Engineering Fracture Mechanics 75 (2008) 4439–4449

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Engineering Fracture Mechanics

journal homepage: www.elsevier .com/locate /engfracmech

A statistical model for crack growth based on tensionand compression Wöhler fields

Enrique Castillo a,*, Alfonso Fernández-Canteli b, Hernán Pinto a, María Luisa Ruiz-Ripoll a

a Department of Applied Mathematics and Computational Sciences, University of Cantabria, Spainb Department of Construction and Manufacturing Engineering, University of Oviedo, Spain

a r t i c l e i n f o

Article history:Received 30 August 2007Received in revised form 9 March 2008Accepted 29 April 2008Available online 7 May 2008

Keywords:Fatigue modelingFatigue designWöhler fieldStress levelCompatibilityCrack growth

0013-7944/$ - see front matter � 2008 Elsevier Ltddoi:10.1016/j.engfracmech.2008.04.011

* Corresponding author.E-mail addresses: castie@unican.es (E. Castillo), af

Ruiz-Ripoll).

a b s t r a c t

First, the general form of a physically valid crack growth model is derived based on func-tional equations. It results that only a single argument function is required to define themodel, and that models not satisfying this condition are incompatible. Second, a statisticalcrack growth model valid for any combination of rmin, rmax is presented. The model isbased on an existing fatigue model based on physical, statistical and compatibility condi-tions, which predicts the Wöhler fields for any constant load test. It is shown how standardfatigue tests combined with one single crack growth test, can be used to derive a generalformula for fatigue growth. This model is applied to some real data to illustrate its appli-cability to practical problems, and the results seem to be very promising.

� 2008 Elsevier Ltd. All rights reserved.

1. Introduction and motivation

Due to the successive improvement in mechanical design, manufacturer concurrence and materials quality, fatigue is of-ten becoming the determining criterion in mechanical and structural integrity design. Three major approaches are availablefor estimating fatigue lives of mechanical and structural components (see [1–3] and Fig. 1): the stress-based, the strain-based and the fracture mechanics-based approaches. Each one of them has specific advantages and limitations in practicalapplications, which justifies the co-existence of the three approaches depending on the particular area of practical applica-tion. The stress-based method is generally preferred in life estimation in the structural domain due to its simplicity and itsuseful applications to damage accumulation analysis based on the Miner approach. Where local yielding is present, as is thecase of mechanical components with notches, the strain-based method is applied, as a more complicated but more generalmethod than the stress-based approach. Finally, due to the phenomenological nature of the former two approaches, sometouch of scientific superiority is generally conferred to the fracture mechanics-based approach. This is, for instance, the casein analyzing damage tolerance, as a typical case of aeronautical design. Despite its importance, the fracture mechanics ap-proach should not be the unique source to fatigue analysis. In fact, since all the three approaches are different ways to facethe same problem, there must be close connections among them, so that any advances in one particular approach could pro-duce advances in the other two. Considerable efforts have been applied to the probabilistic fatigue life analysis using thestress-based method and significative advances have been achieved in the past (see [4–7]). This predominantly phenome-nological approach utilizes S–N curves together with a damage accumulation model, as the Palmgren–Miner rule, for assess-ing cumulative damage under varying load. Though this method could be considered as obsolete when compared to the

. All rights reserved.

c@uniovi.es (A. Fernández-Canteli), hernan.pinto@unican.es (H. Pinto), ripollmluisa@hotmail.com (M.L.

Nomenclature

a0 random maximum effective crack size present in a pieceac the crack size producing failure in the pieceA C0 þ C1r�m þ C2r�MB C5r�m þ C6r�MCi parameters of the modelCi maximum likelihood parameter estimatesf0ða0Þ probability density function (pdf) of a0

Fa0=ac ða0=acÞ cumulative distribution function of a0=ac

hða0=ac;N=N0Þ function giving the new crack size ratio a=ac in terms of the initial crack size ratio a0=ac and the lifetimeN=N0

I0 set of run-outsI1 set of non-run-outsK /�1ðac=a0ÞL likelihoodN number of cycles to failure (lifetime)N0 reference number of cyclesN� dimensionless number of cycles (N� ¼ N=N0)p probability of failure (p 2 ½0;1�)R stress ratio R ¼ rmin=rmax

s K � N=N0a /�1ða0=acÞDr stress amplitude ðDr ¼ rmax � rminÞDrm0 asymptotic value of Dr for constant rminDrM0 asymptotic value of Dr for constant rmax

c the Euler–Mascheroni constant ðc ¼ 0:57772Þrmin minimum stressrmax maximum stressrmean mean stressrm minimum stressrM maximum stressr�m dimensionless minimum stress ðr�m ¼ rm=r0Þr�M dimensionless maximum stress ðr�M ¼ rM=r0Þr0 reference stress/ðÞ arbitrary invertible function

Fracture mechanics-based approach

Stress-based approach Strain-based approach

Fig. 1. Schematic representation of the common available approaches for fatigue live estimation.

4440 E. Castillo et al. / Engineering Fracture Mechanics 75 (2008) 4439–4449

fracture mechanics-based approach, this is not the case, because the statistical treatment inherent to this approach providesa valuable information on the micromechanical processes involved that help to clarify some of the unsolved questions in thecrack growth problem. In particular, it allows us: (a) to get a reliable estimate of the fatigue limit from the statistical S–Nfield model, that could be advantageously used as an alternative to the direct method used in fracture mechanics (see[8]); (b) to get indirect information about the crack growth process starting from the initial crack size (micro- and meso-sco-pic stages), by means of the quantile curves, which can be associated with the effective crack size of the S–N curves; (c) toperform a probabilistic cumulative damage calculation, thus enriching the conventional deterministic crack growth ap-proach using the da=dN vs. DK curves; and (d) to define analytical fatigue models, not based on empirical considerations,able to take into account the mean stress influence, as the most relevant parameter in fatigue life after the stress range.

Applying the stress-based method in conjunction with the fracture mechanics-method is a pending challenge that hasbeen already suggested by [9,10], from a micromechanical perspective, though without considering their statistical aspects.

On the other hand, fatigue design of structures subject to varying loading is not possible without some basic material fa-tigue characterization related to the stress range and stress level, as the main and secondary parameters, respectively. This

E. Castillo et al. / Engineering Fracture Mechanics 75 (2008) 4439–4449 4441

information allows us the evaluation of the damage caused by the loading cycles associated with any given load spectrum(see [11–13] or [14]). Generally, this important information is obtained from tests conducted at constant stress level, eitherrmax;rmin, rmean or R ¼ rmin=rmax, though those Wöhler curves obtained from tests run at rmean ¼ 0, i.e. for completelyreversed stress, are often preferred.

The work in [7,15] provides a statistical model permitting extrapolating these particular results to any combination ofðrmin;rmaxÞ pair. The authors of this paper, based on the compatibility conditions of the Wöhler field together with statisticaland physical conditions and solving a system of functional equations, proposed for the first time a general fatigue regressionmodel that includes the consideration of the mean effect, without the need of resorting to empirical relations. This modelallows us to evaluate the probabilities of failure for any ðrmin;rmaxÞ pair.

Since the fatigue failure is produced by crack growth, standard fatigue tests contain a lot of information about this inter-esting process, and even more important, about the statistical properties of the crack sizes. Unfortunately, this relation hasnot been recognized yet, and even in some cases, discussed.

In this paper, it is shown that there is a close connection between the crack growth curves and the information providedby the Wöhler fields. In particular, a probabilistic model for crack growth is proposed by establishing the relation betweenthe cdfs of the effective initial crack size and the number of cycles at failure from the S–N curves. The scatter of the fatiguelifetime is assumed to result from the variability of the initial crack size followed by a deterministic crack growth accordingto a unique crack growth function, denoted /, thus proving the equivalence and/or complementariness between the stress-based and the fracture mechanics-based approaches.

Herewith, a correspondence between the effective crack sizes and the quantile Wöhler curves is assumed so that by usingthe crack growth function, /, the initial effective cracks fail after application of a number of cycles in accordance with thefatigue life distribution experimentally obtained.

In addition, it is shown that the crack growth curves, giving the evolution of the crack size with the number of cycles, iscompletely determined as soon as the / function is known. This has important implications, because the crack growth func-tion cannot be chosen arbitrarily to be a valid crack growth function.

The paper is organized as follows. The basic general fatigue Weibull model and the Gumbel model in particular are brieflypresented in Section 2, including the required constraints, some properties and a couple of parameter estimation methods. InSection 3 the general structure of crack growth models is obtained using functional equations, and the particular model ispresented and derived from Wöhler fields. In Section 4 an example of application is used to illustrate the proposed methodsand models and to show their practical implications. Finally, Section 5 summarizes the conclusions.

2. The fatigue model

Castillo et al. [7] consider a fatigue test conducted at alternating constant minimum and maximum stresses denoted forsimplicity rm and rM, respectively, and determine the cumulative distribution function of the random lifetime N (number ofcycles to failure) associated with the test, as a Weibull or Gumbel family of models able to reproduce not only the wholeWöhler field, but any combination of minimum and maximum stresses. Though, for simplicity, in this paper we refer tothe simplest Gumbel model, the results can be easily extended to the general Weibull model.

Contrary to other empirical models (see [11,16,12,13]), the main advantage of the model derived in [7,15], is that it usesas less arbitrary assumptions as possible. A complete and detailed analysis is in [17–20,5,21,22].

In summary, they use the following conditions:

(1) The Buckingham P theorem to obtain the simplest possible and a physically valid model (written in terms of dimen-sionless variables).

(2) The weakest link principle.(3) Limited range N and Dr.(4) Compatibility conditions (of life and stress range).(5) Statistical conditions, such as stability and limit behavior.(6) Extreme value analysis properties.(7) Compatibility conditions of the model for different values of rm and rM.

and they obtain the following Gumbel model:

FðN�Þ ¼ 1� exp � exp C0 þ C1r�m þ C2r�M þ C3r�mr�M þ C4 þ C5r�m þ C6r�M þ C7r�mr�M� �

log N�� �� �

; ð1Þ

where F is the cdf of N�, r�m ¼ rm=r0, r�M ¼ rM=r0, N� ¼ N=N0, where r0 and N0 are some reference stress and number ofcycles, respectively, to make the formulas dimensionless, and C1 to C7 are dimensionless constants.

For convenience, in this paper we use the simplified model:

FðN�Þ ¼ 1� exp � exp C0 þ C1r�m þ C2r�M þ ðC5r�m þ C6r�MÞ log N�� �� �

; ð2Þ

corresponding to the particular case C3 ¼ C4 ¼ C7 ¼ 0.

Log N*

Δσ*

σmax= σmax1

σmin= σmin1

*

*

σmax = σmax2< σmax1

*

σmin = σmin2< σmin1

*

Fig. 2. Schematic Wöhler curves for percentiles {0.01,0.05,0.5,0.95,0.99} for r�max ¼ r�max1and r�max ¼ r�max2

, and r�min ¼ r�min1and r�min ¼ r�min2

, withr�min2

< r�min1and r�max2

< r�max1, illustrating the compatibility condition. Dashed lines refer to Wöhler curves for constant r�min, and continuous lines refer to

Wöhler curves for constant r�max.

4442 E. Castillo et al. / Engineering Fracture Mechanics 75 (2008) 4439–4449

The model (1) is a powerful model that allows us to interpolate or extrapolate fatigue results to any combination of rm

and rM, and guarantees the compatibility of all the resulting Wöhler fields for all these combinations, as illustrated in Fig. 2,where the Wöhler fields for four combinations of rm and rM are given and their compatibility becomes apparent throughtheir horizontal straight line intersections. In addition, it has the advantage of having one parameter less than the Weibullmodel, and even more important, that the range of definition for log N is ð�1;1Þ. This avoids deciding whether or not weare in the allowable region.

2.1. Constraints of the model

As indicated in [7], for the model to be physically and statistically valid its parameters must satisfy the followingconstraints:

(1) The asymptotic value Drm0 for N !1 must be non-negative, i.e.

Drm0 ¼ �rmðC5 þ C6Þ

C6P 0: ð3Þ

(2) The asymptotic value Drm0, for N !1 due to physical reasons, must be non-increasing in rm, that is

C6ðC6 þ C5ÞP 0: ð4Þ

(3) The asymptotic value DrM0 for N !1 must be non-negative, i.e.

DrM0 ¼rMðC5 þ C6Þ

C5P 0: ð5Þ

(4) The asymptotic value DrM0 for N !1 must be non-increasing in rM:

C5ðC5 þ C6Þ 6 0: ð6Þ

(5) The cdf in (2) must be non-decreasing in log N:

C5rm þ C6rM > 0; rm0 6 rm 6 rM 6 rM0; ð7Þ

where rm0 and rM0 are stress bounds defining the region where the model is to be used. Eq. (7) implies

C5rm0 þ C6rM0 > 0; C5rm0 þ C6rm0 > 0; C5rM0 þ C6rM0 > 0: ð8Þ

(6) The cdf in (2) must be non-increasing in rm:

C1 þ C5 log N 6 0; N0 6 N; rm0 6 rM 6 rM0; ð9Þ

(7) The cdf in (2) must be non-decreasing in rM:

C2 þ C6 log N P 0; N0 6 N; rm0 6 rm 6 rM0; ð10Þ

E. Castillo et al. / Engineering Fracture Mechanics 75 (2008) 4439–4449 4443

(8) The curvature of the zero-percentile of ðlog N;DrÞ for constant rmin must be non-negative, that is, o2Droðlog NÞ2

P 0, whichleads to

C6 rmðC1C6Þ � C3C5r2m � C2C5rm

� �þ ðC0 � ð� logðlogð1� pÞÞÞÞC2

6 6 0: ð11Þ

It is sufficient to force the positivity at one point, such as p ¼ 0:5.

(9) The curvature of the zero-percentile of ðlog N;DrÞ for constant rmax must be non-negative, that is, o2Dr

oðlog NÞ2P 0, leading

to

C5 rMðC2C5Þ � C1C6rMð Þ þ ðC0 � ðlogð� logð1� pÞÞÞC25 6 0: ð12Þ

2.2. Some properties of the model

The p percentile of the log N � R field for a given rmax and stress ratio R ¼ rm=rM can be derived from expression (1) byreplacing rmin by Rrmax:

log N ¼ logð� log½1� p�Þ � ðC0 þ C2rmax þ C1RrmaxÞC6rmax þ C5Rrmax

; ð13Þ

and since it is a common practice using a regression equation to fit the rmax � log N field, the regression model resulting fromthe Gumbel models is

log N ¼ C0 þ C2rmax þ C1Rrmaxr2max þ c

C6rmax þ C5Rrmax; ð14Þ

where c ¼ 0:57772 is the Euler–Mascheroni number.The importance of Eqs. (13) and (14) is that they have been derived from the indicated properties, and not arbitrarily cho-

sen. This regression model can be used to fit the data, and will be used below for estimation purposes.

2.3. Parameter estimation

To estimate the parameters by the maximum likelihood method, one can maximize the log likelihood:

L ¼Xi2I1

HðNiÞ þ log C5rmiþ C6rMi

� �� logðNiÞ

� ��X

i2I1[I0

expðHðNiÞÞ; ð15Þ

where I0 and I1 are the sets of run-outs and non-run-outs, respectively, and

HðNiÞ ¼ C0 þ C1rmiþ C2rMi

þ C5rmiþ C6rMi

� �log Ni;

subject to the set of constraints (3)–(12).The asymptotic covariance matrix of the bC0; bC1; bC2; bC5; bC6 estimates can be calculated using the well known formula:

Covar ¼ � oL2

oCioCj

!������1

bC � ; ð16Þ

where bC � are the maximum likelihood parameter estimates. This matrix is the basic tool to determine confidence intervals ofother related variables, as percentiles for example.

Confidence intervals for finite sample populations can be obtained by the Bootstrap method (see [23] or [24]).Another possibility for estimating the parameters consists of using the regression model (14), i.e., minimize the following

sum of squares:

Q ¼Xn

i¼1

log N þ C0 þ C2rmax þ C1Rrmax þ cC6rmax þ C5Rrmax

2

; ð17Þ

subject to the constraints (3)–(12).The treatment of the runout data can be handled by iteration. Initially, the runout data are ignored in the first iteration,

and once the parameters have been obtained, one assigns the run-outs to their expected values. Next, the process is repeateduntil convergence. In order to avoid repetition, we do not include the details here. The interested reader is referred to [25].We end this section by saying that there are many other estimation methods (see, for example, [26,27], and the references inthese two papers).

3. Derivation of the crack growth model

This section is organized in two parts. In the first, the general form of crack growth functions is derived. In the second, theparticular form of this function is obtained in terms of the pdf of the initial crack sizes or the / function.

4444 E. Castillo et al. / Engineering Fracture Mechanics 75 (2008) 4439–4449

3.1. General form of crack growth models

Let a0 be the random maximum effective crack size present in a piece, that for simplicity is assumed to be the one pro-ducing failure (after increasing up to the failure size), and let f0ða0Þ be the corresponding probability density function (pdf).Assume that the piece is subject to N cycles of alternating constant stresses ranging from rm and rM. For the sake of rigor, wedeal with dimensionless variables and consider a0=ac and N=N0 instead of a0 and N, respectively. The crack size ac is thatproducing failure in the piece for the stress level and ranges being considered.

Then, the crack size a=ac increases with N=N0, and the resulting pdf after N=N0 changes. The aims of this section consistsof:

(1) Deriving a formula hða=ac;N=N0Þ giving the new crack size ratio a=ac in terms of the initial crack size ratio a0=ac,assumed this to be deterministic, and the number of cycles ratio N=N0.

(2) Determining the pdf of a=ac in terms of the pdf of a0=ac for given N=N0.

Assume that

aac¼ h

a0

ac;

NN0

; ð18Þ

is a function that gives the crack size after N fatigue cycles when the initial (for N ¼ 0 cycles) crack size is a0.To obtain the form of the function a=ac ¼ hða0=ac;N=N0Þ functional equations theory is used considering the following

property that this function must satisfy:The formula must be invariant with respect to the number of cycles used. In other words, if the formula is directly used for

N1 þ N2 cycles starting from an initial crack size a0, one must obtain the same result than using first the formula for N1 cyclesand initial crack size a0, and, based on the resultant crack size as the initial for the second part, derive the new crack sizecorresponding to N2 cycles.

More precisely, if the piece of initial maximum crack size ratio a0=ac is subject to N=N0 ¼ ðN1 þ N2Þ=N0 cycles, the result-ing final crack, according to the definition of the function hð�; �Þ, will be aN=ac ¼ hða0=ac; ðN1 þ N2Þ=N0Þ. However, this valueaN=ac can also be obtained in the following way: first the piece is subject to N1=N0 cycles, and then the maximum crack sizewill be hða0;N1Þ, and next, the resulting piece, with initial crack size hða0=ac;N1=N0Þ is subject to N2=N0 extra cycles, thus,using the hð�; �Þ function definition again, one obtains the final maximum crack size hðhða0=ac;N1=N0Þ;N2=N0Þ. Since both val-ues must provide the same result, no matter the values of N1=N0 and N2=N0, one gets the functional equation:

aac¼ h

a0

ac;N1 þ N2

N0

¼ h h

a0

ac;N1

N0

;N2

N0

; ð19Þ

which is the well known translation equation (see [28–30]), whose unknown is function h, and whose general solution is

aac¼ h

a0

ac;

NN0

¼ / /�1 a0

ac

þ N

N0

; ð20Þ

where /ð�Þ is an arbitrary invertible function, from which one gets:

a0

ac¼ / /�1 a

ac

� N

N0

; ð21Þ

and

NN0¼ /�1 a

ac

� /�1 a0

ac

: ð22Þ

If failure is assumed to occur at a ¼ ac, then Eq. (22) becomes

Nc

N0¼ /�1 ac

ac

� /�1 a0

ac

¼ K � /�1 a0

ac

; ð23Þ

where K ¼ /�1ðacacÞ ¼ /�1ð1Þ.

This beautiful and powerful result states that one cannot choose an arbitrary function of two arguments hð�; �Þ to representour problem, and that the only degree of freedom is an invertible function /ð�Þ. In other words, the function hð�; �Þ, whichinitially appears as a two argument function can be written in terms of a single argument function /ð�Þ as in (20). This isan important result, because if function hð�; �Þ, cannot be represented as in (20), then it is not a valid crack growth function.So, one should revise the existing literature to see if all the proposed crack growth functions are really valid functions in thissense.

The way of obtaining the function / in terms of h is as follows. Let a ¼ /�1ða0=acÞ. Then, from (20) one getshða0=ac;N=N0Þ ¼ /ðaþ N=N0Þ, that is

/ðxÞ ¼ hða0=ac; x� aÞ; /ðaÞ ¼ a0=ac: ð24Þ

E. Castillo et al. / Engineering Fracture Mechanics 75 (2008) 4439–4449 4445

To determine the pdf of a=ac in terms of the pdf of a0=ac one just need to perform a change of variable. Thus, the pdf ofa=ac becomes:

fa=ac ða=acÞ ¼fa0=ac ð/ð/�1ða=acÞ � N=N0ÞÞ/0ð/�1ða=acÞ � N=N0Þ

/0ða=acÞ: ð25Þ

For example, assume that fa0=ac ða=acÞ is a Beta density bð2;2Þ and that /ðxÞ ¼ x2. Then, the densities of the crack sizes forN=N0 ¼ 0;0:15;0:30; . . ., 0.9 are shown in Fig. 3.

It is interesting to analyze Fig. 3 after including the critical crack size ac in it. Initially, in the first stages, that is, whenN 6 N2, the possible values of the crack sizes are below the critical crack size ac, and then, the probability of failure is null.This explains how at the first stages of the loading process none of the samples fails. However, afterwards, that is, whenN > N2, the probability of failure is positive, and failure can occur. In the figure, the probabilities of failure p3 and p5 havebeen illustrated, as the areas of the shadowed regions, associated with N3 and N5 cycles, respectively. Note how the largestcrack sizes grow at a greater speed than the small crack sizes by comparing the same percentile values of the densities.

3.2. The crack growth model derived from the Wöhler fields

In this section it is shown how the crack growth function h in (20) or equivalently, the / function, can be obtained basedon the pdf or cdf of the crack sizes, and viceversa.

If one relates the probability p of failure of the Castillo et al. method:

p ¼ 1� exp � exp C0 þ C1r�m þ C2r�M þ C5r�m þ C6r�M� �

log N�� �� �

; ð26Þ

with this probability, one gets

1� Fða0=acÞ ¼ 1� exp � exp C0 þ C1r�m þ C2r�M þ ðC5r�m þ C6r�MÞ logð/�1ðac=acÞ � /�1ða0=acÞÞ�� �� �

; ð27Þ

which gives the cdf Fa0=ac ða0=acÞ of a0=ac

Fa0=ac ða0=acÞ ¼ exp � exp C0 þ C1r�m þ C2r�M þ ðC5r�m þ C6r�MÞ logð/�1ð1Þ � /�1ða0=acÞÞ�� �� �

; ð28Þ

an interesting and practical function, which gives the cdf of the initial crack size a0=ac in terms of the cdf of N=N0 and the /function.

For notation simplicity, in the following we use:

A ¼ C0 þ C1r�m þ C2r�M; ð29ÞB ¼ C5r�m þ C6r�M: ð30Þ

The random variable NcN0

has cdf

FNc=N0

NN0

¼ Pr

Nc

N06

NN0

� �¼ Pr K � /�1 a0

ac

6

NN0

� �¼ Pr �/�1 a0

ac

6

NN0� K

� �

Fig. 3. Evolution of the pdf fa=ac ða=acÞ with the number of cycles and critical crack size.

4446 E. Castillo et al. / Engineering Fracture Mechanics 75 (2008) 4439–4449

¼ Pr /�1 a0

ac

P K � N

N0

� �¼ 1� Pr /�1 a0

ac

6 K � N

N0

� �¼ 1� Pr

a0

ac6 / K � N

N0

� �¼ 1� Fa0=ac / K � N

N0

; ð31Þ

from which, taking into account (2), one gets

/ K � NN0

¼ F�1

a0=acexp � exp Aþ B log

NN0

� � � �; ð32Þ

and making s ¼ K � NN0

one obtains

/ðsÞ ¼ F�1a0=ac½expð� exp½Aþ B logðK � sÞ�Þ�; ð33Þ

an interesting and practical function, which gives the / function in terms of the cdf of the initial crack size ratio a0=ac.In addition, from (33) one gets:

Fa0=ac ða0=acÞ ¼ exp � exp Aþ B log K � /�1 a0

ac

� � �; ð34Þ

which provides the cdf of the initial crack sizes in terms of the / function, and

/�1ðuÞ ¼ K � explogð� log Fa0=acðuÞÞ � A

B

� �; ð35Þ

and from (20), (33) and (35) one obtains

aac¼ / /�1 a0

ac

þ N

N0

¼ F�1

a0=acexp � exp Aþ B log exp

log � log Fa0=aca0ac

� �� �� A

B

24 35� NN0

24 350@ 1A0@ 1A24 35; ð36Þ

where you can note that the constant K has cancelled out.Note that Eq. (36) provides the crack growth curve for any possible combination of rm and rM (see (29) and (30)), in terms

of the cdf of the initial crack size. In other words, this model allows us determining the crack growth resulting from any loadhistory, that is, the values rm and rM need not be fixed nor have a fixed ratio, as it happens with the traditional lognormaland Weibull models which are used for fracture probability analysis.

Replacing in (36):

Fa0=ac

a0

ac

¼ p; ð37Þ

one obtains the equations of the crack growth curves associated with the percentiles p.

4. Example of application

To check the model performance and to illustrate its use in practical cases, the model is applied in this section to a realexample. The methodology proposed in Section 4 is applied to one practical case, as follows.

Since no experimental data was available, we have used some data published in the existing literature. In our case, thefatigue results from the MIL-HDBK-5G have been chosen. In particular, fatigue sample data from specimens made of notchedInconel 718 bars including three stress ratios ðR ¼ �0:50;0:10;0:50Þ. Because the numerical values concerning rmax andnumbers of cycles to failure were not explicitly supplied in this reference, they were directly estimated from the graphicrepresentations.

The normalizing variables log N0 and r0 have been chosen as log N0 ¼ 0 (N0 ¼ 1 cycle) and r0 ¼ 1000 MPa. and normal-ized the data to dimensionless form by dividing the lifetime and stresses by 1 cycle and 1000 MPa, respectively.

Since the data point with the smallest lifetime appeared as an outlier suspected to belong to low-cycle fatigue and thereare physical reasons to justify it, this data point was not considered. Thus, the model parameters have been estimated usingthe maximum likelihood method discussed in Section 2 and fitted with all data but one outlier and including the run-outs.

The resulting parameter estimates were

C0 ¼ �12:1504; C1 ¼ 49:975; C2 ¼ �42:9144; C5 ¼ �6:34962; C6 ¼ 6:34962;

S-N field

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

1000 10000 100000 1000000 10000000

Log N

Max

R= -0.5

R= 0.1

R= 0.5

R= -0.5

R= 0.1

R= 0.5

R= -0.5

R= 0.1

R= 0.5

σ

Fig. 4. Data and median S–N curves resulting from the fitted model (continuous lines) and resulting S–N curves by fitting three independent Weibullmodels to data corresponding to the three different stress ratios.

E. Castillo et al. / Engineering Fracture Mechanics 75 (2008) 4439–4449 4447

and the median S–N curves resulting from the fitted model and the data have been shown in Fig. 4 (continuous lines). Thismodel corresponds to the Gumbel version of the model (2) with fixed asymptotes including all the constraints.

For the sake of comparison, we have also fitted three independent Weibull models (see [4,22,27]) to the data correspond-ing to the three different stress ratios (R ¼ �0:5;0:1 and 0.5), and the resulting median S–N curves are also shown in Fig. 4(dashed lines). Note that the differences are small. Note also that our model covers all cases, while the standard Weibull orlognormal models require three different models, one per each stress ratio. This illustrates the power of the proposedmethod which is able to cover all cases of stress ratios.

In absence of information about the crack growth curves, we have assumed that the relative initial crack sizes a0=ac followa normal distribution Nð0:4;0:1Þ, which is sufficient together with the Wöhler field model (2) to determine the crack growthcurves.

The proposed models have been used for two combinations of rm and rM, that is, rm ¼ 0:2;rM ¼ 0:6, andrm ¼ �0:7;rM ¼ 0:7, and the results are shown in Fig. 5, where the crack growth curves corresponding with the percentiles

500 1000 5000 10000 50000 100000

0.2

0.4

0.6

0.8

1.0

0.0

Log N/N0

a/ac

Fig. 5. Relation between the densities of the crack sizes and lifetime for two different load tests, corresponding to rm ¼ 0:2;rM ¼ 0:6, andrm ¼ �0:7;rM ¼ 0:7 (dashed curves). The crack growth curves corresponding with the percentiles {0.01,0.25,0.50, 0.75,0.99} have been shown.

0 0.2 0.4 0.6 0.8 1a0 / ac

0

1

2

3

4

5

6

Fig. 6. pdf of the relative initial crack sizes a0=ac based on the assumed / function in (38).

4448 E. Castillo et al. / Engineering Fracture Mechanics 75 (2008) 4439–4449

{0.01,0.25,0.50,0.750.99} have been shown, together with the corresponding densities for N=N0 and the initial crack sizesa0=ac. The left pdf corresponds to the initial crack sizes, and the two top densities correspond to the lifetimes associated withthe two load tests considered, respectively. The percentile crack growth curves have been obtained using (36) with (37). Thedata values in Fig. 4 and the corresponding crack sizes are also indicated in Fig. 5.

The interesting thing is that only one single function, either the cdf of the effective crack sizes (initial or after a givennumber of cycles) or the / function are required to obtain knowledge of the relation between the Wöhler fields (for all com-binations of rm and rM) and the crack growth curves for the same loading conditions.

Alternatively, the / function defining the crack growth could be assumed, and the density of the relative initial crack sizesa0=ac obtained.

For example, if the function / is assumed to be

/ðNÞ ¼ aN þ bN2 () /�1ðaÞ ¼ �aþffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia2 þ 4ba

p2b

; ð38Þ

with a ¼ 4:15918� 10�5 and b ¼ 5:29349� 10�9, the resulting pdf of the relative initial crack sizes a0=ac can be obtainedusing (34). The resulting pdf is shown in Fig. 6.

5. Conclusions

A general model has been given which allows deriving the crack growth curves for any constant load test, that is, alter-nating fatigue load from rm to rM, from the family of Wöhler curves associated with the same load conditions, based on thepdf of the initial crack sizes or on a single crack growth experiment to determine the / function.

This has important practical implications, because these curves can be obtained by running conventional fatigue tests andcomplementing them with simple crack growth data.

The model allows extrapolation to any load conditions, that is when the rm and rM values change with the number ofcycles, using damage accumulation methods as, for example, those given in [31].

Acknowledgements

The authors are indebted to the Spanish Ministry of Science and Technology (Project No. BIA2005-07802-C02-01) and tothe Empa (Dübendorf) for partial support.

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