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.

中文翻译：

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用于超弹性几何精确梁本构建模的物理增强神经网络


我们提出了基于神经网络的超弹性几何精确梁本构模型。所提出的模型是物理增强的，即通过构建满足重要的机械条件，从而提高了准确性和泛化性。光束的应变和曲率用作表示有效超弹性光束电位的前馈神经网络的输入。力和力矩作为束势的梯度接收，确保热力学一致性。通过附加投影项考虑归一化条件。对称条件是通过基于不变的横向各向同性方法和更灵活的点对称约束来实现的，该约束包含在横向各向同性中，但对本构响应的限制较少。此外，提出了一种数据增强方法来改善模型在不同横截面半径下的缩放行为。此外，我们引入了一个带有标量参数的参数化，以表示内半径和外半径之间具有不同比率的环形截面。将光束势公式化为神经网络提供了一个高度灵活的模型。这使得具有非线性材料行为和横截面变形的几何精确梁能够实现高效的本构代理建模，否则需要计算成本高得多的方法。这些模型使用在不同内半径和外半径下具有圆形和环形超弹性可变形截面的梁生成的数据进行校准和测试，显示出出色的精度和泛化性。通过在梁仿真中的应用，进一步证明了所提出的点对称模型的适用性。 在所有研究的案例中，所提出的模型都表现出优异的性能。