Post on 25-May-2019
Research ArticleCrack Closure Effects on Fatigue Crack Propagation RatesApplication of a Proposed Theoretical Model
Joseacute A F O Correia1 Abiacutelio M P De Jesus12
Pedro M G P Moreira1 and Paulo J S Tavares1
1 Institute of Science and Innovation in Mechanical and Industrial Engineering University of Porto Rua Dr Roberto Frias4200-465 Porto Portugal2Department of Mechanical Engineering Faculty of Engineering University of Porto Rua Dr Roberto Frias 4200-465 Porto Portugal
Correspondence should be addressed to Jose A F O Correia jacorreiainegiuppt
Received 10 December 2015 Accepted 18 February 2016
Academic Editor Antonio Riveiro
Copyright copy 2016 Jose A F O Correia et al This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited
Structural design taking into account fatigue damage requires a thorough knowledge of the behaviour of materials In addition tothemonotonic behaviour of thematerials it is also important to assess their cyclic response and fatigue crack propagation behaviourunder constant and variable amplitude loading Materials whenever subjected to fatigue cracking may exhibit mean stress effectsas well as crack closure effects In this paper a theoretical model based on the same initial assumptions of the analytical modelsproposed by Hudak and Davidson and Ellyin is proposed to estimate the influence of the crack closure effects This proposal basedfurther onWalkerrsquos propagation lawwas applied to the P355NL1 steel using an inverse analysis (back-extrapolation) of experimentalfatigue crack propagation results Based on this proposed model it is possible to estimate the crack opening stress intensity factor119870op the relationship between 119880 = Δ119870effΔ119870 quantity and the stress intensity factor the crack length and the stress ratio Thisallows the evaluation of the influence of the crack closure effects for different stress ratio levels in the fatigue crack propagation ratesFinally a good agreement is found between the proposed theoretical model and the analytical models presented in the literature
1 Introduction
In the early seventies Elber [1 2] introduced the crack closureand opening concepts Initially Elber [1 2] suggested thatcrack closure was associated with the effective stress intensityrange using 119880 (= Δ119870effΔ119870) parameter which was assumedas a linear function of stress ratio119877 independent of119870max andΔ119870 Later studies determined that the quantitative parameter119880 is dependent on 119870max After the proposal developed byElber [1 2] other proposals appeared based on elastoplasticanalytical concepts [3ndash7] or the near-threshold fatigue crackgrowth regimes [8ndash12] All approaches are important contri-butions of the crack closure and opening effects which occurin materials when subjected to constant amplitude loadingparticularly in the near-threshold regime of the fatigue crackgrowth All analytical techniques were supported by mea-surements of the crack-tip opening load Other approacheshave been implemented based on elastoplastic analysis usingfinite element methods [13 14]
This paper proposes a theoretical model based onWalkerrsquos propagation law [15] to estimate the quantitativeparameter 119880 important for the evaluation of the fatiguecrack opening stress intensity factor119870op taking into accountthe stress ratio and crack closure and opening effectsThe cur-rent study is applied to the P355NL1 steel using experimentalfatigue crack propagation test results under constant ampli-tude loading [16] The proposed model is applied using aninverse analysis of the experimental results Various analyticalmodels for crack closure are also compared in this study
2 Overview of the Crack Closure and OpeningConcepts and Existing Models
The crack closure and opening concept has been discussed inthe scientific community since the first contributions by Elber[1 2] Elber suggested [1 2] that crack closure and openingeffects could be characterized in terms of the effective stress
Hindawi Publishing CorporationAdvances in Materials Science and EngineeringVolume 2016 Article ID 3026745 11 pageshttpdxdoiorg10115520163026745
2 Advances in Materials Science and Engineering
K
Time
Crack closed
Crack openedKmax
Kop
Kmin
ΔK
ΔKeff
Figure 1 Definition of the effective and applied stress intensityfactor ranges
intensity factor range which is normalized by the appliedstress intensity factor range resulting in the 119880 ratio with thefollowing form
119880 =Δ119870effΔ119870
(1)
where Δ119870eff is the effective stress intensity factor rangeand Δ119870 is the applied stress intensity factor range Elberrsquosformulation is supported by the following assumptions forthe 119880 parameter (i) linear function of stress ratio (119877) and(ii) being relatively independent of maximum stress intensityfactor (119870max) or stress intensity factor range (Δ119870) Theeffective stress intensity factor range is defined as
Δ119870eff = 119870max minus 119870op (2)
where 119870max is the maximum stress intensity factor and 119870opis the crack opening stress intensity factor The applied stressintensity factor range Δ119870 is presented by
Δ119870 = 119870max minus 119870min (3)
with119870min being the minimum stress intensity factor Figure 1illustrates the previous parameters definitions
According to the approach of Elber [1 2] the fatiguecrack growth rate 119889119886119889119873 is a function of the effective stressintensity factor range according to the following generalexpression
119889119886
119889119873= 119891 (Δ119870eff) (4)
According to studies conducted by Elber for the 2024-T3aluminum alloy using stress ratios 119877 between minus01 and 07119880 depends on119877The experimental results led to the followinglinear relationship
119880 = 05 + 04119877 (5)
The quantitative parameter 119880 of (5) can be substituted into(1) and using (2) results in the following relation for the119870op119870max ratio
119870op
119870max= 05 + 01119877 + 04119877
2 (6)
Subsequent studies using various fatigue loading variablesclearly showed that 119880 and 119870max are related variablesTherefore Elber [1 2] approach appears to be inconsis-tentincomplete
An improved function of (5) was proposed by Schijve[9] with a more realistic behaviour to account for the crackclosure and opening effects for minus1 le 119877 le 1
119880 = 055 + 033119877 + 0121198772 (7)
Therefore it is possible to obtain a relation to the crackopening stress intensity factor as follows
119870op
119870max= 045 + 022119877 + 021119877
2+ 012119877
3 (8)
Another proposal was presented by ASTM [10 11] togetherwith a set of conditions
Δ119870 = 119870max for 119877 lt 0
Δ119870 = 119870max minus 119870min for 119877 ge 0
119880 = 0576 + 0015119877 + 04091198772
119870op
119870max= 0424 + 0561119877 minus 0394119877
2+ 0409119877
3
(9)
Newman Jr [3] proposed a general crack opening stressequation for constant amplitude loading function of stressratio 119877 stress level 120590 and four nondimensional constants
120590op
120590max= 1198600+ 1198601119877 + 119860
21198772+ 11986031198773 for 119877 ge 0
120590op
120590max= 1198600+ 1198601119877 for minus 1 le 119877 lt 0
(10)
when 120590op ge 120590max The coefficients 1198600 1198601 1198602 and 119860
3are as
follows
1198600= (0825 minus 034120572 + 005120572
2) [cos(
120587120590max21205900
)]
1120572
1198601= (0415 minus 0071120572)
120590max1205900
1198602= 1 minus 119860
0minus 1198601minus 1198603
1198603= 21198600+ 1198601minus 1
(11)
where 1205900is the flow stress (average between the uniaxial yield
stress and uniaxial ultimate tensile strength of the material)For plane stress conditions 120572 = 1 and for plane strainconditions 120572 = 3 The normalized crack opening stress isobtained using the stress ratio for 120590max = 120590
03
Advances in Materials Science and Engineering 3
This approach proposed by Newman Jr [3] may be usedto correlate fatigue crack growth rate data for other materialsand thicknesses under constant amplitude loading once theproper constraint factor has been determined
The crack closure effects may occur due to the surfaceroughness of the material in the presence of shear defor-mation at the crack tip [17 18] Microstructural studies [18]indicate that fracture surfaces demonstrate the possibilityof crack propagation by the shear mechanism primarilyin the near-threshold regime of Δ119870 [17] In this fatigueregime the crack closure effects have the greatest influenceConsequently Hudak and Davidson [8] proposed a modeldefined by the following expression
119880 = 1 minus119870119900
119870max= 1 minus
119870119900(1 minus 119877)
Δ119870 (12)
where119870119900is a constant related to the pureMode I fatigue crack
growth thresholdThe experimental results from fatigue crack propagation
tests can be represented by the following generalized relations(see Figure 2)
119880 = 120574(1 minus119870119900
119870max) 119870max le 119870
119871 (13)
119880 = 1 119870max ge 119870119871 (14)
where 119870119871
is the limiting 119870max value above which nodetectable closure occurs and 120574 is a parameter that can beobtained using the following expression
120574 = (1 minus119870119900
119870119871
) (15)
The 119870119900
and 119870119871
constants are not material propertiesin the strict sense since they depend on measurementslocation and measurements sensitivity resulting from thecrack propagation tests aiming at evaluating the crack clo-sureopening effects Note that in the limiting condition of119870119900119870119871approaching zero (13) reduces to (12) thus describing
the local measurementsEllyin [12] proposed an approach to define the effective
stress intensity range Δ119870eff taking into account the stressratio 119877 and the threshold value of the stress intensity factorrange Δ119870th with constant amplitude loading (see Figure 3)This approach is amodified version of proposal byHudak andDavidson [8] Generally the crack opening stress intensityfactor range Δ119870op is smaller than the threshold stressintensity factor range Δ119870th that is Δ119870op lt Δ119870th Besidesthe crack opening or closure stress intensities factors are notthe same that is Δ119870cl lt Δ119870op lt Δ119870th Ellyin [12] proposedthe following approach to obtain Δ119870eff (see Figure 3)
Δ119870eff 0 = (Δ1198702minus Δ1198702
th)12
asymp Δ119870[1 minus1
2(Δ119870thΔ119870
)
2
]
gt Δ119870 minus Δ119870th for 119877 asymp 0
1
00
1
U = ΔKeffΔK
U = 1
U = 120574(1 minusKmax)
120574Ko
120574
1KL 1Ko
Ko
1Kmax
Figure 2 Functional relationship between 119880 and 119870max Hudak andDavidson [8]
K
Time
ΔK
ΔKth Kcl ΔKop
ΔKeff
ΔKeff = [(ΔK)2 minus (ΔKth)2]12
Kmax
Kop
Kth
Kmin
Figure 3 Definition of the effective stress intensity range accordingwith Ellyin [12]
Δ119870eff =Δ119870eff 0
[1 minus (1205901198981205901015840
119891)]
=Δ119870eff 0
[1 minus ((1 + 119877) 120590max21205901015840
119891)]
for 119877 = 0
(16)
where 120590119898is the mean stress 120590max is the maximum stress and
1205901015840
119891is the fatigue strength coefficientThe crack closure and opening effects can be estimated
using an elastoplastic analysis based on analytical [3 19]or numerical [13 14] approaches Vormwald [4 5] andSavaidis et al [6 7] estimated the crack closure and openingeffects proposing the use of an analytical elastoplastic analysisdeveloped by Neuber [19] and themodified crack closure andopening model suggested by Newman Jr [3] The numericalelastoplastic analysis using the finite element method isproposed by McClung and Sehitoglu [13] and Nakagaki andAtluri [14] Based on finite element analysis performed byMcClung and Sehitoglu [13] it has been shown that the resultsobtained for 120590op may deviate from the results obtained usingthe crack closure and opening model by Newman Jr [3]
4 Advances in Materials Science and Engineering
Nakagaki and Atluri [14] proposed an elastoplastic finiteelement procedure to account for arbitrary strain hardeningmaterial behaviour In order to determine the crack closureand opening stresses Nakagaki and Atluri [14] proposed thefollowing procedure (1) calculating the displacements of thenodes in the crack axis before the closure (and after opening)(2) determining by extrapolating to zero the restraining forceat the respective nodal displacement in the correspondingnode just after crack closure (and before the opening) byextrapolation against the load level In all the cases studiedthese two sets of extrapolated values for 120590op and 120590cl werefound to correlate excellently
3 Proposed Theoretical Model
In this paper a theoretical model to obtain the effective stressintensity range Δ119870eff that takes into account the effects ofthe mean stress and the crack closure and opening effectsis proposed The starting point for the proposed theoreticalmodel is the initial assumptions proposed by Elber [1 2] anddefined in (1) and (4)The basic fatigue crack propagation lawin regime II was initially proposed by Paris and Erdogan [20]which has the following form
119889119886
119889119873= 119862 (Δ119870)
119898 (17)
where 119862 and 119898 are material constants However this modeldoes not include the other propagation regimes and doesnot take into account the effects of mean stress Many otherfatigue propagation models have been proposed in literaturewith wider range of application and complexity requiring asignificant number of constants [21] Nevertheless the simplemodification of the Paris relation in order to take into accountthe stress ratio effects is assumed as proposed by Dowling[15]
119889119886
119889119873= 119862119908(
Δ119870
(1 minus 119877)1minus120574
)
119898119908
= 119862119908(Δ119870)
119898119908
(18)
where 119862119908 119898119908 and 120574 are constants Besides an extension of
the Paris relation is adopted to account for crack propagationregime I having the following configuration
119889119886
119889119873= 119862119908(Δ119870 minus Δ119870th)
119898119908
(19)
Based on the foregoing assumptions the effective stressintensity factor range is given by the following expression
Δ119870eff = (Δ119870 minus Δ119870th) = (Δ119870
(1 minus 119877)1minus120574
minus Δ119870th)
= Δ119870(1
(1 minus 119877)1minus120574
minusΔ119870thΔ119870
)
(20)
1
00
1
U = 1
1Kmax1KL
U = ΔKeffΔK
(1 minus R)120574minus1
(1 minus R)120574minus1 middot ΔKth0
U = (1 minus ΔKth0Kmax)(1 minus R)120574minus1
1ΔKth0
Figure 4 Functional relationship between119880 and119870max for proposedtheoretical model
Thus the expression that allows obtaining the quantitativeparameter 119880 is given by the following formula
119880 =Δ119870effΔ119870
= (1
(1 minus 119877)1minus120574
minusΔ119870thΔ119870
)
= (1 minus 119877)120574minus1
minusΔ119870thΔ119870
(21)
If the material parameter 120574 is equal to 1 this means thatthe material is not influenced by the stress ratio Thus (21)is simplified and assumes the same form as (12) proposed byHudak and Davidson [8] and is similar to the one proposedby Ellyin [12]
Equation (21) can be presented in another form
119880 = (1 minus 119877)120574minus1
minusΔ119870th
(1 minus 119877)119870max (22)
The threshold value of the stress intensity factor range can bedescribed by a general equation using the following form
Δ119870th = Δ119870th0 sdot 119891 (119877) (23)
where Δ119870th is the threshold value of stress intensity range fora given stress ratio and Δ119870th0 is the threshold value of stressintensity range for 119877 = 0 Klesnil and Lukas [22] proposeda relation between the threshold stress intensity factor rangeΔ119870th and the stress ratio 119877 which is well known
Δ119870th = Δ119870th0 sdot (1 minus 119877)120574 (24)
The validity of (24) is for 119877 ge 0 Replacing this result in (22)results in
119880 = (1 minus 119877)120574minus1
minusΔ119870th0 sdot (1 minus 119877)
120574
(1 minus 119877)119870max (25)
Finally it is possible to display (22) in a more simplified form(see Figure 4)
119880 = (1 minusΔ119870th0
119870max) (1 minus 119877)
120574minus1 for 119870max le 119870119871
119880 = 1 for 119870max ge 119870119871
(26)
Advances in Materials Science and Engineering 5
Table 1 Cyclic elastoplastic and strain-life properties of the P355NL1 steel 119877120576= minus1 and 119877
120576= 0
Material 1205901015840
119891119887 120576
1015840
119891119888 119870
10158401198991015840
MPa mdash mdash mdash MPa mdashP355NL1 100550 minus01033 03678 minus05475 94835 01533
where 119870119871is the limiting 119870max For stress ratio 119877 equal to 0
and material parameter 120574 equal to 1 (22) reduces to
119880 = 1 minusΔ119870th0
119870max for 119877 = 0 120574 = 1 (27)
In summary a theoretical model based on Walkerrsquos propa-gation law [15] was proposed which relates the quantitativeparameter 119880 and Δ119870 (or 119870max) and the stress ratio Theproposed model is based on the same initial assumptions ofthe analytical models proposed by Hudak and Davidson [8]and Ellyin [12]
4 Basic Fatigue Data of the InvestigatedP355NL1 Steel
TheP355NL1 steel is a pressure vessel steel and is a normalizedfine grain low alloy carbon steel Its fatigue behaviour hasbeen investigated [16 23 24] A steel plate 5mm thick wasused to prepare specimens for experimental testing Thissection presents the strain-life fatigue data and the fatiguecrack propagation data obtained for the P355NL1 steel [16]
41 Strain-Life Behaviour The strain-life behaviour of theP355NL1 steel was evaluated through fatigue tests of smoothspecimens carried out under strain controlled conditionsaccording to theASTME606 standard [25]TheRamberg andOsgood [26] relation as shown in the following equation wasfitted to the stabilized cyclic stress-strain data
Δ120576
2=Δ120576119864
2+Δ120576119875
2=Δ120590
2119864+ (
Δ120590
21198701015840)
11198991015840
(28)
where1198701015840 and 1198991015840 are the cyclic strain hardening coefficient andexponent respectively Δ120576119875 is plastic strain range and Δ120590 isthe stress range
Low-cycle fatigue test results are very often representedusing the relation between the strain amplitude and thenumber of reversals to failure 2119873
119891 usually assumed to
correspond to the initiation of a macroscopic crack Morrow[27] suggested a general equation valid for low- and high-cycle fatigue regimes
Δ120576
2=Δ120576119864
2+Δ120576119875
2=
1205901015840
119891
119864(2119873119891)119887
+ 1205761015840
119891(2119873119891)119888
(29)
where 1205761015840119891and 119888 are respectively the fatigue ductility coeffi-
cient and fatigue ductility exponent 1205901015840119891is the fatigue strength
coefficient 119887 is the fatigue strength exponent and 119864 is theYoung modulus
Table 2 Elastic and tensile properties of the P355NL1
Material 119864 ] 119891119906
119891119910
GPa mdash MPa MPaP355NL1 20520 0275 56811 41806
Alternatively to the Morrow relation the Smith-Watson-Topper fatigue damage parameter [28] can be used whichshows the following form
120590max sdotΔ120576
2= SWT
=
(1205901015840
119891)2
119864(2119873119891)2119887
+ 1205901015840
119891sdot 1205761015840
119891sdot (2119873119891)119887+119888
(30)
where 120590max is themaximum stress of the cycle and SWT is thedamage parameter Both Morrow and Smith-Watson-Toppermodels are deterministicmodels and are used to represent theaverage fatigue behaviour of the bridgematerials based on theavailable experimental data
Two series of specimens were tested under distinct strainratios 119877
120576= 0 (19 specimens) and minus1 (24 specimens)
Figure 5 shows a plot of the experimental fatigue data in theform of strain-life and SWT-life curves for the two strainratios The cyclic Ramberg-Osgood and Morrow strain-lifeparameters of the P355NL1 steel are summarized in Table 1for the conjunction of both strain ratios [16 23 24] Thecyclic curve is shown in Figure 5(a) including the influenceof the experimental results fromboth test seriesThis researchadopted the values obtained by combining the results of thetwo test series together since the strain ratio effects are con-sidered negligible (tests performed under strain controlledconditions)This could be explained by the cyclic mean stressrelaxation phenomenon and also by a lower sensitivity ofthe material to the mean stress In addition Table 2 presentsthe elastic and monotonic tensile properties of this pressurevessel steel under investigation The elastic and monotonictensile properties are represented respectively by the Youngmodulus 119864 Poisson ratio ] the ultimate tensile strength 119891
119906
and upper yield stress 119891119910
42 Fatigue Crack Propagation Rates Fatigue crack growthrates of the P355NL1 pressure vessel steel were evaluated forseveral stress ratios using compact tension (CT) specimensfollowing the recommendations of the ASTM E647 standard[10] The CT specimens of P355NL1 steel were manufacturedwith a width 119882 = 40mm and a thickness 119861 = 45mm [16]All tests were performed in air at room temperature under asinusoidal waveform at a maximum frequency of 20Hz Thecrack growth was measured on both faces of the specimens
6 Advances in Materials Science and Engineering
0
100
200
300
400
500
600
0 05 1 15 2
Cyclic curve
Δ120590
(MPa
)
Δ1205762 ()
R120576 = 0
R120576 = minus1
(a)
R120576 = 0
R120576 = minus1
10E minus 01
10E minus 02
10E minus 03
10E minus 04
Δ120576
2(mdash
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
Δ120576
2=
10055
E(2Nf)minus01013 + 03678(2Nf)minus05475
(b)
01
10
100
SWT
(MPa
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
R120576 = 0
R120576 = minus1
(c)
Figure 5 Fatigue data obtained for the P355NL1 steel 119877120576= minus1 and 119877
120576= 0 (a) cyclic curve (b) strain-life data (c) SWT-life data
by visual inspection using two travelling microscopes withan accuracy of 0001mm
Figures 6(a) and 6(b) represent the experimental dataand the Paris law correlations for each tested stress ratio ofthe P355NL1 steel 119877 = 00 119877 = 05 and 119877 = 07 Thematerial constants of the Paris crack propagation relation arepresented in Table 3 The crack propagation rates are onlyslightly influenced by the stress ratio Higher stress ratiosresult in higher crack growth ratesThe lines representing theParis law for 119877 = 00 and 119877 = 05 are approximately parallelto each other
On the other hand the line representing the Paris law for119877 = 07 converges to the other lines as the stress intensityfactor range increases In general the stress ratio effectsare more noticeable for lower ranges of the stress intensityfactors For higher stress intensity factor ranges the stressratio effect tends to vanish
The stress ratio effects on fatigue crack growth rates maybe attributed to the crack closure Crack closure increases
Table 3 Material constants concerning the Paris crack propagationrelation
119877 119862lowast
119898 Δ119870max
mdash mdash mdash Nsdotmmminus15
001 71945119864 minus 15 34993 1498050 62806119864 minus 15 35548 1013070 20370119864 minus 13 30031 687lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
with the reduction in the stress ratio leading to lower fatiguecrack growth rates For higher stress ratios such as 119877 =
07 the crack closure is very likely negligible which meansthat applied stress intensity factor range is fully effectiveAlso the fatigue crack propagation lines for each stress ratioare not fully parallel to each other which means that crackclosure also depends on the stress intensity factor rangelevel
Advances in Materials Science and Engineering 7
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
MB06 (R = 07)MB05 (R = 05)
MB03 (R = 05)MB04 (R = 00)MB02 (R = 00)
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
(a)
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
R = 0
R = 05
R = 07
R2 = 09850
R2 = 09840
R2 = 09960
7195E minus 15 times ΔK3499da
dN=
= 2037E minus 13 times ΔK3003da
dN=
6281E minus 15 times ΔK3555da
dN=
(b)
Figure 6 Fatigue crack propagation data obtained for the P355NL1 steel (a) experimental data (b) Paris correlations for each stress ratio 119877
Table 4Walker constants for the fatigue crack propagation relationof the P355NL1 steel
119862lowast
119898 120574
71945119864 minus 15 34993 092lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
5 Application and Discussion of a ProposedTheoretical Model
The fatigue crack growth data presented in Figure 6 wascorrelated using the Walker relation (see (18)) The materialconstants of the Walker crack propagation relation for theP355NL1 steel are presented in Table 4 In addition theWalker modified relation (see (19)) was used to modelthe fatigue crack propagation on region I and results arepresented in Figure 7 Equation (19) requires the definitionof the propagation threshold which was estimated using thefollowing linear relation
Δ119870th = 152 minus 90252 sdot 119877 (31)
where Δ119870th is the propagation threshold defined inNsdotmmminus15 119877 is the stress ratio and constant value 152corresponds to the propagation threshold for 119877 = 0 (inNsdotmmminus15) the other constant (90252) also shows thesame units of the propagation threshold The propagationthreshold for 119877 = 0 is very consistent with the thresholdvalue for the mild steel with yield stress of 366MPa Thecrack propagation threshold Δ119870th is itself influenced by thestress ratio [29] In [29] for a similar material to the P355NL1steel the relationship betweenΔ119870th and119877 is given showing alinear relation between the crack propagation threshold andthe stress ratio Despite using two distinct crack propagationlaws they gave a continuous representation of the crackpropagation data at the crack propagation I-II regimes
transition Figure 7 illustrates the fatigue propagation lawsadopted in this investigation for each tested stress ratio
The proposed theoretical model is applied using aninverse analysis of the experimental fatigue crack propagationrates available for the P355NL1 steel Using the inverseanalysis of the proposed theoretical model it is possibleto obtain the relationship between the effective stress ratio119877eff and the applied stress intensity factor range Δ119870app119877eff accounts for minimum stress intensity factors higherthan crack closure stress intensity factor In this case theeffective stress intensity factor results from the relationshipbetween the crack opening stress intensity factor 119870op andthe maximum stress intensity factor 119870max Figure 8 showsthat for stress ratios above 05 the crack closure and openingeffects are not significant For the stress ratio equal to 0 asignificant influence on the effective stress ratio of the appliedstress intensity factor range is observed Such influence ismore significant for the initial propagation phase of regionII decreasing significantly for higher applied stress intensityranges Thus it is concluded that the crack closure andopening effects are more significant for the initial crackpropagation phase in region II for higher levels of the stressintensity factor ranges along the crack length Figure 9 showsthe results obtained for the relationship between 119877eff and 119877using the inverse analysis of the proposed theoretical modelThis figure helps to understand the influence of the crackclosure and opening effects as a function of stress ratios Theeffective stress ratio 119877eff shown in Figure 9 is the average ofthe values obtained as a function of Δ119870 according to what isevidenced in Figure 8
The proposed relationship between the quantitativeparameter 119880 and the maximum stress intensity 119870max isalso considered Figure 10 shows the relationship between 119880
and 119870max for the studied stress ratios 0 05 and 07 Therelationship between 119880 and (Δ119870th0119870max) for the studiedstress ratios is shown in Figure 11 The proposed theoretical
8 Advances in Materials Science and Engineering
100 500 15001000
Thresholdregion Paris region
MB02 (R = 00)MB04 (R = 00)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
665Eminus13 middot (ΔK minus ΔKth)31250da
dN=
(a) 119877 = 0
MB03 (R = 05)MB05 (R = 05)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 05
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
315Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(b) 119877 = 05
MB06 (R = 07)Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 07
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
245Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(c) 119877 = 07
Figure 7 Fatigue crack growth data correlations for regimes I and II using two relations
model is able to describe the influence of the crack closureand opening effects on different stress ratios
Figure 12 shows the results for the relationship between119877eff and 119877 using various analytical models available in theliterature These model results are presented together withthe results of the proposed theoretical model Based onthe analysis of Figure 12 it can be noted that the resultsobtained by applying the proposed theoretical model arebetween the results obtained using the Hudak and Davidsonmodel (based on the same assumptions) and Savaidis model(with assumptions supported on plastic deformation of thematerial) The curves of Hudak and Davidson and Savaidisdemonstrate being below the other models Other modelsare based on direct polynomial relationships of orders 2 and3 except Newman model that is based on a polynomialrelationship of order 3 and uses properties of the plasticdeformation of the material
Figure 13 shows the fatigue crack propagation rates asa function of effective intensity range using the proposed
method The results obtained using this theoretical modelare similar to results presented in the literature [3]
6 Conclusions
An interpretation of the crack closure and opening effects ispossible without resorting to an excessive computation timeresulting from complex approaches
Fatigue crack propagation tests allowed the estimation ofthe relationship between the effective stress ratios 119877eff andthe applied stress ratios 119877 without the need for extensiveexperimental program
The evaluation of the influence of the crack closureeffects was possible using the inverse identification procedureapplied with the proposed theoretical model In this studyit was observed that for high applied stress ratios 119877 05 le
119877 le 07 the crack closure effect has little significanceBut for applied stress ratios 119877 0 le 119877 lt 05 there is asignificant influence of the crack closure and opening effects
Advances in Materials Science and Engineering 9
0
02
04
06
08
1
0 200 400 600 800 1000 1200
Ref
f(mdash
)
R = 001
R = 05
R = 07
ΔK (Nmiddotmmminus32)
Figure 8 Relationship between effective stress ratio 119877eff andapplied stress intensity range Δ119870 (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Experimental testsProposed model prediction
No closure effects
Closure effects
Ref
favg
(KopK
max
)
R (KminKmax)
Figure 9 Effective stress ratio 119877eff as a function of stress ratio(P355NL1 steel)
0
02
04
06
08
1
0 0002 0004 0006 0008
Qua
ntita
tive p
aram
eter
U
R = 001R = 05R = 07
1Kmax
Figure 10 Influence of the stress ratio on 119880 versus 1119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
U(mdash
)
ΔKth0Kmax (1Nmiddotmmminus32)
R = 001
R = 05
R = 07
Figure 11 Influence of the stress ratio on 119880 versus Δ119870th0119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Elberrsquos modelSchijversquos modelASTM E647Newmanrsquos model
Savaidis modelHudak et alProposed model
No closure effects
Closure effectsK
opK
max
Stress R-ratio R
Figure 12 Relationship between 119877eff = 119870op119870max and stress ratiofor several models (P355NL1 steel)
Note that for applied stress ratios within 0 le 119877 lt 05 theinfluence of the crack closure and opening effects decreaseswith increasing of the crack length or stress intensity factorrange
The results of the proposed theoreticalmodel proved to beconsistentwhen comparedwith othermodels proposed in theliterature [1ndash12 19] It was possible to estimate the constants119898and 119862 of the modified Paris law (119889119886119889119873 versus Δ119870eff ) Theseparameters are important for fatigue life prediction studies ofstructural details based on fracture mechanics
Other fatigue programs under constant and variableamplitude loading with othermaterials should be considered
10 Advances in Materials Science and Engineering
100 500 15001000
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
R = 05
R = 07
ΔKeff (Nmiddotmmminus15)
da
dN= 39897Eminus12(ΔKeff)
26105
Figure 13 Correlation of fatigue crack growth rates using the crackclosure analysis (P355NL1 steel)
through the use of full field optical techniques to experimen-tally estimate119870op in order to confirm the results obtained inthe inverse analysis of the proposed theoretical model
Competing Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge the Portuguese Science Founda-tion (FCT) for the financial support through the postdoc-toral Grant SFRHBPD1078252015 The authors gratefullyacknowledge the funding of SciTech Science and Technologyfor Competitive and Sustainable Industries RampD projectcofinanced by Programa Operacional Regional do Norte(NORTE2020) throughFundoEuropeu deDesenvolvimentoRegional (FEDER)
References
[1] W Elber ldquoFatigue crack closure under cyclic tensionrdquo Engineer-ing Fracture Mechanics vol 2 no 1 pp 37ndash45 1970
[2] W Elber ldquoThe significance of fatigue crack closure DamageTolerance in Aircraft Structuresrdquo ASTM STP 486 AmericaSociety for Testing and Materials 1971
[3] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984
[4] M Vormwald and T Seeger ldquoThe consequences of short crackclosure on fatigue crack growth under variable amplitudeloadingrdquo Fatigue and Fracture of Engineering Materials andStructures vol 14 no 2-3 pp 205ndash225 1991
[5] M Vormwald ldquoEffect of cyclic plastic strain on fatigue crackgrowthrdquo International Journal of Fatigue vol 82 part 1 pp 80ndash88 2016
[6] G Savaidis M Dankert and T Seeger ldquoAnalytical procedurefor predicting opening loads of cracks at notchesrdquo Fatigue andFracture of Engineering Materials and Structures vol 18 no 4pp 425ndash442 1995
[7] G Savaidis A Savaidis P Zerres and M Vormwald ldquoModeI fatigue crack growth at notches considering crack closurerdquoInternational Journal of Fatigue vol 32 no 10 pp 1543ndash15582010
[8] S J Hudak and D L Davidson ldquoThe dependence of crackclosure on fatigue loading variablesrdquo in Mechanics of ClosureASTM STP 982 J C Newman Jr and W Elber Eds pp 121ndash138America Society for Testing andMaterials Philadelphia PaUSA 1988
[9] J Schijve Fatigue of Structures andMaterials Kluwer AcademicPublishers Burlington Mass USA 2004
[10] American Society for Testing and Materials (ASTM) ldquoASTME647 standard test method for measurement of fatigue crackgrowth ratesrdquo inAnnual Book of ASTMStandards vol 0301 pp591ndash630 American Society for Testing and Materials (ASTM)West Conshohocken Pa USA 1999
[11] N Geerlofs J Zuidema and J Sietsma ldquoOn the relationshipbetween microstructure and fracture toughness in trip-assistedmultiphase steelsrdquo in Proceedings of the 15th European Con-ference of Fracture (ECF15 rsquo04) Stockholm Sweden August2004 httpwwwgruppofratturaitocsindexphpesisECF15paperview8758
[12] F Ellyin Fatigue Damage Crack Growth and Life PredictionChapman amp Hall New York NY USA 1997
[13] R C McClung and H Sehitoglu ldquoCharacterization of fatiguecrack growth in intermediate and large scale yieldingrdquo Journalof Engineering Materials and Technology vol 113 no 1 pp 15ndash22 1991
[14] M Nakagaki and S N Atluri ldquoFatigue crack closure and delayeffects under mode I spectrum loading an efficient elastic-plastic analysis procedurerdquo Fatigue of EngineeringMaterials andStructures vol 1 no 4 pp 421ndash429 1979
[15] N E DowlingMechanical Behaviour ofMaterialsmdashEngineeringMethods for Deformation Fracture and Fatigue Prentice-HallUpper Saddle River NJ USA 2nd edition 1998
[16] H F S G Pereira AM P De Jesus A S Ribeiro andA A Fer-nandes ldquoFatigue damage behavior of a structural componentmade of P355NL1 steel under block loadingrdquo Journal of PressureVessel Technology vol 131 no 2 Article ID021407 9 pages 2009
[17] O N Romaniv G N Nikiforchin and B N Andrusiv ldquoEffectof crack closure and evaluation of the cyclic crack resistance ofconstructional alloysrdquo Soviet Materials Science vol 19 no 3 pp212ndash225 1983
[18] J R LloydTheeffect of residual stress and crack closure on fatiguecrack growth [PhD thesis] Department of Materials Engineer-ing University of Wollongong Wollongong Australia 1999httprouoweduautheses1525
[19] H Neuber ldquoTheory of stress concentration for shear-strainedprismatical bodies with arbitrary nonlinear stress-strain lawrdquoJournal of Applied Mechanic vol 28 pp 544ndash550 1961
[20] P C Paris and F Erdogan ldquoA critical analysis of crack propaga-tion lawsrdquo Journal of Basic Engineering vol 85 no 4 pp 528ndash534 1963
[21] S M Beden S Abdullah and A K Ariffin ldquoReview of fatiguecrack propagation models for metallic componentsrdquo EuropeanJournal of Scientific Research vol 28 no 3 pp 364ndash397 2009
[22] M Klesnil and P Lukas Fatigue of Metallic Materials ElsevierScience Amsterdam The Netherlands 2nd edition 1992
Advances in Materials Science and Engineering 11
[23] A M P De Jesus and J A F O Correia ldquoCritical assessmentof a local strain-based fatigue crack growth model usingexperimental data available for the P355NL1 steelrdquo Journal ofPressure Vessel Technology vol 135 no 1 Article ID 011404 9pages 2013
[24] J A F O Correia A M P De Jesus and A Fernandez-Canteli ldquoLocal unified probabilistic model for fatigue crackinitiation and propagation application to a notched geometryrdquoEngineering Structures vol 52 pp 394ndash407 2013
[25] ASTMmdashAmerican Society for Testing and Materials ldquoASTME606-92 standard practice for strain-controlled fatigue testingrdquoin Annual Book of ASTM Standards part 10 pp 557ndash571 1998
[26] W Ramberg and W R Osgood ldquoDescription of stress-straincurves by three parametersrdquo NACATechnical Note 902 NACA1943 httpntrsnasagovsearchjspR=19930081614
[27] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoInternal Friction Damping and Cyclic Plasticity (ASTM STP378) ASTM International West Conshohocken Pa USA 1965
[28] K N Smith P Watson and T H Topper ldquoA stress-strainfunction for the fatigue of metalrdquo Journal of Materials vol 5no 4 pp 767ndash778 1970
[29] M Szata and G Lesiuk ldquoAlgorithms for the estimation offatigue crack growth using energymethodrdquoArchives of Civil andMechanical Engineering vol 9 no 1 pp 119ndash134 2009
Submit your manuscripts athttpwwwhindawicom
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Biomaterials
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Nano
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Journal ofNanomaterials
2 Advances in Materials Science and Engineering
K
Time
Crack closed
Crack openedKmax
Kop
Kmin
ΔK
ΔKeff
Figure 1 Definition of the effective and applied stress intensityfactor ranges
intensity factor range which is normalized by the appliedstress intensity factor range resulting in the 119880 ratio with thefollowing form
119880 =Δ119870effΔ119870
(1)
where Δ119870eff is the effective stress intensity factor rangeand Δ119870 is the applied stress intensity factor range Elberrsquosformulation is supported by the following assumptions forthe 119880 parameter (i) linear function of stress ratio (119877) and(ii) being relatively independent of maximum stress intensityfactor (119870max) or stress intensity factor range (Δ119870) Theeffective stress intensity factor range is defined as
Δ119870eff = 119870max minus 119870op (2)
where 119870max is the maximum stress intensity factor and 119870opis the crack opening stress intensity factor The applied stressintensity factor range Δ119870 is presented by
Δ119870 = 119870max minus 119870min (3)
with119870min being the minimum stress intensity factor Figure 1illustrates the previous parameters definitions
According to the approach of Elber [1 2] the fatiguecrack growth rate 119889119886119889119873 is a function of the effective stressintensity factor range according to the following generalexpression
119889119886
119889119873= 119891 (Δ119870eff) (4)
According to studies conducted by Elber for the 2024-T3aluminum alloy using stress ratios 119877 between minus01 and 07119880 depends on119877The experimental results led to the followinglinear relationship
119880 = 05 + 04119877 (5)
The quantitative parameter 119880 of (5) can be substituted into(1) and using (2) results in the following relation for the119870op119870max ratio
119870op
119870max= 05 + 01119877 + 04119877
2 (6)
Subsequent studies using various fatigue loading variablesclearly showed that 119880 and 119870max are related variablesTherefore Elber [1 2] approach appears to be inconsis-tentincomplete
An improved function of (5) was proposed by Schijve[9] with a more realistic behaviour to account for the crackclosure and opening effects for minus1 le 119877 le 1
119880 = 055 + 033119877 + 0121198772 (7)
Therefore it is possible to obtain a relation to the crackopening stress intensity factor as follows
119870op
119870max= 045 + 022119877 + 021119877
2+ 012119877
3 (8)
Another proposal was presented by ASTM [10 11] togetherwith a set of conditions
Δ119870 = 119870max for 119877 lt 0
Δ119870 = 119870max minus 119870min for 119877 ge 0
119880 = 0576 + 0015119877 + 04091198772
119870op
119870max= 0424 + 0561119877 minus 0394119877
2+ 0409119877
3
(9)
Newman Jr [3] proposed a general crack opening stressequation for constant amplitude loading function of stressratio 119877 stress level 120590 and four nondimensional constants
120590op
120590max= 1198600+ 1198601119877 + 119860
21198772+ 11986031198773 for 119877 ge 0
120590op
120590max= 1198600+ 1198601119877 for minus 1 le 119877 lt 0
(10)
when 120590op ge 120590max The coefficients 1198600 1198601 1198602 and 119860
3are as
follows
1198600= (0825 minus 034120572 + 005120572
2) [cos(
120587120590max21205900
)]
1120572
1198601= (0415 minus 0071120572)
120590max1205900
1198602= 1 minus 119860
0minus 1198601minus 1198603
1198603= 21198600+ 1198601minus 1
(11)
where 1205900is the flow stress (average between the uniaxial yield
stress and uniaxial ultimate tensile strength of the material)For plane stress conditions 120572 = 1 and for plane strainconditions 120572 = 3 The normalized crack opening stress isobtained using the stress ratio for 120590max = 120590
03
Advances in Materials Science and Engineering 3
This approach proposed by Newman Jr [3] may be usedto correlate fatigue crack growth rate data for other materialsand thicknesses under constant amplitude loading once theproper constraint factor has been determined
The crack closure effects may occur due to the surfaceroughness of the material in the presence of shear defor-mation at the crack tip [17 18] Microstructural studies [18]indicate that fracture surfaces demonstrate the possibilityof crack propagation by the shear mechanism primarilyin the near-threshold regime of Δ119870 [17] In this fatigueregime the crack closure effects have the greatest influenceConsequently Hudak and Davidson [8] proposed a modeldefined by the following expression
119880 = 1 minus119870119900
119870max= 1 minus
119870119900(1 minus 119877)
Δ119870 (12)
where119870119900is a constant related to the pureMode I fatigue crack
growth thresholdThe experimental results from fatigue crack propagation
tests can be represented by the following generalized relations(see Figure 2)
119880 = 120574(1 minus119870119900
119870max) 119870max le 119870
119871 (13)
119880 = 1 119870max ge 119870119871 (14)
where 119870119871
is the limiting 119870max value above which nodetectable closure occurs and 120574 is a parameter that can beobtained using the following expression
120574 = (1 minus119870119900
119870119871
) (15)
The 119870119900
and 119870119871
constants are not material propertiesin the strict sense since they depend on measurementslocation and measurements sensitivity resulting from thecrack propagation tests aiming at evaluating the crack clo-sureopening effects Note that in the limiting condition of119870119900119870119871approaching zero (13) reduces to (12) thus describing
the local measurementsEllyin [12] proposed an approach to define the effective
stress intensity range Δ119870eff taking into account the stressratio 119877 and the threshold value of the stress intensity factorrange Δ119870th with constant amplitude loading (see Figure 3)This approach is amodified version of proposal byHudak andDavidson [8] Generally the crack opening stress intensityfactor range Δ119870op is smaller than the threshold stressintensity factor range Δ119870th that is Δ119870op lt Δ119870th Besidesthe crack opening or closure stress intensities factors are notthe same that is Δ119870cl lt Δ119870op lt Δ119870th Ellyin [12] proposedthe following approach to obtain Δ119870eff (see Figure 3)
Δ119870eff 0 = (Δ1198702minus Δ1198702
th)12
asymp Δ119870[1 minus1
2(Δ119870thΔ119870
)
2
]
gt Δ119870 minus Δ119870th for 119877 asymp 0
1
00
1
U = ΔKeffΔK
U = 1
U = 120574(1 minusKmax)
120574Ko
120574
1KL 1Ko
Ko
1Kmax
Figure 2 Functional relationship between 119880 and 119870max Hudak andDavidson [8]
K
Time
ΔK
ΔKth Kcl ΔKop
ΔKeff
ΔKeff = [(ΔK)2 minus (ΔKth)2]12
Kmax
Kop
Kth
Kmin
Figure 3 Definition of the effective stress intensity range accordingwith Ellyin [12]
Δ119870eff =Δ119870eff 0
[1 minus (1205901198981205901015840
119891)]
=Δ119870eff 0
[1 minus ((1 + 119877) 120590max21205901015840
119891)]
for 119877 = 0
(16)
where 120590119898is the mean stress 120590max is the maximum stress and
1205901015840
119891is the fatigue strength coefficientThe crack closure and opening effects can be estimated
using an elastoplastic analysis based on analytical [3 19]or numerical [13 14] approaches Vormwald [4 5] andSavaidis et al [6 7] estimated the crack closure and openingeffects proposing the use of an analytical elastoplastic analysisdeveloped by Neuber [19] and themodified crack closure andopening model suggested by Newman Jr [3] The numericalelastoplastic analysis using the finite element method isproposed by McClung and Sehitoglu [13] and Nakagaki andAtluri [14] Based on finite element analysis performed byMcClung and Sehitoglu [13] it has been shown that the resultsobtained for 120590op may deviate from the results obtained usingthe crack closure and opening model by Newman Jr [3]
4 Advances in Materials Science and Engineering
Nakagaki and Atluri [14] proposed an elastoplastic finiteelement procedure to account for arbitrary strain hardeningmaterial behaviour In order to determine the crack closureand opening stresses Nakagaki and Atluri [14] proposed thefollowing procedure (1) calculating the displacements of thenodes in the crack axis before the closure (and after opening)(2) determining by extrapolating to zero the restraining forceat the respective nodal displacement in the correspondingnode just after crack closure (and before the opening) byextrapolation against the load level In all the cases studiedthese two sets of extrapolated values for 120590op and 120590cl werefound to correlate excellently
3 Proposed Theoretical Model
In this paper a theoretical model to obtain the effective stressintensity range Δ119870eff that takes into account the effects ofthe mean stress and the crack closure and opening effectsis proposed The starting point for the proposed theoreticalmodel is the initial assumptions proposed by Elber [1 2] anddefined in (1) and (4)The basic fatigue crack propagation lawin regime II was initially proposed by Paris and Erdogan [20]which has the following form
119889119886
119889119873= 119862 (Δ119870)
119898 (17)
where 119862 and 119898 are material constants However this modeldoes not include the other propagation regimes and doesnot take into account the effects of mean stress Many otherfatigue propagation models have been proposed in literaturewith wider range of application and complexity requiring asignificant number of constants [21] Nevertheless the simplemodification of the Paris relation in order to take into accountthe stress ratio effects is assumed as proposed by Dowling[15]
119889119886
119889119873= 119862119908(
Δ119870
(1 minus 119877)1minus120574
)
119898119908
= 119862119908(Δ119870)
119898119908
(18)
where 119862119908 119898119908 and 120574 are constants Besides an extension of
the Paris relation is adopted to account for crack propagationregime I having the following configuration
119889119886
119889119873= 119862119908(Δ119870 minus Δ119870th)
119898119908
(19)
Based on the foregoing assumptions the effective stressintensity factor range is given by the following expression
Δ119870eff = (Δ119870 minus Δ119870th) = (Δ119870
(1 minus 119877)1minus120574
minus Δ119870th)
= Δ119870(1
(1 minus 119877)1minus120574
minusΔ119870thΔ119870
)
(20)
1
00
1
U = 1
1Kmax1KL
U = ΔKeffΔK
(1 minus R)120574minus1
(1 minus R)120574minus1 middot ΔKth0
U = (1 minus ΔKth0Kmax)(1 minus R)120574minus1
1ΔKth0
Figure 4 Functional relationship between119880 and119870max for proposedtheoretical model
Thus the expression that allows obtaining the quantitativeparameter 119880 is given by the following formula
119880 =Δ119870effΔ119870
= (1
(1 minus 119877)1minus120574
minusΔ119870thΔ119870
)
= (1 minus 119877)120574minus1
minusΔ119870thΔ119870
(21)
If the material parameter 120574 is equal to 1 this means thatthe material is not influenced by the stress ratio Thus (21)is simplified and assumes the same form as (12) proposed byHudak and Davidson [8] and is similar to the one proposedby Ellyin [12]
Equation (21) can be presented in another form
119880 = (1 minus 119877)120574minus1
minusΔ119870th
(1 minus 119877)119870max (22)
The threshold value of the stress intensity factor range can bedescribed by a general equation using the following form
Δ119870th = Δ119870th0 sdot 119891 (119877) (23)
where Δ119870th is the threshold value of stress intensity range fora given stress ratio and Δ119870th0 is the threshold value of stressintensity range for 119877 = 0 Klesnil and Lukas [22] proposeda relation between the threshold stress intensity factor rangeΔ119870th and the stress ratio 119877 which is well known
Δ119870th = Δ119870th0 sdot (1 minus 119877)120574 (24)
The validity of (24) is for 119877 ge 0 Replacing this result in (22)results in
119880 = (1 minus 119877)120574minus1
minusΔ119870th0 sdot (1 minus 119877)
120574
(1 minus 119877)119870max (25)
Finally it is possible to display (22) in a more simplified form(see Figure 4)
119880 = (1 minusΔ119870th0
119870max) (1 minus 119877)
120574minus1 for 119870max le 119870119871
119880 = 1 for 119870max ge 119870119871
(26)
Advances in Materials Science and Engineering 5
Table 1 Cyclic elastoplastic and strain-life properties of the P355NL1 steel 119877120576= minus1 and 119877
120576= 0
Material 1205901015840
119891119887 120576
1015840
119891119888 119870
10158401198991015840
MPa mdash mdash mdash MPa mdashP355NL1 100550 minus01033 03678 minus05475 94835 01533
where 119870119871is the limiting 119870max For stress ratio 119877 equal to 0
and material parameter 120574 equal to 1 (22) reduces to
119880 = 1 minusΔ119870th0
119870max for 119877 = 0 120574 = 1 (27)
In summary a theoretical model based on Walkerrsquos propa-gation law [15] was proposed which relates the quantitativeparameter 119880 and Δ119870 (or 119870max) and the stress ratio Theproposed model is based on the same initial assumptions ofthe analytical models proposed by Hudak and Davidson [8]and Ellyin [12]
4 Basic Fatigue Data of the InvestigatedP355NL1 Steel
TheP355NL1 steel is a pressure vessel steel and is a normalizedfine grain low alloy carbon steel Its fatigue behaviour hasbeen investigated [16 23 24] A steel plate 5mm thick wasused to prepare specimens for experimental testing Thissection presents the strain-life fatigue data and the fatiguecrack propagation data obtained for the P355NL1 steel [16]
41 Strain-Life Behaviour The strain-life behaviour of theP355NL1 steel was evaluated through fatigue tests of smoothspecimens carried out under strain controlled conditionsaccording to theASTME606 standard [25]TheRamberg andOsgood [26] relation as shown in the following equation wasfitted to the stabilized cyclic stress-strain data
Δ120576
2=Δ120576119864
2+Δ120576119875
2=Δ120590
2119864+ (
Δ120590
21198701015840)
11198991015840
(28)
where1198701015840 and 1198991015840 are the cyclic strain hardening coefficient andexponent respectively Δ120576119875 is plastic strain range and Δ120590 isthe stress range
Low-cycle fatigue test results are very often representedusing the relation between the strain amplitude and thenumber of reversals to failure 2119873
119891 usually assumed to
correspond to the initiation of a macroscopic crack Morrow[27] suggested a general equation valid for low- and high-cycle fatigue regimes
Δ120576
2=Δ120576119864
2+Δ120576119875
2=
1205901015840
119891
119864(2119873119891)119887
+ 1205761015840
119891(2119873119891)119888
(29)
where 1205761015840119891and 119888 are respectively the fatigue ductility coeffi-
cient and fatigue ductility exponent 1205901015840119891is the fatigue strength
coefficient 119887 is the fatigue strength exponent and 119864 is theYoung modulus
Table 2 Elastic and tensile properties of the P355NL1
Material 119864 ] 119891119906
119891119910
GPa mdash MPa MPaP355NL1 20520 0275 56811 41806
Alternatively to the Morrow relation the Smith-Watson-Topper fatigue damage parameter [28] can be used whichshows the following form
120590max sdotΔ120576
2= SWT
=
(1205901015840
119891)2
119864(2119873119891)2119887
+ 1205901015840
119891sdot 1205761015840
119891sdot (2119873119891)119887+119888
(30)
where 120590max is themaximum stress of the cycle and SWT is thedamage parameter Both Morrow and Smith-Watson-Toppermodels are deterministicmodels and are used to represent theaverage fatigue behaviour of the bridgematerials based on theavailable experimental data
Two series of specimens were tested under distinct strainratios 119877
120576= 0 (19 specimens) and minus1 (24 specimens)
Figure 5 shows a plot of the experimental fatigue data in theform of strain-life and SWT-life curves for the two strainratios The cyclic Ramberg-Osgood and Morrow strain-lifeparameters of the P355NL1 steel are summarized in Table 1for the conjunction of both strain ratios [16 23 24] Thecyclic curve is shown in Figure 5(a) including the influenceof the experimental results fromboth test seriesThis researchadopted the values obtained by combining the results of thetwo test series together since the strain ratio effects are con-sidered negligible (tests performed under strain controlledconditions)This could be explained by the cyclic mean stressrelaxation phenomenon and also by a lower sensitivity ofthe material to the mean stress In addition Table 2 presentsthe elastic and monotonic tensile properties of this pressurevessel steel under investigation The elastic and monotonictensile properties are represented respectively by the Youngmodulus 119864 Poisson ratio ] the ultimate tensile strength 119891
119906
and upper yield stress 119891119910
42 Fatigue Crack Propagation Rates Fatigue crack growthrates of the P355NL1 pressure vessel steel were evaluated forseveral stress ratios using compact tension (CT) specimensfollowing the recommendations of the ASTM E647 standard[10] The CT specimens of P355NL1 steel were manufacturedwith a width 119882 = 40mm and a thickness 119861 = 45mm [16]All tests were performed in air at room temperature under asinusoidal waveform at a maximum frequency of 20Hz Thecrack growth was measured on both faces of the specimens
6 Advances in Materials Science and Engineering
0
100
200
300
400
500
600
0 05 1 15 2
Cyclic curve
Δ120590
(MPa
)
Δ1205762 ()
R120576 = 0
R120576 = minus1
(a)
R120576 = 0
R120576 = minus1
10E minus 01
10E minus 02
10E minus 03
10E minus 04
Δ120576
2(mdash
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
Δ120576
2=
10055
E(2Nf)minus01013 + 03678(2Nf)minus05475
(b)
01
10
100
SWT
(MPa
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
R120576 = 0
R120576 = minus1
(c)
Figure 5 Fatigue data obtained for the P355NL1 steel 119877120576= minus1 and 119877
120576= 0 (a) cyclic curve (b) strain-life data (c) SWT-life data
by visual inspection using two travelling microscopes withan accuracy of 0001mm
Figures 6(a) and 6(b) represent the experimental dataand the Paris law correlations for each tested stress ratio ofthe P355NL1 steel 119877 = 00 119877 = 05 and 119877 = 07 Thematerial constants of the Paris crack propagation relation arepresented in Table 3 The crack propagation rates are onlyslightly influenced by the stress ratio Higher stress ratiosresult in higher crack growth ratesThe lines representing theParis law for 119877 = 00 and 119877 = 05 are approximately parallelto each other
On the other hand the line representing the Paris law for119877 = 07 converges to the other lines as the stress intensityfactor range increases In general the stress ratio effectsare more noticeable for lower ranges of the stress intensityfactors For higher stress intensity factor ranges the stressratio effect tends to vanish
The stress ratio effects on fatigue crack growth rates maybe attributed to the crack closure Crack closure increases
Table 3 Material constants concerning the Paris crack propagationrelation
119877 119862lowast
119898 Δ119870max
mdash mdash mdash Nsdotmmminus15
001 71945119864 minus 15 34993 1498050 62806119864 minus 15 35548 1013070 20370119864 minus 13 30031 687lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
with the reduction in the stress ratio leading to lower fatiguecrack growth rates For higher stress ratios such as 119877 =
07 the crack closure is very likely negligible which meansthat applied stress intensity factor range is fully effectiveAlso the fatigue crack propagation lines for each stress ratioare not fully parallel to each other which means that crackclosure also depends on the stress intensity factor rangelevel
Advances in Materials Science and Engineering 7
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
MB06 (R = 07)MB05 (R = 05)
MB03 (R = 05)MB04 (R = 00)MB02 (R = 00)
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
(a)
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
R = 0
R = 05
R = 07
R2 = 09850
R2 = 09840
R2 = 09960
7195E minus 15 times ΔK3499da
dN=
= 2037E minus 13 times ΔK3003da
dN=
6281E minus 15 times ΔK3555da
dN=
(b)
Figure 6 Fatigue crack propagation data obtained for the P355NL1 steel (a) experimental data (b) Paris correlations for each stress ratio 119877
Table 4Walker constants for the fatigue crack propagation relationof the P355NL1 steel
119862lowast
119898 120574
71945119864 minus 15 34993 092lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
5 Application and Discussion of a ProposedTheoretical Model
The fatigue crack growth data presented in Figure 6 wascorrelated using the Walker relation (see (18)) The materialconstants of the Walker crack propagation relation for theP355NL1 steel are presented in Table 4 In addition theWalker modified relation (see (19)) was used to modelthe fatigue crack propagation on region I and results arepresented in Figure 7 Equation (19) requires the definitionof the propagation threshold which was estimated using thefollowing linear relation
Δ119870th = 152 minus 90252 sdot 119877 (31)
where Δ119870th is the propagation threshold defined inNsdotmmminus15 119877 is the stress ratio and constant value 152corresponds to the propagation threshold for 119877 = 0 (inNsdotmmminus15) the other constant (90252) also shows thesame units of the propagation threshold The propagationthreshold for 119877 = 0 is very consistent with the thresholdvalue for the mild steel with yield stress of 366MPa Thecrack propagation threshold Δ119870th is itself influenced by thestress ratio [29] In [29] for a similar material to the P355NL1steel the relationship betweenΔ119870th and119877 is given showing alinear relation between the crack propagation threshold andthe stress ratio Despite using two distinct crack propagationlaws they gave a continuous representation of the crackpropagation data at the crack propagation I-II regimes
transition Figure 7 illustrates the fatigue propagation lawsadopted in this investigation for each tested stress ratio
The proposed theoretical model is applied using aninverse analysis of the experimental fatigue crack propagationrates available for the P355NL1 steel Using the inverseanalysis of the proposed theoretical model it is possibleto obtain the relationship between the effective stress ratio119877eff and the applied stress intensity factor range Δ119870app119877eff accounts for minimum stress intensity factors higherthan crack closure stress intensity factor In this case theeffective stress intensity factor results from the relationshipbetween the crack opening stress intensity factor 119870op andthe maximum stress intensity factor 119870max Figure 8 showsthat for stress ratios above 05 the crack closure and openingeffects are not significant For the stress ratio equal to 0 asignificant influence on the effective stress ratio of the appliedstress intensity factor range is observed Such influence ismore significant for the initial propagation phase of regionII decreasing significantly for higher applied stress intensityranges Thus it is concluded that the crack closure andopening effects are more significant for the initial crackpropagation phase in region II for higher levels of the stressintensity factor ranges along the crack length Figure 9 showsthe results obtained for the relationship between 119877eff and 119877using the inverse analysis of the proposed theoretical modelThis figure helps to understand the influence of the crackclosure and opening effects as a function of stress ratios Theeffective stress ratio 119877eff shown in Figure 9 is the average ofthe values obtained as a function of Δ119870 according to what isevidenced in Figure 8
The proposed relationship between the quantitativeparameter 119880 and the maximum stress intensity 119870max isalso considered Figure 10 shows the relationship between 119880
and 119870max for the studied stress ratios 0 05 and 07 Therelationship between 119880 and (Δ119870th0119870max) for the studiedstress ratios is shown in Figure 11 The proposed theoretical
8 Advances in Materials Science and Engineering
100 500 15001000
Thresholdregion Paris region
MB02 (R = 00)MB04 (R = 00)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
665Eminus13 middot (ΔK minus ΔKth)31250da
dN=
(a) 119877 = 0
MB03 (R = 05)MB05 (R = 05)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 05
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
315Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(b) 119877 = 05
MB06 (R = 07)Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 07
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
245Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(c) 119877 = 07
Figure 7 Fatigue crack growth data correlations for regimes I and II using two relations
model is able to describe the influence of the crack closureand opening effects on different stress ratios
Figure 12 shows the results for the relationship between119877eff and 119877 using various analytical models available in theliterature These model results are presented together withthe results of the proposed theoretical model Based onthe analysis of Figure 12 it can be noted that the resultsobtained by applying the proposed theoretical model arebetween the results obtained using the Hudak and Davidsonmodel (based on the same assumptions) and Savaidis model(with assumptions supported on plastic deformation of thematerial) The curves of Hudak and Davidson and Savaidisdemonstrate being below the other models Other modelsare based on direct polynomial relationships of orders 2 and3 except Newman model that is based on a polynomialrelationship of order 3 and uses properties of the plasticdeformation of the material
Figure 13 shows the fatigue crack propagation rates asa function of effective intensity range using the proposed
method The results obtained using this theoretical modelare similar to results presented in the literature [3]
6 Conclusions
An interpretation of the crack closure and opening effects ispossible without resorting to an excessive computation timeresulting from complex approaches
Fatigue crack propagation tests allowed the estimation ofthe relationship between the effective stress ratios 119877eff andthe applied stress ratios 119877 without the need for extensiveexperimental program
The evaluation of the influence of the crack closureeffects was possible using the inverse identification procedureapplied with the proposed theoretical model In this studyit was observed that for high applied stress ratios 119877 05 le
119877 le 07 the crack closure effect has little significanceBut for applied stress ratios 119877 0 le 119877 lt 05 there is asignificant influence of the crack closure and opening effects
Advances in Materials Science and Engineering 9
0
02
04
06
08
1
0 200 400 600 800 1000 1200
Ref
f(mdash
)
R = 001
R = 05
R = 07
ΔK (Nmiddotmmminus32)
Figure 8 Relationship between effective stress ratio 119877eff andapplied stress intensity range Δ119870 (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Experimental testsProposed model prediction
No closure effects
Closure effects
Ref
favg
(KopK
max
)
R (KminKmax)
Figure 9 Effective stress ratio 119877eff as a function of stress ratio(P355NL1 steel)
0
02
04
06
08
1
0 0002 0004 0006 0008
Qua
ntita
tive p
aram
eter
U
R = 001R = 05R = 07
1Kmax
Figure 10 Influence of the stress ratio on 119880 versus 1119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
U(mdash
)
ΔKth0Kmax (1Nmiddotmmminus32)
R = 001
R = 05
R = 07
Figure 11 Influence of the stress ratio on 119880 versus Δ119870th0119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Elberrsquos modelSchijversquos modelASTM E647Newmanrsquos model
Savaidis modelHudak et alProposed model
No closure effects
Closure effectsK
opK
max
Stress R-ratio R
Figure 12 Relationship between 119877eff = 119870op119870max and stress ratiofor several models (P355NL1 steel)
Note that for applied stress ratios within 0 le 119877 lt 05 theinfluence of the crack closure and opening effects decreaseswith increasing of the crack length or stress intensity factorrange
The results of the proposed theoreticalmodel proved to beconsistentwhen comparedwith othermodels proposed in theliterature [1ndash12 19] It was possible to estimate the constants119898and 119862 of the modified Paris law (119889119886119889119873 versus Δ119870eff ) Theseparameters are important for fatigue life prediction studies ofstructural details based on fracture mechanics
Other fatigue programs under constant and variableamplitude loading with othermaterials should be considered
10 Advances in Materials Science and Engineering
100 500 15001000
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
R = 05
R = 07
ΔKeff (Nmiddotmmminus15)
da
dN= 39897Eminus12(ΔKeff)
26105
Figure 13 Correlation of fatigue crack growth rates using the crackclosure analysis (P355NL1 steel)
through the use of full field optical techniques to experimen-tally estimate119870op in order to confirm the results obtained inthe inverse analysis of the proposed theoretical model
Competing Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge the Portuguese Science Founda-tion (FCT) for the financial support through the postdoc-toral Grant SFRHBPD1078252015 The authors gratefullyacknowledge the funding of SciTech Science and Technologyfor Competitive and Sustainable Industries RampD projectcofinanced by Programa Operacional Regional do Norte(NORTE2020) throughFundoEuropeu deDesenvolvimentoRegional (FEDER)
References
[1] W Elber ldquoFatigue crack closure under cyclic tensionrdquo Engineer-ing Fracture Mechanics vol 2 no 1 pp 37ndash45 1970
[2] W Elber ldquoThe significance of fatigue crack closure DamageTolerance in Aircraft Structuresrdquo ASTM STP 486 AmericaSociety for Testing and Materials 1971
[3] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984
[4] M Vormwald and T Seeger ldquoThe consequences of short crackclosure on fatigue crack growth under variable amplitudeloadingrdquo Fatigue and Fracture of Engineering Materials andStructures vol 14 no 2-3 pp 205ndash225 1991
[5] M Vormwald ldquoEffect of cyclic plastic strain on fatigue crackgrowthrdquo International Journal of Fatigue vol 82 part 1 pp 80ndash88 2016
[6] G Savaidis M Dankert and T Seeger ldquoAnalytical procedurefor predicting opening loads of cracks at notchesrdquo Fatigue andFracture of Engineering Materials and Structures vol 18 no 4pp 425ndash442 1995
[7] G Savaidis A Savaidis P Zerres and M Vormwald ldquoModeI fatigue crack growth at notches considering crack closurerdquoInternational Journal of Fatigue vol 32 no 10 pp 1543ndash15582010
[8] S J Hudak and D L Davidson ldquoThe dependence of crackclosure on fatigue loading variablesrdquo in Mechanics of ClosureASTM STP 982 J C Newman Jr and W Elber Eds pp 121ndash138America Society for Testing andMaterials Philadelphia PaUSA 1988
[9] J Schijve Fatigue of Structures andMaterials Kluwer AcademicPublishers Burlington Mass USA 2004
[10] American Society for Testing and Materials (ASTM) ldquoASTME647 standard test method for measurement of fatigue crackgrowth ratesrdquo inAnnual Book of ASTMStandards vol 0301 pp591ndash630 American Society for Testing and Materials (ASTM)West Conshohocken Pa USA 1999
[11] N Geerlofs J Zuidema and J Sietsma ldquoOn the relationshipbetween microstructure and fracture toughness in trip-assistedmultiphase steelsrdquo in Proceedings of the 15th European Con-ference of Fracture (ECF15 rsquo04) Stockholm Sweden August2004 httpwwwgruppofratturaitocsindexphpesisECF15paperview8758
[12] F Ellyin Fatigue Damage Crack Growth and Life PredictionChapman amp Hall New York NY USA 1997
[13] R C McClung and H Sehitoglu ldquoCharacterization of fatiguecrack growth in intermediate and large scale yieldingrdquo Journalof Engineering Materials and Technology vol 113 no 1 pp 15ndash22 1991
[14] M Nakagaki and S N Atluri ldquoFatigue crack closure and delayeffects under mode I spectrum loading an efficient elastic-plastic analysis procedurerdquo Fatigue of EngineeringMaterials andStructures vol 1 no 4 pp 421ndash429 1979
[15] N E DowlingMechanical Behaviour ofMaterialsmdashEngineeringMethods for Deformation Fracture and Fatigue Prentice-HallUpper Saddle River NJ USA 2nd edition 1998
[16] H F S G Pereira AM P De Jesus A S Ribeiro andA A Fer-nandes ldquoFatigue damage behavior of a structural componentmade of P355NL1 steel under block loadingrdquo Journal of PressureVessel Technology vol 131 no 2 Article ID021407 9 pages 2009
[17] O N Romaniv G N Nikiforchin and B N Andrusiv ldquoEffectof crack closure and evaluation of the cyclic crack resistance ofconstructional alloysrdquo Soviet Materials Science vol 19 no 3 pp212ndash225 1983
[18] J R LloydTheeffect of residual stress and crack closure on fatiguecrack growth [PhD thesis] Department of Materials Engineer-ing University of Wollongong Wollongong Australia 1999httprouoweduautheses1525
[19] H Neuber ldquoTheory of stress concentration for shear-strainedprismatical bodies with arbitrary nonlinear stress-strain lawrdquoJournal of Applied Mechanic vol 28 pp 544ndash550 1961
[20] P C Paris and F Erdogan ldquoA critical analysis of crack propaga-tion lawsrdquo Journal of Basic Engineering vol 85 no 4 pp 528ndash534 1963
[21] S M Beden S Abdullah and A K Ariffin ldquoReview of fatiguecrack propagation models for metallic componentsrdquo EuropeanJournal of Scientific Research vol 28 no 3 pp 364ndash397 2009
[22] M Klesnil and P Lukas Fatigue of Metallic Materials ElsevierScience Amsterdam The Netherlands 2nd edition 1992
Advances in Materials Science and Engineering 11
[23] A M P De Jesus and J A F O Correia ldquoCritical assessmentof a local strain-based fatigue crack growth model usingexperimental data available for the P355NL1 steelrdquo Journal ofPressure Vessel Technology vol 135 no 1 Article ID 011404 9pages 2013
[24] J A F O Correia A M P De Jesus and A Fernandez-Canteli ldquoLocal unified probabilistic model for fatigue crackinitiation and propagation application to a notched geometryrdquoEngineering Structures vol 52 pp 394ndash407 2013
[25] ASTMmdashAmerican Society for Testing and Materials ldquoASTME606-92 standard practice for strain-controlled fatigue testingrdquoin Annual Book of ASTM Standards part 10 pp 557ndash571 1998
[26] W Ramberg and W R Osgood ldquoDescription of stress-straincurves by three parametersrdquo NACATechnical Note 902 NACA1943 httpntrsnasagovsearchjspR=19930081614
[27] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoInternal Friction Damping and Cyclic Plasticity (ASTM STP378) ASTM International West Conshohocken Pa USA 1965
[28] K N Smith P Watson and T H Topper ldquoA stress-strainfunction for the fatigue of metalrdquo Journal of Materials vol 5no 4 pp 767ndash778 1970
[29] M Szata and G Lesiuk ldquoAlgorithms for the estimation offatigue crack growth using energymethodrdquoArchives of Civil andMechanical Engineering vol 9 no 1 pp 119ndash134 2009
Submit your manuscripts athttpwwwhindawicom
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Biomaterials
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Advances in
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MaterialsJournal of
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Nano
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Journal ofNanomaterials
Advances in Materials Science and Engineering 3
This approach proposed by Newman Jr [3] may be usedto correlate fatigue crack growth rate data for other materialsand thicknesses under constant amplitude loading once theproper constraint factor has been determined
The crack closure effects may occur due to the surfaceroughness of the material in the presence of shear defor-mation at the crack tip [17 18] Microstructural studies [18]indicate that fracture surfaces demonstrate the possibilityof crack propagation by the shear mechanism primarilyin the near-threshold regime of Δ119870 [17] In this fatigueregime the crack closure effects have the greatest influenceConsequently Hudak and Davidson [8] proposed a modeldefined by the following expression
119880 = 1 minus119870119900
119870max= 1 minus
119870119900(1 minus 119877)
Δ119870 (12)
where119870119900is a constant related to the pureMode I fatigue crack
growth thresholdThe experimental results from fatigue crack propagation
tests can be represented by the following generalized relations(see Figure 2)
119880 = 120574(1 minus119870119900
119870max) 119870max le 119870
119871 (13)
119880 = 1 119870max ge 119870119871 (14)
where 119870119871
is the limiting 119870max value above which nodetectable closure occurs and 120574 is a parameter that can beobtained using the following expression
120574 = (1 minus119870119900
119870119871
) (15)
The 119870119900
and 119870119871
constants are not material propertiesin the strict sense since they depend on measurementslocation and measurements sensitivity resulting from thecrack propagation tests aiming at evaluating the crack clo-sureopening effects Note that in the limiting condition of119870119900119870119871approaching zero (13) reduces to (12) thus describing
the local measurementsEllyin [12] proposed an approach to define the effective
stress intensity range Δ119870eff taking into account the stressratio 119877 and the threshold value of the stress intensity factorrange Δ119870th with constant amplitude loading (see Figure 3)This approach is amodified version of proposal byHudak andDavidson [8] Generally the crack opening stress intensityfactor range Δ119870op is smaller than the threshold stressintensity factor range Δ119870th that is Δ119870op lt Δ119870th Besidesthe crack opening or closure stress intensities factors are notthe same that is Δ119870cl lt Δ119870op lt Δ119870th Ellyin [12] proposedthe following approach to obtain Δ119870eff (see Figure 3)
Δ119870eff 0 = (Δ1198702minus Δ1198702
th)12
asymp Δ119870[1 minus1
2(Δ119870thΔ119870
)
2
]
gt Δ119870 minus Δ119870th for 119877 asymp 0
1
00
1
U = ΔKeffΔK
U = 1
U = 120574(1 minusKmax)
120574Ko
120574
1KL 1Ko
Ko
1Kmax
Figure 2 Functional relationship between 119880 and 119870max Hudak andDavidson [8]
K
Time
ΔK
ΔKth Kcl ΔKop
ΔKeff
ΔKeff = [(ΔK)2 minus (ΔKth)2]12
Kmax
Kop
Kth
Kmin
Figure 3 Definition of the effective stress intensity range accordingwith Ellyin [12]
Δ119870eff =Δ119870eff 0
[1 minus (1205901198981205901015840
119891)]
=Δ119870eff 0
[1 minus ((1 + 119877) 120590max21205901015840
119891)]
for 119877 = 0
(16)
where 120590119898is the mean stress 120590max is the maximum stress and
1205901015840
119891is the fatigue strength coefficientThe crack closure and opening effects can be estimated
using an elastoplastic analysis based on analytical [3 19]or numerical [13 14] approaches Vormwald [4 5] andSavaidis et al [6 7] estimated the crack closure and openingeffects proposing the use of an analytical elastoplastic analysisdeveloped by Neuber [19] and themodified crack closure andopening model suggested by Newman Jr [3] The numericalelastoplastic analysis using the finite element method isproposed by McClung and Sehitoglu [13] and Nakagaki andAtluri [14] Based on finite element analysis performed byMcClung and Sehitoglu [13] it has been shown that the resultsobtained for 120590op may deviate from the results obtained usingthe crack closure and opening model by Newman Jr [3]
4 Advances in Materials Science and Engineering
Nakagaki and Atluri [14] proposed an elastoplastic finiteelement procedure to account for arbitrary strain hardeningmaterial behaviour In order to determine the crack closureand opening stresses Nakagaki and Atluri [14] proposed thefollowing procedure (1) calculating the displacements of thenodes in the crack axis before the closure (and after opening)(2) determining by extrapolating to zero the restraining forceat the respective nodal displacement in the correspondingnode just after crack closure (and before the opening) byextrapolation against the load level In all the cases studiedthese two sets of extrapolated values for 120590op and 120590cl werefound to correlate excellently
3 Proposed Theoretical Model
In this paper a theoretical model to obtain the effective stressintensity range Δ119870eff that takes into account the effects ofthe mean stress and the crack closure and opening effectsis proposed The starting point for the proposed theoreticalmodel is the initial assumptions proposed by Elber [1 2] anddefined in (1) and (4)The basic fatigue crack propagation lawin regime II was initially proposed by Paris and Erdogan [20]which has the following form
119889119886
119889119873= 119862 (Δ119870)
119898 (17)
where 119862 and 119898 are material constants However this modeldoes not include the other propagation regimes and doesnot take into account the effects of mean stress Many otherfatigue propagation models have been proposed in literaturewith wider range of application and complexity requiring asignificant number of constants [21] Nevertheless the simplemodification of the Paris relation in order to take into accountthe stress ratio effects is assumed as proposed by Dowling[15]
119889119886
119889119873= 119862119908(
Δ119870
(1 minus 119877)1minus120574
)
119898119908
= 119862119908(Δ119870)
119898119908
(18)
where 119862119908 119898119908 and 120574 are constants Besides an extension of
the Paris relation is adopted to account for crack propagationregime I having the following configuration
119889119886
119889119873= 119862119908(Δ119870 minus Δ119870th)
119898119908
(19)
Based on the foregoing assumptions the effective stressintensity factor range is given by the following expression
Δ119870eff = (Δ119870 minus Δ119870th) = (Δ119870
(1 minus 119877)1minus120574
minus Δ119870th)
= Δ119870(1
(1 minus 119877)1minus120574
minusΔ119870thΔ119870
)
(20)
1
00
1
U = 1
1Kmax1KL
U = ΔKeffΔK
(1 minus R)120574minus1
(1 minus R)120574minus1 middot ΔKth0
U = (1 minus ΔKth0Kmax)(1 minus R)120574minus1
1ΔKth0
Figure 4 Functional relationship between119880 and119870max for proposedtheoretical model
Thus the expression that allows obtaining the quantitativeparameter 119880 is given by the following formula
119880 =Δ119870effΔ119870
= (1
(1 minus 119877)1minus120574
minusΔ119870thΔ119870
)
= (1 minus 119877)120574minus1
minusΔ119870thΔ119870
(21)
If the material parameter 120574 is equal to 1 this means thatthe material is not influenced by the stress ratio Thus (21)is simplified and assumes the same form as (12) proposed byHudak and Davidson [8] and is similar to the one proposedby Ellyin [12]
Equation (21) can be presented in another form
119880 = (1 minus 119877)120574minus1
minusΔ119870th
(1 minus 119877)119870max (22)
The threshold value of the stress intensity factor range can bedescribed by a general equation using the following form
Δ119870th = Δ119870th0 sdot 119891 (119877) (23)
where Δ119870th is the threshold value of stress intensity range fora given stress ratio and Δ119870th0 is the threshold value of stressintensity range for 119877 = 0 Klesnil and Lukas [22] proposeda relation between the threshold stress intensity factor rangeΔ119870th and the stress ratio 119877 which is well known
Δ119870th = Δ119870th0 sdot (1 minus 119877)120574 (24)
The validity of (24) is for 119877 ge 0 Replacing this result in (22)results in
119880 = (1 minus 119877)120574minus1
minusΔ119870th0 sdot (1 minus 119877)
120574
(1 minus 119877)119870max (25)
Finally it is possible to display (22) in a more simplified form(see Figure 4)
119880 = (1 minusΔ119870th0
119870max) (1 minus 119877)
120574minus1 for 119870max le 119870119871
119880 = 1 for 119870max ge 119870119871
(26)
Advances in Materials Science and Engineering 5
Table 1 Cyclic elastoplastic and strain-life properties of the P355NL1 steel 119877120576= minus1 and 119877
120576= 0
Material 1205901015840
119891119887 120576
1015840
119891119888 119870
10158401198991015840
MPa mdash mdash mdash MPa mdashP355NL1 100550 minus01033 03678 minus05475 94835 01533
where 119870119871is the limiting 119870max For stress ratio 119877 equal to 0
and material parameter 120574 equal to 1 (22) reduces to
119880 = 1 minusΔ119870th0
119870max for 119877 = 0 120574 = 1 (27)
In summary a theoretical model based on Walkerrsquos propa-gation law [15] was proposed which relates the quantitativeparameter 119880 and Δ119870 (or 119870max) and the stress ratio Theproposed model is based on the same initial assumptions ofthe analytical models proposed by Hudak and Davidson [8]and Ellyin [12]
4 Basic Fatigue Data of the InvestigatedP355NL1 Steel
TheP355NL1 steel is a pressure vessel steel and is a normalizedfine grain low alloy carbon steel Its fatigue behaviour hasbeen investigated [16 23 24] A steel plate 5mm thick wasused to prepare specimens for experimental testing Thissection presents the strain-life fatigue data and the fatiguecrack propagation data obtained for the P355NL1 steel [16]
41 Strain-Life Behaviour The strain-life behaviour of theP355NL1 steel was evaluated through fatigue tests of smoothspecimens carried out under strain controlled conditionsaccording to theASTME606 standard [25]TheRamberg andOsgood [26] relation as shown in the following equation wasfitted to the stabilized cyclic stress-strain data
Δ120576
2=Δ120576119864
2+Δ120576119875
2=Δ120590
2119864+ (
Δ120590
21198701015840)
11198991015840
(28)
where1198701015840 and 1198991015840 are the cyclic strain hardening coefficient andexponent respectively Δ120576119875 is plastic strain range and Δ120590 isthe stress range
Low-cycle fatigue test results are very often representedusing the relation between the strain amplitude and thenumber of reversals to failure 2119873
119891 usually assumed to
correspond to the initiation of a macroscopic crack Morrow[27] suggested a general equation valid for low- and high-cycle fatigue regimes
Δ120576
2=Δ120576119864
2+Δ120576119875
2=
1205901015840
119891
119864(2119873119891)119887
+ 1205761015840
119891(2119873119891)119888
(29)
where 1205761015840119891and 119888 are respectively the fatigue ductility coeffi-
cient and fatigue ductility exponent 1205901015840119891is the fatigue strength
coefficient 119887 is the fatigue strength exponent and 119864 is theYoung modulus
Table 2 Elastic and tensile properties of the P355NL1
Material 119864 ] 119891119906
119891119910
GPa mdash MPa MPaP355NL1 20520 0275 56811 41806
Alternatively to the Morrow relation the Smith-Watson-Topper fatigue damage parameter [28] can be used whichshows the following form
120590max sdotΔ120576
2= SWT
=
(1205901015840
119891)2
119864(2119873119891)2119887
+ 1205901015840
119891sdot 1205761015840
119891sdot (2119873119891)119887+119888
(30)
where 120590max is themaximum stress of the cycle and SWT is thedamage parameter Both Morrow and Smith-Watson-Toppermodels are deterministicmodels and are used to represent theaverage fatigue behaviour of the bridgematerials based on theavailable experimental data
Two series of specimens were tested under distinct strainratios 119877
120576= 0 (19 specimens) and minus1 (24 specimens)
Figure 5 shows a plot of the experimental fatigue data in theform of strain-life and SWT-life curves for the two strainratios The cyclic Ramberg-Osgood and Morrow strain-lifeparameters of the P355NL1 steel are summarized in Table 1for the conjunction of both strain ratios [16 23 24] Thecyclic curve is shown in Figure 5(a) including the influenceof the experimental results fromboth test seriesThis researchadopted the values obtained by combining the results of thetwo test series together since the strain ratio effects are con-sidered negligible (tests performed under strain controlledconditions)This could be explained by the cyclic mean stressrelaxation phenomenon and also by a lower sensitivity ofthe material to the mean stress In addition Table 2 presentsthe elastic and monotonic tensile properties of this pressurevessel steel under investigation The elastic and monotonictensile properties are represented respectively by the Youngmodulus 119864 Poisson ratio ] the ultimate tensile strength 119891
119906
and upper yield stress 119891119910
42 Fatigue Crack Propagation Rates Fatigue crack growthrates of the P355NL1 pressure vessel steel were evaluated forseveral stress ratios using compact tension (CT) specimensfollowing the recommendations of the ASTM E647 standard[10] The CT specimens of P355NL1 steel were manufacturedwith a width 119882 = 40mm and a thickness 119861 = 45mm [16]All tests were performed in air at room temperature under asinusoidal waveform at a maximum frequency of 20Hz Thecrack growth was measured on both faces of the specimens
6 Advances in Materials Science and Engineering
0
100
200
300
400
500
600
0 05 1 15 2
Cyclic curve
Δ120590
(MPa
)
Δ1205762 ()
R120576 = 0
R120576 = minus1
(a)
R120576 = 0
R120576 = minus1
10E minus 01
10E minus 02
10E minus 03
10E minus 04
Δ120576
2(mdash
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
Δ120576
2=
10055
E(2Nf)minus01013 + 03678(2Nf)minus05475
(b)
01
10
100
SWT
(MPa
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
R120576 = 0
R120576 = minus1
(c)
Figure 5 Fatigue data obtained for the P355NL1 steel 119877120576= minus1 and 119877
120576= 0 (a) cyclic curve (b) strain-life data (c) SWT-life data
by visual inspection using two travelling microscopes withan accuracy of 0001mm
Figures 6(a) and 6(b) represent the experimental dataand the Paris law correlations for each tested stress ratio ofthe P355NL1 steel 119877 = 00 119877 = 05 and 119877 = 07 Thematerial constants of the Paris crack propagation relation arepresented in Table 3 The crack propagation rates are onlyslightly influenced by the stress ratio Higher stress ratiosresult in higher crack growth ratesThe lines representing theParis law for 119877 = 00 and 119877 = 05 are approximately parallelto each other
On the other hand the line representing the Paris law for119877 = 07 converges to the other lines as the stress intensityfactor range increases In general the stress ratio effectsare more noticeable for lower ranges of the stress intensityfactors For higher stress intensity factor ranges the stressratio effect tends to vanish
The stress ratio effects on fatigue crack growth rates maybe attributed to the crack closure Crack closure increases
Table 3 Material constants concerning the Paris crack propagationrelation
119877 119862lowast
119898 Δ119870max
mdash mdash mdash Nsdotmmminus15
001 71945119864 minus 15 34993 1498050 62806119864 minus 15 35548 1013070 20370119864 minus 13 30031 687lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
with the reduction in the stress ratio leading to lower fatiguecrack growth rates For higher stress ratios such as 119877 =
07 the crack closure is very likely negligible which meansthat applied stress intensity factor range is fully effectiveAlso the fatigue crack propagation lines for each stress ratioare not fully parallel to each other which means that crackclosure also depends on the stress intensity factor rangelevel
Advances in Materials Science and Engineering 7
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
MB06 (R = 07)MB05 (R = 05)
MB03 (R = 05)MB04 (R = 00)MB02 (R = 00)
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
(a)
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
R = 0
R = 05
R = 07
R2 = 09850
R2 = 09840
R2 = 09960
7195E minus 15 times ΔK3499da
dN=
= 2037E minus 13 times ΔK3003da
dN=
6281E minus 15 times ΔK3555da
dN=
(b)
Figure 6 Fatigue crack propagation data obtained for the P355NL1 steel (a) experimental data (b) Paris correlations for each stress ratio 119877
Table 4Walker constants for the fatigue crack propagation relationof the P355NL1 steel
119862lowast
119898 120574
71945119864 minus 15 34993 092lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
5 Application and Discussion of a ProposedTheoretical Model
The fatigue crack growth data presented in Figure 6 wascorrelated using the Walker relation (see (18)) The materialconstants of the Walker crack propagation relation for theP355NL1 steel are presented in Table 4 In addition theWalker modified relation (see (19)) was used to modelthe fatigue crack propagation on region I and results arepresented in Figure 7 Equation (19) requires the definitionof the propagation threshold which was estimated using thefollowing linear relation
Δ119870th = 152 minus 90252 sdot 119877 (31)
where Δ119870th is the propagation threshold defined inNsdotmmminus15 119877 is the stress ratio and constant value 152corresponds to the propagation threshold for 119877 = 0 (inNsdotmmminus15) the other constant (90252) also shows thesame units of the propagation threshold The propagationthreshold for 119877 = 0 is very consistent with the thresholdvalue for the mild steel with yield stress of 366MPa Thecrack propagation threshold Δ119870th is itself influenced by thestress ratio [29] In [29] for a similar material to the P355NL1steel the relationship betweenΔ119870th and119877 is given showing alinear relation between the crack propagation threshold andthe stress ratio Despite using two distinct crack propagationlaws they gave a continuous representation of the crackpropagation data at the crack propagation I-II regimes
transition Figure 7 illustrates the fatigue propagation lawsadopted in this investigation for each tested stress ratio
The proposed theoretical model is applied using aninverse analysis of the experimental fatigue crack propagationrates available for the P355NL1 steel Using the inverseanalysis of the proposed theoretical model it is possibleto obtain the relationship between the effective stress ratio119877eff and the applied stress intensity factor range Δ119870app119877eff accounts for minimum stress intensity factors higherthan crack closure stress intensity factor In this case theeffective stress intensity factor results from the relationshipbetween the crack opening stress intensity factor 119870op andthe maximum stress intensity factor 119870max Figure 8 showsthat for stress ratios above 05 the crack closure and openingeffects are not significant For the stress ratio equal to 0 asignificant influence on the effective stress ratio of the appliedstress intensity factor range is observed Such influence ismore significant for the initial propagation phase of regionII decreasing significantly for higher applied stress intensityranges Thus it is concluded that the crack closure andopening effects are more significant for the initial crackpropagation phase in region II for higher levels of the stressintensity factor ranges along the crack length Figure 9 showsthe results obtained for the relationship between 119877eff and 119877using the inverse analysis of the proposed theoretical modelThis figure helps to understand the influence of the crackclosure and opening effects as a function of stress ratios Theeffective stress ratio 119877eff shown in Figure 9 is the average ofthe values obtained as a function of Δ119870 according to what isevidenced in Figure 8
The proposed relationship between the quantitativeparameter 119880 and the maximum stress intensity 119870max isalso considered Figure 10 shows the relationship between 119880
and 119870max for the studied stress ratios 0 05 and 07 Therelationship between 119880 and (Δ119870th0119870max) for the studiedstress ratios is shown in Figure 11 The proposed theoretical
8 Advances in Materials Science and Engineering
100 500 15001000
Thresholdregion Paris region
MB02 (R = 00)MB04 (R = 00)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
665Eminus13 middot (ΔK minus ΔKth)31250da
dN=
(a) 119877 = 0
MB03 (R = 05)MB05 (R = 05)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 05
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
315Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(b) 119877 = 05
MB06 (R = 07)Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 07
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
245Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(c) 119877 = 07
Figure 7 Fatigue crack growth data correlations for regimes I and II using two relations
model is able to describe the influence of the crack closureand opening effects on different stress ratios
Figure 12 shows the results for the relationship between119877eff and 119877 using various analytical models available in theliterature These model results are presented together withthe results of the proposed theoretical model Based onthe analysis of Figure 12 it can be noted that the resultsobtained by applying the proposed theoretical model arebetween the results obtained using the Hudak and Davidsonmodel (based on the same assumptions) and Savaidis model(with assumptions supported on plastic deformation of thematerial) The curves of Hudak and Davidson and Savaidisdemonstrate being below the other models Other modelsare based on direct polynomial relationships of orders 2 and3 except Newman model that is based on a polynomialrelationship of order 3 and uses properties of the plasticdeformation of the material
Figure 13 shows the fatigue crack propagation rates asa function of effective intensity range using the proposed
method The results obtained using this theoretical modelare similar to results presented in the literature [3]
6 Conclusions
An interpretation of the crack closure and opening effects ispossible without resorting to an excessive computation timeresulting from complex approaches
Fatigue crack propagation tests allowed the estimation ofthe relationship between the effective stress ratios 119877eff andthe applied stress ratios 119877 without the need for extensiveexperimental program
The evaluation of the influence of the crack closureeffects was possible using the inverse identification procedureapplied with the proposed theoretical model In this studyit was observed that for high applied stress ratios 119877 05 le
119877 le 07 the crack closure effect has little significanceBut for applied stress ratios 119877 0 le 119877 lt 05 there is asignificant influence of the crack closure and opening effects
Advances in Materials Science and Engineering 9
0
02
04
06
08
1
0 200 400 600 800 1000 1200
Ref
f(mdash
)
R = 001
R = 05
R = 07
ΔK (Nmiddotmmminus32)
Figure 8 Relationship between effective stress ratio 119877eff andapplied stress intensity range Δ119870 (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Experimental testsProposed model prediction
No closure effects
Closure effects
Ref
favg
(KopK
max
)
R (KminKmax)
Figure 9 Effective stress ratio 119877eff as a function of stress ratio(P355NL1 steel)
0
02
04
06
08
1
0 0002 0004 0006 0008
Qua
ntita
tive p
aram
eter
U
R = 001R = 05R = 07
1Kmax
Figure 10 Influence of the stress ratio on 119880 versus 1119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
U(mdash
)
ΔKth0Kmax (1Nmiddotmmminus32)
R = 001
R = 05
R = 07
Figure 11 Influence of the stress ratio on 119880 versus Δ119870th0119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Elberrsquos modelSchijversquos modelASTM E647Newmanrsquos model
Savaidis modelHudak et alProposed model
No closure effects
Closure effectsK
opK
max
Stress R-ratio R
Figure 12 Relationship between 119877eff = 119870op119870max and stress ratiofor several models (P355NL1 steel)
Note that for applied stress ratios within 0 le 119877 lt 05 theinfluence of the crack closure and opening effects decreaseswith increasing of the crack length or stress intensity factorrange
The results of the proposed theoreticalmodel proved to beconsistentwhen comparedwith othermodels proposed in theliterature [1ndash12 19] It was possible to estimate the constants119898and 119862 of the modified Paris law (119889119886119889119873 versus Δ119870eff ) Theseparameters are important for fatigue life prediction studies ofstructural details based on fracture mechanics
Other fatigue programs under constant and variableamplitude loading with othermaterials should be considered
10 Advances in Materials Science and Engineering
100 500 15001000
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
R = 05
R = 07
ΔKeff (Nmiddotmmminus15)
da
dN= 39897Eminus12(ΔKeff)
26105
Figure 13 Correlation of fatigue crack growth rates using the crackclosure analysis (P355NL1 steel)
through the use of full field optical techniques to experimen-tally estimate119870op in order to confirm the results obtained inthe inverse analysis of the proposed theoretical model
Competing Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge the Portuguese Science Founda-tion (FCT) for the financial support through the postdoc-toral Grant SFRHBPD1078252015 The authors gratefullyacknowledge the funding of SciTech Science and Technologyfor Competitive and Sustainable Industries RampD projectcofinanced by Programa Operacional Regional do Norte(NORTE2020) throughFundoEuropeu deDesenvolvimentoRegional (FEDER)
References
[1] W Elber ldquoFatigue crack closure under cyclic tensionrdquo Engineer-ing Fracture Mechanics vol 2 no 1 pp 37ndash45 1970
[2] W Elber ldquoThe significance of fatigue crack closure DamageTolerance in Aircraft Structuresrdquo ASTM STP 486 AmericaSociety for Testing and Materials 1971
[3] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984
[4] M Vormwald and T Seeger ldquoThe consequences of short crackclosure on fatigue crack growth under variable amplitudeloadingrdquo Fatigue and Fracture of Engineering Materials andStructures vol 14 no 2-3 pp 205ndash225 1991
[5] M Vormwald ldquoEffect of cyclic plastic strain on fatigue crackgrowthrdquo International Journal of Fatigue vol 82 part 1 pp 80ndash88 2016
[6] G Savaidis M Dankert and T Seeger ldquoAnalytical procedurefor predicting opening loads of cracks at notchesrdquo Fatigue andFracture of Engineering Materials and Structures vol 18 no 4pp 425ndash442 1995
[7] G Savaidis A Savaidis P Zerres and M Vormwald ldquoModeI fatigue crack growth at notches considering crack closurerdquoInternational Journal of Fatigue vol 32 no 10 pp 1543ndash15582010
[8] S J Hudak and D L Davidson ldquoThe dependence of crackclosure on fatigue loading variablesrdquo in Mechanics of ClosureASTM STP 982 J C Newman Jr and W Elber Eds pp 121ndash138America Society for Testing andMaterials Philadelphia PaUSA 1988
[9] J Schijve Fatigue of Structures andMaterials Kluwer AcademicPublishers Burlington Mass USA 2004
[10] American Society for Testing and Materials (ASTM) ldquoASTME647 standard test method for measurement of fatigue crackgrowth ratesrdquo inAnnual Book of ASTMStandards vol 0301 pp591ndash630 American Society for Testing and Materials (ASTM)West Conshohocken Pa USA 1999
[11] N Geerlofs J Zuidema and J Sietsma ldquoOn the relationshipbetween microstructure and fracture toughness in trip-assistedmultiphase steelsrdquo in Proceedings of the 15th European Con-ference of Fracture (ECF15 rsquo04) Stockholm Sweden August2004 httpwwwgruppofratturaitocsindexphpesisECF15paperview8758
[12] F Ellyin Fatigue Damage Crack Growth and Life PredictionChapman amp Hall New York NY USA 1997
[13] R C McClung and H Sehitoglu ldquoCharacterization of fatiguecrack growth in intermediate and large scale yieldingrdquo Journalof Engineering Materials and Technology vol 113 no 1 pp 15ndash22 1991
[14] M Nakagaki and S N Atluri ldquoFatigue crack closure and delayeffects under mode I spectrum loading an efficient elastic-plastic analysis procedurerdquo Fatigue of EngineeringMaterials andStructures vol 1 no 4 pp 421ndash429 1979
[15] N E DowlingMechanical Behaviour ofMaterialsmdashEngineeringMethods for Deformation Fracture and Fatigue Prentice-HallUpper Saddle River NJ USA 2nd edition 1998
[16] H F S G Pereira AM P De Jesus A S Ribeiro andA A Fer-nandes ldquoFatigue damage behavior of a structural componentmade of P355NL1 steel under block loadingrdquo Journal of PressureVessel Technology vol 131 no 2 Article ID021407 9 pages 2009
[17] O N Romaniv G N Nikiforchin and B N Andrusiv ldquoEffectof crack closure and evaluation of the cyclic crack resistance ofconstructional alloysrdquo Soviet Materials Science vol 19 no 3 pp212ndash225 1983
[18] J R LloydTheeffect of residual stress and crack closure on fatiguecrack growth [PhD thesis] Department of Materials Engineer-ing University of Wollongong Wollongong Australia 1999httprouoweduautheses1525
[19] H Neuber ldquoTheory of stress concentration for shear-strainedprismatical bodies with arbitrary nonlinear stress-strain lawrdquoJournal of Applied Mechanic vol 28 pp 544ndash550 1961
[20] P C Paris and F Erdogan ldquoA critical analysis of crack propaga-tion lawsrdquo Journal of Basic Engineering vol 85 no 4 pp 528ndash534 1963
[21] S M Beden S Abdullah and A K Ariffin ldquoReview of fatiguecrack propagation models for metallic componentsrdquo EuropeanJournal of Scientific Research vol 28 no 3 pp 364ndash397 2009
[22] M Klesnil and P Lukas Fatigue of Metallic Materials ElsevierScience Amsterdam The Netherlands 2nd edition 1992
Advances in Materials Science and Engineering 11
[23] A M P De Jesus and J A F O Correia ldquoCritical assessmentof a local strain-based fatigue crack growth model usingexperimental data available for the P355NL1 steelrdquo Journal ofPressure Vessel Technology vol 135 no 1 Article ID 011404 9pages 2013
[24] J A F O Correia A M P De Jesus and A Fernandez-Canteli ldquoLocal unified probabilistic model for fatigue crackinitiation and propagation application to a notched geometryrdquoEngineering Structures vol 52 pp 394ndash407 2013
[25] ASTMmdashAmerican Society for Testing and Materials ldquoASTME606-92 standard practice for strain-controlled fatigue testingrdquoin Annual Book of ASTM Standards part 10 pp 557ndash571 1998
[26] W Ramberg and W R Osgood ldquoDescription of stress-straincurves by three parametersrdquo NACATechnical Note 902 NACA1943 httpntrsnasagovsearchjspR=19930081614
[27] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoInternal Friction Damping and Cyclic Plasticity (ASTM STP378) ASTM International West Conshohocken Pa USA 1965
[28] K N Smith P Watson and T H Topper ldquoA stress-strainfunction for the fatigue of metalrdquo Journal of Materials vol 5no 4 pp 767ndash778 1970
[29] M Szata and G Lesiuk ldquoAlgorithms for the estimation offatigue crack growth using energymethodrdquoArchives of Civil andMechanical Engineering vol 9 no 1 pp 119ndash134 2009
Submit your manuscripts athttpwwwhindawicom
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Biomaterials
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MaterialsJournal of
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Nano
materials
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Journal ofNanomaterials
4 Advances in Materials Science and Engineering
Nakagaki and Atluri [14] proposed an elastoplastic finiteelement procedure to account for arbitrary strain hardeningmaterial behaviour In order to determine the crack closureand opening stresses Nakagaki and Atluri [14] proposed thefollowing procedure (1) calculating the displacements of thenodes in the crack axis before the closure (and after opening)(2) determining by extrapolating to zero the restraining forceat the respective nodal displacement in the correspondingnode just after crack closure (and before the opening) byextrapolation against the load level In all the cases studiedthese two sets of extrapolated values for 120590op and 120590cl werefound to correlate excellently
3 Proposed Theoretical Model
In this paper a theoretical model to obtain the effective stressintensity range Δ119870eff that takes into account the effects ofthe mean stress and the crack closure and opening effectsis proposed The starting point for the proposed theoreticalmodel is the initial assumptions proposed by Elber [1 2] anddefined in (1) and (4)The basic fatigue crack propagation lawin regime II was initially proposed by Paris and Erdogan [20]which has the following form
119889119886
119889119873= 119862 (Δ119870)
119898 (17)
where 119862 and 119898 are material constants However this modeldoes not include the other propagation regimes and doesnot take into account the effects of mean stress Many otherfatigue propagation models have been proposed in literaturewith wider range of application and complexity requiring asignificant number of constants [21] Nevertheless the simplemodification of the Paris relation in order to take into accountthe stress ratio effects is assumed as proposed by Dowling[15]
119889119886
119889119873= 119862119908(
Δ119870
(1 minus 119877)1minus120574
)
119898119908
= 119862119908(Δ119870)
119898119908
(18)
where 119862119908 119898119908 and 120574 are constants Besides an extension of
the Paris relation is adopted to account for crack propagationregime I having the following configuration
119889119886
119889119873= 119862119908(Δ119870 minus Δ119870th)
119898119908
(19)
Based on the foregoing assumptions the effective stressintensity factor range is given by the following expression
Δ119870eff = (Δ119870 minus Δ119870th) = (Δ119870
(1 minus 119877)1minus120574
minus Δ119870th)
= Δ119870(1
(1 minus 119877)1minus120574
minusΔ119870thΔ119870
)
(20)
1
00
1
U = 1
1Kmax1KL
U = ΔKeffΔK
(1 minus R)120574minus1
(1 minus R)120574minus1 middot ΔKth0
U = (1 minus ΔKth0Kmax)(1 minus R)120574minus1
1ΔKth0
Figure 4 Functional relationship between119880 and119870max for proposedtheoretical model
Thus the expression that allows obtaining the quantitativeparameter 119880 is given by the following formula
119880 =Δ119870effΔ119870
= (1
(1 minus 119877)1minus120574
minusΔ119870thΔ119870
)
= (1 minus 119877)120574minus1
minusΔ119870thΔ119870
(21)
If the material parameter 120574 is equal to 1 this means thatthe material is not influenced by the stress ratio Thus (21)is simplified and assumes the same form as (12) proposed byHudak and Davidson [8] and is similar to the one proposedby Ellyin [12]
Equation (21) can be presented in another form
119880 = (1 minus 119877)120574minus1
minusΔ119870th
(1 minus 119877)119870max (22)
The threshold value of the stress intensity factor range can bedescribed by a general equation using the following form
Δ119870th = Δ119870th0 sdot 119891 (119877) (23)
where Δ119870th is the threshold value of stress intensity range fora given stress ratio and Δ119870th0 is the threshold value of stressintensity range for 119877 = 0 Klesnil and Lukas [22] proposeda relation between the threshold stress intensity factor rangeΔ119870th and the stress ratio 119877 which is well known
Δ119870th = Δ119870th0 sdot (1 minus 119877)120574 (24)
The validity of (24) is for 119877 ge 0 Replacing this result in (22)results in
119880 = (1 minus 119877)120574minus1
minusΔ119870th0 sdot (1 minus 119877)
120574
(1 minus 119877)119870max (25)
Finally it is possible to display (22) in a more simplified form(see Figure 4)
119880 = (1 minusΔ119870th0
119870max) (1 minus 119877)
120574minus1 for 119870max le 119870119871
119880 = 1 for 119870max ge 119870119871
(26)
Advances in Materials Science and Engineering 5
Table 1 Cyclic elastoplastic and strain-life properties of the P355NL1 steel 119877120576= minus1 and 119877
120576= 0
Material 1205901015840
119891119887 120576
1015840
119891119888 119870
10158401198991015840
MPa mdash mdash mdash MPa mdashP355NL1 100550 minus01033 03678 minus05475 94835 01533
where 119870119871is the limiting 119870max For stress ratio 119877 equal to 0
and material parameter 120574 equal to 1 (22) reduces to
119880 = 1 minusΔ119870th0
119870max for 119877 = 0 120574 = 1 (27)
In summary a theoretical model based on Walkerrsquos propa-gation law [15] was proposed which relates the quantitativeparameter 119880 and Δ119870 (or 119870max) and the stress ratio Theproposed model is based on the same initial assumptions ofthe analytical models proposed by Hudak and Davidson [8]and Ellyin [12]
4 Basic Fatigue Data of the InvestigatedP355NL1 Steel
TheP355NL1 steel is a pressure vessel steel and is a normalizedfine grain low alloy carbon steel Its fatigue behaviour hasbeen investigated [16 23 24] A steel plate 5mm thick wasused to prepare specimens for experimental testing Thissection presents the strain-life fatigue data and the fatiguecrack propagation data obtained for the P355NL1 steel [16]
41 Strain-Life Behaviour The strain-life behaviour of theP355NL1 steel was evaluated through fatigue tests of smoothspecimens carried out under strain controlled conditionsaccording to theASTME606 standard [25]TheRamberg andOsgood [26] relation as shown in the following equation wasfitted to the stabilized cyclic stress-strain data
Δ120576
2=Δ120576119864
2+Δ120576119875
2=Δ120590
2119864+ (
Δ120590
21198701015840)
11198991015840
(28)
where1198701015840 and 1198991015840 are the cyclic strain hardening coefficient andexponent respectively Δ120576119875 is plastic strain range and Δ120590 isthe stress range
Low-cycle fatigue test results are very often representedusing the relation between the strain amplitude and thenumber of reversals to failure 2119873
119891 usually assumed to
correspond to the initiation of a macroscopic crack Morrow[27] suggested a general equation valid for low- and high-cycle fatigue regimes
Δ120576
2=Δ120576119864
2+Δ120576119875
2=
1205901015840
119891
119864(2119873119891)119887
+ 1205761015840
119891(2119873119891)119888
(29)
where 1205761015840119891and 119888 are respectively the fatigue ductility coeffi-
cient and fatigue ductility exponent 1205901015840119891is the fatigue strength
coefficient 119887 is the fatigue strength exponent and 119864 is theYoung modulus
Table 2 Elastic and tensile properties of the P355NL1
Material 119864 ] 119891119906
119891119910
GPa mdash MPa MPaP355NL1 20520 0275 56811 41806
Alternatively to the Morrow relation the Smith-Watson-Topper fatigue damage parameter [28] can be used whichshows the following form
120590max sdotΔ120576
2= SWT
=
(1205901015840
119891)2
119864(2119873119891)2119887
+ 1205901015840
119891sdot 1205761015840
119891sdot (2119873119891)119887+119888
(30)
where 120590max is themaximum stress of the cycle and SWT is thedamage parameter Both Morrow and Smith-Watson-Toppermodels are deterministicmodels and are used to represent theaverage fatigue behaviour of the bridgematerials based on theavailable experimental data
Two series of specimens were tested under distinct strainratios 119877
120576= 0 (19 specimens) and minus1 (24 specimens)
Figure 5 shows a plot of the experimental fatigue data in theform of strain-life and SWT-life curves for the two strainratios The cyclic Ramberg-Osgood and Morrow strain-lifeparameters of the P355NL1 steel are summarized in Table 1for the conjunction of both strain ratios [16 23 24] Thecyclic curve is shown in Figure 5(a) including the influenceof the experimental results fromboth test seriesThis researchadopted the values obtained by combining the results of thetwo test series together since the strain ratio effects are con-sidered negligible (tests performed under strain controlledconditions)This could be explained by the cyclic mean stressrelaxation phenomenon and also by a lower sensitivity ofthe material to the mean stress In addition Table 2 presentsthe elastic and monotonic tensile properties of this pressurevessel steel under investigation The elastic and monotonictensile properties are represented respectively by the Youngmodulus 119864 Poisson ratio ] the ultimate tensile strength 119891
119906
and upper yield stress 119891119910
42 Fatigue Crack Propagation Rates Fatigue crack growthrates of the P355NL1 pressure vessel steel were evaluated forseveral stress ratios using compact tension (CT) specimensfollowing the recommendations of the ASTM E647 standard[10] The CT specimens of P355NL1 steel were manufacturedwith a width 119882 = 40mm and a thickness 119861 = 45mm [16]All tests were performed in air at room temperature under asinusoidal waveform at a maximum frequency of 20Hz Thecrack growth was measured on both faces of the specimens
6 Advances in Materials Science and Engineering
0
100
200
300
400
500
600
0 05 1 15 2
Cyclic curve
Δ120590
(MPa
)
Δ1205762 ()
R120576 = 0
R120576 = minus1
(a)
R120576 = 0
R120576 = minus1
10E minus 01
10E minus 02
10E minus 03
10E minus 04
Δ120576
2(mdash
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
Δ120576
2=
10055
E(2Nf)minus01013 + 03678(2Nf)minus05475
(b)
01
10
100
SWT
(MPa
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
R120576 = 0
R120576 = minus1
(c)
Figure 5 Fatigue data obtained for the P355NL1 steel 119877120576= minus1 and 119877
120576= 0 (a) cyclic curve (b) strain-life data (c) SWT-life data
by visual inspection using two travelling microscopes withan accuracy of 0001mm
Figures 6(a) and 6(b) represent the experimental dataand the Paris law correlations for each tested stress ratio ofthe P355NL1 steel 119877 = 00 119877 = 05 and 119877 = 07 Thematerial constants of the Paris crack propagation relation arepresented in Table 3 The crack propagation rates are onlyslightly influenced by the stress ratio Higher stress ratiosresult in higher crack growth ratesThe lines representing theParis law for 119877 = 00 and 119877 = 05 are approximately parallelto each other
On the other hand the line representing the Paris law for119877 = 07 converges to the other lines as the stress intensityfactor range increases In general the stress ratio effectsare more noticeable for lower ranges of the stress intensityfactors For higher stress intensity factor ranges the stressratio effect tends to vanish
The stress ratio effects on fatigue crack growth rates maybe attributed to the crack closure Crack closure increases
Table 3 Material constants concerning the Paris crack propagationrelation
119877 119862lowast
119898 Δ119870max
mdash mdash mdash Nsdotmmminus15
001 71945119864 minus 15 34993 1498050 62806119864 minus 15 35548 1013070 20370119864 minus 13 30031 687lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
with the reduction in the stress ratio leading to lower fatiguecrack growth rates For higher stress ratios such as 119877 =
07 the crack closure is very likely negligible which meansthat applied stress intensity factor range is fully effectiveAlso the fatigue crack propagation lines for each stress ratioare not fully parallel to each other which means that crackclosure also depends on the stress intensity factor rangelevel
Advances in Materials Science and Engineering 7
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
MB06 (R = 07)MB05 (R = 05)
MB03 (R = 05)MB04 (R = 00)MB02 (R = 00)
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
(a)
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
R = 0
R = 05
R = 07
R2 = 09850
R2 = 09840
R2 = 09960
7195E minus 15 times ΔK3499da
dN=
= 2037E minus 13 times ΔK3003da
dN=
6281E minus 15 times ΔK3555da
dN=
(b)
Figure 6 Fatigue crack propagation data obtained for the P355NL1 steel (a) experimental data (b) Paris correlations for each stress ratio 119877
Table 4Walker constants for the fatigue crack propagation relationof the P355NL1 steel
119862lowast
119898 120574
71945119864 minus 15 34993 092lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
5 Application and Discussion of a ProposedTheoretical Model
The fatigue crack growth data presented in Figure 6 wascorrelated using the Walker relation (see (18)) The materialconstants of the Walker crack propagation relation for theP355NL1 steel are presented in Table 4 In addition theWalker modified relation (see (19)) was used to modelthe fatigue crack propagation on region I and results arepresented in Figure 7 Equation (19) requires the definitionof the propagation threshold which was estimated using thefollowing linear relation
Δ119870th = 152 minus 90252 sdot 119877 (31)
where Δ119870th is the propagation threshold defined inNsdotmmminus15 119877 is the stress ratio and constant value 152corresponds to the propagation threshold for 119877 = 0 (inNsdotmmminus15) the other constant (90252) also shows thesame units of the propagation threshold The propagationthreshold for 119877 = 0 is very consistent with the thresholdvalue for the mild steel with yield stress of 366MPa Thecrack propagation threshold Δ119870th is itself influenced by thestress ratio [29] In [29] for a similar material to the P355NL1steel the relationship betweenΔ119870th and119877 is given showing alinear relation between the crack propagation threshold andthe stress ratio Despite using two distinct crack propagationlaws they gave a continuous representation of the crackpropagation data at the crack propagation I-II regimes
transition Figure 7 illustrates the fatigue propagation lawsadopted in this investigation for each tested stress ratio
The proposed theoretical model is applied using aninverse analysis of the experimental fatigue crack propagationrates available for the P355NL1 steel Using the inverseanalysis of the proposed theoretical model it is possibleto obtain the relationship between the effective stress ratio119877eff and the applied stress intensity factor range Δ119870app119877eff accounts for minimum stress intensity factors higherthan crack closure stress intensity factor In this case theeffective stress intensity factor results from the relationshipbetween the crack opening stress intensity factor 119870op andthe maximum stress intensity factor 119870max Figure 8 showsthat for stress ratios above 05 the crack closure and openingeffects are not significant For the stress ratio equal to 0 asignificant influence on the effective stress ratio of the appliedstress intensity factor range is observed Such influence ismore significant for the initial propagation phase of regionII decreasing significantly for higher applied stress intensityranges Thus it is concluded that the crack closure andopening effects are more significant for the initial crackpropagation phase in region II for higher levels of the stressintensity factor ranges along the crack length Figure 9 showsthe results obtained for the relationship between 119877eff and 119877using the inverse analysis of the proposed theoretical modelThis figure helps to understand the influence of the crackclosure and opening effects as a function of stress ratios Theeffective stress ratio 119877eff shown in Figure 9 is the average ofthe values obtained as a function of Δ119870 according to what isevidenced in Figure 8
The proposed relationship between the quantitativeparameter 119880 and the maximum stress intensity 119870max isalso considered Figure 10 shows the relationship between 119880
and 119870max for the studied stress ratios 0 05 and 07 Therelationship between 119880 and (Δ119870th0119870max) for the studiedstress ratios is shown in Figure 11 The proposed theoretical
8 Advances in Materials Science and Engineering
100 500 15001000
Thresholdregion Paris region
MB02 (R = 00)MB04 (R = 00)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
665Eminus13 middot (ΔK minus ΔKth)31250da
dN=
(a) 119877 = 0
MB03 (R = 05)MB05 (R = 05)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 05
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
315Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(b) 119877 = 05
MB06 (R = 07)Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 07
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
245Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(c) 119877 = 07
Figure 7 Fatigue crack growth data correlations for regimes I and II using two relations
model is able to describe the influence of the crack closureand opening effects on different stress ratios
Figure 12 shows the results for the relationship between119877eff and 119877 using various analytical models available in theliterature These model results are presented together withthe results of the proposed theoretical model Based onthe analysis of Figure 12 it can be noted that the resultsobtained by applying the proposed theoretical model arebetween the results obtained using the Hudak and Davidsonmodel (based on the same assumptions) and Savaidis model(with assumptions supported on plastic deformation of thematerial) The curves of Hudak and Davidson and Savaidisdemonstrate being below the other models Other modelsare based on direct polynomial relationships of orders 2 and3 except Newman model that is based on a polynomialrelationship of order 3 and uses properties of the plasticdeformation of the material
Figure 13 shows the fatigue crack propagation rates asa function of effective intensity range using the proposed
method The results obtained using this theoretical modelare similar to results presented in the literature [3]
6 Conclusions
An interpretation of the crack closure and opening effects ispossible without resorting to an excessive computation timeresulting from complex approaches
Fatigue crack propagation tests allowed the estimation ofthe relationship between the effective stress ratios 119877eff andthe applied stress ratios 119877 without the need for extensiveexperimental program
The evaluation of the influence of the crack closureeffects was possible using the inverse identification procedureapplied with the proposed theoretical model In this studyit was observed that for high applied stress ratios 119877 05 le
119877 le 07 the crack closure effect has little significanceBut for applied stress ratios 119877 0 le 119877 lt 05 there is asignificant influence of the crack closure and opening effects
Advances in Materials Science and Engineering 9
0
02
04
06
08
1
0 200 400 600 800 1000 1200
Ref
f(mdash
)
R = 001
R = 05
R = 07
ΔK (Nmiddotmmminus32)
Figure 8 Relationship between effective stress ratio 119877eff andapplied stress intensity range Δ119870 (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Experimental testsProposed model prediction
No closure effects
Closure effects
Ref
favg
(KopK
max
)
R (KminKmax)
Figure 9 Effective stress ratio 119877eff as a function of stress ratio(P355NL1 steel)
0
02
04
06
08
1
0 0002 0004 0006 0008
Qua
ntita
tive p
aram
eter
U
R = 001R = 05R = 07
1Kmax
Figure 10 Influence of the stress ratio on 119880 versus 1119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
U(mdash
)
ΔKth0Kmax (1Nmiddotmmminus32)
R = 001
R = 05
R = 07
Figure 11 Influence of the stress ratio on 119880 versus Δ119870th0119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Elberrsquos modelSchijversquos modelASTM E647Newmanrsquos model
Savaidis modelHudak et alProposed model
No closure effects
Closure effectsK
opK
max
Stress R-ratio R
Figure 12 Relationship between 119877eff = 119870op119870max and stress ratiofor several models (P355NL1 steel)
Note that for applied stress ratios within 0 le 119877 lt 05 theinfluence of the crack closure and opening effects decreaseswith increasing of the crack length or stress intensity factorrange
The results of the proposed theoreticalmodel proved to beconsistentwhen comparedwith othermodels proposed in theliterature [1ndash12 19] It was possible to estimate the constants119898and 119862 of the modified Paris law (119889119886119889119873 versus Δ119870eff ) Theseparameters are important for fatigue life prediction studies ofstructural details based on fracture mechanics
Other fatigue programs under constant and variableamplitude loading with othermaterials should be considered
10 Advances in Materials Science and Engineering
100 500 15001000
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
R = 05
R = 07
ΔKeff (Nmiddotmmminus15)
da
dN= 39897Eminus12(ΔKeff)
26105
Figure 13 Correlation of fatigue crack growth rates using the crackclosure analysis (P355NL1 steel)
through the use of full field optical techniques to experimen-tally estimate119870op in order to confirm the results obtained inthe inverse analysis of the proposed theoretical model
Competing Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge the Portuguese Science Founda-tion (FCT) for the financial support through the postdoc-toral Grant SFRHBPD1078252015 The authors gratefullyacknowledge the funding of SciTech Science and Technologyfor Competitive and Sustainable Industries RampD projectcofinanced by Programa Operacional Regional do Norte(NORTE2020) throughFundoEuropeu deDesenvolvimentoRegional (FEDER)
References
[1] W Elber ldquoFatigue crack closure under cyclic tensionrdquo Engineer-ing Fracture Mechanics vol 2 no 1 pp 37ndash45 1970
[2] W Elber ldquoThe significance of fatigue crack closure DamageTolerance in Aircraft Structuresrdquo ASTM STP 486 AmericaSociety for Testing and Materials 1971
[3] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984
[4] M Vormwald and T Seeger ldquoThe consequences of short crackclosure on fatigue crack growth under variable amplitudeloadingrdquo Fatigue and Fracture of Engineering Materials andStructures vol 14 no 2-3 pp 205ndash225 1991
[5] M Vormwald ldquoEffect of cyclic plastic strain on fatigue crackgrowthrdquo International Journal of Fatigue vol 82 part 1 pp 80ndash88 2016
[6] G Savaidis M Dankert and T Seeger ldquoAnalytical procedurefor predicting opening loads of cracks at notchesrdquo Fatigue andFracture of Engineering Materials and Structures vol 18 no 4pp 425ndash442 1995
[7] G Savaidis A Savaidis P Zerres and M Vormwald ldquoModeI fatigue crack growth at notches considering crack closurerdquoInternational Journal of Fatigue vol 32 no 10 pp 1543ndash15582010
[8] S J Hudak and D L Davidson ldquoThe dependence of crackclosure on fatigue loading variablesrdquo in Mechanics of ClosureASTM STP 982 J C Newman Jr and W Elber Eds pp 121ndash138America Society for Testing andMaterials Philadelphia PaUSA 1988
[9] J Schijve Fatigue of Structures andMaterials Kluwer AcademicPublishers Burlington Mass USA 2004
[10] American Society for Testing and Materials (ASTM) ldquoASTME647 standard test method for measurement of fatigue crackgrowth ratesrdquo inAnnual Book of ASTMStandards vol 0301 pp591ndash630 American Society for Testing and Materials (ASTM)West Conshohocken Pa USA 1999
[11] N Geerlofs J Zuidema and J Sietsma ldquoOn the relationshipbetween microstructure and fracture toughness in trip-assistedmultiphase steelsrdquo in Proceedings of the 15th European Con-ference of Fracture (ECF15 rsquo04) Stockholm Sweden August2004 httpwwwgruppofratturaitocsindexphpesisECF15paperview8758
[12] F Ellyin Fatigue Damage Crack Growth and Life PredictionChapman amp Hall New York NY USA 1997
[13] R C McClung and H Sehitoglu ldquoCharacterization of fatiguecrack growth in intermediate and large scale yieldingrdquo Journalof Engineering Materials and Technology vol 113 no 1 pp 15ndash22 1991
[14] M Nakagaki and S N Atluri ldquoFatigue crack closure and delayeffects under mode I spectrum loading an efficient elastic-plastic analysis procedurerdquo Fatigue of EngineeringMaterials andStructures vol 1 no 4 pp 421ndash429 1979
[15] N E DowlingMechanical Behaviour ofMaterialsmdashEngineeringMethods for Deformation Fracture and Fatigue Prentice-HallUpper Saddle River NJ USA 2nd edition 1998
[16] H F S G Pereira AM P De Jesus A S Ribeiro andA A Fer-nandes ldquoFatigue damage behavior of a structural componentmade of P355NL1 steel under block loadingrdquo Journal of PressureVessel Technology vol 131 no 2 Article ID021407 9 pages 2009
[17] O N Romaniv G N Nikiforchin and B N Andrusiv ldquoEffectof crack closure and evaluation of the cyclic crack resistance ofconstructional alloysrdquo Soviet Materials Science vol 19 no 3 pp212ndash225 1983
[18] J R LloydTheeffect of residual stress and crack closure on fatiguecrack growth [PhD thesis] Department of Materials Engineer-ing University of Wollongong Wollongong Australia 1999httprouoweduautheses1525
[19] H Neuber ldquoTheory of stress concentration for shear-strainedprismatical bodies with arbitrary nonlinear stress-strain lawrdquoJournal of Applied Mechanic vol 28 pp 544ndash550 1961
[20] P C Paris and F Erdogan ldquoA critical analysis of crack propaga-tion lawsrdquo Journal of Basic Engineering vol 85 no 4 pp 528ndash534 1963
[21] S M Beden S Abdullah and A K Ariffin ldquoReview of fatiguecrack propagation models for metallic componentsrdquo EuropeanJournal of Scientific Research vol 28 no 3 pp 364ndash397 2009
[22] M Klesnil and P Lukas Fatigue of Metallic Materials ElsevierScience Amsterdam The Netherlands 2nd edition 1992
Advances in Materials Science and Engineering 11
[23] A M P De Jesus and J A F O Correia ldquoCritical assessmentof a local strain-based fatigue crack growth model usingexperimental data available for the P355NL1 steelrdquo Journal ofPressure Vessel Technology vol 135 no 1 Article ID 011404 9pages 2013
[24] J A F O Correia A M P De Jesus and A Fernandez-Canteli ldquoLocal unified probabilistic model for fatigue crackinitiation and propagation application to a notched geometryrdquoEngineering Structures vol 52 pp 394ndash407 2013
[25] ASTMmdashAmerican Society for Testing and Materials ldquoASTME606-92 standard practice for strain-controlled fatigue testingrdquoin Annual Book of ASTM Standards part 10 pp 557ndash571 1998
[26] W Ramberg and W R Osgood ldquoDescription of stress-straincurves by three parametersrdquo NACATechnical Note 902 NACA1943 httpntrsnasagovsearchjspR=19930081614
[27] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoInternal Friction Damping and Cyclic Plasticity (ASTM STP378) ASTM International West Conshohocken Pa USA 1965
[28] K N Smith P Watson and T H Topper ldquoA stress-strainfunction for the fatigue of metalrdquo Journal of Materials vol 5no 4 pp 767ndash778 1970
[29] M Szata and G Lesiuk ldquoAlgorithms for the estimation offatigue crack growth using energymethodrdquoArchives of Civil andMechanical Engineering vol 9 no 1 pp 119ndash134 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
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CeramicsJournal of
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CompositesJournal of
NanoparticlesJournal of
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International Journal of
Biomaterials
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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
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The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
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CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
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MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Advances in Materials Science and Engineering 5
Table 1 Cyclic elastoplastic and strain-life properties of the P355NL1 steel 119877120576= minus1 and 119877
120576= 0
Material 1205901015840
119891119887 120576
1015840
119891119888 119870
10158401198991015840
MPa mdash mdash mdash MPa mdashP355NL1 100550 minus01033 03678 minus05475 94835 01533
where 119870119871is the limiting 119870max For stress ratio 119877 equal to 0
and material parameter 120574 equal to 1 (22) reduces to
119880 = 1 minusΔ119870th0
119870max for 119877 = 0 120574 = 1 (27)
In summary a theoretical model based on Walkerrsquos propa-gation law [15] was proposed which relates the quantitativeparameter 119880 and Δ119870 (or 119870max) and the stress ratio Theproposed model is based on the same initial assumptions ofthe analytical models proposed by Hudak and Davidson [8]and Ellyin [12]
4 Basic Fatigue Data of the InvestigatedP355NL1 Steel
TheP355NL1 steel is a pressure vessel steel and is a normalizedfine grain low alloy carbon steel Its fatigue behaviour hasbeen investigated [16 23 24] A steel plate 5mm thick wasused to prepare specimens for experimental testing Thissection presents the strain-life fatigue data and the fatiguecrack propagation data obtained for the P355NL1 steel [16]
41 Strain-Life Behaviour The strain-life behaviour of theP355NL1 steel was evaluated through fatigue tests of smoothspecimens carried out under strain controlled conditionsaccording to theASTME606 standard [25]TheRamberg andOsgood [26] relation as shown in the following equation wasfitted to the stabilized cyclic stress-strain data
Δ120576
2=Δ120576119864
2+Δ120576119875
2=Δ120590
2119864+ (
Δ120590
21198701015840)
11198991015840
(28)
where1198701015840 and 1198991015840 are the cyclic strain hardening coefficient andexponent respectively Δ120576119875 is plastic strain range and Δ120590 isthe stress range
Low-cycle fatigue test results are very often representedusing the relation between the strain amplitude and thenumber of reversals to failure 2119873
119891 usually assumed to
correspond to the initiation of a macroscopic crack Morrow[27] suggested a general equation valid for low- and high-cycle fatigue regimes
Δ120576
2=Δ120576119864
2+Δ120576119875
2=
1205901015840
119891
119864(2119873119891)119887
+ 1205761015840
119891(2119873119891)119888
(29)
where 1205761015840119891and 119888 are respectively the fatigue ductility coeffi-
cient and fatigue ductility exponent 1205901015840119891is the fatigue strength
coefficient 119887 is the fatigue strength exponent and 119864 is theYoung modulus
Table 2 Elastic and tensile properties of the P355NL1
Material 119864 ] 119891119906
119891119910
GPa mdash MPa MPaP355NL1 20520 0275 56811 41806
Alternatively to the Morrow relation the Smith-Watson-Topper fatigue damage parameter [28] can be used whichshows the following form
120590max sdotΔ120576
2= SWT
=
(1205901015840
119891)2
119864(2119873119891)2119887
+ 1205901015840
119891sdot 1205761015840
119891sdot (2119873119891)119887+119888
(30)
where 120590max is themaximum stress of the cycle and SWT is thedamage parameter Both Morrow and Smith-Watson-Toppermodels are deterministicmodels and are used to represent theaverage fatigue behaviour of the bridgematerials based on theavailable experimental data
Two series of specimens were tested under distinct strainratios 119877
120576= 0 (19 specimens) and minus1 (24 specimens)
Figure 5 shows a plot of the experimental fatigue data in theform of strain-life and SWT-life curves for the two strainratios The cyclic Ramberg-Osgood and Morrow strain-lifeparameters of the P355NL1 steel are summarized in Table 1for the conjunction of both strain ratios [16 23 24] Thecyclic curve is shown in Figure 5(a) including the influenceof the experimental results fromboth test seriesThis researchadopted the values obtained by combining the results of thetwo test series together since the strain ratio effects are con-sidered negligible (tests performed under strain controlledconditions)This could be explained by the cyclic mean stressrelaxation phenomenon and also by a lower sensitivity ofthe material to the mean stress In addition Table 2 presentsthe elastic and monotonic tensile properties of this pressurevessel steel under investigation The elastic and monotonictensile properties are represented respectively by the Youngmodulus 119864 Poisson ratio ] the ultimate tensile strength 119891
119906
and upper yield stress 119891119910
42 Fatigue Crack Propagation Rates Fatigue crack growthrates of the P355NL1 pressure vessel steel were evaluated forseveral stress ratios using compact tension (CT) specimensfollowing the recommendations of the ASTM E647 standard[10] The CT specimens of P355NL1 steel were manufacturedwith a width 119882 = 40mm and a thickness 119861 = 45mm [16]All tests were performed in air at room temperature under asinusoidal waveform at a maximum frequency of 20Hz Thecrack growth was measured on both faces of the specimens
6 Advances in Materials Science and Engineering
0
100
200
300
400
500
600
0 05 1 15 2
Cyclic curve
Δ120590
(MPa
)
Δ1205762 ()
R120576 = 0
R120576 = minus1
(a)
R120576 = 0
R120576 = minus1
10E minus 01
10E minus 02
10E minus 03
10E minus 04
Δ120576
2(mdash
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
Δ120576
2=
10055
E(2Nf)minus01013 + 03678(2Nf)minus05475
(b)
01
10
100
SWT
(MPa
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
R120576 = 0
R120576 = minus1
(c)
Figure 5 Fatigue data obtained for the P355NL1 steel 119877120576= minus1 and 119877
120576= 0 (a) cyclic curve (b) strain-life data (c) SWT-life data
by visual inspection using two travelling microscopes withan accuracy of 0001mm
Figures 6(a) and 6(b) represent the experimental dataand the Paris law correlations for each tested stress ratio ofthe P355NL1 steel 119877 = 00 119877 = 05 and 119877 = 07 Thematerial constants of the Paris crack propagation relation arepresented in Table 3 The crack propagation rates are onlyslightly influenced by the stress ratio Higher stress ratiosresult in higher crack growth ratesThe lines representing theParis law for 119877 = 00 and 119877 = 05 are approximately parallelto each other
On the other hand the line representing the Paris law for119877 = 07 converges to the other lines as the stress intensityfactor range increases In general the stress ratio effectsare more noticeable for lower ranges of the stress intensityfactors For higher stress intensity factor ranges the stressratio effect tends to vanish
The stress ratio effects on fatigue crack growth rates maybe attributed to the crack closure Crack closure increases
Table 3 Material constants concerning the Paris crack propagationrelation
119877 119862lowast
119898 Δ119870max
mdash mdash mdash Nsdotmmminus15
001 71945119864 minus 15 34993 1498050 62806119864 minus 15 35548 1013070 20370119864 minus 13 30031 687lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
with the reduction in the stress ratio leading to lower fatiguecrack growth rates For higher stress ratios such as 119877 =
07 the crack closure is very likely negligible which meansthat applied stress intensity factor range is fully effectiveAlso the fatigue crack propagation lines for each stress ratioare not fully parallel to each other which means that crackclosure also depends on the stress intensity factor rangelevel
Advances in Materials Science and Engineering 7
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
MB06 (R = 07)MB05 (R = 05)
MB03 (R = 05)MB04 (R = 00)MB02 (R = 00)
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
(a)
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
R = 0
R = 05
R = 07
R2 = 09850
R2 = 09840
R2 = 09960
7195E minus 15 times ΔK3499da
dN=
= 2037E minus 13 times ΔK3003da
dN=
6281E minus 15 times ΔK3555da
dN=
(b)
Figure 6 Fatigue crack propagation data obtained for the P355NL1 steel (a) experimental data (b) Paris correlations for each stress ratio 119877
Table 4Walker constants for the fatigue crack propagation relationof the P355NL1 steel
119862lowast
119898 120574
71945119864 minus 15 34993 092lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
5 Application and Discussion of a ProposedTheoretical Model
The fatigue crack growth data presented in Figure 6 wascorrelated using the Walker relation (see (18)) The materialconstants of the Walker crack propagation relation for theP355NL1 steel are presented in Table 4 In addition theWalker modified relation (see (19)) was used to modelthe fatigue crack propagation on region I and results arepresented in Figure 7 Equation (19) requires the definitionof the propagation threshold which was estimated using thefollowing linear relation
Δ119870th = 152 minus 90252 sdot 119877 (31)
where Δ119870th is the propagation threshold defined inNsdotmmminus15 119877 is the stress ratio and constant value 152corresponds to the propagation threshold for 119877 = 0 (inNsdotmmminus15) the other constant (90252) also shows thesame units of the propagation threshold The propagationthreshold for 119877 = 0 is very consistent with the thresholdvalue for the mild steel with yield stress of 366MPa Thecrack propagation threshold Δ119870th is itself influenced by thestress ratio [29] In [29] for a similar material to the P355NL1steel the relationship betweenΔ119870th and119877 is given showing alinear relation between the crack propagation threshold andthe stress ratio Despite using two distinct crack propagationlaws they gave a continuous representation of the crackpropagation data at the crack propagation I-II regimes
transition Figure 7 illustrates the fatigue propagation lawsadopted in this investigation for each tested stress ratio
The proposed theoretical model is applied using aninverse analysis of the experimental fatigue crack propagationrates available for the P355NL1 steel Using the inverseanalysis of the proposed theoretical model it is possibleto obtain the relationship between the effective stress ratio119877eff and the applied stress intensity factor range Δ119870app119877eff accounts for minimum stress intensity factors higherthan crack closure stress intensity factor In this case theeffective stress intensity factor results from the relationshipbetween the crack opening stress intensity factor 119870op andthe maximum stress intensity factor 119870max Figure 8 showsthat for stress ratios above 05 the crack closure and openingeffects are not significant For the stress ratio equal to 0 asignificant influence on the effective stress ratio of the appliedstress intensity factor range is observed Such influence ismore significant for the initial propagation phase of regionII decreasing significantly for higher applied stress intensityranges Thus it is concluded that the crack closure andopening effects are more significant for the initial crackpropagation phase in region II for higher levels of the stressintensity factor ranges along the crack length Figure 9 showsthe results obtained for the relationship between 119877eff and 119877using the inverse analysis of the proposed theoretical modelThis figure helps to understand the influence of the crackclosure and opening effects as a function of stress ratios Theeffective stress ratio 119877eff shown in Figure 9 is the average ofthe values obtained as a function of Δ119870 according to what isevidenced in Figure 8
The proposed relationship between the quantitativeparameter 119880 and the maximum stress intensity 119870max isalso considered Figure 10 shows the relationship between 119880
and 119870max for the studied stress ratios 0 05 and 07 Therelationship between 119880 and (Δ119870th0119870max) for the studiedstress ratios is shown in Figure 11 The proposed theoretical
8 Advances in Materials Science and Engineering
100 500 15001000
Thresholdregion Paris region
MB02 (R = 00)MB04 (R = 00)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
665Eminus13 middot (ΔK minus ΔKth)31250da
dN=
(a) 119877 = 0
MB03 (R = 05)MB05 (R = 05)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 05
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
315Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(b) 119877 = 05
MB06 (R = 07)Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 07
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
245Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(c) 119877 = 07
Figure 7 Fatigue crack growth data correlations for regimes I and II using two relations
model is able to describe the influence of the crack closureand opening effects on different stress ratios
Figure 12 shows the results for the relationship between119877eff and 119877 using various analytical models available in theliterature These model results are presented together withthe results of the proposed theoretical model Based onthe analysis of Figure 12 it can be noted that the resultsobtained by applying the proposed theoretical model arebetween the results obtained using the Hudak and Davidsonmodel (based on the same assumptions) and Savaidis model(with assumptions supported on plastic deformation of thematerial) The curves of Hudak and Davidson and Savaidisdemonstrate being below the other models Other modelsare based on direct polynomial relationships of orders 2 and3 except Newman model that is based on a polynomialrelationship of order 3 and uses properties of the plasticdeformation of the material
Figure 13 shows the fatigue crack propagation rates asa function of effective intensity range using the proposed
method The results obtained using this theoretical modelare similar to results presented in the literature [3]
6 Conclusions
An interpretation of the crack closure and opening effects ispossible without resorting to an excessive computation timeresulting from complex approaches
Fatigue crack propagation tests allowed the estimation ofthe relationship between the effective stress ratios 119877eff andthe applied stress ratios 119877 without the need for extensiveexperimental program
The evaluation of the influence of the crack closureeffects was possible using the inverse identification procedureapplied with the proposed theoretical model In this studyit was observed that for high applied stress ratios 119877 05 le
119877 le 07 the crack closure effect has little significanceBut for applied stress ratios 119877 0 le 119877 lt 05 there is asignificant influence of the crack closure and opening effects
Advances in Materials Science and Engineering 9
0
02
04
06
08
1
0 200 400 600 800 1000 1200
Ref
f(mdash
)
R = 001
R = 05
R = 07
ΔK (Nmiddotmmminus32)
Figure 8 Relationship between effective stress ratio 119877eff andapplied stress intensity range Δ119870 (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Experimental testsProposed model prediction
No closure effects
Closure effects
Ref
favg
(KopK
max
)
R (KminKmax)
Figure 9 Effective stress ratio 119877eff as a function of stress ratio(P355NL1 steel)
0
02
04
06
08
1
0 0002 0004 0006 0008
Qua
ntita
tive p
aram
eter
U
R = 001R = 05R = 07
1Kmax
Figure 10 Influence of the stress ratio on 119880 versus 1119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
U(mdash
)
ΔKth0Kmax (1Nmiddotmmminus32)
R = 001
R = 05
R = 07
Figure 11 Influence of the stress ratio on 119880 versus Δ119870th0119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Elberrsquos modelSchijversquos modelASTM E647Newmanrsquos model
Savaidis modelHudak et alProposed model
No closure effects
Closure effectsK
opK
max
Stress R-ratio R
Figure 12 Relationship between 119877eff = 119870op119870max and stress ratiofor several models (P355NL1 steel)
Note that for applied stress ratios within 0 le 119877 lt 05 theinfluence of the crack closure and opening effects decreaseswith increasing of the crack length or stress intensity factorrange
The results of the proposed theoreticalmodel proved to beconsistentwhen comparedwith othermodels proposed in theliterature [1ndash12 19] It was possible to estimate the constants119898and 119862 of the modified Paris law (119889119886119889119873 versus Δ119870eff ) Theseparameters are important for fatigue life prediction studies ofstructural details based on fracture mechanics
Other fatigue programs under constant and variableamplitude loading with othermaterials should be considered
10 Advances in Materials Science and Engineering
100 500 15001000
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
R = 05
R = 07
ΔKeff (Nmiddotmmminus15)
da
dN= 39897Eminus12(ΔKeff)
26105
Figure 13 Correlation of fatigue crack growth rates using the crackclosure analysis (P355NL1 steel)
through the use of full field optical techniques to experimen-tally estimate119870op in order to confirm the results obtained inthe inverse analysis of the proposed theoretical model
Competing Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge the Portuguese Science Founda-tion (FCT) for the financial support through the postdoc-toral Grant SFRHBPD1078252015 The authors gratefullyacknowledge the funding of SciTech Science and Technologyfor Competitive and Sustainable Industries RampD projectcofinanced by Programa Operacional Regional do Norte(NORTE2020) throughFundoEuropeu deDesenvolvimentoRegional (FEDER)
References
[1] W Elber ldquoFatigue crack closure under cyclic tensionrdquo Engineer-ing Fracture Mechanics vol 2 no 1 pp 37ndash45 1970
[2] W Elber ldquoThe significance of fatigue crack closure DamageTolerance in Aircraft Structuresrdquo ASTM STP 486 AmericaSociety for Testing and Materials 1971
[3] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984
[4] M Vormwald and T Seeger ldquoThe consequences of short crackclosure on fatigue crack growth under variable amplitudeloadingrdquo Fatigue and Fracture of Engineering Materials andStructures vol 14 no 2-3 pp 205ndash225 1991
[5] M Vormwald ldquoEffect of cyclic plastic strain on fatigue crackgrowthrdquo International Journal of Fatigue vol 82 part 1 pp 80ndash88 2016
[6] G Savaidis M Dankert and T Seeger ldquoAnalytical procedurefor predicting opening loads of cracks at notchesrdquo Fatigue andFracture of Engineering Materials and Structures vol 18 no 4pp 425ndash442 1995
[7] G Savaidis A Savaidis P Zerres and M Vormwald ldquoModeI fatigue crack growth at notches considering crack closurerdquoInternational Journal of Fatigue vol 32 no 10 pp 1543ndash15582010
[8] S J Hudak and D L Davidson ldquoThe dependence of crackclosure on fatigue loading variablesrdquo in Mechanics of ClosureASTM STP 982 J C Newman Jr and W Elber Eds pp 121ndash138America Society for Testing andMaterials Philadelphia PaUSA 1988
[9] J Schijve Fatigue of Structures andMaterials Kluwer AcademicPublishers Burlington Mass USA 2004
[10] American Society for Testing and Materials (ASTM) ldquoASTME647 standard test method for measurement of fatigue crackgrowth ratesrdquo inAnnual Book of ASTMStandards vol 0301 pp591ndash630 American Society for Testing and Materials (ASTM)West Conshohocken Pa USA 1999
[11] N Geerlofs J Zuidema and J Sietsma ldquoOn the relationshipbetween microstructure and fracture toughness in trip-assistedmultiphase steelsrdquo in Proceedings of the 15th European Con-ference of Fracture (ECF15 rsquo04) Stockholm Sweden August2004 httpwwwgruppofratturaitocsindexphpesisECF15paperview8758
[12] F Ellyin Fatigue Damage Crack Growth and Life PredictionChapman amp Hall New York NY USA 1997
[13] R C McClung and H Sehitoglu ldquoCharacterization of fatiguecrack growth in intermediate and large scale yieldingrdquo Journalof Engineering Materials and Technology vol 113 no 1 pp 15ndash22 1991
[14] M Nakagaki and S N Atluri ldquoFatigue crack closure and delayeffects under mode I spectrum loading an efficient elastic-plastic analysis procedurerdquo Fatigue of EngineeringMaterials andStructures vol 1 no 4 pp 421ndash429 1979
[15] N E DowlingMechanical Behaviour ofMaterialsmdashEngineeringMethods for Deformation Fracture and Fatigue Prentice-HallUpper Saddle River NJ USA 2nd edition 1998
[16] H F S G Pereira AM P De Jesus A S Ribeiro andA A Fer-nandes ldquoFatigue damage behavior of a structural componentmade of P355NL1 steel under block loadingrdquo Journal of PressureVessel Technology vol 131 no 2 Article ID021407 9 pages 2009
[17] O N Romaniv G N Nikiforchin and B N Andrusiv ldquoEffectof crack closure and evaluation of the cyclic crack resistance ofconstructional alloysrdquo Soviet Materials Science vol 19 no 3 pp212ndash225 1983
[18] J R LloydTheeffect of residual stress and crack closure on fatiguecrack growth [PhD thesis] Department of Materials Engineer-ing University of Wollongong Wollongong Australia 1999httprouoweduautheses1525
[19] H Neuber ldquoTheory of stress concentration for shear-strainedprismatical bodies with arbitrary nonlinear stress-strain lawrdquoJournal of Applied Mechanic vol 28 pp 544ndash550 1961
[20] P C Paris and F Erdogan ldquoA critical analysis of crack propaga-tion lawsrdquo Journal of Basic Engineering vol 85 no 4 pp 528ndash534 1963
[21] S M Beden S Abdullah and A K Ariffin ldquoReview of fatiguecrack propagation models for metallic componentsrdquo EuropeanJournal of Scientific Research vol 28 no 3 pp 364ndash397 2009
[22] M Klesnil and P Lukas Fatigue of Metallic Materials ElsevierScience Amsterdam The Netherlands 2nd edition 1992
Advances in Materials Science and Engineering 11
[23] A M P De Jesus and J A F O Correia ldquoCritical assessmentof a local strain-based fatigue crack growth model usingexperimental data available for the P355NL1 steelrdquo Journal ofPressure Vessel Technology vol 135 no 1 Article ID 011404 9pages 2013
[24] J A F O Correia A M P De Jesus and A Fernandez-Canteli ldquoLocal unified probabilistic model for fatigue crackinitiation and propagation application to a notched geometryrdquoEngineering Structures vol 52 pp 394ndash407 2013
[25] ASTMmdashAmerican Society for Testing and Materials ldquoASTME606-92 standard practice for strain-controlled fatigue testingrdquoin Annual Book of ASTM Standards part 10 pp 557ndash571 1998
[26] W Ramberg and W R Osgood ldquoDescription of stress-straincurves by three parametersrdquo NACATechnical Note 902 NACA1943 httpntrsnasagovsearchjspR=19930081614
[27] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoInternal Friction Damping and Cyclic Plasticity (ASTM STP378) ASTM International West Conshohocken Pa USA 1965
[28] K N Smith P Watson and T H Topper ldquoA stress-strainfunction for the fatigue of metalrdquo Journal of Materials vol 5no 4 pp 767ndash778 1970
[29] M Szata and G Lesiuk ldquoAlgorithms for the estimation offatigue crack growth using energymethodrdquoArchives of Civil andMechanical Engineering vol 9 no 1 pp 119ndash134 2009
Submit your manuscripts athttpwwwhindawicom
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NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
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TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
6 Advances in Materials Science and Engineering
0
100
200
300
400
500
600
0 05 1 15 2
Cyclic curve
Δ120590
(MPa
)
Δ1205762 ()
R120576 = 0
R120576 = minus1
(a)
R120576 = 0
R120576 = minus1
10E minus 01
10E minus 02
10E minus 03
10E minus 04
Δ120576
2(mdash
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
Δ120576
2=
10055
E(2Nf)minus01013 + 03678(2Nf)minus05475
(b)
01
10
100
SWT
(MPa
)
10E + 02 10E + 03 10E + 04 10E + 05 10E + 06 10E + 07
Reversals to failure 2Nf
R120576 = 0
R120576 = minus1
(c)
Figure 5 Fatigue data obtained for the P355NL1 steel 119877120576= minus1 and 119877
120576= 0 (a) cyclic curve (b) strain-life data (c) SWT-life data
by visual inspection using two travelling microscopes withan accuracy of 0001mm
Figures 6(a) and 6(b) represent the experimental dataand the Paris law correlations for each tested stress ratio ofthe P355NL1 steel 119877 = 00 119877 = 05 and 119877 = 07 Thematerial constants of the Paris crack propagation relation arepresented in Table 3 The crack propagation rates are onlyslightly influenced by the stress ratio Higher stress ratiosresult in higher crack growth ratesThe lines representing theParis law for 119877 = 00 and 119877 = 05 are approximately parallelto each other
On the other hand the line representing the Paris law for119877 = 07 converges to the other lines as the stress intensityfactor range increases In general the stress ratio effectsare more noticeable for lower ranges of the stress intensityfactors For higher stress intensity factor ranges the stressratio effect tends to vanish
The stress ratio effects on fatigue crack growth rates maybe attributed to the crack closure Crack closure increases
Table 3 Material constants concerning the Paris crack propagationrelation
119877 119862lowast
119898 Δ119870max
mdash mdash mdash Nsdotmmminus15
001 71945119864 minus 15 34993 1498050 62806119864 minus 15 35548 1013070 20370119864 minus 13 30031 687lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
with the reduction in the stress ratio leading to lower fatiguecrack growth rates For higher stress ratios such as 119877 =
07 the crack closure is very likely negligible which meansthat applied stress intensity factor range is fully effectiveAlso the fatigue crack propagation lines for each stress ratioare not fully parallel to each other which means that crackclosure also depends on the stress intensity factor rangelevel
Advances in Materials Science and Engineering 7
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
MB06 (R = 07)MB05 (R = 05)
MB03 (R = 05)MB04 (R = 00)MB02 (R = 00)
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
(a)
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
R = 0
R = 05
R = 07
R2 = 09850
R2 = 09840
R2 = 09960
7195E minus 15 times ΔK3499da
dN=
= 2037E minus 13 times ΔK3003da
dN=
6281E minus 15 times ΔK3555da
dN=
(b)
Figure 6 Fatigue crack propagation data obtained for the P355NL1 steel (a) experimental data (b) Paris correlations for each stress ratio 119877
Table 4Walker constants for the fatigue crack propagation relationof the P355NL1 steel
119862lowast
119898 120574
71945119864 minus 15 34993 092lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
5 Application and Discussion of a ProposedTheoretical Model
The fatigue crack growth data presented in Figure 6 wascorrelated using the Walker relation (see (18)) The materialconstants of the Walker crack propagation relation for theP355NL1 steel are presented in Table 4 In addition theWalker modified relation (see (19)) was used to modelthe fatigue crack propagation on region I and results arepresented in Figure 7 Equation (19) requires the definitionof the propagation threshold which was estimated using thefollowing linear relation
Δ119870th = 152 minus 90252 sdot 119877 (31)
where Δ119870th is the propagation threshold defined inNsdotmmminus15 119877 is the stress ratio and constant value 152corresponds to the propagation threshold for 119877 = 0 (inNsdotmmminus15) the other constant (90252) also shows thesame units of the propagation threshold The propagationthreshold for 119877 = 0 is very consistent with the thresholdvalue for the mild steel with yield stress of 366MPa Thecrack propagation threshold Δ119870th is itself influenced by thestress ratio [29] In [29] for a similar material to the P355NL1steel the relationship betweenΔ119870th and119877 is given showing alinear relation between the crack propagation threshold andthe stress ratio Despite using two distinct crack propagationlaws they gave a continuous representation of the crackpropagation data at the crack propagation I-II regimes
transition Figure 7 illustrates the fatigue propagation lawsadopted in this investigation for each tested stress ratio
The proposed theoretical model is applied using aninverse analysis of the experimental fatigue crack propagationrates available for the P355NL1 steel Using the inverseanalysis of the proposed theoretical model it is possibleto obtain the relationship between the effective stress ratio119877eff and the applied stress intensity factor range Δ119870app119877eff accounts for minimum stress intensity factors higherthan crack closure stress intensity factor In this case theeffective stress intensity factor results from the relationshipbetween the crack opening stress intensity factor 119870op andthe maximum stress intensity factor 119870max Figure 8 showsthat for stress ratios above 05 the crack closure and openingeffects are not significant For the stress ratio equal to 0 asignificant influence on the effective stress ratio of the appliedstress intensity factor range is observed Such influence ismore significant for the initial propagation phase of regionII decreasing significantly for higher applied stress intensityranges Thus it is concluded that the crack closure andopening effects are more significant for the initial crackpropagation phase in region II for higher levels of the stressintensity factor ranges along the crack length Figure 9 showsthe results obtained for the relationship between 119877eff and 119877using the inverse analysis of the proposed theoretical modelThis figure helps to understand the influence of the crackclosure and opening effects as a function of stress ratios Theeffective stress ratio 119877eff shown in Figure 9 is the average ofthe values obtained as a function of Δ119870 according to what isevidenced in Figure 8
The proposed relationship between the quantitativeparameter 119880 and the maximum stress intensity 119870max isalso considered Figure 10 shows the relationship between 119880
and 119870max for the studied stress ratios 0 05 and 07 Therelationship between 119880 and (Δ119870th0119870max) for the studiedstress ratios is shown in Figure 11 The proposed theoretical
8 Advances in Materials Science and Engineering
100 500 15001000
Thresholdregion Paris region
MB02 (R = 00)MB04 (R = 00)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
665Eminus13 middot (ΔK minus ΔKth)31250da
dN=
(a) 119877 = 0
MB03 (R = 05)MB05 (R = 05)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 05
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
315Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(b) 119877 = 05
MB06 (R = 07)Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 07
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
245Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(c) 119877 = 07
Figure 7 Fatigue crack growth data correlations for regimes I and II using two relations
model is able to describe the influence of the crack closureand opening effects on different stress ratios
Figure 12 shows the results for the relationship between119877eff and 119877 using various analytical models available in theliterature These model results are presented together withthe results of the proposed theoretical model Based onthe analysis of Figure 12 it can be noted that the resultsobtained by applying the proposed theoretical model arebetween the results obtained using the Hudak and Davidsonmodel (based on the same assumptions) and Savaidis model(with assumptions supported on plastic deformation of thematerial) The curves of Hudak and Davidson and Savaidisdemonstrate being below the other models Other modelsare based on direct polynomial relationships of orders 2 and3 except Newman model that is based on a polynomialrelationship of order 3 and uses properties of the plasticdeformation of the material
Figure 13 shows the fatigue crack propagation rates asa function of effective intensity range using the proposed
method The results obtained using this theoretical modelare similar to results presented in the literature [3]
6 Conclusions
An interpretation of the crack closure and opening effects ispossible without resorting to an excessive computation timeresulting from complex approaches
Fatigue crack propagation tests allowed the estimation ofthe relationship between the effective stress ratios 119877eff andthe applied stress ratios 119877 without the need for extensiveexperimental program
The evaluation of the influence of the crack closureeffects was possible using the inverse identification procedureapplied with the proposed theoretical model In this studyit was observed that for high applied stress ratios 119877 05 le
119877 le 07 the crack closure effect has little significanceBut for applied stress ratios 119877 0 le 119877 lt 05 there is asignificant influence of the crack closure and opening effects
Advances in Materials Science and Engineering 9
0
02
04
06
08
1
0 200 400 600 800 1000 1200
Ref
f(mdash
)
R = 001
R = 05
R = 07
ΔK (Nmiddotmmminus32)
Figure 8 Relationship between effective stress ratio 119877eff andapplied stress intensity range Δ119870 (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Experimental testsProposed model prediction
No closure effects
Closure effects
Ref
favg
(KopK
max
)
R (KminKmax)
Figure 9 Effective stress ratio 119877eff as a function of stress ratio(P355NL1 steel)
0
02
04
06
08
1
0 0002 0004 0006 0008
Qua
ntita
tive p
aram
eter
U
R = 001R = 05R = 07
1Kmax
Figure 10 Influence of the stress ratio on 119880 versus 1119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
U(mdash
)
ΔKth0Kmax (1Nmiddotmmminus32)
R = 001
R = 05
R = 07
Figure 11 Influence of the stress ratio on 119880 versus Δ119870th0119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Elberrsquos modelSchijversquos modelASTM E647Newmanrsquos model
Savaidis modelHudak et alProposed model
No closure effects
Closure effectsK
opK
max
Stress R-ratio R
Figure 12 Relationship between 119877eff = 119870op119870max and stress ratiofor several models (P355NL1 steel)
Note that for applied stress ratios within 0 le 119877 lt 05 theinfluence of the crack closure and opening effects decreaseswith increasing of the crack length or stress intensity factorrange
The results of the proposed theoreticalmodel proved to beconsistentwhen comparedwith othermodels proposed in theliterature [1ndash12 19] It was possible to estimate the constants119898and 119862 of the modified Paris law (119889119886119889119873 versus Δ119870eff ) Theseparameters are important for fatigue life prediction studies ofstructural details based on fracture mechanics
Other fatigue programs under constant and variableamplitude loading with othermaterials should be considered
10 Advances in Materials Science and Engineering
100 500 15001000
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
R = 05
R = 07
ΔKeff (Nmiddotmmminus15)
da
dN= 39897Eminus12(ΔKeff)
26105
Figure 13 Correlation of fatigue crack growth rates using the crackclosure analysis (P355NL1 steel)
through the use of full field optical techniques to experimen-tally estimate119870op in order to confirm the results obtained inthe inverse analysis of the proposed theoretical model
Competing Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge the Portuguese Science Founda-tion (FCT) for the financial support through the postdoc-toral Grant SFRHBPD1078252015 The authors gratefullyacknowledge the funding of SciTech Science and Technologyfor Competitive and Sustainable Industries RampD projectcofinanced by Programa Operacional Regional do Norte(NORTE2020) throughFundoEuropeu deDesenvolvimentoRegional (FEDER)
References
[1] W Elber ldquoFatigue crack closure under cyclic tensionrdquo Engineer-ing Fracture Mechanics vol 2 no 1 pp 37ndash45 1970
[2] W Elber ldquoThe significance of fatigue crack closure DamageTolerance in Aircraft Structuresrdquo ASTM STP 486 AmericaSociety for Testing and Materials 1971
[3] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984
[4] M Vormwald and T Seeger ldquoThe consequences of short crackclosure on fatigue crack growth under variable amplitudeloadingrdquo Fatigue and Fracture of Engineering Materials andStructures vol 14 no 2-3 pp 205ndash225 1991
[5] M Vormwald ldquoEffect of cyclic plastic strain on fatigue crackgrowthrdquo International Journal of Fatigue vol 82 part 1 pp 80ndash88 2016
[6] G Savaidis M Dankert and T Seeger ldquoAnalytical procedurefor predicting opening loads of cracks at notchesrdquo Fatigue andFracture of Engineering Materials and Structures vol 18 no 4pp 425ndash442 1995
[7] G Savaidis A Savaidis P Zerres and M Vormwald ldquoModeI fatigue crack growth at notches considering crack closurerdquoInternational Journal of Fatigue vol 32 no 10 pp 1543ndash15582010
[8] S J Hudak and D L Davidson ldquoThe dependence of crackclosure on fatigue loading variablesrdquo in Mechanics of ClosureASTM STP 982 J C Newman Jr and W Elber Eds pp 121ndash138America Society for Testing andMaterials Philadelphia PaUSA 1988
[9] J Schijve Fatigue of Structures andMaterials Kluwer AcademicPublishers Burlington Mass USA 2004
[10] American Society for Testing and Materials (ASTM) ldquoASTME647 standard test method for measurement of fatigue crackgrowth ratesrdquo inAnnual Book of ASTMStandards vol 0301 pp591ndash630 American Society for Testing and Materials (ASTM)West Conshohocken Pa USA 1999
[11] N Geerlofs J Zuidema and J Sietsma ldquoOn the relationshipbetween microstructure and fracture toughness in trip-assistedmultiphase steelsrdquo in Proceedings of the 15th European Con-ference of Fracture (ECF15 rsquo04) Stockholm Sweden August2004 httpwwwgruppofratturaitocsindexphpesisECF15paperview8758
[12] F Ellyin Fatigue Damage Crack Growth and Life PredictionChapman amp Hall New York NY USA 1997
[13] R C McClung and H Sehitoglu ldquoCharacterization of fatiguecrack growth in intermediate and large scale yieldingrdquo Journalof Engineering Materials and Technology vol 113 no 1 pp 15ndash22 1991
[14] M Nakagaki and S N Atluri ldquoFatigue crack closure and delayeffects under mode I spectrum loading an efficient elastic-plastic analysis procedurerdquo Fatigue of EngineeringMaterials andStructures vol 1 no 4 pp 421ndash429 1979
[15] N E DowlingMechanical Behaviour ofMaterialsmdashEngineeringMethods for Deformation Fracture and Fatigue Prentice-HallUpper Saddle River NJ USA 2nd edition 1998
[16] H F S G Pereira AM P De Jesus A S Ribeiro andA A Fer-nandes ldquoFatigue damage behavior of a structural componentmade of P355NL1 steel under block loadingrdquo Journal of PressureVessel Technology vol 131 no 2 Article ID021407 9 pages 2009
[17] O N Romaniv G N Nikiforchin and B N Andrusiv ldquoEffectof crack closure and evaluation of the cyclic crack resistance ofconstructional alloysrdquo Soviet Materials Science vol 19 no 3 pp212ndash225 1983
[18] J R LloydTheeffect of residual stress and crack closure on fatiguecrack growth [PhD thesis] Department of Materials Engineer-ing University of Wollongong Wollongong Australia 1999httprouoweduautheses1525
[19] H Neuber ldquoTheory of stress concentration for shear-strainedprismatical bodies with arbitrary nonlinear stress-strain lawrdquoJournal of Applied Mechanic vol 28 pp 544ndash550 1961
[20] P C Paris and F Erdogan ldquoA critical analysis of crack propaga-tion lawsrdquo Journal of Basic Engineering vol 85 no 4 pp 528ndash534 1963
[21] S M Beden S Abdullah and A K Ariffin ldquoReview of fatiguecrack propagation models for metallic componentsrdquo EuropeanJournal of Scientific Research vol 28 no 3 pp 364ndash397 2009
[22] M Klesnil and P Lukas Fatigue of Metallic Materials ElsevierScience Amsterdam The Netherlands 2nd edition 1992
Advances in Materials Science and Engineering 11
[23] A M P De Jesus and J A F O Correia ldquoCritical assessmentof a local strain-based fatigue crack growth model usingexperimental data available for the P355NL1 steelrdquo Journal ofPressure Vessel Technology vol 135 no 1 Article ID 011404 9pages 2013
[24] J A F O Correia A M P De Jesus and A Fernandez-Canteli ldquoLocal unified probabilistic model for fatigue crackinitiation and propagation application to a notched geometryrdquoEngineering Structures vol 52 pp 394ndash407 2013
[25] ASTMmdashAmerican Society for Testing and Materials ldquoASTME606-92 standard practice for strain-controlled fatigue testingrdquoin Annual Book of ASTM Standards part 10 pp 557ndash571 1998
[26] W Ramberg and W R Osgood ldquoDescription of stress-straincurves by three parametersrdquo NACATechnical Note 902 NACA1943 httpntrsnasagovsearchjspR=19930081614
[27] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoInternal Friction Damping and Cyclic Plasticity (ASTM STP378) ASTM International West Conshohocken Pa USA 1965
[28] K N Smith P Watson and T H Topper ldquoA stress-strainfunction for the fatigue of metalrdquo Journal of Materials vol 5no 4 pp 767ndash778 1970
[29] M Szata and G Lesiuk ldquoAlgorithms for the estimation offatigue crack growth using energymethodrdquoArchives of Civil andMechanical Engineering vol 9 no 1 pp 119ndash134 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Advances in Materials Science and Engineering 7
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
MB06 (R = 07)MB05 (R = 05)
MB03 (R = 05)MB04 (R = 00)MB02 (R = 00)
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
(a)
250 1000 1500500
10E minus 5
10E minus 6
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
ΔK (Nmiddotmmminus15)
R = 0
R = 05
R = 07
R2 = 09850
R2 = 09840
R2 = 09960
7195E minus 15 times ΔK3499da
dN=
= 2037E minus 13 times ΔK3003da
dN=
6281E minus 15 times ΔK3555da
dN=
(b)
Figure 6 Fatigue crack propagation data obtained for the P355NL1 steel (a) experimental data (b) Paris correlations for each stress ratio 119877
Table 4Walker constants for the fatigue crack propagation relationof the P355NL1 steel
119862lowast
119898 120574
71945119864 minus 15 34993 092lowast119889119886119889119873 in mmcycle and Δ119870 in Nsdotmmminus15
5 Application and Discussion of a ProposedTheoretical Model
The fatigue crack growth data presented in Figure 6 wascorrelated using the Walker relation (see (18)) The materialconstants of the Walker crack propagation relation for theP355NL1 steel are presented in Table 4 In addition theWalker modified relation (see (19)) was used to modelthe fatigue crack propagation on region I and results arepresented in Figure 7 Equation (19) requires the definitionof the propagation threshold which was estimated using thefollowing linear relation
Δ119870th = 152 minus 90252 sdot 119877 (31)
where Δ119870th is the propagation threshold defined inNsdotmmminus15 119877 is the stress ratio and constant value 152corresponds to the propagation threshold for 119877 = 0 (inNsdotmmminus15) the other constant (90252) also shows thesame units of the propagation threshold The propagationthreshold for 119877 = 0 is very consistent with the thresholdvalue for the mild steel with yield stress of 366MPa Thecrack propagation threshold Δ119870th is itself influenced by thestress ratio [29] In [29] for a similar material to the P355NL1steel the relationship betweenΔ119870th and119877 is given showing alinear relation between the crack propagation threshold andthe stress ratio Despite using two distinct crack propagationlaws they gave a continuous representation of the crackpropagation data at the crack propagation I-II regimes
transition Figure 7 illustrates the fatigue propagation lawsadopted in this investigation for each tested stress ratio
The proposed theoretical model is applied using aninverse analysis of the experimental fatigue crack propagationrates available for the P355NL1 steel Using the inverseanalysis of the proposed theoretical model it is possibleto obtain the relationship between the effective stress ratio119877eff and the applied stress intensity factor range Δ119870app119877eff accounts for minimum stress intensity factors higherthan crack closure stress intensity factor In this case theeffective stress intensity factor results from the relationshipbetween the crack opening stress intensity factor 119870op andthe maximum stress intensity factor 119870max Figure 8 showsthat for stress ratios above 05 the crack closure and openingeffects are not significant For the stress ratio equal to 0 asignificant influence on the effective stress ratio of the appliedstress intensity factor range is observed Such influence ismore significant for the initial propagation phase of regionII decreasing significantly for higher applied stress intensityranges Thus it is concluded that the crack closure andopening effects are more significant for the initial crackpropagation phase in region II for higher levels of the stressintensity factor ranges along the crack length Figure 9 showsthe results obtained for the relationship between 119877eff and 119877using the inverse analysis of the proposed theoretical modelThis figure helps to understand the influence of the crackclosure and opening effects as a function of stress ratios Theeffective stress ratio 119877eff shown in Figure 9 is the average ofthe values obtained as a function of Δ119870 according to what isevidenced in Figure 8
The proposed relationship between the quantitativeparameter 119880 and the maximum stress intensity 119870max isalso considered Figure 10 shows the relationship between 119880
and 119870max for the studied stress ratios 0 05 and 07 Therelationship between 119880 and (Δ119870th0119870max) for the studiedstress ratios is shown in Figure 11 The proposed theoretical
8 Advances in Materials Science and Engineering
100 500 15001000
Thresholdregion Paris region
MB02 (R = 00)MB04 (R = 00)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
665Eminus13 middot (ΔK minus ΔKth)31250da
dN=
(a) 119877 = 0
MB03 (R = 05)MB05 (R = 05)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 05
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
315Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(b) 119877 = 05
MB06 (R = 07)Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 07
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
245Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(c) 119877 = 07
Figure 7 Fatigue crack growth data correlations for regimes I and II using two relations
model is able to describe the influence of the crack closureand opening effects on different stress ratios
Figure 12 shows the results for the relationship between119877eff and 119877 using various analytical models available in theliterature These model results are presented together withthe results of the proposed theoretical model Based onthe analysis of Figure 12 it can be noted that the resultsobtained by applying the proposed theoretical model arebetween the results obtained using the Hudak and Davidsonmodel (based on the same assumptions) and Savaidis model(with assumptions supported on plastic deformation of thematerial) The curves of Hudak and Davidson and Savaidisdemonstrate being below the other models Other modelsare based on direct polynomial relationships of orders 2 and3 except Newman model that is based on a polynomialrelationship of order 3 and uses properties of the plasticdeformation of the material
Figure 13 shows the fatigue crack propagation rates asa function of effective intensity range using the proposed
method The results obtained using this theoretical modelare similar to results presented in the literature [3]
6 Conclusions
An interpretation of the crack closure and opening effects ispossible without resorting to an excessive computation timeresulting from complex approaches
Fatigue crack propagation tests allowed the estimation ofthe relationship between the effective stress ratios 119877eff andthe applied stress ratios 119877 without the need for extensiveexperimental program
The evaluation of the influence of the crack closureeffects was possible using the inverse identification procedureapplied with the proposed theoretical model In this studyit was observed that for high applied stress ratios 119877 05 le
119877 le 07 the crack closure effect has little significanceBut for applied stress ratios 119877 0 le 119877 lt 05 there is asignificant influence of the crack closure and opening effects
Advances in Materials Science and Engineering 9
0
02
04
06
08
1
0 200 400 600 800 1000 1200
Ref
f(mdash
)
R = 001
R = 05
R = 07
ΔK (Nmiddotmmminus32)
Figure 8 Relationship between effective stress ratio 119877eff andapplied stress intensity range Δ119870 (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Experimental testsProposed model prediction
No closure effects
Closure effects
Ref
favg
(KopK
max
)
R (KminKmax)
Figure 9 Effective stress ratio 119877eff as a function of stress ratio(P355NL1 steel)
0
02
04
06
08
1
0 0002 0004 0006 0008
Qua
ntita
tive p
aram
eter
U
R = 001R = 05R = 07
1Kmax
Figure 10 Influence of the stress ratio on 119880 versus 1119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
U(mdash
)
ΔKth0Kmax (1Nmiddotmmminus32)
R = 001
R = 05
R = 07
Figure 11 Influence of the stress ratio on 119880 versus Δ119870th0119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Elberrsquos modelSchijversquos modelASTM E647Newmanrsquos model
Savaidis modelHudak et alProposed model
No closure effects
Closure effectsK
opK
max
Stress R-ratio R
Figure 12 Relationship between 119877eff = 119870op119870max and stress ratiofor several models (P355NL1 steel)
Note that for applied stress ratios within 0 le 119877 lt 05 theinfluence of the crack closure and opening effects decreaseswith increasing of the crack length or stress intensity factorrange
The results of the proposed theoreticalmodel proved to beconsistentwhen comparedwith othermodels proposed in theliterature [1ndash12 19] It was possible to estimate the constants119898and 119862 of the modified Paris law (119889119886119889119873 versus Δ119870eff ) Theseparameters are important for fatigue life prediction studies ofstructural details based on fracture mechanics
Other fatigue programs under constant and variableamplitude loading with othermaterials should be considered
10 Advances in Materials Science and Engineering
100 500 15001000
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
R = 05
R = 07
ΔKeff (Nmiddotmmminus15)
da
dN= 39897Eminus12(ΔKeff)
26105
Figure 13 Correlation of fatigue crack growth rates using the crackclosure analysis (P355NL1 steel)
through the use of full field optical techniques to experimen-tally estimate119870op in order to confirm the results obtained inthe inverse analysis of the proposed theoretical model
Competing Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge the Portuguese Science Founda-tion (FCT) for the financial support through the postdoc-toral Grant SFRHBPD1078252015 The authors gratefullyacknowledge the funding of SciTech Science and Technologyfor Competitive and Sustainable Industries RampD projectcofinanced by Programa Operacional Regional do Norte(NORTE2020) throughFundoEuropeu deDesenvolvimentoRegional (FEDER)
References
[1] W Elber ldquoFatigue crack closure under cyclic tensionrdquo Engineer-ing Fracture Mechanics vol 2 no 1 pp 37ndash45 1970
[2] W Elber ldquoThe significance of fatigue crack closure DamageTolerance in Aircraft Structuresrdquo ASTM STP 486 AmericaSociety for Testing and Materials 1971
[3] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984
[4] M Vormwald and T Seeger ldquoThe consequences of short crackclosure on fatigue crack growth under variable amplitudeloadingrdquo Fatigue and Fracture of Engineering Materials andStructures vol 14 no 2-3 pp 205ndash225 1991
[5] M Vormwald ldquoEffect of cyclic plastic strain on fatigue crackgrowthrdquo International Journal of Fatigue vol 82 part 1 pp 80ndash88 2016
[6] G Savaidis M Dankert and T Seeger ldquoAnalytical procedurefor predicting opening loads of cracks at notchesrdquo Fatigue andFracture of Engineering Materials and Structures vol 18 no 4pp 425ndash442 1995
[7] G Savaidis A Savaidis P Zerres and M Vormwald ldquoModeI fatigue crack growth at notches considering crack closurerdquoInternational Journal of Fatigue vol 32 no 10 pp 1543ndash15582010
[8] S J Hudak and D L Davidson ldquoThe dependence of crackclosure on fatigue loading variablesrdquo in Mechanics of ClosureASTM STP 982 J C Newman Jr and W Elber Eds pp 121ndash138America Society for Testing andMaterials Philadelphia PaUSA 1988
[9] J Schijve Fatigue of Structures andMaterials Kluwer AcademicPublishers Burlington Mass USA 2004
[10] American Society for Testing and Materials (ASTM) ldquoASTME647 standard test method for measurement of fatigue crackgrowth ratesrdquo inAnnual Book of ASTMStandards vol 0301 pp591ndash630 American Society for Testing and Materials (ASTM)West Conshohocken Pa USA 1999
[11] N Geerlofs J Zuidema and J Sietsma ldquoOn the relationshipbetween microstructure and fracture toughness in trip-assistedmultiphase steelsrdquo in Proceedings of the 15th European Con-ference of Fracture (ECF15 rsquo04) Stockholm Sweden August2004 httpwwwgruppofratturaitocsindexphpesisECF15paperview8758
[12] F Ellyin Fatigue Damage Crack Growth and Life PredictionChapman amp Hall New York NY USA 1997
[13] R C McClung and H Sehitoglu ldquoCharacterization of fatiguecrack growth in intermediate and large scale yieldingrdquo Journalof Engineering Materials and Technology vol 113 no 1 pp 15ndash22 1991
[14] M Nakagaki and S N Atluri ldquoFatigue crack closure and delayeffects under mode I spectrum loading an efficient elastic-plastic analysis procedurerdquo Fatigue of EngineeringMaterials andStructures vol 1 no 4 pp 421ndash429 1979
[15] N E DowlingMechanical Behaviour ofMaterialsmdashEngineeringMethods for Deformation Fracture and Fatigue Prentice-HallUpper Saddle River NJ USA 2nd edition 1998
[16] H F S G Pereira AM P De Jesus A S Ribeiro andA A Fer-nandes ldquoFatigue damage behavior of a structural componentmade of P355NL1 steel under block loadingrdquo Journal of PressureVessel Technology vol 131 no 2 Article ID021407 9 pages 2009
[17] O N Romaniv G N Nikiforchin and B N Andrusiv ldquoEffectof crack closure and evaluation of the cyclic crack resistance ofconstructional alloysrdquo Soviet Materials Science vol 19 no 3 pp212ndash225 1983
[18] J R LloydTheeffect of residual stress and crack closure on fatiguecrack growth [PhD thesis] Department of Materials Engineer-ing University of Wollongong Wollongong Australia 1999httprouoweduautheses1525
[19] H Neuber ldquoTheory of stress concentration for shear-strainedprismatical bodies with arbitrary nonlinear stress-strain lawrdquoJournal of Applied Mechanic vol 28 pp 544ndash550 1961
[20] P C Paris and F Erdogan ldquoA critical analysis of crack propaga-tion lawsrdquo Journal of Basic Engineering vol 85 no 4 pp 528ndash534 1963
[21] S M Beden S Abdullah and A K Ariffin ldquoReview of fatiguecrack propagation models for metallic componentsrdquo EuropeanJournal of Scientific Research vol 28 no 3 pp 364ndash397 2009
[22] M Klesnil and P Lukas Fatigue of Metallic Materials ElsevierScience Amsterdam The Netherlands 2nd edition 1992
Advances in Materials Science and Engineering 11
[23] A M P De Jesus and J A F O Correia ldquoCritical assessmentof a local strain-based fatigue crack growth model usingexperimental data available for the P355NL1 steelrdquo Journal ofPressure Vessel Technology vol 135 no 1 Article ID 011404 9pages 2013
[24] J A F O Correia A M P De Jesus and A Fernandez-Canteli ldquoLocal unified probabilistic model for fatigue crackinitiation and propagation application to a notched geometryrdquoEngineering Structures vol 52 pp 394ndash407 2013
[25] ASTMmdashAmerican Society for Testing and Materials ldquoASTME606-92 standard practice for strain-controlled fatigue testingrdquoin Annual Book of ASTM Standards part 10 pp 557ndash571 1998
[26] W Ramberg and W R Osgood ldquoDescription of stress-straincurves by three parametersrdquo NACATechnical Note 902 NACA1943 httpntrsnasagovsearchjspR=19930081614
[27] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoInternal Friction Damping and Cyclic Plasticity (ASTM STP378) ASTM International West Conshohocken Pa USA 1965
[28] K N Smith P Watson and T H Topper ldquoA stress-strainfunction for the fatigue of metalrdquo Journal of Materials vol 5no 4 pp 767ndash778 1970
[29] M Szata and G Lesiuk ldquoAlgorithms for the estimation offatigue crack growth using energymethodrdquoArchives of Civil andMechanical Engineering vol 9 no 1 pp 119ndash134 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
8 Advances in Materials Science and Engineering
100 500 15001000
Thresholdregion Paris region
MB02 (R = 00)MB04 (R = 00)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
665Eminus13 middot (ΔK minus ΔKth)31250da
dN=
(a) 119877 = 0
MB03 (R = 05)MB05 (R = 05)
Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 05
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
315Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(b) 119877 = 05
MB06 (R = 07)Extended Paris lawmdashthreshold regionParis lawmdashParis region
100 500 15001000
Thresholdregion Paris region
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 07
ΔK (Nmiddotmmminus15)
71945E minus 15 middot (ΔK)34993da
dN=
245Eminus13 middot (ΔK minus ΔKth)31150da
dN=
(c) 119877 = 07
Figure 7 Fatigue crack growth data correlations for regimes I and II using two relations
model is able to describe the influence of the crack closureand opening effects on different stress ratios
Figure 12 shows the results for the relationship between119877eff and 119877 using various analytical models available in theliterature These model results are presented together withthe results of the proposed theoretical model Based onthe analysis of Figure 12 it can be noted that the resultsobtained by applying the proposed theoretical model arebetween the results obtained using the Hudak and Davidsonmodel (based on the same assumptions) and Savaidis model(with assumptions supported on plastic deformation of thematerial) The curves of Hudak and Davidson and Savaidisdemonstrate being below the other models Other modelsare based on direct polynomial relationships of orders 2 and3 except Newman model that is based on a polynomialrelationship of order 3 and uses properties of the plasticdeformation of the material
Figure 13 shows the fatigue crack propagation rates asa function of effective intensity range using the proposed
method The results obtained using this theoretical modelare similar to results presented in the literature [3]
6 Conclusions
An interpretation of the crack closure and opening effects ispossible without resorting to an excessive computation timeresulting from complex approaches
Fatigue crack propagation tests allowed the estimation ofthe relationship between the effective stress ratios 119877eff andthe applied stress ratios 119877 without the need for extensiveexperimental program
The evaluation of the influence of the crack closureeffects was possible using the inverse identification procedureapplied with the proposed theoretical model In this studyit was observed that for high applied stress ratios 119877 05 le
119877 le 07 the crack closure effect has little significanceBut for applied stress ratios 119877 0 le 119877 lt 05 there is asignificant influence of the crack closure and opening effects
Advances in Materials Science and Engineering 9
0
02
04
06
08
1
0 200 400 600 800 1000 1200
Ref
f(mdash
)
R = 001
R = 05
R = 07
ΔK (Nmiddotmmminus32)
Figure 8 Relationship between effective stress ratio 119877eff andapplied stress intensity range Δ119870 (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Experimental testsProposed model prediction
No closure effects
Closure effects
Ref
favg
(KopK
max
)
R (KminKmax)
Figure 9 Effective stress ratio 119877eff as a function of stress ratio(P355NL1 steel)
0
02
04
06
08
1
0 0002 0004 0006 0008
Qua
ntita
tive p
aram
eter
U
R = 001R = 05R = 07
1Kmax
Figure 10 Influence of the stress ratio on 119880 versus 1119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
U(mdash
)
ΔKth0Kmax (1Nmiddotmmminus32)
R = 001
R = 05
R = 07
Figure 11 Influence of the stress ratio on 119880 versus Δ119870th0119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Elberrsquos modelSchijversquos modelASTM E647Newmanrsquos model
Savaidis modelHudak et alProposed model
No closure effects
Closure effectsK
opK
max
Stress R-ratio R
Figure 12 Relationship between 119877eff = 119870op119870max and stress ratiofor several models (P355NL1 steel)
Note that for applied stress ratios within 0 le 119877 lt 05 theinfluence of the crack closure and opening effects decreaseswith increasing of the crack length or stress intensity factorrange
The results of the proposed theoreticalmodel proved to beconsistentwhen comparedwith othermodels proposed in theliterature [1ndash12 19] It was possible to estimate the constants119898and 119862 of the modified Paris law (119889119886119889119873 versus Δ119870eff ) Theseparameters are important for fatigue life prediction studies ofstructural details based on fracture mechanics
Other fatigue programs under constant and variableamplitude loading with othermaterials should be considered
10 Advances in Materials Science and Engineering
100 500 15001000
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
R = 05
R = 07
ΔKeff (Nmiddotmmminus15)
da
dN= 39897Eminus12(ΔKeff)
26105
Figure 13 Correlation of fatigue crack growth rates using the crackclosure analysis (P355NL1 steel)
through the use of full field optical techniques to experimen-tally estimate119870op in order to confirm the results obtained inthe inverse analysis of the proposed theoretical model
Competing Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge the Portuguese Science Founda-tion (FCT) for the financial support through the postdoc-toral Grant SFRHBPD1078252015 The authors gratefullyacknowledge the funding of SciTech Science and Technologyfor Competitive and Sustainable Industries RampD projectcofinanced by Programa Operacional Regional do Norte(NORTE2020) throughFundoEuropeu deDesenvolvimentoRegional (FEDER)
References
[1] W Elber ldquoFatigue crack closure under cyclic tensionrdquo Engineer-ing Fracture Mechanics vol 2 no 1 pp 37ndash45 1970
[2] W Elber ldquoThe significance of fatigue crack closure DamageTolerance in Aircraft Structuresrdquo ASTM STP 486 AmericaSociety for Testing and Materials 1971
[3] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984
[4] M Vormwald and T Seeger ldquoThe consequences of short crackclosure on fatigue crack growth under variable amplitudeloadingrdquo Fatigue and Fracture of Engineering Materials andStructures vol 14 no 2-3 pp 205ndash225 1991
[5] M Vormwald ldquoEffect of cyclic plastic strain on fatigue crackgrowthrdquo International Journal of Fatigue vol 82 part 1 pp 80ndash88 2016
[6] G Savaidis M Dankert and T Seeger ldquoAnalytical procedurefor predicting opening loads of cracks at notchesrdquo Fatigue andFracture of Engineering Materials and Structures vol 18 no 4pp 425ndash442 1995
[7] G Savaidis A Savaidis P Zerres and M Vormwald ldquoModeI fatigue crack growth at notches considering crack closurerdquoInternational Journal of Fatigue vol 32 no 10 pp 1543ndash15582010
[8] S J Hudak and D L Davidson ldquoThe dependence of crackclosure on fatigue loading variablesrdquo in Mechanics of ClosureASTM STP 982 J C Newman Jr and W Elber Eds pp 121ndash138America Society for Testing andMaterials Philadelphia PaUSA 1988
[9] J Schijve Fatigue of Structures andMaterials Kluwer AcademicPublishers Burlington Mass USA 2004
[10] American Society for Testing and Materials (ASTM) ldquoASTME647 standard test method for measurement of fatigue crackgrowth ratesrdquo inAnnual Book of ASTMStandards vol 0301 pp591ndash630 American Society for Testing and Materials (ASTM)West Conshohocken Pa USA 1999
[11] N Geerlofs J Zuidema and J Sietsma ldquoOn the relationshipbetween microstructure and fracture toughness in trip-assistedmultiphase steelsrdquo in Proceedings of the 15th European Con-ference of Fracture (ECF15 rsquo04) Stockholm Sweden August2004 httpwwwgruppofratturaitocsindexphpesisECF15paperview8758
[12] F Ellyin Fatigue Damage Crack Growth and Life PredictionChapman amp Hall New York NY USA 1997
[13] R C McClung and H Sehitoglu ldquoCharacterization of fatiguecrack growth in intermediate and large scale yieldingrdquo Journalof Engineering Materials and Technology vol 113 no 1 pp 15ndash22 1991
[14] M Nakagaki and S N Atluri ldquoFatigue crack closure and delayeffects under mode I spectrum loading an efficient elastic-plastic analysis procedurerdquo Fatigue of EngineeringMaterials andStructures vol 1 no 4 pp 421ndash429 1979
[15] N E DowlingMechanical Behaviour ofMaterialsmdashEngineeringMethods for Deformation Fracture and Fatigue Prentice-HallUpper Saddle River NJ USA 2nd edition 1998
[16] H F S G Pereira AM P De Jesus A S Ribeiro andA A Fer-nandes ldquoFatigue damage behavior of a structural componentmade of P355NL1 steel under block loadingrdquo Journal of PressureVessel Technology vol 131 no 2 Article ID021407 9 pages 2009
[17] O N Romaniv G N Nikiforchin and B N Andrusiv ldquoEffectof crack closure and evaluation of the cyclic crack resistance ofconstructional alloysrdquo Soviet Materials Science vol 19 no 3 pp212ndash225 1983
[18] J R LloydTheeffect of residual stress and crack closure on fatiguecrack growth [PhD thesis] Department of Materials Engineer-ing University of Wollongong Wollongong Australia 1999httprouoweduautheses1525
[19] H Neuber ldquoTheory of stress concentration for shear-strainedprismatical bodies with arbitrary nonlinear stress-strain lawrdquoJournal of Applied Mechanic vol 28 pp 544ndash550 1961
[20] P C Paris and F Erdogan ldquoA critical analysis of crack propaga-tion lawsrdquo Journal of Basic Engineering vol 85 no 4 pp 528ndash534 1963
[21] S M Beden S Abdullah and A K Ariffin ldquoReview of fatiguecrack propagation models for metallic componentsrdquo EuropeanJournal of Scientific Research vol 28 no 3 pp 364ndash397 2009
[22] M Klesnil and P Lukas Fatigue of Metallic Materials ElsevierScience Amsterdam The Netherlands 2nd edition 1992
Advances in Materials Science and Engineering 11
[23] A M P De Jesus and J A F O Correia ldquoCritical assessmentof a local strain-based fatigue crack growth model usingexperimental data available for the P355NL1 steelrdquo Journal ofPressure Vessel Technology vol 135 no 1 Article ID 011404 9pages 2013
[24] J A F O Correia A M P De Jesus and A Fernandez-Canteli ldquoLocal unified probabilistic model for fatigue crackinitiation and propagation application to a notched geometryrdquoEngineering Structures vol 52 pp 394ndash407 2013
[25] ASTMmdashAmerican Society for Testing and Materials ldquoASTME606-92 standard practice for strain-controlled fatigue testingrdquoin Annual Book of ASTM Standards part 10 pp 557ndash571 1998
[26] W Ramberg and W R Osgood ldquoDescription of stress-straincurves by three parametersrdquo NACATechnical Note 902 NACA1943 httpntrsnasagovsearchjspR=19930081614
[27] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoInternal Friction Damping and Cyclic Plasticity (ASTM STP378) ASTM International West Conshohocken Pa USA 1965
[28] K N Smith P Watson and T H Topper ldquoA stress-strainfunction for the fatigue of metalrdquo Journal of Materials vol 5no 4 pp 767ndash778 1970
[29] M Szata and G Lesiuk ldquoAlgorithms for the estimation offatigue crack growth using energymethodrdquoArchives of Civil andMechanical Engineering vol 9 no 1 pp 119ndash134 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Advances in Materials Science and Engineering 9
0
02
04
06
08
1
0 200 400 600 800 1000 1200
Ref
f(mdash
)
R = 001
R = 05
R = 07
ΔK (Nmiddotmmminus32)
Figure 8 Relationship between effective stress ratio 119877eff andapplied stress intensity range Δ119870 (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Experimental testsProposed model prediction
No closure effects
Closure effects
Ref
favg
(KopK
max
)
R (KminKmax)
Figure 9 Effective stress ratio 119877eff as a function of stress ratio(P355NL1 steel)
0
02
04
06
08
1
0 0002 0004 0006 0008
Qua
ntita
tive p
aram
eter
U
R = 001R = 05R = 07
1Kmax
Figure 10 Influence of the stress ratio on 119880 versus 1119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
U(mdash
)
ΔKth0Kmax (1Nmiddotmmminus32)
R = 001
R = 05
R = 07
Figure 11 Influence of the stress ratio on 119880 versus Δ119870th0119870maxrelationship (P355NL1 steel)
0
02
04
06
08
1
0 02 04 06 08 1
Elberrsquos modelSchijversquos modelASTM E647Newmanrsquos model
Savaidis modelHudak et alProposed model
No closure effects
Closure effectsK
opK
max
Stress R-ratio R
Figure 12 Relationship between 119877eff = 119870op119870max and stress ratiofor several models (P355NL1 steel)
Note that for applied stress ratios within 0 le 119877 lt 05 theinfluence of the crack closure and opening effects decreaseswith increasing of the crack length or stress intensity factorrange
The results of the proposed theoreticalmodel proved to beconsistentwhen comparedwith othermodels proposed in theliterature [1ndash12 19] It was possible to estimate the constants119898and 119862 of the modified Paris law (119889119886119889119873 versus Δ119870eff ) Theseparameters are important for fatigue life prediction studies ofstructural details based on fracture mechanics
Other fatigue programs under constant and variableamplitude loading with othermaterials should be considered
10 Advances in Materials Science and Engineering
100 500 15001000
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
R = 05
R = 07
ΔKeff (Nmiddotmmminus15)
da
dN= 39897Eminus12(ΔKeff)
26105
Figure 13 Correlation of fatigue crack growth rates using the crackclosure analysis (P355NL1 steel)
through the use of full field optical techniques to experimen-tally estimate119870op in order to confirm the results obtained inthe inverse analysis of the proposed theoretical model
Competing Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge the Portuguese Science Founda-tion (FCT) for the financial support through the postdoc-toral Grant SFRHBPD1078252015 The authors gratefullyacknowledge the funding of SciTech Science and Technologyfor Competitive and Sustainable Industries RampD projectcofinanced by Programa Operacional Regional do Norte(NORTE2020) throughFundoEuropeu deDesenvolvimentoRegional (FEDER)
References
[1] W Elber ldquoFatigue crack closure under cyclic tensionrdquo Engineer-ing Fracture Mechanics vol 2 no 1 pp 37ndash45 1970
[2] W Elber ldquoThe significance of fatigue crack closure DamageTolerance in Aircraft Structuresrdquo ASTM STP 486 AmericaSociety for Testing and Materials 1971
[3] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984
[4] M Vormwald and T Seeger ldquoThe consequences of short crackclosure on fatigue crack growth under variable amplitudeloadingrdquo Fatigue and Fracture of Engineering Materials andStructures vol 14 no 2-3 pp 205ndash225 1991
[5] M Vormwald ldquoEffect of cyclic plastic strain on fatigue crackgrowthrdquo International Journal of Fatigue vol 82 part 1 pp 80ndash88 2016
[6] G Savaidis M Dankert and T Seeger ldquoAnalytical procedurefor predicting opening loads of cracks at notchesrdquo Fatigue andFracture of Engineering Materials and Structures vol 18 no 4pp 425ndash442 1995
[7] G Savaidis A Savaidis P Zerres and M Vormwald ldquoModeI fatigue crack growth at notches considering crack closurerdquoInternational Journal of Fatigue vol 32 no 10 pp 1543ndash15582010
[8] S J Hudak and D L Davidson ldquoThe dependence of crackclosure on fatigue loading variablesrdquo in Mechanics of ClosureASTM STP 982 J C Newman Jr and W Elber Eds pp 121ndash138America Society for Testing andMaterials Philadelphia PaUSA 1988
[9] J Schijve Fatigue of Structures andMaterials Kluwer AcademicPublishers Burlington Mass USA 2004
[10] American Society for Testing and Materials (ASTM) ldquoASTME647 standard test method for measurement of fatigue crackgrowth ratesrdquo inAnnual Book of ASTMStandards vol 0301 pp591ndash630 American Society for Testing and Materials (ASTM)West Conshohocken Pa USA 1999
[11] N Geerlofs J Zuidema and J Sietsma ldquoOn the relationshipbetween microstructure and fracture toughness in trip-assistedmultiphase steelsrdquo in Proceedings of the 15th European Con-ference of Fracture (ECF15 rsquo04) Stockholm Sweden August2004 httpwwwgruppofratturaitocsindexphpesisECF15paperview8758
[12] F Ellyin Fatigue Damage Crack Growth and Life PredictionChapman amp Hall New York NY USA 1997
[13] R C McClung and H Sehitoglu ldquoCharacterization of fatiguecrack growth in intermediate and large scale yieldingrdquo Journalof Engineering Materials and Technology vol 113 no 1 pp 15ndash22 1991
[14] M Nakagaki and S N Atluri ldquoFatigue crack closure and delayeffects under mode I spectrum loading an efficient elastic-plastic analysis procedurerdquo Fatigue of EngineeringMaterials andStructures vol 1 no 4 pp 421ndash429 1979
[15] N E DowlingMechanical Behaviour ofMaterialsmdashEngineeringMethods for Deformation Fracture and Fatigue Prentice-HallUpper Saddle River NJ USA 2nd edition 1998
[16] H F S G Pereira AM P De Jesus A S Ribeiro andA A Fer-nandes ldquoFatigue damage behavior of a structural componentmade of P355NL1 steel under block loadingrdquo Journal of PressureVessel Technology vol 131 no 2 Article ID021407 9 pages 2009
[17] O N Romaniv G N Nikiforchin and B N Andrusiv ldquoEffectof crack closure and evaluation of the cyclic crack resistance ofconstructional alloysrdquo Soviet Materials Science vol 19 no 3 pp212ndash225 1983
[18] J R LloydTheeffect of residual stress and crack closure on fatiguecrack growth [PhD thesis] Department of Materials Engineer-ing University of Wollongong Wollongong Australia 1999httprouoweduautheses1525
[19] H Neuber ldquoTheory of stress concentration for shear-strainedprismatical bodies with arbitrary nonlinear stress-strain lawrdquoJournal of Applied Mechanic vol 28 pp 544ndash550 1961
[20] P C Paris and F Erdogan ldquoA critical analysis of crack propaga-tion lawsrdquo Journal of Basic Engineering vol 85 no 4 pp 528ndash534 1963
[21] S M Beden S Abdullah and A K Ariffin ldquoReview of fatiguecrack propagation models for metallic componentsrdquo EuropeanJournal of Scientific Research vol 28 no 3 pp 364ndash397 2009
[22] M Klesnil and P Lukas Fatigue of Metallic Materials ElsevierScience Amsterdam The Netherlands 2nd edition 1992
Advances in Materials Science and Engineering 11
[23] A M P De Jesus and J A F O Correia ldquoCritical assessmentof a local strain-based fatigue crack growth model usingexperimental data available for the P355NL1 steelrdquo Journal ofPressure Vessel Technology vol 135 no 1 Article ID 011404 9pages 2013
[24] J A F O Correia A M P De Jesus and A Fernandez-Canteli ldquoLocal unified probabilistic model for fatigue crackinitiation and propagation application to a notched geometryrdquoEngineering Structures vol 52 pp 394ndash407 2013
[25] ASTMmdashAmerican Society for Testing and Materials ldquoASTME606-92 standard practice for strain-controlled fatigue testingrdquoin Annual Book of ASTM Standards part 10 pp 557ndash571 1998
[26] W Ramberg and W R Osgood ldquoDescription of stress-straincurves by three parametersrdquo NACATechnical Note 902 NACA1943 httpntrsnasagovsearchjspR=19930081614
[27] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoInternal Friction Damping and Cyclic Plasticity (ASTM STP378) ASTM International West Conshohocken Pa USA 1965
[28] K N Smith P Watson and T H Topper ldquoA stress-strainfunction for the fatigue of metalrdquo Journal of Materials vol 5no 4 pp 767ndash778 1970
[29] M Szata and G Lesiuk ldquoAlgorithms for the estimation offatigue crack growth using energymethodrdquoArchives of Civil andMechanical Engineering vol 9 no 1 pp 119ndash134 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
10 Advances in Materials Science and Engineering
100 500 15001000
10E minus 5
10E minus 6
10E minus 7
10E minus 8
10E minus 9
10E minus 2
10E minus 3
10E minus 4
dadN
(mm
cyc
le)
R = 00
R = 05
R = 07
ΔKeff (Nmiddotmmminus15)
da
dN= 39897Eminus12(ΔKeff)
26105
Figure 13 Correlation of fatigue crack growth rates using the crackclosure analysis (P355NL1 steel)
through the use of full field optical techniques to experimen-tally estimate119870op in order to confirm the results obtained inthe inverse analysis of the proposed theoretical model
Competing Interests
The authors declare that they have no competing interestsregarding the publication of this paper
Acknowledgments
The authors acknowledge the Portuguese Science Founda-tion (FCT) for the financial support through the postdoc-toral Grant SFRHBPD1078252015 The authors gratefullyacknowledge the funding of SciTech Science and Technologyfor Competitive and Sustainable Industries RampD projectcofinanced by Programa Operacional Regional do Norte(NORTE2020) throughFundoEuropeu deDesenvolvimentoRegional (FEDER)
References
[1] W Elber ldquoFatigue crack closure under cyclic tensionrdquo Engineer-ing Fracture Mechanics vol 2 no 1 pp 37ndash45 1970
[2] W Elber ldquoThe significance of fatigue crack closure DamageTolerance in Aircraft Structuresrdquo ASTM STP 486 AmericaSociety for Testing and Materials 1971
[3] J C Newman Jr ldquoA crack opening stress equation for fatiguecrack growthrdquo International Journal of Fracture vol 24 no 4pp R131ndashR135 1984
[4] M Vormwald and T Seeger ldquoThe consequences of short crackclosure on fatigue crack growth under variable amplitudeloadingrdquo Fatigue and Fracture of Engineering Materials andStructures vol 14 no 2-3 pp 205ndash225 1991
[5] M Vormwald ldquoEffect of cyclic plastic strain on fatigue crackgrowthrdquo International Journal of Fatigue vol 82 part 1 pp 80ndash88 2016
[6] G Savaidis M Dankert and T Seeger ldquoAnalytical procedurefor predicting opening loads of cracks at notchesrdquo Fatigue andFracture of Engineering Materials and Structures vol 18 no 4pp 425ndash442 1995
[7] G Savaidis A Savaidis P Zerres and M Vormwald ldquoModeI fatigue crack growth at notches considering crack closurerdquoInternational Journal of Fatigue vol 32 no 10 pp 1543ndash15582010
[8] S J Hudak and D L Davidson ldquoThe dependence of crackclosure on fatigue loading variablesrdquo in Mechanics of ClosureASTM STP 982 J C Newman Jr and W Elber Eds pp 121ndash138America Society for Testing andMaterials Philadelphia PaUSA 1988
[9] J Schijve Fatigue of Structures andMaterials Kluwer AcademicPublishers Burlington Mass USA 2004
[10] American Society for Testing and Materials (ASTM) ldquoASTME647 standard test method for measurement of fatigue crackgrowth ratesrdquo inAnnual Book of ASTMStandards vol 0301 pp591ndash630 American Society for Testing and Materials (ASTM)West Conshohocken Pa USA 1999
[11] N Geerlofs J Zuidema and J Sietsma ldquoOn the relationshipbetween microstructure and fracture toughness in trip-assistedmultiphase steelsrdquo in Proceedings of the 15th European Con-ference of Fracture (ECF15 rsquo04) Stockholm Sweden August2004 httpwwwgruppofratturaitocsindexphpesisECF15paperview8758
[12] F Ellyin Fatigue Damage Crack Growth and Life PredictionChapman amp Hall New York NY USA 1997
[13] R C McClung and H Sehitoglu ldquoCharacterization of fatiguecrack growth in intermediate and large scale yieldingrdquo Journalof Engineering Materials and Technology vol 113 no 1 pp 15ndash22 1991
[14] M Nakagaki and S N Atluri ldquoFatigue crack closure and delayeffects under mode I spectrum loading an efficient elastic-plastic analysis procedurerdquo Fatigue of EngineeringMaterials andStructures vol 1 no 4 pp 421ndash429 1979
[15] N E DowlingMechanical Behaviour ofMaterialsmdashEngineeringMethods for Deformation Fracture and Fatigue Prentice-HallUpper Saddle River NJ USA 2nd edition 1998
[16] H F S G Pereira AM P De Jesus A S Ribeiro andA A Fer-nandes ldquoFatigue damage behavior of a structural componentmade of P355NL1 steel under block loadingrdquo Journal of PressureVessel Technology vol 131 no 2 Article ID021407 9 pages 2009
[17] O N Romaniv G N Nikiforchin and B N Andrusiv ldquoEffectof crack closure and evaluation of the cyclic crack resistance ofconstructional alloysrdquo Soviet Materials Science vol 19 no 3 pp212ndash225 1983
[18] J R LloydTheeffect of residual stress and crack closure on fatiguecrack growth [PhD thesis] Department of Materials Engineer-ing University of Wollongong Wollongong Australia 1999httprouoweduautheses1525
[19] H Neuber ldquoTheory of stress concentration for shear-strainedprismatical bodies with arbitrary nonlinear stress-strain lawrdquoJournal of Applied Mechanic vol 28 pp 544ndash550 1961
[20] P C Paris and F Erdogan ldquoA critical analysis of crack propaga-tion lawsrdquo Journal of Basic Engineering vol 85 no 4 pp 528ndash534 1963
[21] S M Beden S Abdullah and A K Ariffin ldquoReview of fatiguecrack propagation models for metallic componentsrdquo EuropeanJournal of Scientific Research vol 28 no 3 pp 364ndash397 2009
[22] M Klesnil and P Lukas Fatigue of Metallic Materials ElsevierScience Amsterdam The Netherlands 2nd edition 1992
Advances in Materials Science and Engineering 11
[23] A M P De Jesus and J A F O Correia ldquoCritical assessmentof a local strain-based fatigue crack growth model usingexperimental data available for the P355NL1 steelrdquo Journal ofPressure Vessel Technology vol 135 no 1 Article ID 011404 9pages 2013
[24] J A F O Correia A M P De Jesus and A Fernandez-Canteli ldquoLocal unified probabilistic model for fatigue crackinitiation and propagation application to a notched geometryrdquoEngineering Structures vol 52 pp 394ndash407 2013
[25] ASTMmdashAmerican Society for Testing and Materials ldquoASTME606-92 standard practice for strain-controlled fatigue testingrdquoin Annual Book of ASTM Standards part 10 pp 557ndash571 1998
[26] W Ramberg and W R Osgood ldquoDescription of stress-straincurves by three parametersrdquo NACATechnical Note 902 NACA1943 httpntrsnasagovsearchjspR=19930081614
[27] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoInternal Friction Damping and Cyclic Plasticity (ASTM STP378) ASTM International West Conshohocken Pa USA 1965
[28] K N Smith P Watson and T H Topper ldquoA stress-strainfunction for the fatigue of metalrdquo Journal of Materials vol 5no 4 pp 767ndash778 1970
[29] M Szata and G Lesiuk ldquoAlgorithms for the estimation offatigue crack growth using energymethodrdquoArchives of Civil andMechanical Engineering vol 9 no 1 pp 119ndash134 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Advances in Materials Science and Engineering 11
[23] A M P De Jesus and J A F O Correia ldquoCritical assessmentof a local strain-based fatigue crack growth model usingexperimental data available for the P355NL1 steelrdquo Journal ofPressure Vessel Technology vol 135 no 1 Article ID 011404 9pages 2013
[24] J A F O Correia A M P De Jesus and A Fernandez-Canteli ldquoLocal unified probabilistic model for fatigue crackinitiation and propagation application to a notched geometryrdquoEngineering Structures vol 52 pp 394ndash407 2013
[25] ASTMmdashAmerican Society for Testing and Materials ldquoASTME606-92 standard practice for strain-controlled fatigue testingrdquoin Annual Book of ASTM Standards part 10 pp 557ndash571 1998
[26] W Ramberg and W R Osgood ldquoDescription of stress-straincurves by three parametersrdquo NACATechnical Note 902 NACA1943 httpntrsnasagovsearchjspR=19930081614
[27] J DMorrow ldquoCyclic plastic strain energy and fatigue ofmetalsrdquoInternal Friction Damping and Cyclic Plasticity (ASTM STP378) ASTM International West Conshohocken Pa USA 1965
[28] K N Smith P Watson and T H Topper ldquoA stress-strainfunction for the fatigue of metalrdquo Journal of Materials vol 5no 4 pp 767ndash778 1970
[29] M Szata and G Lesiuk ldquoAlgorithms for the estimation offatigue crack growth using energymethodrdquoArchives of Civil andMechanical Engineering vol 9 no 1 pp 119ndash134 2009
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials
Submit your manuscripts athttpwwwhindawicom
ScientificaHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CorrosionInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Polymer ScienceInternational Journal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CeramicsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CompositesJournal of
NanoparticlesJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
International Journal of
Biomaterials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
NanoscienceJournal of
TextilesHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Journal of
NanotechnologyHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal of
CrystallographyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
CoatingsJournal of
Advances in
Materials Science and EngineeringHindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Smart Materials Research
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
MetallurgyJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
BioMed Research International
MaterialsJournal of
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Nano
materials
Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014
Journal ofNanomaterials