The mathematical formulation of constitutive models to describe the path-dependent, i.e., inelastic, behavior of materials is a challenging task and has been a focus in mechanics research for several decades. There have been increased efforts to facilitate or automate this task through data-driven techniques, impelled in particular by the recent revival of neural networks (NNs) in computational mechanics. However, it seems questionable to simply not consider fundamental findings of constitutive modeling originating from the last decades research within NN-based approaches. Herein, we propose a comparative study on different feedforward and recurrent neural network architectures to model inelasticity. Within this study, we divide the models into three basic classes: black box NNs, NNs enforcing physics in a weak form, and NNs enforcing physics in a strong form. Thereby, the first class of networks can learn constitutive relations from data while the underlying physics are completely ignored, whereas the latter two are constructed such that they can account for fundamental physics, where special attention is paid to the second law of thermodynamics in this work. Conventional linear and nonlinear viscoelastic as well as elastoplastic models are used for training data generation and, later on, as reference. After training with random walk time sequences containing information on stress, strain, and, for some models, internal variables, the NN-based models are compared to the reference solution, whereby interpolation and extrapolation are considered. Besides the quality of the stress prediction, the related free energy and dissipation rate are analyzed to evaluate the models. Overall, the presented study enables a clear recording of the advantages and disadvantages of different NN architectures to model inelasticity and gives guidance on how to train and apply these models.

翻译：描述材料路径依赖（即非弹性）行为的本构模型数学公式化是一项具有挑战性的任务，且数十年来一直是力学研究的重点。受计算力学领域神经网络最新复兴的推动，通过数据驱动技术促进或自动化这一任务的努力日益增加。然而，在基于神经网络的方法中完全忽略过去几十年本构建模研究的基本发现似乎存在疑问。本文对不同前馈和循环神经网络架构在模拟非弹性行为方面进行了比较研究。我们将模型分为三类：黑箱神经网络、弱形式物理约束神经网络和强形式物理约束神经网络。第一类网络可从数据中学习本构关系，但完全忽略底层物理机制；而后两类网络则设计为能够考虑基本物理原理，其中特别关注热力学第二定律。采用经典线性和非线性粘弹性模型以及弹塑性模型生成训练数据，并作为基准参考。在使用包含应力、应变及（部分模型中）内变量的随机游走时间序列进行训练后，将基于神经网络的模型与基准解进行对比，同时考虑插值与外推情况。除应力预测质量外，还分析相关的自由能和耗散率以评估模型性能。总体而言，本研究清晰记录了不同神经网络架构在模拟非弹性行为方面的优缺点，并为这些模型的训练与应用提供了指导。