基于深度学习的粘弹性本构建模

粘弹性材料在工程中有着广泛的应用，构建精确且高效的描述材料本构关系的粘弹性模型是非常重要的。粘弹性材料具有高度的历史依赖性，变形过程与加载速率相关，应力响应取决于变形的历史过程。尽管过去的研究中已经建立大量的传统模型，但是复杂的计算和非线性行为的捕获依旧是研究者面临的关键挑战。很多学者开始将机器学习和深度学习技术应用于本构建模，开发基于数据驱动的计算方法来代替传统的本构建模方法，直接由数据建立材料的应力-应变本构关系，从而预测材料的力学行为。

根据循环神经网络的原理和粘弹性材料的历史依赖性的相似性，本文提出了一种基于门控循环单元（GRU）的网络模型用于预测材料的本构关系：利用高斯分布插值方法和理论求解生成训练数据集，采用Z-score方法对数据集归一化处理，使用均方误差（MSE）损失函数和Adam优化算法迭代训练神经网络模型。设计单轴拉伸、双轴拉伸和纯剪切的数值算例，验证了GRU模型对未知数据的预测能力较好，平均误差在10%以内，是实现粘弹性本构建模的新方法。

基于迁移学习的策略，以GRU模型为预训练模型并冻结参数，在此基础上增加循环层，权重微调得到迁移模型，对应变-能量介导的非线性粘弹性行为进行训练和预测。该方法可以降低神经网络对数据量的要求及模型训练的复杂度，增强网络模型的复用性。研究表明，基于深度学习的神经网络模型为粘弹性本构建模提供了新的可行思路，未来可以进一步与仿真软件和有限元计算相结合，提高计算效率和精度。

Viscoelastic materials have a wide range of applications in engineering, and it is crucial to construct accurate and efficient models to describe their constitutive relationships. Viscoelastic materials exhibit a high degree of history dependence, where the deformation process is related to the loading rate, and the stress response depends on the history of deformation. Despite the establishment of numerous traditional models in past research, complex computations and the capture of nonlinear behavior remain key challenges for researchers. Many scholars have begun to apply machine learning and deep learning techniques to constitutive modeling, developing data-driven computational methods to replace traditional constitutive modeling methods, directly establishing the stress-strain constitutive relationship of materials from data, thereby predicting the mechanical behavior of materials.

Based on the principles of recurrent neural networks and the similarity of the history dependence of viscoelastic materials, this paper proposes a network model based on Gated Recurrent Units (GRUs) for predicting the constitutive relationship of materials: using Gaussian distribution interpolation methods and theoretical solutions to generate datasets, using the Z-score method to normalize the dataset, and using the Mean Squared Error (MSE) loss function and Adam optimization algorithm to iteratively train the neural network model. Numerical examples of uniaxial stretching, biaxial stretching and pure shear are designed, and it is verified that the GRU model has a better prediction ability for unknown data with an average error within 10%, which is a new method to realize the viscoelastic intrinsic constructive model.

Based on the strategy of transfer learning, the GRU model is used as a pre-training model with its parameters frozen. On this basis, a recurrent layer is added and the weights are fine-tuned to obtain the transfer model, which is trained and predicted for the nonlinear viscoelastic behavior guided by strain-energy. This method can reduce the data requirements of the neural network and the complexity of model training, enhancing the reusability of the network model. The research shows that the neural network model based on deep learning provides a new feasible idea for viscoelastic constitutive modeling, and it can be further combined with simulation software and finite element calculations in the future to improve computational efficiency and accuracy.

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