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Article(id=1241029732632940958, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241029724655383270, articleNumber=null, orderNo=null, doi=10.16579/j.issn.1001.9669.2025.02.007, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1688400000000, receivedDateStr=2023-07-04, revisedDate=1691337600000, revisedDateStr=2023-08-07, acceptedDate=null, acceptedDateStr=null, onlineDate=1773814145477, onlineDateStr=2026-03-18, pubDate=1739548800000, pubDateStr=2025-02-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773814145477, onlineIssueDateStr=2026-03-18, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773814145477, creator=13701087609, updateTime=1773814145477, updator=13701087609, issue=Issue{id=1241029724655383270, tenantId=1146029695717560320, journalId=1227999626482147330, year='2025', volume='47', issue='2', pageStart='1', pageEnd='158', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=null, createTime=1773814143575, creator=13701087609, updateTime=1773840259815, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241139264176574915, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241029724655383270, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241139264176574916, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241029724655383270, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=53, endPage=60, ext={EN=ArticleExt(id=1241029732955902376, articleId=1241029732632940958, tenantId=1146029695717560320, journalId=1227999626482147330, language=EN, title=Forecast of bending fatigue life for gears considering the influence of residual stress and hardness, columnId=1241029725523604201, journalTitle=Journal of Mechanical Strength, columnName=Fatigue∙Damage∙Fracture∙Failure Analysis, runingTitle=null, highlight=null, articleAbstract=

In order to study the influence of residual stress and hardness on bending fatigue performance of gear, the 20MnCrS5 steel gear with carburizing heat treatment was taken as the research object, and composite small diameter shot peening strengthening treatment was carried out to realize the gear with different hardness and residual stress states of the same material.Based on the maximum principal strain criterion, incorporating separate factors for residual stress influence and residual stress-hardness coupling influence were introduced respectively to establish the fatigue life prediction model.Through shot-peened gear bending fatigue tests, optimal values for both the residual stress influence coefficient and correction coefficient were determined.The two models’ accuracy for life prediction was contrasted, and the accuracy of these models was further verified through unshot peened gear bending fatigue test.The results show that considering only residual stress influence yielded an optimal value of 0.09 for the residual stress influence coefficient, the model achieving high life predictive accuracy.Whereas considering the effects of residual stresses and hardness, it requires a correction coefficient with an optimal value of 0.04, the model achieve even higher predictive accuracy.

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CHEN Yong, E-mail：
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为研究残余应力和硬度对齿轮弯曲疲劳性能的影响，以渗碳热处理20MnCrS5钢制齿轮为研究对象，进行复合小径喷丸强化处理，进而实现同一材料的齿轮具有不同硬度和残余应力状态。以最大主应变准则为基础，分别引入残余应力影响项以及残余应力-硬度耦合影响项建立疲劳寿命预测模型。通过喷丸齿轮弯曲疲劳试验确定最优残余应力影响系数和修正系数，对比两种模型的寿命预测精度；通过未喷丸齿轮弯曲疲劳试验验证模型的准确性。结果表明，当单一考虑残余应力影响时，残余应力影响系数最优值为0.09，模型可实现较高的寿命预测精度。同时考虑残余应力和硬度影响，修正系数最优值为0.04，模型具有更高的预测精度。

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陈勇（通信作者），男，1954年生，北京人，教授，博士研究生导师；主要研究方向为汽车系统动力学、高强度齿轮磨损摩擦学；E-mail：。
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赵鹏博，男，1996年生，河北保定人，硕士研究生；主要研究方向为齿轮弯曲疲劳；E-mail：。

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赵鹏博，男，1996年生，河北保定人，硕士研究生；主要研究方向为齿轮弯曲疲劳；E-mail：。

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赵鹏博，男，1996年生，河北保定人，硕士研究生；主要研究方向为齿轮弯曲疲劳；E-mail：。

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journalId=1227999626482147330, articleId=1241029732632940958, language=EN, label=Fig.5, caption=Detection result of the hardness, figureFileSmall=vFwkEAeGrpnrc5jnsIsIcA==, figureFileBig=FC4x5C11jpopdSk0nxfBDQ==, tableContent=null), ArticleFig(id=1241029747837292594, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029732632940958, language=CN, label=图5, caption=硬度检测结果, figureFileSmall=vFwkEAeGrpnrc5jnsIsIcA==, figureFileBig=FC4x5C11jpopdSk0nxfBDQ==, tableContent=null), ArticleFig(id=1241029747937955898, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029732632940958, language=EN, label=Fig.6, caption=Finite element model of gears, figureFileSmall=8h6EfqNgL820Dic5/xKHqA==, figureFileBig=IVhIqbQTS3vz+Rghe2819g==, tableContent=null), ArticleFig(id=1241029748021841982, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029732632940958, language=CN, label=图6, caption=齿轮有限元模型, 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Mechanical property of 20MnCrS5

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项目 Project数值 Numerical value
密度 Density/(g/cm3)7.838 0
弹性模量 Modulus of elasticity/GPa209.266
泊松比 Poisson ratio0.292 5
抗拉强度 Yield strength/MPa1 232
), ArticleFig(id=1241029750920106129, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029732632940958, language=CN, label=表1, caption=

20MnCrS5的力学性能

, figureFileSmall=null, figureFileBig=null, tableContent=
项目 Project数值 Numerical value
密度 Density/(g/cm3)7.838 0
弹性模量 Modulus of elasticity/GPa209.266
泊松比 Poisson ratio0.292 5
抗拉强度 Yield strength/MPa1 232
), ArticleFig(id=1241029751029158038, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029732632940958, language=EN, label=Tab.2, caption=

Simulation result of bending stress and strain for the tooth root

, figureFileSmall=null, figureFileBig=null, tableContent=
载荷
Load/kN		弯曲应力云图
Bending stress nephogram		应变云图
Strain nephogram
8
13
), ArticleFig(id=1241029751117238429, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029732632940958, language=CN, label=表2, caption=

齿根弯曲应力和应变模拟结果

, figureFileSmall=null, figureFileBig=null, tableContent=
载荷
Load/kN		弯曲应力云图
Bending stress nephogram		应变云图
Strain nephogram
8
13
), ArticleFig(id=1241029751196930210, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029732632940958, language=EN, label=Tab.3, caption=

Experimental result of bending fatigue life for not shot peened gears

, figureFileSmall=null, figureFileBig=null, tableContent=
载荷
Load Fn/kN		寿命
Life Nf		载荷
Load Fn/kN		寿命
Life Nf
6162 618951 754
241 35752 135
265 16360 021
307 31563 561
328 41364 900
360 192102 356
792 746138 963
95 65410 259
120 20513 435
162 74717 350
193 41718 788
252 67419 063
), ArticleFig(id=1241029751276621992, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241029732632940958, language=CN, label=表3, caption=

未喷丸齿轮弯曲疲劳寿命试验结果

, figureFileSmall=null, figureFileBig=null, tableContent=
载荷
Load Fn/kN		寿命
Life Nf		载荷
Load Fn/kN		寿命
Life Nf
6162 618951 754
241 35752 135
265 16360 021
307 31563 561
328 41364 900
360 192102 356
792 746138 963
95 65410 259
120 20513 435
162 74717 350
193 41718 788
252 67419 063
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考虑残余应力和硬度影响的齿轮弯曲疲劳寿命预估
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赵鹏博 1 , 陈勇 2 , 尤晖 2 , 周兴云 3 , 李光鑫 1 , 武一民 1 , 刘祥环 4
机械强度 | 疲劳·损伤·断裂·失效分析 2025,47(2): 53-60
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机械强度 | 疲劳·损伤·断裂·失效分析 2025, 47(2): 53-60
考虑残余应力和硬度影响的齿轮弯曲疲劳寿命预估
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赵鹏博1 , 陈勇2 , 尤晖2, 周兴云3, 李光鑫1, 武一民1, 刘祥环4
作者信息
  • 1.河北工业大学 天津市新能源汽车动力传动与安全技术重点实验室，天津　300401
  • 2.广西大学 机械工程学院，南宁　530004
  • 3.浙江义利汽车零部件有限公司，义乌　322000
  • 4.株洲齿轮有限责任公司，株洲　412000

    • 赵鹏博，男，1996年生，河北保定人，硕士研究生；主要研究方向为齿轮弯曲疲劳；E-mail：。

通讯作者:

陈勇（通信作者），男，1954年生，北京人，教授，博士研究生导师；主要研究方向为汽车系统动力学、高强度齿轮磨损摩擦学；E-mail：。
Forecast of bending fatigue life for gears considering the influence of residual stress and hardness
Pengbo ZHAO1 , Yong CHEN2 , Hui YOU2, Xingyun ZHOU3, Guangxin LI1, Yimin WU1, Xianghuan LIU4
Affiliations
  • 1.Tianjin Key Laboratory of Power Transmission and Safety Technology for New Energy Vehicles, Hebei University of Technology,Tianjin 300401, China
  • 2.School of Mechanical Engineering, Guangxi University, Nanning 530004, China
  • 3.Zhejiang Yili Auto Parts Co., Ltd., Yiwu 322000, China
  • 4.Zhuzhou Gear Co., Ltd., Zhuzhou 412000, China
出版时间: 2025-02-15 doi: 10.16579/j.issn.1001.9669.2025.02.007
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摘要
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为研究残余应力和硬度对齿轮弯曲疲劳性能的影响，以渗碳热处理20MnCrS5钢制齿轮为研究对象，进行复合小径喷丸强化处理，进而实现同一材料的齿轮具有不同硬度和残余应力状态。以最大主应变准则为基础，分别引入残余应力影响项以及残余应力-硬度耦合影响项建立疲劳寿命预测模型。通过喷丸齿轮弯曲疲劳试验确定最优残余应力影响系数和修正系数，对比两种模型的寿命预测精度；通过未喷丸齿轮弯曲疲劳试验验证模型的准确性。结果表明，当单一考虑残余应力影响时，残余应力影响系数最优值为0.09，模型可实现较高的寿命预测精度。同时考虑残余应力和硬度影响，修正系数最优值为0.04，模型具有更高的预测精度。

20MnCrS5渗碳齿轮  /  弯曲疲劳  /  复合小径喷丸  /  寿命预估  /  最大主应变准则

In order to study the influence of residual stress and hardness on bending fatigue performance of gear, the 20MnCrS5 steel gear with carburizing heat treatment was taken as the research object, and composite small diameter shot peening strengthening treatment was carried out to realize the gear with different hardness and residual stress states of the same material.Based on the maximum principal strain criterion, incorporating separate factors for residual stress influence and residual stress-hardness coupling influence were introduced respectively to establish the fatigue life prediction model.Through shot-peened gear bending fatigue tests, optimal values for both the residual stress influence coefficient and correction coefficient were determined.The two models’ accuracy for life prediction was contrasted, and the accuracy of these models was further verified through unshot peened gear bending fatigue test.The results show that considering only residual stress influence yielded an optimal value of 0.09 for the residual stress influence coefficient, the model achieving high life predictive accuracy.Whereas considering the effects of residual stresses and hardness, it requires a correction coefficient with an optimal value of 0.04, the model achieve even higher predictive accuracy.

20MnCrS5 carburized gear  /  Bending fatigue  /  Composite small diameter shot peening  /  Life forecast  /  Maximum principal strain criterion
赵鹏博, 陈勇, 尤晖, 周兴云, 李光鑫, 武一民, 刘祥环. 考虑残余应力和硬度影响的齿轮弯曲疲劳寿命预估. 机械强度, 2025 , 47 (2) : 53 -60 . DOI: 10.16579/j.issn.1001.9669.2025.02.007
Pengbo ZHAO, Yong CHEN, Hui YOU, Xingyun ZHOU, Guangxin LI, Yimin WU, Xianghuan LIU. Forecast of bending fatigue life for gears considering the influence of residual stress and hardness[J]. Journal of Mechanical Strength, 2025 , 47 (2) : 53 -60 . DOI: 10.16579/j.issn.1001.9669.2025.02.007
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0　引言
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齿轮作为机械传动系统中的重要零部件，在服役环境中承受循环载荷作用，在运转过程中会发生接触疲劳失效或弯曲疲劳失效。接触疲劳失效在整个失效问题中占比达到74%[1]，弯曲疲劳失效占比较低。然而，不断的循环应力作用使裂纹在齿轮内部形成和扩展，在很短时间内就会导致轮齿的断裂，齿轮传动系统立刻失效，造成灾难性的后果。因此，在高载荷和轻量化的发展趋势下，对齿轮的弯曲疲劳强度提出更高的要求，对齿轮弯曲疲劳寿命进行精确预测也至关重要。
齿轮弯曲疲劳性能的主要影响因素为几何结构[2]、材料[3]以及表面完整性因素[4]等。热处理、精加工、喷丸等处理后会在材料表面形成残余压应力层[5]，这一应力层的残余压应力对提高其抵抗裂纹萌生和扩展的能力具有积极的作用。齿轮在经过渗碳处理后，由表及里形成具有较高硬度和残余压应力的硬化层，可以显著提高齿轮等机械零部件的力学性能和疲劳强度[6-8]。喷丸强化等工艺可以为齿轮引入较高的硬度和残余压应力，从而有效抑制裂纹的产生和扩展[9]。高残余应力可以显著提升齿轮弯曲疲劳性能已成为共识。针对残余应力对齿轮弯曲疲劳性能的影响已有许多学者进行了研究。杨庆祥等[10]对渗碳及渗碳喷丸复合处理的齿轮进行了弯曲疲劳极限的定量分析，发现喷丸可以使齿轮的弯曲疲劳极限提高约38%，其主要原因为喷丸引入了较高的残余压应力。李明睿等[11]以喷丸强化后DD6单晶合金圆棒试件为研究对象，开展低周疲劳试验，研究结果表明，喷丸可以引入残余应力，从而有效提高DD6低周疲劳寿命，建立的低周疲劳寿命预测模型具有较好的准确性。陈海宽等[12]针对喷丸对18CrNiMo7-6钢硬度的影响进行了有限元分析，发现喷丸覆盖率增大，材料硬度逐渐增大，丸粒半径和速度增加，材料的硬化层深度逐渐增加。WINKLER等[13]对不同模数渗碳齿轮和喷丸齿轮的齿根弯曲疲劳强度和疲劳断裂特性进行了深入研究，通过脉动型试验和运转型试验研究发现，喷丸可以提高齿轮的弯曲疲劳强度，且对裂纹萌生的诱发因素进行了深入分析。
齿轮弯曲疲劳性能体现为某一极限载荷下具有无限寿命。载荷在齿轮齿根处表现为弯曲应力。因此，精确计算齿根弯曲应力是进行疲劳性能分析的首要条件。目前齿根弯曲应力计算方法主要依据悬臂梁理论，即将齿轮看做悬臂梁进行求解。由此衍生出多种计算标准并得到广泛应用[14][15]7-23。进行弯曲疲劳试验来分析齿轮弯曲疲劳性能是最准确的方法，但一个完整的弯曲疲劳试验需要消耗很大的财力和物力。因此，许多学者探求有限元模拟的方法进行齿轮弯曲疲劳性能分析。张云超等[16]基于渐开线直齿轮建立裂纹扩展有限元模型，计算了裂纹扩展不同阶段的裂纹尖端应力强度因子，运用Paris公式构建裂纹扩展速率模型，实现了含有齿根裂纹的齿轮弯曲疲劳寿命预测。熊勋等[17]使用Franc3D和Abaqus软件进行带预制裂纹的Q235钢的CT试样裂纹扩展及寿命预估，设计相应的CT试样裂纹扩展试验，结果表明仿真值与试验值较为接近。何龙龙等[18]对含中心穿透斜裂纹平板和带初始裂纹的加筋翼梁典型结构进行疲劳裂纹扩展路径及寿命预测，并进行试验验证，发现预测的裂纹扩展路径和试验结果具有较强的一致性。王建明等[19]建立含有喷丸后残余应力场的四点弯曲有限元模型，并且建立考虑裂纹闭合效应的有限元模型，发现残余压应力对提高弯曲疲劳寿命起着主导作用。高云等[20]采用有限元法对渐开线圆柱齿轮齿根疲劳裂纹扩展进行数值模拟，分析了载荷大小和初始裂纹长度对齿轮剩余寿命的影响。
目前喷丸工艺大多采用大直径丸粒（丸粒直径为0.6~0.8 mm）进行喷涂，在引入高硬度和残余压应力的同时不可避免地会增大材料表面粗糙度，进而降低了表面质量[21-22]。复合小径喷丸作为一种特殊的喷丸工艺，采用直径为0.2~0.25 mm的弹丸进行第一次喷涂，可以在材料表面引入高硬度和高残余压应力。采用直径为0.1~0.15 mm的弹丸进行第二次喷涂，进一步引入残余压应力和硬度的同时产生较小的表面粗糙度，从而减缓了表面疲劳裂纹的产生，为齿轮等零部件表面强化提供了新方法。许任宗等[23]发现复合小径喷丸处理可以提高齿轮表层硬度和残余压应力，降低齿面粗糙度，且弹丸冲击坑具有明显的储油、润滑功能，从而显著提高齿轮的接触疲劳性能。但有关复合小径喷丸对齿轮弯曲疲劳性能的影响研究尚有不足。目前，针对残余应力与硬度对齿轮弯曲疲劳性能的影响研究大多采用试验方法。试验装置主要为旋转式和脉动式弯曲疲劳试验机。尽管脉动式弯曲疲劳试验机可以快速形成齿轮弯曲疲劳S-N曲线，但往往也是以月为单位。同时，同种处理方式的齿轮弯曲疲劳研究结果很难与其他同材料、不同处理工艺的齿轮形成联系。
本文对齿轮与传动系统研究中心（FZG）标准齿轮进行复合小径喷丸处理，探究其对齿轮弯曲疲劳性能的提升效果。同时，利用Abaqus软件进行齿根弯曲应力和应变模拟。以最大主应变准则为基础，引入硬度和残余应力影响项，形成考虑硬度和残余应力的寿命预测模型。结合齿轮弯曲疲劳试验数据得到最优影响系数，对不同硬度和残余应力状态的齿轮进行高精度的疲劳寿命预测，为齿轮类零件的表面强化和寿命预测提供参考。
1　试验齿轮制备与试验方法
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1.1　材料及试件加工
本文选用20MnCrS5低碳合金钢作为试验齿轮的制造材料，该材料具有高强度和高韧性的特点，广泛应用于汽车、工程机械的传动部件。材料的力学性能如表1所示。
试验齿轮的齿数z=24，齿宽B=14 mm，模数mn=4.5 mm，均由同一批次锻造的20MnCrS5毛坯加工制造。
采用的热处理工艺路线如图1所示，使用与氮气混合的甲醇和丙烷作为渗碳剂。所有试验齿轮均同炉进行热处理，处理后表面硬度为58~62 HRC，理论有效硬化层深可以达到0.7~0.9 mm。
将渗碳后的齿轮进行复合小径喷丸处理。进而实现齿轮硬度及残余应力差异。第1次喷涂时丸粒直径为0.2~0.25 mm，喷丸强度为0.16 mmA。第2次喷涂时丸粒直径为0.1~0.15 mm，喷丸强度为0.21 mmN。两次喷丸的喷射角度均为90°，表面覆盖率均为200%。两次喷涂所用钢丸材质相同，硬度为58~64 HRC。
1.2　弯曲疲劳试验方法
完整的齿轮弯曲疲劳测试包括疲劳极限确定以及基于疲劳极限的S-N曲线绘制。详细的试验过程可参见GB/T 14230—2021[24]11-13。严格按照标准规定的试验方法进行试验。
采用图2（a）所示的Letry M6311型高频疲劳试验机进行齿轮弯曲疲劳试验。加载方式为单齿脉动加载。试验机稳定工作频率约为85 Hz。装夹位置如图2（c）所示，为保证下部夹具在上下往复冲击时不会触碰非试验齿，基于齿轮参数，采用跨4齿加载方法。上部夹具连接传感器。对下部夹具进行上、下往复的高频冲击。所用的夹具严格按照齿轮的参数进行设计制造，确保齿轮可以被牢固地固定在夹具上，且轮齿不影响冲头的行程，轮齿所受负载方向与基圆相切。
试验过程中出现下列两种情况之一均判断为齿轮疲劳失效：
1）轮齿齿根出现可见疲劳裂纹或轮齿沿齿根断齿。
2）因齿根出现疲劳裂纹而引起载荷值或加载频率下降5%~10%（失载）。
使用便携式残余应力检测仪进行残余应力的检测。检测时采用X射线衍射法，并用电解腐蚀的方法进行剥层，进而实现不同深度的测量。检测位置如图3（a）所示，测试现场如图3（b）所示。检测位置为齿根处，在同一检测点进行重复5次的测量，均值视为该位置残余应力。共设置7个检测点，检测深度为80 μm，前5个点的深度间隔为10 μm，后2个点的深度间隔为20 μm。残余应力检测结果如图4所示，取残余应力最大值作为齿轮初始残余应力。
使用HMV-G-XY-S型显微维氏硬度计测量试验齿的硬度。测试位置如图3（c）所示，测试现场如图3（d）所示。检测位置为齿根处，试验力为9.807 N，保荷时间为10 s，检测方向为垂直齿根过渡圆切线。在同一检测点进行重复5次的测量，均值视为该位置硬度。设置10个检测点，总计检测深度为1 mm，检测点间隔为0.1 mm。硬度检测结果如图5所示，取硬度最大值为齿轮初始硬度值。
1.3　齿根弯曲应力计算
基于GB/T 14230—2021[24]11-13计算不同载荷下的齿根弯曲应力，计算式为
式中，Fn为加载时的载荷值；α为加载点处压力角；YF为尺寸系数；YS为载荷作用下的应力修正系数；Yβ为螺旋角系数；YST为与试验齿轮尺寸有关的应力修正系数；YδrelT为相对齿根圆角敏感系数；YRrelT为静强度下的相对齿根圆角敏感系数；YX为尺寸系数。
进行弯曲疲劳试验时，必须预先施加一个较小的静态预紧力，采用的应力比（循环特性系数）r=0.05。因此，需要将齿根弯曲应力转换为r=0的对应值，转换式为
式中，Rm为20MnCrS5材料的抗拉强度，Rm=1 482 MPa。
2　弯曲疲劳仿真
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2.1　齿轮弯曲疲劳有限元模型
严格按照齿轮参数建立试验齿轮的有限元模型。根据夹头位置及形状建立的齿轮有限元模型如图6（a）所示。轮齿部分采用细化网格，最小尺寸为0.01 mm。同时，为增加运算效率，夹具、齿轮轮缘区域采用尺寸较大的渐疏网格。单元类型均为C3D8R，总体的网格数量为250 348。本文采用跨4齿的对称式加载方法，加载位置如图6（b）所示，加载点半径rE为56.25 mm。
2.2　考虑残余应力与硬度的寿命预测准则
常见的齿轮弯曲疲劳寿命预测准则主要包括最大主应变准则、Smith-Watson-Topper准则、等效主应变准则等。何海风等[25]研究发现最大主应变准则在进行齿轮弯曲疲劳寿命预测时的精度较高。因此，本文采用考虑Morrow平均应力修正的最大主应变准则来进行疲劳寿命预测。
最大主应变准则认为决定齿根疲劳寿命的因素为齿根处的最大主应变。最大主应变准则计算式为
式中，Δε1为齿根处最大主应变；Rmean为单载荷循环中的平均应力；为疲劳强度系数；b为强度指数；为疲劳延性系数；c为延性指数；Nf为疲劳寿命。根据Seeger理论可计算式中的各项参数，计算式为
式中，RUTS为拉伸应力极限；ψ为无量纲系数，其值取决于RUTS与E的关系[若RUTS/E小于0.003，取值为1，否则ψ=1.375-125（RUTS/E）]。各参数计算结果分别为
齿轮在经过热处理、磨削等加工后不可避免地会引入残余应力。同时，提高残余应力也是喷丸处理最主要的目的。上述准则在预测寿命时并没有考虑残余应力这一因素。并且鲜有文献将残余应力参量引入齿轮疲劳寿命预测中。为了在寿命预测中增加材料自身残余应力要素，对上述准则进行残余应力修正。引入残余应力影响系数mr，形成的疲劳寿命预测模型如下：
喷丸为齿轮引入残余应力的同时也提高了齿轮的表面硬度。因此，在式（6）的基础上，加入硬度因素，形成的疲劳寿命预测模型如下：
式中，β为修正系数；mh为硬度值。
3　结果与讨论
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3.1　残余应力及硬度修正系数确定
本文使用Abaqus软件进行了齿根弯曲应力与应变仿真。同时根据式（2）进行了齿根弯曲应力计算。计算值与模拟值对比结果如图7所示，计算值均比仿真值要小。随着载荷逐渐增加，两者的误差出现不同程度的增加，但是误差值很小，因此在进行有限元仿真时忽略不计。
载荷为8、13 kN时，齿根弯曲应力云图与应变云图对比结果如表2所示。不同载荷下齿根区域最大弯曲应力与最大应变的位置均相同，证明了仿真的准确性，可以使用此有限元模型进行最大主应变的求解。
不考虑残余应力和硬度时的最大主应变准则预测结果（Sample 1）和试验结果对比如图8所示。当载荷为13、14 kN时，两者表现出较好的一致性。当载荷小于13 kN时，随着载荷逐渐降低，两者的差异逐步增大。这是因为最大主应力准则预测模型并没有考虑材料自身的硬度和残余应力因素。但是，硬度和残余压应力可以明显提升齿轮的高周疲劳寿命。在低周疲劳阶段，较高的载荷使齿轮的齿根区域在高载荷作用下极易发生塑性变形，从而产生应力松弛。因此，硬度和残余应力对齿轮的低周疲劳寿命影响程度较低。进而使最大主应变准则预测模型在齿轮高周疲劳阶段的预测值与试验值相似。
基于最大主应变准则，结合应变模拟结果和弯曲疲劳试验结果计算最优拟合参数。考虑残余应力因素，mr=0.09时表现出最高的精度。图8展示了mr=0.09时单一考虑残余应力的模型预测寿命（Sample 2）与试验寿命结果对比。单一考虑残余应力影响，当载荷为13、14 kN时，预测寿命偏高，误差值分别为25.8%、26.2%。但是，当载荷小于13 kN时，预测模型表现出极高的预测精度。当载荷为12 kN时，拟合曲线对应的寿命为48 382次，模型预测值为49 125次，相对误差仅为1.54%，可以忽略不计。
添加硬度影响因素后，β=0.04时显示出最高的精度。图8展示了β=0.04时同时考虑残余应力和硬度的模型预测结果与试验寿命结果对比（Sample 3）。同时考虑残余应力和硬度影响，模型在载荷为13、14 kN的预测值明显减小，更加接近于试验曲线值。同时高周疲劳区的预测寿命有较小程度的降低。但降低幅度很小，可以忽略不计。由此可见，同时考虑残余应力和硬度的预测模型精度更高。
图9、图10分别为单一考虑残余应力的仿真寿命与试验寿命分散带图和同时考虑残余应力和硬度的仿真寿命与试验寿命分散带图。由图9、图10可知，预测结果和试验结果均在三倍分散带以内，证明了残余应力影响系数和修正系数取值的合理性和准确性。
3.2　修正系数验证
前文已经确定，mr=0.09，β=0.04两种情况的最大主应变疲劳寿命预测准则具有较好的精度。
但是，前文的结果只是基于喷丸材料的试验数据。相同材料但是不同残余应力、硬度的齿轮应同时适用于这一准则，进而需要对修正系数进行准确性验证。为此，使用未喷丸的齿轮进行弯曲疲劳试验，实现相同材料的齿轮具有不同的硬度和残余应力状态。试验载荷选取6、7、9、13 kN。表3所示为未喷丸齿轮的弯曲疲劳寿命试验结果。
单一考虑残余应力的模型预测结果和试验结果对比如图11所示，同时考虑残余应力和硬度的模型预测结果和试验结果对比如图12所示。由图11、图12可知，预测寿命和试验寿命的结果均在三倍分散带以内，进一步证明了残余应力影响系数和修正系数取值的准确性。值得注意的是，在最大主应变准则中直接进行影响项的加减并不具有物理意义。因此，本文展现的添加修正系数所得的预测模型仅为经验公式。
根据式（6）、式（7），齿根弯曲应力最大的位置即为疲劳裂纹的萌生位置。有限元模拟与实际轮齿断裂位置对比如图13所示。图13（a）所示为典型的裂纹扩展路径，图13（b）所示为不同载荷下裂纹初始萌生位置。不同载荷下的轮齿断裂路径基本相同，初始裂纹萌生位置略有不同，但均分布在齿根切线方向29.5°~30.2°范围内。如图13（c）所示，依据GB/T 3480.3—2021标准[15]7-23，齿根最大弯曲应力为齿根切线方向30°处，此区域即为裂纹萌生位置。图13（d）为有限元模拟裂纹扩展云图。模拟结果的初始裂纹出现位置为齿根切线方向30.1°。国家标准经验结果、有限元模拟结果和实际断裂结果高度吻合，这进一步验证了有限元模拟的准确性。
4　结论
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通过试验与有限元模拟相结合的方式研究了残余应力和硬度对20MnCrS5渗碳齿轮弯曲疲劳性能的影响。以最大主应变准则为基础，分别引入残余应力影响项和残余应力-硬度耦合影响项，基于复合小径喷丸齿轮的弯曲疲劳试验数据，确定了残余应力影响系数以及耦合修正系数。并使用未喷丸齿轮验证参数选取的准确性。主要结论如下：
1）单一考虑残余应力影响时，残余应力影响系数的最优取值为0.09。针对不同残余应力状态的齿轮均可实现较高精度的弯曲疲劳寿命预测，当载荷为12 kN时，拟合曲线对应的寿命为48 382次，模型预测值为49 125次，相对误差仅为1.54%，可以忽略不计，且预测寿命均在三倍分散带内。
2）在单一考虑残余应力的模型基础上耦合添加硬度影响项进行修正，耦合修正系数的最优取值为0.04。相比单一考虑残余应力的模型展现出更高的预测精度，预测结果更加接近寿命拟合曲线，且预测寿命均在三倍分散带内。
3）使用未喷丸齿轮进行修正系数验证，预测寿命结果均在三倍分散带内，验证了系数取值的准确性。
本文所提出的预测模型仅需要仿真齿轮的最大主应变，以及测量齿轮的硬度和残余应力，即可实现较高精度的齿轮弯曲疲劳寿命预测。但尚有不足之处，所得的修正系数仅适用于20MnCrS5材料的FZG标准齿轮，是否适用于其他材料、不同处理工艺的齿轮需要进一步验证。
基金
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  • 国家重点研发计划(2018YFE0207000)
  • 河北省全职引进高端人才项目(20181228)
文献
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2025年第47卷第2期
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doi: 10.16579/j.issn.1001.9669.2025.02.007
  • 接收时间：2023-07-04
  • 首发时间：2026-03-18
  • 出版时间：2025-02-15
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出版历史
  • 收稿日期：2023-07-04
  • 修回日期：2023-08-07
基金
National Key R&D Program of China(2018YFE0207000)
国家重点研发计划(2018YFE0207000)
Hebei Province Full-Time Introduction of High-End Talent Project(20181228)
河北省全职引进高端人才项目(20181228)
作者信息
    1.河北工业大学 天津市新能源汽车动力传动与安全技术重点实验室，天津　300401
    2.广西大学 机械工程学院，南宁　530004
    3.浙江义利汽车零部件有限公司，义乌　322000
    4.株洲齿轮有限责任公司，株洲　412000

通讯作者:

陈勇（通信作者），男，1954年生，北京人，教授，博士研究生导师；主要研究方向为汽车系统动力学、高强度齿轮磨损摩擦学；E-mail：。
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https://castjournals.cast.org.cn/joweb/jxqd/CN/10.16579/j.issn.1001.9669.2025.02.007
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2种不同金属材料的力学参数
科
Family		 属数
Number of
genus		 种数
Number of
species		 占总种数比例
Percentage of
total species (%)		 属
Genus		 种数
Number of
species		 占总种数比例
Percentage of total
species (%)
鹅膏菌科Amanitaceae 2 11 5.26 鹅膏菌属 Amanita 10 4.78
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
多孔菌科 Polyporaceae 8 14 6.70 蜡蘑属 Laccaria 5 2.39
红菇科 Russulaceae 3 23 11.00 小皮伞属 Marasmius 6 2.87
小菇属 Mycena 11 5.26
光柄菇属 Pluteus 5 2.39
红菇属 Russula 17 8.13
栓菌属 Trametes 5 2.39
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