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. The normalization terms are formulated for both isotropic and transversely isotropic material behavior and do not violate polyconvexity. 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 potential is numerically verified 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. 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 extrapolation capability is compared to models with reduced physical background. Within all numerical examples, excellent and physically meaningful predictions have been achieved with the proposed physics-augmented neural network.

翻译：本研究提出一种基于神经网络构建的超弹性本构模型，该模型通过构造方式满足所有常见本构条件，尤其适用于可压缩材料行为。通过采用不同的不变量集合作为输入，将超弹性势能函数构建为凸神经网络，从而满足应力张量对称性、客观性、材料对称性、多凸性及热力学一致性。进一步利用解析增长项与归一化项确保模型具有物理合理的应力响应特性，其中归一化项可保证未变形状态为无应力零能状态。该归一化项针对各向同性及横观各向同性材料行为分别构建，且不破坏多凸性条件。通过精确满足上述所有条件，所提出的物理增强模型将完备的力学基础与神经网络提供的非凡灵活性相结合，从而实现了近几十年发展的超弹性理论与现代机器学习技术的有机统一。此外，采用可变形状态空间采样的数值方法验证超弹性势能函数的非负性——据作者所知，这是针对所考虑的非线性可压缩模型唯一可行的验证方案。通过解析势能函数生成的数据进行模型标定，进而开展有限元仿真应用，验证了模型的适用性。同时展示了模型对含噪声数据的适应能力，并将其外推能力与物理背景简化的模型进行对比。所有数值算例表明，所提出的物理增强神经网络均取得了卓越且具有物理意义的预测结果。