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Article(id=1242151087256449440, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1242151082701430843, articleNumber=null, orderNo=null, doi=10.3969/j.issn.1007-7294.2024.06.011, pmid=null, cstr=null, oa=null, hot=null, price=null, onlineType=0, articleFormat=0, articleType=null, articleTypeStr=null, receivedDate=1703260800000, receivedDateStr=2023-12-23, revisedDate=null, revisedDateStr=null, acceptedDate=null, acceptedDateStr=null, onlineDate=1774081497253, onlineDateStr=2026-03-21, pubDate=1718812800000, pubDateStr=2024-06-20, doiRegisterDate=null, doiRegisterDateStr=null, onlineIssueDate=1774081497253, onlineIssueDateStr=2026-03-21, onlineJustAcceptDate=null, onlineJustAcceptDateStr=null, onlineFirstDate=null, onlineFirstDateStr=null, sourceXml=null, magXml=null, createTime=1774081497253, creator=13701087609, updateTime=1774081497253, updator=13701087609, issue=Issue{id=1242151082701430843, tenantId=1146029695717560320, journalId=1240685776644648972, year='2024', volume='28', issue='6', pageStart='803', pageEnd='966', issueExtLink='null', onlineDate='null', pubDate='null', beforeIssueId=null, nextIssueId=null, price=null, status=1, issueComplete=1, articleOrder=1, issueType=-1, specialIssue=0, createTime=1774081496167, creator=13701087609, updateTime=1774081719798, updator=13701087609, preIssue=null, nextIssue=null, ext={EN=IssueExt(id=1242152020732354696, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1242151082701430843, language=EN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=), CN=IssueExt(id=1242152020732354697, tenantId=1146029695717560320, journalId=1240685776644648972, issueId=1242151082701430843, language=CN, specialIssueTitle=, coverIllustrator=null, specialIssueEditor=, specialIssueAbout=)}, issueFiles=null}, startPage=917, endPage=924, ext={EN=ArticleExt(id=1242151087466164647, articleId=1242151087256449440, tenantId=1146029695717560320, journalId=1240685776644648972, language=EN, title=Fatigue life prediction considering the additional damage under multiaxial non-proportional loading, columnId=1242129251223274417, journalTitle=Journal of Ship Mechanics, columnName=Structural Mechanics, runingTitle=null, highlight=null, articleAbstract=

The significant life reduction of mechanical components during non-proportional loading compared to proportional loading results from the additional damage. In this study, the factors affecting the fatigue life under non-proportional loading were analyzed, and a new non-proportional damage factor was proposed. A novel fatigue life prediction model based on equivalent strain was established with the maximum shear plane as the critical plane. The new multiaxial fatigue life prediction model not only considers the effect of normal stress on the critical plane on the material crack initiation and crack propagation, but also reflects the non-proportional loading on the multiaxial fatigue life by introducing the non-proportional hardening coefficient of the materials and non-proportionality of the load paths. Experimental data under multiaxial loading of five materials were used to verify the accuracy of the novel model, and were also compared with two classical models. The results show that the prediction accuracy of the proposed model is higher.

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与比例载荷加载相比，在非比例载荷加载时由于附加损伤的产生，机械构件会出现明显的寿命缩短。本文通过分析非比例载荷作用下影响疲劳寿命的因素，提出一个新的非比例损伤因子，以最大剪切面为临界面，建立新的基于等效应变的疲劳寿命预测模型。新的多轴疲劳寿命预测模型不仅考虑法向应力对材料裂纹萌生与裂纹扩展的影响，而且通过引入材料附加强化系数与载荷路径的非比例度反映非比例载荷加载对多轴疲劳寿命的影响。采用5种材料在多轴载荷下的实验数据来验证本文模型的准确性，同时与两种经典模型进行对比，验证与对比分析说明，本文所提模型预测精度较好。

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通讯作者，E-mail：
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朱鹏年（1998-），男，硕士研究生

高建雄（1988-），男，副教授，通讯作者，E-mail：。

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朱鹏年（1998-），男，硕士研究生

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朱鹏年（1998-），男，硕士研究生

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高建雄（1988-），男，副教授，通讯作者，E-mail：。

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高建雄（1988-），男，副教授，通讯作者，E-mail：。

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tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1242151087256449440, language=EN, label=Fig.6, caption=Box plot of model prediction errors, figureFileSmall=8NEQbVz6hMCij8fRchcAGQ==, figureFileBig=UPGjyuWNfdA600PLTxd7WA==, tableContent=null), ArticleFig(id=1242151100267180927, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1242151087256449440, language=CN, label=图6, caption=模型预测误差的箱形图, figureFileSmall=8NEQbVz6hMCij8fRchcAGQ==, figureFileBig=UPGjyuWNfdA600PLTxd7WA==, tableContent=null), ArticleFig(id=1242151100346872709, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1242151087256449440, language=EN, label=Tab.1, caption=

Non-proportionality factor for each loading path

, figureFileSmall=null, figureFileBig=null, tableContent=
加载路径ABCD
fnp00.770.771
), ArticleFig(id=1242151100472701831, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1242151087256449440, language=CN, label=表1, caption=

各加载路径下的非比例度因子

, figureFileSmall=null, figureFileBig=null, tableContent=
加载路径ABCD
fnp00.770.771
), ArticleFig(id=1242151100548199306, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1242151087256449440, language=EN, label=Tab.2, caption=

Performance parameters of materials

, figureFileSmall=null, figureFileBig=null, tableContent=
材料 E/GPa σb/MPa σy/MPa α b c
16MnR212.5544.5324.40.2966.40.842-0.101-0.618
BT-91181080910011800.278-0.025-0.665
Pure-Ti11255847506470.548-0.033-0.646
Q235B204390.92690.26407.60.81-0.042-0.59
S460N208.56435000.258340.157-0.079-0.493
), ArticleFig(id=1242151100640474000, tenantId=1146029695717560320, journalId=1240685776644648972, articleId=1242151087256449440, language=CN, label=表2, caption=

材料性能参数

, figureFileSmall=null, figureFileBig=null, tableContent=
材料 E/GPa σb/MPa σy/MPa α b c
16MnR212.5544.5324.40.2966.40.842-0.101-0.618
BT-91181080910011800.278-0.025-0.665
Pure-Ti11255847506470.548-0.033-0.646
Q235B204390.92690.26407.60.81-0.042-0.59
S460N208.56435000.258340.157-0.079-0.493
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多轴非比例加载下考虑附加损伤的疲劳寿命预测
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朱鹏年 , 高建雄 , 袁逸萍 , 徐蓉霞 , 吴志峯
船舶力学 | 结构力学 2024,28(6): 917-924
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船舶力学 | 结构力学 2024, 28(6): 917-924
多轴非比例加载下考虑附加损伤的疲劳寿命预测
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朱鹏年, 高建雄 , 袁逸萍, 徐蓉霞, 吴志峯
作者信息
  • 新疆大学　机械工程学院，乌鲁木齐　830046

    • 朱鹏年（1998-），男，硕士研究生

    高建雄（1988-），男，副教授，通讯作者，E-mail：。

通讯作者:

通讯作者，E-mail：
Fatigue life prediction considering the additional damage under multiaxial non-proportional loading
Peng-nian ZHU, Jian-xiong GAO , Yi-ping YUAN, Rong-xia XU, Zhi-feng WU
Affiliations
  • School of Mechanical Engineering, Xinjiang University, Urumqi 830046, China
出版时间: 2024-06-20 doi: 10.3969/j.issn.1007-7294.2024.06.011
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摘要
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与比例载荷加载相比，在非比例载荷加载时由于附加损伤的产生，机械构件会出现明显的寿命缩短。本文通过分析非比例载荷作用下影响疲劳寿命的因素，提出一个新的非比例损伤因子，以最大剪切面为临界面，建立新的基于等效应变的疲劳寿命预测模型。新的多轴疲劳寿命预测模型不仅考虑法向应力对材料裂纹萌生与裂纹扩展的影响，而且通过引入材料附加强化系数与载荷路径的非比例度反映非比例载荷加载对多轴疲劳寿命的影响。采用5种材料在多轴载荷下的实验数据来验证本文模型的准确性，同时与两种经典模型进行对比，验证与对比分析说明，本文所提模型预测精度较好。

多轴疲劳  /  非比例度  /  临界面法  /  等效应变  /  附加损伤

The significant life reduction of mechanical components during non-proportional loading compared to proportional loading results from the additional damage. In this study, the factors affecting the fatigue life under non-proportional loading were analyzed, and a new non-proportional damage factor was proposed. A novel fatigue life prediction model based on equivalent strain was established with the maximum shear plane as the critical plane. The new multiaxial fatigue life prediction model not only considers the effect of normal stress on the critical plane on the material crack initiation and crack propagation, but also reflects the non-proportional loading on the multiaxial fatigue life by introducing the non-proportional hardening coefficient of the materials and non-proportionality of the load paths. Experimental data under multiaxial loading of five materials were used to verify the accuracy of the novel model, and were also compared with two classical models. The results show that the prediction accuracy of the proposed model is higher.

multiaxial fatigue  /  non-proportionality  /  critical plane method  /  equivalent strain  /  additional damage
朱鹏年, 高建雄, 袁逸萍, 徐蓉霞, 吴志峯. 多轴非比例加载下考虑附加损伤的疲劳寿命预测. 船舶力学, 2024 , 28 (6) : 917 -924 . DOI: 10.3969/j.issn.1007-7294.2024.06.011
Peng-nian ZHU, Jian-xiong GAO, Yi-ping YUAN, Rong-xia XU, Zhi-feng WU. Fatigue life prediction considering the additional damage under multiaxial non-proportional loading[J]. Journal of Ship Mechanics, 2024 , 28 (6) : 917 -924 . DOI: 10.3969/j.issn.1007-7294.2024.06.011
正文
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0　引言
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在实际工程中，大多数构件都是处于多轴应力-应变状态，而非单轴应力-应变状态，例如压力容器、燃气轮机、车辆曲轴等，所以多轴疲劳成为机械构件失效的主要形式[1-2]。相比于单轴疲劳，多轴疲劳失效机制更加复杂，如材料的微观结构、化学成分、应力集中和平均应力都对构件的疲劳寿命有极大的影响[3-4]，这给多轴载荷作用下疲劳寿命预测带来了巨大的挑战。多轴疲劳失效存在多重影响因素，然而多轴疲劳的研究往往是基于有限的实验数据，现有多轴疲劳寿命预测模型对于疲劳寿命与失效形式评估的准确性需要进一步研究[5]。根据现有的研究，多轴疲劳寿命预测模型可分为三类：等效应力/应变模型、基于能量的模型和临界平面法[6-8]。
等效应力/应变模型在比例载荷下可给出较为满意的预测结果[9]，然而并不适用于非比例载荷下的寿命预测。临界平面法认为损伤的发生与特定的平面有关[10]，将临界平面上的应力、应变参数与裂纹萌生及扩展机理结合起来，对于疲劳寿命预测能给出较好的预测结果，该方法被国内外学者广泛使用。近年来，基于能量的方法开始被广泛研究，然而能量是标量，不能反映疲劳破坏机理，因此大多数的能量模型都耦合了临界平面理论。
针对非比例载荷作用下的非比例附加损伤，学者们提出了采用不同的非比例系数来进行量化。一些学者通过分析应力/应变参量对于裂纹萌生与扩展的贡献，建立包含应力/应变的参量来描述附加损伤，Fatemi等[11]通过实验研究定义了包含临界面上的最大法向应力的参数，认为法向应力加速了裂纹扩展；李静等[12]定义了包含临界平面上的虚法向应力的参数，反映虚法向应力对于裂纹生长的影响；Shang等[13]采用临界面上的法向应变变程反映其对于裂纹脱聚行为的影响来描述附加损伤；Zhao等[14]定义了亚临界平面上的应变参数以量化非比例附加损伤。但是以应力/应变作为参量的非比例系数没有考虑加载路径的影响，也不能体现材料本身对于附加损伤的影响，附加损伤现象还与加载路径相关。为此，研究者基于不同路径下的转动惯量[15]、塑性应变能的大小[16]、应变值的变化[17]定义了不同的路径非比例度，同时结合材料附加强化系数定义了考虑加载路径的非比例系数。从加载路径角度来考虑附加损伤，弥补了等效应变模型未考虑加载路径的不足，然而对裂纹萌生与扩展机理的解释不够完善。非比例载荷下的附加损伤与材料特性和加载路径密切相关，在此基础上考虑应力/应变参数对于疲劳裂纹的形成机制与扩展模式的影响，有助于建立更加普适的寿命预测模型，提升多轴疲劳寿命预测的准确性。
本文对于非比例载荷下疲劳寿命缩短的现象展开研究，探讨材料硬化常数与路径非比例度对附加损伤的影响，并给出基于静强度参数评估材料硬化常数的近似计算方法，引入最大法向应力，建立新的非比例损伤因子，结合等效应变模型建立新的寿命预测模型。该模型建立在最大剪切面上，通过对模型中参量的研究，分析参量对于裂纹萌生和扩展的影响。采用5种材料的实验数据进行验证，并与FS模型与SWT模型进行对比分析。
1　经典的多轴疲劳模型
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1.1　FS模型
由于BM模型仅使用应变参数，无法反映异相加载所产生的附加强化效应，Fatemi和Socie[11]在BM模型的基础上引入最大法向应力来描述附加强化效应的影响。经过不同加载条件下对于裂纹萌生和扩展的研究，最大法向应力促进裂纹扩展，以最大剪切应变为第一控制参数，利用最大法向应力代替BM模型中的法向应变作为第二控制参数，建立了如下FS模型：
式中，Δγmax为临界面上的最大剪切应变幅，σn，max为临界平面上的最大法向应力，n为材料常数，σy为屈服强度，νe为弹性泊松比，为疲劳强度系数，E为弹性模量，b为疲劳强度指数，νp为塑性泊松比，为疲劳延性系数，c为疲劳强度指数。
1.2　SWT模型
SWT模型[18]用于考虑平均应力对疲劳寿命的影响，可用于拉伸裂纹破坏的寿命预测。SWT参数由最大拉应力σmax与最大主应变幅εa构成，SWT模型如下：
SWT参数σmaxεa可解释为应变能，其可以较好地评估低周和高周疲劳中的平均应力效应。
2　新的非比例损伤因子
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基于临界面理论，采用剪切应变与正应变控制疲劳过程，基于Von-Mises准则确定等效应变，等效应变模型可由下式给出：
式中，Δεeq为等效应变幅，Δεn为法向应变幅。
当以等效应变模型作为主要参量进行寿命预测时，采用文献[19]数据，给出了Pure-Ti[19]在不同加载条件下的等效应变幅值-寿命图，由图1可以看出，等效应变模型对于不同载荷条件给出了相同的等效应变，进而给出相同的寿命预测结果，无法反映非比例载荷作用下材料的附加损伤情况，对于寿命预测结果容易带来较大误差。影响附加损伤的因素是多方面的，主要受材料特性、加载路径和应力水平的影响[20]。本文通过对上述影响因素进行研究，提出了新的非比例损伤因子用于疲劳寿命预测。
2.1　材料附加强化的影响
由于不同材料的组成成分与微观结构不同，在非比例载荷作用下应力-应变主轴旋转，使更多的晶粒到达有利于滑移的位置[21]，滑移系统相互作用，使位错运动发生积塞，导致微裂纹的萌生，这是非比例载荷导致附加损伤的根本原因[22]。此外，附加损伤强化的程度与材料的位错能相关，低位错能的材料，如不锈钢等表现出较大的附加强化现象，而高位错能的铝合金材料附加强化效果不明显。文献[23]提出非比例附加强化系数，用于表征材料对于非比例附加强化现象的敏感性：
式中，σOP为稳定塑性变形阶段90°圆形路径下的等效应力，σIP为相同等效应变下、同相载荷作用下的等效应力。
采用式（4）计算材料附加强化系数时，需要得到稳定塑性应变下的应力参数，然而材料的塑性变形需要经历较长时间才能达到稳定，利用该方法评估材料附加硬化系数较困难。文献[24]经过试验研究发现，附加强化现象与材料的静态变形密切相关，给出了利用材料静态拉伸常数屈服强度σy与抗拉强度σb来近似评估附加强化系数的方法，可表示为
式中，S为材料常数，β为静强度强化系数。
2.2　路径非比例度的影响
由于不同加载路径下，材料内部形成的微观结构不同，导致附加强化程度不同，所以采用路径非比例度来反映加载路径对附加强化的影响。对于不同非比例的定义已经开展了大量的研究，Chen等[25]在对42CrMo钢的实验研究中，非比例附加强化与应变路径相关，附加强化程度为圆形路径>方形路径>椭圆路径>扭转路径，利用极坐标下最大剪切应变与相位角构成的阴影面积定义了非比例度；何国求等[26]通过对不同路径下316L不锈钢的实验研究，采用位错自由运动间距来描述非比例度；Itoh等[17]研究表明，主应变的方向变化与变化后应变的大小，显著影响附加强化的程度，基于应变历史定义了如下非比例度因子fnp：
式中，ε1（t）为t时刻应变的最大值，ε1，max为循环周期内ε1（t）的最大值，ξ（t）为ε1（t）与ε1，max的夹角，T为一个循环周期。本文所使用的加载路径如图2所示，根据式（6）计算的各路径下的非比例度如表1所示。
考虑材料本身与路径非比例度的影响，本文定义了一个新的附加强化参数k，
式中，fnp为路径非比例度，由式（6）计算得到。
该强化参数考虑了材料循环强化系数α与路径非比例度fnp来反映材料附加强化现象与路径非比例度对疲劳寿命的影响。
2.3　最大法向应力的影响
文献[27]的研究表明：一些材料如Pure-Ti与1050QT，不存在明显的附加强化效应（α = 0），但在非比例载荷作用下，疲劳寿命也较比例载荷缩短明显，以式（7）定义的非比例因子为例，仅能从路径非比例角度反映寿命缩短情况，在相同加载路径下，对于不同材料给出相同的强化参数，会导致预测误差[28]。所以本文引入最大法向应力，综合考虑附加强化对疲劳寿命的影响。图3为Pure-Ti在不同加载条件下临界平面上最大法向应力与寿命的关系图。
由图3可知，最大法向应力在比例载荷下变化不明显，在非比例载荷下，应力水平变化显著；相同加载路径下，随着寿命的减低而增加，可以反映非比例附加强化效果。定义最大方向应力的参数κ如下：
式中，σn，max为临界平面上的最大法向应力，为疲劳强度系数，该参数可认为是FS模型参数的修正形式[29]。
从裂纹扩展机理角度出发，最大法向应力有助于裂纹的打开、加速裂纹的扩展，进而导致疲劳寿命缩短。其中最大剪切面上的最大法向应力可以通过轴向应力幅值来估计，或者表示为关于疲劳寿命的函数[30]：
式中，σa为轴向应力幅值。
综合考虑路径因子、材料特性以及最大法向应力，本文提出了新的非比例损伤因子μ，可表示为
新的非比例损伤因子综合考虑了载荷条件、材料特性的影响，对于部分附加强化效应不明显的材料，可以同时考虑路径非比例度与最大法向应力反映疲劳寿命的缩短情况。
由图4可知，本文所提非比例因子伴随着疲劳寿命的减低而增加，符合实际情况。
3　新的多轴疲劳寿命预测模型
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等效应变模型主要是以剪切应变与法向应变作为主要参数。从疲劳破坏的物理机理出发，疲劳裂纹的萌生主要是由剪切应变控制，裂纹的萌生发生在最大剪切面上，正应变主要影响位错的流动性，影响着疲劳延性进而控制裂纹扩展过程[10]。所以本文以等效应变模型为基础，将临界平面建立在最大剪切面上是合理的。
在非比例载荷作用时，非比例损伤因子与法向应变随着相位差的增大而增大，能够反映疲劳寿命随相位差增大而减小的情况，所以选取法向应变与非比例损伤因子为主要控制参数，由式（10）与式（3），本文所建立的新的多轴疲劳寿命预测模型如下：
4　实验验证与对比
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本文选用了五种材料对本文模型进行准确性比较。五种材料分别为16MnR钢[31]、BT-9钛合金、Pure-Ti[19]、Q235B钢[32]和S460N钢[33]，试样均为薄壁圆管试件，材料的力学性能与疲劳性能参数由表2给出。本文所提模型的寿命预测结果如图5（a）所示，FS模型与SWT模型用来进行对比分析，结果如图5（b）和5（c）所示。
从图5可以看出，本文所提模型预测结果大多处于三倍误差分散带以内，只有S460N钢与16MnR钢的少数点位于三倍误差分散带之外，本文模型的预测结果较好，对于Pure-Ti给出了保守的预测结果；FS模型对于Pure-Ti的预测结果更好，其对于其他四种材料的寿命预测结果是不保守的；而SWT模型给出的预测大多处在三倍分散带之外，预测结果趋于危险，可能的原因是SWT对于剪切作用造成破坏的材料，预测结果偏差较大。
上述预测结果表明：本文通过考虑材料硬化系数、路径非比例度和临界平面上的最大法向应力，综合考虑影响疲劳寿命的不同因素所建立的新模型，对于本文所选材料的预测结果更准确，对于部分附加强化效应不明显的材料，通过考虑加载路径与最大法向应力影响，对于相同路径给出的疲劳寿命预测结果也较为准确。
5　结果与讨论
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为了定量分析与比较模型的预测精度，采用数学统计方法对所选模型进行评估，引入概率分析误差来进行评估，可由下式给出：
式中，Ne为试验寿命，Np为预测寿命。
本文采用箱式图和正态曲线来评估所选模型，如图6所示。对于所选材料在不同模型下的计算结果，从图6可以看出：本文模型的计算结果标准差较小，均值与中值比较靠近Error = 0的水平线，正态分布曲线也较为集中，表现出较好的预测精度；FS模型也有较小的偏差，但其标准差相较于本文模型大，所以正态分布曲线较分散；SWT模型的偏差较大，正态分布较为分散，预测精度较低。以上分析表明，本文模型综合考虑了材料附加强化系数、路径非比例度和最大法向应力对疲劳寿命的影响，建立的新的疲劳寿命预测模型有较好的预测精度。
6　结论
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本文通过考虑非比例载荷作用所产生的附加损伤效果，基于等效应变模型建立了新的多轴疲劳寿命预测模型，主要结论如下：
（1）通过综合考虑加载路径、材料特性和临界平面上的最大法向应力建立的新的非比例强化因子，反映非比例载荷作用下疲劳寿命缩短的现象，结合非比例损伤因子，建立了新的基于等效应变的多轴疲劳寿命预测模型。
（2）由预测结果与定量分析可知，本文模型预测结果相较于两种经典模型FS模型与SWT模型，本文预测更为准确，表明引入非比例强化因子确实可以提高疲劳寿命预测精度。
基金
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  • 国家自然科学基金资助项目(52065062)
  • 新疆维吾尔自治区自然科学基金资助项目(2020D01C056)
  • 新疆大学博士科研启动项目(2020bs12)
  • 新疆维吾尔自治区重点研发项目(2021B01003)
文献
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参考文献 引证文献
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2024年第28卷第6期
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doi: 10.3969/j.issn.1007-7294.2024.06.011
  • 接收时间：2023-12-23
  • 首发时间：2026-03-21
  • 出版时间：2024-06-20
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  • 收稿日期：2023-12-23
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国家自然科学基金资助项目(52065062)
新疆维吾尔自治区自然科学基金资助项目(2020D01C056)
新疆大学博士科研启动项目(2020bs12)
新疆维吾尔自治区重点研发项目(2021B01003)
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    新疆大学　机械工程学院，乌鲁木齐　830046

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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
小菇科 Mycenaceae 2 12 5.74 丝盖伞属 Inocybe 5 2.39
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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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