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数据驱动和基于机器学习的方法目前正在推动材料建模的快速发展。从单轴拉伸数据的简单回归开始，已经迅速扩展到高维和基于大数据的替代建模，基本上涵盖了所有类型的技术兴趣材料，包括金属、聚合物、复合材料等。与传统材料建模不同，此方法放弃使用分析本构律，避免了由于如基于经验的建模假设和实验（或计算）测试的选择过于严格而无法描述真实物理引起的建模误差。尽管该方法已取得了不错进展，但目前仍存在许多问题。绕过或替代材料模型的最新技术植根于监督学习或曲线拟合设置，其需要大量由输入-输出组成的数据，即应变-应力对。无论材料模型应力-应变关系数据是来自实验还是多尺度模拟，其不可解释性和数据负载都是一个长期挑战。
来自苏黎世联邦理工学院的Moritz Flaschel等，提出了一种数据驱动的材料定律自动发现方法，称为EUCLID（高效无监督本构定律识别和发现）。EUCLID不需要对材料模型进行先验选择，可以灵活地描述各种不同的材料行为，并且它仅依赖于未标记的数据，即全场位移和全局反作用力，而不依赖于单个实验生成的应力数据。该方法通过建立一个可解释的候选材料模型库，仅使用物理约束根据给定的未标记数据，自动发现库中最相关的特征。材料模型库通过傅里叶级数扩展屈服函数来构建，而各向同性和运动硬化则通过假设依赖于内部历史变量的屈服函数来引入，该屈服函数随塑性变形演变。为了选择最相关的傅里叶模式和识别硬化机制，EUCLID利用物理知识，即基于线性动量平衡来确定支配发现的优化问题，以补偿应力数据的不可用性。结果表明，EUCLID无需使用任何应力数据，能够仅从位移和净反作用力数据中发现可解释的塑性模型。该方法为受监督的数据驱动和机器学习方法提供了一种物理约束、数据高效的替代方案。该文近期发表于npj Computational Materials 8：91(2022)，英文标题与摘要如下，点击左下角“阅读原文”可以自由获取论文PDF。
Discovering plasticity models without stress data
Moritz Flaschel, Siddhant Kumar & Laura De Lorenzis
We propose an approach for data-driven automated discovery of material laws, which we call EUCLID (Efficient Unsupervised Constitutive Law Identification and Discovery), and we apply it here to the discovery of plasticity models, including arbitrarily shaped yield surfaces and isotropic and/or kinematic hardening laws. The approach is unsupervised, i.e., it requires no stress data but only full-field displacement and global force data; it delivers interpretable models, i.e., models that are embodied by parsimonious mathematical expressions discovered through sparse regression of a potentially large catalog of candidate functions; it is one-shot, i.e., discovery only needs one experiment. The material model library is constructed by expanding the yield function with a Fourier series, whereas isotropic and kinematic hardening is introduced by assuming a yield function dependency on internal history variables that evolve with the plastic deformation. For selecting the most relevant Fourier modes and identifying the hardening behavior, EUCLID employs physics knowledge, i.e., the optimization problem that governs the discovery enforces the equilibrium constraints in the bulk and at the loaded boundary of the domain. Sparsity promoting regularization is deployed to generate a set of solutions out of which a solution with low cost and high parsimony is automatically selected. Through virtual experiments, we demonstrate the ability of EUCLID to accurately discover several plastic yield surfaces and hardening mechanisms of different complexity.