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

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.