Research Article Crack Closure Effects on Fatigue Crack...

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Research Article Crack Closure Effects on Fatigue Crack Propagation Rates: Application of a Proposed Theoretical Model José A. F. O. Correia, 1 Abílio M. P. De Jesus, 1,2 Pedro M. G. P. Moreira, 1 and Paulo J. S. Tavares 1 1 Institute of Science and Innovation in Mechanical and Industrial Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal 2 Department of Mechanical Engineering, Faculty of Engineering, University of Porto, Rua Dr. Roberto Frias, 4200-465 Porto, Portugal Correspondence should be addressed to Jos´ e A. F. O. Correia; [email protected] Received 10 December 2015; Accepted 18 February 2016 Academic Editor: Antonio Riveiro Copyright © 2016 Jos´ e A. F. O. Correia et al. is is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Structural design taking into account fatigue damage requires a thorough knowledge of the behaviour of materials. In addition to the monotonic behaviour of the materials, it is also important to assess their cyclic response and fatigue crack propagation behaviour under constant and variable amplitude loading. Materials whenever subjected to fatigue cracking may exhibit mean stress effects as well as crack closure effects. In this paper, a theoretical model based on the same initial assumptions of the analytical models proposed by Hudak and Davidson and Ellyin is proposed to estimate the influence of the crack closure effects. is proposal based further on Walker’s propagation law was applied to the P355NL1 steel using an inverse analysis (back-extrapolation) of experimental fatigue crack propagation results. Based on this proposed model it is possible to estimate the crack opening stress intensity factor, op , the relationship between = Δ eff quantity and the stress intensity factor, the crack length, and the stress ratio. is allows the evaluation of the influence of the crack closure effects for different stress ratio levels, in the fatigue crack propagation rates. Finally, 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 closure and opening concepts. Initially Elber [1, 2] suggested that crack closure was associated with the effective stress intensity range, using (= Δ eff /Δ) parameter which was assumed as a linear function of stress ratio, , independent of max and Δ. Later studies determined that the quantitative parameter is dependent on max . Aſter the proposal developed by Elber [1, 2], other proposals appeared, based on elastoplastic analytical concepts [3–7] or the near-threshold fatigue crack growth regimes [8–12]. All approaches are important contri- butions of the crack closure and opening effects, which occur in materials when subjected to constant amplitude loading, particularly in the near-threshold regime of the fatigue crack growth. All analytical techniques were supported by mea- surements of the crack-tip opening load. Other approaches have been implemented based on elastoplastic analysis, using finite element methods [13, 14]. is paper proposes a theoretical model based on Walker’s propagation law [15] to estimate the quantitative parameter, , important for the evaluation of the fatigue crack opening stress intensity factor, op , taking into account the stress ratio and crack closure and opening effects. e cur- rent study is applied to the P355NL1 steel using experimental fatigue crack propagation test results under constant ampli- tude loading [16]. e proposed model is applied using an inverse analysis of the experimental results. Various analytical models for crack closure are also compared in this study. 2. Overview of the Crack Closure and Opening Concepts and Existing Models e crack closure and opening concept has been discussed in the scientific community since the first contributions by Elber [1, 2]. Elber suggested [1, 2] that crack closure and opening effects could be characterized in terms of the effective stress Hindawi Publishing Corporation Advances in Materials Science and Engineering Volume 2016, Article ID 3026745, 11 pages http://dx.doi.org/10.1155/2016/3026745

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Research ArticleCrack Closure Effects on Fatigue Crack Propagation Rates:Application of a Proposed Theoretical Model

Jos A. F. O. Correia,1 Ablio M. P. De Jesus,1,2

Pedro M. G. P. Moreira,1 and Paulo J. S. Tavares1

1 Institute of Science and Innovation in Mechanical and Industrial Engineering, University of Porto, Rua Dr. Roberto Frias,4200-465 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; [email protected]

Received 10 December 2015; Accepted 18 February 2016

Academic Editor: Antonio Riveiro

Copyright 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 onWalkers 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 factor,op, the relationship between = eff/ 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 rates.Finally, 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 (= eff/) parameter which was assumedas a linear function of stress ratio,, independent ofmax and. Later studies determined that the quantitative parameter is dependent on max. After the proposal developed byElber [1, 2], other proposals appeared, based on elastoplasticanalytical concepts [37] or the near-threshold fatigue crackgrowth regimes [812]. All approaches are important contri-butions of the crack closure and opening effects, which occurin materials when subjected to constant amplitude loading,particularly 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 onWalkers propagation law [15] to estimate the quantitativeparameter, , important for the evaluation of the fatiguecrack opening stress intensity factor,op, taking into accountthe stress ratio and crack closure and opening effects.The 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 pageshttp://dx.doi.org/10.1155/2016/3026745

2 Advances in Materials Science and Engineering

K

Time

Crack closed

Crack openedKmax

Kop

Kmin

K

Keff

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 ratio, with thefollowing form:

=eff

, (1)

where eff is the effective stress intensity factor rangeand is the applied stress intensity factor range. Elbersformulation is supported by the following assumptions forthe parameter: (i) linear function of stress ratio () and(ii) being relatively independent of maximum stress intensityfactor (max) or stress intensity factor range (). Theeffective stress intensity factor range is defined as

eff = max op, (2)

where max is the maximum stress intensity factor and opis the crack opening stress intensity factor. The applied stressintensity factor range, , is presented by

= max min (3)

withmin being the minimum stress intensity factor. Figure 1illustrates the previous parameters definitions.

According to the approach of Elber [1, 2], the fatiguecrack growth rate, /, is a function of the effective stressintensity factor range, according to the following generalexpression:

= (eff) . (4)

According to studies conducted by Elber for the 2024-T3aluminum alloy, using stress ratios, , between 0.1 and 0.7, depends on.The experimental results led to the followinglinear relationship:

= 0.5 + 0.4. (5)

The quantitative parameter of (5) can be substituted into(1) and using (2) results in the following relation for theop/max ratio:

op

max= 0.5 + 0.1 + 0.4

2. (6)

Subsequent studies using various fatigue loading variablesclearly showed that and max are related variables.Therefore, Elber [1, 2] approach appears to be inconsis-tent/incomplete.

An improved function of (5) was proposed by Schijve[9] with a more realistic behaviour to account for the crackclosure and opening effects for 1 1:

= 0.55 + 0.33 + 0.122. (7)

Therefore, it is possible to obtain a relation to the crackopening stress intensity factor as follows:

op

max= 0.45 + 0.22 + 0.21

2+ 0.12

3. (8)

Another proposal was presented by ASTM [10, 11] togetherwith a set of conditions:

= max for < 0,

= max min for 0,

= 0.576 + 0.015 + 0.4092,

op

max= 0.424 + 0.561 0.394

2+ 0.409

3.

(9)

Newman Jr. [3] proposed a general crack opening stressequation for constant amplitude loading, function of stressratio, , stress level, , and four nondimensional constants:

op

max= 0+ 1 +

22+ 33 for 0,

op

max= 0+ 1 for 1 < 0,

(10)

when op max. The coefficients 0, 1, 2, and 3 are asfollows:

0= (0.825 0.34 + 0.05

2) [cos(

max20

)]

1/

,

1= (0.415 0.071)

max0

,

2= 1

0 1 3,

3= 20+ 1 1,

(11)

where 0is the flow stress (average between the uniaxial yield

stress and uniaxial ultimate tensile strength of the material).For plane stress conditions, = 1, and for plane strainconditions, = 3. The normalized crack opening stress isobtained using the stress ratio for max = 0/3.

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 [17]. In this fatigueregime, the crack closure effects have the greatest influence.Consequently, Hudak and Davidson [8] proposed a modeldefined by the following expression:

= 1

max= 1

(1 )

, (12)

whereis a constant related to the pureMode I fatigue crack

growth threshold.The experimental results from fatigue crack propagation

tests can be represented by the following generalized relations(see Figure 2):

= (1

max) , max , (13)

= 1, max , (14)

where

is the limiting max value above which nodetectable closure occurs and is a parameter that can beobtained using the following expression:

= (1

) . (15)

The

and

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-sure/opening effects. Note that in the limiting condition of/approaching zero (13) reduces to (12) thus describing

the local measurements.Ellyin [12] proposed an approach to define the effective

stress intensity range, eff , taking into account the stressratio, , and the threshold value of the stress intensity factorrange, th, with constant amplitude loading (see Figure 3).This approach is amodified version of proposal byHudak andDavidson [8]. Generally, the crack opening stress intensityfactor range, op, is smaller than the threshold stressintensity factor range, th; that is, op < th. Besides,the crack opening or closure stress intensities factors are notthe same; that is, cl < op < th. Ellyin [12] proposedthe following approach to obtain eff (see Figure 3):

eff ,0 = (2 2

th)1/2

[1 1

2(th

)

2

]

> th for 0,

1

00

1

U = Keff/K

U = 1

U = (1 Kmax)

Ko

1/KL 1/Ko

Ko

1/Kmax

Figure 2: Functional relationship between and max, Hudak andDavidson [8].

K

Time

K

Kth Kcl Kop

Keff

Keff = [(K)2 (Kth)2]1/2

Kmax

Kop

Kth

Kmin

Figure 3: Definition of the effective stress intensity range accordingwith Ellyin [12].

eff =eff ,0

[1 (/

)]

=eff ,0

[1 ((1 + ) max/2

)]

for = 0,

(16)

where is the mean stress, max is the maximum stress, and

is the fatigue strength coefficient.The 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 op 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 studied,these two sets of extrapolated values for op and cl werefound to correlate excellently.

3. Proposed Theoretical Model

In this paper, a theoretical model to obtain the effective stressintensity range, eff , 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:

= ()

, (17)

where and 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 literature,with 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]:

= (

(1 )1

)

= ()

, (18)

where , , and are constants. Besides, an extension of

the Paris relation is adopted to account for crack propagationregime I, having the following configuration:

= ( th)

. (19)

Based on the foregoing assumptions, the effective stressintensity factor range is given by the following expression:

eff = ( th) = (

(1 )1

th)

= (1

(1 )1

th

) .

(20)

1

00

1

U = 1

1/Kmax1/KL

U = Keff/K

(1 R)1

(1 R)1 Kth,0

U = (1 Kth,0Kmax)(1 R)1

1/Kth,0

Figure 4: Functional relationship between andmax for proposedtheoretical model.

Thus, the expression that allows obtaining the quantitativeparameter, , is given by the following formula:

=eff

= (1

(1 )1

th

)

= (1 )1

th

.

(21)

If the material parameter, , 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:

= (1 )1

th

(1 )max. (22)

The threshold value of the stress intensity factor range can bedescribed by a general equation, using the following form:

th = th,0 () , (23)

where th is the threshold value of stress intensity range fora given stress ratio and th,0 is the threshold value of stressintensity range, for = 0. Klesnil and Lukas [22] proposeda relation between the threshold stress intensity factor range,th, and the stress ratio, , which is well known:

th = th,0 (1 ). (24)

The validity of (24) is for 0. Replacing this result in (22)results in

= (1 )1

th,0 (1 )

(1 )max. (25)

Finally, it is possible to display (22) in a more simplified form(see Figure 4):

= (1 th,0

max) (1 )

1 for max ,

= 1 for max ,(26)

Advances in Materials Science and Engineering 5

Table 1: Cyclic elastoplastic and strain-life properties of the P355NL1 steel, = 1 and

= 0.

Material

MPa MPa P355NL1 1005.50 0.1033 0.3678 0.5475 948.35 0.1533

where is the limiting max. For stress ratio, , equal to 0

and material parameter, , equal to 1, (22) reduces to

= 1 th,0

max, for = 0, = 1. (27)

In summary, a theoretical model based on Walkers propa-gation law [15] was proposed, which relates the quantitativeparameter and (or max) 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].

4.1. Strain-Life Behaviour. The strain-life behaviour of theP355NL1 steel was evaluated through fatigue tests of smoothspecimens, carried out under strain controlled conditions,according to theASTME606 standard [25].TheRamberg andOsgood [26] relation, as shown in the following equation, wasfitted to the stabilized cyclic stress-strain data:

2=

2+

2=

2+ (

2)

1/

, (28)

where and are the cyclic strain hardening coefficient andexponent, respectively; is plastic strain range; and isthe stress range.

Low-cycle fatigue test results are very often representedusing the relation between the strain amplitude and thenumber of reversals to failure, 2

, 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:

2=

2+

2=

(2)

+

(2)

, (29)

where and are, respectively, the fatigue ductility coeffi-

cient and fatigue ductility exponent; is the fatigue strength

coefficient, is the fatigue strength exponent, and is theYoung modulus.

Table 2: Elastic and tensile properties of the P355NL1.

Material ] GPa MPa MPa

P355NL1 205.20 0.275 568.11 418.06

Alternatively to the Morrow relation, the Smith-Watson-Topper fatigue damage parameter [28] can be used, whichshows the following form:

max

2= SWT

=

(

)2

(2)2

+

(2)+

,

(30)

where max 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,

= 0 (19 specimens) and 1 (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 1,for 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 series.This 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, , Poisson ratio, ], the ultimate tensile strength,

,

and upper yield stress, .

4.2. Fatigue Crack Propagation Rates. Fatigue crack growthrates of the P355NL1 pressure vessel steel were evaluated, forseveral stress ratios, using compact tension (CT) specimens,following the recommendations of the ASTM E647 standard[10]. The CT specimens of P355NL1 steel were manufacturedwith a width = 40mm and a thickness = 4.5mm [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 0.5 1 1.5 2

Cyclic curve

(MPa

)

/2 (%)

R = 0

R = 1

(a)

R = 0

R = 1

1.0E 01

1.0E 02

1.0E 03

1.0E 04

/

2(

)

1.0E + 02 1.0E + 03 1.0E + 04 1.0E + 05 1.0E + 06 1.0E + 07

Reversals to failure, 2Nf

2=

1005.5

E(2Nf)0.1013 + 0.3678(2Nf)0.5475

(b)

0.1

1.0

10.0

SWT

(MPa

)

1.0E + 02 1.0E + 03 1.0E + 04 1.0E + 05 1.0E + 06 1.0E + 07

Reversals to failure, 2NfR = 0

R = 1

(c)

Figure 5: Fatigue data obtained for the P355NL1 steel, = 1 and

= 0: (a) cyclic curve; (b) strain-life data; (c) SWT-life data.

by visual inspection, using two travelling microscopes withan accuracy of 0.001mm.

Figures 6(a) and 6(b) represent the experimental dataand the Paris law correlations for each tested stress ratio ofthe P355NL1 steel, = 0.0, = 0.5, and = 0.7. 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 rates.The lines representing theParis law, for = 0.0 and = 0.5, are approximately parallelto each other.

On the other hand, the line representing the Paris law for = 0.7 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.

max

Nmm1.5

0.01 7.1945 15 3.4993 14980.50 6.2806 15 3.5548 10130.70 2.0370 13 3.0031 687/ in mm/cycle and in Nmm1.5.

with the reduction in the stress ratio leading to lower fatiguecrack growth rates. For higher stress ratios, such as =0.7, the crack closure is very likely negligible, which meansthat applied stress intensity factor range is fully effective.Also, 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

1.0E 5

1.0E 6

1.0E 2

1.0E 3

1.0E 4

MB06 (R = 0.7)MB05 (R = 0.5)

MB03 (R = 0.5)MB04 (R = 0.0)MB02 (R = 0.0)

da/dN

(mm

/cyc

le)

K (Nmm1.5)

(a)

250 1000 1500500

1.0E 5

1.0E 6

1.0E 2

1.0E 3

1.0E 4

da/dN

(mm

/cyc

le)

K (Nmm1.5)

R = 0

R = 0.5

R = 0.7

R2 = 0.9850

R2 = 0.9840

R2 = 0.9960

7.195E 15 K3.499da

dN=

= 2.037E 13 K3.003da

dN=

6.281E 15 K3.555da

dN=

(b)

Figure 6: Fatigue crack propagation data obtained for the P355NL1 steel: (a) experimental data; (b) Paris correlations for each stress ratio, .

Table 4:Walker constants for the fatigue crack propagation relationof the P355NL1 steel.

7.1945 15 3.4993 0.92/ in mm/cycle and in Nmm1.5.

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:

th = 152 90.252 , (31)

where th is the propagation threshold defined inNmm1.5, is the stress ratio, and constant value 152corresponds to the propagation threshold for = 0 (inNmm1.5); the other constant (90.252) also shows thesame units of the propagation threshold. The propagationthreshold for = 0 is very consistent with the thresholdvalue for the mild steel with yield stress of 366MPa. Thecrack propagation threshold th is itself influenced by thestress ratio [29]. In [29], for a similar material to the P355NL1steel, the relationship betweenth and 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 ratio,eff , and the applied stress intensity factor range, app.eff 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, op, andthe maximum stress intensity factor, max. Figure 8 showsthat for stress ratios above 0.5 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 eff and ,using the inverse analysis of the proposed theoretical model.This figure helps to understand the influence of the crackclosure and opening effects, as a function of stress ratios. Theeffective stress ratio, eff , shown in Figure 9, is the average ofthe values obtained as a function of , according to what isevidenced in Figure 8.

The proposed relationship between the quantitativeparameter, , and the maximum stress intensity, max, isalso considered. Figure 10 shows the relationship between and max for the studied stress ratios, 0, 0.5, and 0.7. Therelationship between and (th,0/max) 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 = 0.0)MB04 (R = 0.0)

Extended Paris lawthreshold regionParis lawParis region

1.0E 5

1.0E 6

1.0E 7

1.0E 8

1.0E 9

1.0E 2

1.0E 3

1.0E 4

da/dN

(mm

/cyc

le)

R = 0.0

K (Nmm1.5)

7.1945E 15 (K)3.4993dadN

=

6.65E13 (K Kth)3.1250dadN =

(a) = 0

MB03 (R = 0.5)MB05 (R = 0.5)

Extended Paris lawthreshold regionParis lawParis region

100 500 15001000

Thresholdregion Paris region

1.0E 5

1.0E 6

1.0E 7

1.0E 8

1.0E 9

1.0E 2

1.0E 3

1.0E 4

da/dN

(mm

/cyc

le)

R = 0.5

K (Nmm1.5)

7.1945E 15 (K)3.4993dadN

=

3.15E13 (K Kth)3.1150dadN =

(b) = 0.5

MB06 (R = 0.7)Extended Paris lawthreshold regionParis lawParis region

100 500 15001000

Thresholdregion Paris region

1.0E 5

1.0E 6

1.0E 7

1.0E 8

1.0E 9

1.0E 2

1.0E 3

1.0E 4

da/dN

(mm

/cyc

le)

R = 0.7

K (Nmm1.5)

7.1945E 15 (K)3.4993dadN

=

2.45E13 (K Kth)3.1150dadN =

(c) = 0.7

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 betweeneff and 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 model,are 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, eff , andthe applied stress ratios, , 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 study,it was observed that, for high applied stress ratios, , 0.5 0.7, the crack closure effect has little significance.But for applied stress ratios, , 0 < 0.5, there is asignificant influence of the crack closure and opening effects.

Advances in Materials Science and Engineering 9

0

0.2

0.4

0.6

0.8

1

0 200 400 600 800 1000 1200

Ref

f(

)

R = 0.01

R = 0.5

R = 0.7

K (Nmm3/2)

Figure 8: Relationship between effective stress ratio, eff , andapplied stress intensity range, (P355NL1 steel).

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Experimental testsProposed model prediction

No closure effects

Closure effects

Ref

f,avg

(Kop/K

max

)

R (Kmin/Kmax)

Figure 9: Effective stress ratio, eff , as a function of stress ratio(P355NL1 steel).

0

0.2

0.4

0.6

0.8

1

0 0.002 0.004 0.006 0.008

Qua

ntita

tive p

aram

eter

, U

R = 0.01R = 0.5R = 0.7

1/Kmax

Figure 10: Influence of the stress ratio on versus 1/maxrelationship (P355NL1 steel).

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

U(

)

Kth,0/Kmax (1/Nmm3/2)

R = 0.01

R = 0.5

R = 0.7

Figure 11: Influence of the stress ratio on versus th,0/maxrelationship (P355NL1 steel).

0

0.2

0.4

0.6

0.8

1

0 0.2 0.4 0.6 0.8 1

Elbers modelSchijves modelASTM E647Newmans model

Savaidis modelHudak et al.Proposed model

No closure effects

Closure effectsK

op/K

max

Stress R-ratio, R

Figure 12: Relationship between eff = op/max and stress ratiofor several models (P355NL1 steel).

Note that, for applied stress ratios within 0 < 0.5, 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 [112, 19]. It was possible to estimate the constantsand of the modified Paris law (/ versus eff ). 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

1.0E 5

1.0E 6

1.0E 7

1.0E 8

1.0E 9

1.0E 2

1.0E 3

1.0E 4

da/dN

(mm

/cyc

le)

R = 0.0

R = 0.5

R = 0.7

Keff (Nmm1.5)

da

dN= 3.9897E12(Keff)

2.6105

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 estimateop, 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 SFRH/BPD/107825/2015. The authors gratefullyacknowledge the funding of SciTech: Science and Technologyfor Competitive and Sustainable Industries, R&D projectcofinanced by Programa Operacional Regional do Norte(NORTE2020), throughFundoEuropeu deDesenvolvimentoRegional (FEDER).

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