Constitutive modelling is crucial for engineering design and simulations to accurately describe material behavior. However, traditional phenomenological models often struggle to capture the complexities of real materials under varying stress conditions due to their fixed forms and limited parameters. While recent advances in deep learning have addressed some limitations of classical models, purely data-driven methods tend to require large datasets, lack interpretability, and struggle to generalize beyond their training data. To tackle these issues, we introduce "Fusion-based Constitutive model (FuCe): Towards model-data augmentation in constitutive modelling". This approach combines established phenomenological models with an ICNN architecture, designed to train on the limited and noisy force-displacement data typically available in practical applications. The hybrid model inherently adheres to necessary constitutive conditions. During inference, Monte Carlo dropout is employed to generate Bayesian predictions, providing mean values and confidence intervals that quantify uncertainty. We demonstrate the model's effectiveness by learning two isotropic constitutive models and one anisotropic model with a single fibre direction, across six different stress states. The framework's applicability is also showcased in finite element simulations across three geometries of varying complexities. Our results highlight the framework's superior extrapolation capabilities, even when trained on limited and noisy data, delivering accurate and physically meaningful predictions across all numerical examples.

翻译：本构建模对于工程设计和仿真至关重要，用以精确描述材料行为。然而，传统的唯象模型由于其固定形式和有限参数，往往难以捕捉真实材料在多变应力条件下的复杂性。尽管深度学习的最新进展解决了经典模型的一些局限性，但纯数据驱动的方法往往需要大量数据集，缺乏可解释性，并且难以泛化到训练数据之外。为解决这些问题，我们提出了"基于融合的本构模型（FuCe）：迈向本构建模中的模型-数据增强"。该方法将成熟的唯象模型与ICNN架构相结合，旨在利用实际应用中通常可获得的有限且含噪声的力-位移数据进行训练。该混合模型本质上遵循必要的本构条件。在推理过程中，采用蒙特卡洛Dropout生成贝叶斯预测，提供量化不确定性的均值与置信区间。我们通过针对六种不同应力状态，学习两个各向同性本构模型和一个具有单一纤维方向的各向异性模型，证明了该模型的有效性。该框架的适用性还在三种不同复杂度的几何结构的有限元仿真中得到了展示。我们的结果突显了该框架卓越的外推能力，即使在有限且含噪声的数据上训练，也能在所有数值示例中提供准确且具有物理意义的预测。