Learning constitutive laws under explicit strain limits: An interpretable strain-limiting elasticity--Kolmogorov Arnold neural network framework

TL;DR

The paper tackles the challenge of modeling strain-saturating materials by combining a physics-based strain-limiting elasticity backbone with interpretable Kolmogorov--Arnol d Networks to learn smooth residual corrections. By enforcing odd symmetry, bounded strain, and vanishing tangent modulus at large stresses, the framework ensures mechanical admissibility while retaining data-driven flexibility. The KAN component uses a spline-based, sign-aware mapping to learn residuals, providing transparent parameters that correspond to strain magnitudes at stress knots and their local tangents. Across synthetic benchmarks and Treloar rubber data, the SLE-KAN approach achieves high accuracy, preserves physical constraints, and demonstrates a transparent trade-off between data fidelity and mechanical admissibility. The work offers a principled, interpretable alternative to black-box models for constitutive modeling under large deformations with potential extensions to multidimensional and thermodynamically consistent formulations.

Abstract

A physically consistent framework for modeling materials with saturating deformation, such as elastomers and biological tissues, is provided by strain-limiting elasticity. Fundamental limitations of classical elasticity are addressed through the enforcement of bounded strains; however, significant challenges for data-driven learning are posed by the strong nonlinearity of these laws. In this work, an interpretable hybrid constitutive modeling framework integrating strain-limiting elasticity (SLE) with Kolmogorov-Arnold Networks (KANs) is proposed to balance mechanical admissibility with data-driven flexibility. The dominant nonlinear response is captured by the SLE backbone, while smooth residual corrections are learned exclusively via a KAN. Essential mechanical principles-including symmetry, monotonicity, and bounded strain-are embedded directly into the model structure to ensure physical admissibility. The framework is assessed on synthetic benchmarks, where near-exact recovery is achieved in smooth regimes and consistency is retained under sharp transitions. Application to Treloar's rubber elasticity data demonstrates systematic improvement in stress-stretch agreement while preserving explicit strain limits. A regime-based analysis reveals a transparent trade-off between data fidelity and mechanical admissibility, demonstrating that deviations arise from deliberately imposed physical restrictions rather than unconstrained model expressivity. This SLE-KAN framework offers a robust, physics-consistent alternative to black-box neural networks for constitutive modeling.

Figures (14)

• Figure 1: Representative stress--strain response exhibiting strain-limiting behavior. The response is approximately linear at small stresses and asymptotically approaches a finite strain bound as the applied stress increases.
• Figure 2: General structure of a Kolmogorov--Arnold Network (KAN). The input is decomposed into scalar components, transformed through interpretable nonlinear functions, and recombined to form the output.
• Figure 3: Spline-based representation of the nonlinear constitutive function $g(|\tau|)$. Trainable coefficients control the strain magnitude at discrete stress knots, while the piecewise-linear structure ensures continuity and controlled local behavior.
• Figure 4: Comparison between the learned KAN constitutive response and the analytical strain-limiting model. The KAN reproduces the linear regime, transition region, and asymptotic saturation behavior while remaining consistent with the imposed physical constraints.
• Figure 5: Conceptual comparison between classical neural networks and Kolmogorov--Arnold Networks. Classical networks rely on distributed nonlinear representations, whereas KANs employ structured and interpretable functional decompositions aligned with constitutive principles.