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

Abstract

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.

Figures (15)

• Figure 1: The geometrically exact beam with deformable cross-section as a multiscale problem. A warping problem on the cross-section scale is solved at each quadrature point on the beam scale. Modified schematic adopted from herrnbockTwoscaleOffandOnline2023.
• Figure 2: The geometrically exact beam in its initial (left) and current configuration (right).
• Figure 3: Schematic visualization of the cross-sectional warping problem. The inputs are the strain measures $\boldsymbol{\epsilon}$ and $\boldsymbol{\kappa}$ and a discretization of the cross-section. The outputs are the beam potential $\psi$ and the stress resultants $\boldsymbol{n}$ and $\boldsymbol{m}$.
• Figure 4: Physics-augmented neural network-based beam constitutive model. The inputs are the strains and curvatures $\boldsymbol{p} = (\boldsymbol{\epsilon}, \boldsymbol{\kappa})$, which are converted to the FFNN inputs $\boldsymbol{\mathcal{I}}$ -- in the simplest case $\boldsymbol{\mathcal{I}} = \boldsymbol{p}$. The forces and moments $\boldsymbol{q} = (\boldsymbol{n}, \boldsymbol{m})$ are derived from the beam strain energy $\psi^\text{PANN}$, the model output.
• Figure 5: Ring-parameterized PANN-based constitutive model ${\mathcal{W}}_{P}$. Note that the FFNN parts of the models are complemented with normalization terms, cf. \ref{['fig:PANN-beam-model']}.

Theorems & Definitions (2)

• Remark 2.1
• Remark 3.1