Research Article A Modified Nonlinear Damage Accumulation...
8
Research Article A Modified Nonlinear Damage Accumulation Model for Fatigue Life Prediction Considering Load Interaction Effects Huiying Gao, Hong-Zhong Huang, Shun-Peng Zhu, Yan-Feng Li, and Rong Yuan School of Mechanical, Electronic, and Industrial Engineering, University of Electronic Science and Technology of China, No. 2006, Xiyuan Avenue, West Hi-Tech Zone, Chengdu, Sichuan 611731, China Correspondence should be addressed to Hong-Zhong Huang; [email protected] Received 28 August 2013; Accepted 24 October 2013; Published 20 January 2014 Academic Editors: F. Berto and K. Dincer Copyright © 2014 Huiying Gao et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Many structures are subjected to variable amplitude loading in engineering practice. e foundation of fatigue life prediction under variable amplitude loading is how to deal with the fatigue damage accumulation. A nonlinear fatigue damage accumulation model to consider the effects of load sequences was proposed in earlier literature, but the model cannot consider the load interaction effects, and sometimes it makes a major error. A modified nonlinear damage accumulation model is proposed in this paper to account for the load interaction effects. Experimental data of two metallic materials are used to validate the proposed model. e agreement between the model prediction and experimental data is observed, and the predictions by proposed model are more possibly in accordance with experimental data than that by primary model and Miner’s rule. Comparison between the predicted cumulative damage by the proposed model and an existing model shows that the proposed model predictions can meet the accuracy requirement of the engineering project and it can be used to predict the fatigue life of welded aluminum alloy joint of Electric Multiple Units (EMU); meanwhile, the accuracy of approximation can be obtained from the proposed model though more simple computing process and less material parameters calling for extensive testing than the existing model. 1. Introduction Many mechanical components experiencing cyclic loading with variable amplitude are prone to fail due to fatigue; thus, fatigue life prediction of these mechanical components has become a focal research issue. Fatigue life prediction sub- jected to variable amplitude loading is a complex problem in engineering practices. Compared to constant amplitude load- ing, it is much more intractable to deal with this sort of prob- lem. Among the problems of fatigue life prediction, one of the most important and rudimentary ones is the modeling of fatigue damage accumulation [1]. Currently, the models used to describe fatigue damage accumulation can be classified into two categories: the linear and nonlinear approaches. Palmgren-Miner rule (just the Miner’s rule for short) is the epitome of linear damage accu- mulation approach and receives extensive usage in engineer- ing machinery due to its simplicity [2]. e drawback of the Miner’s rule is the hypothesis that damage accumulation has nothing to do with the load conditions, the load sequences, the interaction between various loads, and the damage induced by stresses below the fatigue limit [3]. To remedy the drawback of the Miner’s rule, many fatigue damage accumu- lation methods have been proposed and a majority of these models are based on nonlinear accumulation laws. e nonlinear fatigue damage accumulation models can be classified into the following categories: damage curve based approaches [4], continuum damage mechanics models [5–8], interaction between various loads considered models [9–11], energy based methods [12–16], physical properties degradation based model [14, 17, 18], ductility exhaustion based methods [19, 20], and thermodynamic entropy based theories [21, 22]. Detailed comments on these models can be found in [23]. In general, load sequences and interaction effects are two important issues in the fatigue damage accumulation. A non- linear fatigue damage accumulation model which is on the basis of damage curve approach explains the influence of load sequences very well, but there is little illustration about load interaction effects. e purpose of this paper is to propose Hindawi Publishing Corporation e Scientific World Journal Volume 2014, Article ID 164378, 7 pages http://dx.doi.org/10.1155/2014/164378
Transcript of Research Article A Modified Nonlinear Damage Accumulation...
Research ArticleA Modified Nonlinear Damage Accumulation Model forFatigue Life Prediction Considering Load Interaction Effects
Huiying Gao Hong-Zhong Huang Shun-Peng Zhu Yan-Feng Li and Rong Yuan
School of Mechanical Electronic and Industrial Engineering University of Electronic Science and Technology of ChinaNo 2006 Xiyuan Avenue West Hi-Tech Zone Chengdu Sichuan 611731 China
Correspondence should be addressed to Hong-Zhong Huang hzhuanguestceducn
Received 28 August 2013 Accepted 24 October 2013 Published 20 January 2014
Academic Editors F Berto and K Dincer
Copyright copy 2014 Huiying Gao et al This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Many structures are subjected to variable amplitude loading in engineering practiceThe foundation of fatigue life prediction undervariable amplitude loading is how to deal with the fatigue damage accumulation A nonlinear fatigue damage accumulation modelto consider the effects of load sequences was proposed in earlier literature but the model cannot consider the load interactioneffects and sometimes it makes a major error A modified nonlinear damage accumulation model is proposed in this paper toaccount for the load interaction effects Experimental data of two metallic materials are used to validate the proposed model Theagreement between the model prediction and experimental data is observed and the predictions by proposed model are morepossibly in accordance with experimental data than that by primary model and Minerrsquos rule Comparison between the predictedcumulative damage by the proposedmodel and an existingmodel shows that the proposedmodel predictions canmeet the accuracyrequirement of the engineering project and it can be used to predict the fatigue life of welded aluminum alloy joint of ElectricMultiple Units (EMU) meanwhile the accuracy of approximation can be obtained from the proposed model though more simplecomputing process and less material parameters calling for extensive testing than the existing model
1 Introduction
Many mechanical components experiencing cyclic loadingwith variable amplitude are prone to fail due to fatigue thusfatigue life prediction of these mechanical components hasbecome a focal research issue Fatigue life prediction sub-jected to variable amplitude loading is a complex problem inengineering practices Compared to constant amplitude load-ing it is much more intractable to deal with this sort of prob-lem Among the problems of fatigue life prediction one of themost important and rudimentary ones is the modeling offatigue damage accumulation [1]
Currently the models used to describe fatigue damageaccumulation can be classified into two categories the linearand nonlinear approaches Palmgren-Miner rule (just theMinerrsquos rule for short) is the epitome of linear damage accu-mulation approach and receives extensive usage in engineer-ing machinery due to its simplicity [2] The drawback of theMinerrsquos rule is the hypothesis that damage accumulation hasnothing to do with the load conditions the load sequences
the interaction between various loads and the damageinduced by stresses below the fatigue limit [3] To remedy thedrawback of the Minerrsquos rule many fatigue damage accumu-lation methods have been proposed and a majority of thesemodels are based on nonlinear accumulation laws
The nonlinear fatigue damage accumulation models canbe classified into the following categories damage curvebased approaches [4] continuum damage mechanics models[5ndash8] interaction between various loads considered models[9ndash11] energy based methods [12ndash16] physical propertiesdegradation based model [14 17 18] ductility exhaustionbased methods [19 20] and thermodynamic entropy basedtheories [21 22] Detailed comments on these models can befound in [23]
In general load sequences and interaction effects are twoimportant issues in the fatigue damage accumulation A non-linear fatigue damage accumulation model which is on thebasis of damage curve approach explains the influence of loadsequences very well but there is little illustration about loadinteraction effects The purpose of this paper is to propose
Hindawi Publishing Corporatione Scientific World JournalVolume 2014 Article ID 164378 7 pageshttpdxdoiorg1011552014164378
2 The Scientific World Journal
a modified nonlinear fatigue damage accumulation modelbased ondamage curve approach to consider the load interac-tion effectsThe structure of this paper is organized as followsA nonlinear fatigue damage accumulation model proposedby Manson and Halford [4] is briefly introduced and thecomparison between predicted results through this modeland experimental data of two metallic materials is made toprovide a fundamental basis for proposing amodifiedmodelThen the same sets of experimental data are used to validatethe proposed model under two-level load conditions Finallycomparison analysis of predictions by the proposed modeland two existing models is carried out for further validatingthe accuracy of the modified model
2 Nonlinear Fatigue Damage AccumulationModel Based on Damage Curve Approach
21 Nonlinear Fatigue Damage Accumulation Model Based onDamage Curve Approach Early in 1954 Marco and Starkey[24] proposed a nonlinear fatigue damage accumulationmodel subsequently a lot of research work had been car-ried out and the approaches also were continuously beingimproved A damage accumulation model based on damagecurve approach proposed by Manson and Halford (just theManson-Halford model for short) is investigated in thissection the effects of load sequences under two-level loadingare explained very well by this modelThe detailed derivationprocess can be found in [4] only a brief introduction is givenin this section
Manson and Halford obtained the expression of cracklength through detailed deducing which can be expressed by
119886 = 1198860+ (018 minus 119886
0) (
119899119886
119873119891
)
(23)11987304
119891
(1)
Then the damage is given as a function of crack length 1198860and
cycle ratio
119863 =
1
018
[
[
1198860+ (018 minus 119886
0) (
119899119886
119873119891
)
(23)11987304
119891
]
]
(2)
where 119899119886is the number of cycles to reach a crack length
of 119886 119873119891is the number of cycles to failure and 119886
0is the
characteristic defect length of the material when 119899119886119873119891= 0
The nonlinear fatigue damage accumulationmodel undermultilevel loading can be described as follows Firstly sup-pose if a loading stress 120590
1is applied to the material for 119899
1
cycles and the damage increases from 0 to point119860 along witha damage curve Γ
1 then a different loading stress 120590
2is applied
for 1198992cycles and the damage will increase from point 119861 to 119862
along with another damage curve Γ2 Using (2) the damage
at points 119860 and 119861 can be obtained as
119863119860=
1
018
[
[
1198860+ (018 minus 119886
0) (
1198991
1198731198911
)
(23)11987304
119891
]
]
119863119861=
1
018
[
[
1198860+ (018 minus 119886
0) (
1198991015840
2
1198731198912
)
(23)11987304
119891
]
]
(3)
According to the damage characteristic of materials theaccumulated damage at point 119860 is equal to that at point 119861Through equating (3) the following expression can beobtained that is tomake the damage increase from 0 to point119861 along with the damage curve Γ
2 the following cycle ratio is
needed
1198991015840
2
1198731198912
= (
1198991
1198731198911
)
(11987311989111198731198912)04
(4)
and the cycle ratio after the loading cycles at 1205902becomes
1198991015840
2
1198731198912
+
1198992
1198731198912
= (
1198991
1198731198911
)
(11987311989111198731198912)04
+
1198992
1198731198912
(5)
Similarly the total cycle ratio after the action of theloading stress 120590
3for 1198993cycles is obtained that is
(
1198991
1198731198911
)
(11987311989111198731198912)04
+
1198992
1198731198912
(11987311989121198731198913)04
+
1198993
1198731198913
(6)
Assuming that the sum of (6) is equal to unity fatiguefailure occurs By analogy the damage accumulation ruleunder multilevel loading can be described as
[[[(
1198991
1198731198911
)
12057212
+
1198992
1198731198912
]
12057223
+
1198993
1198731198913
]
12057234
+ sdot sdot sdot +
119899119894minus1
119873119891(119894minus1)
]
120572119894minus1119894
+
119899119894
119873119891119894
= 1
(7)
where
120572119894minus1119894
= (
119873119891(119894minus1)
119873119891119894
)
04
(8)
where the subscripts 1 2 3 119894 minus 1 119894 are the sequencenumbers of loading stress 119899
1 1198992 1198993 119899
119894minus1 119899119894are the cycle
numbers under different loading stress and 1198731198911 1198731198912 1198731198913
119873119891(119894minus1)
119873119891119894
represent the fatigue life under 1205901 1205902 1205903
120590119894minus1 120590119894 respectively
When the loading stress applied at all levels is equal toeach other 119873
1198911= 1198731198912
= 1198731198913
= sdot sdot sdot = 119873119891(119894minus1)
= 119873119891119894 thus
12057212
= 12057223
= 12057234
= sdot sdot sdot = 120572119894minus1119894
= 1 therefore (7) and (8) canbe simplified into the Minerrsquos rule
For Manson-Halford model the exponent parameter 120572 isa key factor which can produce more accurate results if 120572 isdetermined adequately The detailed deterministic process ofexponent parameter120572 and thematerial constant 04was givenby Manson and Halford based on the concept of effectivemicrocosmic crack growth [4]
22 Comparison of Experimental Data and the Model Predic-tion Results In this section the predicted damage results oftwo different materials are obtained using Manson-Halfordmodel Twomaterials that is 45 and 16Mn steels respectivelyare used and tests are carried out under two-level loading
The Scientific World Journal 3
Table 1 Experimental data and the predicted results of 45 steel
Loading stress levelMpa Load sequences 1198991
11989911198731198911
1198992
11989921198731198912
(E) 11989921198731198912
(P) Error
33146ndash2844 High-low12500 0250 250400 05008 04241 minus153225000 0500 168300 03366 02411 minus283737500 0750 64500 01290 01082 minus1612
2844ndash33146 Low-high12500 0250 37900 07580 09693 2788250000 0500 38900 07780 08247 600375000 0750 43400 08680 05145 minus4073
Table 2 Experimental data and the predicted results of 16Mn steel
Loading stress levelMpa Load sequences 1198991
11989911198731198911
1198992
11989921198731198912
(E) 11989921198731198912
(P) Error
5629ndash3923 High-low1000 02520 56300 07154 03411 minus37431700 04284 476000 06048 02263 minus62572450 06174 22900 02910 01358 minus5333
37265ndash3923 Low-high64400 0240 62800 07980 09028 1313116000 0433 62900 07990 07449 minus677150000 0560 23300 02960 06118 10669
that is high-low and low-high loading For 45 steel thehigh-low loading spectrum is 33146ndash2844Mpa while thelow-high loading spectrum is 2844ndash33146Mpa For 16Mnsteel the high-low and low-high loading spectra are 5629ndash3923Mpa and 37265ndash3923Mpa respectively More detailscan be found in [25ndash27]
According to (7) and (8) the damage accumulationmodelunder two-level loading can be expressed as
(
1198991
1198731198911
)
120572
+
1198992
1198731198912
= 1 (9)
or
1198992
1198731198912
= 1 minus (
1198991
1198731198911
)
120572
120572 = (
1198731198911
1198731198912
)
04
(10)
where 1198991 1198992indicate the number of cycles at the 1st and 2nd
loading stress levels and 1198731198911
and 1198731198912
represent the fatiguefailure life at the corresponding load levels respectively
Thus the cycle ratio predictions at the 2nd loading stresslevel can be obtained from (9) and (10)The experimental dataand the model prediction results are listed in Tables 1 and 2The abbreviations in these two tables are explained as followsthe ldquoErdquo and ldquoPrdquo represent experimental andmodel predictionresults respectively
Comparing the experimental data with themodel predic-tion results of Manson-Halfordmodel it is obvious that mostpredicted results are close to the experimental data As shownin Figures 1 and 2 it should be noted that they are relativelyclose to practical situation It needs to be pointed out that inthis paper the ldquo119860-119861rdquo-shaped format in the figures indicatesthat the loading stress level changes from ldquo119860Mpardquo to ldquo119861Mpardquo
Pred
icte
d re
sults
(Man
son-
Hal
ford
mod
el)
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
33146ndash28442844ndash33146
Figure 1 Comparison of Manson-Halford model prediction resultsand experimental data for 45 steel
3 A Modified Nonlinear FatigueDamage Accumulation Model Consideringthe Load Interaction Effects
As shown in Tables 1 and 2 the Manson-Halford modelprediction results are relatively close to the experimental dataHowever it should be noted that there is a large differencebetween the experimental data and predicted value for 16Mnsteel Meanwhile under high-low loading conditions thepredicted results are lower than experiment data and mostpredictions are larger than practical value under low-highloading conditions This may be caused by consideringload sequences only without laying enough emphasis on
4 The Scientific World JournalPr
edic
ted
resu
lts (M
anso
n-H
alfo
rd m
odel
)
0
01
02
03
04
05
06
07
08
09
1
0 02 04 06 08 1
Experimental data
5629ndash392337265ndash3923
Figure 2 Comparison ofManson-Halfordmodel prediction resultsand experimental data for 16Mn steel
the influence of load interactionTherefore Manson-Halfordmodel will be modified in this paper to consider the loadinteraction effects and aforementioned problems
Based onManson-Halfordmodel Xu et al [28] suggestedthat the exponent parameter 120572
119894minus1119894should be modified to
include load amplitude and effective stress related to loadingpath whereas in (7)-(8) only the effects of load sequenceshad been taken into accountMoreover in view of the currentsituation that some existing models such as Corten-Dolanmodel and Freudenthal-Heller model are in the form ofload amplitude ratio to consider the load interaction effects[10 29] hence refer to the models in [10 28 29] to considerthe effects of load interaction and error distribution theexponent parameter 120572
119894minus1119894can be modified as
120572119894minus1119894
= (
119873119891(119894minus1)
119873119891119894
)
04sdotmin120590119894minus1120590119894 120590119894120590119894minus1
(11)
Then the damage accumulation model under two-levelloading can be described as
(
1198991
1198731198911
)
120572
+
1198992
1198731198912
= 1 (12)
where
120572 = (
1198731198911
1198731198912
)
04sdotmin12059011205902 12059021205901
(13)
For high-low loading conditions 0 lt 11987311989111198731198912
lt 1 then0 lt 120572 lt 1 therefore the damage caused by the 2nd loadingstress level should meet the following expression
1198992
1198731198912
= 1 minus (
1198991
1198731198911
)
120572
lt 1 minus
1198991
1198731198911
(14)
Hence the cumulative damage under high-low loading con-ditions is
1198991
1198731198911
+
1198992
1198731198912
=
1198991
1198731198911
+ 1 minus (
1198991
1198731198911
)
120572
lt 1 (15)
Similarly for low-high loading conditions11987311989111198731198912
gt 1120572 gt 1 then 119899
21198731198912
= 1minus (11989911198731198911)120572gt 1minus (119899
11198731198911) thus the
cumulative damage under low-high loading conditions is
1198991
1198731198911
+
1198992
1198731198912
=
1198991
1198731198911
+ 1 minus (
1198991
1198731198911
)
120572
gt 1 (16)
Therefore the model mentioned above reflects the non-linearity of damage accumulation and takes the effects of loadsequences and load interaction into account simultaneously
4 Validation of the Proposed Model
41 Validation Study 1 Theexperimental data adopted here isstill the data sets used in Section 2The comparison of experi-mental data and the model prediction results by theManson-Halford model and proposed model can be seen in Figures3 and 4 (ldquoM-H modelrdquo refers to Manson-Halford model andldquoP modelrdquo represents the proposed model) The results showthat nearly 80 of proposed model predictions are betterthan that by the Manson-Halford model and the inaccuracyunder high-low and low-high loading conditions has beenboth reduced this indicates that the predictions by proposedmodel aremore possibly in accordance with the experimentaldata Furthermore Figure 3 shows the predicted results bythe Minerrsquos rule for 45 steel it can be seen that the errorsbetween the prediction results by proposed model andexperimental value are smaller than that by Minerrsquos ruleThus the proposed model has a better prediction than theManson-Halford model and Minerrsquos rule
42 Validation Study 2 Nowadays the requirements of highspeed are the prospects and development trends of railwaypassenger and the high-speed railway also has become animportant symbol of modernization of national railway Asa kind of green transportation the safety and reliabilityproblems of high-speed train are concerned by researchersalthough the advantages are well known to everyone Atpresent all high-speed passenger car bogie frame and carbody have steel and aluminum alloy welded structures sincethe self-weight of train structure needs to be considerablyreduced but due to the harsh load conditions and the inher-ent weld defect such as geometric irregularity nonmetallicinclusion residual stress and heat-affected zone (HAZ)welded joints have been turned into the major failure areas ofhigh-speed train structures [30] Aluminum alloy materialsare widely used in the structures of trains ships construc-tions and so forth because of their low density high strengthand inoxidability Therefore it is of great significance forbetter fatigue-life prediction of high-speed train and makingsure of its safe operation to figure out an appropriate fatiguedamage accumulation method for welded aluminum alloyjoint
The Scientific World Journal 5
Table 3 Experimental data and comparison results of cumulative damage predictions by the proposed model and existing model for buttjoint
Load mode 1205901Mpa 120590
2Mpa 119899
1103 119899
2103 119873
11989111198731198912
119863 (by the existing model [31]) 119863 (by the proposed model)Mode 1 104 74 1099 7976 549300 1540100 09260 08988Mode 2 89 74 1761 10292 880500 1540100 10810 09372Mode 3 74 89 7701 5456 1540100 880500 09290 10660Mode 4 74 104 7701 4189 1540100 549300 10140 11053
Table 4 Experimental data and comparison results of cumulative damage predictions by the proposed model and existing model for filletjoint
Load mode 1205901Mpa 120590
2Mpa 119899
1103 119899
2103 119873
11989111198731198912
119863 (by the existing model in [31]) 119863 (by the proposed model)Mode 5 93 73 3099 5875 619800 1546100 10140 09056Mode 6 83 73 4761 6811 952300 1546100 10270 09426Mode 7 73 83 5092 7082 1546100 952300 09930 10614Mode 8 73 93 7730 4264 1546100 619800 10670 11029
Pred
icte
d re
sults
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
33146ndash2844 (M-H model)2844ndash33146 (M-H model)33146ndash2844 (P model)
2844ndash33146 (P model)33146ndash2844 (Minerrsquos rule)2844ndash33146 (Minerrsquos rule)
Figure 3 Comparison of prediction results of the proposed modelMinerrsquos rule and experimental data for 45 steel
In the light of the above experimental data of weldedaluminum alloy joint of Electric Multiple Units (EMU) areused in this section to verify the applicability of the proposedmodel in the fields of high-speed train There are two sortsof welded aluminum alloy joint used in this section that isbutt joint and fillet joint and the tests are also carried outunder two-level loading The experimental data of EMU arelisted in Tables 3 and 4 In addition comparison between theexperimental data and predictions by proposed model andan existing model is carried out for further validating theaccuracy of the proposed model
According to the existing model proposed in [31] thefatigue damage curve is
119863 = (
119899
119873
)
1+(log12(120590120590119904))
119905
(17)
Pred
icte
d re
sults
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
5629ndash3923 (M-H model)372653ndash3923 (M-H model)
5629ndash3923 (proposed model)37265ndash3923 (proposed model)
Figure 4 Comparison of prediction results of the proposed modeland experimental data for 16Mn steel
Hence the damage accumulation model under two-levelloading is given by
119863 =
2
sum
119894=1
119863119894= [
1198992
1198731198912
+ (
1198991
1198731198911
)
11988611198862
]
1198862
1198861= 1 + (log
12
1205901
120590119904
)
119905
1198862= 1 + (log
12
1205902
120590119904
)
119905
(18)
where 119863 is the cumulative damage 119886119894is the life damage
exponent under the 119894th level load 1198991 1198992and 119873
1198911 1198731198912
represent the same meaning as mentioned above 120590119904refers to
material yield strength and 119905 reflects the influence degree of
6 The Scientific World Journal
load sequences effects of the specimen its determination isby means of fitting according to experimental data
As aformentioned the cumulative damage can be cal-culated in different way unlike the modified model Thecomparison results of cumulative damage predictions by theexistingmodel and proposedmodel are shown inTables 3 and4
From Tables 3 and 4 it is obvious that the cumulativedamage predictions by the proposed model exceed unityunder low-high loading conditions andwhen the load ampli-tude changes from high to low level the value is less thanunity this verifies the nonlinearity effect of damage accumu-lation Moreover note that the prediction errors of the pro-posed model are within the range of 20 this can meet theaccuracy requirement of the engineering project so the pro-posed model can accurately predict the value of critical dam-age and it can be used to determine the fatigue-life of weldedaluminum alloy joint of Electric Multiple Units (EMU)Furthermore the prediction results of cumulative damagecalculated by these two models are relatively close to eachother thus the inaccuracy of these two models predictionsis also much close This shows that cumulative damage pre-dictions by the proposedmodel correspond approximately tothe prediction results by the existing model but do not needsome precise parameters related to material property andrequired experimental data regression Therefore using theproposed model fatigue life prediction can be obtained withhigh-precision simple computing process and less materialparameters than the existing model
5 Conclusion and Discussion
A modified model is presented in this paper for consideringload interaction and load sequences effects on the basis ofa nonlinear cumulative damage model that is Manson-Halford model The main achievements and conclusions canbe summarized as follows
(1) The exponent parameter in Manson-Halford modelhas beenmodified to consider the effects of load inter-action which can be characterized by introducing theratio of applied load amplitude
(2) The experimental data of 45 steel and 16Mn steelare used to validate the accuracy of the modifiedmodel through comparing with the predicted resultsof Manson-Halford model and the proposed modelThrough comparative analysis it is worth notingthat the inaccuracy of the proposed model has beenreduced not only under high-low loading conditionsbut also to the contrary and nearly 80 of proposedmodel predictions are better than that by Manson-Halford model On the other hand the inaccuracycaused by the proposed model is smaller than thatby Minerrsquos rule for 45 steel therefore fatigue lifeprediction by the proposed model is more possibly inaccordance with the practical situation thanManson-Halford model and Minerrsquos rule
(3) Comparing cumulative damage predictions by theproposed model with the results through an existing
model it can be found that the prediction results ofthe proposed model can reflect the nonlinearity ofdamage accumulation and this proposed model inthis paper is applicable to determine the fatigue lifeof welded aluminum alloy joint of Electric MultipleUnits (EMU) because it can be used to accuratelypredict the value of critical damage Meanwhile thereis good consistency among these two models thatis fatigue life prediction can be obtained with high-precision simple computing process and lessmaterialparameters than the existing model
Although the results are quite close to the experimentaldata all validations are carried out under two-level loadingthus there is a requirement for demonstrating the validationunder multi-level and random loading conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the partial supportprovided by the National Natural Science Foundation ofChina under the Contract no 11272082 and the FundamentalResearch Funds for the Central Universities under the Con-tract no E022050205
References
[1] K Ni and S K Zhang ldquoFatigue reliability analysis under twostage loadingrdquo ACTA Mechanica Sinica vol 31 no 1 pp 106ndash112 1999
[2] M AMiner ldquoCumulative damage in fatiguerdquo Journal of AppliedMechanics vol 12 no 3 pp 159ndash164 1945
[3] S-P ZhuH-ZHuang andZ-LWang ldquoFatigue life estimationconsidering damaging and strengthening of low amplitudeloads under different load sequences using fuzzy sets approachrdquoInternational Journal of Damage Mechanics vol 20 no 6 pp876ndash899 2011
[4] S S Manson and G R Halford ldquoPractical implementation ofthe double linear damage rule and damage curve approach fortreating cumulative fatigue damagerdquo International Journal ofFracture vol 17 no 2 pp 169ndash192 1981
[5] J L Chaboche and P M Lesne ldquoA non-linear continuousfatigue damage modelrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 11 no 1 pp 1ndash17 1988
[6] G Cheng and A Plumtree ldquoA fatigue damage accumulationmodel based on continuum damage mechanics and ductilityexhaustionrdquo International Journal of Fatigue vol 20 no 7 pp495ndash501 1998
[7] V Dattoma S Giancane R Nobile and F W Panella ldquoFatiguelife prediction under variable loading based on a newnon-linearcontinuum damage mechanics modelrdquo International Journal ofFatigue vol 28 no 2 pp 89ndash95 2006
[8] J Besson ldquoContinuum models of ductile fracture a reviewrdquoInternational Journal of Damage Mechanics vol 19 no 1 pp 3ndash52 2010
The Scientific World Journal 7
[9] H Xu Fatigue Strength Higher Education Press Beijing China1988
[10] A M Freudenthal and R A Heller ldquoOn stress interaction infatigue and cumulative damage rulerdquo Journal of the AerospaceScience vol 26 no 7 pp 431ndash442 1959
[11] S P Zhu H Z Huang Y Liu et al ldquoA practical method fordetermining the Corten-Dolan exponent and its application tofatigue life predictionrdquo International Journal of Turbo and JetEngines vol 29 no 2 pp 79ndash87 2012
[12] G R Halford ldquoThe energy required for fatiguerdquo Journal ofMaterials vol 1 no 1 pp 3ndash18 1966
[13] N Xiaode L Guangxia and LHao ldquoHardening law and fatiguedamage of a cyclic hardening metalrdquo Engineering FractureMechanics vol 26 no 2 pp 163ndash170 1987
[14] S-P Zhu and H-Z Huang ldquoA generalized frequency separa-tion-strain energy damage functionmodel for low cycle fatigue-creep life predictionrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 33 no 4 pp 227ndash237 2010
[15] S P Zhu H Z Huang L P He et al ldquoA generalized energy-based fatigue-creep damage parameter for life prediction ofturbine disk alloysrdquo Fracture Mechanics vol 90 pp 89ndash1002012
[16] S P Zhu H Z Huang V Ontiveros L P He and MModarres ldquoProbabilistic low cycle fatigue life prediction usingan energy-based damage parameter and accounting for modeluncertaintyrdquo International Journal of DamageMechanics vol 21no 8 pp 1128ndash1153 2012
[17] Y Duyi and W Zhenlin ldquoA new approach to low-cycle fatiguedamage based on exhaustion of static toughness and dissipationof cyclic plastic strain energy during fatiguerdquo InternationalJournal of Fatigue vol 23 no 8 pp 679ndash687 2001
[18] S Zhu HHuang and L Xie ldquoNonlinear fatigue damage cumu-lative model and the analysis of strength degradation basedon the double parameter fatigue criterionrdquo China MechanicalEngineering vol 19 no 22 pp 2753ndash2761 2008
[19] S-P Zhu H-Z Huang H Li R Sun and M J Zuo ldquoA newductility exhaustion model for high temperature low cyclefatigue life prediction of turbine disk alloysrdquo InternationalJournal of Turbo and Jet Engines vol 28 no 2 pp 119ndash131 2011
[20] S P Zhu H Z Huang Y Liu et al ldquoAn efficient life predictionmethodology for low cycle fatigue-creep based on ductilityexhaustion theoryrdquo International Journal of Damage Mechanicsvol 22 no 4 pp 556ndash571 2013
[21] A Risitano and G Risitano ldquoCumulative damage evaluationof steel using infrared thermographyrdquo Theoretical and AppliedFracture Mechanics vol 54 no 2 pp 82ndash90 2010
[22] M Naderi M Amiri and M M Khonsari ldquoOn the thermo-dynamic entropy of fatigue fracturerdquo Proceedings of the RoyalSociety A vol 466 no 2114 pp 423ndash438 2010
[23] XHYangWX Yao andCMDuan ldquoThe reviewof ascertain-able fatigue cumulative damage rulerdquo Engineering Science vol5 no 4 pp 82ndash87 2003
[24] S M Marco and W L Starkey ldquoA concept of fatigue damagerdquoTransaction of the ASME vol 76 pp 627ndash632 1954
[25] D G Shang and W X Yao ldquoStudy on nonlinear continuousdamage cumulative model for uniaxial fatiguerdquo Acta Aeronau-tica Et Astronautica Sinica vol 19 no 6 pp 647ndash656 1998
[26] D Y Ye Variation in fatigue properties of structural steel andresearch of new approach for fatigue life prediction [PhD thesis]Northeastern University 1996
[27] D-G Shang and W-X Yao ldquoA nonlinear damage cumulativemodel for uniaxial fatiguerdquo International Journal of Fatigue vol21 no 2 pp 187ndash194 1999
[28] J Xu D G Shang G Q Sun H Chen and E T Liu ldquoFatiguelife prediction for GH4169 superalloy under multiaxial variableamplitude loadingrdquo Journal of Beijing University of Technologyvol 38 no 10 pp 1462ndash1466 2012
[29] H T Corten and T J Dolon ldquoCumulative fatigue damagerdquo inProceedings of the International Conference on Fatigue of Metalspp 235ndash246 1956
[30] Y H Lu Study on fatigue reliability of welded bogie frame forraiway vehicle [PhD thesis] Southwest Jiaotong University2011
[31] J Tian Z-M Liu and R He ldquoNonlinear fatigue-cumulativedamage model for welded aluminum alloy joint of EMUrdquoJournal of the China Railway Society vol 34 no 3 pp 40ndash432012
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2 The Scientific World Journal
a modified nonlinear fatigue damage accumulation modelbased ondamage curve approach to consider the load interac-tion effectsThe structure of this paper is organized as followsA nonlinear fatigue damage accumulation model proposedby Manson and Halford [4] is briefly introduced and thecomparison between predicted results through this modeland experimental data of two metallic materials is made toprovide a fundamental basis for proposing amodifiedmodelThen the same sets of experimental data are used to validatethe proposed model under two-level load conditions Finallycomparison analysis of predictions by the proposed modeland two existing models is carried out for further validatingthe accuracy of the modified model
2 Nonlinear Fatigue Damage AccumulationModel Based on Damage Curve Approach
21 Nonlinear Fatigue Damage Accumulation Model Based onDamage Curve Approach Early in 1954 Marco and Starkey[24] proposed a nonlinear fatigue damage accumulationmodel subsequently a lot of research work had been car-ried out and the approaches also were continuously beingimproved A damage accumulation model based on damagecurve approach proposed by Manson and Halford (just theManson-Halford model for short) is investigated in thissection the effects of load sequences under two-level loadingare explained very well by this modelThe detailed derivationprocess can be found in [4] only a brief introduction is givenin this section
Manson and Halford obtained the expression of cracklength through detailed deducing which can be expressed by
119886 = 1198860+ (018 minus 119886
0) (
119899119886
119873119891
)
(23)11987304
119891
(1)
Then the damage is given as a function of crack length 1198860and
cycle ratio
119863 =
1
018
[
[
1198860+ (018 minus 119886
0) (
119899119886
119873119891
)
(23)11987304
119891
]
]
(2)
where 119899119886is the number of cycles to reach a crack length
of 119886 119873119891is the number of cycles to failure and 119886
0is the
characteristic defect length of the material when 119899119886119873119891= 0
The nonlinear fatigue damage accumulationmodel undermultilevel loading can be described as follows Firstly sup-pose if a loading stress 120590
1is applied to the material for 119899
1
cycles and the damage increases from 0 to point119860 along witha damage curve Γ
1 then a different loading stress 120590
2is applied
for 1198992cycles and the damage will increase from point 119861 to 119862
along with another damage curve Γ2 Using (2) the damage
at points 119860 and 119861 can be obtained as
119863119860=
1
018
[
[
1198860+ (018 minus 119886
0) (
1198991
1198731198911
)
(23)11987304
119891
]
]
119863119861=
1
018
[
[
1198860+ (018 minus 119886
0) (
1198991015840
2
1198731198912
)
(23)11987304
119891
]
]
(3)
According to the damage characteristic of materials theaccumulated damage at point 119860 is equal to that at point 119861Through equating (3) the following expression can beobtained that is tomake the damage increase from 0 to point119861 along with the damage curve Γ
2 the following cycle ratio is
needed
1198991015840
2
1198731198912
= (
1198991
1198731198911
)
(11987311989111198731198912)04
(4)
and the cycle ratio after the loading cycles at 1205902becomes
1198991015840
2
1198731198912
+
1198992
1198731198912
= (
1198991
1198731198911
)
(11987311989111198731198912)04
+
1198992
1198731198912
(5)
Similarly the total cycle ratio after the action of theloading stress 120590
3for 1198993cycles is obtained that is
(
1198991
1198731198911
)
(11987311989111198731198912)04
+
1198992
1198731198912
(11987311989121198731198913)04
+
1198993
1198731198913
(6)
Assuming that the sum of (6) is equal to unity fatiguefailure occurs By analogy the damage accumulation ruleunder multilevel loading can be described as
[[[(
1198991
1198731198911
)
12057212
+
1198992
1198731198912
]
12057223
+
1198993
1198731198913
]
12057234
+ sdot sdot sdot +
119899119894minus1
119873119891(119894minus1)
]
120572119894minus1119894
+
119899119894
119873119891119894
= 1
(7)
where
120572119894minus1119894
= (
119873119891(119894minus1)
119873119891119894
)
04
(8)
where the subscripts 1 2 3 119894 minus 1 119894 are the sequencenumbers of loading stress 119899
1 1198992 1198993 119899
119894minus1 119899119894are the cycle
numbers under different loading stress and 1198731198911 1198731198912 1198731198913
119873119891(119894minus1)
119873119891119894
represent the fatigue life under 1205901 1205902 1205903
120590119894minus1 120590119894 respectively
When the loading stress applied at all levels is equal toeach other 119873
1198911= 1198731198912
= 1198731198913
= sdot sdot sdot = 119873119891(119894minus1)
= 119873119891119894 thus
12057212
= 12057223
= 12057234
= sdot sdot sdot = 120572119894minus1119894
= 1 therefore (7) and (8) canbe simplified into the Minerrsquos rule
For Manson-Halford model the exponent parameter 120572 isa key factor which can produce more accurate results if 120572 isdetermined adequately The detailed deterministic process ofexponent parameter120572 and thematerial constant 04was givenby Manson and Halford based on the concept of effectivemicrocosmic crack growth [4]
22 Comparison of Experimental Data and the Model Predic-tion Results In this section the predicted damage results oftwo different materials are obtained using Manson-Halfordmodel Twomaterials that is 45 and 16Mn steels respectivelyare used and tests are carried out under two-level loading
The Scientific World Journal 3
Table 1 Experimental data and the predicted results of 45 steel
Loading stress levelMpa Load sequences 1198991
11989911198731198911
1198992
11989921198731198912
(E) 11989921198731198912
(P) Error
33146ndash2844 High-low12500 0250 250400 05008 04241 minus153225000 0500 168300 03366 02411 minus283737500 0750 64500 01290 01082 minus1612
2844ndash33146 Low-high12500 0250 37900 07580 09693 2788250000 0500 38900 07780 08247 600375000 0750 43400 08680 05145 minus4073
Table 2 Experimental data and the predicted results of 16Mn steel
Loading stress levelMpa Load sequences 1198991
11989911198731198911
1198992
11989921198731198912
(E) 11989921198731198912
(P) Error
5629ndash3923 High-low1000 02520 56300 07154 03411 minus37431700 04284 476000 06048 02263 minus62572450 06174 22900 02910 01358 minus5333
37265ndash3923 Low-high64400 0240 62800 07980 09028 1313116000 0433 62900 07990 07449 minus677150000 0560 23300 02960 06118 10669
that is high-low and low-high loading For 45 steel thehigh-low loading spectrum is 33146ndash2844Mpa while thelow-high loading spectrum is 2844ndash33146Mpa For 16Mnsteel the high-low and low-high loading spectra are 5629ndash3923Mpa and 37265ndash3923Mpa respectively More detailscan be found in [25ndash27]
According to (7) and (8) the damage accumulationmodelunder two-level loading can be expressed as
(
1198991
1198731198911
)
120572
+
1198992
1198731198912
= 1 (9)
or
1198992
1198731198912
= 1 minus (
1198991
1198731198911
)
120572
120572 = (
1198731198911
1198731198912
)
04
(10)
where 1198991 1198992indicate the number of cycles at the 1st and 2nd
loading stress levels and 1198731198911
and 1198731198912
represent the fatiguefailure life at the corresponding load levels respectively
Thus the cycle ratio predictions at the 2nd loading stresslevel can be obtained from (9) and (10)The experimental dataand the model prediction results are listed in Tables 1 and 2The abbreviations in these two tables are explained as followsthe ldquoErdquo and ldquoPrdquo represent experimental andmodel predictionresults respectively
Comparing the experimental data with themodel predic-tion results of Manson-Halfordmodel it is obvious that mostpredicted results are close to the experimental data As shownin Figures 1 and 2 it should be noted that they are relativelyclose to practical situation It needs to be pointed out that inthis paper the ldquo119860-119861rdquo-shaped format in the figures indicatesthat the loading stress level changes from ldquo119860Mpardquo to ldquo119861Mpardquo
Pred
icte
d re
sults
(Man
son-
Hal
ford
mod
el)
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
33146ndash28442844ndash33146
Figure 1 Comparison of Manson-Halford model prediction resultsand experimental data for 45 steel
3 A Modified Nonlinear FatigueDamage Accumulation Model Consideringthe Load Interaction Effects
As shown in Tables 1 and 2 the Manson-Halford modelprediction results are relatively close to the experimental dataHowever it should be noted that there is a large differencebetween the experimental data and predicted value for 16Mnsteel Meanwhile under high-low loading conditions thepredicted results are lower than experiment data and mostpredictions are larger than practical value under low-highloading conditions This may be caused by consideringload sequences only without laying enough emphasis on
4 The Scientific World JournalPr
edic
ted
resu
lts (M
anso
n-H
alfo
rd m
odel
)
0
01
02
03
04
05
06
07
08
09
1
0 02 04 06 08 1
Experimental data
5629ndash392337265ndash3923
Figure 2 Comparison ofManson-Halfordmodel prediction resultsand experimental data for 16Mn steel
the influence of load interactionTherefore Manson-Halfordmodel will be modified in this paper to consider the loadinteraction effects and aforementioned problems
Based onManson-Halfordmodel Xu et al [28] suggestedthat the exponent parameter 120572
119894minus1119894should be modified to
include load amplitude and effective stress related to loadingpath whereas in (7)-(8) only the effects of load sequenceshad been taken into accountMoreover in view of the currentsituation that some existing models such as Corten-Dolanmodel and Freudenthal-Heller model are in the form ofload amplitude ratio to consider the load interaction effects[10 29] hence refer to the models in [10 28 29] to considerthe effects of load interaction and error distribution theexponent parameter 120572
119894minus1119894can be modified as
120572119894minus1119894
= (
119873119891(119894minus1)
119873119891119894
)
04sdotmin120590119894minus1120590119894 120590119894120590119894minus1
(11)
Then the damage accumulation model under two-levelloading can be described as
(
1198991
1198731198911
)
120572
+
1198992
1198731198912
= 1 (12)
where
120572 = (
1198731198911
1198731198912
)
04sdotmin12059011205902 12059021205901
(13)
For high-low loading conditions 0 lt 11987311989111198731198912
lt 1 then0 lt 120572 lt 1 therefore the damage caused by the 2nd loadingstress level should meet the following expression
1198992
1198731198912
= 1 minus (
1198991
1198731198911
)
120572
lt 1 minus
1198991
1198731198911
(14)
Hence the cumulative damage under high-low loading con-ditions is
1198991
1198731198911
+
1198992
1198731198912
=
1198991
1198731198911
+ 1 minus (
1198991
1198731198911
)
120572
lt 1 (15)
Similarly for low-high loading conditions11987311989111198731198912
gt 1120572 gt 1 then 119899
21198731198912
= 1minus (11989911198731198911)120572gt 1minus (119899
11198731198911) thus the
cumulative damage under low-high loading conditions is
1198991
1198731198911
+
1198992
1198731198912
=
1198991
1198731198911
+ 1 minus (
1198991
1198731198911
)
120572
gt 1 (16)
Therefore the model mentioned above reflects the non-linearity of damage accumulation and takes the effects of loadsequences and load interaction into account simultaneously
4 Validation of the Proposed Model
41 Validation Study 1 Theexperimental data adopted here isstill the data sets used in Section 2The comparison of experi-mental data and the model prediction results by theManson-Halford model and proposed model can be seen in Figures3 and 4 (ldquoM-H modelrdquo refers to Manson-Halford model andldquoP modelrdquo represents the proposed model) The results showthat nearly 80 of proposed model predictions are betterthan that by the Manson-Halford model and the inaccuracyunder high-low and low-high loading conditions has beenboth reduced this indicates that the predictions by proposedmodel aremore possibly in accordance with the experimentaldata Furthermore Figure 3 shows the predicted results bythe Minerrsquos rule for 45 steel it can be seen that the errorsbetween the prediction results by proposed model andexperimental value are smaller than that by Minerrsquos ruleThus the proposed model has a better prediction than theManson-Halford model and Minerrsquos rule
42 Validation Study 2 Nowadays the requirements of highspeed are the prospects and development trends of railwaypassenger and the high-speed railway also has become animportant symbol of modernization of national railway Asa kind of green transportation the safety and reliabilityproblems of high-speed train are concerned by researchersalthough the advantages are well known to everyone Atpresent all high-speed passenger car bogie frame and carbody have steel and aluminum alloy welded structures sincethe self-weight of train structure needs to be considerablyreduced but due to the harsh load conditions and the inher-ent weld defect such as geometric irregularity nonmetallicinclusion residual stress and heat-affected zone (HAZ)welded joints have been turned into the major failure areas ofhigh-speed train structures [30] Aluminum alloy materialsare widely used in the structures of trains ships construc-tions and so forth because of their low density high strengthand inoxidability Therefore it is of great significance forbetter fatigue-life prediction of high-speed train and makingsure of its safe operation to figure out an appropriate fatiguedamage accumulation method for welded aluminum alloyjoint
The Scientific World Journal 5
Table 3 Experimental data and comparison results of cumulative damage predictions by the proposed model and existing model for buttjoint
Load mode 1205901Mpa 120590
2Mpa 119899
1103 119899
2103 119873
11989111198731198912
119863 (by the existing model [31]) 119863 (by the proposed model)Mode 1 104 74 1099 7976 549300 1540100 09260 08988Mode 2 89 74 1761 10292 880500 1540100 10810 09372Mode 3 74 89 7701 5456 1540100 880500 09290 10660Mode 4 74 104 7701 4189 1540100 549300 10140 11053
Table 4 Experimental data and comparison results of cumulative damage predictions by the proposed model and existing model for filletjoint
Load mode 1205901Mpa 120590
2Mpa 119899
1103 119899
2103 119873
11989111198731198912
119863 (by the existing model in [31]) 119863 (by the proposed model)Mode 5 93 73 3099 5875 619800 1546100 10140 09056Mode 6 83 73 4761 6811 952300 1546100 10270 09426Mode 7 73 83 5092 7082 1546100 952300 09930 10614Mode 8 73 93 7730 4264 1546100 619800 10670 11029
Pred
icte
d re
sults
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
33146ndash2844 (M-H model)2844ndash33146 (M-H model)33146ndash2844 (P model)
2844ndash33146 (P model)33146ndash2844 (Minerrsquos rule)2844ndash33146 (Minerrsquos rule)
Figure 3 Comparison of prediction results of the proposed modelMinerrsquos rule and experimental data for 45 steel
In the light of the above experimental data of weldedaluminum alloy joint of Electric Multiple Units (EMU) areused in this section to verify the applicability of the proposedmodel in the fields of high-speed train There are two sortsof welded aluminum alloy joint used in this section that isbutt joint and fillet joint and the tests are also carried outunder two-level loading The experimental data of EMU arelisted in Tables 3 and 4 In addition comparison between theexperimental data and predictions by proposed model andan existing model is carried out for further validating theaccuracy of the proposed model
According to the existing model proposed in [31] thefatigue damage curve is
119863 = (
119899
119873
)
1+(log12(120590120590119904))
119905
(17)
Pred
icte
d re
sults
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
5629ndash3923 (M-H model)372653ndash3923 (M-H model)
5629ndash3923 (proposed model)37265ndash3923 (proposed model)
Figure 4 Comparison of prediction results of the proposed modeland experimental data for 16Mn steel
Hence the damage accumulation model under two-levelloading is given by
119863 =
2
sum
119894=1
119863119894= [
1198992
1198731198912
+ (
1198991
1198731198911
)
11988611198862
]
1198862
1198861= 1 + (log
12
1205901
120590119904
)
119905
1198862= 1 + (log
12
1205902
120590119904
)
119905
(18)
where 119863 is the cumulative damage 119886119894is the life damage
exponent under the 119894th level load 1198991 1198992and 119873
1198911 1198731198912
represent the same meaning as mentioned above 120590119904refers to
material yield strength and 119905 reflects the influence degree of
6 The Scientific World Journal
load sequences effects of the specimen its determination isby means of fitting according to experimental data
As aformentioned the cumulative damage can be cal-culated in different way unlike the modified model Thecomparison results of cumulative damage predictions by theexistingmodel and proposedmodel are shown inTables 3 and4
From Tables 3 and 4 it is obvious that the cumulativedamage predictions by the proposed model exceed unityunder low-high loading conditions andwhen the load ampli-tude changes from high to low level the value is less thanunity this verifies the nonlinearity effect of damage accumu-lation Moreover note that the prediction errors of the pro-posed model are within the range of 20 this can meet theaccuracy requirement of the engineering project so the pro-posed model can accurately predict the value of critical dam-age and it can be used to determine the fatigue-life of weldedaluminum alloy joint of Electric Multiple Units (EMU)Furthermore the prediction results of cumulative damagecalculated by these two models are relatively close to eachother thus the inaccuracy of these two models predictionsis also much close This shows that cumulative damage pre-dictions by the proposedmodel correspond approximately tothe prediction results by the existing model but do not needsome precise parameters related to material property andrequired experimental data regression Therefore using theproposed model fatigue life prediction can be obtained withhigh-precision simple computing process and less materialparameters than the existing model
5 Conclusion and Discussion
A modified model is presented in this paper for consideringload interaction and load sequences effects on the basis ofa nonlinear cumulative damage model that is Manson-Halford model The main achievements and conclusions canbe summarized as follows
(1) The exponent parameter in Manson-Halford modelhas beenmodified to consider the effects of load inter-action which can be characterized by introducing theratio of applied load amplitude
(2) The experimental data of 45 steel and 16Mn steelare used to validate the accuracy of the modifiedmodel through comparing with the predicted resultsof Manson-Halford model and the proposed modelThrough comparative analysis it is worth notingthat the inaccuracy of the proposed model has beenreduced not only under high-low loading conditionsbut also to the contrary and nearly 80 of proposedmodel predictions are better than that by Manson-Halford model On the other hand the inaccuracycaused by the proposed model is smaller than thatby Minerrsquos rule for 45 steel therefore fatigue lifeprediction by the proposed model is more possibly inaccordance with the practical situation thanManson-Halford model and Minerrsquos rule
(3) Comparing cumulative damage predictions by theproposed model with the results through an existing
model it can be found that the prediction results ofthe proposed model can reflect the nonlinearity ofdamage accumulation and this proposed model inthis paper is applicable to determine the fatigue lifeof welded aluminum alloy joint of Electric MultipleUnits (EMU) because it can be used to accuratelypredict the value of critical damage Meanwhile thereis good consistency among these two models thatis fatigue life prediction can be obtained with high-precision simple computing process and lessmaterialparameters than the existing model
Although the results are quite close to the experimentaldata all validations are carried out under two-level loadingthus there is a requirement for demonstrating the validationunder multi-level and random loading conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the partial supportprovided by the National Natural Science Foundation ofChina under the Contract no 11272082 and the FundamentalResearch Funds for the Central Universities under the Con-tract no E022050205
References
[1] K Ni and S K Zhang ldquoFatigue reliability analysis under twostage loadingrdquo ACTA Mechanica Sinica vol 31 no 1 pp 106ndash112 1999
[2] M AMiner ldquoCumulative damage in fatiguerdquo Journal of AppliedMechanics vol 12 no 3 pp 159ndash164 1945
[3] S-P ZhuH-ZHuang andZ-LWang ldquoFatigue life estimationconsidering damaging and strengthening of low amplitudeloads under different load sequences using fuzzy sets approachrdquoInternational Journal of Damage Mechanics vol 20 no 6 pp876ndash899 2011
[4] S S Manson and G R Halford ldquoPractical implementation ofthe double linear damage rule and damage curve approach fortreating cumulative fatigue damagerdquo International Journal ofFracture vol 17 no 2 pp 169ndash192 1981
[5] J L Chaboche and P M Lesne ldquoA non-linear continuousfatigue damage modelrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 11 no 1 pp 1ndash17 1988
[6] G Cheng and A Plumtree ldquoA fatigue damage accumulationmodel based on continuum damage mechanics and ductilityexhaustionrdquo International Journal of Fatigue vol 20 no 7 pp495ndash501 1998
[7] V Dattoma S Giancane R Nobile and F W Panella ldquoFatiguelife prediction under variable loading based on a newnon-linearcontinuum damage mechanics modelrdquo International Journal ofFatigue vol 28 no 2 pp 89ndash95 2006
[8] J Besson ldquoContinuum models of ductile fracture a reviewrdquoInternational Journal of Damage Mechanics vol 19 no 1 pp 3ndash52 2010
The Scientific World Journal 7
[9] H Xu Fatigue Strength Higher Education Press Beijing China1988
[10] A M Freudenthal and R A Heller ldquoOn stress interaction infatigue and cumulative damage rulerdquo Journal of the AerospaceScience vol 26 no 7 pp 431ndash442 1959
[11] S P Zhu H Z Huang Y Liu et al ldquoA practical method fordetermining the Corten-Dolan exponent and its application tofatigue life predictionrdquo International Journal of Turbo and JetEngines vol 29 no 2 pp 79ndash87 2012
[12] G R Halford ldquoThe energy required for fatiguerdquo Journal ofMaterials vol 1 no 1 pp 3ndash18 1966
[13] N Xiaode L Guangxia and LHao ldquoHardening law and fatiguedamage of a cyclic hardening metalrdquo Engineering FractureMechanics vol 26 no 2 pp 163ndash170 1987
[14] S-P Zhu and H-Z Huang ldquoA generalized frequency separa-tion-strain energy damage functionmodel for low cycle fatigue-creep life predictionrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 33 no 4 pp 227ndash237 2010
[15] S P Zhu H Z Huang L P He et al ldquoA generalized energy-based fatigue-creep damage parameter for life prediction ofturbine disk alloysrdquo Fracture Mechanics vol 90 pp 89ndash1002012
[16] S P Zhu H Z Huang V Ontiveros L P He and MModarres ldquoProbabilistic low cycle fatigue life prediction usingan energy-based damage parameter and accounting for modeluncertaintyrdquo International Journal of DamageMechanics vol 21no 8 pp 1128ndash1153 2012
[17] Y Duyi and W Zhenlin ldquoA new approach to low-cycle fatiguedamage based on exhaustion of static toughness and dissipationof cyclic plastic strain energy during fatiguerdquo InternationalJournal of Fatigue vol 23 no 8 pp 679ndash687 2001
[18] S Zhu HHuang and L Xie ldquoNonlinear fatigue damage cumu-lative model and the analysis of strength degradation basedon the double parameter fatigue criterionrdquo China MechanicalEngineering vol 19 no 22 pp 2753ndash2761 2008
[19] S-P Zhu H-Z Huang H Li R Sun and M J Zuo ldquoA newductility exhaustion model for high temperature low cyclefatigue life prediction of turbine disk alloysrdquo InternationalJournal of Turbo and Jet Engines vol 28 no 2 pp 119ndash131 2011
[20] S P Zhu H Z Huang Y Liu et al ldquoAn efficient life predictionmethodology for low cycle fatigue-creep based on ductilityexhaustion theoryrdquo International Journal of Damage Mechanicsvol 22 no 4 pp 556ndash571 2013
[21] A Risitano and G Risitano ldquoCumulative damage evaluationof steel using infrared thermographyrdquo Theoretical and AppliedFracture Mechanics vol 54 no 2 pp 82ndash90 2010
[22] M Naderi M Amiri and M M Khonsari ldquoOn the thermo-dynamic entropy of fatigue fracturerdquo Proceedings of the RoyalSociety A vol 466 no 2114 pp 423ndash438 2010
[23] XHYangWX Yao andCMDuan ldquoThe reviewof ascertain-able fatigue cumulative damage rulerdquo Engineering Science vol5 no 4 pp 82ndash87 2003
[24] S M Marco and W L Starkey ldquoA concept of fatigue damagerdquoTransaction of the ASME vol 76 pp 627ndash632 1954
[25] D G Shang and W X Yao ldquoStudy on nonlinear continuousdamage cumulative model for uniaxial fatiguerdquo Acta Aeronau-tica Et Astronautica Sinica vol 19 no 6 pp 647ndash656 1998
[26] D Y Ye Variation in fatigue properties of structural steel andresearch of new approach for fatigue life prediction [PhD thesis]Northeastern University 1996
[27] D-G Shang and W-X Yao ldquoA nonlinear damage cumulativemodel for uniaxial fatiguerdquo International Journal of Fatigue vol21 no 2 pp 187ndash194 1999
[28] J Xu D G Shang G Q Sun H Chen and E T Liu ldquoFatiguelife prediction for GH4169 superalloy under multiaxial variableamplitude loadingrdquo Journal of Beijing University of Technologyvol 38 no 10 pp 1462ndash1466 2012
[29] H T Corten and T J Dolon ldquoCumulative fatigue damagerdquo inProceedings of the International Conference on Fatigue of Metalspp 235ndash246 1956
[30] Y H Lu Study on fatigue reliability of welded bogie frame forraiway vehicle [PhD thesis] Southwest Jiaotong University2011
[31] J Tian Z-M Liu and R He ldquoNonlinear fatigue-cumulativedamage model for welded aluminum alloy joint of EMUrdquoJournal of the China Railway Society vol 34 no 3 pp 40ndash432012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 3
Table 1 Experimental data and the predicted results of 45 steel
Loading stress levelMpa Load sequences 1198991
11989911198731198911
1198992
11989921198731198912
(E) 11989921198731198912
(P) Error
33146ndash2844 High-low12500 0250 250400 05008 04241 minus153225000 0500 168300 03366 02411 minus283737500 0750 64500 01290 01082 minus1612
2844ndash33146 Low-high12500 0250 37900 07580 09693 2788250000 0500 38900 07780 08247 600375000 0750 43400 08680 05145 minus4073
Table 2 Experimental data and the predicted results of 16Mn steel
Loading stress levelMpa Load sequences 1198991
11989911198731198911
1198992
11989921198731198912
(E) 11989921198731198912
(P) Error
5629ndash3923 High-low1000 02520 56300 07154 03411 minus37431700 04284 476000 06048 02263 minus62572450 06174 22900 02910 01358 minus5333
37265ndash3923 Low-high64400 0240 62800 07980 09028 1313116000 0433 62900 07990 07449 minus677150000 0560 23300 02960 06118 10669
that is high-low and low-high loading For 45 steel thehigh-low loading spectrum is 33146ndash2844Mpa while thelow-high loading spectrum is 2844ndash33146Mpa For 16Mnsteel the high-low and low-high loading spectra are 5629ndash3923Mpa and 37265ndash3923Mpa respectively More detailscan be found in [25ndash27]
According to (7) and (8) the damage accumulationmodelunder two-level loading can be expressed as
(
1198991
1198731198911
)
120572
+
1198992
1198731198912
= 1 (9)
or
1198992
1198731198912
= 1 minus (
1198991
1198731198911
)
120572
120572 = (
1198731198911
1198731198912
)
04
(10)
where 1198991 1198992indicate the number of cycles at the 1st and 2nd
loading stress levels and 1198731198911
and 1198731198912
represent the fatiguefailure life at the corresponding load levels respectively
Thus the cycle ratio predictions at the 2nd loading stresslevel can be obtained from (9) and (10)The experimental dataand the model prediction results are listed in Tables 1 and 2The abbreviations in these two tables are explained as followsthe ldquoErdquo and ldquoPrdquo represent experimental andmodel predictionresults respectively
Comparing the experimental data with themodel predic-tion results of Manson-Halfordmodel it is obvious that mostpredicted results are close to the experimental data As shownin Figures 1 and 2 it should be noted that they are relativelyclose to practical situation It needs to be pointed out that inthis paper the ldquo119860-119861rdquo-shaped format in the figures indicatesthat the loading stress level changes from ldquo119860Mpardquo to ldquo119861Mpardquo
Pred
icte
d re
sults
(Man
son-
Hal
ford
mod
el)
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
33146ndash28442844ndash33146
Figure 1 Comparison of Manson-Halford model prediction resultsand experimental data for 45 steel
3 A Modified Nonlinear FatigueDamage Accumulation Model Consideringthe Load Interaction Effects
As shown in Tables 1 and 2 the Manson-Halford modelprediction results are relatively close to the experimental dataHowever it should be noted that there is a large differencebetween the experimental data and predicted value for 16Mnsteel Meanwhile under high-low loading conditions thepredicted results are lower than experiment data and mostpredictions are larger than practical value under low-highloading conditions This may be caused by consideringload sequences only without laying enough emphasis on
4 The Scientific World JournalPr
edic
ted
resu
lts (M
anso
n-H
alfo
rd m
odel
)
0
01
02
03
04
05
06
07
08
09
1
0 02 04 06 08 1
Experimental data
5629ndash392337265ndash3923
Figure 2 Comparison ofManson-Halfordmodel prediction resultsand experimental data for 16Mn steel
the influence of load interactionTherefore Manson-Halfordmodel will be modified in this paper to consider the loadinteraction effects and aforementioned problems
Based onManson-Halfordmodel Xu et al [28] suggestedthat the exponent parameter 120572
119894minus1119894should be modified to
include load amplitude and effective stress related to loadingpath whereas in (7)-(8) only the effects of load sequenceshad been taken into accountMoreover in view of the currentsituation that some existing models such as Corten-Dolanmodel and Freudenthal-Heller model are in the form ofload amplitude ratio to consider the load interaction effects[10 29] hence refer to the models in [10 28 29] to considerthe effects of load interaction and error distribution theexponent parameter 120572
119894minus1119894can be modified as
120572119894minus1119894
= (
119873119891(119894minus1)
119873119891119894
)
04sdotmin120590119894minus1120590119894 120590119894120590119894minus1
(11)
Then the damage accumulation model under two-levelloading can be described as
(
1198991
1198731198911
)
120572
+
1198992
1198731198912
= 1 (12)
where
120572 = (
1198731198911
1198731198912
)
04sdotmin12059011205902 12059021205901
(13)
For high-low loading conditions 0 lt 11987311989111198731198912
lt 1 then0 lt 120572 lt 1 therefore the damage caused by the 2nd loadingstress level should meet the following expression
1198992
1198731198912
= 1 minus (
1198991
1198731198911
)
120572
lt 1 minus
1198991
1198731198911
(14)
Hence the cumulative damage under high-low loading con-ditions is
1198991
1198731198911
+
1198992
1198731198912
=
1198991
1198731198911
+ 1 minus (
1198991
1198731198911
)
120572
lt 1 (15)
Similarly for low-high loading conditions11987311989111198731198912
gt 1120572 gt 1 then 119899
21198731198912
= 1minus (11989911198731198911)120572gt 1minus (119899
11198731198911) thus the
cumulative damage under low-high loading conditions is
1198991
1198731198911
+
1198992
1198731198912
=
1198991
1198731198911
+ 1 minus (
1198991
1198731198911
)
120572
gt 1 (16)
Therefore the model mentioned above reflects the non-linearity of damage accumulation and takes the effects of loadsequences and load interaction into account simultaneously
4 Validation of the Proposed Model
41 Validation Study 1 Theexperimental data adopted here isstill the data sets used in Section 2The comparison of experi-mental data and the model prediction results by theManson-Halford model and proposed model can be seen in Figures3 and 4 (ldquoM-H modelrdquo refers to Manson-Halford model andldquoP modelrdquo represents the proposed model) The results showthat nearly 80 of proposed model predictions are betterthan that by the Manson-Halford model and the inaccuracyunder high-low and low-high loading conditions has beenboth reduced this indicates that the predictions by proposedmodel aremore possibly in accordance with the experimentaldata Furthermore Figure 3 shows the predicted results bythe Minerrsquos rule for 45 steel it can be seen that the errorsbetween the prediction results by proposed model andexperimental value are smaller than that by Minerrsquos ruleThus the proposed model has a better prediction than theManson-Halford model and Minerrsquos rule
42 Validation Study 2 Nowadays the requirements of highspeed are the prospects and development trends of railwaypassenger and the high-speed railway also has become animportant symbol of modernization of national railway Asa kind of green transportation the safety and reliabilityproblems of high-speed train are concerned by researchersalthough the advantages are well known to everyone Atpresent all high-speed passenger car bogie frame and carbody have steel and aluminum alloy welded structures sincethe self-weight of train structure needs to be considerablyreduced but due to the harsh load conditions and the inher-ent weld defect such as geometric irregularity nonmetallicinclusion residual stress and heat-affected zone (HAZ)welded joints have been turned into the major failure areas ofhigh-speed train structures [30] Aluminum alloy materialsare widely used in the structures of trains ships construc-tions and so forth because of their low density high strengthand inoxidability Therefore it is of great significance forbetter fatigue-life prediction of high-speed train and makingsure of its safe operation to figure out an appropriate fatiguedamage accumulation method for welded aluminum alloyjoint
The Scientific World Journal 5
Table 3 Experimental data and comparison results of cumulative damage predictions by the proposed model and existing model for buttjoint
Load mode 1205901Mpa 120590
2Mpa 119899
1103 119899
2103 119873
11989111198731198912
119863 (by the existing model [31]) 119863 (by the proposed model)Mode 1 104 74 1099 7976 549300 1540100 09260 08988Mode 2 89 74 1761 10292 880500 1540100 10810 09372Mode 3 74 89 7701 5456 1540100 880500 09290 10660Mode 4 74 104 7701 4189 1540100 549300 10140 11053
Table 4 Experimental data and comparison results of cumulative damage predictions by the proposed model and existing model for filletjoint
Load mode 1205901Mpa 120590
2Mpa 119899
1103 119899
2103 119873
11989111198731198912
119863 (by the existing model in [31]) 119863 (by the proposed model)Mode 5 93 73 3099 5875 619800 1546100 10140 09056Mode 6 83 73 4761 6811 952300 1546100 10270 09426Mode 7 73 83 5092 7082 1546100 952300 09930 10614Mode 8 73 93 7730 4264 1546100 619800 10670 11029
Pred
icte
d re
sults
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
33146ndash2844 (M-H model)2844ndash33146 (M-H model)33146ndash2844 (P model)
2844ndash33146 (P model)33146ndash2844 (Minerrsquos rule)2844ndash33146 (Minerrsquos rule)
Figure 3 Comparison of prediction results of the proposed modelMinerrsquos rule and experimental data for 45 steel
In the light of the above experimental data of weldedaluminum alloy joint of Electric Multiple Units (EMU) areused in this section to verify the applicability of the proposedmodel in the fields of high-speed train There are two sortsof welded aluminum alloy joint used in this section that isbutt joint and fillet joint and the tests are also carried outunder two-level loading The experimental data of EMU arelisted in Tables 3 and 4 In addition comparison between theexperimental data and predictions by proposed model andan existing model is carried out for further validating theaccuracy of the proposed model
According to the existing model proposed in [31] thefatigue damage curve is
119863 = (
119899
119873
)
1+(log12(120590120590119904))
119905
(17)
Pred
icte
d re
sults
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
5629ndash3923 (M-H model)372653ndash3923 (M-H model)
5629ndash3923 (proposed model)37265ndash3923 (proposed model)
Figure 4 Comparison of prediction results of the proposed modeland experimental data for 16Mn steel
Hence the damage accumulation model under two-levelloading is given by
119863 =
2
sum
119894=1
119863119894= [
1198992
1198731198912
+ (
1198991
1198731198911
)
11988611198862
]
1198862
1198861= 1 + (log
12
1205901
120590119904
)
119905
1198862= 1 + (log
12
1205902
120590119904
)
119905
(18)
where 119863 is the cumulative damage 119886119894is the life damage
exponent under the 119894th level load 1198991 1198992and 119873
1198911 1198731198912
represent the same meaning as mentioned above 120590119904refers to
material yield strength and 119905 reflects the influence degree of
6 The Scientific World Journal
load sequences effects of the specimen its determination isby means of fitting according to experimental data
As aformentioned the cumulative damage can be cal-culated in different way unlike the modified model Thecomparison results of cumulative damage predictions by theexistingmodel and proposedmodel are shown inTables 3 and4
From Tables 3 and 4 it is obvious that the cumulativedamage predictions by the proposed model exceed unityunder low-high loading conditions andwhen the load ampli-tude changes from high to low level the value is less thanunity this verifies the nonlinearity effect of damage accumu-lation Moreover note that the prediction errors of the pro-posed model are within the range of 20 this can meet theaccuracy requirement of the engineering project so the pro-posed model can accurately predict the value of critical dam-age and it can be used to determine the fatigue-life of weldedaluminum alloy joint of Electric Multiple Units (EMU)Furthermore the prediction results of cumulative damagecalculated by these two models are relatively close to eachother thus the inaccuracy of these two models predictionsis also much close This shows that cumulative damage pre-dictions by the proposedmodel correspond approximately tothe prediction results by the existing model but do not needsome precise parameters related to material property andrequired experimental data regression Therefore using theproposed model fatigue life prediction can be obtained withhigh-precision simple computing process and less materialparameters than the existing model
5 Conclusion and Discussion
A modified model is presented in this paper for consideringload interaction and load sequences effects on the basis ofa nonlinear cumulative damage model that is Manson-Halford model The main achievements and conclusions canbe summarized as follows
(1) The exponent parameter in Manson-Halford modelhas beenmodified to consider the effects of load inter-action which can be characterized by introducing theratio of applied load amplitude
(2) The experimental data of 45 steel and 16Mn steelare used to validate the accuracy of the modifiedmodel through comparing with the predicted resultsof Manson-Halford model and the proposed modelThrough comparative analysis it is worth notingthat the inaccuracy of the proposed model has beenreduced not only under high-low loading conditionsbut also to the contrary and nearly 80 of proposedmodel predictions are better than that by Manson-Halford model On the other hand the inaccuracycaused by the proposed model is smaller than thatby Minerrsquos rule for 45 steel therefore fatigue lifeprediction by the proposed model is more possibly inaccordance with the practical situation thanManson-Halford model and Minerrsquos rule
(3) Comparing cumulative damage predictions by theproposed model with the results through an existing
model it can be found that the prediction results ofthe proposed model can reflect the nonlinearity ofdamage accumulation and this proposed model inthis paper is applicable to determine the fatigue lifeof welded aluminum alloy joint of Electric MultipleUnits (EMU) because it can be used to accuratelypredict the value of critical damage Meanwhile thereis good consistency among these two models thatis fatigue life prediction can be obtained with high-precision simple computing process and lessmaterialparameters than the existing model
Although the results are quite close to the experimentaldata all validations are carried out under two-level loadingthus there is a requirement for demonstrating the validationunder multi-level and random loading conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the partial supportprovided by the National Natural Science Foundation ofChina under the Contract no 11272082 and the FundamentalResearch Funds for the Central Universities under the Con-tract no E022050205
References
[1] K Ni and S K Zhang ldquoFatigue reliability analysis under twostage loadingrdquo ACTA Mechanica Sinica vol 31 no 1 pp 106ndash112 1999
[2] M AMiner ldquoCumulative damage in fatiguerdquo Journal of AppliedMechanics vol 12 no 3 pp 159ndash164 1945
[3] S-P ZhuH-ZHuang andZ-LWang ldquoFatigue life estimationconsidering damaging and strengthening of low amplitudeloads under different load sequences using fuzzy sets approachrdquoInternational Journal of Damage Mechanics vol 20 no 6 pp876ndash899 2011
[4] S S Manson and G R Halford ldquoPractical implementation ofthe double linear damage rule and damage curve approach fortreating cumulative fatigue damagerdquo International Journal ofFracture vol 17 no 2 pp 169ndash192 1981
[5] J L Chaboche and P M Lesne ldquoA non-linear continuousfatigue damage modelrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 11 no 1 pp 1ndash17 1988
[6] G Cheng and A Plumtree ldquoA fatigue damage accumulationmodel based on continuum damage mechanics and ductilityexhaustionrdquo International Journal of Fatigue vol 20 no 7 pp495ndash501 1998
[7] V Dattoma S Giancane R Nobile and F W Panella ldquoFatiguelife prediction under variable loading based on a newnon-linearcontinuum damage mechanics modelrdquo International Journal ofFatigue vol 28 no 2 pp 89ndash95 2006
[8] J Besson ldquoContinuum models of ductile fracture a reviewrdquoInternational Journal of Damage Mechanics vol 19 no 1 pp 3ndash52 2010
The Scientific World Journal 7
[9] H Xu Fatigue Strength Higher Education Press Beijing China1988
[10] A M Freudenthal and R A Heller ldquoOn stress interaction infatigue and cumulative damage rulerdquo Journal of the AerospaceScience vol 26 no 7 pp 431ndash442 1959
[11] S P Zhu H Z Huang Y Liu et al ldquoA practical method fordetermining the Corten-Dolan exponent and its application tofatigue life predictionrdquo International Journal of Turbo and JetEngines vol 29 no 2 pp 79ndash87 2012
[12] G R Halford ldquoThe energy required for fatiguerdquo Journal ofMaterials vol 1 no 1 pp 3ndash18 1966
[13] N Xiaode L Guangxia and LHao ldquoHardening law and fatiguedamage of a cyclic hardening metalrdquo Engineering FractureMechanics vol 26 no 2 pp 163ndash170 1987
[14] S-P Zhu and H-Z Huang ldquoA generalized frequency separa-tion-strain energy damage functionmodel for low cycle fatigue-creep life predictionrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 33 no 4 pp 227ndash237 2010
[15] S P Zhu H Z Huang L P He et al ldquoA generalized energy-based fatigue-creep damage parameter for life prediction ofturbine disk alloysrdquo Fracture Mechanics vol 90 pp 89ndash1002012
[16] S P Zhu H Z Huang V Ontiveros L P He and MModarres ldquoProbabilistic low cycle fatigue life prediction usingan energy-based damage parameter and accounting for modeluncertaintyrdquo International Journal of DamageMechanics vol 21no 8 pp 1128ndash1153 2012
[17] Y Duyi and W Zhenlin ldquoA new approach to low-cycle fatiguedamage based on exhaustion of static toughness and dissipationof cyclic plastic strain energy during fatiguerdquo InternationalJournal of Fatigue vol 23 no 8 pp 679ndash687 2001
[18] S Zhu HHuang and L Xie ldquoNonlinear fatigue damage cumu-lative model and the analysis of strength degradation basedon the double parameter fatigue criterionrdquo China MechanicalEngineering vol 19 no 22 pp 2753ndash2761 2008
[19] S-P Zhu H-Z Huang H Li R Sun and M J Zuo ldquoA newductility exhaustion model for high temperature low cyclefatigue life prediction of turbine disk alloysrdquo InternationalJournal of Turbo and Jet Engines vol 28 no 2 pp 119ndash131 2011
[20] S P Zhu H Z Huang Y Liu et al ldquoAn efficient life predictionmethodology for low cycle fatigue-creep based on ductilityexhaustion theoryrdquo International Journal of Damage Mechanicsvol 22 no 4 pp 556ndash571 2013
[21] A Risitano and G Risitano ldquoCumulative damage evaluationof steel using infrared thermographyrdquo Theoretical and AppliedFracture Mechanics vol 54 no 2 pp 82ndash90 2010
[22] M Naderi M Amiri and M M Khonsari ldquoOn the thermo-dynamic entropy of fatigue fracturerdquo Proceedings of the RoyalSociety A vol 466 no 2114 pp 423ndash438 2010
[23] XHYangWX Yao andCMDuan ldquoThe reviewof ascertain-able fatigue cumulative damage rulerdquo Engineering Science vol5 no 4 pp 82ndash87 2003
[24] S M Marco and W L Starkey ldquoA concept of fatigue damagerdquoTransaction of the ASME vol 76 pp 627ndash632 1954
[25] D G Shang and W X Yao ldquoStudy on nonlinear continuousdamage cumulative model for uniaxial fatiguerdquo Acta Aeronau-tica Et Astronautica Sinica vol 19 no 6 pp 647ndash656 1998
[26] D Y Ye Variation in fatigue properties of structural steel andresearch of new approach for fatigue life prediction [PhD thesis]Northeastern University 1996
[27] D-G Shang and W-X Yao ldquoA nonlinear damage cumulativemodel for uniaxial fatiguerdquo International Journal of Fatigue vol21 no 2 pp 187ndash194 1999
[28] J Xu D G Shang G Q Sun H Chen and E T Liu ldquoFatiguelife prediction for GH4169 superalloy under multiaxial variableamplitude loadingrdquo Journal of Beijing University of Technologyvol 38 no 10 pp 1462ndash1466 2012
[29] H T Corten and T J Dolon ldquoCumulative fatigue damagerdquo inProceedings of the International Conference on Fatigue of Metalspp 235ndash246 1956
[30] Y H Lu Study on fatigue reliability of welded bogie frame forraiway vehicle [PhD thesis] Southwest Jiaotong University2011
[31] J Tian Z-M Liu and R He ldquoNonlinear fatigue-cumulativedamage model for welded aluminum alloy joint of EMUrdquoJournal of the China Railway Society vol 34 no 3 pp 40ndash432012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
4 The Scientific World JournalPr
edic
ted
resu
lts (M
anso
n-H
alfo
rd m
odel
)
0
01
02
03
04
05
06
07
08
09
1
0 02 04 06 08 1
Experimental data
5629ndash392337265ndash3923
Figure 2 Comparison ofManson-Halfordmodel prediction resultsand experimental data for 16Mn steel
the influence of load interactionTherefore Manson-Halfordmodel will be modified in this paper to consider the loadinteraction effects and aforementioned problems
Based onManson-Halfordmodel Xu et al [28] suggestedthat the exponent parameter 120572
119894minus1119894should be modified to
include load amplitude and effective stress related to loadingpath whereas in (7)-(8) only the effects of load sequenceshad been taken into accountMoreover in view of the currentsituation that some existing models such as Corten-Dolanmodel and Freudenthal-Heller model are in the form ofload amplitude ratio to consider the load interaction effects[10 29] hence refer to the models in [10 28 29] to considerthe effects of load interaction and error distribution theexponent parameter 120572
119894minus1119894can be modified as
120572119894minus1119894
= (
119873119891(119894minus1)
119873119891119894
)
04sdotmin120590119894minus1120590119894 120590119894120590119894minus1
(11)
Then the damage accumulation model under two-levelloading can be described as
(
1198991
1198731198911
)
120572
+
1198992
1198731198912
= 1 (12)
where
120572 = (
1198731198911
1198731198912
)
04sdotmin12059011205902 12059021205901
(13)
For high-low loading conditions 0 lt 11987311989111198731198912
lt 1 then0 lt 120572 lt 1 therefore the damage caused by the 2nd loadingstress level should meet the following expression
1198992
1198731198912
= 1 minus (
1198991
1198731198911
)
120572
lt 1 minus
1198991
1198731198911
(14)
Hence the cumulative damage under high-low loading con-ditions is
1198991
1198731198911
+
1198992
1198731198912
=
1198991
1198731198911
+ 1 minus (
1198991
1198731198911
)
120572
lt 1 (15)
Similarly for low-high loading conditions11987311989111198731198912
gt 1120572 gt 1 then 119899
21198731198912
= 1minus (11989911198731198911)120572gt 1minus (119899
11198731198911) thus the
cumulative damage under low-high loading conditions is
1198991
1198731198911
+
1198992
1198731198912
=
1198991
1198731198911
+ 1 minus (
1198991
1198731198911
)
120572
gt 1 (16)
Therefore the model mentioned above reflects the non-linearity of damage accumulation and takes the effects of loadsequences and load interaction into account simultaneously
4 Validation of the Proposed Model
41 Validation Study 1 Theexperimental data adopted here isstill the data sets used in Section 2The comparison of experi-mental data and the model prediction results by theManson-Halford model and proposed model can be seen in Figures3 and 4 (ldquoM-H modelrdquo refers to Manson-Halford model andldquoP modelrdquo represents the proposed model) The results showthat nearly 80 of proposed model predictions are betterthan that by the Manson-Halford model and the inaccuracyunder high-low and low-high loading conditions has beenboth reduced this indicates that the predictions by proposedmodel aremore possibly in accordance with the experimentaldata Furthermore Figure 3 shows the predicted results bythe Minerrsquos rule for 45 steel it can be seen that the errorsbetween the prediction results by proposed model andexperimental value are smaller than that by Minerrsquos ruleThus the proposed model has a better prediction than theManson-Halford model and Minerrsquos rule
42 Validation Study 2 Nowadays the requirements of highspeed are the prospects and development trends of railwaypassenger and the high-speed railway also has become animportant symbol of modernization of national railway Asa kind of green transportation the safety and reliabilityproblems of high-speed train are concerned by researchersalthough the advantages are well known to everyone Atpresent all high-speed passenger car bogie frame and carbody have steel and aluminum alloy welded structures sincethe self-weight of train structure needs to be considerablyreduced but due to the harsh load conditions and the inher-ent weld defect such as geometric irregularity nonmetallicinclusion residual stress and heat-affected zone (HAZ)welded joints have been turned into the major failure areas ofhigh-speed train structures [30] Aluminum alloy materialsare widely used in the structures of trains ships construc-tions and so forth because of their low density high strengthand inoxidability Therefore it is of great significance forbetter fatigue-life prediction of high-speed train and makingsure of its safe operation to figure out an appropriate fatiguedamage accumulation method for welded aluminum alloyjoint
The Scientific World Journal 5
Table 3 Experimental data and comparison results of cumulative damage predictions by the proposed model and existing model for buttjoint
Load mode 1205901Mpa 120590
2Mpa 119899
1103 119899
2103 119873
11989111198731198912
119863 (by the existing model [31]) 119863 (by the proposed model)Mode 1 104 74 1099 7976 549300 1540100 09260 08988Mode 2 89 74 1761 10292 880500 1540100 10810 09372Mode 3 74 89 7701 5456 1540100 880500 09290 10660Mode 4 74 104 7701 4189 1540100 549300 10140 11053
Table 4 Experimental data and comparison results of cumulative damage predictions by the proposed model and existing model for filletjoint
Load mode 1205901Mpa 120590
2Mpa 119899
1103 119899
2103 119873
11989111198731198912
119863 (by the existing model in [31]) 119863 (by the proposed model)Mode 5 93 73 3099 5875 619800 1546100 10140 09056Mode 6 83 73 4761 6811 952300 1546100 10270 09426Mode 7 73 83 5092 7082 1546100 952300 09930 10614Mode 8 73 93 7730 4264 1546100 619800 10670 11029
Pred
icte
d re
sults
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
33146ndash2844 (M-H model)2844ndash33146 (M-H model)33146ndash2844 (P model)
2844ndash33146 (P model)33146ndash2844 (Minerrsquos rule)2844ndash33146 (Minerrsquos rule)
Figure 3 Comparison of prediction results of the proposed modelMinerrsquos rule and experimental data for 45 steel
In the light of the above experimental data of weldedaluminum alloy joint of Electric Multiple Units (EMU) areused in this section to verify the applicability of the proposedmodel in the fields of high-speed train There are two sortsof welded aluminum alloy joint used in this section that isbutt joint and fillet joint and the tests are also carried outunder two-level loading The experimental data of EMU arelisted in Tables 3 and 4 In addition comparison between theexperimental data and predictions by proposed model andan existing model is carried out for further validating theaccuracy of the proposed model
According to the existing model proposed in [31] thefatigue damage curve is
119863 = (
119899
119873
)
1+(log12(120590120590119904))
119905
(17)
Pred
icte
d re
sults
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
5629ndash3923 (M-H model)372653ndash3923 (M-H model)
5629ndash3923 (proposed model)37265ndash3923 (proposed model)
Figure 4 Comparison of prediction results of the proposed modeland experimental data for 16Mn steel
Hence the damage accumulation model under two-levelloading is given by
119863 =
2
sum
119894=1
119863119894= [
1198992
1198731198912
+ (
1198991
1198731198911
)
11988611198862
]
1198862
1198861= 1 + (log
12
1205901
120590119904
)
119905
1198862= 1 + (log
12
1205902
120590119904
)
119905
(18)
where 119863 is the cumulative damage 119886119894is the life damage
exponent under the 119894th level load 1198991 1198992and 119873
1198911 1198731198912
represent the same meaning as mentioned above 120590119904refers to
material yield strength and 119905 reflects the influence degree of
6 The Scientific World Journal
load sequences effects of the specimen its determination isby means of fitting according to experimental data
As aformentioned the cumulative damage can be cal-culated in different way unlike the modified model Thecomparison results of cumulative damage predictions by theexistingmodel and proposedmodel are shown inTables 3 and4
From Tables 3 and 4 it is obvious that the cumulativedamage predictions by the proposed model exceed unityunder low-high loading conditions andwhen the load ampli-tude changes from high to low level the value is less thanunity this verifies the nonlinearity effect of damage accumu-lation Moreover note that the prediction errors of the pro-posed model are within the range of 20 this can meet theaccuracy requirement of the engineering project so the pro-posed model can accurately predict the value of critical dam-age and it can be used to determine the fatigue-life of weldedaluminum alloy joint of Electric Multiple Units (EMU)Furthermore the prediction results of cumulative damagecalculated by these two models are relatively close to eachother thus the inaccuracy of these two models predictionsis also much close This shows that cumulative damage pre-dictions by the proposedmodel correspond approximately tothe prediction results by the existing model but do not needsome precise parameters related to material property andrequired experimental data regression Therefore using theproposed model fatigue life prediction can be obtained withhigh-precision simple computing process and less materialparameters than the existing model
5 Conclusion and Discussion
A modified model is presented in this paper for consideringload interaction and load sequences effects on the basis ofa nonlinear cumulative damage model that is Manson-Halford model The main achievements and conclusions canbe summarized as follows
(1) The exponent parameter in Manson-Halford modelhas beenmodified to consider the effects of load inter-action which can be characterized by introducing theratio of applied load amplitude
(2) The experimental data of 45 steel and 16Mn steelare used to validate the accuracy of the modifiedmodel through comparing with the predicted resultsof Manson-Halford model and the proposed modelThrough comparative analysis it is worth notingthat the inaccuracy of the proposed model has beenreduced not only under high-low loading conditionsbut also to the contrary and nearly 80 of proposedmodel predictions are better than that by Manson-Halford model On the other hand the inaccuracycaused by the proposed model is smaller than thatby Minerrsquos rule for 45 steel therefore fatigue lifeprediction by the proposed model is more possibly inaccordance with the practical situation thanManson-Halford model and Minerrsquos rule
(3) Comparing cumulative damage predictions by theproposed model with the results through an existing
model it can be found that the prediction results ofthe proposed model can reflect the nonlinearity ofdamage accumulation and this proposed model inthis paper is applicable to determine the fatigue lifeof welded aluminum alloy joint of Electric MultipleUnits (EMU) because it can be used to accuratelypredict the value of critical damage Meanwhile thereis good consistency among these two models thatis fatigue life prediction can be obtained with high-precision simple computing process and lessmaterialparameters than the existing model
Although the results are quite close to the experimentaldata all validations are carried out under two-level loadingthus there is a requirement for demonstrating the validationunder multi-level and random loading conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the partial supportprovided by the National Natural Science Foundation ofChina under the Contract no 11272082 and the FundamentalResearch Funds for the Central Universities under the Con-tract no E022050205
References
[1] K Ni and S K Zhang ldquoFatigue reliability analysis under twostage loadingrdquo ACTA Mechanica Sinica vol 31 no 1 pp 106ndash112 1999
[2] M AMiner ldquoCumulative damage in fatiguerdquo Journal of AppliedMechanics vol 12 no 3 pp 159ndash164 1945
[3] S-P ZhuH-ZHuang andZ-LWang ldquoFatigue life estimationconsidering damaging and strengthening of low amplitudeloads under different load sequences using fuzzy sets approachrdquoInternational Journal of Damage Mechanics vol 20 no 6 pp876ndash899 2011
[4] S S Manson and G R Halford ldquoPractical implementation ofthe double linear damage rule and damage curve approach fortreating cumulative fatigue damagerdquo International Journal ofFracture vol 17 no 2 pp 169ndash192 1981
[5] J L Chaboche and P M Lesne ldquoA non-linear continuousfatigue damage modelrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 11 no 1 pp 1ndash17 1988
[6] G Cheng and A Plumtree ldquoA fatigue damage accumulationmodel based on continuum damage mechanics and ductilityexhaustionrdquo International Journal of Fatigue vol 20 no 7 pp495ndash501 1998
[7] V Dattoma S Giancane R Nobile and F W Panella ldquoFatiguelife prediction under variable loading based on a newnon-linearcontinuum damage mechanics modelrdquo International Journal ofFatigue vol 28 no 2 pp 89ndash95 2006
[8] J Besson ldquoContinuum models of ductile fracture a reviewrdquoInternational Journal of Damage Mechanics vol 19 no 1 pp 3ndash52 2010
The Scientific World Journal 7
[9] H Xu Fatigue Strength Higher Education Press Beijing China1988
[10] A M Freudenthal and R A Heller ldquoOn stress interaction infatigue and cumulative damage rulerdquo Journal of the AerospaceScience vol 26 no 7 pp 431ndash442 1959
[11] S P Zhu H Z Huang Y Liu et al ldquoA practical method fordetermining the Corten-Dolan exponent and its application tofatigue life predictionrdquo International Journal of Turbo and JetEngines vol 29 no 2 pp 79ndash87 2012
[12] G R Halford ldquoThe energy required for fatiguerdquo Journal ofMaterials vol 1 no 1 pp 3ndash18 1966
[13] N Xiaode L Guangxia and LHao ldquoHardening law and fatiguedamage of a cyclic hardening metalrdquo Engineering FractureMechanics vol 26 no 2 pp 163ndash170 1987
[14] S-P Zhu and H-Z Huang ldquoA generalized frequency separa-tion-strain energy damage functionmodel for low cycle fatigue-creep life predictionrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 33 no 4 pp 227ndash237 2010
[15] S P Zhu H Z Huang L P He et al ldquoA generalized energy-based fatigue-creep damage parameter for life prediction ofturbine disk alloysrdquo Fracture Mechanics vol 90 pp 89ndash1002012
[16] S P Zhu H Z Huang V Ontiveros L P He and MModarres ldquoProbabilistic low cycle fatigue life prediction usingan energy-based damage parameter and accounting for modeluncertaintyrdquo International Journal of DamageMechanics vol 21no 8 pp 1128ndash1153 2012
[17] Y Duyi and W Zhenlin ldquoA new approach to low-cycle fatiguedamage based on exhaustion of static toughness and dissipationof cyclic plastic strain energy during fatiguerdquo InternationalJournal of Fatigue vol 23 no 8 pp 679ndash687 2001
[18] S Zhu HHuang and L Xie ldquoNonlinear fatigue damage cumu-lative model and the analysis of strength degradation basedon the double parameter fatigue criterionrdquo China MechanicalEngineering vol 19 no 22 pp 2753ndash2761 2008
[19] S-P Zhu H-Z Huang H Li R Sun and M J Zuo ldquoA newductility exhaustion model for high temperature low cyclefatigue life prediction of turbine disk alloysrdquo InternationalJournal of Turbo and Jet Engines vol 28 no 2 pp 119ndash131 2011
[20] S P Zhu H Z Huang Y Liu et al ldquoAn efficient life predictionmethodology for low cycle fatigue-creep based on ductilityexhaustion theoryrdquo International Journal of Damage Mechanicsvol 22 no 4 pp 556ndash571 2013
[21] A Risitano and G Risitano ldquoCumulative damage evaluationof steel using infrared thermographyrdquo Theoretical and AppliedFracture Mechanics vol 54 no 2 pp 82ndash90 2010
[22] M Naderi M Amiri and M M Khonsari ldquoOn the thermo-dynamic entropy of fatigue fracturerdquo Proceedings of the RoyalSociety A vol 466 no 2114 pp 423ndash438 2010
[23] XHYangWX Yao andCMDuan ldquoThe reviewof ascertain-able fatigue cumulative damage rulerdquo Engineering Science vol5 no 4 pp 82ndash87 2003
[24] S M Marco and W L Starkey ldquoA concept of fatigue damagerdquoTransaction of the ASME vol 76 pp 627ndash632 1954
[25] D G Shang and W X Yao ldquoStudy on nonlinear continuousdamage cumulative model for uniaxial fatiguerdquo Acta Aeronau-tica Et Astronautica Sinica vol 19 no 6 pp 647ndash656 1998
[26] D Y Ye Variation in fatigue properties of structural steel andresearch of new approach for fatigue life prediction [PhD thesis]Northeastern University 1996
[27] D-G Shang and W-X Yao ldquoA nonlinear damage cumulativemodel for uniaxial fatiguerdquo International Journal of Fatigue vol21 no 2 pp 187ndash194 1999
[28] J Xu D G Shang G Q Sun H Chen and E T Liu ldquoFatiguelife prediction for GH4169 superalloy under multiaxial variableamplitude loadingrdquo Journal of Beijing University of Technologyvol 38 no 10 pp 1462ndash1466 2012
[29] H T Corten and T J Dolon ldquoCumulative fatigue damagerdquo inProceedings of the International Conference on Fatigue of Metalspp 235ndash246 1956
[30] Y H Lu Study on fatigue reliability of welded bogie frame forraiway vehicle [PhD thesis] Southwest Jiaotong University2011
[31] J Tian Z-M Liu and R He ldquoNonlinear fatigue-cumulativedamage model for welded aluminum alloy joint of EMUrdquoJournal of the China Railway Society vol 34 no 3 pp 40ndash432012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 5
Table 3 Experimental data and comparison results of cumulative damage predictions by the proposed model and existing model for buttjoint
Load mode 1205901Mpa 120590
2Mpa 119899
1103 119899
2103 119873
11989111198731198912
119863 (by the existing model [31]) 119863 (by the proposed model)Mode 1 104 74 1099 7976 549300 1540100 09260 08988Mode 2 89 74 1761 10292 880500 1540100 10810 09372Mode 3 74 89 7701 5456 1540100 880500 09290 10660Mode 4 74 104 7701 4189 1540100 549300 10140 11053
Table 4 Experimental data and comparison results of cumulative damage predictions by the proposed model and existing model for filletjoint
Load mode 1205901Mpa 120590
2Mpa 119899
1103 119899
2103 119873
11989111198731198912
119863 (by the existing model in [31]) 119863 (by the proposed model)Mode 5 93 73 3099 5875 619800 1546100 10140 09056Mode 6 83 73 4761 6811 952300 1546100 10270 09426Mode 7 73 83 5092 7082 1546100 952300 09930 10614Mode 8 73 93 7730 4264 1546100 619800 10670 11029
Pred
icte
d re
sults
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
33146ndash2844 (M-H model)2844ndash33146 (M-H model)33146ndash2844 (P model)
2844ndash33146 (P model)33146ndash2844 (Minerrsquos rule)2844ndash33146 (Minerrsquos rule)
Figure 3 Comparison of prediction results of the proposed modelMinerrsquos rule and experimental data for 45 steel
In the light of the above experimental data of weldedaluminum alloy joint of Electric Multiple Units (EMU) areused in this section to verify the applicability of the proposedmodel in the fields of high-speed train There are two sortsof welded aluminum alloy joint used in this section that isbutt joint and fillet joint and the tests are also carried outunder two-level loading The experimental data of EMU arelisted in Tables 3 and 4 In addition comparison between theexperimental data and predictions by proposed model andan existing model is carried out for further validating theaccuracy of the proposed model
According to the existing model proposed in [31] thefatigue damage curve is
119863 = (
119899
119873
)
1+(log12(120590120590119904))
119905
(17)
Pred
icte
d re
sults
0 02 04 06 08 10
01
02
03
04
05
06
07
08
09
1
Experimental data
5629ndash3923 (M-H model)372653ndash3923 (M-H model)
5629ndash3923 (proposed model)37265ndash3923 (proposed model)
Figure 4 Comparison of prediction results of the proposed modeland experimental data for 16Mn steel
Hence the damage accumulation model under two-levelloading is given by
119863 =
2
sum
119894=1
119863119894= [
1198992
1198731198912
+ (
1198991
1198731198911
)
11988611198862
]
1198862
1198861= 1 + (log
12
1205901
120590119904
)
119905
1198862= 1 + (log
12
1205902
120590119904
)
119905
(18)
where 119863 is the cumulative damage 119886119894is the life damage
exponent under the 119894th level load 1198991 1198992and 119873
1198911 1198731198912
represent the same meaning as mentioned above 120590119904refers to
material yield strength and 119905 reflects the influence degree of
6 The Scientific World Journal
load sequences effects of the specimen its determination isby means of fitting according to experimental data
As aformentioned the cumulative damage can be cal-culated in different way unlike the modified model Thecomparison results of cumulative damage predictions by theexistingmodel and proposedmodel are shown inTables 3 and4
From Tables 3 and 4 it is obvious that the cumulativedamage predictions by the proposed model exceed unityunder low-high loading conditions andwhen the load ampli-tude changes from high to low level the value is less thanunity this verifies the nonlinearity effect of damage accumu-lation Moreover note that the prediction errors of the pro-posed model are within the range of 20 this can meet theaccuracy requirement of the engineering project so the pro-posed model can accurately predict the value of critical dam-age and it can be used to determine the fatigue-life of weldedaluminum alloy joint of Electric Multiple Units (EMU)Furthermore the prediction results of cumulative damagecalculated by these two models are relatively close to eachother thus the inaccuracy of these two models predictionsis also much close This shows that cumulative damage pre-dictions by the proposedmodel correspond approximately tothe prediction results by the existing model but do not needsome precise parameters related to material property andrequired experimental data regression Therefore using theproposed model fatigue life prediction can be obtained withhigh-precision simple computing process and less materialparameters than the existing model
5 Conclusion and Discussion
A modified model is presented in this paper for consideringload interaction and load sequences effects on the basis ofa nonlinear cumulative damage model that is Manson-Halford model The main achievements and conclusions canbe summarized as follows
(1) The exponent parameter in Manson-Halford modelhas beenmodified to consider the effects of load inter-action which can be characterized by introducing theratio of applied load amplitude
(2) The experimental data of 45 steel and 16Mn steelare used to validate the accuracy of the modifiedmodel through comparing with the predicted resultsof Manson-Halford model and the proposed modelThrough comparative analysis it is worth notingthat the inaccuracy of the proposed model has beenreduced not only under high-low loading conditionsbut also to the contrary and nearly 80 of proposedmodel predictions are better than that by Manson-Halford model On the other hand the inaccuracycaused by the proposed model is smaller than thatby Minerrsquos rule for 45 steel therefore fatigue lifeprediction by the proposed model is more possibly inaccordance with the practical situation thanManson-Halford model and Minerrsquos rule
(3) Comparing cumulative damage predictions by theproposed model with the results through an existing
model it can be found that the prediction results ofthe proposed model can reflect the nonlinearity ofdamage accumulation and this proposed model inthis paper is applicable to determine the fatigue lifeof welded aluminum alloy joint of Electric MultipleUnits (EMU) because it can be used to accuratelypredict the value of critical damage Meanwhile thereis good consistency among these two models thatis fatigue life prediction can be obtained with high-precision simple computing process and lessmaterialparameters than the existing model
Although the results are quite close to the experimentaldata all validations are carried out under two-level loadingthus there is a requirement for demonstrating the validationunder multi-level and random loading conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the partial supportprovided by the National Natural Science Foundation ofChina under the Contract no 11272082 and the FundamentalResearch Funds for the Central Universities under the Con-tract no E022050205
References
[1] K Ni and S K Zhang ldquoFatigue reliability analysis under twostage loadingrdquo ACTA Mechanica Sinica vol 31 no 1 pp 106ndash112 1999
[2] M AMiner ldquoCumulative damage in fatiguerdquo Journal of AppliedMechanics vol 12 no 3 pp 159ndash164 1945
[3] S-P ZhuH-ZHuang andZ-LWang ldquoFatigue life estimationconsidering damaging and strengthening of low amplitudeloads under different load sequences using fuzzy sets approachrdquoInternational Journal of Damage Mechanics vol 20 no 6 pp876ndash899 2011
[4] S S Manson and G R Halford ldquoPractical implementation ofthe double linear damage rule and damage curve approach fortreating cumulative fatigue damagerdquo International Journal ofFracture vol 17 no 2 pp 169ndash192 1981
[5] J L Chaboche and P M Lesne ldquoA non-linear continuousfatigue damage modelrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 11 no 1 pp 1ndash17 1988
[6] G Cheng and A Plumtree ldquoA fatigue damage accumulationmodel based on continuum damage mechanics and ductilityexhaustionrdquo International Journal of Fatigue vol 20 no 7 pp495ndash501 1998
[7] V Dattoma S Giancane R Nobile and F W Panella ldquoFatiguelife prediction under variable loading based on a newnon-linearcontinuum damage mechanics modelrdquo International Journal ofFatigue vol 28 no 2 pp 89ndash95 2006
[8] J Besson ldquoContinuum models of ductile fracture a reviewrdquoInternational Journal of Damage Mechanics vol 19 no 1 pp 3ndash52 2010
The Scientific World Journal 7
[9] H Xu Fatigue Strength Higher Education Press Beijing China1988
[10] A M Freudenthal and R A Heller ldquoOn stress interaction infatigue and cumulative damage rulerdquo Journal of the AerospaceScience vol 26 no 7 pp 431ndash442 1959
[11] S P Zhu H Z Huang Y Liu et al ldquoA practical method fordetermining the Corten-Dolan exponent and its application tofatigue life predictionrdquo International Journal of Turbo and JetEngines vol 29 no 2 pp 79ndash87 2012
[12] G R Halford ldquoThe energy required for fatiguerdquo Journal ofMaterials vol 1 no 1 pp 3ndash18 1966
[13] N Xiaode L Guangxia and LHao ldquoHardening law and fatiguedamage of a cyclic hardening metalrdquo Engineering FractureMechanics vol 26 no 2 pp 163ndash170 1987
[14] S-P Zhu and H-Z Huang ldquoA generalized frequency separa-tion-strain energy damage functionmodel for low cycle fatigue-creep life predictionrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 33 no 4 pp 227ndash237 2010
[15] S P Zhu H Z Huang L P He et al ldquoA generalized energy-based fatigue-creep damage parameter for life prediction ofturbine disk alloysrdquo Fracture Mechanics vol 90 pp 89ndash1002012
[16] S P Zhu H Z Huang V Ontiveros L P He and MModarres ldquoProbabilistic low cycle fatigue life prediction usingan energy-based damage parameter and accounting for modeluncertaintyrdquo International Journal of DamageMechanics vol 21no 8 pp 1128ndash1153 2012
[17] Y Duyi and W Zhenlin ldquoA new approach to low-cycle fatiguedamage based on exhaustion of static toughness and dissipationof cyclic plastic strain energy during fatiguerdquo InternationalJournal of Fatigue vol 23 no 8 pp 679ndash687 2001
[18] S Zhu HHuang and L Xie ldquoNonlinear fatigue damage cumu-lative model and the analysis of strength degradation basedon the double parameter fatigue criterionrdquo China MechanicalEngineering vol 19 no 22 pp 2753ndash2761 2008
[19] S-P Zhu H-Z Huang H Li R Sun and M J Zuo ldquoA newductility exhaustion model for high temperature low cyclefatigue life prediction of turbine disk alloysrdquo InternationalJournal of Turbo and Jet Engines vol 28 no 2 pp 119ndash131 2011
[20] S P Zhu H Z Huang Y Liu et al ldquoAn efficient life predictionmethodology for low cycle fatigue-creep based on ductilityexhaustion theoryrdquo International Journal of Damage Mechanicsvol 22 no 4 pp 556ndash571 2013
[21] A Risitano and G Risitano ldquoCumulative damage evaluationof steel using infrared thermographyrdquo Theoretical and AppliedFracture Mechanics vol 54 no 2 pp 82ndash90 2010
[22] M Naderi M Amiri and M M Khonsari ldquoOn the thermo-dynamic entropy of fatigue fracturerdquo Proceedings of the RoyalSociety A vol 466 no 2114 pp 423ndash438 2010
[23] XHYangWX Yao andCMDuan ldquoThe reviewof ascertain-able fatigue cumulative damage rulerdquo Engineering Science vol5 no 4 pp 82ndash87 2003
[24] S M Marco and W L Starkey ldquoA concept of fatigue damagerdquoTransaction of the ASME vol 76 pp 627ndash632 1954
[25] D G Shang and W X Yao ldquoStudy on nonlinear continuousdamage cumulative model for uniaxial fatiguerdquo Acta Aeronau-tica Et Astronautica Sinica vol 19 no 6 pp 647ndash656 1998
[26] D Y Ye Variation in fatigue properties of structural steel andresearch of new approach for fatigue life prediction [PhD thesis]Northeastern University 1996
[27] D-G Shang and W-X Yao ldquoA nonlinear damage cumulativemodel for uniaxial fatiguerdquo International Journal of Fatigue vol21 no 2 pp 187ndash194 1999
[28] J Xu D G Shang G Q Sun H Chen and E T Liu ldquoFatiguelife prediction for GH4169 superalloy under multiaxial variableamplitude loadingrdquo Journal of Beijing University of Technologyvol 38 no 10 pp 1462ndash1466 2012
[29] H T Corten and T J Dolon ldquoCumulative fatigue damagerdquo inProceedings of the International Conference on Fatigue of Metalspp 235ndash246 1956
[30] Y H Lu Study on fatigue reliability of welded bogie frame forraiway vehicle [PhD thesis] Southwest Jiaotong University2011
[31] J Tian Z-M Liu and R He ldquoNonlinear fatigue-cumulativedamage model for welded aluminum alloy joint of EMUrdquoJournal of the China Railway Society vol 34 no 3 pp 40ndash432012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
6 The Scientific World Journal
load sequences effects of the specimen its determination isby means of fitting according to experimental data
As aformentioned the cumulative damage can be cal-culated in different way unlike the modified model Thecomparison results of cumulative damage predictions by theexistingmodel and proposedmodel are shown inTables 3 and4
From Tables 3 and 4 it is obvious that the cumulativedamage predictions by the proposed model exceed unityunder low-high loading conditions andwhen the load ampli-tude changes from high to low level the value is less thanunity this verifies the nonlinearity effect of damage accumu-lation Moreover note that the prediction errors of the pro-posed model are within the range of 20 this can meet theaccuracy requirement of the engineering project so the pro-posed model can accurately predict the value of critical dam-age and it can be used to determine the fatigue-life of weldedaluminum alloy joint of Electric Multiple Units (EMU)Furthermore the prediction results of cumulative damagecalculated by these two models are relatively close to eachother thus the inaccuracy of these two models predictionsis also much close This shows that cumulative damage pre-dictions by the proposedmodel correspond approximately tothe prediction results by the existing model but do not needsome precise parameters related to material property andrequired experimental data regression Therefore using theproposed model fatigue life prediction can be obtained withhigh-precision simple computing process and less materialparameters than the existing model
5 Conclusion and Discussion
A modified model is presented in this paper for consideringload interaction and load sequences effects on the basis ofa nonlinear cumulative damage model that is Manson-Halford model The main achievements and conclusions canbe summarized as follows
(1) The exponent parameter in Manson-Halford modelhas beenmodified to consider the effects of load inter-action which can be characterized by introducing theratio of applied load amplitude
(2) The experimental data of 45 steel and 16Mn steelare used to validate the accuracy of the modifiedmodel through comparing with the predicted resultsof Manson-Halford model and the proposed modelThrough comparative analysis it is worth notingthat the inaccuracy of the proposed model has beenreduced not only under high-low loading conditionsbut also to the contrary and nearly 80 of proposedmodel predictions are better than that by Manson-Halford model On the other hand the inaccuracycaused by the proposed model is smaller than thatby Minerrsquos rule for 45 steel therefore fatigue lifeprediction by the proposed model is more possibly inaccordance with the practical situation thanManson-Halford model and Minerrsquos rule
(3) Comparing cumulative damage predictions by theproposed model with the results through an existing
model it can be found that the prediction results ofthe proposed model can reflect the nonlinearity ofdamage accumulation and this proposed model inthis paper is applicable to determine the fatigue lifeof welded aluminum alloy joint of Electric MultipleUnits (EMU) because it can be used to accuratelypredict the value of critical damage Meanwhile thereis good consistency among these two models thatis fatigue life prediction can be obtained with high-precision simple computing process and lessmaterialparameters than the existing model
Although the results are quite close to the experimentaldata all validations are carried out under two-level loadingthus there is a requirement for demonstrating the validationunder multi-level and random loading conditions
Conflict of Interests
The authors declare that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
The authors would like to acknowledge the partial supportprovided by the National Natural Science Foundation ofChina under the Contract no 11272082 and the FundamentalResearch Funds for the Central Universities under the Con-tract no E022050205
References
[1] K Ni and S K Zhang ldquoFatigue reliability analysis under twostage loadingrdquo ACTA Mechanica Sinica vol 31 no 1 pp 106ndash112 1999
[2] M AMiner ldquoCumulative damage in fatiguerdquo Journal of AppliedMechanics vol 12 no 3 pp 159ndash164 1945
[3] S-P ZhuH-ZHuang andZ-LWang ldquoFatigue life estimationconsidering damaging and strengthening of low amplitudeloads under different load sequences using fuzzy sets approachrdquoInternational Journal of Damage Mechanics vol 20 no 6 pp876ndash899 2011
[4] S S Manson and G R Halford ldquoPractical implementation ofthe double linear damage rule and damage curve approach fortreating cumulative fatigue damagerdquo International Journal ofFracture vol 17 no 2 pp 169ndash192 1981
[5] J L Chaboche and P M Lesne ldquoA non-linear continuousfatigue damage modelrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 11 no 1 pp 1ndash17 1988
[6] G Cheng and A Plumtree ldquoA fatigue damage accumulationmodel based on continuum damage mechanics and ductilityexhaustionrdquo International Journal of Fatigue vol 20 no 7 pp495ndash501 1998
[7] V Dattoma S Giancane R Nobile and F W Panella ldquoFatiguelife prediction under variable loading based on a newnon-linearcontinuum damage mechanics modelrdquo International Journal ofFatigue vol 28 no 2 pp 89ndash95 2006
[8] J Besson ldquoContinuum models of ductile fracture a reviewrdquoInternational Journal of Damage Mechanics vol 19 no 1 pp 3ndash52 2010
The Scientific World Journal 7
[9] H Xu Fatigue Strength Higher Education Press Beijing China1988
[10] A M Freudenthal and R A Heller ldquoOn stress interaction infatigue and cumulative damage rulerdquo Journal of the AerospaceScience vol 26 no 7 pp 431ndash442 1959
[11] S P Zhu H Z Huang Y Liu et al ldquoA practical method fordetermining the Corten-Dolan exponent and its application tofatigue life predictionrdquo International Journal of Turbo and JetEngines vol 29 no 2 pp 79ndash87 2012
[12] G R Halford ldquoThe energy required for fatiguerdquo Journal ofMaterials vol 1 no 1 pp 3ndash18 1966
[13] N Xiaode L Guangxia and LHao ldquoHardening law and fatiguedamage of a cyclic hardening metalrdquo Engineering FractureMechanics vol 26 no 2 pp 163ndash170 1987
[14] S-P Zhu and H-Z Huang ldquoA generalized frequency separa-tion-strain energy damage functionmodel for low cycle fatigue-creep life predictionrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 33 no 4 pp 227ndash237 2010
[15] S P Zhu H Z Huang L P He et al ldquoA generalized energy-based fatigue-creep damage parameter for life prediction ofturbine disk alloysrdquo Fracture Mechanics vol 90 pp 89ndash1002012
[16] S P Zhu H Z Huang V Ontiveros L P He and MModarres ldquoProbabilistic low cycle fatigue life prediction usingan energy-based damage parameter and accounting for modeluncertaintyrdquo International Journal of DamageMechanics vol 21no 8 pp 1128ndash1153 2012
[17] Y Duyi and W Zhenlin ldquoA new approach to low-cycle fatiguedamage based on exhaustion of static toughness and dissipationof cyclic plastic strain energy during fatiguerdquo InternationalJournal of Fatigue vol 23 no 8 pp 679ndash687 2001
[18] S Zhu HHuang and L Xie ldquoNonlinear fatigue damage cumu-lative model and the analysis of strength degradation basedon the double parameter fatigue criterionrdquo China MechanicalEngineering vol 19 no 22 pp 2753ndash2761 2008
[19] S-P Zhu H-Z Huang H Li R Sun and M J Zuo ldquoA newductility exhaustion model for high temperature low cyclefatigue life prediction of turbine disk alloysrdquo InternationalJournal of Turbo and Jet Engines vol 28 no 2 pp 119ndash131 2011
[20] S P Zhu H Z Huang Y Liu et al ldquoAn efficient life predictionmethodology for low cycle fatigue-creep based on ductilityexhaustion theoryrdquo International Journal of Damage Mechanicsvol 22 no 4 pp 556ndash571 2013
[21] A Risitano and G Risitano ldquoCumulative damage evaluationof steel using infrared thermographyrdquo Theoretical and AppliedFracture Mechanics vol 54 no 2 pp 82ndash90 2010
[22] M Naderi M Amiri and M M Khonsari ldquoOn the thermo-dynamic entropy of fatigue fracturerdquo Proceedings of the RoyalSociety A vol 466 no 2114 pp 423ndash438 2010
[23] XHYangWX Yao andCMDuan ldquoThe reviewof ascertain-able fatigue cumulative damage rulerdquo Engineering Science vol5 no 4 pp 82ndash87 2003
[24] S M Marco and W L Starkey ldquoA concept of fatigue damagerdquoTransaction of the ASME vol 76 pp 627ndash632 1954
[25] D G Shang and W X Yao ldquoStudy on nonlinear continuousdamage cumulative model for uniaxial fatiguerdquo Acta Aeronau-tica Et Astronautica Sinica vol 19 no 6 pp 647ndash656 1998
[26] D Y Ye Variation in fatigue properties of structural steel andresearch of new approach for fatigue life prediction [PhD thesis]Northeastern University 1996
[27] D-G Shang and W-X Yao ldquoA nonlinear damage cumulativemodel for uniaxial fatiguerdquo International Journal of Fatigue vol21 no 2 pp 187ndash194 1999
[28] J Xu D G Shang G Q Sun H Chen and E T Liu ldquoFatiguelife prediction for GH4169 superalloy under multiaxial variableamplitude loadingrdquo Journal of Beijing University of Technologyvol 38 no 10 pp 1462ndash1466 2012
[29] H T Corten and T J Dolon ldquoCumulative fatigue damagerdquo inProceedings of the International Conference on Fatigue of Metalspp 235ndash246 1956
[30] Y H Lu Study on fatigue reliability of welded bogie frame forraiway vehicle [PhD thesis] Southwest Jiaotong University2011
[31] J Tian Z-M Liu and R He ldquoNonlinear fatigue-cumulativedamage model for welded aluminum alloy joint of EMUrdquoJournal of the China Railway Society vol 34 no 3 pp 40ndash432012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
The Scientific World Journal 7
[9] H Xu Fatigue Strength Higher Education Press Beijing China1988
[10] A M Freudenthal and R A Heller ldquoOn stress interaction infatigue and cumulative damage rulerdquo Journal of the AerospaceScience vol 26 no 7 pp 431ndash442 1959
[11] S P Zhu H Z Huang Y Liu et al ldquoA practical method fordetermining the Corten-Dolan exponent and its application tofatigue life predictionrdquo International Journal of Turbo and JetEngines vol 29 no 2 pp 79ndash87 2012
[12] G R Halford ldquoThe energy required for fatiguerdquo Journal ofMaterials vol 1 no 1 pp 3ndash18 1966
[13] N Xiaode L Guangxia and LHao ldquoHardening law and fatiguedamage of a cyclic hardening metalrdquo Engineering FractureMechanics vol 26 no 2 pp 163ndash170 1987
[14] S-P Zhu and H-Z Huang ldquoA generalized frequency separa-tion-strain energy damage functionmodel for low cycle fatigue-creep life predictionrdquo Fatigue and Fracture of EngineeringMaterials and Structures vol 33 no 4 pp 227ndash237 2010
[15] S P Zhu H Z Huang L P He et al ldquoA generalized energy-based fatigue-creep damage parameter for life prediction ofturbine disk alloysrdquo Fracture Mechanics vol 90 pp 89ndash1002012
[16] S P Zhu H Z Huang V Ontiveros L P He and MModarres ldquoProbabilistic low cycle fatigue life prediction usingan energy-based damage parameter and accounting for modeluncertaintyrdquo International Journal of DamageMechanics vol 21no 8 pp 1128ndash1153 2012
[17] Y Duyi and W Zhenlin ldquoA new approach to low-cycle fatiguedamage based on exhaustion of static toughness and dissipationof cyclic plastic strain energy during fatiguerdquo InternationalJournal of Fatigue vol 23 no 8 pp 679ndash687 2001
[18] S Zhu HHuang and L Xie ldquoNonlinear fatigue damage cumu-lative model and the analysis of strength degradation basedon the double parameter fatigue criterionrdquo China MechanicalEngineering vol 19 no 22 pp 2753ndash2761 2008
[19] S-P Zhu H-Z Huang H Li R Sun and M J Zuo ldquoA newductility exhaustion model for high temperature low cyclefatigue life prediction of turbine disk alloysrdquo InternationalJournal of Turbo and Jet Engines vol 28 no 2 pp 119ndash131 2011
[20] S P Zhu H Z Huang Y Liu et al ldquoAn efficient life predictionmethodology for low cycle fatigue-creep based on ductilityexhaustion theoryrdquo International Journal of Damage Mechanicsvol 22 no 4 pp 556ndash571 2013
[21] A Risitano and G Risitano ldquoCumulative damage evaluationof steel using infrared thermographyrdquo Theoretical and AppliedFracture Mechanics vol 54 no 2 pp 82ndash90 2010
[22] M Naderi M Amiri and M M Khonsari ldquoOn the thermo-dynamic entropy of fatigue fracturerdquo Proceedings of the RoyalSociety A vol 466 no 2114 pp 423ndash438 2010
[23] XHYangWX Yao andCMDuan ldquoThe reviewof ascertain-able fatigue cumulative damage rulerdquo Engineering Science vol5 no 4 pp 82ndash87 2003
[24] S M Marco and W L Starkey ldquoA concept of fatigue damagerdquoTransaction of the ASME vol 76 pp 627ndash632 1954
[25] D G Shang and W X Yao ldquoStudy on nonlinear continuousdamage cumulative model for uniaxial fatiguerdquo Acta Aeronau-tica Et Astronautica Sinica vol 19 no 6 pp 647ndash656 1998
[26] D Y Ye Variation in fatigue properties of structural steel andresearch of new approach for fatigue life prediction [PhD thesis]Northeastern University 1996
[27] D-G Shang and W-X Yao ldquoA nonlinear damage cumulativemodel for uniaxial fatiguerdquo International Journal of Fatigue vol21 no 2 pp 187ndash194 1999
[28] J Xu D G Shang G Q Sun H Chen and E T Liu ldquoFatiguelife prediction for GH4169 superalloy under multiaxial variableamplitude loadingrdquo Journal of Beijing University of Technologyvol 38 no 10 pp 1462ndash1466 2012
[29] H T Corten and T J Dolon ldquoCumulative fatigue damagerdquo inProceedings of the International Conference on Fatigue of Metalspp 235ndash246 1956
[30] Y H Lu Study on fatigue reliability of welded bogie frame forraiway vehicle [PhD thesis] Southwest Jiaotong University2011
[31] J Tian Z-M Liu and R He ldquoNonlinear fatigue-cumulativedamage model for welded aluminum alloy joint of EMUrdquoJournal of the China Railway Society vol 34 no 3 pp 40ndash432012
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
International Journal of
AerospaceEngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
RoboticsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Active and Passive Electronic Components
Control Scienceand Engineering
Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
RotatingMachinery
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporation httpwwwhindawicom
Journal ofEngineeringVolume 2014
Submit your manuscripts athttpwwwhindawicom
VLSI Design
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Shock and Vibration
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Civil EngineeringAdvances in
Acoustics and VibrationAdvances in
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Electrical and Computer Engineering
Journal of
Advances inOptoElectronics
Hindawi Publishing Corporation httpwwwhindawicom
Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
SensorsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Modelling amp Simulation in EngineeringHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Chemical EngineeringInternational Journal of Antennas and
Propagation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Navigation and Observation
International Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
DistributedSensor Networks
International Journal of
