  We present a framework for the multiscale modeling of finite strain
magneto-elasticity based on physics-augmented neural networks (NNs). By using a
set of problem specific invariants as input, an energy functional as the output
and by adding several non-trainable expressions to the overall total energy
density functional, the model fulfills multiple physical principles by
construction, e.g., thermodynamic consistency and material symmetry. Three
NN-based models with varying requirements in terms of an extended polyconvexity
condition of the magneto-elastic potential are presented. First, polyconvexity,
which is a global concept, is enforced via input convex neural networks
(ICNNs). Afterwards, we formulate a relaxed local version of the polyconvexity
and fulfill it in a weak sense by adding a tailored loss term. As an
alternative, a loss term to enforce the weaker requirement of strong
ellipticity locally is proposed, which can be favorable to obtain a better
trade-off between compatibility with data and physical constraints. Databases
for training of the models are generated via computational homogenization for
both compressible and quasi-incompressible magneto-active polymers (MAPs).
Thereby, to reduce the computational cost, 2D statistical volume elements and
an invariant-based sampling technique for the pre-selection of relevant states
are used. All models are calibrated by using the database, whereby
interpolation and extrapolation are considered separately. Furthermore, the
performance of the NN models is compared to a conventional model from the
literature. The numerical study suggests that the proposed physics-augmented NN
approach is advantageous over the conventional model for MAPs. Thereby, the two
more flexible NN models in combination with the weakly enforced local
polyconvexity lead to good results, whereas the model based only on ICNNs has
proven to be too restrictive.
