  In the present work, a machine learning based constitutive model for
electro-mechanically coupled material behavior at finite deformations is
proposed. Using different sets of invariants as inputs, an internal energy
density is formulated as a convex neural network. In this way, the model
fulfills the polyconvexity condition which ensures material stability, as well
as thermodynamic consistency, objectivity, material symmetry, and growth
conditions. Depending on the considered invariants, this physics-augmented
machine learning model can either be applied for compressible or nearly
incompressible material behavior, as well as for arbitrary material symmetry
classes. The applicability and versatility of the approach is demonstrated by
calibrating it on transversely isotropic data generated with an analytical
potential, as well as for the effective constitutive modeling of an
analytically homogenized, transversely isotropic rank-one laminate composite
and a numerically homogenized cubic metamaterial. These examinations show the
excellent generalization properties that physics-augmented neural networks
offer also for multi-physical material modeling such as nonlinear
electro-elasticity.
