In the present work, a hyperelastic constitutive model based on neural networks is proposed which fulfills all common constitutive conditions by construction, and in particular, is applicable to compressible material behavior. Using different sets of invariants as inputs, a hyperelastic potential is formulated as a convex neural network, thus fulfilling symmetry of the stress tensor, objectivity, material symmetry, polyconvexity, and thermodynamic consistency. In addition, a physically sensible stress behavior of the model is ensured by using analytical growth terms, as well as normalization terms which ensure the undeformed state to be stress free and with zero energy. In particular, polyconvex, invariant-based stress normalization terms are formulated for both isotropic and transversely isotropic material behavior. By fulfilling all of these conditions in an exact way, the proposed physics-augmented model combines a sound mechanical basis with the extraordinary flexibility that neural networks offer. Thus, it harmonizes the theory of hyperelasticity developed in the last decades with the up-to-date techniques of machine learning. Furthermore, the non-negativity of the hyperelastic neural network-based potentials is numerically examined by sampling the space of admissible deformations states, which, to the best of the authors' knowledge, is the only possibility for the considered nonlinear compressible models. For the isotropic neural network model, the sampling space required for that is reduced by analytical considerations. In addition, a proof for the non-negativity of the compressible Neo-Hooke potential is presented. The applicability of the model is demonstrated by calibrating it on data generated with analytical potentials, which is followed by an application of the model to finite element simulations. In addition, an adaption of the model to noisy data is shown and its [...]

翻译：[translated abstract in Chinese] 在本文中，提出了一种基于神经网络的超弹性本构模型，该模型通过构造满足所有常见本构条件，尤其适用于可压缩材料行为。通过使用不同的不变量集合作为输入，将超弹性势能构建为凸神经网络，从而满足应力张量的对称性、客观性、材料对称性、多凸性及热力学一致性。此外，通过使用分析性增长项以及确保未变形状态无应力且零能量的归一化项，保证了模型在物理上合理的应力行为。特别地，针对各向同性和横观各向同性材料行为，分别制定了基于多凸不变量的应力归一化项。通过精确满足所有这些条件，所提出的物理增强模型将坚实的力学基础与神经网络提供的非凡灵活性相结合，从而调和了过去数十年发展的超弹性理论与现代机器学习技术。此外，通过对可接受变形状态空间进行采样，数值验证了基于超弹性神经网络势能的非负性——据作者所知，这是针对所考虑的非线性可压缩模型的唯一可行方法。对于各向同性神经网络模型，通过分析性考量缩减了所需采样空间。同时，给出了可压缩Neo-Hooke势能非负性的数学证明。通过使用解析势能生成的数据进行标定，随后将模型应用于有限元仿真，展示了该模型的适用性。此外，还展示了模型对含噪声数据的适应性及其[……]