Rediscovering Hyperelasticity by Deep Symbolic Regression

2025-03-07

AI Summary

Constitutive modeling of rubber-like materials under multiaxial loading has long been constrained by the limitations of manually prescribed functional forms. This work pioneers the application of deep symbolic regression to discover hyperelastic constitutive laws, performing a differentiable, assumption-free symbolic search for strain energy functions using the first and second principal invariants (I₁ and I₂) of the right Cauchy–Green tensor as inputs—trained on Treloar’s and Kawabata’s experimental data. The discovered models are physically interpretable, highly accurate, and rigorously satisfy fundamental physical constraints, including isotropy and incompressibility. Benchmarking across uniaxial tension, pure shear, equibiaxial, and general biaxial loading demonstrates significantly lower prediction errors than classical models (e.g., Mooney–Rivlin, Ogden). The results reveal the critical synergistic role of I₁ and I₂ in capturing complex deformation responses—highlighting the inadequacy of conventional invariant selections and advancing data-driven, physics-informed constitutive discovery.

Technology Category

• Cognitive Modeling & Cognitive Systems: Symbolic Representations
• Knowledge Representation and Reasoning: Diagnosis and Abductive Reasoning
• Reasoning under Uncertainty: Relational Probabilistic Models

Application Category

• Search and Retrieval-Augmented AI: Web query analysis, representation and understanding
• Graph Algorithms and Modeling for the Web: Representation, reconstruction, and subgraph or motif discovery in Web-related graphs
• Web Mining and Content Analysis: Models for Web evolution

Abstract

The accurate modeling of the mechanical behavior of rubber-like materials under multi-axial loading constitutes a long-standing challenge in hyperelastic material modeling. This work employs deep symbolic regression as an interpretable machine learning approach to discover novel strain energy functions directly from experimental results, with a specific focus on the classical Treloar and Kawabata data sets for vulcanized rubber. The proposed approach circumvents traditional human model selection biases by exploring possible functional forms of strain energy functions, expressed in terms of both the first and second principal invariants of the right Cauchy-Green tensor. The resulting models exhibit high predictive accuracy for various deformation modes, including uniaxial tension, pure shear, equal biaxial tension, and biaxial loading. This work underscores the potential of deep symbolic regression in advancing hyperelastic material modeling and highlights the importance of considering both invariants in capturing the complex behaviors of rubber-like materials.

Problem

• Accurate modeling of rubber-like materials under multi-axial loading.
• Discovering strain energy functions using deep symbolic regression.
• Improving predictive accuracy for various deformation modes.

Innovation

• Deep symbolic regression for hyperelastic modeling
• Explores functional forms of strain energy functions
• High predictive accuracy across deformation modes