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Article(id=1241446332515274802, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241446328396476740, articleNumber=null, orderNo=null, doi=10.16579/j.issn.1001.9669.2025.07.006, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1699545600000, receivedDateStr=2023-11-10, revisedDate=1703088000000, revisedDateStr=2023-12-21, acceptedDate=null, acceptedDateStr=null, onlineDate=1773913470628, onlineDateStr=2026-03-19, pubDate=1752508800000, pubDateStr=2025-07-15, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1773913470628, onlineIssueDateStr=2026-03-19, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1773913470628, creator=13701087609, updateTime=1773913470628, updator=13701087609, issue=Issue{id=1241446328396476740, tenantId=1146029695717560320, journalId=1227999626482147330, year='2025', volume='47', issue='7', 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=1773913469647, creator=13701087609, updateTime=1773916972123, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1241461018921062969, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241446328396476740, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1241461018921062970, tenantId=1146029695717560320, journalId=1227999626482147330, issueId=1241446328396476740, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=49, endPage=55, ext={EN=ArticleExt(id=1241446332758544438, articleId=1241446332515274802, tenantId=1146029695717560320, journalId=1227999626482147330, language=EN, title=Analysis of disc spring stiffness degradation induced by fatigue damage, columnId=1241446330749481285, journalTitle=Journal of Mechanical Strength, columnName=·Fatigue·Damage·Fracture·Failure Analysis·, runingTitle=null, highlight=null, articleAbstract=

The disc spring will bear the cyclic displacement load during using, resulting in the fatigue damage and the stiffness degradation of the disc spring, which causes irreversible influence on the compensation function when accumulated sufficiently to cause fracture. Therefore, the disc spring material which occurred internal fatigue and structural stiffness degradation was studied under the cyclic load. By considering characteristics of the geometric nonlinearity and the action of the cyclic load, based on the traditional stiffness degradation model, a stiffness degradation model fitting for disc springs was established. The force change of the structural system, the law of stiffness degradation of the disc spring, and the stiffness degradation model were analyzed and verified with the finite element software. The model was modified based on the test data to obtain the model that can be used to calculate the degradation of the disc spring stiffness. This model can predict the deformation of disc springs’ structure in service, and determine the fatigue damage and performance degradation degree,which can provide some basis and reference for the application of disc springs.

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HAO Xinrui, E-mail：
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碟形弹簧在使用过程中会承受循环位移载荷，造成疲劳损伤，导致碟形弹簧发生刚度退化，当累积到足以导致断裂的程度，对补偿功能造成不可逆的影响。因此，研究了在循环载荷下，碟形弹簧材料内部发生疲劳损伤及其结构的刚度退化情况。在传统的刚度退化模型的基础上考虑碟形弹簧几何非线性的特点以及循环载荷的作用，建立一个适用于碟形弹簧的刚度退化模型。分析了在循环载荷作用后碟形弹簧的结构体系受力性能的变化情况以及刚度退化规律，运用有限元软件对模型进行了验证。基于试验数据对模型进行修正，得到可用于计算碟形弹簧刚度退化程度的模型。该模型可以预测碟形弹簧结构在服役过程中的变形及结构疲劳损伤、性能退化程度判定，可为碟形弹簧的应用提供一定的依据和参考。

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郝鑫瑞，男，1997年生，辽宁沈阳人，硕士研究生；主要研究方向为CAD/CAE/CAM集成技术；E-mail：。
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张晓光，男，1977年生，辽宁铁岭人，硕士，教授，硕士研究生导师；主要研究方向为CAD/CAE/CAM集成技术；E-mail：。

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张晓光，男，1977年生，辽宁铁岭人，硕士，教授，硕士研究生导师；主要研究方向为CAD/CAE/CAM集成技术；E-mail：。

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张晓光，男，1977年生，辽宁铁岭人，硕士，教授，硕士研究生导师；主要研究方向为CAD/CAE/CAM集成技术；E-mail：。

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diagram of the disc spring, figureFileSmall=D8wDJ/ttBrAGZGaAmP5WcA==, figureFileBig=ABRQ3KKXGHzd8sK+MJVOpQ==, tableContent=null), ArticleFig(id=1241446346176123315, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241446332515274802, language=CN, label=图10, caption=碟形弹簧损伤图, figureFileSmall=D8wDJ/ttBrAGZGaAmP5WcA==, figureFileBig=ABRQ3KKXGHzd8sK+MJVOpQ==, tableContent=null), ArticleFig(id=1241446346301952438, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241446332515274802, language=EN, label=Tab. 1, caption=

Simulation stress value of the disc spring

, figureFileSmall=null, figureFileBig=null, tableContent=
序号
No.		fRmax/MPa
理论值
Theoretical value		仿真值
Simulation value
10.75h02 1422 166
20.65h01 8811 882
30.55h01 6131 640
), ArticleFig(id=1241446346373255609, tenantId=1146029695717560320, journalId=1227999626482147330, articleId=1241446332515274802, language=CN, label=表1, caption=

碟形弹簧仿真应力值

, figureFileSmall=null, figureFileBig=null, tableContent=
序号
No.		fRmax/MPa
理论值
Theoretical value		仿真值
Simulation value
10.75h02 1422 166
20.65h01 8811 882
30.55h01 6131 640
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疲劳损伤诱发的碟形弹簧刚度退化分析
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张晓光 , 郝鑫瑞 , 韩绍年
机械强度 | ·疲劳·损伤·断裂·失效分析· 2025,47(7): 49-55
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机械强度 | ·疲劳·损伤·断裂·失效分析· 2025, 47(7): 49-55
疲劳损伤诱发的碟形弹簧刚度退化分析
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张晓光 , 郝鑫瑞 , 韩绍年
作者信息
  • 辽宁工业大学 机械工程与自动化学院，锦州　121001

    • 张晓光，男，1977年生，辽宁铁岭人，硕士，教授，硕士研究生导师；主要研究方向为CAD/CAE/CAM集成技术；E-mail：。

通讯作者:

郝鑫瑞，男，1997年生，辽宁沈阳人，硕士研究生；主要研究方向为CAD/CAE/CAM集成技术；E-mail：。
Analysis of disc spring stiffness degradation induced by fatigue damage
Xiaoguang ZHANG , Xinrui HAO , Shaonian HAN
Affiliations
  • College of Mechanical Engineering and Automation, Liaoning University of Technology, Jinzhou 121001, China
出版时间: 2025-07-15 doi: 10.16579/j.issn.1001.9669.2025.07.006
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摘要
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碟形弹簧在使用过程中会承受循环位移载荷，造成疲劳损伤，导致碟形弹簧发生刚度退化，当累积到足以导致断裂的程度，对补偿功能造成不可逆的影响。因此，研究了在循环载荷下，碟形弹簧材料内部发生疲劳损伤及其结构的刚度退化情况。在传统的刚度退化模型的基础上考虑碟形弹簧几何非线性的特点以及循环载荷的作用，建立一个适用于碟形弹簧的刚度退化模型。分析了在循环载荷作用后碟形弹簧的结构体系受力性能的变化情况以及刚度退化规律，运用有限元软件对模型进行了验证。基于试验数据对模型进行修正，得到可用于计算碟形弹簧刚度退化程度的模型。该模型可以预测碟形弹簧结构在服役过程中的变形及结构疲劳损伤、性能退化程度判定，可为碟形弹簧的应用提供一定的依据和参考。

位移载荷  /  碟形弹簧  /  疲劳损伤  /  刚度退化  /  弹塑性

The disc spring will bear the cyclic displacement load during using, resulting in the fatigue damage and the stiffness degradation of the disc spring, which causes irreversible influence on the compensation function when accumulated sufficiently to cause fracture. Therefore, the disc spring material which occurred internal fatigue and structural stiffness degradation was studied under the cyclic load. By considering characteristics of the geometric nonlinearity and the action of the cyclic load, based on the traditional stiffness degradation model, a stiffness degradation model fitting for disc springs was established. The force change of the structural system, the law of stiffness degradation of the disc spring, and the stiffness degradation model were analyzed and verified with the finite element software. The model was modified based on the test data to obtain the model that can be used to calculate the degradation of the disc spring stiffness. This model can predict the deformation of disc springs’ structure in service, and determine the fatigue damage and performance degradation degree,which can provide some basis and reference for the application of disc springs.

Displacement load  /  Disk spring  /  Fatigue damage  /  Stiffness degradation  /  Elasto-plasticity
张晓光, 郝鑫瑞, 韩绍年. 疲劳损伤诱发的碟形弹簧刚度退化分析. 机械强度, 2025 , 47 (7) : 49 -55 . DOI: 10.16579/j.issn.1001.9669.2025.07.006
Xiaoguang ZHANG, Xinrui HAO, Shaonian HAN. Analysis of disc spring stiffness degradation induced by fatigue damage[J]. Journal of Mechanical Strength, 2025 , 47 (7) : 49 -55 . DOI: 10.16579/j.issn.1001.9669.2025.07.006
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0　引言
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碟形弹簧[1]是一种结构简单、尺寸紧凑、体积较小的盘状弹性元件，如图1所示。其本身具有较高的承载能力及一定的缓冲性能[2-4]。随受载变形量的增加，碟形弹簧非线性弹性的特点愈发明显。这一结构特点使碟形弹簧广泛应用在航空航天、汽车、机械、建筑等领域。
通常会根据实际需求，采用叠合、对合及复合等不同的组合方式来构成一种非线性的弹性元件或者装置，达到补偿结构加载或者缓冲平衡的目的。如图2（a）所示，碟形弹簧载荷与位移的特性曲线呈现一种非线性的关系。在碟形弹簧的结构参数内径、外径、厚度以及材料一致时，其对应的特性曲线也会受h0/t的影响（h0是碟形弹簧的内锥高度，也叫极限位移）。而易先忠等[5]的研究表明，h0/t处于不同范围时，非线性特性曲线也将会有各不相同的特点。如图2（b）所示，碟形弹簧在不同阶段的高厚比之下，其本身的刚度呈现非线性的特点。因此，在实际的工程应用中，都会控制碟形弹簧在对应的界限以内。而随着循环次数的增加，不同的碟形弹簧结构受位移载荷而产生损伤的程度也大不相同。王申明等[6-8]指出，碟形弹簧的主要失效形式为疲劳失效，在经过长时间的循环载荷后其刚度开始逐渐降低，无法达到预期的补偿功能。
目前，国内外学者针对刚度变化进行了大量研究。赵磊等[9-10]研究了在长期载荷作用下，高速铁路无砟轨道-桥梁结构刚度退化规律，并进行了多级变幅长期列车载荷试验，分析了轨道和桥梁结构体系的刚度演化规律，研究了该结构体系的刚度退化情况并预测了其服役寿命。MEDINA等[11-12]研究以形状记忆合金（Shape Memory Alloy, SMA）为材料的补偿结构时发现，在发生刚度退化时，材料也受到残余应变的影响，建立了最大应变与机械性能退化（刚度和残余应变）之间相关的刚度退化模型。魏程风等[13]研究了传统的钢筋本构模型，在传统的Clough本构模型的基础上，为了量化循环加载下的刚度影响因素而引入修正因子，探究了浇节点刚度退化与构件中钢筋模量退化间的关系。SHIRI等[14]发现，部分描述疲劳损伤过程的刚度退化模型没有考虑载荷的多级加载顺序，为了解决这一问题，建立了一种添加对数表达式来考虑载荷顺序效应的刚度退化模型，并且还加入了加权系数来提高模型的准确性。KASHTALYAN等[15]发现在多轴载荷作用下，多向层合板在基体开裂后仍能继续承载载荷，但层合板的刚度会受到损伤层板刚度的影响，据此提出了一种充分考虑损伤模式相互影响的预测方法，该方法可以预测多向层合板之间的刚度退化性能。
综上所述，诸多学者在研究构件的刚度时，选择了不同的损伤参量与传统的刚度退化模型结合，从而获取在相应条件下刚度随载荷次数变化的规律。但是，众多模型在对于构件本身具有材料非线性属性方面考虑得较少。因此，本文以碟形弹簧在位移载荷下的刚度损伤为基础，通过对碟形弹簧加载过程进行分段考虑，重点研究了碟形弹簧在进入弹塑性阶段后，随着载荷次数增加刚度发生退化的规律。通过引入Chaboche非线性连续损伤（Nonlinear Continuum Damage, NLCD）模型，建立一个适用于碟形弹簧的刚度退化模型。通过选择不同位移载荷，对组合碟形弹簧模型进行数值模拟和相应的疲劳试验，分析了在进入塑性阶段后不同位移载荷对碟形弹簧载荷特性、刚度特性的影响规律，并且通过公式得到在一定循环次数之后碟形弹簧的剩余刚度值，为碟形弹簧产品的工程应用提供依据。
1　碟形弹簧的刚度退化模型
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传统的Almen-Laszlo法简化了碟形弹簧计算模型，只能计算碟形弹簧在未加载时的初始状态数值。而在实际的使用过程中，碟形弹簧因承受位移载荷而产生变形，不同的位移载荷对各项性能存在一定影响。绝大多数的碟形弹簧结构在工作时，存在着局部区域进入塑性阶段，在静载情况下碟形弹簧受到不同位移载荷影响，局部最大应力会大于材料的屈服应力的情况。有关试验发现，在经过首次压平以后，后续加载和卸载曲线差异很小，可以认为已经消除材料的塑性影响，结构进入了平稳状态，整体上表现为非线性弹性。当碟形弹簧处于弹性阶段时，其弹性模量为206 000 MPa，应力应变的关系为线性。但碟形弹簧往往会在加载区域进入塑性阶段。考虑到碟形弹簧材料为性能较好的金属韧性材料，在此采用Mises屈服准则，即令结构的应力强度为Mises等效应力，硬化准则定义塑性变形后的屈服条件。为了简化过程，采用双线性各向同性模型，即
式中，E为弹性模量；Re为屈服强度；Eτ为切线模量；R为应力；ε为应变；εs为达到屈服强度时的应变。
因此，简化后的应力-应变曲线如图3所示。当碟形弹簧的最大应力超过屈服强度时，用Miner累积损伤理论预测的碟形弹簧的刚度退化行为将会存在偏差。碟形弹簧随循环次数发生塑性变形，会导致碟簧整体在前期的循环次数里发生循环硬化现象，并且当硬化到一定程度就会停止，随后会随着循环次数的增加发生软化。因此，对碟形弹簧的加载历程属于由低到高的分级变载荷的加载情况，如图4所示。
为了描述碟形弹簧在塑性阶段的刚度损伤过程，引入一种NLCD模型[16-17]，即
式中，Rmax、Rmean均为恒定。通过对式（2）进行积分，可以得到对应的损伤演化方程：
式中，β为材料拟合常数；α为一个由最大应力Rmax和平均应力Rmean确定的函数。
将式（3）与传统的刚度模型联立，并且将联立后的刚度模型公式进行简化，即
式中，α为一个由最大应力Rmax确定的函数式。得到如式（5）所示的修正：
材料参数需要通过对应材料的疲劳试验来确定。
在恒幅加载条件下，结构未发生破坏时，N =0，D = 0；当结构发生失效时，N = Nf，D = 1。因此可得
将各函数α(Rmax )、M (Rmean )用等式替换，变换化简后的双线性工程应力-应变为
通过取特殊值Rmean = 0将式（7）变形，得
利用本材料在单轴对称加载情况下的S-N曲线即可得到对应的材料参数，即H、h、M0、β。
由文献［18］得到，60Si2Mn弹簧钢可靠度为99%的安全寿命公式为
式中，Rmax为最大应力值。
碟形弹簧材料为60Si2Mn，其在常温状态下的机械性能[19-20]为Re = 1 545 MPa，Rm = 1 680 MPa，R-1 =786 MPa，υ = 0.3，E = 206 000 MPa。由S-N曲线拟合得到材料参数H = 15.428，h = 2. 23，M0 = 5 299. 826，β = 0.313 1。将参数H、h、β的拟合值代入式（4），即
根据国家标准GB/T 1972.2—2023中的相关公式对A71型碟形弹簧进行计算。当Re = 1 545 MPa时，对应碟形弹簧的变形量f = 0.54h0；而f = 0.75h0，为碟形弹簧的使用界限。因此，选择0.55h0、0.65h0、0.75h0这3种位移条件下的力值、刚度以及最大应力进行计算。
通过理论计算得到的刚度值分别为32 322、31 627、31 220 N/mm。将对应的数值代入式（10）可以得到理论的刚度退化曲线，如图5所示。
2　碟形弹簧有限元模型的建立及仿真计算
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2.1　碟形弹簧的结构参数
常见的标准碟形弹簧会根据高厚比h0/t分为A、B、C 3个系列，在此基础上也会根据簧片的厚度分为无支撑面碟簧和有支撑面碟簧。而本文研究的是A系列无支撑面碟簧，它是一种常见而广泛应用的碟形弹簧，结构参数为外径D=50 mm、内径d=25.4 mm、厚度t=3 mm，其材料为60Si2Mn，对应的材料属性：弹性模量为206 000 MPa，泊松比为0.3，屈服强度为1 545 MPa，密度为7 850 kg/m3。碟形弹簧在承载的过程中只受压不受拉，其应力比为0，而对应产生的最大应力和应力幅均为Rmax/2。当外部载荷和支撑面上的反作用力均沿内、外圆周分布时，由于碟形弹簧在承载过程中，每一片的受力均等，因此，为了简化计算节约时间，选取其中的一组碟形弹簧（两片）作为仿真对象。经过一系列参数的调整，选取0.5 mm来进行网格的划分。
2.2　碟形弹簧的弹塑性有限元分析
当A50型碟形弹簧上边缘产生的最大应力未超过屈服应力时，根据有限元仿真在碟形弹簧下边缘处产生的力值成线性变化，在此区间里，碟形弹簧的变形可以视为完全恢复。在碟形弹簧的弹性范围内，进行多种不同的位移加载（0.05h0、0.15h0、0.25h0、0.35h0、0.45h0、0.54h0），所得的应力云图如图6所示。
根据碟形弹簧在弹性范围内的损伤情况仅与对应的加载次数有关，可以得到，A50型碟形弹簧在弹性范围内的刚度退化规律为随着载荷次数的增加呈线性退化的趋势。
当碟形弹簧最大应力超过屈服应力并进入塑性阶段时，对3种不同位移情况下的应力进行分析发现，应力最大的情况均呈现在碟形弹簧的上边缘处。随着碟簧循环次数的增加，上边缘处将成为最危险的破坏点，即疲劳断裂点，控制着碟形弹簧的疲劳寿命。
经过仿真计算，在0.55h0、0.65h0、0.75h0三种位移载荷下的应力云图和应力值分别如图7和表1所示。
可见，与理论计算数值相贴近，并且下边缘处与底座的支反力分别为18 754、21 330、24 596 N，与理论计算出的载荷17 959、20 978、23 953 N的吻合程度较好。
理论所得的最大应力与仿真所得数据的误差均在5%以内，证明了此有限元模型的可靠性。将对应的数值代入式（10）可以得到仿真刚度退化曲线，如图8所示。
3　刚度退化模型的修正
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通过实际试验发现，碟形弹簧在使用过程中刚度会随载荷次数的增加，呈现先增后减的趋势。这是由于碟形弹簧在使用过程中经历了循环应变硬化，其上边缘区域对变形的抵抗能力增强，然而当碟形弹簧的硬化达到饱和后开始衰减。因此，在弹塑性阶段仅用式（10）来描述刚度的退化是不全面的，需要对式（10）进行修正。
通过试验可以看出，碟形弹簧均受力情况在前0.1Nf时得到了增大。在碟形弹簧的初始加载阶段，应力水平较低，经过强化阶段后转变为在较高的应力水平下加载，由此变成了两级加载的形式。因此在使用碟形弹簧刚度退化模型时，需要进行分段。对碟形弹簧的第一级加载，其寿命为n≤0.1Nf，而在第一阶段，碟簧刚度呈上升趋势。由式（10）可以得到，在第一级加载的损伤情况为
当碟形弹簧的循环次数n > 0.1Nf时，碟形弹簧的刚度开始随着循环次数的增加而衰减，由此可以得到，第二级的损伤情况为
将仿真数值与理论数值分别代入新公式，由此得到的新刚度退化曲线如图9所示。根据刚度退化规律及刚度退化计算式进行仿真计算，可判断承受反复载荷作用的碟形弹簧组的变形、疲劳性能退化及损伤情况。利用式（11）、式（12）可对反复载荷作用下的结构刚度退化情况进行预测，预测结构变形，并且判定结构性能退化情况。在该结构使用一定的年限后，通过载荷试验可测得结构的初始刚度，再根据式（11）或式（12）即可计算得到K (n)/K0值，据此判断出结构的损伤状态，为工程结构的维修、拆除等决策提供技术依据。
通过试验所得数据及碟形弹簧的损伤情况可以看出，碟形弹簧叠合磨损出现在接触面以及边缘处，如图10所示。而碟形弹簧的受力情况也会随着循环次数的变化而改变。
4　结论
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通过对碟形弹簧刚度变化情况的研究，发现了碟形弹簧的剩余刚度在循环载荷作用下变化的规律，建立了在疲劳载荷的作用下适用于碟形弹簧的刚度模型，对模型的适用性进行了验证。主要结论如下：
1）在循环位移过程中，碟形弹簧的刚度先在一定程度上增加约10%，随后逐渐减小。如果对补偿结构进行力值补偿，则在循环一定周次后，补偿效果将显著降低。因此，在使用碟形弹簧组件时，建议采取以下措施：添加石墨等减少摩擦、选择合适预压缩来控制碟形弹簧的刚度，以防止刚度过大造成损坏，并定期调整平衡点以改善碟形弹簧组件的平衡效果。
2）根据刚度退化规律和试验所得的数据进行拟合，对模型进行了修正，得到了可用于计算碟形弹簧刚度退化程度的模型。利用碟形弹簧刚度退化计算公式，可以预测结构在服役过程中的变形发展情况，也可以进行结构疲劳损伤、性能退化程度判定，从而为碟形弹簧的后续应用提供一定的依据和参考。
基金
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  • 辽宁省博士科研启动基金计划项目(2021-BS-261)
文献
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2025年第47卷第7期
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doi: 10.16579/j.issn.1001.9669.2025.07.006
  • 接收时间：2023-11-10
  • 首发时间：2026-03-19
  • 出版时间：2025-07-15
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  • 收稿日期：2023-11-10
  • 修回日期：2023-12-21
基金
Liaoning Provincial Doctoral Research Start-up Program(2021-BS-261)
辽宁省博士科研启动基金计划项目(2021-BS-261)
作者信息
    辽宁工业大学 机械工程与自动化学院，锦州　121001

通讯作者:

郝鑫瑞，男，1997年生，辽宁沈阳人，硕士研究生；主要研究方向为CAD/CAE/CAM集成技术；E-mail：。
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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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