International Journal of Fatigue -...

International Journal of Fatigue 42 (2012) 172–182

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International Journal of Fatigue

journal homepage: www.elsevier .com/locate / i j fa t igue

Prediction of fatigue crack initiation lives at elongated notch roots using shortcrack concepts

Jaime Tupiassú Pinho de Castro a,⇑, Marco Antonio Meggiolaro a, Antonio Carlos de Oliveira Miranda b,Hao Wu c, Abdellatif Imad d, Noureddine Benseddiq d

a Mechanical Engineering Department, Pontifical Catholic University of Rio de Janeiro (PUC-Rio), Rio de Janeiro, RJ 22451-900, Brazilb Department of Civil and Ambiental Engineering, University of Brasília, SG-12 Building, Darcy Ribeiro Campus, DF 70910-900, Brazilc The Parading Center of Mechanics Laboratory of Ministry of Education, Tongji University, Shanghai 200092, Chinad Laboratoire de Mécanique de Lille (CNRS UMR 8107), Université Lille1, Villeneuve D’Ascq, France

a r t i c l e i n f o

Article history:Available online 7 November 2011

Keywords:Notch sensitivityShort cracksNon-propagating cracksFatigue life predictionShort crack tolerance

0142-1123/$ - see front matter � 2011 Elsevier Ltd. Adoi:10.1016/j.ijfatigue.2011.10.010

⇑ Corresponding author. Tel.: +55 21 3527 1642; faE-mail addresses: [email protected] (J.T.P. Cast

Meggiolaro), [email protected] (A.C.O. Miranda), [email protected] (A. Imad), noured(N. Benseddiq).

a b s t r a c t

Re-initiation lives of fatigue cracks departing from stop-holes roots, previously introduced at the tip ofdeep cracks on modified SE(T) specimens, have been satisfactorily predicted using their properly calcu-lated notch sensitivity factor q, considering the notch tip stress gradient influence on the fatigue behaviorof mechanically short cracks. This is an indispensable detail, since traditional q estimates are only appli-cable to semi-circular notches, whereas elongated slits can have q values which also depend on theirshape, not only on their tip radius. Based on this experimental evidence, a criterion for acceptance ofshort cracks is proposed.

� 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The notch sensitivity factor 0 6 q 6 1 is widely used in struc-tural design to quantify the difference between Kt, the linear elasticstress concentration factor (SCF) of a notch, and Kf, its correspond-ing fatigue SCF, which quantifies the actual notch effect on the fa-tigue strength of structural components [1]. The SCF Kt is equal tormax/rn, where rmax is the maximum (linear elastic) stress at thenotch root caused by rn, and rn is the nominal stress that wouldact at that point if the notch did not affect the stress field aroundit. The fatigue SCF is usually defined by

Kf ¼ 1þ qðKt � 1Þ ¼ SL=SLntc ð1Þ

where SL and SLntc are the material fatigue limits (or their fatiguestrengths at a convenient very long life) measured on standard(smooth and polished) and on notched test specimens, respectively.But, as the fatigue process depends on two parameters, Eq. (1) canbe generalized considering that Kf may depend e.g. on R = rmin/rmax,by writing Kf(R) = SL(R)/SLntc(R).

It is well known that q can be associated with the relatively fastgeneration of tiny non-propagating fatigue cracks at notch roots,

ll rights reserved.

x: +55 21 3527 1665.ro), [email protected] ([email protected] (H. Wu),[email protected]

see Fig. 1. Indeed, according to Frost et al. [2], early experimentalevidence that small non-propagating fatigue cracks are found atnotch roots when SL/Kt < rn < SL/Kf goes back as far as 1949. Hence,it is certainly reasonable to expect that such tiny cracks can beused to quantitatively explain why Kf 6 Kt and, consequently, thatthe notch sensitivity can be analytically predictable from thefatigue propagation behavior of short cracks emanating fromthe notch tip. It is demonstrated below that in fact this can bedone using relatively simple but sound mechanical principles,which do not require heuristic arguments, or arbitrary fittingparameters.

To associate the notch sensitivity to the transition between thenon-propagating and propagating states of short fatigue cracks,first the influence of stress field gradients around notch roots ontheir propagation behavior is studied. Knowing that for stress anal-ysis purposes the behavior of notches with depth b and tip radius qcan be well simulated by elliptical notches with semi-axes b and cand tip radius q = c2/b, it is shown that, for any given material, qdepends not only on the notch tip radius q, but also on its depthb [3,4]. In other words, the crack shape, characterized by its tip ra-dius to depth q/c ratio (or by the semi-axes c/b ratio), has a stronginfluence on its effect on fatigue strength. This means that shallowand elongated notches of same tip radius q may have quite differ-ent notch sensitivities q.

It is also shown that the material influence on the notch sensi-tivity depends on its propagation threshold for (long) fatiguecracks DKth(R) and on its fatigue limit for crack initiation from a

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Fig. 1. Classical data showing that non-propagating fatigue cracks are generated atthe notch roots if SL/Kt < rn < SL/Kf [2].

Fig. 2. Kitagawa–Takahashi plot describing the crack size influence on the fatiguepropagation of short and long cracks under R = 0 in a HT80 steel withDK0 ¼ 11:2 MPa

pm and DS0 = 575 MPa [6]: long cracks with a� a0 stop when

Dr 6 DK0=pðpaÞ, very short cracks with a� a0 stop when Dr 6 DS0, and the ETS

curve predicts crack stop when Dr 6 DK0=p

pðaþ a0Þ.

J.T.P. Castro et al. / International Journal of Fatigue 42 (2012) 172–182 173

smooth surface DSL(R) = 2SL(R). Note that, from the physical pointof view, it is certainly more reasonable to use the stress rangeand maxima Dr and rmax as the crack driving forces, but this oldDr–R approach has some operational advantages for structuralcalculations.

The predicted q values are verified by fatigue testing several Al6082 T6 notched specimens. Finally, it should be noted that ‘‘shortcrack’’ here means ‘‘mechanically short crack’’ not ‘‘microstructur-ally short crack’’, since material isotropy is assumed in the model-ing. Note, however, that this simplified hypothesis is corroboratedby the measured experimental results.

2. The propagation of (mechanically) short fatigue cracks

Short cracks must behave differently from long cracks, sincetheir fatigue crack propagation (FCP) threshold must be smallerthan the long crack threshold DKth(R), otherwise the stress rangeDr required to propagate them would be higher than the materialfatigue limit DSL(R). Indeed, assuming as usual that the FCP processis primarily controlled by the stress intensity factor (SIF) range, forthe cases where the SIF range grows as the crack size a increases,DK / Dr(pa), if short cracks with a ? 0 had the same DKth(R)threshold of long cracks, their propagation by fatigue would re-quire Dr ?1, a physical non-sense [5].

The crack size influence on the propagation threshold of shortfatigue cracks under pulsating loads DKth(a, R = 0) can be satisfac-torily modeled using the short crack characteristic size a0 proposedby El Haddad-Topper-Smith (ETS), which is estimated fromDS0 = DSL(R = 0) and DK0 = DKth(R = 0) [6]. This clever trick repro-duces well the Kitagawa–Takahashi [7] plot trend, see Fig. 2, usinga modified SIF range DK0 to describe the fatigue propagation of anycrack, short or long,

DK 0 ¼ Drffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðaþ a0Þ

p; where a0 ¼ ð1=pÞðDK0=DS0Þ2 ð2Þ

Using this a0 trick, it is indeed possible to reproduce the ex-pected limits DKth(a ?1) = DK0 and Dr(a ? 0) = DS0, see Fig. 2.Knowing that steels typically have 6 < DK0 < 12 MPa

pm, ultimate

tensile strength 400 < SU < 2000 MPa, and fatigue limit 200 < SL <1000 MPa (because the very clean high-strength steels tend tomaintain the SL/SUffi0.5 trend of lower strength steels under fullyalternated loads, with R = �1); and estimating by Goodman thepulsating (R = 0) fatigue limit DS0 ¼ 2SUSL=ðSU þ SLÞ ) 260 < DS0

< 1300 MPa; it can be expected that the maximum range of theETS short crack characteristic size a0 for steels is

ð1=pÞðDK0min=DS0max Þ

2 ffi 7 < a0 < 700 lm

ffi ð1=pÞðDK0max=DS0minÞ2 ð3Þ

This a0 range may be overestimated, since the minimum thresholdDK0min is not necessarily associated with the maximum fatiguecrack initiation limit DS0max, nor DK0max is always associated withDS0min. But it nevertheless justifies the ‘‘short crack’’ denominationused for cracks of a similar small size, and highlights the short crackdependence on the FCP threshold and on the fatigue limit of thematerial. In other words, it can be expected that cracks up to afew millimeters may still behave as short cracks in some steels,meaning that they may have smaller propagation thresholds thanthose measured by testing ‘‘long’’ cracks, which have a� a0.

Similarly, since the typical strengths of wrought aluminum al-loys are 70 < SU < 600 MPa, 30 < SL < 230 MPa, 40 < DS0 < 330 MPa,and 1.2 < DK0 < 5 MPam, their maximum a0 (over)estimated range,and thus their short crack influence scale, is wider than the steelsrange, 1 lm < a0 < 5 mm.

Since the ETS DK0 has been deduced using Griffith’s plate SIF,DK = Dr(pa), Yu et al. [8] and Atzori et al. [9] used the non-dimen-sional geometry factor g(a/w) of the general SIF expressionDK = Dr(pa) � g(a/w) to deal with other geometries, re-defining

DK 0 ¼ gða=wÞ � Drffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðaþ a0Þ

p; where a0

¼ ð1=pÞ½DK0=ðgða=wÞ � DS0Þ�2 ð4Þ

This expression implies that (under pulsating loads) the tolera-ble stress range Dr tends to the fatigue limit DS0 when a ? 0. Thisis only true when Dr is the notch root stress range, instead of thenominal stress. However, the geometry factor g(a/w) found in SIFtables usually includes the notch root SCF, thus use Dr instead ofDrn as the notch tip nominal stress range. Hence, a clearer wayto define the short crack characteristic size a0 when it departs froma notch root is to explicitly recognize this practice, separating thegeometry factor g(a/w) into two parts: g(a/w) = g � u(a), whereu(a) describes the stress gradient ahead of the notch tip, whichtends to the notch root SCF as the crack length a ? 0, whereas gencompasses the remaining terms, such as the free surfacecorrection:

DK 0 ¼ g �uðaÞ � Drffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffipðaþ a0Þ

p; where a0

¼ ð1=pÞ½DK0=ðg � DS0Þ�2 ð5Þ

In other words, the first factor u(a) does not appear in thisexpression for a0 because for very small cracks the notch rootstress range u(a ? 0) � Dr should tend to DS0.

Alternatively, from an operational point of view, the short crackproblem can be probably more clearly modeled by letting the SIFrange DK retain its original equation, while the FCP threshold

Fig. 3. Ratio between short and long crack propagation thresholds as a function ofa/a0.

174 J.T.P. Castro et al. / International Journal of Fatigue 42 (2012) 172–182

expression (under pulsating loads) is modified to become a func-tion of the crack length a, namely DK0(a), resulting in

DK0ðaÞ ¼ DK0 �ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia=ðaþ a0Þ

pð6Þ

The El Haddad–Topper–Smith’s equation can be seen as onepossible asymptotic match between the short and long crackbehaviors. Following Bazant’s [10] reasoning, a more general equa-tion can be used introducing an adjustable parameter c to fitexperimental data

DK0ðaÞ ¼ DK0 � ½1þ ða0=aÞc=2��1=c ð7Þ

Eqs. (2)–(6) are obtained from Eq. (7) when c = 2.0. The bi-linearlimit estimate, given in Fig. 2 by Dr(a 6 a0) = DS0 for short cracks,and by DK0(a P a0) for long ones, is obtained wheng(a/w) = g�u(a) = 1 and c ?1. The fitting parameter c allows thisDK0(a) estimate to better fit short crack propagation data fromTanaka et al. [11] and Livieri and Tovo [12], see Fig. 3. Most datain this figure is bounded by the short crack threshold curves gener-ated using c = 1.5 and c = 8.

In the following sections, these ideas are first applied to predictthe propagation behavior of short cracks emanating from circularholes, and then extended to describe the behavior of cracks whichdepart from semi-elliptical notches, resulting in improved esti-mates of the notch sensitivity q and of the largest non-propagatingcrack size tolerated at such notch tips.

3. The fatigue behavior of short cracks which depart fromcircular holes

The FCP behavior of short cracks emanating from circular holesin Kirsch (infinite) plates is now evaluated. The SIF of a single crackwith length a emanating from a circular hole with radius q in aKirsch plate loaded by a tensile stress range Dr is expressed by

DK ¼ 1:12 �uða=qÞ � Drffiffiffiffiffiffipap

ð8Þwhere the factor u(a/q) � u(x), which describes the hole stress con-centration and gradient effects on the SIF, is given (within 1%, [13])by

uðxÞ ¼ 1þ 0:2ð1þ xÞ þ

0:3

ð1þ xÞ6

!

� 2� 2:354x

1þ xþ 1:2056

x1þ x

� �2

� 0:2211x

1þ x

� �3 !

; x � aq

ð9Þ

Note that, when the crack size a tends to zero, Eq. (8) becomes

lima!0

DK ¼ 1:12 � 3 � Drffiffiffiffiffiffipap

ð10Þ

as expected, since it combines the solution for an edge crack in asemi-infinite plate with the SCF of a circular Kirsch hole, whichhas Kt = u(x = 0) = 3. Note also that the other limit, for the longcracks with a� a0, results in

lima!1

DK ¼ Drffiffiffiffiffiffiffiffiffiffiffiffipa=2

pð11Þ

which is the SIF for a long crack with length a in a Griffith (infinite)plate, since the crack tip is so far away from the hole border that itdoes not suffer anymore the notch influence on its surroundingstress field. Note that the actual crack length is a + 2q, however,as a ?1, the q value disappears from this equation. Thus, the SIFof a long crack which departs from a Kirsch hole has u(x ?1)= 1/1.12

p2 ffi 0.63.

Using Eq. (7) to express the FCP threshold, it can then be statedthat any crack departing from a Kirsch hole under pulsating loadswill propagate when

DK ¼ g �uða=qÞ � Drffiffiffiffiffiffipap

> DK0ðaÞ

¼ DK0 � ½1þ ða0=aÞc=2��1=c ð12Þ

where g = 1.12 is the free surface correction. Knowing that thethreshold DK0(a) � DK0 for a long crack, then the crack lengthparameter a0 from the above equation is

a0 ¼ ð1=pÞ½DK0=ð1:12 � DS0Þ�2 ð13Þ

As discussed before, the stress gradient component of thegeometry factor u(a/q) does not appear in this definition of a0.The crack propagation criterion given by Eq. (12) can then be re-stated using two dimensionless functions, one related to the notchstress gradient u(a/q), and the other to g(a/q, DS0/Dr, DK0/DS0

pq,

c) which depends on the crack size, on the fatigue resistances, andon the crack driving force [14], and re-written as

uaq

� �>

ðDK0=DS0ffiffiffiffiqp Þ � ðDS0=DrÞ

gffiffiffiffiffiffiffiffiffiffiffiffipa=q

p� �cþ ðDK0=DS0

ffiffiffiffiqp Þch i1=c

� gaq;DS0

Dr;

DK0

DS0ffiffiffiffiqp ; c

� �ð14Þ

In other words, if x � a/q and j � DK0/DS0p

q = g � p(pa0)2/q), afatigue crack departing from a Kirsch hole under pulsating loadsgrows whenever u(x)/g(x, DS0/Dr, j, c) > 1. Fig. 4 plots some u/gfunctions for several fatigue strength to loading stress range ratiosDS0/Dr as a function of the normalized crack length x, assuming anotch root radius and short crack characteristic size combinationwith j = 1.5, and a material with c = 6 [15].

For high applied stress ranges Dr, the strength to load DS0/Drratio is small, and the corresponding u/g curve is always higherthan 1, meaning that cracks will initiate and propagate from theKirsch hole border, without stopping during this process. Oneexample of such a case is the upper curve in Fig. 4, which showsthe function u/g1.4 obtained for DS0/Dr = 1.4. On the other hand,small stress ranges Dr with load ratios DS0/Dr P Kt = 3 have u/gfunctions which are smaller than 1, meaning that no crack will ini-tiate from the Kirsch hole, and that small enough cracks will notpropagate from it at such low loads. This is illustrated by curvesu/g3, associated with the limit case where the local stress rangeequals the material fatigue strength DS0/Dr = 3, and u/g4, associ-ated with a still smaller load, DS0/Dr = 4.

But three other curves must be analyzed in Fig. 4. The first onecrosses the u/g = 1 line once, see the u/g2.3 curve. This means thatsuch a pulsating intermediate load range can initiate and propa-gate a fatigue crack from the notch border, until the decreasing(because of the stress gradient ahead of the notch root) u/g2.3 ratioreaches 1, where the crack stops. Hence, this Dr = DS0/2.3 loading

Fig. 4. The fatigue stress concentration factor Kf can be obtained by finding thefunction u/g which is tangent to the u/g = 1 line, thus in this (j = 1.5, c = 6) caseKf = 1.64.


level generates a non-propagating fatigue crack at this hole border,with a size given by the corresponding a/q abscissa.

The second is the u/g1.85 curve, which intersects the u/g = 1 linetwice. This pulsating load level will also generate a fatigue crack atthe Kirsch hole border, which will propagate until reaching themaximum size obtained from the abscissa of the first intersectionpoint (on the left), where the crack stops. Moreover, cracks longerthan the size defined by the abscissa of that second intersectionpoint will re-start propagating by fatigue under Dr = DS0/1.85, un-til eventually fracturing the Kirsch plate. However, the crack initi-ated by fatigue under such a pulsating intermediate load rangecannot propagate between these two intersection points by fatiguealone (assuming the load range Dr and R = 0 ratio remain con-stant.) Hence, it can only grow in this region if helped by a differentdamage mechanism, such as corrosion or creep, for example.

Note that the FCP behavior of these two curves seems differentin Fig. 4, yet they are similar. Indeed, the u/g2.3 curve crosses the u/g = 1 line twice if the graph is extended to include larger cracks, seeFig. 5, because a sufficiently long crack can always propagate by fa-tigue under any given (even if small) Dr range whenever its SIFrange DK = g(a/w) � Dr(pa) grows with the crack size a, as in thisKirsch plate. In fact, all u/g curves become higher than 1 for suffi-ciently large a/q values, even those that cannot initiate a crack byfatigue, such as u/g4.

Finally, the u/g1.64 curve is tangent to the u/g = 1 line in Figs. 4and 5. This means that this stress range Dr = DS0/1.64 is the small-est one that can cause crack initiation and propagation (without ar-rest) from the notch border by fatigue alone. Thus, the fatigue SCFof this Kirsh hole (with j = g � p(pa0/q) = 1.5 and c = 6) is, by defi-

Fig. 5. As after leaving the region affected by the stress gradient near the Kirsh hole borcracks), any stress range Dr can propagate it by fatigue when it is sufficiently long.

nition, Kf = 1.64. Moreover, the tangency point (between the u/g1.64

curve and the u/g = 1 line) abscissa gives the largest non-propagating crack size that can arise from it by fatigue alone,amax = xmax � q. For any other c and j = g � p(pa0/q) combination,Kf and amax can always be found by solving the system

u=g ¼ 1@ðu=gÞ=@x ¼ 0

)� uðxmaxÞ ¼ gðxmax;Kf ;j; cÞ

@uðxmaxÞ=@x ¼ @gðxmax;Kf ;j; cÞ=@x

�ð15Þ

This system can be numerically solved for several combinationsof j and c, to obtain this notch sensitivity factor q behavior from

qðj; cÞ � ðKf ðj; cÞ � 1Þ=ðKt � 1Þ ð16Þ

Fixing c = 6, the notch sensitivity factor calculated in this way iscompared in Fig. 6 to the classical q-curves proposed by Peterson along time ago (those which are reproduced in most machine ele-ments books), showing that this prediction is indeed reasonable.

Thus, the notch sensitivity q can be calculated using appropriateanalytical procedures, without appealing to semi-empirical heuris-tic arguments, by quantifying how the stress gradient at the notchroot affects the short crack propagation, including the material-dependent data fit parameter c influence on DK0(a). Moreover, thisapproach can be easily extended to semi-elliptical notches, whichcan be modeled in the same way, as shown in the followingsections.

Since fatigue is a phenomenon which depends on two drivingforces, Dr and rmax, before ending this section it must be men-tioned that Eqs. (7) and (13) can be easily extended to considerthe rmax (here indirectly modeled by the R-ratio) influence in theshort crack behavior. First, the short crack characteristic sizeshould be re-defined as

aR ¼ ð1=pÞ½DKR=ð1:12 � DSRÞ�2 ð17Þ

where DKR = DKth(a� aR, R) is the FCP threshold for long cracks andDSR is the fatigue limit, both measured (or properly estimated) atthe desired R-ratio. Likewise, the corresponding short crack FCPthreshold should be re-written as

DKRðaÞ ¼ DKR � ½1þ ðaR=aÞc=2��1=c ð18Þ

All these details are important when such models are used tomake predictions in real life situations, since they do influencethe calculation results. In particular, neglecting the rmax effect onfatigue can lead to severe non-conservative life estimations, thusthis potentially dangerous practice cannot be accepted for mechan-ical design or analysis purposes.

der, this crack SIF steadily grows as its size a increases (as usual for far field loaded

Fig. 6. Notch sensitivity q as a function of the Kirsh hole radius q, estimated usingthe median values of oK0, oS0 and SU from 450 steels and aluminum alloys,estimated for c = 6.

Fig. 7. The ratio K1.2 = ry(x/b = 1.2, 0)/rn at just b/5 ahead of the tip of ellipticalInglis holes is almost independent of its linear elastic SCF Kt ¼ 1þ 2b=c ¼ 1þ2pðb=qÞ.


4. The physical behavior of short cracks which depart fromelongated notches

Before jumping into more elaborated mechanics, it is worth topresent a simple and unambiguous mechanical explanation to jus-tify why a crack can start from a sharp notch root and propagate fora while before stopping and becoming non-propagating (underfixed loading conditions). This can help to understand why in thesecases q must depend not only on q and on the material, but also onthe notch shape, given by its root radius to depth ratio q/b.

A very reasonable estimate for the SIF of a small crack a whichdeparts from the tip of an elliptical notch in an Inglis plate, withsemi-axes b and c (b� a), root radius q = c2/b, and with the 2b axiscentered at the x axis origin, is given by KIðaÞ ffi rn �

pðpaÞ�f1ða; b; cÞ � f2ðfree surfaceÞ, where rn is the nominal stress (perpen-dicular to a and b); f1(a, b, c) ffi ry(x)/rn; ry(x) is the stress whichacts at the point (x = b + a, y = 0) in front of the notch root whenthere is no crack; and f2 = 1.12. The distribution of the ry(x = b + a,y = 0) stress ahead of the elliptical notch tip is given by [16]:

f1 ¼ryðx; y ¼ 0Þ

rn

¼ 1þðb2 � 2bcÞ x�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � b2 þ c2

p� �ðx2 � b2 þ c2Þ þ bc2ðb� cÞx

ðb� cÞ2ðx2 � b2 þ c2Þffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffix2 � b2 þ c2

pð19Þ

Note that the slender the elliptical notches are, meaning thesmaller their semi-axes c/b and tip radius to depth q/b ratios are,the higher are their SCF. But higher Kt imply in steeper stress gra-dients @ryðx; y ¼ 0Þ=@x around the notch tip. In fact, the linearelastic SCF induced by any elliptical hole with b P c drops fromKt = 1 + 2b/c = 1 + 2

p(b/q) = ry(1)/rn P 3 at its tip border to a value

1.82 < K1.2 = ry(1.2)/rn < 2.11 at a point just b/5 ahead of it, mean-ing their Saint Venant’s controlling distance is associated withtheir depth b, not with their tip radii q, see Fig. 7.

This is the cause for the peculiar growth of short cracks whichdepart from elongated notch roots. Since the stresses become al-most independent on elliptical tip radii after just about b/5 fromthem, the SIF of short cracks departing from slender, high Kt ellip-tical notches, which in principle should tend to increase with theirlength a = x � b, may instead decrease after they grow for a shortwhile. Indeed, the Kt affected stress gradient effect in the short

crack STF KI ffi 1:12 � rnpðpaÞ � f1 may decrease sharply due the

high stress drop close to the notch tip, overcompensating the crackgrowth effect, see Fig. 8.

This KI(a) estimate can be used to evaluate the sizes of non-propagating fatigue cracks tolerable at notch roots, using the shortcrack FCP behavior. A simple numerical example clarifies thispoint: if a large steel plate with SU = 600 MPa, SL = 200 MPa andDK0 = 9 MPa�m works under a purely alternated load (stress) rangeDrn = 100 MPa at R = �1, let’s verify if it is possible to change anoriginally circular d = 20 mm central hole by an elliptical one with2b = 20 mm (perpendicular to rn) and 2c = 2 mm, without inducingthe plate to fail by fatigue.

As SL < 0.5SU/2, the specified plate fatigue strength probably al-ready considers surface roughness and similar effects. Neglectingthe buckling problem, important in thin plates, the circular holeis certainly safe, since it has a safety factor against fatigue crack ini-tiation uF = SL/Kf�rn = 200/150 ffi 1.33, because this large hole hasKf ffi Kt = 3. But the elongated elliptical hole would not be admissi-ble by traditional SN design routines, since it has a very highKt = 1 + 2b/c = 21 and a small tip radius q = c2/b = 0.1 mm: its notchsensitivity estimated from the usual Peterson q plot [1] would beq ffi 0.32)Kf = 1 + q�(Kt-1) = 7.33, thus it would induce a maximumload amplitude Kf�rn = 376 MPa > SL. However, as this Kf value isconsiderably higher than typical values reported in the literature[15–18], it is worth to re-study this simple problem consideringthe short crack FCP behavior.

Supposing, as usual, that DKth(R < 0) ffi DK0, and that S0L ¼ 0:5SU

(S0L is the material not the plate fatigue limit, as the modifying factorsrequired to estimate the component fatigue limit SL are not neededfor FCP modeling); using ETS to estimate the FCP threshold byDK0(a) = K0/[1 + (a0/a)]�0.5, and Goodman to estimate DS0 = SU/1.5and a0 = (1/p)(1.5DK0/1.12�SU)2 ffi 0.13 mm; then the SIF rangesDKI(a) for short cracks that depart from the two holes borders arecompared to DK0(a) in Fig. 9.

The SIF DKI(a) curve for cracks departing from the circular holeremains below the DK0(a) FCP threshold curve (which considersthe short crack behavior) up to a ffi 1.54 mm. Thus, if (say) a smallsurface scratch locally augments the stress range at that hole bor-der up to the point it initiates a tiny crack, this crack would notpropagate under this (fixed) Drn = 100 MPa and R = �1 load. Thisconfirms its ‘‘safe’’ prediction by traditional SN fatigue designprocedures. Only if a crack with a > 1.54 mm is introduced at thisKirsch hole border by any other means, it would then be able topropagate by fatigue under those otherwise safe loadingconditions.

On the other hand, under these same loading conditions, theDKI(a) curve for the elliptical hole starts above DK0(a), thus a crack

Fig. 8. The estimate KI ffi 1:12 � rnpðpaÞ � f1ðKt ; aÞ for the cracks which depart from

the tips of an Inglis elliptical hole with b = 10 mm illustrates how the derivative oKI/oa may decrease sharply just after the cracks initiate there.

Fig. 9. Cracks do not initiate at the circular hole, which tolerates cracksa < 1.54 mm, while the crack which initiates at the elliptical notch tip stops afterreaching a ffi 0.33 mm.

Fig. 10. The circular hole is more robust than the elliptical notch: if the load isincreased by just 10%, the crack initiated at the elliptical notch tip does not stopanymore, while the circular hole still tolerates a crack of size a < �1 mm departingfrom its border.


should initiate at its border, as expected from its high Kt. But as thistiny crack propagates through the high stress gradient ahead of thenotch root, it sees rapidly diminishing stresses at its tip during itsearly growth, which overcompensate the increasing crack size ef-fect on DKI(a). Eventually this crack SIF becomes smaller thanDK0(a) at a ffi 0.33 mm, when it stops and becomes non-propagat-ing (if the nominal Drn and R loading remain fixed), see Fig. 9. Asfatigue failures include crack initiation and growth up to fracture,in this sense both notches could be considered safe for this serviceloading. However, the non-propagating crack at the elliptical notchtip, a clear evidence of fatigue damage, renders it much less robustthan the circular one. For example, a small 10% increment in Drn

could make the crack initiate and propagate from the ellipticalnotch until the plate fractures, while the circular notch would re-main safe, still tolerating a crack a ffi 1 mm, as shown in Fig. 10.

These conclusions are quite interesting, but they are based onsimple estimates, thus cannot of course be used for analysis pur-poses. Nevertheless, since these reasonable estimation proceduresare based on clear and sound mechanical hypotheses, which do notrequire heuristic arguments such as ill-defined material-depen-dent characteristic distances, they do justify the development ofthe more precise calculations presented in the following section.

5. The analysis of short cracks which depart from elongatednotches

The logical reasoning used to model the notch sensitivity of thecircular Kirsch hole can now be extended to model elliptical

notches, which in turn can be used to model most notches thathave the same depth and tip radius. The SIF range of a single crackwith length a emanating from a semi-elliptical notch with semi-axes b and c (where b is in the same direction as a) at the edgeof a very large plate loaded in mode I can be written as

DKI ¼ g � Fða=b; c=bÞ � Drffiffiffiffiffiffipap

ð20Þ

where g = 1.12 is the free surface correction (FSC), and F(a/b, c/b) isa geometry factor associated with the notch stress concentration,which can be expressed as a function of the dimensionless param-eter s = a/(b + a) and of the notch SCF, given by [14]

Kt ¼ ½1þ 2ðb=cÞ� � f1þ ½0:12=ð1þ c=bÞ2:5�g ð21Þ

To obtain expressions for F, Finite Element (FE) calculationswere performed using the Quebra2D program [19,20] consideringseveral cracked semi-elliptical notch configurations. The numericalresults, which agreed well with standard solutions [13], were fittedwithin 3% using empirical equations, resulting in

Fða=b; c=bÞ � f ðKt ; sÞ ¼ Kt

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½1� expð�sK2

t Þ�=sK2t

q; c 6 b and

s ¼ a=ðbþ aÞ ð22Þ

F 0ða=b; c=bÞ � f 0ðKt ; sÞ ¼ Kt½1� expð�K2t Þ��s=2

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi½1� expð�sK2

t Þ�=sK2t

q; c b ð23Þ

Fig. 11 shows how well Eq. (22) fits the F(a/b, c/b) points gener-ated by the FE calculations. Similar results are found for Eq. (23)[14].

Note that the SIF expression for cracks departing from semi-elliptical notches include the effect of their Kt (through these F orF0 functions) and also retain their g free surface correction, seeEq. (20), since as the parameter s ? 0 when a ? 0, the maximumstress at their notch tip is given by rmax ? F(0, c/b) � rn = Kt � rn.Thus, just the g factor, but not the F(a/b,c/b) part of KI, should beconsidered in the short surface crack characteristic size a0 defini-tion, as done in Eq. (13).

Note also that the semi-elliptical Kt expression includes a term[1 + 0.12/(1 + c/b)2.5], which could be interpreted as the notch freesurface correction. Therefore, when the semi-elliptical notch tendsto a crack as c/b ? 0, then its SCF Kt ? 1.12�2p(b/q). This 1.12factor expresses the free surface correction for the notch, not forthe crack. Indeed, when c/b ? 0, this 1.12 factor disappears fromF, which in this case is given by F(a/b, 0) = 1/

ps)DKI =

Fig. 11. Finite Element calculations and proposed analytical fit for the geometryfactor of semi-elliptical notches with c 6 b.

Fig. 12. Notch sensitivity q as a function of the semi-elliptical notch root radiusq = c2/b for aluminum alloys having a0 = 0.26 mm (SU ffi 225 MPa andDK0 = 2.9 MPa

pm.).


g�F�Dr�p(p�a) = g�Dr�[p�(a + b)]0.5, as expected, since the resultingcrack for c ? 0 would have length a + b.

Traditional notch sensitivity estimates, based on the fitting ofquestionable semi-empirical equations to a few experimental datapoints, assume that q depends only on the notch root q and on thematerial ultimate strength SU. Thus, similar materials with thesame SU but different DK0 should have identical notch sensitivities,according to those simplistic estimates. The same should occurwith shallow and deep elongated notches of identical tip radii.However, whereas well established empirical relations relate thefatigue limit DS0 to tensile strength SU of many materials, thereare no such relations between their FCP threshold DK0 and SU.Moreover, it is also important to point out that the q estimationfor elongated notches by the traditional procedures can generateunrealistic Kf values, as exemplified above. In conclusion, such tra-ditional estimates should not be taken for granted.

The proposed model, on the other hand, is based on the FCPmechanics of short cracks which depart from elliptical notch roots,recognizing that their q values are associated with their toleranceto non-propagating cracks. It shows that their notch sensitivities,besides depending on q, DS0, DK0 and c, are also strongly depen-dent on their shape, given by their c/b ratio, see Fig. 12. This figurecurves are calculated for median aluminum alloy properties:tensile strength SU = 225 MPa, fatigue limit SL = 90 MPa (thus, forpulsating loads by Goodman, DS0 = 2SLSR/(SL + SR) = 129 MPa), andlong crack FCP threshold DK0 = 2.9 MPa

pm, which give

a0 = 0.26 mm, supposing c = 6. Similar curves for steels are pre-sented in [14]. Their corresponding Peterson’s curve is wellapproximated by the semi-circular c/b = 1 notch, but this curve isnot applicable for much different c/b ratios. Therefore, the pro-posed predictions indicate that these traditional notch sensitivityestimates should not be used for elongated notches, a forecastexperimentally verified, as discussed in the following section.

6. Experimental verification of the elongated notch sensitivitypredictions

Several fatigue tests were carried out on modified SE(T) speci-mens of thickness B = 8 mm and width W = 80 mm, to find thenumber of cycles required to re-initiate the crack after drilling astop-hole of radius q centered at its tip, generating an elongatedslit with b = 27.5 mm, as detailed in [3,4]. The original objectiveof those tests was to study the life improvement obtained by thestop-hole repair technique, but these tests can also be used to sup-port the validity of the q calculations. The test specimens weremade from an aluminum alloy 6082 T6, with SY = 280 MPa,SU = 327 MPa, and Young’s modulus E = 68 GPa. The particularlycareful tests were performed at 30 Hz under constant load rangeat R = 0.57, to avoid any eventual crack closure influence on the

FCP behavior. The specimens were first pre-cracked until reachingthe required crack size. Then they were removed to introduce thestop-holes in a milling machine, using a slight under-size drill pre-cisely centered at their crack tips. Finally, the holes were enlargedto reach their correct size using a reamer. The stop-hole sizes werelarge enough to remove the plastic zones which followed the pre-vious crack tips.

The fatigue crack re-initiation lives (at the tip of the resultingelongated notch), obtained after repairing the original crack, canbe modeled by eN procedures using (i) the Al alloy Coffin–Manson’sparameters [21] r0f ¼ 485 MPa, b = �0.0695, e0f ¼ 0:733 andc = �0.827, and Ramberg–Osgood’s coefficient and exponent ofthe cyclic stress–strain curve, H = 443 MPa and h = 0.064; (ii) thenominal stress range and R-ratio; and finally (iii) the stress concen-tration factor of the notches generated after repairing the cracks bya stop-hole at their tips, which can be estimated from Eq. (21).Hence, e.g. for a stop-hole radius q = 1, the resulting SCF isKt ffi 12.38. This slender notch Kt can also be estimated by Inglis,Kt ffi 1 + 2

p(a/q) = 11.49. The SCF calculated by Finite Elements

(FE) are Kt = 11.8, 8.1, and 7.6 for the 3 stop-hole radii, q = 1, 2.5,and 3 mm, respectively.

The life improvement introduced by the stop-holes can thus beestimated by calculating the stress and strain maxima and rangesat their root borders by Neuber’s rule, and by using them to calcu-late the crack re-initiation lives by an appropriate De N rule, con-sidering the influence of the mean loads. Neglecting this effectcould lead to severely non-conservative predictions, as the R-ratioused in the tests was high (and in fact Coffin-Manson predictionsare highly non-conservative, thus useless in this case). Figs. 13–15 show that the lives predicted by the elastic version of Morrow’sequation (which is an extension of the classical Goodman line) andby the Smith–Watson–Topper (SWT) equation are similar in thiscase. But it is worth to emphasize that such a similarity cannotbe assumed beforehand, since in many other cases these rulescan predict very different fatigue lives. Note that these curves arecorrected versions of similar curves presented in [4], which lamen-tably reproduced some imprecise calculations.

The lives predicted for the two larger holes reproduced reason-ably well the tests results, see Figs. 13 and 14. But the predictionsfor the q = 1 mm hole shown in Fig. 15 are too conservative in com-parison to the measured data. Before proceeding, please note thatbetter-than-predicted fatigue lives do not mean that the smallerhole is more efficient than the larger ones. Indeed, as it could be ex-pected, the larger stop-holes are associated with longer fatiguecrack re-initiation lives for a given load (in these graphs given by

Fig. 13. Predicted and measured fatigue crack re-initiation lives at the q = 3.0 mmstop-hole borders, using traditional eN procedures: the Kt of the resultingb = 27.5 mm elongated slit in Neuber’s rule, and Morrow elastic and SWT equationsto properly account for the mean or maximum stresses associated with the highR = 0.57 loading.

Fig. 14. Measured and similarly predicted fatigue crack re-initiation lives at theq = 2.5 mm stop-hole borders (using their Kt and appropriate eN procedures).

Fig. 15. Measured and similarly predicted fatigue crack re-initiation lives for theq = 1.0 mm stop-holes (using their Kt and appropriate eN procedures). Note howconservative the predictions in this case are, when compared with the better lifepredictions obtained for the larger stop-holes.

Fig. 16. Predicted and measured fatigue crack re-initiation lives after centering anddrilling the stop-holes with radii q = 3.0 mm at the tip of the crack previouslypropagated in a SE(T) specimen, using modified eN procedures: the resultingb = 27.5 mm elongated slit fatigue SCF Kf estimated using the procedures proposedin this paper (instead of its Kt) in Neuber’s rule, and Morrow elastic and SWTequations to properly account for the mean or maximum stresses associated withthe high R = 0.57 loading. These predictions are similar to those presented in Fig. 13,since these holes q ffi 1.


the apparent SIF range DKapp applied after the repaired specimenswere remounted on the fatigue testing machine, calculated treat-ing the resulting a = 27.5 mm slits as if they were cracks). From amodeling point of view, the main result obtained from Figs. 13–15 analysis is that eN life predictions made using traditional proce-dures based on Kt, Neuber, and Morrow or SWT were satisfactoryfor the larger stop-holes, but severely underestimated the re-initi-ation lives for the smaller one.

There are few mechanical reasons which can explain the betterthan expected fatigue lives obtained from the specimens with thesmaller stop-holes. One of them would be the presence of signifi-cant compressive residual stresses at the q = 1 mm stop-hole tips.But all the stop-holes were drilled and reamed following identicalprocedures. Thus, it is reasonable to assume that drilling and ream-ing processes left the same sub-surface residual stress state aroundall hole borders, since all hole sizes were large enough to removethe previous crack tip plastic zones, leaving only virgin materialahead of their roots. Moreover, as the bigger stop-hole lives werequite well predicted supposing rres = 0, it is difficult to justifywhy high compressive residual stresses would be present only atthe q = 1 mm stop-hole roots. The same can be said about the sur-face finish of the stop-holes. However, the smaller stop-holes gen-erate elongated notches with a quite larger Kt than the biggerholes, thus with a much steeper stress gradient near their roots.As discussed above, this effect can significantly affect the growthof short cracks and, consequently, the stop-hole fatigue notch sen-sitivity, possibly providing a sound mechanical explanation for themeasured behavior.

Indeed, when using the properly calculated fatigue SCF Kf in-stead of Kt with the traditional eN procedures, considering theelongated notch sensitivity q by the method proposed here, allthe estimated fatigue crack re-initiation lives reproduce quite wellthe measured results, see Figs. 16–18. The Al 6082 T6 fatigue limitand fatigue crack propagation threshold under pulsating loads re-quired to calculate Kf are estimated as DK0 = 4.8 MPa

pm and

DS0 = 110 MPa, following traditional structural design practices.The c exponent was chosen as c = 6, as recommended by [14].

Note that Figs. 15 and 16 present as good predictions as Figs. 13and 14, since the larger stop-holes notch sensitivity q ffi 1, whereasthe life predictions for the smaller q = 1 mm stop-holes presentedin Fig. 18 are significantly better than the Kt-based predictionsshown in Fig. 15.

Note also that the term ‘‘prediction’’ can in fact be used here,since the curves presented in Figs. 13–18 result from re-initiationlife estimations calculated using only mechanical principles andmaterial data obtained from literature [21], without consideringany of the measured data points. Thus they are indeed predicted,not data-fitted curves. Moreover, an additional test made afterthese calculations predicted that the q = 1 mm stop-hole could tol-erate a DKapp = 7 MPa

pm, as indeed it did, see Fig. 18.

Fig. 17. Measured and similarly predicted fatigue crack re-initiation lives for theq = 2.5 mm stop-holes (using their properly calculated Kf and appropriate eNprocedures). As q ffi 1 for these holes too, these predictions are similar to thosepresented in Fig. 14.

Fig. 18. Measured and similarly predicted fatigue crack re-initiation lives for theq = 1.0 mm stop-holes (using their properly calculated Kf instead of their Kt inNeuber’s rule, and appropriate eN procedures). Note the significant improvement onthese life predictions in relation to the Kt-based predictions shown in Fig. 15.


7. An acceptance criterion for short cracks

Based on the encouraging life estimations for these fatiguecrack re-initiation data, the reverse path can be followed, assumingthe methodology presented here can be used to generate an unam-biguous acceptance criterion for small cracks, a potentially muchuseful tool for practical applications. Most structural componentsare designed against fatigue crack initiation, using eN or SN proce-dures which do not recognize cracks. Hence, their ‘‘infinite life’’predictions may become unreliable when such cracks are intro-duced by any means, say by an accident during manufacturing oroperation, and not quickly detected and properly removed. Largecracks may be easily detected and dealt with, but small cracksmay pass unnoticed even in careful inspections, if they are smallerthan the detection threshold of the inspection method used toidentify them. Thus, structural components designed for very longfatigue lives should be designed to be tolerant to short cracks.

However, this self-evident requirement is still not usually in-cluded in fatigue design routines, as most long-life designs just in-tend to maintain the stress range at the structural componentcritical point below its fatigue limit, guaranteeing that Dr < SR//F,where /F is a suitable safety factor. Of course such calculations canbecome quite involved when designing e.g. against fatigue damagecaused by random non-proportional loads, but their philosophy re-mains the same. Nevertheless, most long-life designs work justfine, which means that they are somehow tolerant to undetectableor to functionally admissible short cracks. But the question ‘‘howmuch tolerant’’ cannot be answered by SN or eN procedures alone.

Such problem can be avoided by adding Eqs. (15) and (16) to the‘‘infinite’’ life design criterion which, to tolerate a crack of size ain its simplest version, should be written as

Dr < DKR=fffiffiffiffiffiffipap

� gða=wÞ � ½1þ ðaR=aÞc=2�1=cg; where

aR ¼ ð1=pÞ � ½DKR=gDSR�2 ð24Þ

As the fatigue limit DSL(R) = DSR already considers the influence ofmicrostructural defects inherent to the structural component mate-rial, Eq. (24) complements it considering the component toleranceto cracks. A simple case study can clarify how useful this conceptcan be, as discussed next.

Due to an unusual manufacturing problem, a batch of a veryimportant component left the factory with small surface cracks,causing some unexpected expensive failures. The task was to esti-mate the effects of such cracks in the stress range that those steelcomponents could tolerate under uniaxial fatigue loads, knowingthat they could be treated as uniaxially loaded strips with 2 mmby 3.4 mm rectangular cross section; that their measured fatiguelimit under R = �1 is SL = 246 MPa; and their that ultimate strengthis SU = 990 MPa. As this SL < SU/2, it may include some surfaceroughness effects which should not affect the cracks. But, in the ab-sence of reliable information, the only safe option is to use themeasured SL value to estimate SR and aR. Therefore, by Goodman

SR ¼ ½SLSUð1� RÞ�=½SUð1� RÞ þ SLð1þ RÞ� ð25Þ

The mode I stress range Dr tolerable by this component when ithas a uniaxial surface crack of depth a can thus be expressed by

Dr <DKR=uF

ffiffiffiffiffiffipap

0:752þ 2:02 awþ 0:37ð1� sin pa

2w Þ3

h isec pa

2w

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2wpa tan pa

2w

q� 1þ aR

a

� c=2h i1=c

ð26Þ

where w = 3.4 mm was the component width, and its g(a/w)geometry factor was obtained from [13].

Fig. 19 and 20 plot the maximum tolerable stress ranges (for/F = 1) under several R values. As the FCP threshold of thesestructural components was not available, it had to be estimatedby traditional engineering receipts to generate the curves pre-sented in those figures: Hence, their threshold has been assumedwithin the typical range for steels, 6 < DK0 < 12 MPa

pm [15,22],

supposing, as usual, that DKR ffi DK0 for R < 0 (since their reportedload histories did not include severe underloads). The lower limitFCP threshold estimations used for positive R loads areDKth(0 < R 6 0.17) = 6 MPa

pm, and DKth(R > 0.17) = 7�(1 – 0.85R)

MPap

m. Using g = 1.12 and DK0 = 6 MPap

m, these componentsshort crack characteristic value is thus estimated as a0 = 59 lm.In this way, Fig. 19 shows that if these components work e.g. underDr = 286 MPa and R = �0.12, they could tolerate cracks up toa ffi 105 lm. Similarly, under Dr = 176 MPa and R = 0.44, theywould tolerate cracks up to a ffi 150 lm. Fig. 20 uses semi-log coor-dinates to enhance this component small tolerance to short cracks.

Therefore, this simple (but sensible) model indicates that those(small) components are not too tolerant to 1D surface cracks. How-ever, as this conclusion is based on estimated properties, it is worthto study its sensibility to the assumed property values. Fig. 21shows the prediction range associated with the typical interval ex-pected for the estimated properties, enhancing how important it isto measure them to obtain less scattered predictions. Such rangeestimates can be quite useful for designers and quality controlengineers, for example they can help decide what to do when aproduction or operational accident damages the surface of other-wise well-behaved components.

However, this model has some limitations which must beconsidered. First, it can only be used to describe the behavior ofmacroscopic short cracks, since it is based in isotropic continuum

Fig. 19. Effect of a surface crack of size a in the largest stress range DrR(a) tolerableby a strip of width w = 3.4 mm loaded in mode I, for various R-ratios (it is assumedthat it has DK0 ¼ 6 MPa

pm and c = 6, thus a0 = 59 and a0.8 = 55 lm).

Fig. 20. Similar to Fig. 19, but with semi-log scale to enhance the short cracktolerance of this 3.4 mm wide strip. Note that small cracks with a < 30 lm havepractically no effect in its fatigue resistance.

Fig. 21. Typical steel threshold 6 < DK0 < 12 MPap

m and c exponent 1.5 < c < 8ranges influence in the largest mode I stress ranges Dr0 tolerated by thew = 3.4 mm strip, as a function of the 1D superficial crack size a.


mechanics and uses measured (or estimated) macroscopic materialproperties. Therefore, it can only be applied to short cracks (com-pared to aR) which are large in relation to the characteristic sizeof the intrinsic material anisotropy (e.g. its grain size). Smallercracks grow in a size scale where the material is anisotropic andusually inhomogeneous, thus their FCP is also affected at least bymicrostructural barriers, such as second phase particles or grainboundaries, meaning that their growth under fatigue cannot be as-sumed independent of such microstructural features. However, asgrains and second phase particles cannot be mapped in most prac-tical structural components anyway, such problems, in spite oftheir potential academic interest, are not really a major barrierfrom the fatigue design or structural integrity evaluation pointsof view.

But this model has another limitation which may be moreimportant for practical applications: Eq. (24) assumes that the(short) crack is unidimensional (1D) and grows without changingthe original plane of its faces, thus that it can be completely char-acterized solely by its depth a. But most short cracks are surfaceor corner cracks, which cannot be completely described in thissimplified way. These cracks tend to grow by fatigue at least intwo directions, maintaining their original plane when they areloaded under pure mode I conditions. In these cases, they canbe modeled as bidimensional (2D) cracks which grow both indepth and width. In reality, both long and short cracks (thesemeaning cracks not much larger than aR) only behave as 1D cracksafter having cut all the component width to become a throughcrack, with a more or less straight front which propagates in anapproximately uniform way. Thus, Eq. (24) must be adapted toconsider this fact.

Therefore, assuming that: (i) the cracks are loaded in pure modeI, under quasi-constant Dr and R conditions, with no major over-loads; (ii) material properties measured (or estimated) testing 1Dspecimens may be used to simulate the FCP behavior of 2D cracks;and (iii) 2D surface or corner cracks can be well modeled as havingan approximately elliptical front, then their SIF can be described bythe classical Newman–Raju equations [15,23]. In this case, it can beexpected that the component tolerance to cracks be given by:

Dr <DKR=f

ffiffiffiffiffiffipap

�Uaða; c;w; tÞ � ½1þ ðaR=aÞc=2�1=cgDKR=f

ffiffiffiffiffiffipcp

�Ucða; c;w; tÞ � ½1þ ðaR=cÞc=2�1=cg

(ð27Þ

For semi-elliptical surface cracks in a plate of thickness t, the SIFin the depth a and width c directions, KI,a = r

p(pa)�Ua and KI,c =

rp

(pc)�Uc, are given by:

KI;a ¼ rffiffiffiffiffiffipap

�Ua ¼ rffiffiffiffiffiffipap

� F �M=ffiffiffiffiQp

KI;c ¼ rffiffiffiffiffiffipap

�Uc ¼ rffiffiffiffiffiffipcp

� F � ðM=ffiffiffiffiQpÞ � a=c � G

Fðc=w; a=tÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffisec½ðpc=2wÞ

ffiffiffiffiffiffiffia=t

p�

q� ½1� 0:025½ðc=wÞ

ffiffiffiffiffiffiffia=t

p�2

þ0:06½ðc=wÞffiffiffiffiffiffiffia=t

p�4�

M ¼

1:13� 0:09 ac þ 0:89

0:2þa=c � 0:54h i

a2

t2

þ 0:5� 10:65þa=c þ 14ð1� a

c Þ24

h ia4

t4 ; a 6 c

c=aþ 0:04ðc=aÞ2 þ ðc=aÞ4:5ða=tÞ2 0:2� 0:11ða=tÞ2h i

; a > c

8>>>>><>>>>>:

Q ¼1þ 1:464ða=cÞ1:65

; a 6 c

1þ 1:464ðc=aÞ1:65; a > c

(

G ¼1:1þ 0:35ða=tÞ2; a 6 c

1:1þ 0:35ða=tÞ2ðc=aÞ; a > c

(

8>>>>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð28Þ

For quarter-elliptical corner cracks, the Ua and Uc geometryfunctions are:


Ua ¼ sec pc2w

ffiffiat

p� 0:752þ 2:02 c

w

ffiffiat

pþ 0:37 1� sin pc

2w

ffiffiat

p� � 3h i

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2wpc

ffiffita

qtanðpc

2w

ffiffiat

pÞ

r

1:08� 0:03 ac þ �0:44þ 1:06

0:3þa=c

� �at

� 2�þ �0:5þ 0:25 a

c þ 14:8 1� ac

� 15h i

at

� 4�

1:08þ0:15 atð Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ1:464ða=cÞ1:65p

Uc ¼ secðpc2w

ffiffiat

pÞ½0:752þ 2:02 c

w

ffiffiat

pþ 0:37ð1� sinðpc

2w

ffiffiat

pÞÞ3�

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi2wpc

ffiffita

qtanðpc

2w

ffiffiat

pÞ

r1:08� 0:03 a

c þ �0:44þ 1:060:3þa=c

� �at

� 2�

þ �0:5þ 0:25 ac þ 14:8ð1� a

c Þ15

h iðat Þ

4�

1:08þ0:4 atð Þ2ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

1þ1:464ða=cÞ1:65p

8>>>>>>>>>>>>>>>>>>>>>>>>>><>>>>>>>>>>>>>>>>>>>>>>>>>>:

ð29Þ

These complicated SIF functions enhance the operational advantageof treating the FCP threshold as a function of the crack size, DKth(a),as claimed above. For structural calculations and mechanical designpurposes, it is indeed relatively simple to use either Eqs. (24) and(27) to evaluate the influence of surface cracks on the componentfatigue strength. Moreover, it is not too difficult to adapt the 2Dequations to include notch effects. However, as this task is not ele-mentary either, it will be treated in a subsequent paper.

8. Conclusions

A generalized El Haddad–Topper–Smith’s parameter was usedto model the crack size dependence of the threshold stressintensity range for short cracks, as well as the behavior of non-propagating fatigue cracks. This dependence was used to estimatethe notch sensitivity factor q of semi-elliptical notches, from study-ing the propagation behavior of short non-propagating cracks thatmay initiate from their tips. The predicted notch sensitivitiesreproduced well the classical Peterson’s q estimates for circularholes or approximately semi-circular notches, but it was foundthat the notch sensitivity of elongated slits has a very strongdependence on the notch aspect ratio, defined by the ratio c/b ofthe semi-elliptical notch that approximates the slit shape havingthe same tip radius. These predictions were confirmed by experi-mental measurements of the re-initiation life of long fatigue cracksrepaired by introducing a stop-hole at their tips, using their calcu-lated Kf and appropriate N procedures. Based on this promisingperformance, a criterion to evaluate the influence of small or largesurface cracks in the fatigue resistance was proposed.

Acknowledgements

CNPq, the Brazilian Research Council, has provided researchscholarships for the Brazilian authors.

References

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[4] Wu H, Imad A, Noureddine B, Castro JTP, Meggiolaro MA. On the prediction ofthe residual fatigue life of cracked structures repaired by the stop-holemethod. Int J Fatigue 2010;32:670–7.

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