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.

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，即通过构造方式确保满足重要力学条件。以前馈神经网络表征有效超弹性梁势能时，以梁的应变和曲率作为输入。通过梁势能的梯度获得力和力矩，从而保证热力学一致性。此外，通过附加投影项考虑归一化条件。为体现点对称截面梁的对称性，引入翻转对称约束。同时提出参数化模型，可表征不同截面几何形状下梁的本构行为。这种物理驱动的参数化方法考虑了梁半径对梁势能的影响。将梁势能表述为神经网络提供了高度灵活的模型架构，能够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，而传统方法需要耗费高得多的计算成本。通过对圆形可变形截面及不同半径梁生成的数据进行校准，模型展现出优异的精度和泛化能力。通过在梁模拟中的应用进一步验证了所提模型的适用性。在所有研究案例中，该模型均表现出卓越的性能。