Polyconvex Physics-Augmented Neural Network Constitutive Models in Principal Stretches

2025-03-01

AI Summary

Addressing the challenge of simultaneously ensuring physical consistency and predictive accuracy in constitutive modeling of soft materials, this paper proposes a physics-guided deep learning framework based on principal stretches. Methodologically, it introduces the eigenvalues of the right stretch tensor ( mathbf{U} ) (i.e., principal stretches) as direct inputs to an input-convex neural network (ICNN), and constructs a hierarchical convex potential using ( mathbf{U} ), ( operatorname{cof} mathbf{U} ), and the Jacobian determinant ( J ), thereby rigorously enforcing polynomial convexity, frame-indifference, and thermodynamic consistency. Crucially, physical constraints are embedded intrinsically into the network architecture—eliminating the need for post-hoc regularization. Validation on both synthetic and experimental data demonstrates that the model achieves high predictive accuracy across diverse hyperelastic materials, significantly outperforming conventional invariant-based models while inherently guaranteeing physically admissible constitutive responses.

Technology Category

• Computer Vision: Low Level & Physics-based Vision
• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Systems and Infrastructure for Web, Mobile and WoT: Applied ML and AI for Web-based mobile applications
• Responsible Web: Consent frameworks and practices on the web

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 $ ext{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 $ ext{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 $ ext{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.

Problem

• Develops polyconvex neural network models for soft materials.
• Uses principal stretches instead of Cauchy-Green tensor invariants.
• Demonstrates model accuracy with synthetic and experimental data.

Innovation

• Polyconvex functions using principal stretches
• Physics-aware deep-learning framework
• ICNN for strain energy computation