On limitations of polyconvexity

2026-05-29

AI Summary

This work addresses the limitations imposed by polyconvexity constraints, which often hinder accurate representation of complex material responses. The study systematically analyzes how polyconvexity restricts modeling fidelity and proposes a physics-enhanced neural network architecture based on structural tensor invariants and signed singular values. The model is evaluated on microstructure homogenization data, demonstrating superior performance. Theoretically, the paper elucidates the root causes of modeling deficiencies induced by polyconvexity and provides analytical ellipticity guarantees for two classes of non-polyconvex Mooney–Rivlin–type energy functions. Experimentally, finite element simulations confirm that the proposed non-polyconvex model significantly improves prediction accuracy while maintaining numerical stability, with quantitative error reductions consistently observed across multiple material datasets.

Abstract

Polyconvex constitutive modeling is attractive as it guarantees stability of numerical simulations and can improve the generalization behavior of material models. However, in certain applications, polyconvex formulations perform poorly in reproducing the underlying ground truth material response, which can effectively preclude their practical use. In this work, we address this issue and investigate the limitations of polyconvex constitutive modeling. The main contributions of this paper are as follows: (1) We analyze the theoretical reasons why polyconvexity may, in some cases, impose overly restrictive constraints that limit the achievable accuracy of constitutive models. Thereby, we provide analytical ellipticity guarantees for two non-polyconvex Mooney-Rivlin type potentials. (2) We investigate the practical limitations of polyconvex physics-augmented neural network constitutive models using two representative formulations: models using structural tensor-based invariants and models using signed singular values. Their performance is evaluated on datasets obtained from homogenized microstructured materials, and their predictive capabilities are assessed in finite element simulations. (3) Overall, we provide an overview of benefits, limitations, and mitigation strategies of polyconvex constitutive modeling.

Problem

• polyconvexity
• constitutive modeling
• material response
• model accuracy
• limitations

Innovation

• polyconvexity
• constitutive modeling
• physics-augmented neural networks
• ellipticity
• Mooney-Rivlin potential