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.

翻译：多凸本构建模因其能保证数值模拟的稳定性并改善材料模型的泛化行为而备受青睐。然而，在某些应用中，多凸公式在再现底层真实材料响应方面表现不佳，这实际上可能限制其实用性。本文针对这一问题，探讨了多凸本构建模的局限性。主要贡献如下：（1）从理论层面分析了多凸性在某些情况下可能施加过度严格约束，从而限制本构模型可达精度的原因。据此，我们为两种非多凸的Mooney-Rivlin型势函数提供了解析椭圆性保证。（2）通过两种代表性公式——基于结构张量不变量的模型和基于有符号奇异值的模型——研究了多凸物理增强神经网络本构模型的实践局限性。利用均质化微结构材料数据集评估其性能，并在有限元模拟中评估其预测能力。（3）总体而言，本文概述了多凸本构建模的优势、局限及缓解策略。