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Article(id=1246028560700387521, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246028557319783390, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2023.053, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1698163200000, receivedDateStr=2023-10-25, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775005958962, onlineDateStr=2026-04-01, pubDate=1719244800000, pubDateStr=2024-06-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775005958962, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775005958962, creator=13701087609, updateTime=1775005958962, updator=13701087609, issue=Issue{id=1246028557319783390, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='3', pageStart='289', pageEnd='426', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775005958156, creator=13701087609, updateTime=1775006058227, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246028977123471371, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246028557319783390, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246028977123471372, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246028557319783390, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=302, endPage=312, ext={EN=ArticleExt(id=1246028560973017288, articleId=1246028560700387521, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=A Neural Network-assisted Theoretical Constitutive Model to Predict the High Temperature Flow Behavior of High-entropy Alloys, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

Metals and alloys are widely used in industry due to their excellent mechanical properties. Researchers have been continuously searching new materials with better properties or mechanisms to enhance existing ones. In the metal and alloy forming process, hot deformation can effectively refine the grain and improve mechanical properties such as yield strength and tensile strength. Therefore, it is necessary to study the deformation behavior of metal and alloy materials at high temperatures. The hyperbolic-sinusoidal Arrhenius-type model has been widely used by researchers because of its good simulation effect at high temperatures. In this paper, the building process of the model is studied, and the modeling process is optimized with the help of a neural network model. A neural network model is constructed to efficiently determine the hyperbolic-sinusoidal Arrhenius-type equations, based on which the flow stress of high-entropy alloys (HEAs) for different high temperatures and strain rates can be well predicted. The reported hot deformation behaviors of Al0.3CoCrFeNi HEAs are examined by current model. The results show that the coefficients obtained by the neural network method can better describe the experimental hot flow stress, especially at high strain rate or low temperature conditions. The root-mean-square error (RMSE) and the correlation coefficient R are used to assess the degree of difference between the results. The RMSE and R of the neural network method at total data are 27.7 and 0.985, respectively, which are better than 33.1 and 0.979 of the traditional method. To show the general applicability of the model, the hot deformation behaviors of (CoCrNi)94Ti3Al3, FeCrCuNi2Mn2, and AlCrCuFeNi are analyzed by the model. The research work presented in this paper can improve the efficiency and accuracy of the hyperbolic-sinusoidal Arrhenius-type model and reduce the difficulty of establishing the model, and is of positive significance for the wide use of the model.

, correspAuthors=Miaolin Feng, authorNote=null, correspAuthorsNote=null, copyrightStatement=null, copyrightOwner=null, extLink=null, articleAbsUrl=null, sourceXml=null, magXml=null, pdfUrl=null, pdf=null, pdfFileSize=null, pdfExtLink=null, richHtmlUrl=null, mobilePdfUrl=null, reviewReport=null, pdfFirstPage=null, abstractGraph=null, abstractGraphContent=null, abstractVideo=null, citation=null, cebUrl=null, magXmlContent=null, mapNumber=null, authorCompany=null, fund=null, authors=null, authorsList=Jian Jiang, Tao Hu, Sanshao Zhuang, Miaolin Feng), CN=ArticleExt(id=1246028567033786737, articleId=1246028560700387521, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=神经网络辅助理论本构模型预测高熵合金高温流动应力行为, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

本文建立了一套确定双曲正弦Arrhenius型方程系数的神经网络模型，选取了高熵合金不同高温和应变速率下的流动应力预测来验证模型. 首先采用Al0.3CoCrFeNi高熵合金进行检验，并与传统方法进行比较，结果表明，神经网络方法在高应变速率和低温条件下得到的系数能够更好地描述试验热流应力. 进一步采用均方根误差（RMSE）和相关系数（R）对模型结果和试验结果进行评估，神经网络方法在整体数据下的RMSE和R分别为27.7和0.985，优于传统方法的33.1和0.979. 最后，利用该神经网络模型研究了其他高熵合金，如（CoCrNi）94 Ti3Al3、FeCrCuNi2Mn2和AlCrCuFeNi的热变形行为，神经网络预测结果与试验结果吻合好，表明该神经网络模型具有较好的普遍适用性.

, correspAuthors=冯淼林, authorNote=null, correspAuthorsNote=
** E-mail：.
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language=EN, label=Fig.10, caption=Predicted flow stress-strain curves vs. experimental true stress-strain curves for FeCrCuNi2Mn2 at these temperatures, figureFileSmall=yg3SvZi3x1eht3mR6PFwAw==, figureFileBig=vJdC8rQ1nL7nJo4IsMHwvA==, tableContent=null), ArticleFig(id=1246028571395863081, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028560700387521, language=CN, label=图10, caption=FeCrCuNi2Mn2在不同温度下的应力-应变和相应的预测结果, figureFileSmall=yg3SvZi3x1eht3mR6PFwAw==, figureFileBig=vJdC8rQ1nL7nJo4IsMHwvA==, tableContent=null), ArticleFig(id=1246028571496526383, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028560700387521, language=EN, label=Fig.11, caption=Predicted flow stress-strain curves vs. experimental true stress-strain curves for AlCrCuFeNi at these temperatures, figureFileSmall=/w5Wik+9rhdjAg7FclGNqw==, figureFileBig=3VsUM9n/XDrXQl/3pgnV8g==, tableContent=null), ArticleFig(id=1246028571580412470, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028560700387521, language=CN, label=图11, caption=AlCrCuFeNi在不同温度下的应力-应变和相应的预测结果, figureFileSmall=/w5Wik+9rhdjAg7FclGNqw==, figureFileBig=3VsUM9n/XDrXQl/3pgnV8g==, tableContent=null), ArticleFig(id=1246028571664298552, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028560700387521, language=EN, label=Table 1, caption=

RMSE and R calculated by neural network method and traditional method

, figureFileSmall=null, figureFileBig=null, tableContent=
方法RMSER
训练集验证集总数据训练集验证集总数据
神经网络方法27.128.927.70.9850.9850.985
传统方法（平均斜率法）——33.1——0.979
), ArticleFig(id=1246028571769156159, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246028560700387521, language=CN, label=表1, caption=

神经网络方法和传统方法应力预测结果与试验值的RMSE和R

, figureFileSmall=null, figureFileBig=null, tableContent=
方法RMSER
训练集验证集总数据训练集验证集总数据
神经网络方法27.128.927.70.9850.9850.985
传统方法（平均斜率法）——33.1——0.979
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神经网络辅助理论本构模型预测高熵合金高温流动应力行为
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姜健 , 胡涛 , 庄三少 , 冯淼林 **
固体力学学报 | 研究论文 2024,45(3): 302-312
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固体力学学报 | 研究论文 2024, 45(3): 302-312
神经网络辅助理论本构模型预测高熵合金高温流动应力行为
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姜健, 胡涛, 庄三少, 冯淼林**
作者信息
  • 上海交通大学船舶海洋与建筑工程学院工程力学系（海洋工程国家重点实验室），上海，200240

通讯作者:

** E-mail：.
A Neural Network-assisted Theoretical Constitutive Model to Predict the High Temperature Flow Behavior of High-entropy Alloys
Jian Jiang, Tao Hu, Sanshao Zhuang, Miaolin Feng**
Affiliations
  • State Key Laboratory of Ocean Engineering, Department of Engineering Mechanics, School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, 200240
出版时间: 2024-06-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.053
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摘要
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本文建立了一套确定双曲正弦Arrhenius型方程系数的神经网络模型，选取了高熵合金不同高温和应变速率下的流动应力预测来验证模型. 首先采用Al0.3CoCrFeNi高熵合金进行检验，并与传统方法进行比较，结果表明，神经网络方法在高应变速率和低温条件下得到的系数能够更好地描述试验热流应力. 进一步采用均方根误差（RMSE）和相关系数（R）对模型结果和试验结果进行评估，神经网络方法在整体数据下的RMSE和R分别为27.7和0.985，优于传统方法的33.1和0.979. 最后，利用该神经网络模型研究了其他高熵合金，如（CoCrNi）94 Ti3Al3、FeCrCuNi2Mn2和AlCrCuFeNi的热变形行为，神经网络预测结果与试验结果吻合好，表明该神经网络模型具有较好的普遍适用性.

高熵合金  /  高温变形  /  神经网络  /  本构方程

Metals and alloys are widely used in industry due to their excellent mechanical properties. Researchers have been continuously searching new materials with better properties or mechanisms to enhance existing ones. In the metal and alloy forming process, hot deformation can effectively refine the grain and improve mechanical properties such as yield strength and tensile strength. Therefore, it is necessary to study the deformation behavior of metal and alloy materials at high temperatures. The hyperbolic-sinusoidal Arrhenius-type model has been widely used by researchers because of its good simulation effect at high temperatures. In this paper, the building process of the model is studied, and the modeling process is optimized with the help of a neural network model. A neural network model is constructed to efficiently determine the hyperbolic-sinusoidal Arrhenius-type equations, based on which the flow stress of high-entropy alloys (HEAs) for different high temperatures and strain rates can be well predicted. The reported hot deformation behaviors of Al0.3CoCrFeNi HEAs are examined by current model. The results show that the coefficients obtained by the neural network method can better describe the experimental hot flow stress, especially at high strain rate or low temperature conditions. The root-mean-square error (RMSE) and the correlation coefficient R are used to assess the degree of difference between the results. The RMSE and R of the neural network method at total data are 27.7 and 0.985, respectively, which are better than 33.1 and 0.979 of the traditional method. To show the general applicability of the model, the hot deformation behaviors of (CoCrNi)94Ti3Al3, FeCrCuNi2Mn2, and AlCrCuFeNi are analyzed by the model. The research work presented in this paper can improve the efficiency and accuracy of the hyperbolic-sinusoidal Arrhenius-type model and reduce the difficulty of establishing the model, and is of positive significance for the wide use of the model.

high-entropy alloys  /  high-temperature deformation  /  neural network  /  constitutive equation
姜健, 胡涛, 庄三少, 冯淼林. 神经网络辅助理论本构模型预测高熵合金高温流动应力行为. 固体力学学报, 2024 , 45 (3) : 302 -312 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.053
Jian Jiang, Tao Hu, Sanshao Zhuang, Miaolin Feng. A Neural Network-assisted Theoretical Constitutive Model to Predict the High Temperature Flow Behavior of High-entropy Alloys[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (3) : 302 -312 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2023.053
正文
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0　引言
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Yeh[1]和Cantor[2]于2004年将包含5种或以上原子比例相等或接近相等的主元元素组成的合金定义为高熵合金（HEAs）. 相较于传统合金，高熵合金具有高强度[3]、高硬度[4]、高温稳定性[5]和耐腐蚀[6,7]等优异性能. 因此，高熵合金在科学研究及工业应用领域受到了广泛关注[8].
金属材料在再结晶温度以上的塑性变形被称作热变形. 在金属材料成型过程中，热变形可以有效地细化晶粒，从而提高母材的屈服强度和抗拉强度等力学性能[9,10]. 因此，有必要研究金属材料在高温下的变形行为，尤其是高温变形过程中流动应力的演化. 众多学者研究了高熵合金流动行为，并构建了一系列本构模型，主要包括Zelili-Armstrong模型[11]、Johnson-Cook模型[12]和双曲正弦Arrhenius模型[13]等. Brown[14]研究了CoCrFeMnNi合金应变速率为0.01～1 s-1，温度为200-800 ℃的热压缩行为，采用上述的本构模型分别进行应力预测，结果表明，在高温条件下，双曲正弦Arrhenius模型的预测效果最理想. Patnamsetty[15]采用双曲正弦Arrhenius模型研究了Al0.3CoCrFeNi在1023～1423 K温度范围和10-3～10 s-1应变速率范围内的流动应力，结果表明该模型能较好地描述材料在实验条件下的热变形行为，但在高应变速率和低温条件下，预测应力与试验应力存在较大差异.
近年来，随着计算机技术的普及和发展，机器学习方法在科学研究领域得到了广泛的应用，其中人工神经网络是得到广泛应用的模型之一. 人工神经网络可以通过机器学习大量的试验数据来建立或优化的材料本构模型，已经被用于复合材料本构建模[16]和金属材料的J2塑性本构优化[17]等. 人工神经网络模型也被直接用于预测流动应力[18,19]，通过输入应变、温度和应变速率直接输出流动应力，结果较为准确，但神经网络的泛化能力尚未得到充分的讨论和认可.
本文通过人工神经网络模型来确定双曲正弦Arrhenius型模型的相关系数，从而来计算材料的流动应力. 通过一些公开的高熵合金热变形数据，预测了不同高温和应变速率下材料的流动应力，并将该模型得到的流动应力与传统计算系数方法得到的结果进行了比较，验证了模型的准确性和适用性.
1　方法
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1.1　本构方程
双曲正弦Arrhenius模型是由Sellars和McTegart[13]在Zener和Hollomon[20]工作的基础上提出的一种半经验本构模型. 该模型认为材料的流动应力（σ）满足如下方程：
其中A和n是材料常数. Zener-Hollomon参数Z为温度补偿的应变率因子，可以表示为：
其中Q为热变形的表观活化能，为应变速率，R为通用气体常数. 式（1）中α为调节应力参数（MPa-1），可以表示为，
其中β和n1为应力指数[20-22]. 上式中的β和n1可以通过下式确定：
方程（4）和（5）分别适用于低应力（ασ<0.8）和高应力（ασ>1.2）的材料流动应力计算. 双曲正弦函数方程（1）不仅适用于低应力也适用于高应力，本文将采用双曲正弦函数来进行高熵合金材料的热变形行为研究.
为计算这些方程中的系数，传统的方法是将式（2）代入式（4）（5）后取对数将方程改写为：
由式（6）和（7）可知，分别线性拟合不同温度下的和lnσ以及和σ可得到斜率n1和β的值，然后可以通过式（3）计算得到α. 进一步将式（2）代入式（1）取对数将方程改写为：
由上式可知，分别线性拟合不同温度下的和ln[sinh（ασ）]可得到式（8）中的斜率n，在不同应变率下线性拟合ln[sinh（ασ）]和1/T可得到斜率Q/nR. 最后直接对式（1）取对数将方程改写为：
根据式（9），通过线性拟合lnZ和ln[sinh（ασ）]可得到A的值.
通过上述步骤得到系数后，可由下式进行流动应力计算，该式由（1）和（2）式转换而来：
1.2　人工神经网络
本文构建了一个具有两个隐藏层的全连接神经网络，用于确定双曲正弦Arrhenius型模型系数与应变之间的关系. 所使用的神经网络示意图如图1所示，网络的输入为应变，输出为模型的系数，神经网络的隐藏层可以增加网络的复杂程度，考虑到传统方法常用四阶或以上多项式拟合系数与应变的关系，我们选择加入两层隐藏层，每层均含有10个神经元，激活函数选择Elu函数，优化器选择常用的Adam优化器. 对于反向传播过程所需的损失函数，我们将神经网络输出的系数与温度及应变速率相结合，计算双曲正弦Arrhenius模型的应力，然后与实际应力进行比较，将模型得到的应力和实际应力的均方根误差（RMSE）作为损失函数. 整个过程如图2所示. 我们将使用三个结构相同的神经网络来完成我们的工作. 神经网络1和2分别根据式（4）和（5）预测n1和β，则α可由式（3）确定. 网络3根据得到的α和式（1）预测其他系数A、Q和n.
2　结果与讨论
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2.1　双曲正弦Arrhenius型模型系数
为了验证我们的模型，我们使用Al0.3CoCrFe-Ni高熵合金在热变形下的应力-应变数据，采用我们的方法计算本构模型的系数，并与传统方法（平均斜率法）进行比较. 训练和验证的数据由图4、图5中实线表示的试验曲线中取点得到. 训练集含有温度为1023、1123、1173、1273、1323和1423 K的试验曲线中的1614个数据点，验证集含有温度为1073、1223和1373 K的试验曲线中的869个数据点，验证集数据占总体数据的35%.
采用神经网络方法和传统方法得到的Al0.3 CoCrFeNi双曲正弦Arrhenius型模型系数如图3所示. 两种方法得到的系数数量级接近，但具体数值存在差异，这是由于传统方法在计算过程中存在误差造成的. 传统求系数方法需要计算拟合直线的平均斜率，但获得这些拟合直线的数据点较少，且在不同温度或不同应变速率下斜率变化较大. Sajadi[23]采用双曲正弦Arrhenius型模型预测了FeCrCu-Ni2Mn2的流动应力，他们利用不同温度下四条拟合直线的斜率平均值计算β值时，4条斜率值分别为0.067、0.10、0.028和0.019，平均值为0.054. 可以看出，原始斜率值相差很大，平均值确定系数是一种粗略的近似方法. 神经网络选择系数计算的应力与试验应力的差值作为损失函数，参数训练以损失最小为目标，使系数结果能更好地描述试验应力（见3.2节）. 利用我们建立的神经网络来计算系数是简单高效的，对应力-应变关系及其对应的温度和应变速率进行训练后，可以直接给出系数与应变的关系来描述流动应力.
2.2　流动应力预测
将神经网络得到的系数代入双曲正弦Arrhenius型模型中，即可计算出预测应力. 利用3.1节中得到的系数进行应力计算，并对神经网络方法和传统方法得到的结果进行分析.
图4和图5为应力-应变预测结果（虚线）与试验结果（实线）的对比，其中（a1）-（i1）为传统方法预测的结果，（a2）-（i2）为神经网络方法预测结果. 从图4可以看出，神经网络方法得到的应力-应变曲线更接近试验的应力-应变曲线. 传统方法对高应变速率（10 s-1）下的流动应力预测偏差较大，特别是在1023-1123 K较低温度区域. 从图5可以看出，在没有神经网络训练的温度下，预测的应力-应变曲线与试验应力-应变曲线吻合较好，预测曲线甚至优于传统方法得到的曲线，尤其是在低温下（图5（g1）-（g2））. 结果表明，神经网络辅助的双曲正弦Arrhenius型模型能够很好地描述未训练温度下的热流动应力，具有较好的泛化能力. 两种方法的试验应力与预测应力的关系如图6所示. 可以看出，机器学习方法得到的结果明显更接近最佳拟合曲线，特别是在大于500 MPa的高应力下. 神经网络训练过程的损失曲线如图7所示，其中第一、二、三个神经网络分别对应方程（4）的幂指数型本构、方程（5）的指数型本构和方程（1）的双曲正弦型本构模型. 可以看出三个神经网络模型的损失曲线随着训练过程趋向稳定，其中双曲正弦型模型的表现最好.
采用均方根误差（RMSE）和相关系数（R）来评价预测值与试验值之间的差异程度，其定义如下：
式中，E为试验流动应力，P为预测的流动应力，N为选取的数据量，和分别为E和P的平均值.
表1给出了传统方法和神经网络方法预测的应力结果的RMSE和R值. 神经网络方法在总数据集下的RMSE和R分别为27.7和0.985，优于传统方法的33.1和0.979，这表明通过神经网络方法得到的系数能够较好地描述试验应力-应变关系. 验证数据的RMSE和R分别为28.9和0.985，反映了神经网络辅助模型的泛化能力和准确性.
从上述结果分析可以发现，对神经网络进行训练，得到理论模型的系数，神经网络在这里只是一个可以优化的函数生成器. 如果撇开本构理论模型，而直接使用神经网络输出流动应力，则需要对训练好的神经网络的适用条件和泛化能力进行充分验证，避免神经网络通常对结果的过拟合，难以保证其在未知区域的预测效果. 本文采用的神经网络辅助本构理论模型，克服了直接使用神经网络预测流动应力的过拟合缺陷，其结果的泛化能力由理论模型决定.
2.3　模型的普遍适用性
对于任意给定的高熵合金热变形数据，我们建立的模型应该能够直接对模型的系数进行分析，而不需要调整神经网络的超参数. 为了表明该模型的普遍适用性，我们采用该模型分析了（CoCrNi）94 Ti3Al3[8]、FeCrCuNi2Mn2[23]和AlCrCuFeNi[24]的热变形行为，预测应力与试验应力的关系如图8所示. 图9，图10，图11分别展示了在不同温度、应变率的应力应变曲线和相应的神经网络预测结果.
这三种合金数据中包含的温度和应变率均比上节采用的Al0.3CoCrFeNi高熵合金数据少. 但模型在预测应力方面的表现仍然良好，其与试验结果的RMSE值分别为12.4、19.9和10.0，与试验数据吻合较好. 预测结果和试验数据的最小二乘拟合R分别为0.982、0.97和0.98，表明预测的流动应力与试验数据具有较好的相关性. 结果表明，神经网络辅助模型满足描述不同合金热变形行为的要求.
3　结论
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本文建立了神经网络辅助双曲正弦Arrhenius型模型来确定高熵合金在高温下的流动应力. 主要结论如下：
（1）神经网络辅助双曲正弦Arrhenius型模型能较好地模拟试验流动应力，特别是在高应变速率及低温条件下. 神经网络方法对总数据的RMSE和R分别为27.7和0.985，优于传统方法的33.1和0.979.
（2）神经网络辅助双曲正弦Arrhenius型模型得到的系数能够较好地预测未训练温度下的应力，验证集的RMSE和R分别为28.9和0.985，表明该方法具有较好的泛化能力.
（3）神经网络辅助双曲正弦Arrhenius型模型具有良好的普遍适用性，分析得到（CoCrNi）94 Ti3Al3、FeCrCuNi2Mn2和AlCrCuFeNi的R值分别为0.982、0.97和0.98.
基金
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  • 国家自然科学基金项目(U2067220; 52371284)
  • 中国核工业集团领创科研项目
文献
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2024年第45卷第3期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2023.053
  • 接收时间：2023-10-25
  • 首发时间：2026-04-01
  • 出版时间：2024-06-25
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  • 收稿日期：2023-10-25
基金
国家自然科学基金项目(U2067220; 52371284)
中国核工业集团领创科研项目
作者信息
    上海交通大学船舶海洋与建筑工程学院工程力学系（海洋工程国家重点实验室），上海，200240

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2种不同金属材料的力学参数
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鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
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