This work investigates different sufficient and necessary criteria for hyperelastic, isotropic polyconvex material models, focusing on neural network implementations for incompressible materials. Furthermore, the expressiveness, accuracy, simplicity as well as the efficiency of those models is analyzed. This also enables an assessment of the practical applicability of the models. Convex Signed Singular Value Neural Networks (CSSV-NNs) are applied to compressible materials and tailored to incompressibility (inc-CSSV-NNs), resulting in a universal approximation for frame-indifferent, isotropic polyconvex energies for the compressible as well as incompressible case. While other existing approaches also guarantee frame-indifference, isotropy and polyconvexity, they impose too restrictive constraints and thus limit the expressiveness of the model since they are only based on sufficient but not necessary criteria. This is further substantiated by numerical examples of several, well-established classical models (Neo--Hooke, Mooney--Rivlin, Gent and Arruda--Boyce) and Treloar's experimental data. Moreover, the numerical examples include an explicitly constructed energy function that cannot be approximated by neural networks constrained by Ball's criterion for polyconvexity. This substantiates that Ball's criterion, though sufficient, is not necessary for polyconvexity.

翻译：本文研究了超弹性、各向同性多凸材料模型的不同充分必要条件，重点关注不可压缩材料的神经网络实现。此外，分析了这些模型的表现力、精度、简洁性及效率，从而评估了模型的工程适用性。凸符号奇异值神经网络（CSSV-NNs）应用于可压缩材料，并针对不可压缩性进行定制化设计（inc-CSSV-NNs），为可压缩与不可压缩情形下满足框架不变性、各向同性和多凸性的能量函数提供了通用近似。尽管其他现有方法也能保证框架不变性、各向同性与多凸性，但它们仅基于充分条件而非必要条件，施加了过于严格的约束，限制了模型的表现力。多个经典模型（Neo-Hooke、Mooney-Rivlin、Gent和Arruda-Boyce）以及Treloar实验数据的数值算例进一步验证了此观点。此外，数值算例包含一个显式构造的能量函数，该函数无法被受Ball多凸性判据约束的神经网络近似，这证实虽然Ball判据是充分的，但并非多凸性的必要条件。