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Hyperelastic material models are frequently used in the engineering sciences. Application examples include biomechanics and the modeling of the finite elastoplasticity of metallic alloys. Given the rising importance of machine learning approaches, hyperelastic models have also been approximated successfully by neural networks, see [1]. In the present work, the focus is on such approximations for isotropic energies. They are often defined in terms of invariants of the right Cauchy-Green tensor. In this way, the energies are a priori isotropic and also fulfill the principle of material frame indifference. Moreover, it can be shown in a straightforward way that convex neural networks can automatically enforce the polyconvexity of such energies - a property that is important for proving the existence of solutions. However, it is also known that these neural networks rely only on a sufficient criterion for polyconvexity, but not a necessary one. Accordingly, they cannot capture all hyperelastic models. In this paper, the limitations of these networks are carefully elaborated, and improved models are proposed to address the shortcomings.
[1] L. Linden et al., Neural networks meet hyperelasticity: A guide to enforcing physics, J. Mech. Phys. Solids, 179 (2023), 105363.
