as calibrating the model using data generated with analytical potentials and applying it to finite element (FE) simulations. Its extrapolation capability is compared to models with reduced physical background, showing excellent and physically meaningful predictions with the proposed PANN.

[1] Linden, L., Klein, D. K., Kalina, K. A., Brummund, J., Weeger, O. and Kästner, M., Neural networks meet hyperelasticity: A guide to enforcing physics, (submitted 2023).

[2] Kalina, K. A., Linden, L., Brummund, J. and Kästner, M., FEANN - An efficient data-driven multiscale approach based on physics-constrained neural networks and automated data mining, Comput. Mech. (2023).

[3] Truesdell, C. and Noll, W., The Non-Linear Field Theories of Mechanics. 3rd ed. Springer Berlin Heidelberg, 2004.