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目录 contents

    摘要

    基于连续损伤力学理论,对2A12铝合金冲击局部损伤及后继疲劳寿命进行了研究。首先采用有限元方法模拟了冲击过程,获得冲击凹坑局部的残余应力场与塑性应变场;然后根据残余应力状态选取疲劳破坏的危险点,通过Lermaitre塑性损伤模型得到危险点的初始损伤;最后根据修正的Chaboche疲劳损伤模型估算疲劳寿命。通过与相关的试验结果进行对比验证,表明连续损伤力学结合有限元分析方法可有效评估金属结构冲击损伤及后继疲劳寿命。

    Abstract

    Based on the continuum damage mechanics, the impact local damage and consequent fatigue life of 2A12 aluminum alloy are investigated. Firstly, finite element simulation of impact is conducted to obtain the local residual stress and plastic strain fields around the impact pit. Then the dangerous points are selected according to the residual stress state, and the initial damages are obtained based on the Lemaitre’s plastic damage model. Finally, the fatigue life of the impacted specimen is calculated using the modified Chaboche’s fatigue damage model. Compared with experimental data, the combination of continuum damage mechanics and finite element method(FEM) is proven feasible in evaluating the impact damage and consequent fatigue life of metal materials or structures.

    承受交变载荷的金属结构(如直升机的动部件)的主要破坏形式之一是疲劳断裂,因此对其疲劳寿命的预测一直是一个关键技术。各类损伤(缺陷)往往是金属结构的疲劳破坏源,而实际结构中出现各类缺陷在所难免。传统的安全寿命法评估寿命时缺少对实际存在缺陷(损伤)的针对性分[1],实际应用中既可能过于保守,也可能偏危险。因此,基于具体缺陷(损伤)的疲劳寿命分析是发展更合理的疲劳寿命评估方法(如缺陷容限法)的前提。

    近年来国内外发展了一些基于局部应力集中分析的疲劳寿命预测方法,比如局部应力应变[2]、临界平面法、应力场强[3,4]等。局部应力应变法是在缺口应变分析和低周疲劳基础上发展起来的一种疲劳寿命估算方法,它结合了缺口部位的应力应变历程和材料疲劳寿命曲线,使用简单方便,但其对于高周疲劳适用性较差;临界平面法将损伤的概念引入多轴疲劳寿命分析中,假定疲劳破坏产生于疲劳损伤参量最大的平面。但该方法中的这些参量缺乏明确的物理意义,并且在复杂加载情况下缺口处的应力-应变响应难以由解析方法获[5]。姚卫[6,7,8]提出了考虑局部损伤区应力场影响的应力场强法,该方法克服了局部应力应变法无法考虑尺寸效应等缺陷,对缺口效应给出了更好的解释。但实际应用中其损伤区域的大小需要由大量试验确定,而且场强的计算也颇为复杂。对于带有微缺陷的结构疲劳问题,日本学者Yukitaka等做了一系列研究工[9,10],他们结合断裂力学思想,以一系列含人工微缺陷的试样疲劳试验为基础,提出一种缺陷等效面积模型。

    冲击凹坑是实际工程中常见的一类缺陷,其对于构件疲劳寿命的影响主要有3个因[11,12,13]:(1)冲击形成的凹坑导致的局部应力集中;(2)冲击产生的残余应力;(3)由于局部塑性变形造成的冲击损伤。对于后两个因素,上述各种疲劳寿命理论都不能直接反映。损伤力学通过引入损伤变量来研究材料在载荷作用下的损伤演化过程,具有比较完备的物理、力学体系,损伤变量通常具有相对明确的物理意义,逐渐成为预测构件疲劳寿命的有效方法之一。詹志新[14,15,16]采用连续损伤力学方法对含有冲击缺陷钛合金板进行疲劳寿命分析,综合考虑了缺陷处冲击残余应力、冲击初始损伤、凹坑应力集中和疲劳损伤的共同作用。

    本文以连续损伤力学为基础,结合有限元分析对2A12铝合金试样冲击凹坑缺陷的初始冲击损伤和高周疲劳寿命进行了分析,并与相关试验结果进行对比,验证方法的准确性。

  • 1 理论模型

    1
  • 1.1 塑性损伤模型

    1.1

    在各向同性损伤的假设下,对于各向同性材料,损伤变量D可以用材料的模量退化来表征,即

    D=E-EDE
    (1)

    式中:EED分别为材料完好和损伤后的弹性模量;D的变化范围为0~1。金属材料在较大的塑性变形条件下会导致晶界断裂或二相粒子的脱粘,这些因素可以归纳为塑性损伤。 Lemaitre提出了一个正比于等效塑性应变率且与应力状态和当前损伤程度相关的塑性损伤率模[17]

    D˙=σeq2Rv2ES0(1-D)2s0p˙
    (2)

    式中:σeq为等效应力;p˙为累积塑性应变率;S0s0为材料参数;Rv为应力三轴函数,其表达式为

    Rv=23(1+ν)+3(1-2ν)σHσeq2
    (3)

    式中:σH为静水应力;ν为泊松比。在单轴应力状态下,Rv=1。

    将式(2)对整个塑性变形历程进行积分,即可得到冲击凹坑缺陷由于塑性变形引起的初始损伤,如下

    D0=σeqmax2Rv2ES0s0Δp
    (4)

    式中Δp为在这段时间历程内的累积塑性应变值。

  • 1.2 疲劳损伤模型

    1.2

    Chaboche和Lesne[18]提出一个非线性单轴应力疲劳损伤模型

    D˙=dDdN=[1-(1-D)β+1]ασaM(σm)(1-D)β
    (5)

    式中:N为循环次数;σaσm分别为疲劳载荷的应力幅值和平均应力;β为材料常数;αM(σm)的定义分别为

    α=1-aσa-σ0σb-σa
    (6)
    M(σm)=M0(1-bσm)
    (7)

    式中:σb为材料的抗拉强度极限;σ0为材料的疲劳极限;aM0b均为材料常数,对于金属材料,Dattoma[19]得出,a=0.080 1;σ为麦考利括号(Macaulay brackets)。

    林有智[20]根据畸变能密度理论将上述模型修正为可考虑多轴应力的情况

    D˙=dDdN=[1-(1-D)β+1]ασTM(σm)(1-D)β
    (8)

    式中

    σT=12[(σ1-σ2)2+(σ2-σ3)2+(σ3-σ1)2]
    (9)

    式中:σT是构件的等效总应力;σ1σ2σ3表示构件3个方向的主应力。

    如疲劳加载前(N=0)的初始损伤为D00D01),对式(8)积分可以得到单级疲劳载荷下的寿命Nf(对应D=1

    Nf=1(1-α)(1+β)M(σm)σTβ1-[1-(1-D0)β+1]1-α
    (10)

    单级疲劳载荷作用N次(NNf)后的损伤量为

    D=1-1-[1-(1-D0)1+β]1-α+σM)αN(1+β)[1-α]σTM(σm)βα-1-β-1
    (11)

    对于多级疲劳载荷,可以利用式(10)和式(11)逐级累积得到最终的疲劳寿命。

  • 2 铝合金冲击损伤和后继疲劳寿命预估

    2
  • 2.1 2A12铝合金板的冲击和疲劳试验

    2.1

    试验件形状尺寸如图1所示。采用落锤装置制作冲击凹坑缺陷,通过标定,获得半径 r=3mm,深度d=0.4mm的凹坑,随后利用高频疲劳机对带凹坑缺陷的试样进行应力比为R=-1,最大名义应力分别为200 MPa和 230 MPa的两组疲劳试验,试验结果列于表1中。

    图1
                            2A12铝合金试样几何尺寸和冲击位置示意图

    图1 2A12铝合金试样几何尺寸和冲击位置示意图

    Fig.1 Geometrical dimension of 2A12 aluminum alloy specimen and impact position

    表1 疲劳试验结果

    Tab.1 Results of fatigue test

    名义应力/ MPa试验件个数

    疲劳寿命/

    cycle

    平均寿命/

    cycle

    标准差
    230356 10072 83311 833
    81 300
    81 100
    2005220 200252 34034 967
    232 000
    287 500
    220 500
    301 500

    2为带0.4 mm冲击凹坑缺陷的2A12试验件疲劳破坏后的断面图,疲劳断裂位置在凹坑处,说明冲击凹坑缺陷降低了材料的疲劳性能;观察试验件的断口,可以发现疲劳源出现在凹坑边缘处。

    图2
                            2A12铝合金试验件疲劳断口

    图2 2A12铝合金试验件疲劳断口

    Fig.2 Fatigue section of 2A12 aluminum alloy specimen

  • 2.2 冲击过程有限元模拟

    2.2

    利用ABAQUS软件建立与疲劳试验对应的有限元模型,如图3所示。在冲击位置附近对网格进行局部加密,模型共有单元数15 596个,单元类型为8节点减缩积分实体单元(C3D8R)。将冲头设置为刚体,铝合金板为弹塑性材料,材料参数由拉伸试验获得,见表2。冲击能量为2.64 J时,模拟获得的凹坑深度与试验符合。冲击凹坑处有很大的局部变形,残余应力很大,因此考虑了应力松弛效[21‑22]

    图3
                            2A12铝合金试样冲击模拟有限元模型

    图3 2A12铝合金试样冲击模拟有限元模型

    Fig.3 Finite element impact model of 2A12 aluminum alloy

    表2 2A12铝合金相关力学性能参数

    Tab.2 Mechanical parameters of 2A12 aluminum alloy

    E / MPaνσy/ MPaσb/MPaρ/(kgm-3)
    68 0000.334105472 780

    冲击凹坑处的残余应力分布如图4所示,可以看出,最大残余应力出现在凹坑底部和边缘。图5给出了残余应力σxx为试验件长度方向)从凹坑底部开始沿z方向(厚度方向)的变化规律,σx在凹坑底部附近主要表现为压应力,且在距离坑底表面约0.6 mm深度处最大;此后随着离凹坑底部距离的增大残余应力逐渐减小,并在接近试样底部位置时表现为拉应力状态。凹坑附近的等效塑性应变分布情况见图6,等效塑性应变主要集中在凹坑内部,且在凹坑底部位置最大。

    图4
                            冲击凹坑附近残余应力分布云图

    图4 冲击凹坑附近残余应力分布云图

    Fig.4 Distribution of residual stress around the impact pit

    图5
                            残余应力σx在凹坑底部沿z方向(厚度方向)的分布曲线

    图5 残余应力σx在凹坑底部沿z方向(厚度方向)的分布曲线

    Fig.5 Distribution curve of residual stress σx along z direction (thickness direction) at bottom of impact pit

    图6
                            冲击凹坑附近等效塑性应变分布云图

    图6 冲击凹坑附近等效塑性应变分布云图

    Fig.6 Equivalent plastic strain distribution around impact pit

  • 2.3 疲劳载荷下凹坑局部应力模拟及疲劳寿命预测

    2.3

    选择图4AB两点作为危险点,其中A点的残余拉应力最大, B点虽然是残余压应力,但该点等效塑性应变最大,并且是疲劳载荷应力集中的突出部位。这两点分别处于凹坑底部与边缘,也与实际中疲劳裂纹通常的萌生点吻合。将AB两点冲击过程中的等效应力、等效塑性应变等数据代入式(4),即可得到危险点的初始塑性损伤,见表3。塑性损伤模型和疲劳损伤模型中的材料参数参考Chen等的研[23],相关参数取值见表4

    表3 冲击凹坑缺陷危险点初始塑性损伤计算

    Tab.3 Plastic damage calculation at dangerous points of impact pit

    危险点σeq/ MPaRvΔpD0
    A496.1960.8870.1090.094
    B384.6061.4090.4860.362

    表4 塑性损伤和疲劳损伤模型中的相关材料参数

    Tab.4 Material parameters of plastic damage and fatigue damage models

    S0/ MPas0σ0/ MPaaβM0b
    1.6931370.080 11.95114 3390.003

    后继疲劳载荷的应力比为R=-1,最大名义应力分别为200 MPa和230 MPa,有限元模拟疲劳加载的最大拉伸应力状态,即分别对冲击后的模型施加200 MPa和230 MPa的拉伸载荷。图7为凹坑局部x方向(疲劳拉伸载荷方向)应力分量的分布云图,可以看出A点为拉应力状态,而B点仍处于压应力状态,因此A点为疲劳破坏点,与试验中的疲劳源位置相同。将该点的等效应力、初始损伤及相关数据代入式(10),分别计算出试样在最大名义应力200 MPa和230 MPa疲劳载荷下的预测寿命,列于表5。对比理论预测值与试验结果可以看出,基于连续损伤力学模型的预测寿命与试验结果相吻合。

    图7
                            冲击凹坑在后继疲劳最大拉伸载荷作用下的应力分布

    图7 冲击凹坑在后继疲劳最大拉伸载荷作用下的应力分布

    Fig.7 Stress distribution around impact pit under conseque-nt maximum fatigue tensile load

    表5 疲劳试验结果与损伤模型预测值对比

    Tab.5 Comparison of fatigue test results and damage model predictions

    名义应力/ MPa试验平均寿命/cycle

    预测寿命/

    cycle

    误差/ %
    200252 340309 19322.53
    23072 83383 10814.11
  • 3 结 论

    3

    基于连续损伤力学理论,对2A12铝合金的冲击损伤和后继疲劳寿命进行了评估,并结合疲劳试验验证寿命预测结果的准确性。

    (1)综合考虑了冲击造成的初始损伤、残余应力以及应力集中等多种因素对疲劳寿命的影响,预测的疲劳源位置与试验结果一致,疲劳寿命与试验值吻合,证明了损伤力学理论结合有限元分析方法可以有效评估金属材料的冲击损伤和后继疲劳寿命。

    (2)研究发现冲击凹坑缺陷的疲劳断裂危险点在凹坑的边缘处,该处的冲击残余应力以及后继加载过程中拉应力占主导地位。而凹坑底部在冲击过程中产生较大的残余压应力,对疲劳破坏有一定的抑制作用。

  • 参考文献

    • 1

      顾文标, 喻溅鉴, 邹静, 等. 直升机金属结构缺陷容限验证技术研究 [J]. 直升机技术, 2013(1):20-25.

      GU Wenbiao, YU Jianjian, ZOU Jing, et al. The study on flaw tolerance substantiation of helicopter metallic structure[J]. Journal of Helicopter Technique, 2013(1):20-25.

    • 2

      董月香, 高增梁. 疲劳寿命预测方法综述[J]. 大型铸锻件, 2006(3):39-42.

      DONG Yuexiang, GAO Zengliang. General description of the fatigue life prediction method[J]. Journal of Heavy Casting and Forging, 2006(3):39-42.

    • 3

      张莉, 程靳, 李新刚. 基于临界平面法的缺口件疲劳寿命预测方法[J]. 宇航学报, 2007, 28(4):824-826.

      ZHANG Li, CHENG Jin, LI Xingang. The fatigue life prediction method of notched specimen based on the critical plane[J]. Journal of Astronautics, 2007, 28(4):824-826.

    • 4

      GOUGH H, POLLARD H. The strength of metals under combined alternating stresses[J]. Proceedings of the Institution of Mechanical Engineers, 1935, 131(1): 3-103.

    • 5

      STULEN F, CUMMINGS H. A failure criterion for multi-axial fatigue stresses[J]. Proceedings American Society for Testing and Materials, 1954(2):822-835.

    • 6

      姚卫星. 结构疲劳寿命分析[M]. 北京: 国防工业出版社, 2002.

      YAO Weixing. Fatigue life prediction of structures[M]. Beijing: National Defense Industry Press, 2002.

    • 7

      YAO W X. Stress field intensity approach for predicting fatigue life[J]. International Journal of Fatigue, 1993, 15(3):234-245.

    • 8

      YAO W X. A verification of the assumption of anti-fatigue design[J]. International Journal of Fatigue, 2001, 23(3):271-277.

    • 9

      YUKITAKA M, TAKAHASHI K, TAKADA M, et al. Quantitative evaluation of effect of artificial small defects on torsional fatigue strength[J].Transactions of the Japan Society of Mechanical Engineers Series A, 1998, 64(618):271-277.

    • 10

      YUKITAKA M. Material defects as the basis of fatigue design[J]. International Journal of Fatigue, 2012, 41(41):2-10.

    • 11

      MARANDI S M, RAHMANI K, TAJDARI M. Foreign object damage on the leading edge of gas turbine blades[J]. Aerospace Science Technology, 2014, 33(1):65-75.

    • 12

      RUSCHAU J J, NICHOLAS T, THOMPSON S R. Influence of foreign object damage (FOD) on the fatigue life of simulated Ti-6Al-4V airfoils[J]. Intern-ational Journal of Impact Engineering, 2001, 25(3):233-250.

    • 13

      PETERS J O, RITCHIE R O. Influence of foreign-object damage on crack initiation and early crack growth during high-cycle fatigue of Ti-6Al-4V[J]. Engineering Fracture Mechanics, 2000, 67(3):193-207.

    • 14

      ZHAN Z X, HU W P, MENG Q C, et al. Continuum damage mechanics-based approach to the fatigue life prediction for 7050-T7451 aluminum alloy with impact pit[J]. International Journal of Damage Mechanics, 2016, 25(7):943-966.

    • 15

      詹志新, 佟阳, 李彬凯, 等. 考虑冲击缺陷的钛合金板的疲劳寿命预估[J]. 航空学报, 2016,37(7):2200-2207.

      ZHAN Zhixin, TONG Yang, LI Binkai, et al. Fatigue life prediction for titanium plate considering impact defect[J]. Journal of Aeronautics, 2016, 37(7):2200-2207.

    • 16

      ZHAN Z X, HU W P, SHEN F. Fatigue life calculation for a specimen with an impact pit considering impact damage, residual stress relaxation and elastic-plastic fatigue damage[J]. International Journal of Fatigue,2017(96):208-223.

    • 17

      LEMAITRE J, DESMORAT R. Engineering damage mechanics: Ductile, creep, fatigue and brittle failures[M]. USA: Springer, 2005.

    • 18

      CHABOCHE J, LESNE P. A non-linear continuous fatigue damage model[J]. Fatigue Fracture Engineering Materials Structures, 1988, 11(1):1-17.

    • 19

      DATTOMA V, GIANCANE S, NOBILE R, et al. Fatigue life prediction under variable loading based on a new non-linear continuum damage mechanics model[J]. International Journal of Fatigue, 2006, 28(2):89-95.

    • 20

      林有智, 傅高升, 李雷, 等. 钛合金非线性疲劳损伤工程模型研究[J]. 船舶力学, 2013, 17(11):1300-1308.

      LIN Youzhi, FU Gaosheng, LI Lei, et al. Study on non-linear fatigue damage engineering model for titanium alloy[J]. Journal of Ship Mechanics, 2013, 17(11):1300-1308.

    • 21

      刘大海, 黎俊初, 熊洪淼. 2A12铝合金焊接筋板件蠕变时效成型有限元分析[J]. 南昌航空航天大学学报, 2012, 26(1):28-34.

      LIU Dahai, LI Junchu, XIONG Hongmiao. Finite element analysis on creep age forming of the welded rib-web panels of 2A12 aluminum alloy[J]. Journal of Nanchang University of Aeronautics and Astronautics, 2012, 26(1):28-34.

    • 22

      侯奕冰. 2A14铝合金蠕变校形工艺数值模拟及实验研究[D]. 哈尔滨: 哈尔滨工业大学, 2014.

      HOU Yibing. Numerical simulation and experimental study creep on sizing process of 2A14 aluminum alloy[D]. Harbin: Harbin Institute of Technology, 2014.

    • 23

      CHEN X M, SUN Q, GUAN D, et al. A continuum damage mechanics approach for fatigue life prediction of open-hole metallic plate[J]. Applied Mechanics and Materials, 2014, 670/671(2):1060-1067.

倪阳

机 构:南京航空航天大学航空学院,南京,210016

Affiliation:College of Aerospace Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing, 210016, China

周储伟

机 构:南京航空航天大学航空学院,南京,210016

Affiliation:College of Aerospace Engineering, Nanjing University of Aeronautics & Astronautics, Nanjing, 210016, China

角 色:通讯作者

Role:Corresponding author

邮 箱:zcw@nuaa.edu.cn

作者简介:周储伟,男,教授,博士生导师,Email:zcw@nuaa.edu.cn。

喻溅鉴

机 构:中国直升机设计研究所,景德镇,333001

Affiliation:China Helicopter Research and Development Institute, Jingdezhen, 333001, China

邹静

机 构:中国直升机设计研究所,景德镇,333001

Affiliation:China Helicopter Research and Development Institute, Jingdezhen, 333001, China

胥橙庭

角 色:中文编辑

Role:Editor

html/njhkht/201901010/alternativeImage/2e8ebb5b-f810-4227-872b-a22cff3740a6-F001.jpg
名义应力/ MPa试验件个数

疲劳寿命/

cycle

平均寿命/

cycle

标准差
230356 10072 83311 833
81 300
81 100
2005220 200252 34034 967
232 000
287 500
220 500
301 500
html/njhkht/201901010/alternativeImage/2e8ebb5b-f810-4227-872b-a22cff3740a6-F002.jpg
html/njhkht/201901010/alternativeImage/2e8ebb5b-f810-4227-872b-a22cff3740a6-F003.jpg
E / MPaνσy/ MPaσb/MPaρ/(kgm-3)
68 0000.334105472 780
html/njhkht/201901010/alternativeImage/2e8ebb5b-f810-4227-872b-a22cff3740a6-F004.jpg
html/njhkht/201901010/alternativeImage/2e8ebb5b-f810-4227-872b-a22cff3740a6-F005.jpg
html/njhkht/201901010/alternativeImage/2e8ebb5b-f810-4227-872b-a22cff3740a6-F006.jpg
html/njhkht/201901010/alternativeImage/2e8ebb5b-f810-4227-872b-a22cff3740a6-F007.jpg
危险点σeq/ MPaRvΔpD0
A496.1960.8870.1090.094
B384.6061.4090.4860.362
S0/ MPas0σ0/ MPaaβM0b
1.6931370.080 11.95114 3390.003
名义应力/ MPa试验平均寿命/cycle

预测寿命/

cycle

误差/ %
200252 340309 19322.53
23072 83383 10814.11

图1 2A12铝合金试样几何尺寸和冲击位置示意图

Fig.1 Geometrical dimension of 2A12 aluminum alloy specimen and impact position

表1 疲劳试验结果

Tab.1 Results of fatigue test

图2 2A12铝合金试验件疲劳断口

Fig.2 Fatigue section of 2A12 aluminum alloy specimen

图3 2A12铝合金试样冲击模拟有限元模型

Fig.3 Finite element impact model of 2A12 aluminum alloy

表2 2A12铝合金相关力学性能参数

Tab.2 Mechanical parameters of 2A12 aluminum alloy

图4 冲击凹坑附近残余应力分布云图

Fig.4 Distribution of residual stress around the impact pit

图5 残余应力σx在凹坑底部沿z方向(厚度方向)的分布曲线

Fig.5 Distribution curve of residual stress σx along z direction (thickness direction) at bottom of impact pit

图6 冲击凹坑附近等效塑性应变分布云图

Fig.6 Equivalent plastic strain distribution around impact pit

图7 冲击凹坑在后继疲劳最大拉伸载荷作用下的应力分布

Fig.7 Stress distribution around impact pit under conseque-nt maximum fatigue tensile load

表3 冲击凹坑缺陷危险点初始塑性损伤计算

Tab.3 Plastic damage calculation at dangerous points of impact pit

表4 塑性损伤和疲劳损伤模型中的相关材料参数

Tab.4 Material parameters of plastic damage and fatigue damage models

表5 疲劳试验结果与损伤模型预测值对比

Tab.5 Comparison of fatigue test results and damage model predictions

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  • 参考文献

    • 1

      顾文标, 喻溅鉴, 邹静, 等. 直升机金属结构缺陷容限验证技术研究 [J]. 直升机技术, 2013(1):20-25.

      GU Wenbiao, YU Jianjian, ZOU Jing, et al. The study on flaw tolerance substantiation of helicopter metallic structure[J]. Journal of Helicopter Technique, 2013(1):20-25.

    • 2

      董月香, 高增梁. 疲劳寿命预测方法综述[J]. 大型铸锻件, 2006(3):39-42.

      DONG Yuexiang, GAO Zengliang. General description of the fatigue life prediction method[J]. Journal of Heavy Casting and Forging, 2006(3):39-42.

    • 3

      张莉, 程靳, 李新刚. 基于临界平面法的缺口件疲劳寿命预测方法[J]. 宇航学报, 2007, 28(4):824-826.

      ZHANG Li, CHENG Jin, LI Xingang. The fatigue life prediction method of notched specimen based on the critical plane[J]. Journal of Astronautics, 2007, 28(4):824-826.

    • 4

      GOUGH H, POLLARD H. The strength of metals under combined alternating stresses[J]. Proceedings of the Institution of Mechanical Engineers, 1935, 131(1): 3-103.

    • 5

      STULEN F, CUMMINGS H. A failure criterion for multi-axial fatigue stresses[J]. Proceedings American Society for Testing and Materials, 1954(2):822-835.

    • 6

      姚卫星. 结构疲劳寿命分析[M]. 北京: 国防工业出版社, 2002.

      YAO Weixing. Fatigue life prediction of structures[M]. Beijing: National Defense Industry Press, 2002.

    • 7

      YAO W X. Stress field intensity approach for predicting fatigue life[J]. International Journal of Fatigue, 1993, 15(3):234-245.

    • 8

      YAO W X. A verification of the assumption of anti-fatigue design[J]. International Journal of Fatigue, 2001, 23(3):271-277.

    • 9

      YUKITAKA M, TAKAHASHI K, TAKADA M, et al. Quantitative evaluation of effect of artificial small defects on torsional fatigue strength[J].Transactions of the Japan Society of Mechanical Engineers Series A, 1998, 64(618):271-277.

    • 10

      YUKITAKA M. Material defects as the basis of fatigue design[J]. International Journal of Fatigue, 2012, 41(41):2-10.

    • 11

      MARANDI S M, RAHMANI K, TAJDARI M. Foreign object damage on the leading edge of gas turbine blades[J]. Aerospace Science Technology, 2014, 33(1):65-75.

    • 12

      RUSCHAU J J, NICHOLAS T, THOMPSON S R. Influence of foreign object damage (FOD) on the fatigue life of simulated Ti-6Al-4V airfoils[J]. Intern-ational Journal of Impact Engineering, 2001, 25(3):233-250.

    • 13

      PETERS J O, RITCHIE R O. Influence of foreign-object damage on crack initiation and early crack growth during high-cycle fatigue of Ti-6Al-4V[J]. Engineering Fracture Mechanics, 2000, 67(3):193-207.

    • 14

      ZHAN Z X, HU W P, MENG Q C, et al. Continuum damage mechanics-based approach to the fatigue life prediction for 7050-T7451 aluminum alloy with impact pit[J]. International Journal of Damage Mechanics, 2016, 25(7):943-966.

    • 15

      詹志新, 佟阳, 李彬凯, 等. 考虑冲击缺陷的钛合金板的疲劳寿命预估[J]. 航空学报, 2016,37(7):2200-2207.

      ZHAN Zhixin, TONG Yang, LI Binkai, et al. Fatigue life prediction for titanium plate considering impact defect[J]. Journal of Aeronautics, 2016, 37(7):2200-2207.

    • 16

      ZHAN Z X, HU W P, SHEN F. Fatigue life calculation for a specimen with an impact pit considering impact damage, residual stress relaxation and elastic-plastic fatigue damage[J]. International Journal of Fatigue,2017(96):208-223.

    • 17

      LEMAITRE J, DESMORAT R. Engineering damage mechanics: Ductile, creep, fatigue and brittle failures[M]. USA: Springer, 2005.

    • 18

      CHABOCHE J, LESNE P. A non-linear continuous fatigue damage model[J]. Fatigue Fracture Engineering Materials Structures, 1988, 11(1):1-17.

    • 19

      DATTOMA V, GIANCANE S, NOBILE R, et al. Fatigue life prediction under variable loading based on a new non-linear continuum damage mechanics model[J]. International Journal of Fatigue, 2006, 28(2):89-95.

    • 20

      林有智, 傅高升, 李雷, 等. 钛合金非线性疲劳损伤工程模型研究[J]. 船舶力学, 2013, 17(11):1300-1308.

      LIN Youzhi, FU Gaosheng, LI Lei, et al. Study on non-linear fatigue damage engineering model for titanium alloy[J]. Journal of Ship Mechanics, 2013, 17(11):1300-1308.

    • 21

      刘大海, 黎俊初, 熊洪淼. 2A12铝合金焊接筋板件蠕变时效成型有限元分析[J]. 南昌航空航天大学学报, 2012, 26(1):28-34.

      LIU Dahai, LI Junchu, XIONG Hongmiao. Finite element analysis on creep age forming of the welded rib-web panels of 2A12 aluminum alloy[J]. Journal of Nanchang University of Aeronautics and Astronautics, 2012, 26(1):28-34.

    • 22

      侯奕冰. 2A14铝合金蠕变校形工艺数值模拟及实验研究[D]. 哈尔滨: 哈尔滨工业大学, 2014.

      HOU Yibing. Numerical simulation and experimental study creep on sizing process of 2A14 aluminum alloy[D]. Harbin: Harbin Institute of Technology, 2014.

    • 23

      CHEN X M, SUN Q, GUAN D, et al. A continuum damage mechanics approach for fatigue life prediction of open-hole metallic plate[J]. Applied Mechanics and Materials, 2014, 670/671(2):1060-1067.

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