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, which improves accuracy and generalization. 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 received as the gradients of the beam potential, ensuring thermodynamic consistency. Normalization conditions are considered via additional projection terms. Symmetry conditions are implemented by an invariant-based approach for transverse isotropy and a more flexible point symmetry constraint, which is included in transverse isotropy but poses fewer restrictions on the constitutive response. Furthermore, a data augmentation approach is proposed to improve the scaling behavior of the models for varying cross-section radii. Additionally, we introduce a parameterization with a scalar parameter to represent ring-shaped cross-sections with different ratios between the inner and outer radii. 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 and tested with data generated for beams with circular and ring-shaped hyperelastic deformable cross-sections at varying inner and outer radii, showing excellent accuracy and generalization. The applicability of the proposed point symmetric model is further demonstrated by applying it in beam simulations. In all studied cases, the proposed model shows excellent performance.

翻译：本文提出基于神经网络的超弹性几何精确梁本构模型。所提模型具有物理增强特性，即通过构造方式满足重要力学条件，从而提升精度与泛化能力。梁的应变与曲率作为前馈神经网络的输入，该网络表征有效的超弹性梁势能。力与力矩通过梁势能的梯度获得，确保热力学一致性。归一化条件通过附加投影项实现。对称条件采用基于不变量的横向各向同性方法及更灵活的点对称约束——该约束包含于横向各向同性中，但对本构响应的限制更少。此外，提出数据增强方法以改进模型对不同截面半径的尺度适应性。同时引入含标量参数的参数化方法，用于表征内径与外径比例各异的环形截面。将梁势能表述为神经网络可构建高度灵活的模型，从而为具有非线性材料行为和截面变形的几何精确梁实现高效的本构代理建模，而传统方法需要昂贵得多的计算成本。通过在不同内外径条件下对圆形与环形超弹性可变形截面梁生成数据进行模型校准与测试，结果表明模型具有优异的精度与泛化能力。进一步将所提点对称模型应用于梁仿真，验证了其适用性。在所有研究案例中，所提模型均表现出卓越的性能。