This work investigates different sufficient and necessary criteria for hyperelastic, isotropic polyconvex material models, focusing on neural network implementations for compressible and 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. 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与必要条件，重点关注可压缩与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、准确性、简洁性及效率，从而评估其实际适用性。凸符号奇异值神经网络（CSSV-NNs）被应用于可压缩材料，并通过定制化处理适应不可压缩性（inc-CSSV-NNs），从而为可压缩与不可压缩情况下的无框架偏好、各向同性多凸能量提供了通用逼近。尽管其他现有方法也保证了无框架偏好、各向同性和多凸性，但施加了过于严格的约束，从而限制了模型的表现力。这一点通过多个经典模型（Neo-Hooke、Mooney-Rivlin、Gent和Arruda-Boyce）的数值算例以及Treloar实验数据得到进一步证实。此外，数值算例还包含一个显式构造的能量函数，该函数无法通过受Ball多凸性准则约束的神经网络进行逼近，这证实了Ball准则虽为多凸性的充分条件，但并非必要条件。