Accurately modeling the mechanical behavior of materials is crucial for numerous engineering applications. The quality of these models depends directly on the accuracy of the constitutive law that defines the stress–strain relation. However, discovering these constitutive material laws remains a significant challenge, in particular when only material deformation data is available. To address this challenge, unsupervised machine learning methods have been proposed to learn the constitutive law from deformation data. Nonetheless, existing approaches have several limitations: they either fail to ensure that the learned constitutive relations are consistent with physical principles, or they rely on boundary force data for training which are unavailable in many in-situ scenarios. Here, we introduce a machine learning approach to learn physics-consistent constitutive relations solely from material deformation without boundary force information. This is achieved by considering a dynamic formulation rather than static equilibrium data and applying an input convex neural network (ICNN). We validate the effectiveness of the proposed method on a diverse range of hyperelastic material laws. We demonstrate that it is robust to a significant level of noise and that it converges to the ground truth with increasing data resolution. We also show that the model can be effectively trained using a displacement field from a subdomain of the test specimen and that the learned constitutive relation from one material sample is transferable to other samples with different geometries. The developed methodology provides an effective tool for discovering constitutive relations. It is, due to its design based on dynamics, particularly suited for applications to strain-rate-dependent materials and situations where constitutive laws need to be inferred from in-situ measurements without access to global force data.

中文翻译：

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从动态位移中学习物理场一致的材料行为

准确模拟材料的机械行为对于许多工程应用至关重要。这些模型的质量直接取决于定义应力-应变关系的本构定律的准确性。然而，发现这些本构材料定律仍然是一个重大挑战，尤其是在只有材料变形数据可用的情况下。为了应对这一挑战，已经提出了无监督机器学习方法，从变形数据中学习本构律。尽管如此，现有方法有几个局限性：它们要么无法确保学习的本构关系与物理原理一致，要么它们依赖于边界力数据进行训练，而这在许多原位场景中是不可用的。在这里，我们介绍了一种机器学习方法，可以仅从材料变形中学习物理一致性本构关系，而无需边界力信息。这是通过考虑动态公式而不是静态平衡数据并应用输入凸神经网络 （ICNN） 来实现的。我们验证了所提出的方法在各种超弹性材料定律上的有效性。我们证明，它对相当高的噪声水平具有鲁棒性，并且随着数据分辨率的提高，它会收敛到地面实况。我们还表明，可以使用来自测试样本子域的位移场有效地训练模型，并且从一个材料样本中学到的本构关系可以转移到具有不同几何形状的其他样本。开发的方法为发现本构关系提供了有效的工具。 由于其基于动力学的设计，它特别适用于应变速率相关材料的应用，以及需要从原位测量中推断本构定律而无法访问全局力数据的情况。