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Theoretical and Applied Fracture Mechanics 44 (2005) 105–115
www.elsevier.com/locate/tafmec
Fatigue crack growth with overload under spectrum loading
X.P. Huang a,*, J.B. Zhang a, W.C. Cui b, J.X. Leng b
a School of Naval Architecture, Ocean and Civil Engineering, Shanghai Jiao Tong University, Shanghai, Chinab China Ship Scientific Research Center, P.O. Box 116, Wuxi, China
Abstract
Load cycle interactions can have a very significant effect in fatigue crack growth under variable amplitude loading.Studying of fatigue crack growth rate and fatigue life calculation under spectrum loading is very important for the reli-able life prediction of engineering structures. In this paper, a fatigue life prediction model under various load spectra,using the strain energy density factor approach and the plastic zone size near crack tip as main parameters in calculatingeffective strain energy density factor, has been proposed. The present model is validated with fatigue crack growth testdata provided by Ray under various variable amplitude and spectrum loading in 7075-T6 and 2024-T3 aluminum alloy,respectively. Predictions of present model are compared with those of the state-space model, FASTRN and AFGROWcodes. The results show that the predicted results agree well with the test data.� 2005 Elsevier Ltd. All rights reserved.
Keywords: Spectrum loading; Crack opening ratio; Effective SEDF range; Fatigue life prediction
1. Introduction
Many engineering structures are subjected torandom loading in service. The fatigue growth lifewill be affected by load sequence. Neglecting theeffect of cycle interaction in fatigue calculationsunder variable amplitude loading can lead to com-pletely invalid life predictions. However, for designpurposes it is particularly difficult to generate analgorithm to quantify these sequence effects on fa-
0167-8442/$ - see front matter � 2005 Elsevier Ltd. All rights reservdoi:10.1016/j.tafmec.2005.06.001
* Corresponding author.E-mail address: [email protected] (X.P. Huang).
tigue crack propagation, due to the number and tothe complexity of the mechanisms involved in thisproblem [1]. One of the theories to explain theseload sequence effect is that the plasticity inducedfatigue crack closure is the primary mechanism[2]. There are many calculating models of crackpropagation life under spectrum loading, such asWheeler model, Willenborg et al. model, basedon plastic zone correction theory in the vicinityof crack tip [3–5], the U � R model based on theconcept of effective stress intensity factor range[6–13] and the model based on strain energy den-sity factor [14–16].
ed.
106 X.P. Huang et al. / Theoretical and Applied Fracture Mechanics 44 (2005) 105–115
In this paper, a feasible study towards the crackpropagation law under various spectra loading hasbeen carried out based on effective strain energydensity factor. A crack growth life predictionmodel under spectrum loading is provided. Predic-tions of present model are compared with testdata, those of the state-space model, FASTRNand AFGROW codes under various variableamplitude and spectrum loading.
2. Fatigue crack growth under spectrum loading
2.1. Fatigue crack propagation rate
It is difficult to give a crack growth life calcula-tion model which can consider all the effect factorssuch as plastic induced closure, residual stressesand strain, strain hardening, crack face roughness,and oxidation of the crack face, et al. The resultsof experimental study on fatigue over the past sev-eral decades provide a knowledge base, and theprimary mechanism under many conditions isplasticity. The purpose of this paper is to addresshow to characterize the effect of load sequence infatigue crack propagation under variable ampli-tude loading.
The fatigue crack propagation will be decreasedor arrested after experiencing overload one ormore times. This phenomenon can be explainedby crack opening ratio U. When it is overloaded,the increased rmax and unchanged rmin lead tothe decrease of stress ratio R = rmin/rmax and U
is accordingly decreased. When it is underloaded,stress ratio R changes similarly while U proves toincrease unexpectedly [10]. How to take accountof the effect during the load cycles after overload-ing and underloading is the main problem.
Paris crack growth law is widely used to calcu-late the fatigue crack propagation life in engineer-ing structures. The expression may not beadequate to analyze the crack growth behaviorof cracked structures under spectrum loading forthe equation dose not involve the mean stress leveland the equation is restricted to cracks propagatednormal to the applied load [16]. The strain energydensity factor (SEDF) approach has been used toanalyze fatigue crack growth behavior of cracked
structures [14,17]. After the effect of load sequencewas discussed by Schijive and Broek and the effec-tive stress intensity factor was proposed by Elber[5], the crack fatigue growth rate was expressedda/dN versus DKeff. A number of load-interactionmodel have been developed to correlate fatiguecrack growth rates and to predict crack growthunder variable amplitude loading during the pastthree decades. In this paper, the effective strainenergy density factor range is used in the crackgrowth rate equation. The crack growth rate canbe expressed as:
dadN
¼ BðDSeffÞm ð1Þ
DSeff ¼ C � DS ð2Þ
where da/dN is the crack growth rate, DS is thestrain energy density factor (SEDF) range, DSeff
is the effective strain energy density factor range,B and m are material constants, C is a correctioncoefficient of plastic zone size in the vicinity ofcrack tip.
2.2. Strain energy density factor
The strain energy density factor S takes the fol-lowing form:
S ¼ a11K21 þ 2a12K1K2 þ a22K2
2 þ a33K23 ð3Þ
in which K1, K2 and K3 are the stress intensityfactors (SIF) to tensile, in-plane shear, and out-of-plane shear loads, aij (i, j = 1,2,3) are the coeffi-cients. In this study, K2 = K3 = 0 and the directionof the crack growth is found by taking oS/oh = 0which gives h = 0. The strain energy density factorS in the case of plane strain is given by
S ¼ ½ð1� 2mÞ=4G� � K21 ð4Þ
The strain energy density factor range is given by
DS ¼ 1� 2m4G
ðK21;max � K2
1;minÞ
¼ 1� 2m4G
ðK1;max þ K1;minÞDK1 ð5Þ
where m is Poisson�s ratio and G is the shearmodulus.
X.P. Huang et al. / Theoretical and Applied Fracture Mechanics 44 (2005) 105–115 107
2.3. Calculation of the effective strain energy
density factor range
Investigations indicate that crack opening ratiocan also be affected bymaximum stress intensity fac-tor and the constrained state of the crack tip. Megg-iolaro pointed out that the crack closure concept
Table 1Geometry parameters and material properties of the specimens
Specimen material ry (MPa) ru (MPa) B
7075-T6 aluminum alloy 520 575 9.92024-T3 aluminum alloy 327.9 473.3 5.08
0
20
40
60
80
100
Stre
ssσ
(MPa
)
p cycles
q cycles
One block of loading
Cra
ck le
ngth
a (m
m)
0 50 100 150 200 250 3000
20
40
60
80
100
120
Cycle N X103(cycles)
=120.66MPa=68.95MPa
p=29, q=1
Present model FASTRN State Space Porter Data AFGROW
σ2σ1
Fig. 1. Comparison of predicted values with test data and those of staratios).
dominated by stress ratio cannot provide rationalexplanation on certain conditions (e.g. crack propa-gation retardation or arrest under high stress ratio)[1]. Thus, in this model, a correction coefficient ofplastic zone size in the vicinity of crack tip due tooverloading or underload is introduced. The effec-tive strain energy density factor range is expressed as
m n t (mm) w (mm)
· 10�4 2.07 0.5 4.1 305· 10�4 2.04 0.5 4.1 229
0 20 40 60 80 100 120 1400
20
40
60
80
100
120
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
=76.54MPa=68.95MPa
p=29, q=1
Cycle N X103(cycles)
Present model FASTRN State Space Porter Data AFGROW
0 50 100 150 200 2500
20
40
60
80
100
Cycle N X103(cycles)
=137.90MPa=68.95MPa
p=29, q=1
Present model FASTRN State Space Porter Data AFGROW
σ2σ1
σ2σ1
te-space, FASTRN and AFGROW codes (at different overload
108 X.P. Huang et al. / Theoretical and Applied Fracture Mechanics 44 (2005) 105–115
DSeff ¼ C � DS ¼ C1� 2m4G
ðK21;max � K2
1;minÞ
¼ 1� 2m4G
ðK1;max þ K1;minÞCDK1 ð6Þ
Lots of research works on retardation responseof overloading, based on the correction of yieldzone vicinity of crack tip, have been reported [3].A fatigue life prediction model, which consideringthe constrain states of the crack tip depend on theapplied stress, yield strength of material and thespecimen thickness, under spectrum loading wasreported in literature [4]. In this paper, the modelin literature [5] is employed and modified that theeffect of underload is included in the present model.A physical explanation of crack retardation due toenlarged plastic zone is presented below.
0 50 100 150 200 2500
20
40
60
80
100
120
Cycle N X103(cycles)
=103.43MPa=68.95MPa
p=29, q=1
Present model FASTRN State Space Porter Data AFGROW
0 50 100 150 200 250 300 350 4000
20
40
60
80
100
120
Cycle N X103(cycles)
=103.43MPa=68.95MPa
p=300, q=1
Present model FASTRN State Space Porter Data AFGROW
σ2σ1
σ2σ1
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
Fig. 2. Comparison of predicted values with test data and those of
The plastic zone size is relatively small underconstant amplitude loading, while the resultingplastic zone becomes larger when a single overloadis applied. Provided that crack length is a (aOL)and the applied stress is rmax ðr�
maxÞ correspondingpresent load and (overload) respectively. A plasticzone with size of ry ðr�OLÞ will appear in the vicinityof crack tip. The size of plastic zone, ry, can be cal-culated using Eqs. (8) and (10) which consideringplate thickness and yield strength given by Voor-wald et al. [4]. Wheeler assumes that overload re-sponse will effect if only a + ry does not exceedthe range of aOL þ r�OL during the stress cycles afterexperiencing an overload. The overload responsewill disappear when a + ry reaches to or exceedaOL þ r�OL due to crack propagation or stressincrease.
0 50 100 150 200 250 3000
20
40
60
80
100
120
Cycle N X103(cycles)
=103.43MPa=68.95MPa
p=50, q=1
Present model FASTRN State Space Porter Data AFGROW
0 50 100 150 200 250 300 350 400 4500
20
40
60
80
100
120
Cycle N X103(cycles)
=103.43MPa=68.95MPa
p=1000, q=1
Present model FASTRN State Space Porter Data AFGROW
σ2σ1
σ1
σ2
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
state-space, FASTRN and AFGROW codes (at different p).
X.P. Huang et al. / Theoretical and Applied Fracture Mechanics 44 (2005) 105–115 109
In the contrast to a single-cycle overload, asingle-cycle underload makes the reverse plas-tic flow and depletion of the resulting plasticzone. The increment of yield zone size caused by
Stre
ssσ
(MPa
)
p cycles q cycles
One block of loading
σ1
σ2
103.43
51.72
Cra
ck le
ngth
a (m
m)
0 20 40 60 80 100 1200
10
20
30
40
50
60
70
80
90
Cycle N X103(cycles)
=155.14MPa=5.17MPa
p=50,q=1
Present model FASTRN State Space Porter Data AFGROW
σ2σ1
0 20 40 60 80 100 1200
10
20
30
40
50
60
70
80
90
Cycle N X103(cycles)
=134.45MPa=5.17MPa
p=50, q=1Cra
ck le
ngth
(mm
)
Present model FASTRN State Space Porter Data AFGROW
σ2σ1
Fig. 3. Comparison of predicted values with test data and those of staat different load ratios).
underload can be quantitively calculated byEq. (9). The correction coefficient of plastic zonesize can be calculated using the following ex-pressions:
0 10 20 30 40 50 60 70 80 900
10
20
30
40
50
60
70
80
90
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
Cycle N X103(cycles)
=103.43MPa=5.17MPa
Present model FASTRN State Space Porter Data AFGROW
No overload only underload p=50, q=1
0 20 40 60 80 100 1200
10
20
30
40
50
60
70
80
90
Cycle N X103(cycles)
=155.14MPa=31.03MPa
p=50, q=1
Present model FASTRN State Space Porter Data AFGROW
σ2σ1
σ2σ1
0 20 40 60 80 1000
10
20
30
40
50
60
70
80
90
100
Cycle N X103(cycles)
=134.45MPa=31.03MPa
p=50, q=1
Present model FASTRN State Space Porter Data AFGROW
σ2σ1
te-space, FASTRN and AFGROW codes (overload–underload
110 X.P. Huang et al. / Theoretical and Applied Fracture Mechanics 44 (2005) 105–115
C¼ry
aOLþ r�OL�a� rD
� �n
ðaþ ry < aOLþ r�OL� rDÞ
1 ðaþ ry P aOLþ r�OL� rDÞ
8<:
ð7Þ
ry ¼bKmax
ry
� �2
ð8Þ
rD ¼ bDKmax
r�y
!2
ð9Þ
b¼
1
6pðtP 2.5ðKmax=ryÞ2Þ
1
pt6
1
pðKmax=ryÞ2
� �1
6pþ 5
6p2.5� tðKmax=ryÞ�2
2.5�p�1
!
1
pKmax
ry
� �2
< t< 2.5Kmax
ry
� �2 !
8>>>>>>>>>>>>><>>>>>>>>>>>>>:
ð10Þ
Stre
ssσ
(MPa
)
p cycles q cycles
One block of loading
σ1
σ2
103.43
51.72
Cra
ck le
ngth
a (m
m)
0 30 60 90 120 1500
10
20
30
40
50
60
70
80
90
Cycle N X103(cycles)
=155.14MPa=30.03MPa
p=50,q=1
Present model FASTRN State Space Porter Data AFGROW
σ2σ1
Fig. 4. Comparison of predicted values with test data and those of staat different load ratios).
DKu ¼ Mffiffiffiffiffiffipa
pðri�1
min � riminÞ ð11Þ
where a, aOL are crack length at present and atprior overloading, ry, r�OL are yield zone size underpresent maximum stress and under maximumstress of prior overloading, rD is increment of yieldzone size caused by underloading, t is plate thick-ness, ry; r�
y are tensile, compressive yield stress,Kmax is maximum stress intensity factor in everyload cycle, ri�1
min; rimin are minimum stresses in load
cycle i � 1 and i, n is material constant determinedby test. n = 0 when load sequence effect can beneglected.
Substituting K�max and Kmax corresponding
to the prior overload and present load respec-tively into Eq. (8), r�OL and ry are determined.Parameter rD is used to consider the effect ofunderload.
0 20 40 60 80 100 1200
10
20
30
40
50
60
70
80
90
100
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
Cycle N X103(cycles)
=155.14MPa=5.17MPa
p=50,n=q
Present model FASTRN State Space Porter Data AFGROW
0 30 60 90 120 1500
10
20
30
40
50
60
70
80
90
100
Cycle N X103(cycles)
=134.45MPa=5.17MPa
p=50,q=1
Present model FASTRN State Space Porter Data AFGROW
σ2σ1
σ1
σ2
te-space, FASTRN and AFGROW codes (underload–overload
X.P. Huang et al. / Theoretical and Applied Fracture Mechanics 44 (2005) 105–115 111
From present model and the correlativeparameters, the overloading retardation res-
0
10
20
30
40
50
60
70
80
90
Stre
ssσ
(MPa
)
One block of loadingProgram
Program
Program
0
10
20
30
40
50
60
70
80
90
Stre
ssσ
(MPa
)
One block loading
0
10
20
30
40
50
60
70
80
90
Stre
ssσ
(MPa
)
One block loading
Fig. 5. Comparison of predicted values with test data and those of sblock loading.
ponses can be expressed by above equations.These effect parameters including maximum and
0 20 40 60 80 100 120 1400
10
20
30
40
50
60
70
80
90
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
Cycle N X103(cycles)
Present model FASTRN State Space Porter Data AFGROW
P1
P2
P3
0 20 40 60 80 100 120 1400
10
20
30
40
50
60
70
80
90
Cycle N X103(cycles)
Present model FASTRN State Space Test Data AFGROW
0 10 20 30 40 50 60 70 80 900
10
20
30
40
50
60
70
80
90
Cycle N X103(cycles)
Present model FASTRN State Space Test Data AFGROW
tate-space, FASTRN and AFGROW codes under six types of
0
20
40
60
80
Stre
ssσ
(MPa
)
One block of loading0 10 20 30 40 50 60 70 80 90
0
10
20
30
40
50
60
70
80
90
Cycle N X103(cycles)
Present model FASTRN State Space Test Data AFGROW
Program P4
Program P5
Program P6
0
10
20
30
40
50
60
70
80
90
Stre
ssσ
(MPa
)
One block of loading0 10 20 30 40 50 60 70 80 90
0
10
20
30
40
50
60
70
80
90
Cycle N X103(cycles)
Present model FASTRN State Space Test Data AFGROW
0
10
20
30
40
50
60
70
80
90
100
Stre
ssσ
(MPa
)
One block of loading0 50 100 150 200 250
0
10
20
30
40
50
60
70
80
90
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
Cycle N X103(cycles)
Present model FASTRN State Space Test Data AFGROW
Fig. 5 (continued)
112 X.P. Huang et al. / Theoretical and Applied Fracture Mechanics 44 (2005) 105–115
minimum stresses of a cycle, yield strength ofmaterial and the plate thickness are consideredin the effective stress intensity factor range cal-culation.
3. Model validation
The fatigue crack propagation model undervariable amplitude loading is presented above.
0
10
20
30
40
50
60
70
80
90
100
Stre
ssσ
(MPa
)
One block of loading0 50 100 150 200 250 300
0
10
20
30
40
50
60
70
80
90
Cycle N X103(cycles)
Present model FASTRN State Space Test Data AFGROW
Program P7
Program P8
Program P9
0
10
20
30
40
50
60
70
80
90
100
Stre
ssσ
(MPa
)
One block of loading0 20 40 60 80 100 120 140 160 180 200
0
10
20
30
40
50
60
70
80
90
Present model FASTRN State Space Test Data AFGROW
Cycle N X103(cycles)
0
10
20
30
40
50
60
70
80
90
Stre
ssσ
(MPa
)
One block loading0 20 40 60 80 100 120 140 160 180 200
0
10
20
30
40
50
60
70
80
90
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
Cra
ck le
ngth
a (m
m)
Cycle N X103(cycles)
Present model FASTRN State Space Test Data AFGROW
Fig. 5 (continued)
X.P. Huang et al. / Theoretical and Applied Fracture Mechanics 44 (2005) 105–115 113
For demonstrating the validation of the model,predictions of present model are compared withtest data, those of the state-space model, FASTRNand AFGROW codes given in Ref. [11].
Porter provided some fatigue test data ofthrough thickness center-cracked 7075-T6 alumi-num alloy plate specimens under variable ampli-tude loading. The geometry parameters and
0
10
20
30
40
50
60
70
80
90
100
110
Stre
ssσ
(MPa
)
One block of loading0 20 40 60 80 100 120 140
0
10
20
30
40
50
60
70
80
90
Cycle N X103(cycles)
Cra
ck le
ngth
a (m
m)
Present model FASTRN State Space Test Data AFGROW
Program P10
Fig. 5 (continued)
114 X.P. Huang et al. / Theoretical and Applied Fracture Mechanics 44 (2005) 105–115
material properties of the specimens are listedin Table 1. Fig. 1 illustrates the cyclic stressexcitation and the comparisons of predicted valueswith Porter data and those of the state-spacemodel, FASTRN and AFGROW codes underdifferent overload ratios (r2/r1 = 76.54/68.95,120.66/68.95 and 137.9/68.95 respectively). Presentmodel can give better prediction at high stressratio. Fig. 2 shows comparisons of predictedvalues with Porter data, those of the state-spacemodel, FASTRN and AFGROW codes, underdifferent number of p (q = 1, p = 29, 50, 300 and1000 respectively). Fig. 3 shows comparisons ofpredicted values with test data and those ofstate-space, FASTRN and AFGROW codesunder overload–underload at different load ratios.Fig. 4 shows comparisons of predicted values withtest data and those of state-space, FASTRN andAFGROW codes under underload–overload atdifferent load ratios. From these comparisons, itcan be seen that the predicted values are in goodagreement with test data.
The model has also been used to predict thefatigue test results on center-cracked 2024-T3aluminum alloy specimens under program loadinggiven by Ray [11]. The geometry parameters andmaterial properties of the specimens are presentedin Table 1. The load spectra and the schematiccomparisons of predicted values with test dataand those of state-space, FASTRN and AFGROWcodes are presented in Fig. 5. It can be seen fromthe comparisons that the predicted values under
10 different program loadings agree well with testdata. It is clear that neglecting the effect of loadsequence in fatigue calculations under variableamplitude loading can lead to completely invalidlife predictions. The present model has beenproved to be rationally applicable for crack prop-agation under variable amplitude loading.
4. Summary and conclusions
Fatigue crack closure is the most used mecha-nism to explain load cycle interactions such as de-lays in or arrests of the crack growth afteroverloads. In fact, neglecting crack closure inmany fatigue life calculations can result in overlyconservative predictions, increasing maintenancecosts by unnecessarily reducing the period betweeninspections. Many models based on effective stressintensity factor take the stress ratio as the main ef-fect factor in considering the effect of load se-quence. It has long been proved to satisfactorilyexplain plane stress crack retardation effects. Butin the procedure based on SEDF, the effect ofmean stress is included in SEDF. For better pre-dicting the delay of the overload, a correction coef-ficient of plastic zone size in the vicinity of cracktip is introduced.
A crack growth life calculation model based oneffective strain energy density factor under variableamplitude loading is presented in this paper. Inthis model, the effect of mean stress is included
X.P. Huang et al. / Theoretical and Applied Fracture Mechanics 44 (2005) 105–115 115
in strain energy density factor and the overloadretardation is corrected by the plastic zone size inthe vicinity of crack tip. Thus the model has dis-tinct advantage to deal with problem of overloadresponse and is appropriate to characterize crackpropagation under variable loading. The modelis used to predict the fatigue crack propagationtest data and compared with those of state-space,FASTRN and AFGROW codes under severaltypes of spectrum loadings. The comparisonsshow that the present model has satisfactoryaccuracy.
Acknowledgement
The authors greatly appreciate Professor G.C.Sih who gave us some useful suggestions andcomments.
References
[1] M.A. Meggiolaro, J.T.P. Castro, On the dominant role ofcrack closure on fatigue crack growth modelling, Int. J.Fatigue 25 (2003) 843–854.
[2] K. Solanki, S.R. Daniewicz, J.C. Newman, Finite elementanalysis of plasticity-induced fatigue crack closure: anoverview, Eng. Fract. Mech. 71 (2004) 149–171.
[3] Z.W. Chen, O.S. Lee, Models for crack growth analysisunder random loading, J. Mech. Strength 22 (4) (2000)245–248 (in Chinese).
[4] H.J.C. Voorwald, M.A.S. Torres, Modelling of fatiguecrack growth following overloads, Int. J. Fatigue 13 (5)(1991) 423–427.
[5] X.P. Huang, D.X. Shi, H.B. Cui, Propagation of fatiguecrack in high strength steel for submarine under randomloading, J. Harbin Eng. Univ. 22 (2) (2001) 6–9 (in Chinese).
[6] W. Elber, The significance of fatigue crack closure infatigue, ASTM STP 486 (1972) 230–242.
[7] J. Schijve, Some formulas for the crack opening stress level,Eng. Fract. Mech. 14 (1981) 461–465.
[8] J.C. Newman, A crack opening stress equation for fatiguecrack growth, Int. J. Fract. 24 (1984) R131–R135.
[9] D.Y. Jeong, G.C. Sih, Evaluation of Elber�s crack closuremodel as an exploration of time load sequence effects oncrack growth rates, Technical Report, US Department ofTransportation Research and Special Program Adminis-tration, Transportation System Center, 1990.
[10] A. Ray, P. Patankar, Fatigue crack growth under variable-amplitude loading: Part I—Model formulation in state-space setting, Appl. Math. Modell. 25 (2001) 979–994.
[11] A. Ray, P. Patankar, Fatigue crack growth under variable-amplitude loading: Part II—code development and modelvalidation, Appl. Math. Modell. 25 (2001) 995–1013.
[12] J.C. Newman, Crack-growth calculations in 7075-T7351aluminum alloy under various load spectra using animproved crack-closure model, Eng. Fract. Mech. 71(2004) 2347–2363.
[13] X.P. Huang, Y. Han, W.C. Cui, et al., Fatigue lifeprediction of weld-joints under variable amplitude fatigueloading, J. Ship Mech. 9 (1) (2005) 89–97 (in Chinese).
[14] G.C. Sih, D.Y. Jeong, Fatigue load sequence effect rankedby critical available energy density, Theor. Appl. Fract.Mech. 14 (1) (1990) 141–151.
[15] E.T. Moyer Jr., G.C. Sih, Fatigue analysis of an edge crackspecimen: hysteresis strain energy density, Eng. Fract.Mech. 19 (1984) 643–652.
[16] G.C. Sih, Mechanics of Fracture Initiation and Propaga-tion, Kluwer Academic Publisher, Boston, 1991.
[17] D.H. Choi, H.Y. Choi, Fatigue life prediction of out-of-plane gusset welded joints using strain energy densityfactor approach, Theor. Appl. Fract. Mech. 44 (1) (2005)16–26.