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Article(id=1246045605819167224, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246045605047415288, articleNumber=null, orderNo=null, doi=10.19636/j.cnki.cjsm42-1250/o3.2024.018, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1714406400000, receivedDateStr=2024-04-30, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1775010022835, onlineDateStr=2026-04-01, pubDate=1729785600000, pubDateStr=2024-10-25, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1775010022835, onlineIssueDateStr=2026-04-01, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1775010022835, creator=13701087609, updateTime=1775010022835, updator=13701087609, issue=Issue{id=1246045605047415288, tenantId=1146029695717560320, journalId=1241755870837649424, year='2024', volume='45', issue='5', pageStart='565', pageEnd='708', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1775010022651, creator=13701087609, updateTime=1775010094858, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1246045907972636933, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246045605047415288, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1246045907972636934, tenantId=1146029695717560320, journalId=1241755870837649424, issueId=1246045605047415288, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=576, endPage=586, ext={EN=ArticleExt(id=1246045606116962810, articleId=1246045605819167224, tenantId=1146029695717560320, journalId=1241755870837649424, language=EN, title=Research on the Mechanism of Transition from Static to Dynamic Microscale Friction Behavior, columnId=1244229834482757770, journalTitle=Chinese Journal of Solid Mechanics, columnName=Research Paper, runingTitle=null, highlight=null, articleAbstract=

Microscale contact and friction behavior are widely present in various important industrial devices and systems. As electromechanical systems become more integrated and miniaturized, the impact of friction on devices becomes increasingly important. At the microscale, friction behavior exhibits a strong dependence on interfacial viscosity and contact size. By developing a series of modifiable potential functions to quantitatively regulate interfacial properties, friction on atomically smooth interfaces with different properties is fully simulated using molecular dynamics methods. The study first examined the influence of various interfacial potential energies on the static friction coefficient, revealing a nonlinear relationship between the static friction coefficient and interfacial potential energy intensity. Furthermore, it was found that this nonlinearity is attributed to the competition between interfacial viscosity and contact stiffness. Additionally, the study investigated the influence of contact size on static friction coefficient. The simulation results showed that as the tangential contact length of the interface increases, the peak static friction force first increases and then stabilizes. By analyzing the contact layer cloud maps obtained through post-processing, interfacial friction is observed as a “nucleation-propagation” process, influenced by different contact sizes which affect the dynamic process and lead to changes in the peak static friction force. This study provides new insights into the effects of interfacial potential energy and contact size on microscale friction through molecular dynamics simulations, it is feasible to regulate friction by changing interfacial potential energy, but attention should be paid to the nonlinear changes in the friction coefficient. Besides, solely increasing the contact size cannot infinitely increase the peak static friction force.

, correspAuthors=Xiaoming Liu, 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=Xin Wu, Jianqiao Hu, Xiaoming Liu), CN=ArticleExt(id=1246045613956117145, articleId=1246045605819167224, tenantId=1146029695717560320, journalId=1241755870837649424, language=CN, title=微尺度静动摩擦转变行为的机理研究, columnId=1241831201896469478, journalTitle=固体力学学报, columnName=研究论文, runingTitle=null, highlight=null, articleAbstract=

微尺度接触与摩擦行为广泛存在于各种重要工业器件与系统中，随着各类机电系统高集成化和微小型化的发展，摩擦对器件的影响越发不可忽视. 在微尺度下，摩擦行为表现出对界面粘性与接触尺寸的强依赖性，认识微尺度下静动摩擦转变对控制摩擦、减小能耗具有重要的意义. 本研究工作采用分子动力学方法对微尺度摩擦行为的静动转变过程开展了研究，通过发展一系列可修正势函数来定量调控摩擦界面物性，进而阐明了界面粘性与界面接触刚度对静摩擦系数的影响规律并揭示了两者的竞争机制. 此外，本工作还研究了微尺度下临界静摩擦系数的尺寸效应，观察到了临界静摩擦力的饱和现象并通过接触层云图揭示了饱和现象的微观机制.

, correspAuthors=刘小明, authorNote=null, correspAuthorsNote=
** E-mail：.
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From concurrent slip to single-dislocation-assisted slip, refAbstract=null)], funds=[Fund(id=1246045620176270217, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, awardId=12022210; 12032001, language=CN, fundingSource=国家自然科学基金项目(12022210; 12032001), fundOrder=null, country=null), Fund(id=1246045620289516429, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, awardId=2023QNRC001, language=CN, fundingSource=青年托举项目(2023QNRC001), fundOrder=null, country=null)], companyList=[AuthorCompany(id=1246045614237135527, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, xref=1, ext=[AuthorCompanyExt(id=1246045614245524138, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, companyId=1246045614237135527, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190), AuthorCompanyExt(id=1246045614253912745, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, companyId=1246045614237135527, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=1中国科学院力学研究所非线性力学国家重点实验室，北京，100190)]), AuthorCompany(id=1246045614367158959, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, xref=2, ext=[AuthorCompanyExt(id=1246045614371353264, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, companyId=1246045614367158959, language=EN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2School of Engineering Science, University of Chinese Academy of Sciences, Beijing, 100049), AuthorCompanyExt(id=1246045614379741873, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, companyId=1246045614367158959, language=CN, country=null, province=null, city=null, postcode=null, companyName=null, departmentName=null, remark=2中国科学院大学工程科学学院，北京，100049)])], figs=[ArticleFig(id=1246045616476893979, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, language=EN, label=Fig.1, caption=(a) Schematic illustration of simulation model. The bright blue part represents the lateral force Fx-apply loading area. When studying the effect of interfacial potential energy, it is loaded in the left end and middle of the block, respectively. When studying the effect of model's length, it is only loaded in the middle of the block; (b) Schematic illustration of model, obtained from OVITO post-processing software[35], figureFileSmall=EruCbWf/GBt8toIGMON9Vw==, figureFileBig=XO4t/9IWJbXHn/0dospWUw==, tableContent=null), ArticleFig(id=1246045616564974371, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, language=CN, label=图1, caption=（a）模型示意图，亮蓝色部分为推力Fx-apply加载区域，当研究界面势能影响时分别在块体左端与中部加载，当讨论模型长度影响时，只在块体中部加载；（b）分子动力学模型图[35], figureFileSmall=EruCbWf/GBt8toIGMON9Vw==, figureFileBig=XO4t/9IWJbXHn/0dospWUw==, tableContent=null), ArticleFig(id=1246045616783078190, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, language=EN, label=Fig.2, caption=Simulation results of changing interface potential energy and viscosity. The horizontal axis is the ratio of U to U0, and the vertical axis is the static friction coefficientμs. 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(a)-(c) Shear stress cloud maps; (d)-(f) Stress ratio cloud maps, figureFileSmall=cFqLNlIpvl2vsfCq4dd2aw==, figureFileBig=9kIxYCSeJYXP5beO9KYoEA==, tableContent=null), ArticleFig(id=1246045619635204979, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, language=CN, label=图8, caption=U为0.001U0、L分别为50a、200a、800a模型的接触层云图，横轴代表模拟时间，纵轴代表模型长度，色温越高代表数值越大. 黑色实线标记达到静摩擦极限Ff-static的时刻. 圆圈标记或三角标记代表该时刻的接触层上的极值（圆圈标记对应x轴正方向传播，三角标记对应x轴负方向传播）.（a）-（c）切应力云图；（d）-（f）应力比云图, figureFileSmall=cFqLNlIpvl2vsfCq4dd2aw==, figureFileBig=9kIxYCSeJYXP5beO9KYoEA==, tableContent=null), ArticleFig(id=1246045619752645496, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, language=EN, label=Fig.9, caption=The shear stress peak and stress ratio peak propagation speed of the contact layer in the model with U=0.001U0. x+(-) represents the positive (negative) direction of x axis, figureFileSmall=36HYwg9jvruxV7RrKCMwpg==, figureFileBig=A0xTOKpQR3xlT8MhHWGCPQ==, tableContent=null), ArticleFig(id=1246045619844920187, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, language=CN, label=图9, caption=U为0.001U0模型的接触层切应力极值与应力比极值传播速度. x+（-）代表x轴正向（负向）, figureFileSmall=36HYwg9jvruxV7RrKCMwpg==, figureFileBig=A0xTOKpQR3xlT8MhHWGCPQ==, tableContent=null), ArticleFig(id=1246045619941389182, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, language=EN, label=Table 1, caption=

Parameters for the simulation

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参数参数值
时间步长（t0）0.001
温度（εkB-1）0.001
Fx-apply加载率10-6
), ArticleFig(id=1246045620033663874, tenantId=1146029695717560320, journalId=1241755870837649424, articleId=1246045605819167224, language=CN, label=表1, caption=

模拟参数

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参数参数值
时间步长（t0）0.001
温度（εkB-1）0.001
Fx-apply加载率10-6
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微尺度静动摩擦转变行为的机理研究
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武鑫 1, 2 , 胡剑桥 1 , 刘小明 1, 2, **
固体力学学报 | 研究论文 2024,45(5): 576-586
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固体力学学报 | 研究论文 2024, 45(5): 576-586
微尺度静动摩擦转变行为的机理研究
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武鑫1, 2, 胡剑桥1, 刘小明1, 2, **
作者信息
  • 1中国科学院力学研究所非线性力学国家重点实验室，北京，100190
  • 2中国科学院大学工程科学学院，北京，100049

通讯作者:

Research on the Mechanism of Transition from Static to Dynamic Microscale Friction Behavior
Xin Wu1, 2, Jianqiao Hu1, Xiaoming Liu1, 2, **
Affiliations
  • 1State Key Laboratory of Nonlinear Mechanics, Institute of Mechanics, Chinese Academy of Sciences, Beijing, 100190
  • 2School of Engineering Science, University of Chinese Academy of Sciences, Beijing, 100049
出版时间: 2024-10-25 doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.018
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摘要
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微尺度接触与摩擦行为广泛存在于各种重要工业器件与系统中，随着各类机电系统高集成化和微小型化的发展，摩擦对器件的影响越发不可忽视. 在微尺度下，摩擦行为表现出对界面粘性与接触尺寸的强依赖性，认识微尺度下静动摩擦转变对控制摩擦、减小能耗具有重要的意义. 本研究工作采用分子动力学方法对微尺度摩擦行为的静动转变过程开展了研究，通过发展一系列可修正势函数来定量调控摩擦界面物性，进而阐明了界面粘性与界面接触刚度对静摩擦系数的影响规律并揭示了两者的竞争机制. 此外，本工作还研究了微尺度下临界静摩擦系数的尺寸效应，观察到了临界静摩擦力的饱和现象并通过接触层云图揭示了饱和现象的微观机制.

摩擦  /  原子级光滑界面  /  分子动力学  /  界面势能  /  模型长度

Microscale contact and friction behavior are widely present in various important industrial devices and systems. As electromechanical systems become more integrated and miniaturized, the impact of friction on devices becomes increasingly important. At the microscale, friction behavior exhibits a strong dependence on interfacial viscosity and contact size. By developing a series of modifiable potential functions to quantitatively regulate interfacial properties, friction on atomically smooth interfaces with different properties is fully simulated using molecular dynamics methods. The study first examined the influence of various interfacial potential energies on the static friction coefficient, revealing a nonlinear relationship between the static friction coefficient and interfacial potential energy intensity. Furthermore, it was found that this nonlinearity is attributed to the competition between interfacial viscosity and contact stiffness. Additionally, the study investigated the influence of contact size on static friction coefficient. The simulation results showed that as the tangential contact length of the interface increases, the peak static friction force first increases and then stabilizes. By analyzing the contact layer cloud maps obtained through post-processing, interfacial friction is observed as a “nucleation-propagation” process, influenced by different contact sizes which affect the dynamic process and lead to changes in the peak static friction force. This study provides new insights into the effects of interfacial potential energy and contact size on microscale friction through molecular dynamics simulations, it is feasible to regulate friction by changing interfacial potential energy, but attention should be paid to the nonlinear changes in the friction coefficient. Besides, solely increasing the contact size cannot infinitely increase the peak static friction force.

friction  /  atomically smooth surface  /  molecular dynamics  /  interfacial potential energy  /  model length
武鑫, 胡剑桥, 刘小明. 微尺度静动摩擦转变行为的机理研究. 固体力学学报, 2024 , 45 (5) : 576 -586 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.018
Xin Wu, Jianqiao Hu, Xiaoming Liu. Research on the Mechanism of Transition from Static to Dynamic Microscale Friction Behavior[J]. Chinese Journal of Solid Mechanics, 2024 , 45 (5) : 576 -586 . DOI: 10.19636/j.cnki.cjsm42-1250/o3.2024.018
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0　引言
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摩擦接触是普遍存在于日常生活和工业生产中的重要界面力学现象. 随着人类生活智能化水平的提高和微纳制造技术的不断发展，许多在环境、交通、健康、生物和航空航天等领域拥有重要潜在应用的新型微纳机电和传感器件不断涌现，由于器件尺寸急剧减小，材料比表面积迅速增大，摩擦和磨损问题成了许多微型器件设计和应用的突出瓶颈. 因此，理解并控制摩擦对提高设备可靠性、高效利用能源等方面有重要的意义[1].
经典库仑摩擦定律认为接触面积与摩擦系数无关，摩擦力与法向压力成正比. 库仑模型能描述很多宏观干摩擦现象，但由于实验手段的限制，相当长历史时间人类都无法清晰观察真实的摩擦界面，因此对摩擦的机理缺乏本质认识. 直到上世纪随着显微技术的发展，研究者发现宏观看似紧密接触的界面其实是由许多尺寸不同的微小突起（asperity）接触对构成的，因此真实接触面积远小于表观接触面积；Bowden和Tabor[2]指出了真实接触面积的重要性，成为摩擦领域继库仑摩擦定律之后又一个重要结论. 到本世纪初，Fineberg等[3]通过设计精巧的光弹性滑动实验发现宏观摩擦界面在滑动前的锋面传播现象，Fineberg等[4-6]还通过进一步的实验说明锋面传播与裂纹扩展的相似性，认为锋面传播动力学也能用断裂力学中裂纹面能量平衡的观点来类比研究. Rubinstein等[7]通过实验研究了界面上不同速度的波各自减少真实接触面积强弱程度；Maegawa等[8]实验探究了不均匀法向载荷对锋面传播的影响；Braun等[9]与Tr ø mborg等[10]通过小块-弹簧模型解释了锋面传播现象；Katano等[11]以锋面传播实验结果为基础，在摩擦系数表达式中添加锋面传播特征长度项，拟合得到新的摩擦规律；Tr ø mborg等[12]与Bar-Sinai等[13]对慢速锋面向快速锋面转变提出了不同的机理，前者认为是局部滑动后发生弛豫导致了转变，后者则认为速度强化机制导致了转变；Bonfanti等[14]与Barras等[15]使用断裂力学观点研究微观摩擦，与实验工作进行了联系. 以上工作全面地研究了锋面传播现象，讨论了宏观界面的摩擦动力学过程. 在微观摩擦领域，研究者主要使用分子动力学模拟与原子力显微镜实验两种手段进行研究，Yifei Mo等[16]使用模拟探究了如何定义原子级接触的真实接触面积；Qunyang Li等[17-19]使用原子力显微镜进行了一系列实验，探讨了摩擦的老化效应（ageing effect）与尺度、速度等对摩擦系数的影响；Jianping Gao等[20]类比气体系统，证明摩擦系统具有可预测的平均值，对库仑摩擦定律给出了微观解释. 以上工作在微观尺度考察了摩擦行为，提供了理解宏观摩擦行为的理论依据.
微观尺度下的摩擦行为与宏观摩擦在形式上相似：都能视为界面间发生了相对滑动. 但机制又非常不同，随着比表面积的减小，界面作用不可忽视. 在静动摩擦转变这方面，Fineberg等[21]通过实验验证了宏观界面静摩擦系数并不是一个常数，而与法向压力相关；Otsuki等[22,23]通过有限元模拟讨论了静摩擦系数随外载和模型尺寸的变化，建立了黏滑阶段的一维模型；Binquan Luan等[24]通过分子动力学模拟研究单个突起与不同粘性界面的接触，主要关注了滑动摩擦力的变化；Jianqiao Hu等[25]通过分子动力学模拟研究了不同尺寸的单个突起的摩擦情形，考察了突起尺寸、滑动速度、晶体取向等的影响；Zhang等[26]使用原子力显微镜探究了界面粘性对动摩擦的影响；Zhao Deng等[27]使用原子力显微镜研究石墨烯的摩擦，观察到了负摩擦系数；Pugno等[28]通过类比粘性接头的解，得到预测石墨烯/石墨烯界面摩擦力的公式，指出当模型在滑动方向足够长时，摩擦力会发生饱和. 这些工作充分体现了界面作用在微观尺度的重要影响与微观摩擦的独特性. 但它们也存在局限性，具体来说，这些研究往往只关注固定物性的摩擦界面，而忽略了不同界面基本物性对摩擦行为的重要影响. 这种局限性可能导致我们对摩擦现象的理解不够全面和深入[21-23、26,27].
如上所述，微观尺度下的静动摩擦整体转变过程还有待研究. 本文基于可调控的Morse势函数，采用分子动力学方法模拟了微观尺度不同物性以及不同接触尺寸的界面静动摩擦转变行为，阐明了界面物性造成的竞争效应与接触尺寸对摩擦动力学的影响.
1　模型与方法
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本文使用LAMMPS[29]进行分子动力学模拟，采用LJ单位制，能量基准单位为ε0、长度基准单位为r0、时间基准单位为t0. 模拟模型如图1所示，由两部分组成：刚性基底与弹性块体. 基底与块体是同种FCC晶体（晶格常数），晶体取向都为x-[100]，y-[010]，z-[001]，块体在x，y，z三个方向的尺寸分别为：L，15a，12a，模拟参数如表1所示. 记体作用势为U0，界面作用势为U，U等于U0乘上一个小于1的系数. U0是修正Morse势函数[30]. 其表达式为：
其中U0（r）是两个相距r的原子之间的势能，ε0是势能阱深度，r0是平衡距离，α0控制原子键刚度，截断半径rc是势能作用最远范围. 参数a0～a3是保证势能与作用力连续性的拟合参数. 使用该修正Morse势函数的优势在于能通过改变截断半径rc来定量调控材料和界面物性[31-34]. 当原子间距大于平衡距离1小于rc时就会互相吸引，这是界面上黏附力的来源，因此通过直接取rc等于1就能得到无黏附力界面（即无粘性界面）. 刚性基底尺寸足够大，厚度足够厚，其边界不会对块体的行为造成影响.
模拟过程中，设定模拟边界为pps，先对系统进行能量最小化，随后设定NVE系综，使用Langevin热浴控制系统温度为0.001ε/kB（kB是玻尔兹曼常量），得到稳定模型后，在块体的上半部分加载负z方向的恒定压强，此过程中系统保持平衡，随后从0开始线性加载x方向推力Fx-apply.
本文对不同界面势能U对摩擦的影响以及x方向长度L对摩擦的影响进行了研究. 在研究U影响时，固定模型长度L为100a，选取8种由弱到强的界面势能U（0.0005U0-0.01U0），计算有粘性情形与无粘性情形，并分别从块体左端与中部进行切向加载. 在研究L影响时，选取三种势能（0.001U0、0.01U0、0.1U0），L从30a到800a之间变化，只考虑粘性界面，从块体中部切向加载.
取块体与基体的x方向作用力为界面摩擦力Fx，Fx的峰值作为极限静摩擦力，在Fx突破极限静摩擦力后模拟停止. 取极限静摩擦力与法向载荷的比值为静摩擦系数μs. 将块体最底部厚度为2a的一层定义为接触层，通过Python后处理得到接触层不同位置的信息.
2　界面粘性与接触刚度的竞争机制
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本小节通过调控界面势能U与界面粘性，阐明了静摩擦系数随界面势能的非线性变化现象. 如图2所示，给出了不同界面势能下不同区域加载的静摩擦系数变化规律.
无论是左端加载还是中部加载，在界面势能强度较低时（U/U0≤0.005），静摩擦系数μs随界面势能强度增加而逐渐减小，且有无界面粘性不影响μs，表现出对界面粘性的弱依赖性；当界面势能强度较高时（U/U0>0.005），无粘性模型的μs几乎不再随界面势能强度变化，而有粘性模型的μs随着界面强度增加而不断增加，无粘性模型与有粘性模型μs的差别界面强度随着增加而越发显著. 图2说明静摩擦系数μs随界面势能U是非线性变化的.
图3是块体与基底平衡时的界面势能面示意图，由于块体与基底都是FCC晶体，其表面的势能面是周期的，当界面受压，块体与基底的势能面会互相嵌入一定距离，设平衡距离为h. 在U不变时，外界压力越大，h越大，势能面越难以发生侧向错动，摩擦力越大，当h达到界面势阱最深处时抗拒稳定势能面滑动的力最大，记为γ. 由于在弹性变形前提下，很难将界面压缩到势阱最深处，可以合理假设γ是常数.
由图3可知，此时摩擦力来源为两部分：势能面抵抗滑动的阻力与界面间吸引力（黏附力）. 势能面抵抗滑动阻力在形式上类似宏观界面几何互锁的影响，然而由于原子间排斥力相当大，在弹性前提下块体与基底之间是不可能有实际接触的. 所以界面上的“互锁”是势能面之间的互锁，可以用界面法向接触刚度来描述互锁深度. 在法向外载恒定时，界面势能U越小，法向接触刚度越小，平衡距离h（即互锁深度）越大，因此势能面抵抗滑动阻力越大，但粘性与U成比例，此时粘性提供黏附力变小. 还要注意，当U改变时，势能面形状会发生改变，本文忽略势能面形状变化影响，因为在弹性范围内模型所受法向外载并不大，势能面被压缩程度非常小.
因此，在平衡界面上摩擦力组成为：抵抗稳定势能面滑动的力λγ（界面法向接触刚度的影响）与抗拒剪切的黏附力η（粘性的影响）.
于是摩擦力可以写成：
则静摩擦系数μs为：
对于无粘性模型有：
其中λ是小于1的、与法向力相关的正数，η与势能成正比. 对于式（2），前人已经提出类似的摩擦力构成[20]，但没有指出接触刚度与粘性之间的竞争机制. 在我们的模型中，法向载荷不变，只改变了界面势能U. 当U很小时，黏附力η很小，摩擦力来源几乎就是λγ，μs约等于，当U缓慢增加，h减小，势能面滑动更容易，λ减小，因此μs会减小，这个阶段粘性几乎不起作用（U/U0≤0.005），接触刚度主导摩擦，因此图2中粘性模型与无粘性模型的μs一致. 当U继续增加到足够大（U/U0>0.005），界面刚度越来越大，平衡距离h稍作改变就能与法向载荷平衡，因此界面法向接触刚度的贡献λγ几乎不变，所以无粘性时μs随U增加几乎不变. 此时黏附力η随U增加，影响不可忽视，界面粘性开始主导摩擦，因此有粘性模型的μs会随U增加而显著增加.
下面分别考虑界面粘性与法向接触刚度的影响来继续论证上述竞争机制. 考虑粘性影响时，对块体施加沿z轴正向准静态增加的拉力，此时势能面错动不会阻碍拉开过程，即排除了界面法向接触刚度的影响，抵抗拉开过程的阻力就是块体与基底的吸引力，它全部由黏附力贡献. 在拉开过程中，吸引力会先增大后减小，用吸引力最大值表征黏附力，将其除以接触面积，就得到该界面势能下单位面积所受黏附力. 图4（a）是单位面积黏附力Fviscosity随U的变化，Fviscosity与U成线性关系. 在U/U0≤0.005时，Fviscosity的影响很小，可以忽略. 由于黏附力只与原子间距相关，与作用方向无关，因此Fviscosity能表征（2）式中粘性对摩擦力的贡献η.
图4（b）是取U为0.001U0，0.01U0，0.1U0，对无粘模型中部加载改变法向压强σzz（即法向外载FN）的模拟结果，此时摩擦系数等于，对所有势能，摩擦系数随着FN增加而减小. 假设γ不随外载变化，这是因为FN增加会使势能面平衡距离h增加，λ随之增加，但λ增加的幅度没有FN大，因此最终摩擦系数随着FN增加而减小. 对U为0.1U0模型，当增加FN足够大时，摩擦系数不再变化，这是因为界面势能已足够大，在压缩到一定程度时平衡距离h稍微改变就能与法向载荷平衡.
本文继续通过如下模拟来讨论势能面法向接触刚度的影响，同时说明γ与势能面形状不变的假设的合理性. 如下图所示.
使用界面势能U=0.001U0的模型来进行模拟，不考虑界面粘性，“VL”代表“Velocity Load”. 此时不加载法向力与切向外载，并限制块体在y、z方向运动，只允许块体在x方向运动；通过恒定速度加载，让块体向x轴正方向滑动. 由于块体在z方向上被严格限制，不允许发生任何运动，这相当于将势能面嵌入到势能阱的最深处，由于界面无黏附力，块体受到的摩擦力完全由法向接触刚度贡献，此时也就达到了势能面抵抗错动阻力的极值.
上图是无粘模型块体与基底不同间距时（间距分别为0.01、0.1、0.3、0.5、0.6、0.8、1）摩擦力随块体滑动位移变化情况. 首先，对于间距大于0.6的模型，块体所受摩擦力一直是0，这是因为模拟中取消了界面粘性，间距大于0.6时块体与基底之间没有吸引力，此时势能面几乎不被压缩，块体滑动不会受到任何阻力. 其次，对于间距小于0.6的模型，块体所受摩擦力完全一致，并表现出周期性（周期为），这是因为约束了块体z方向运动，只要势能面开始嵌入，滑动后就会嵌入到势阱最深处，与初始间距无关. 又由于块体与基底材料晶格常数，则势能面周期也为，这正是摩擦力的周期. 此时摩擦力数值非常大，这说明本文模型所加法向载荷远没有将势能面充分压缩，再次说明假设γ为常数并忽略势能面形状改变的影响是合理的. 还要注意，尽管改变U相当于同时改变了法向接触刚度与粘性，但由于在U比较大时势能面压缩程度非常轻微（因为我们施加的恒定法向外载一直在弹性范围内，相对来说很小），平衡距离h改变也微乎其微，因此法向接触刚度即使变化也不造成势能面压缩情况显著变化. 所以尽管法向接触刚度也在变化，但认为粘性主导了U比较大时的界面摩擦依然是合理的.
模拟结果揭示法向力的变化会影响静摩擦系数，这与Fineberg等[21]的宏观实验结果是一致的，对于原子级光滑界面的摩擦，晶体界面非公度接触时可能会出现结构超滑现象[36]，我们的模拟并不涉及此方面.
3　模型滑动方向的长度对摩擦的影响
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本小节通过选取不同界面势能，调控界面长度，观察到极限静摩擦力的饱和现象. 如图7所示，给出了不同界面势能下不同长度模型的极限静摩擦力与静摩擦系数变化规律.
图7（a）是极限静摩擦力Ff-static的变化，对三种界面势能，Ff-static随着L的增加首先会增加，L增加到一定程度（转变长度Lc）后Ff-static饱和，不再随L增加而变化. 其中0.001U0模型的转变长度Lc在200a附近，0.01U0与0.1U0转变长度在500a附近，说明U较小时Lc随着U增加而增加，当U足够大时Lc不再变化. 图7（b）是静摩擦系数μs的变化. 模拟中保持法向压强σzz不变，因此法向力FN随L线性增加，由于Ff-static增加幅度远不如FN增加幅度，因此μs随L增加不断减小. 通过对接触层进行详细分析，我们认为极限静摩擦力随接触长度饱和是界面摩擦动力学过程随接触长度的变化导致的.
图8展示了不同长度模型接触层切向应力与应力比云图，说明原子级光滑界面的静动摩擦转变是一个“局部形核-扩展”过程. 图8（a）-（c）是接触层切应力τxz云图，对于L较小的模型（L<50a），切应力的极值分布很分散，界面上大部分区域切应力都会增加，大部分界面都是局部滑动形核的区域. 对于L较大的模型（L≥200a），τxz极值只分布在模型中部，即加载区域附近，形成一个切应力局部集中的“核”. 在局部没有达到静摩擦极限时，应力集中使切应力极值分布在加载区域（中部）. 在充分形核后，局部切应力不断增加，接触层中部率先达到局部极值，发生局部滑动，切应力极值分布区域开始向x轴正负方向扩展. 图8（d）-（f）是接触层切应力与法向应力的应力比（τxz/σzz）云图，应力比极值分布同样先集中在加载区域，在形核充分后向两端传播. 应力比形式上与静摩擦系数非常相似，我们可以认为局部滑动前一刻的应力比极值就是局部摩擦系数（注意到每一时刻接触层上都存在应力比极值，但并非每个时刻都可以视为静摩擦系数）.
以上“局部形核-扩展”过程是准静态的，切应力极值传播速度和应力比极值传播速度都非常慢. 图9是U为0.001U0模型切应力极值传播速度与应力比极值传播速度随长度变化的结果. 这些速度都在10-4量级，而体材料的纵波波速vL为：
对比不同L结果，在准静态切向加载条件下，L较小时界面没有足够区域让局部滑动的核传播，在形核不充分时就发生了整体滑动. L增加时，界面局部滑动的核传播的区域不断增加，形核越来越充分，摩擦力也随之增加. 在L增加到足够大时，形核足够充分，局部形核区域的摩擦力提供了绝大部分界面摩擦力，此时L继续增加极限静摩擦力不会增加，发生饱和. 对于极限静摩擦力饱和现象，Pugno等[26]模拟了石墨烯薄片在石墨烯基底上滑动，也得出了薄片长度增长摩擦力会先增加再饱和的结论，但这个解只适用于薄片. 他们预测摩擦力饱和长度Lc的公式中包含界面剪切强度G，认为G越大Lc会越小. 剪切强度G与界面强度U成正比，本研究工作表明Lc与会随着G增加而增加，但不会一直增加，这是模型维度增加所导致的.
Hurtado等[37]讨论了单个asperity的摩擦，也提出了摩擦应力与接触尺寸相关. 他们认为接触边缘的应力集中会导致摩擦滑移的不稳定性，这意味着界面局部会产生位错，从而导致摩擦应力与接触尺寸的依赖性. 在本文的模拟中通过倒圆角规避块体边缘应力集中现象，同时载荷水平比较低，块体变形始终在弹性范围内，因此模拟结果显示的尺寸效应完全是基于界面摩擦动力学的，材料内部不产生位错. 同时本文模拟的是三维块体的摩擦，更侧重接触面整体情况的考量，而Hurtado等讨论的是单个asperity的摩擦，相当于考察一个局部接触. 因此几何上的差异与考察角度的不同也是模型结果不同的重要原因.
另外，Fineberg等[2]的宏观实验中所得到的锋面传播速度要快的多，在低界面强度以及无粘性的一些模拟结果中，我们同样观察到了与材料波速相当的传播现象，粘性可能对局部滑动锋面传播速度有重要影响.
4　结论
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综上所述，我们基于可调控的修正Morse势函数，使用分子动力学方法模拟了原子级光滑界面的静动摩擦转变过程. 主要结论有：
（1）原子级光滑界面的摩擦力来源为法向接触刚度的贡献与界面粘性的贡献. 在法向外载恒定时，刚度与粘性的竞争会让静摩擦系数μs随界面强度U非线性变化. 当U较小时，界面接触刚度会主导摩擦行为，μs会随U增加而减小，粘性影响微弱；当U较大时，界面粘性会主导摩擦行为，μs会随U增加而增加，接触刚度影响微弱.
（2）对原子级光滑接触界面，极限静摩擦力不会随着滑动方向上接触尺寸增加而无限增加，会在接触长度足够长时发生饱和，这是因为接触长度足够长时有充足区域让局部区域“形核”然后传播.
基金
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  • 国家自然科学基金项目(12022210; 12032001)
  • 青年托举项目(2023QNRC001)
文献
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2024年第45卷第5期
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doi: 10.19636/j.cnki.cjsm42-1250/o3.2024.018
  • 接收时间：2024-04-30
  • 首发时间：2026-04-01
  • 出版时间：2024-10-25
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  • 收稿日期：2024-04-30
基金
国家自然科学基金项目(12022210; 12032001)
青年托举项目(2023QNRC001)
作者信息
    1中国科学院力学研究所非线性力学国家重点实验室，北京，100190
    2中国科学院大学工程科学学院，北京，100049

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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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红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
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