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Physics-driven machine learning model on temperature and time-dependent deformation in lithium metal and its finite element implementation
锂金属随温度和时间变化变形的物理驱动机器学习模型及其有限元实现

\author{ \作者{
Jici Wen , Qingrong Zou , Yujie Wei
文继慈 , 邹庆荣 , 魏玉洁

School of Applied Science, Beijing Information Science and Technology University, Beijing 100192, China
北京信息科技大学应用科学学院，中国北京 100192
School of Engineering Sciences, University of Chinese Academy of Sciences, Beijing 100049, China
中国科学院大学工程科学学院，中国北京 100049
}

A R T I C L E I N F O

Keywords: 关键词：

Physics-driven machine learning
物理驱动的机器学习
Lithium-metal anode 锂金属阳极
Creep 蠕变
Finite-element analysis 有限元分析
Constitutive model 构成模型

Abstract 摘要

A B S T R A C T Precise understanding on the temperature and time-dependent deformation in lithium-metal anode is of compelling need for durable service of Li-based batteries. Due to both temporal and spatial intertwined thermal agitations and the scarcity of experiments, faithful deformation map of Li-metal covering a broad range of service condition is still lacking. Here we design a physicsdriven machine learning (PD-ML) algorithm to map the temperature, stress and rate-dependent deformation in Li-metal. We demonstrate that the PD-ML model, fed with limited experimental results, can predict the mechanical response of Li-metal in a wide span of temperature and deformation rate, and help to realize a deformation map of Li-metal with high fidelity. A finite element (FE) procedure based on the PD-ML constitutive model is then developed. The integration of PD-ML with FE procedure inherits the power of FE analysis and the accuracy originated from PD-ML in describing temperature, stress and rate-dependent mechanical response of Limetal. The method introduced here paves a new way for constitutive modelling to capture the complex deformation in solids involving multi-field and multiscale mechanics.
要使锂基电池持久耐用，就必须准确了解锂金属负极随温度和时间变化的变形情况。由于时间和空间上相互交织的热躁动以及实验的稀缺性，目前仍缺乏覆盖广泛使用条件的锂金属忠实变形图。在此，我们设计了一种物理驱动的机器学习（PD-ML）算法来绘制锂金属中与温度、应力和速率相关的变形图。我们证明，在有限的实验结果支持下，PD-ML 模型可以预测锂金属在大跨度温度和变形率条件下的机械响应，并有助于实现高保真的锂金属变形图。然后，基于 PD-ML 构成模型开发了有限元（FE）程序。PD-ML 与 FE 程序的集成继承了 FE 分析的强大功能以及 PD-ML 在描述 Limetal 与温度、应力和速率相关的机械响应时所具有的精确性。这里介绍的方法为构造建模捕捉涉及多场和多尺度力学的固体复杂变形铺平了一条新路。

1. Introduction 1.导言

When cataloguing deformation mechanisms and formulate physically sound and faithful constitutive laws for temperature and rate-sensitive deformation in solids, we ought to have deep understandings about the underlying kinetics and thermodynamics of deformation processes occurring in a huge span of spatial and temporal scales (Argon, 1975; Frost and Ashby, 1982; Meyers et al., 1999; Huang et al., 2018). The lack of systematic experimental data at different temperature, stress, and strain rate often leads to an incomplete understanding about the underlying multi-scale deformation, and consequentially render difficulties for the development of predictive constitutive models with high fidelity. The plasticity and creep deformation in solids, for example, are intertwined with each other and are in general a function of temperature, stress and deformation rate (Johnson and Cook, 1985). A highly predictive constitutive model, which may be contingent on limited experimental observations on the mechanical responses at different temperature and strain rate, is of compelling need for a variety of materials in engineering practice.
在编目形变机理并为固体中的温度和速率敏感形变制定物理上可靠和忠实的构效定律时，我们应该深入了解在巨大的空间和时间尺度跨度中发生的形变过程的基本动力学和热力学（Argon，1975；Frost 和 Ashby，1982；Meyers 等人，1999；Huang 等人，2018）。由于缺乏不同温度、应力和应变率下的系统实验数据，人们往往无法全面了解多尺度变形的基本原理，从而难以开发出高保真的预测构成模型。例如，固体的塑性变形和蠕变变形相互交织，通常是温度、应力和变形率的函数（Johnson 和 Cook，1985 年）。工程实践中的各种材料都迫切需要一个高度预测性的构成模型，该模型可能取决于对不同温度和应变速率下机械响应的有限实验观察。
Li-metal is such a typical example where temperature, stress and deformation rate are involved during its engineering service.
锂金属就是这样一个典型的例子，在其工程服务过程中涉及到温度、应力和变形率。
Fig. 1. The conventional vs the PD-ML based constitutive modelling for Li-metal. (a) Stress-strain curves at different temperature and at a constant strain rate , and (b) those at different strain rate at a constant temperature 298 K (experimental data from LePage et al., 2019, replotted). (c) and (d), modelling strategies: (c) A conventional method with explicit formula for plastic flow as a function of stress and temperature. (d) The PD-ML approach, where plastic flow is learned from the numerical algorithm.
图 1.传统与基于 PD-ML 的锂金属构成模型对比。(a) 不同温度和恒定应变率 下的应力-应变曲线，以及 (b) 恒温 298 K 下不同应变率下的应力-应变曲线（实验数据来自 LePage 等人，2019 年，重新绘制）。(c)和(d)，建模策略：(c) 传统方法，带有作为应力和温度函数的塑性流动显式公式。(d) PD-ML 方法，从数值算法中学习塑性流动。
Because of its highest theoretical specific capacity ( ), lowest density, and most negative potential (about -3.04 V) (Lin et al., 2017; Wood et al., 2017), Li-metal as the preferred anode is used in many new types of battery including the Li-S battery, the Li-Air battery, and the solid-state battery. However, Li-metal anode gives rise to a series of problems in Li-metal batteries (LMBs), including interfacial contact and resistance (Zhang et al., 2020; Chen et al., 2021), dendrite growth (Xu et al., 2014; Porz et al., 2017; Barai et al., 2017; Wang et al., 2020), mossy and dead Li (Chen et al., 2017), and so on. It is now well known that plastic and creep properties of Li-metal anode have great impact on the performance of the solid-state LMBs by influencing the pressure-dependent interfacial contact and resistance (Krauskopf et al., 2019). Mechanical failures of the Li-metal anode also degrade the electrochemical performance of LMBs (Kozen et al., 2017).
由于锂金属具有最高的理论比容量（ ）、最低的密度和最负的电位（约-3.04 V）（Lin 等人，2017 年；Wood 等人，2017 年），锂金属作为首选阳极被用于许多新型电池中，包括锂-S 电池、锂-空气电池和固态电池。然而，锂金属负极在锂金属电池（LMB）中引发了一系列问题，包括界面接触和电阻（Zhang 等人，2020；Chen 等人，2021）、枝晶生长（Xu 等人，2014；Porz 等人，2017；Barai 等人，2017；Wang 等人，2020）、苔藓和死锂（Chen 等人，2017）等。众所周知，锂金属阳极的塑性和蠕变特性会影响压力相关的界面接触和电阻，从而对固态 LMB 的性能产生重大影响（Krauskopf 等人，2019 年）。锂金属阳极的机械故障也会降低 LMB 的电化学性能（Kozen 等人，2017 年）。
Many related investigations have been done to shed light on the reliability of Li-metal anode due to plasticity and creep at different temperature and strain rate. Wang et al. measured the elastic-visco-plastic mechanical properties of Li-metal by nanoindentation tests (Wang and Cheng, 2017; Herbert, et al., 2018a; 2018b). Masias et al. (2019) characterized stress-strain response and creep behavior of Li-metal in tension and compression. Xu et al. (2017) and Fincher et al. (2020) explored the influence of size and strain-rate on plasticity of Li-metal. Its dependence on both strain rate and temperature was further investigated comprehensively by LePage et al. (2019). Narayan and Anand (2018) formulated a large deformation isotropic elastic-viscoplastic constitutive model for Li-metal. Those experimental investigations and modelling deepen our understanding on the mechanical response of Li-metal anode in variant conditions. There however remain challenges to construct a complete deformation map of Li-metal subject to different temperature and strain rate, or creep at different stress and temperature.
为了揭示锂金属阳极在不同温度和应变速率下因塑性和蠕变而产生的可靠性，人们进行了许多相关研究。Wang 等人通过纳米压痕试验测量了锂金属的弹塑性力学性能（Wang 和 Cheng，2017；Herbert 等人，2018a；2018b）。Masias 等人（2019）描述了 Li 金属在拉伸和压缩过程中的应力应变响应和蠕变行为。Xu 等人（2017 年）和 Fincher 等人（2020 年）探讨了尺寸和应变速率对锂金属塑性的影响。LePage 等人（2019 年）进一步全面研究了其对应变率和温度的依赖性。Narayan 和 Anand（2018 年）制定了锂金属的大变形各向同性弹性-粘弹性构成模型。这些实验研究和建模加深了我们对锂金属阳极在不同条件下机械响应的理解。然而，要构建锂金属在不同温度和应变率条件下的完整变形图，或在不同应力和温度条件下的蠕变图，仍然存在挑战。
Recent rapid development of the ML method in engineering problems where patterns or scientific principles may be extracted from big data or from strong nonlinear problems. For example, ML method, fed with data from experiments, can be adopted to describe the constitutive behavior of materials. Mozaffar et al. (2019) and Gorji et al. (2020) offers an effective method to predict path-dependent plasticity through deep learning. ML has also been used for new material design (Shi et al., 2019; Bessa et al., 2017; Curtarolo et al., 2013), structural optimization (Liu et al., 2020), property prediction (Kozuch et al., 2018; Hsu et al., 2020), multiscale modeling (Karapiperis et al., 2021), and its development and applications in special domains are rapidly growing (Li et al., 2018).
近来，从大数据或强非线性问题中提取模式或科学原理的 ML 方法在工程问题中得到迅速发展。例如，ML 方法可以利用实验数据来描述材料的构成行为。Mozaffar 等人（2019 年）和 Gorji 等人（2020 年）提供了一种通过深度学习预测路径依赖塑性的有效方法。ML 还被用于新材料设计（Shi 等人，2019；Bessa 等人，2017；Curtarolo 等人，2013）、结构优化（Liu 等人，2020）、性能预测（Kozuch 等人，2018；Hsu 等人，2020）、多尺度建模（Karapiperis 等人，2021），其在特殊领域的发展和应用也在快速增长（Li 等人，2018）。
In the paper, we introduce a new, robust, and accurate constitutive model by combining physical laws with orchestrated ML algorithm for temperature-, stress-, and rate-dependent deformation in Li-metal. We organize the paper as follows. In Section 2, the PD-
在本文中，我们将物理定律与协调 ML 算法相结合，针对锂金属中与温度、应力和速率有关的变形，介绍了一种新的、稳健且精确的构成模型。本文结构如下。在第 2 节中，PD-
Table 1 表 1
The material parameters of Li-metal, from (Krauskopf et al., 2019; LePage et al., 2019).
锂金属的材料参数，摘自（Krauskopf 等人，2019 年；LePage 等人，2019 年）。
6200 0.401 1.0 6.6 37.0
Fig. 2. Predictability of the conventional model on the response of Li-metal under uniaxial tension. (a) The temperature dependence of curves at a constant strain rate , predictions from the conventional model vs. experiments; (b) Rate effects predicted by the conventional model vs. experiments
图 2.传统模型对单轴拉伸下锂金属 响应的可预测性。(a) 在恒定应变速率 条件下 曲线的温度依赖性，传统模型的预测与实验的对比；(b) 传统模型预测的速率效应与实验的对比 .
ML model of Li-metal and its numerical algorithm is introduced. In Section 3, we verify the PD-ML model using experimental results and demonstrate its robustness and accuracy. The PD-ML model is then implemented in the commercial FE package ABAQUS (DS Simulia Corp., 2020) as a user-material subroutine (UMAT), and we further explore the applicability of the PD-ML-FEM algorithm for the deformation of Li-metal in Section 4. We conclude in Section 5 with final remarks and discussions.
第 2 节介绍了锂金属的 ML 模型及其数值算法。在第 3 节中，我们利用实验结果验证了 PD-ML 模型，并证明了其稳健性和准确性。第 4 节中，我们进一步探讨了 PD-ML-FEM 算法在锂金属变形中的适用性。最后，我们在第 5 节中进行了总结和讨论。

2. Physics-driven machine learning algorithm
2.物理驱动的机器学习算法

To construct a faithful constitutive model which satisfying the thermodynamic principles and conservation laws, we often need systematic experimental observations to shed light on the underlying deformation process, to reveal the dependence of all variates, to calibrate unknown constants in the equations, and to validate the predictability of the constitutive model. As shown in Fig. 1, with experimental data given in Figs. 1a and b (LePage et al., 2019), we may either adopt a conventional modelling approach or a PD-ML algorithm, as to be explained next.
为了构建一个符合热力学原理和守恒定律的忠实的结构模型，我们通常需要系统的实验观测来揭示基本的变形过程，揭示所有变量的依赖关系，校准方程中的未知常数，并验证结构模型的可预测性。如图 1 所示，图 1a 和 b 中给出了实验数据（LePage et al.
For the conventional constitutive model on Li-metal, the dependence of plastic flow on stress and temperature is known as a prior. As seen in Fig. 1a, the stress-strain curves of the bulk Li-metal under uniaxial tension exhibit typical elastic-plastic response of polycrystalline metals, yielding and irrecoverable plastic deformation after the initial elasticity. The isotropic linear elastic response is characterized by a Young's modulus and a Poisson's ratio . After yielding, the Li-metal may sustain a large amount of plastic deformation with tensile failure-strain up to (Hull and Rosenberg, 1959). Li-metal has a low melting temperature ( 453 K ) and low activation energy for self-diffusion ( ) (Messer and Noack, 1975; Hao et al., 2018). The dominant mechanism accounting for the plasticity is due to the coble creep at its working temperature for electric vehicle batteries, in the range of 233 to 323 K (Chen et al., 2020; Wang et al., 2020; Zhang, 2011). It therefore leads to a highly rate-sensitive deformation, as seen in Fig. 1b.
在锂金属的传统构成模型中，塑性流动与应力和温度的关系是已知的先验关系。如图 1a 所示，块状锂金属在单轴拉伸下的应力-应变曲线表现出多晶金属的典型弹塑性响应，即在初始弹性之后出现屈服和不可恢复的塑性变形。各向同性线性弹性响应的特征是杨氏模量 和泊松比 。屈服后，锂金属可能会产生大量塑性变形，拉伸破坏应变可达 （Hull 和 Rosenberg，1959 年）。锂金属具有较低的熔化温度（453 K）和较低的自扩散活化能（ ）（Messer 和 Noack，1975 年；Hao 等人，2018 年）。塑性的主要机制是由于电动汽车电池工作温度（233 至 323 K）范围内的柯布蠕变（Chen 等人，2020 年；Wang 等人，2020 年；Zhang，2011 年）。因此，如图 1b 所示，它会导致对速率高度敏感的变形。
In the conventional approach, we use the von Mises-based visco-plastic theory with isotropic strain hardening. The Prandtl-Reuss laws is adopted for the plastic flow as
在传统方法中，我们使用了基于冯-米塞斯的粘弹性理论和各向同性应变硬化理论。塑性流动采用的普朗特-罗伊斯定律为
where is the plastic strain rate tensor, is the equivalent strain rate, is the von Mises stress, is the deviatoric part of the Cauchy stress , and is the hydrostatic pressure for being the Kronecker delta function ( if , and 0 if ). The plastic strain rate is characterized by a set of creep constants (LePage et al., 2019; Masias et al., 2019),
其中， 是塑性应变率张量， 是等效应变率， 是冯米塞斯应力， 是考奇应力的偏离部分 ， 是静水压力， 是克朗克尔三角函数（如果 为 ，如果 为0）。塑性应变率 由一组蠕变常数表征（LePage 等人，2019 年；Masias 等人，2019 年）、
Fig. 3. The integrated ML algorithm with physical assumptions. (a) Diagram to show the input to and the output from the ML model, and (b) the numerical algorithm for the constitutive model integrating PD-ML.
图 3.带有物理假设的集成 ML 算法。(a) 显示 ML 模型输入和输出的示意图；(b) 构成模型集成 PD-ML 的数值算法。
which includes a creep component , an activation energy , a kinetic constant , the corresponding reference stress , the gas constant , and the absolute temperature .
其中包括蠕变分量 、活化能 、动力学常数 、相应的参考应力 、气体常数 和绝对温度 。
We implemented the conventional constitutive model shown in Fig. 1c in the commercial finite-element (FE) software Abaqus as a user material subroutine (DS Simulia Corp., 2020). With parameters listed in Table 1, we obtain the tensile stress-strain curves at different temperature and a constant strain rate from FE calculations in Fig. 2a, and the rate-sensitivity of the mechanical response at a given temperature in Fig. 2b. We can see that the conventional power-law model fails to capture the stress hardening and rate-dependence, which, in principle, reflects the weakness of the model (see Eq. 2) in predicting the stress- and temperature-dependent creep rate. It is noted that improvements to the fitting may be possible through the use of more complicated hardening or by using series solution in the power-law creep. The hardening behavior in Li-metal may be included in the power-law creep by formulating as , where and are respectively the strain hardening modulus and exponent. We may alternatively employ series solutions and write the collective power-law flow rate as , where , are variables to be fitted. Nevertheless, determining these parameters are not trivial. In most circumstances, they are not universal and cannot be quantified uniquely. It becomes worse when experimental data is limited.
我们在商用有限元（FE）软件 Abaqus 中以用户材料子程序（DS Simulia 公司，2020 年）的形式实现了图 1c 所示的传统构成模型。利用表 1 中列出的参数，我们通过 FE 计算得到了不同温度和恒定应变速率下的拉伸应力-应变曲线（图 2a），以及给定温度下的机械响应速率敏感性（图 2b）。我们可以看到，传统的幂律模型未能捕捉到应力硬化和速率依赖性，这原则上反映了该模型（见公式 2）在预测应力和温度相关蠕变速率方面的弱点。值得注意的是，通过使用更复杂的硬化或在幂律蠕变中使用序列解，可能会改善拟合效果。通过将 表述为 ，其中 和 分别为应变硬化模量和指数，可将锂金属中的硬化行为纳入幂律蠕变。我们也可以采用系列解法，将集体幂律流速写成 ，其中 和 是待拟合的变量。然而，确定这些参数并非易事。在大多数情况下，这些参数并不通用，无法唯一量化。当实验数据有限时，情况会变得更糟。
To better capture the stress- and temperature-dependent plastic flow, we adopt a PD-ML model to supplant the regular power-law flow in Li-metal, as shown in Fig. 3. As the prior physical knowledge (physics driven, PD), those variables ( ), acquired from experimental data, are selected as inputs, where is the equivalent strain rate and is the equivalent strain, and and are the strain rate tensor and time, respectively. The outputs include the plastic strain rate and the strain hardening . In brief, the learning algorithm is a function to realize the following mapping through machine learning (ML)
为了更好地捕捉与应力和温度有关的塑性流动，我们采用了 PD-ML 模型来取代锂金属中的常规幂律流动，如图 3 所示。作为先验物理知识（物理驱动，PD），我们选择从实验数据中获取的变量 ( ) 作为输入，其中 是等效应变率， 是等效应变， 和 分别是应变率张量和时间。输出包括塑性应变率 和应变硬化 。简而言之，学习算法是通过机器学习（ML）实现以下映射的函数
Note that , in contrast to have an explicit expression in terms of its variables, represents a black-box function embedded in the PDML algorithm. The learned will be used for stress update, and is needed for the consistence condition on the yield surface. The plastic flow in Li-metal (see Eq. 3) is multi-factor influenced and highly nonlinear. Given there are only a small amount of experimental data, we choose the tree-based ML algorithm over other ML algorithms (e.g., the support vector machine, the tree-based ML algorithm, and the artificial neural network). A complete description about the learning algorithm is supplied in Appendix A.
需要注意的是， 并不是变量的明确表达式，而是 PDML 算法中嵌入的黑盒函数。学习到的 将用于应力更新，而 则需要用于屈服面上的一致性条件。锂金属中的塑性流动（见公式 3）受多因素影响，具有高度非线性。鉴于只有少量实验数据，我们选择了基于树的 ML 算法，而不是其他 ML 算法（如支持向量机、基于树的 ML 算法和人工神经网络）。关于学习算法的完整描述见附录 A。
Fig. 3 b gives the iterative process to ensure convergence. The numerical procedure composes of standard elastic predictor and visco-plastic updating. From time to , an initial predictor, by assuming the incremental strain being elastic, we obtain a trial
图 3 b 给出了确保收敛的迭代过程。数值计算过程包括标准弹性预测和粘弹性更新。从时间 到 的初始预测器，假设增量应变是弹性的，我们会得到一个试验结果
Fig. 4. Predictability of the conventional model and the PD-ML model on relation at different temperatures. (a) Traditional model vs. experiments. (b) to (d) Trained and predicted results from PD-ML model vs experiments: (b) The predicted curves at temperatures ( 348 and 398 K ) higher than those of the trained ones. (c) The predicted curves at temperatures ( 273 and 298 K ) within the trained temperature range. (d) The predicted curves at temperatures ( 198 and 248 K ) lower than those of the trained ones.
图 4.不同温度下传统模型和 PD-ML 模型对 关系的可预测性。(a) 传统模型与实验对比。(b) 至 (d) PD-ML 模型与实验的训练和预测结果：(b) 温度（348 和 398 K）下的预测曲线高于训练曲线。(c) 温度 ( 273 和 298 K ) 在训练温度范围内的预测曲线。(d) 温度 ( 198 和 248 K ) 比训练曲线低时的预测曲线。
elastic strain at current time step,
当前时间步的弹性应变 、
and update the trial stress as
并将试验应力 更新为
where is the four order elastic constant tensor. Once visco-plastic deformation involves, the implicit return-mapping PD-ML algorithm takes over to update and . With being the first guess, we proceed with an iterative process by taking the current von Mises stress , equivalent strain , temperature , and equivalent strain rate as inputs to the PD-ML and update and , see Fig. 3b for illustration. The standard associative flow rule in Eq. (1) is then used to update the plastic strain rate . Sequentially, we may write the current elastic strain and stress tensor as
其中 是四阶弹性常数张量。一旦发生粘弹性变形，隐式返回映射 PD-ML 算法就会接手更新 和 。以 为第一猜测值，我们将当前的冯米塞斯应力 、等效应变 、温度 和等效应变率 作为 PD-ML 的输入，并更新 和 ，从而进行迭代过程，见图 3b 的说明。然后使用公式 (1) 中的标准关联流规则更新塑性应变率 。依次，我们可以将当前的弹性应变和应力张量写成
and 和
Fig. 5. Deformation map of Li-metal. (a) vs. for a wide range of temperatures. (b) A three-dimensional diagram to show the temperature , stress , and plastic flow relationship. It covers a wide temperature span from 173 to 423 K .
图 5.锂金属的变形图。(a) 各种温度下 与 的关系。(b) 显示温度 、应力 和塑性流动 关系的三维图。它涵盖了 173 至 423 K 的宽温度跨度。
respectively. The iteration continues till satisfying the following criterion. The difference of between this step and the previous one, , is infinitesimal, i.e., for being a small tolerance. With known , we may proceed to update strains at time , and before the next time increment.
分别为迭代一直进行到 满足以下条件为止。这一步与上一步 之间的 差值为无穷小，即 对于 是一个小公差。在已知 的情况下，我们可以在 和 时间更新应变，然后再进行下一次增量。

3. Results with the PD-ML model
3.PD-ML 模型的结果

We now apply both the conventional model and the PD-ML model to the viscoplastic deformation of Li-metal, and demonstrate their predictability in comparison with available experimental data. Both temperature and strain rate factors are taken into account in the two types of models.
现在，我们将传统模型和 PD-ML 模型应用于锂金属的粘塑性变形，并通过与现有实验数据的对比证明了它们的可预测性。两种模型都考虑了温度和应变率因素。

3.1. Temperature-dependence
3.1.温度依赖性

As seen in Fig. 4, we first show the normalized plastic strain rate as a function of stress from experiments at six temperatures (Fig. 4a). It is seen that increases slowly under low stress, followed by a rapid ascending at intermediate to high stress. At the late stage, plastic strain dominates and approaches to the applied strain rate , i.e., . There is a strong nonlinear dependence of and on temperature. The traditional power law model with Eq. (2) assumes intrinsically a linear relationship in the