Generalized invariants meet constitutive neural networks: A novel framework for hyperelastic materials

2025-08-16

AI Summary

Addressing the challenge of a priori selecting appropriate strain invariants and determining the functional form of strain energy in hyperelastic constitutive modeling, this paper proposes a data-driven framework that unifies generalized invariant discovery with constitutive relation learning within a single neural network architecture. The framework jointly learns both the optimal combination of strain invariants and the corresponding nonlinear strain energy mapping directly from experimental data. Designed for isotropic, incompressible materials, it balances physical interpretability with high predictive accuracy. Validation on rubber and brain tissue datasets demonstrates that the method automatically recovers classical stretch-dominated models (e.g., Neo-Hookean, Mooney–Rivlin) and accurately captures the nonlinear shear response of brain tissue under small deformations. Quantitatively, it achieves significantly higher prediction accuracy than conventional phenomenological models and state-of-the-art neural network-based approaches.

Technology Category

• Machine Learning: Learning with Manifolds
• Reasoning under Uncertainty: Relational Probabilistic Models
• Constraint Satisfaction and Optimization: Constraint Learning and Acquisition

Application Category

• Graph Algorithms and Modeling for the Web: Graph neural networks and deep learning approaches for Web-related graphs
• Web Mining and Content Analysis: Large pretrained models with web data
• Search and Retrieval-Augmented AI: Web learning to rank, online learning, and counterfactual learning for ranking

Abstract

The major challenge in determining a hyperelastic model for a given material is the choice of invariants and the selection how the strain energy function depends functionally on these invariants. Here we introduce a new data-driven framework that simultaneously discovers appropriate invariants and constitutive models for isotropic incompressible hyperelastic materials. Our approach identifies both the most suitable invariants in a class of generalized invariants and the corresponding strain energy function directly from experimental observations. Unlike previous methods that rely on fixed invariant choices or sequential fitting procedures, our method integrates the discovery process into a single neural network architecture. By looking at a continuous family of possible invariants, the model can flexibly adapt to different material behaviors. We demonstrate the effectiveness of this approach using popular benchmark datasets for rubber and brain tissue. For rubber, the method recovers a stretch-dominated formulation consistent with classical models. For brain tissue, it identifies a formulation sensitive to small stretches, capturing the nonlinear shear response characteristic of soft biological matter. Compared to traditional and neural-network-based models, our framework provides improved predictive accuracy and interpretability across a wide range of deformation states. This unified strategy offers a robust tool for automated and physically meaningful model discovery in hyperelasticity.

Problem

• Choosing suitable invariants for hyperelastic material models
• Determining functional dependence of strain energy on invariants
• Integrating invariant discovery and model fitting into one framework

Innovation

• Simultaneously discovers invariants and constitutive models
• Integrates discovery into single neural network
• Adapts flexibly to different material behaviors