We propose a new approach for unsupervised learning of hyperelastic constitutive laws with physics-consistent deep neural networks. In contrast to supervised learning, which assumes the availability of stress–strain pairs, the approach only uses realistically measurable full-field displacement and global reaction force data, thus it lies within the scope of our recent framework for Efficient Unsupervised Constitutive Law Identification and Discovery (EUCLID) and we denote it as NN-EUCLID. The absence of stress labels is compensated for by leveraging a physics-motivated loss function based on the conservation of linear momentum to guide the learning process. The constitutive model is based on input-convex neural networks, which are capable of learning a function that is convex with respect to its inputs. By employing a specially designed neural network architecture, multiple physical and thermodynamic constraints for hyperelastic constitutive laws, such as material frame indifference, material stability, and stress-free reference configuration are automatically satisfied. We demonstrate the ability of the approach to accurately learn several hidden isotropic and anisotropic hyperelastic constitutive laws – including e.g., Mooney–Rivlin, Arruda–Boyce, Ogden, and Holzapfel models – without using stress data. For anisotropic hyperelasticity, the unknown anisotropic fiber directions are automatically discovered jointly with the constitutive model. The neural network-based constitutive models show good generalization capability beyond the strain states observed during training and are readily deployable in a general finite element framework for simulating complex mechanical boundary value problems with good accuracy.

我们提出了一种使用物理一致的深度神经网络无监督学习超弹性本构定律的新方法。与假设应力-应变对的可用性的监督学习相比，该方法仅使用实际可测量的全场位移和全局反作用力数据，因此它位于我们最近的有效无监督本构法识别和发现框架的范围内(EUCLID)，我们将其表示为 NN-EUCLID。通过利用基于线性动量守恒的物理驱动损失函数来指导学习过程，可以补偿应力标签的缺失。本构模型基于输入-凸神经网络，该神经网络能够学习相对于其输入是凸的函数。通过采用专门设计的神经网络架构，自动满足超弹性本构定律的多个物理和热力学约束，例如材料框架无差异、材料稳定性和无应力参考配置。我们展示了该方法准确学习几个隐藏的各向同性和各向异性超弹性本构定律的能力——包括例如 Mooney-Rivlin、Arruda-Boyce、Ogden、和 Holzapfel 模型——不使用应力数据。对于各向异性超弹性，未知的各向异性纤维方向与本构模型联合自动发现。基于神经网络的本构模型显示出超出训练期间观察到的应变状态的良好泛化能力，并且易于部署在通用有限元框架中，以高精度模拟复杂的机械边界值问题。