Physics-augmented neural networks for constitutive modeling of hyperelastic geometrically exact beams

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.