摘要
840D车轮曾是我国运用数量最多的货车车轮,在役数量一度达320万片之多。840D车轮辐板孔疲劳裂纹曾是货车车轮故障的主要形式,且曾发生过多起因辐板孔裂纹导致的行车事故,对铁路运输安全带来严重威胁。
本文根据840D车轮辐板孔裂纹调查统计数据分析、轮轨机械载荷和制动热负荷下的裂纹尖端应力和强度因子仿真计算等结果,对裂纹成因、扩展规律、剩余寿命、容限尺寸及运用安全性进行了系统分析研究,主要研究工作包括以下几个方面:
1裂纹成因及其扩展规律的试验研究
对典型辐板孔裂纹的宏、微观形貌及金相组织进行分析,确定裂纹的性质、产生原因及扩展模式;对辐板裂纹萌生寿命进行了试验研究。
2裂纹形成和扩展规律的仿真研究
根据车轮运用中的轮轨机械载荷和制动热负荷条件,采用有限元法进行辐板孔应力应变和裂纹应力强度因子计算,提出轮轨机械载荷和制动热负荷组合作用下的应力叠加方法,研究并建立考虑踏面磨耗量、辐板孔位置等多参数影响的裂纹形成寿命和剩余寿命预测模型,计算出典型制动工况和大秦线全程运用条件下裂纹的形成寿命、裂纹扩展剩余寿命,进而综合分析导致裂纹形成和扩展的载荷条件并探讨踏面磨耗量、辐板孔位等几何因素对裂纹形成和扩展的影响机制。
3复合型裂纹应力强度因子仿真及断裂准则研究
采用1/4节点位移法及子模型技术,建立车轮辐板孔裂纹的三维有限元模型,完成了复合型裂纹应力强度因子仿真并得到不同裂纹尺度下应力强度因子的变化规律;建立辐板孔复合型裂纹的断裂准则,预测了裂纹失稳扩展的临界尺寸。
4对调查样本数据的统计分析
利用长期跟踪大秦线带裂纹车轮试验数据和铁路货车技术管理信息系统(HMIS)记录的数据,根据在全路7家车辆修造单位现场调研取得的20316个辐板孔车轮样本数据,分析辐板孔裂纹的分布规律,得出裂纹发生率与使用时间、轮辋厚度、孔偏位置的关系;采用二参数威布尔分布,建立辐板孔裂纹萌生寿命与扩展速率的概率模型,计算不同可靠度下的裂纹萌生寿命与扩展速率并进而推断840D车轮的安全使用条件。
5辐板孔裂纹研究的工程化应用
在大秦线实际制动运用工况的基础上,补充了直线制动工况和曲线制动工况,分析了踏面制动热输入对车轮强度的影响,为UIC510-5标准车轮强度评定方法进行平均应力的修正提供了依据,初步尝试了货车车轮辐板结构优化设计工作,并对轮辋裂纹扩展研究进行了可行性分析。
The type840D wheel used to be the largest amount of wagon wheels in service, with its number almost up to3.2million. The fatigue crack around the plate hole of840D wheel was the main form of wagon wheel malfunctions, which resulted in a couple of operation accidents, seriously threats the railway transportation safety.
Based on the analysis to abundant investigation and statistic data of the crack around plate hole of840D wheel, plus the simulation to stress and crack intensity factor of wheel-rail load and heat load during braking, this thesis studies the crack initiation condition, crack propagation law, residual life, crackallowance size and serving safety. The primary work and remarks of this thesis are as follows:
1. Experimental research on crack initiation and propagation law
The analysis of macroscopic and microscopic fracture pattern and metallographic character were processed to determine the crack initiation mechanism and propagation model; then crack initiation life of spoke plate were studied.
2. Simulation research on crack initiation and propagation law
The stress and strain of plate hole and crack stress intensity factor have been analyzed by finite element method according to the wheel-rail mechanical load and heat load during braking. The stress superimposing method was put forward for solving the stress superimposition under mechanical and thermal loads. The model was set up to study influence of rim wear and plate hole offset on crack initiation life and residual life, then the crack forming life, expanding life and residual life are calculated under braking condition and other typical conditions on the Datong-Qinhuangdao Railway. Based on this, the load cause of crack forming and expanding was analyzed, the influence mechanism of rim wear, position of plate hole and other geometrical factors on crack forming and expanding were discussed.
3. Simulation of complex crack stress intensity factor and study on breaking rules
The3D FE model of crack around plate hole was set up applying the1/4node displacement method and the sub-model technique. Based on study on complex crack intensity factor, the stress intensity factor variation rules under different crack dimensions were simulated, then breaking rules of complex crack around plate hole were established, critical dimensions of crack expanding was predicted.
4. Statistical analysis on the sample data
According to the sample data both in tracking observation of the operating wheels on the Datong-Qinhuangdao Railway and the management information system of railway wagon (HMIS), which was equal to20316, obtained from7wagon plants. This thesis draw relationships among crack ratio with employ time, rim thickness and hole offset. The probability model that can be used to predict the crack initiation lifespan and expansion velocity was built with the adoption of a2-parameter Weibull distribution, then the crack initiation lifespan and expansion velocity under different reliability were analyzed and the safety states of operating840D wheels could thus be inferred.
5. The engineering application of the plate hole crack study
Based on real braking conditions on the Datong-Qinhuangdao Railway, braking conditions in straight line and curve line were complementary, the influence on wheel strength was simulated by inputting tread braking heat This thesis made a correction to the UIC510-5wheel strength assessment using mean stress method. The structure optimization of wagon wheel spokes and the feasibility analysis to the rim crack expanding study are also put out.
引文
[1]张斌等.铁路车轮、轮箍失效分析及伤损图谱[M].北京.中国铁道出版社.2002
[2]J.G.Gimenez铁路车轮的应力分析—在国际轮轴会议上提出论文的回顾[C].第九届国际轮轴会议论文集.铁道部四方车辆研究所选译汇编.1998
[3]马大炜.车轮技术的研究现状与发展[J].中国铁路.1990.(9).9-12
[4]徐罗平.提速列车车轮失效分析.机车车辆工艺[J].2002年第4期
[5]International Eisenbahnverband. UIC Merkblatt 5154.1993 Eisenbahnfahrzeuge fur den Transportvon Fahrgasten:Laufdrehgestelle-Laufwerk Festigkeitsprufung arm Rahmen von Drehgestellen [S]. York.1992
[6]International Eisenbahnverband. UIC Merkblatt 615-41994. Triebfahr-zeuge:Drehgestelle and Laufwerke "Festigkeitsprufung ard Stukturen von Drehgestellrahmen"[S]. York.1993
[7]UI515-3/1993. Eisenbahnfahrzeuge/Drehgestelle-Laufwerke:Verfahren fur die Berechnung von Radsatzwellen[S].1994
[8]The European Standard. EN13104:2001. Railway applications—Wheel-sets and bogiesPowered axles[M]—Design method[S].2001
[9]Association of American Railroads Procedure for the Analytic Evaluation of Locomotive and Freight Car Wheel Designs.S-660-83. in Manual of Standards and Recommended Practices Section G[M]---Wheels and Axles[S].1998. G31-G34
[10]J.Raison. Thermomechanisches and akustisches Verhalten der Rader von Eisenbahnfahrzeugen[M]. ZEV+DET Glas.Ann.1998.122(9/10)375-384
[11]L.Ramanan, R.Krishna Kumar, R.Sriraman. Thermal-mechanical finite element analysis of a rail wheel[J]. International Journal of Mechanical Sciences.1999.(41).487-505
[12]Williiam C.Vantuono(美)货车车轮设计.国外铁道车辆[J].第四十卷第四期.2003
[13]刘会英.客货车车轮强度有限元计算[J].铁道车辆.1987.7.1-5
[14]余红英,樊永生.火车车轮非线性有限元热应力分析[J].华北工学院学报.2001.22(1).25-28
[15]关莹.谢素明.踏面制动热-机耦合过程的数值仿真[J].大连铁道学院学报.2003.24(3).9-12
[16]关莹.列车制动过程热—机耦合的数值仿真[D].大连铁道学院.2002
[17]Lunden R. Contact region fatigue of a railway under combined mechanical rolling pressure and thermal brake loading[J].Wear.l991.144.57-68
[18]C.H.KnceneB各种制动工况下整体辗钢车轮的温度场、变形及应力[J].国外铁道车辆.1996.(4).18-20
[19]刘云.提速货车车轮的温度场及热应力场的数值模拟[D].北京交通大学.2004
[20]V.C.INOZEMTSEV.车轮摩擦热力学[C].第十一届国际轮轴会议论文集.北京.中国铁道出版社.1997.256-259
[21]Wise S. Railway wheel-sets a critical review[C]. Proceedings of the Mechanical Engineers.1987.201 (D4).257
[22]Fermer M. Optimizationofrailwayfreightcarwheelby useoffractional factorial designmethod[C]. ProceedingsoftheInstituteofMechanical Engineers.1994.208.97-112
[23]MoyarGL.StoneDH,An analysis of the rmal contributions to railway wheels shelling[J].Wear.1991.144.117-123
[24]Edel KO, Schaper M. Fracture mechanics fatigue resistance analysis of the crackdamaged treadofoverbrakedsolidrailwaywheels.RailInternational.1992.35
[25]朱宇宙.钱立和.车轮用钢断裂力学性能研究[C].第九届全国疲劳与断裂学术会议.昆明.1998.北京.航空工业出版社.1998.270-273
[26]孟江英.钱立和.铸钢车轮及其性能研究[J].钢铁.1999.34(10).38-52
[27]郭建.邱之静等.整体铸钢车轮断裂力学性能研究[J].材料工程.1995.13(6).28-30
[28]Markus Diener等.姚昆炎译.货车车轮的断裂机理和金属学的研究[J].国外机车车辆工艺.1993.(2).18-24
[29]K.O.Edel等.韦苏来译.整体车轮的疲劳裂纹扩展特性[J].国外铁道车辆.1991.(1).49-54
[30]K.O.Edel等.卓桃月译.整体车轮踏面裂损的断裂力学疲劳抗力分析[J].国外铁道车辆.1994.(3).33-40
[31]张峰.车轮的损伤容限及寿命估计[C].中国力学学会材料与结构之力学测试国际学术会议.香港.1998.北京.科学出版社.1998.147-151
[32]张峰.车轮断裂韧性与组织和性能的关系[J].理化检验-物理分册.2004.40(4).172-175
[33]Edel K O.Ottlinger P.Fatigue Crack Growth Characteristics of Solid Wheels.Rail International[J].1990.(4).15-22
[34]张明如.车轮钢热-机械疲劳性能的研究[C].第八届全国疲劳学术会议.西安.1997.北京.中国金属学会.1997.269-272
[35]钱立和.朱宇宙.车轮损伤现象的热疲劳和机械疲劳研究[C].第八届全国疲劳学术会议.西安.1997.北京.中国金属学会.1997.273-276
[36]Ekberg A. Rolling Contact Fatigue of Railway Wheels-A Parametric Study [J]. Wear.1997.(211):280-288
[37]PistoriusP J H,MaraisJ J.Thermal Fatigueof Steel Tyreson Urban Railway Systems[J]. International Journal of Fatigue.1995,17(7):471-475
[38]Bowles, C. and D. Broek. On the formation of fatigue striations[J], International Journal of Fracture(1972) 8(1):75-85.
[39]Man, J., T. Vystavel, et al.Study of cyclic strain localization and fatigue crack initiation using FIB technique[J]. International Journal of Fatigue 2012,39(0):44-53.
[40]Mi, B., J. E. Michaels, et al. An ultrasonic method for dynamic monitoring of fatigue crack initiation and growth[J]. The Journal of the Acoustical Society of America 2006,119(1):74-85.
[41]Kondo, Y. Prediction of Fatigue Crack Initiation Life Based on Pit Growth[J]. Corrosion 2009, 65(1):7-11.
[42]Xue, Y., H. El Kadiri, et al.Micromechanisms of multistage fatigue crack growth in a high-strength aluminum alloy[J]. Acta Materialia 2007,55(6):1975-1984.
[43]Lykins, C. D., S. Mall, et al. An evaluation of parameters for predicting fretting fatigue crack initiation[J]. International Journal of Fatigue 2000,22(8):703-716.
[44]Gall, K., M. F. Horstemeyer, et al. On the driving force for fatigue crack formation from inclusions and voids in a cast A356 aluminum alloy[J]. International Journal of Fracture 2001, 108(3):207-233.
[45]Golden, P. J., A. F. Grandt Jr, et al. A comparison of fatigue crack formation at holes in 2024-T3 and 2524-T3 aluminum alloy specimens[J]. International Journal of Fatigue 1999,21, Supplement 1(0):S211-S219.
[46]Girshowich, S. and E. Reinberg. Formation of fatigue cracks at holes:observation and detection[J]. International Journal of Fatigue 2005,7(1):49-54.
[47]Costa, J. D. and J. M. Ferreira. Fatigue crack initiation in notched specimens of 17Mn4 steel[J]. International Journal of Fatigue 1993,15(6):501-507.
[48]Jones, R. L., M. A. Phoplonker, et al.Local strain approach to fatigue crack formation life at notches[J]. International Journal of Fatigue 1989,11 (4):255-259.
[49]Bozek, J. E., J. D. Hochhalter, et al. A geometric approach to modeling microstructurally small fatigue crack formation:Ⅰ. Probabilistic simulation of constituent particle cracking in AA 7075-T651 [J]. Modelling and Simulation in Materials Science and Engineering 2008,16(6):065007.
[50]Bridier, F., P. Villechaise, et al. Slip and fatigue crack formation processes in an α/β titanium alloy in relation to crystallographic texture on different scales[J]. Acta Materialia 2008,56(15): 3951-3962.
[51]McDowell, D. L. and F. P. E. Dunne. Microstructure-sensitive computational modeling of fatigue crack formation[J]. International Journal of Fatigue 2010,32(9):1521-1542.
[52]Oja, M., K. S. Ravi Chandran, et al.Orientation Imaging Microscopy of fatigue crack formation in Waspaloy:Crystallographic conditions for crack nucleation[J]. International Journal of Fatigue 2010,32(3):551-556.
[53]Pook, L. P.The fatigue crack direction and threshold behaviour of mild steel under mixed mode I and III loading[J].1985International Journal of Fatigue 7(1):21-30.
[54]Szolwinski, M. P. and T. N. Farris. Mechanics of fretting fatigue crack formation[J]. Wear 1996, 198(1-2):93-107.
[55]Osterstock, S., C. Robertson, et al.Stage I surface crack formation in thermal fatigue:A predictive multi-scale approach[J]. Materials Science and Engineering:A.2010,528(1):379-390.
[56]Polak, J., J. Man, et al.AFM evidence of surface relief formation and models of fatigue crack nucleation[J]. International Journal of Fatigue 2003,25(9-11):1027-1036.
[57]Shyam, A. and E. Lara-Curzio.A model for the formation of fatigue striations and its relationship with small fatigue crack growth in an aluminum alloy[J]. International Journal of Fatigue2010,32(11):1843-1852.
[58]UIC510-5/2003.Technische Zulassung von Vollraedern, rue Jean Rey[S],2003
[59]施治才.侯卫星.陈明资.宁郁编译.货车转向架疲劳强度规范[M].铁道部四方研究所,1985.
[60]Forschungs-und Versuchsamt des Internationalen Eisenbahnverbandes Frage B12/RP17-1982. Standard is ie rung der Giiterwagen[R].Utrecht:Banth Verlag,1982
[61]Cameron Lonsdale刘敬辉.马大炜译.重载货车车轮热负荷问题的探讨[J].国外铁道车辆.2005.42(6).32-37
[62]马大炜.铁道车辆制动热负荷的计算及应用[J].中国铁道科学.2000.21(4).30-37
[63]铁道部四方车辆研究所研究报告.提速货车车轮热负荷试验研究报告[R].青岛.1999.411.
[64]虞丽娟.机车车轮瞬态温度场的有限元研究[J].铁道学报.1993.15(3).11-16
[65]Lonsdale C. Pilch J. Wheel Thermal Loading Issues for Heavy Haul Service [C].Proceedings of the Air Brake Association. America. Air Brake Association.2003.
[66]中国铁道科学研究院研究报告.货车车轮辐板(孔)的疲劳行为及对策研究[R].北京.2005.1-5
[67]王勖成.邵敏.有限单元法基本原理和数值方法[M].北京.清华大学出版社.2001.421-438
[68]孔祥谦.热应力有限单元法分析[M].上海.上海交通大学出版社.1999.210-215
[69]平修二.热应力与热疲劳[M].北京:国防工业出版社.1984.115-12
[70]唐兴伦,范群波ANSYS工程应用教程(热与电磁学篇)[M].北京.中国铁道出版社.2003.79-80
[71]史建平等.铁道部科学研究院金属与化学研究所,铁路常用材料P-S-N曲线及Goodman图册[R].北京.铁道部科学研究院研究报告.1999
[72]de-Andres, A., J. L. Perez, et al.Elastoplastic finite element analysis of three-dimensional fatigue crack growth in aluminum shafts subjected to axial loading[J]. International Journal of Solids and Structures 1999,36(15):2231-2258.
[73]Sukumar, N., D. L. Chopp, et al.Extended finite element method and fast marching method for three-dimensional fatigue crack propagation[J]. Engineering Fracture Mechanics 2003,70(1):29-48.
[74]Ural, A., G. Heber, et al.Three-dimensional, parallel, finite element simulation of fatigue crack growth in a spiral bevel pinion gear[J]. Engineering Fracture Mechanics 2005.72(8):1148-1170.
[75]Hurley, P. J. and W. J. Evans. A new method for predicting fatigue crack propagation rates[J]. Materials Science and Engineering:A 2007,466(1-2):265-273.
[76]Nguyen, O., E. A. Repetto, et al.A cohesive model of fatigue crack growth[J]. International Journal of Fracture 2001,110(4):351-369.
[77]Ritchie, R. O.Mechanisms of fatigue-crack propagation in ductile and brittle solids[J]. International Journal of Fracture 1999,100(1):55-83.
[78]Roe, K. L. and T. Siegmund. An irreversible cohesive zone model for interface fatigue crack growth simulation[J]. Engineering Fracture Mechanics 2003,70(2):209-232.
[79]Venning, L., S. C. Hogg, et al.Fatigue crack growth and closure in fine-grained aluminium alloys[J]. Materials Science and Engineering:A,2006,428(1-2):247-255.
[80]Hurley, P. J. and W. J. Evans. A methodology for predicting fatigue crack propagation rates in titanium based on damage accumulation[J]. Scripta Materialia 2007,56(8):681-684.
[81]中国航空研究院.应力强度因子手册[M].北京:科学出版社.1981
[82]S.Suresh著.王中光译.材料的疲劳[M].北京:国防工业出版社.1993
[83]徐灏.疲劳强度[M].北京:高等教育出版社.1988
[84]曾春华,邹十践.疲劳分析方法及应用[M].国防工业出版社.1991
[85]R.Fuoco, M.M. Ferreira, C.R.F. Azevedo. Failure analysis of a cast steel railway wheel[J]. Engineering Failure Analysis.2004, (11).817-828
[86]王莺.典型钢种高温疲劳裂纹扩展规律研究[D].浙江工业大学研究生学位论文.2004
[87]张芳.典型钢种高温疲劳裂纹扩展规律的试验研究与计算机模拟[D].浙江工业大学研究生学位论文.2004
[88]Begley, J. and J. Landes. The J-integral as a fracture criterion[J]. Astm Stp 1972,514(514): 1-20.
[89]Batdorf, S. B. and H. L. Heinisch. Weakest Link Theory Reformulated for Arbitrary Fracture Criterion[J]. Journal of the American Ceramic Society 1978,61(7-8):355-358.
[90]Schiffmann, R., J. Heyer, et al. On the application of the damage work density as a new initiation criterion for ductile fracture[J].Engineering Fracture Mechanics 2003,70(12):1543-1551.
[91]Chang, J., J.-q. Xu, et al. A general mixed-mode brittle fracture criterion for cracked materials[J]. Engineering Fracture Mechanics 2006,73(9):1249-1263.
[92]Bian, L. and F. Taheri. Fatigue fracture criteria and microstructures of magnesium alloy plates[J]. Materials Science and Engineering:A,2008,487(1-2):74-85.
[93]Reyes, A., M. Eriksson, et al.Assessment of yield and fracture criteria using shear and bending tests[J]. Materials & ampDesign 2009,30(3):596-608.
[94]Tuler, F. R. and B. M. Butcher. A criterion for the time dependence of dynamic fracture[J]. International Journal of Fracture 1968,4(4):431-437.
[95]Whitney, J. M. and R. J. Nuismer. Stress Fracture Criteria for Laminated Composites Containing Stress Concentrations[J]. Journal of Composite Materials 1974,8(3):253-265.
[96]Seweryn, A. Brittle fracture criterion for structures with sharp notches[J]. Engineering Fracture Mechanics 1994,47(5):673-681.
[97]Toribio, J. A fracture criterion for high-strength steel notched bars[J]. Engineering Fracture Mechanics 1997,57(4):391-404.
[98]Kornev, V. M. and V. D. Kurguzov. Multiparametric sufficient criterion of quasi-brittle fracture for complicated stress state[J]. Engineering Fracture Mechanics 2008,75(5):1099-1113.
[99]Ayatollahi, M. R. and A. R. Torabi. A criterion for brittle fracture in U-notched components under mixed mode loading[J]. Engineering Fracture Mechanics 2009,76(12):1883-1896.
[100]Zhang, X.-m., W.-d. Zeng, et al. Fracture criterion for predicting surface cracking of Ti40 alloy in hot forming processes[J]. Transactions of Nonferrous Metals Society of China 2009,19(2): 267-271.
[101]Khan, A. S. and H. Liu. Strain rate and temperature dependent fracture criteria for isotropic and anisotropic metals[J]. International Journal of Plasticity 2012,37(0):1-15.
[102]Nait-Abdelaziz, M., F. Zairi, et al. J integral as a fracture criterion of rubber-like materials using the intrinsic defect concept[J]. Mechanics of Materials 2012,53(0):80-90.
[103]Seweryn, A. and A. Lukaszewicz. Verification of brittle fracture criteria for elements with V-shaped notches[J]. Engineering Fracture Mechanics 2002,69(13):1487-1510.
[104]Komori, K. Ductile fracture criteria for simulating shear by node separation method[J]. Theoretical and Applied Fracture Mechanics 2005,43(1):101-114.
[105]Yu, H. L. and D. Y. Jeong. Application of a stress triaxiality dependent fracture criterion in the finite element analysis of unnotched Charpy specimens[J]. Theoretical and Applied Fracture Mechanics 2010,54(1):54-62.
[106]Mostafavi, M., D. J. Smith, et al. A micromechanical fracture criterion accounting for in-plane and out-of-plane constraint[J]. Computational Materials Science 2011,50(10):2759-2770.
[107]丁遂栋,孙利民.断裂力学[M].北京:机械工业出版社,1997
[108]王铎.断裂力学(上册)[M].哈尔滨:哈尔滨工业大学出版社,1989
[109]朱伯芳.有限单元法原理与应用(第二版)[M].北京:中国水利水电出版社,1998
[110]小枫工作室.最新经典ANSYS及Workbench教程[M].北京:电子工业出版社,2004
[111]祝效华,余志祥,等.ANSYS高级工程有限元分析范例精选[M].北京:电子工业出版社,2004
[112]S. A. Fawaz. Stress intensity factor solutions for part-elliptical through cracks. Engineering Fracture Mechanics[J],1999,63(2):209-226
[113]S. A.Fawaz. Application of the virtual crack closure technique to calculate stress intensity factors for through cracks with an elliptical crack front[J]. Engineering Fracture Mechanics,1998, 59(3):327~342
[114]陈雷,黄海明,高小勤.840D货车车轮辐板孔边裂纹扩展机理分析[J].中国铁道科学,2007,28(2):76-72
[115]杨晓华,姚卫星.疲劳裂纹随机扩展模型[J].强度与环境,1995,(2):39-50.
[116]陈雷,习年生,王俊彪.840D车轮辐板孔裂纹扩展速率统计分析研究[J].中国铁道科学,2005,26(5):62-67.
[117]陈勃,鲍蕊,张建宇,费斌军.飞机结构耐久性/损伤容限综合分析模型[J].航空学报,2004,25(2):133-136.
[118]徐倩,缪龙秀.货车B级钢侧架经济寿命研究[J].铁道机车机辆,2004,24(5):1-3.
[119]蒋鑫龙,王志智,郑敏仲.冷挤压强化件的耐久性分析[J].机械强度,1999,21(4):296-298.
[120]霍波,刘文珽.耐久性分析技术的发展与改进[J].北京航空航天大学学报,1995,21(1):90-95.
[121]钟群鹏,金星,洪延姬,陶春虎.断裂失效的概率分析和评估基础[M].北京:北京航空航天大学出版社,2000.
[122]军用飞机疲劳·损伤容限·耐久性设计手册(第4册)[M].北京:中国航空研究院,1994,32-33.
[123]J.W. Prowan.概率断裂力学和可靠性[M].北京:航空工业出版社,2001.