  We present neural network-based constitutive models for hyperelastic
geometrically exact beams. The proposed models are physics-augmented, i.e.,
formulated to fulfill important mechanical conditions by construction. Strains
and curvatures of the beam are used as input for feed-forward neural networks
that represent the effective hyperelastic beam potential. Forces and moments
are then received as the gradients of the beam potential, ensuring
thermodynamic consistency. Furthermore, normalization conditions are considered
via additional projection terms. To include the symmetry of beams with
point-symmetric cross-sections, a flip symmetry constraint is introduced.
Additionally, parameterized models are proposed that can represent the beam's
constitutive behavior for varying cross-sectional geometries. The physically
motivated parameterization takes into account the influence of the beam radius
on the beam potential. Formulating the beam potential as a neural network
provides a highly flexible model. This enables efficient constitutive surrogate
modeling for geometrically exact beams with nonlinear material behavior and
cross-sectional deformation, which otherwise would require computationally much
more expensive methods. The models are calibrated to data generated for beams
with circular, deformable cross-sections and varying radii, showing excellent
accuracy and generalization. The applicability of the proposed model is further
demonstrated by applying it in beam simulations. In all studied cases, the
proposed model shows excellent performance.
