Polyconvex Physics-Augmented Neural Network Constitutive Models in Principal Stretches

TL;DR

This work introduces a physics-augmented neural network framework, \u03bb-PANN, for hyperelasticity formulated in terms of principal stretches. By combining convex, permutation-invariant neural components that operate on the eigenvalues of the right stretch tensor $\mathbf{U}$ and its cofactor with an ICNN, the model enforces polyconvexity and objectivity while remaining expressive enough to capture a wide range of materials, including Ogden-type responses. The approach demonstrates strong performance on synthetic Ogden and generalized Ogden/invariant data, matches or exceeds invariant-based models in extrapolation, and accurately fits experimental rubber data, with FE validation showing near-ground-truth predictions in Cook’s membrane. Overall, the paper provides a scalable, physically consistent alternative to invariant-based formulations and lays groundwork for eigenvalue-based constitutive modeling and future anisotropy and dissipation extensions.

Abstract

Accurate constitutive models of soft materials are crucial for understanding their mechanical behavior and ensuring reliable predictions in the design process. To this end, scientific machine learning research has produced flexible and general material model architectures that can capture the behavior of a wide range of materials, reducing the need for expert-constructed closed-form models. The focus has gradually shifted towards embedding physical constraints in the network architecture to regularize these over-parameterized models. Two popular approaches are input convex neural networks (ICNN) and neural ordinary differential equations (NODE). A related alternative has been the generalization of closed-form models, such as sparse regression from a large library. Remarkably, all prior work using ICNN or NODE uses the invariants of the Cauchy-Green tensor and none uses the principal stretches. In this work, we construct general polyconvex functions of the principal stretches in a physics-aware deep-learning framework and offer insights and comparisons to invariant-based formulations. The framework is based on recent developments to characterize polyconvex functions in terms of convex functions of the right stretch tensor $\mathbf{U}$, its cofactor $\text{cof}\mathbf{U}$, and its determinant $J$. Any convex function of a symmetric second-order tensor can be described with a convex and symmetric function of its eigenvalues. Thus, we first describe convex functions of $\mathbf{U}$ and $\text{cof}\mathbf{U}$ in terms of their respective eigenvalues using deep Holder sets composed with ICNN functions. A third ICNN takes as input $J$ and the two convex functions of $\mathbf{U}$ and $\text{cof}\mathbf{U}$, and returns the strain energy as output. The ability of the model to capture arbitrary materials is demonstrated using synthetic and experimental data.

Figures (11)

• Figure 1: Classification of standard constitutive models for soft materials
• Figure 2: Schematic representation of the presented polyconvex physics-augmented neural network model with principal stretch inputs, $\lambda$-PANN.
• Figure 3: Ogden model median losses over $10$ random models as specified in Table \ref{['tb:initialGuessSens']}. a) Training loss; b) Extrapolation loss.
• Figure 4: Visualizing response of $\lambda$-PANNs on Dataset 5 of $\lambda$-Ogden set. Dotted green lines indicate the training domain of $20\%$. Dashed lines denote model predictions and solid lines denote ground truth response.
• Figure 5: Generalized Ogden model median losses over $10$ random models as specified in Table \ref{['tb:paramGeneralOgden']}. a) Training loss; b) Extrapolation loss.