  In the present work, a hyperelastic constitutive model based on neural
networks is proposed which fulfills all common constitutive conditions by
construction, and in particular, is applicable to compressible material
behavior. Using different sets of invariants as inputs, a hyperelastic
potential is formulated as a convex neural network, thus fulfilling symmetry of
the stress tensor, objectivity, material symmetry, polyconvexity, and
thermodynamic consistency. In addition, a physically sensible stress behavior
of the model is ensured by using analytical growth terms, as well as
normalization terms which ensure the undeformed state to be stress free and
with zero energy. In particular, polyconvex, invariant-based stress
normalization terms are formulated for both isotropic and transversely
isotropic material behavior. By fulfilling all of these conditions in an exact
way, the proposed physics-augmented model combines a sound mechanical basis
with the extraordinary flexibility that neural networks offer. Thus, it
harmonizes the theory of hyperelasticity developed in the last decades with the
up-to-date techniques of machine learning. Furthermore, the non-negativity of
the hyperelastic neural network-based potentials is numerically examined by
sampling the space of admissible deformations states, which, to the best of the
authors' knowledge, is the only possibility for the considered nonlinear
compressible models. For the isotropic neural network model, the sampling space
required for that is reduced by analytical considerations. In addition, a proof
for the non-negativity of the compressible Neo-Hooke potential is presented.
The applicability of the model is demonstrated by calibrating it on data
generated with analytical potentials, which is followed by an application of
the model to finite element simulations. In addition, an adaption of the model
to noisy data is shown and its [...]
