International Journal of Fatigue 82 (2016) 540–549

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International Journal of Fatigue

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

From NASGRO to fractals: Representing crack growth in metals

http://dx.doi.org/10.1016/j.ijfatigue.2015.09.0090142-1123/� 2015 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. Tel.: +39 0583 4326 604; fax: +39 0583 4326 565.E-mail address: marco.paggi@imtlucca.it (M. Paggi).

R. Jones a, F. Chen a, S. Pitt a, M. Paggi b,⇑, A. Carpinteri caCentre of Expertise in Structural Mechanics, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Victoria 3800, Australiab IMT Institute for Advanced Studies Lucca, Piazza San Francesco 19, 55100 Lucca, ItalycDepartment of Structural, Geotechnical and Building Engineering, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy

a r t i c l e i n f o a b s t r a c t

Article history:Received 9 June 2015Received in revised form 7 September 2015Accepted 10 September 2015Available online 25 September 2015

Keywords:Fatigue crack growth modelsFractalityFatigue experimentsProfilometric analysisRoughness

This paper presents the results of an extensive experimental analysis of the fractal properties of fatiguecrack rough surfaces. The analysis of the power-spectral density functions of profilometric traces shows apredominance of the box fractal dimension D = 1.2. This result leads to a particularization of the fatiguecrack growth equation based on fractality proposed by the last two authors which is very close to thegeneralized Frost–Dugdale equation proposed by the first three authors. The two approaches, albeit basedon different initial modelling assumptions, are both very effective in predicting the crack growth rate ofshort cracks.

� 2015 Elsevier Ltd. All rights reserved.

1. Introduction

The breakdown of physical similitude in fatigue crack growthhas been pioneeringly revealed in the early work of Barenblattand Botvina [1]. Since then, several attempts have been made inorder to understand the impact of this incomplete similarity onthe Paris and Wöhler representations of fatigue. Experimental con-firmation of size-scale effects has been reported by Ritchie [2] andgeneralized theories of fatigue based on dimensional analysis andincomplete similarity have been proposed [3–6] to interpret thisanomalous phenomenon. Using these concepts, a dependency ofthe Paris’ law on the grain size has also been put into evidenceby Plekhov et al. [7].

Supported by these dimensional analysis considerations, a newfatigue crack growth theory based on fractality of crack profiles hasbeen developed [3,6,8–12] to interpret the (experimentallyevidenced) anomalous crack growth rate of short cracks andcrack-size effects on the fatigue threshold, facts that were not fullyexplained by previous theories. At the same time and indepen-dently, Jones and coworkers [13–17] have shown that the anoma-lous crack growth rate of short cracks can also be captured using ageneralization of the pioneering Frost–Dugdale equation [18]. Ithas also been shown [19] that the NASGRO crack growth equationwith the constants determined form tests on long cracks can be

used to represent the growth of short cracks if closure effects areset to near zero and the threshold is set to a very small value.

In the present contribution, an extensive experimental analysisof the fractal properties of fatigue crack rough surfaces ispresented.

Given the fact proven in the previous publications that the gen-eralized fatigue crack growth equation based on fractality [8] andthe generalized Frost–Dugdale equation [14] are both viablemethods to model and simulate the anomalous fatigue behaviourof short cracks, in this study we attempt at applying the twoapproaches to a set of experimental data whose fatigue curvesare known and from which it is possible to compute the profilefractal dimension, a key parameter in the fractal model. Analysisof the power-spectral density functions of profilometric tracesshows a predominance of the fractal dimension D = 1.2. This resultleads to a particularization of the fatigue crack growth equationbased on fractality which is very close to the generalizedFrost–Dugdale equation, thus explaining why the two approaches,albeit based on different initial modelling assumptions, are bothvery effective in predicting the crack growth rate of short cracks.Finally, by analogy with the generalized Frost–Dugdale equation,a generalization of the fractal approach in case of crack closure isproposed.

2. Crack growth equations

The recent state-of-the-art review of fatigue crack growth anddamage tolerance [19] revealed that the growth of both long and

Fig. 1. Comparison of the various da/dN versus DK test data for AA7050-T7451,from Jones et al. [25].

R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549 541

short cracks could be represented by the NASGRO crack growthequation [20], viz:

da=dN ¼ DDKðm�pÞeff ðDKeff � DKeff ;thrÞp=ð1� Kmax=AÞq ð1Þ

where D is a constant,DKeff is an effective stress intensity factor andthe terms DKthr and A are best interpreted as parameters chosen soas to fit the measured da/dN versus DK data. The term DKeff is gen-erally expressed in the form

DKeff ¼ Kmax � Kop ð2Þwhere Kop is defined as the value of the stress intensity factor atwhich the crack first opens. A range of formulae for determiningKop have been suggested in the related literature. However, themost widely used formulae for Kop was proposed by Newman[21]. This formulation is used in the crack growth computer pro-grams FASTRAN, NASGRO and AFGROW. It was also shown that,as first suggested by McEvily et al. [22], the crack opening stressintensity factor Kop decays to Kmin as the crack length reduces.One way to achieve this is via the McEvily et al. [22] formulation:

DKop ¼ ð1� e�kaÞDKopl ð3Þwhere DKopl is the long crack value of DKop (=Kop � Kmin) and k is amaterial dependent constant.

Jones [19] and Jones et al. [23] have shown that crack growth ina large cross-section of aerospace and rail materials can be cap-tured by Eq. (1) with m = p and q = p/2 and that in such cases poften takes a value that is approximately 2. Jones et al. [24] havealso shown that this approach applies to Mode I, Mode II andMixed Mode I/II disbonding in adhesively bonded structures.

On the other hand Jones et al. [13–17] have shown that thecrack growth can also be captured using the Generalized Frost–Dugdale equation, viz:

da=dN ¼ C�að1�c=2ÞðDjÞc � ðda=dNÞ0 ð4Þwhere Dj is a crack driving force

Dj ¼ DKpðKmaxÞð1�pÞ ð5ÞHere p, c, which in most instances is approximately equal to 3, andC* are constants, and the term (da/dN)0 reflects both the fatiguethreshold and the nature of the notch (defect/discontinuity) fromwhich cracking initiates.

It was also shown that, in numerous instances, crack growth canbe captured equally well by using Eqs. (1) or (4). Figs. 1–5 illustratethis by presenting crack growth data for 7050-T7451 and the MCand MB wheel steels replotted as per Eqs. (1) and (4). This phe-nomenon is also seen by comparing the crack growth representa-tions presented in Jones et al. [25], which presents a crackgrowth data for a range of materials expressed as per Eq. (1), andby Jones et al. [17], which presents crack growth data for a rangeof materials expressed as per Eq. (4).

3. Fatigue crack growth of fractal cracks

Having established that the Generalized Frost–Dugdale equa-tion and the Nasgro equation can effectively be used to describethe phenomenon of fatigue crack growth over a wide range of cracksizes, let us now examine the fractal representation of cracks. Letus consider a simple test such that the crack grows in pure ModeI, i.e. straight ahead, as per Fig. 6.

In this case the near tip stress field is proportional to r�1/2,where r is the radial distance from the crack tip.

Now consider the case when the crack surface is rough androughness is modelled as a fractal, as motivated by micromechan-ical considerations and molecular dynamics simulations, see Figs. 7and 8.

In this case if the crack has a fractal dimension D (1 < D < 2),then the near crack-tip stress field takes the form [27–29]

r / rðD�2Þ=2 ð6Þso that the nature of the near tip stress field is fundamentally chan-ged and no longer has the classical linear elastic fracture mechanics(LEFM) 1/

pr singularity.

The crack tip stress field only equates to the classical linear elas-tic fracture mechanics (LEFM) solution when the crack profile is aline in Euclidean space and thus has a fractal dimension D = 1. Inall other cases the near tip stress field differs from the classical linearelastic fracture mechanics solution. In such cases, Spagnoli [3] andthen Paggi and Carpinteri [8] have shown that crack growth dependsboth on the stress intensity factor K and the crack length viz:

da=dN ¼ C�=Dað1�DÞð1þm=2ÞðDKÞm ð7Þwhere m and C* are constants. This law is formally identical to theclassical Paris law except that the coefficient multiplying thestress-intensity factor range is now dependent on the crack lengtha, whereas in the Paris law [30,31] it was assumed to be a materialconstant. At this stage it should be noted that Bouchaud [32] hasshown that for small cracks the fractal dimension D is slightly vary-ing and is about 1.2 in several natural examples. This finding wassubstantiated by Mandelbrot [33].

Frost and Dugdale [34] were the first to note that for manymaterials the crack growth rate is often (approximately) propor-tional to the cube of the stress amplitude (Dr)3. This has been con-firmed by a range of investigations, viz: Frost et al. [35], Barter et al.[36], Molent et al. [37], Tsouvalis et al. [38], Jones et al. [17], and asa result is now adopted in the Royal Australian Air Force (RAAF)Structural Assessment Manual, Main [39], the RAAF P3C (Orion)repair assessment manual, Duthie and Matricciani [40] and Aylinget al. [41] and the Association of State Highway and Officials [42].

In addition to the assumption of cubic stress dependency, if wecan also experimentally validate that D in Eq. (7) is approximatelyequal to 1.2, then Eq. (7) yields

da=dN ¼ Ca�1=2ðDKÞ3 ð8Þwhich is a subset of the Generalized Frost–Dugdale equation, viz:

da=dN ¼ C�a�1=2½DKpðKmaxÞð1�pÞ�3 � ðda=dNÞ0 ð9Þ

Fig. 2. Nasgro representation of crack growth in 7050-T7451, from Jones et al. [25].

y = 6.1E-12x

1.0E-10

1.0E-09

1.0E-08

1.0E-07

1.0E-06

1.0E-05

10 100 1000 10000 100000 1000000 10000000

da/d

N (m

/cyc

le)

((ΔK/(1−R)0.2)3/a0.5 (MPa3 m)

Short crack Test 2 R = 0.7Short crack Test 2 R =0.1Linear fitDSTO CT R = 0.1DSTO CT R = 0.8DSTO CT R = 0.5DSTO MT R = 0.2Short crack Test 3 R = 0.1Short crack Test 3 R = 0.7Short crack Test 4 R = 0.5

Fig. 3. Generalised Frost–Dugdale representation of crack growth in 7050-T7451,from Jones et al. [16].

y = 1.26E-09x -1.37E-05R² = 9.97E-01

0.0E+00

5.0E-05

1.0E-04

1.5E-04

2.0E-04

0 50000 100000 150000 200000

da/d

N (m

m/c

ycle

)

ΔK3/a1/2 (MPa3 metres)

MC test1

MC test 2

MC test 3

MB test 1

MB test 2

MB test 3

Fig. 4. Generalised Frost–Dugdale representation of crack growth in MC and MBwheel steels, from Jones et al. [17].

y = 3.5e-10 x2.01

R = 0.980

Fig. 5. Nasgro representation of crack growth in MC and MB wheel steels, fromJones et al. [17].

Fig. 6. A classical Mode I fatigue crack growth.

542 R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549

Fig. 7. Models of a fractal like crack, from Saether and Ta’asan [26].

Fig. 8. Plan and surface view of a fractal like cracks, from Saether and Ta’asan [26].

Surface topology of the area:Fractal Box dimension D =1.19

Area scaned:1.231*0.937mm

Fig. 9. A view of the surface profile representative of the failure specimens and a typical local detail on the failure surface.

R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549 543

Noting in fact that Kmax =DK/(1 � R), where R = Kmin/Kmax is theloading ratio, Eq. (9) can be recast as follows:

da=dN ¼ C�a�1=2DK3=ð1� RÞ3ð1�pÞ � ðda=dNÞ0 ð10ÞEq. (10) matches Eq. (8) exactly if p = 1 and if the term (da/dN)0

is vanishing. However, in principle, a power-law dependency fromthe dimensionless number (1 � R) is admissible from dimensionalanalysis considerations, as theoretically demonstrated in Ciavarellaet al. [4] and confirmed by several semi-empirical fatigue crackgrowth models (see e.g. Roberts and Erdogan [43], Klesnil andLukas [44], Hojo et al. [45], Liu and Chen [46], Walker [47], and

Radhakrishnan [48]). Therefore, the more general expression inEq. (10) would appear to be an extension of the fractal crackgrowth law to account for mean stress effects and for cases show-ing a threshold term. The validity of the assumption that D can beapproximated by 1.2 for fatigue rough surfaces is investigated inthe next section according to a profilometric analysis.

4. Fractal dimension measurements

Jones et al. [16] presented the results of crack growth fromsmall near micrometer size initial defects in 7050-T7451, which

Table 1Fractal dimension results for 7050-T7451 aluminium alloys.

Point Specimen number

TD ET D2 E

1 1.182 1.200 1.425a 1.2462 1.220 1.229 1.280 1.2223 1.299 1.267 1.243 1.2554 1.122 1.194 1.199 1.2705 1.264 1.317 1.167 1.2606 1.329a 1.295 1.144 1.2227 1.162 1.294 1.120 1.2638 1.236 1.335 1.218 1.3549 1.107 1.197 1.174 1.286

10 1.245 1.326a 1.163 1.31911 1.299 1.295 1.257 1.30412 1.227 1.332 1.129 1.25413 1.248 1.292 1.183 1.320a

14 1.173 1.315 1.183 1.30515 1.140 1.240 1.277 1.27216 1.159 1.263 1.150 1.30517 1.287 1.296 1.123 1.26318 1.333a 1.180 1.159 1.28119 1.490a 1.194 1.101 1.306a

20 1.176 1.219 1.174 1.30821 1.246 1.314a 1.190 1.31422 1.299 1.218 1.167 1.28023 1.190 1.178 1.128 1.25724 1.184 1.310a 1.140 1.334a

25 1.252 1.256 1.029 1.373a

Ave value of D 1.23 1.26 1.18 1.29

a These values are associated with the transition to rapid crack growth.

8 10

11 12

Fig. 10. The failed surface and the as

544 R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549

is used in both the F/A-18 Classic Hornet and the Super Hornet. Theexperimental setup, which is described in more detail in theAppendix A, found that crack growth in this material could bedescribed by Eq. (4) with p = 0.2.

The failure surfaces of four specimens, designations TD, ET, D2and E, were examined and the variation of the fractal dimensionwas measured as described in the Appendix A. A total of twenty-five equally distributed points on each surface were examined.The size of each of the locations examined was approximatelyranging from 1.2 by 0.9 mm and the results for one such locationare shown in Fig. 9. The resulting values of D obtained from a studyof the profilometric traces are given in Table 1.

Away from the regions where the failure surface transitioned torapid crack growth, the value of Dwas approximately constant andvaried from approximately 1.17 to 1.3. The mean values of D forspecimens TD, ET, D2 and E were 1.23, 1.26, 1.19 and 1.29 whichyields a mean value for all of the specimens of approximately1.24. As such this example supports Bouchard’s hypothesis thatthe fatigue crack growth surfaces have a fractal dimension ofapproximately 2.2, i.e., their profiles have D = 1.2.

Fractal dimension measurements were also performed on an in-service fatigue crack in a Queensland Rail (now Aurizon) wheel,(see Fig. 10). The resultant fractal dimensions are given in Table 2.Here it should be noted that locations 6–10 are associated withrapid crack growth whilst locations 1–5 are associated with slowfatigue crack growth. Again, these results further confirm thatthe fractal dimension of profiles extracted from fatigue crackgrowth surfaces is approximately equal to 1.2.

1 2

3 5

4

Rail Wheel Profilometer Loca�ons

6 7

9

sociated measurement locations.

Table 2Fractal dimension D at various locations on the fatigue crack surface.

Fractal dimension, D

1 2 3 4 5 6a 7 8 9 10b 11b 12c

X 1.19 1.20 1.21 1.17 1.19 1.31 1.34 1.31 1.36 1.21 1.20 1.26Y 1.19 1.19 1.29 1.18 1.24 1.32 1.37 1.34 1.46 1.18 1.25 1.23

a At this point the crack growth is rapid and the striations were widely spaced.b Profilometer reading was taken on an upward slope.c Profilometer reading was taken on a downward slope.

Fig. 11. Compact specimen test geometry.

2 3

5

9

4

6 7 8

10

1

X

X

y

Fig. 12. Locations of the fractal measurements.

Table 3Fractal dimension D at various locations on the surface.

Fractal dimension

1 2 3 4 5 6 7a 8 9 10b

X 1.28 1.21 1.24 1.29 1.24 1.14 1.35 1.35 1.21 1.28Y 1.28 1.32 1.26 1.29 1.29 1.30 1.29 1.23 1.29 1.22

a From this point onwards crack growth was very rapid.b Surface too rough to obtain an accurate reading.

2 564

13

Fig. 13. A fatigue failure in a BHP bolt.

Table 4Fractal dimension associated with the BHP bolt.

Fractal dimension

1 2 3 4 5 6

X 1.232 1.239 1.205 1.168 1.258 1.218Y 1.241 1.193 1.163 1.161 1.225 1.225

R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549 545

The fractal dimension associated with laboratory fatigue cracktests on an ASTM standard compact tension (CT) MC wheel steelspecimens [14] was also measured, see Fig. 11. The locations atwhich the fractal dimension were measured are shown in Fig. 12and the resultant values are given in Table 3 where we see a fractaldimension in the direction of crack growth of approximately 1.28.

As such, the value associated with the lab tests and with the in-service cracking are very close.

To further evaluate the nature of the fractal dimension associ-ated with in-service cracking, tests were performed on a fatiguefailure in a bolt from a BHP mining machine, see Fig. 13. Here

1 2

3 6

5 4

x

Squat

Fig. 14. The first rail squat fatigue surface.

1 2

6 5 4

3

Fig. 15. The second rail squat fatigue surface.

Table 5Fractal dimension associated with the first rail squat.

Fractal dimension, D

1 2 3 4 5 6

X 1.250 1.277 1.17 1.200 1.226 1.190Y 1.260 1.272 1.18 1.292 1.255 1.160

In each case the size of the region investigated was �228 lm by 300 lm.

Table 6Fractal dimension associated with rail squat number 2.

Fractal dimension, D

1 2 3 4a 5 6

X 1.18 1.227 1.215 1.284 1.168 1.171Y 1.19 1.230 1.243 1.334 1.187 1.234

a At this point we are seeing very rapid crack growth.

546 R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549

we again see a near uniform fractal dimension with a mean valueof approximately 1.2, see Table 4. Measurements were also madeon squats in two different head hardened rail specimens, seeFigs. 14 and 15. The associated fractal dimension measurementsare given in Tables 5 and 6.

We thus see that all of the specimens examined have revealed afractal dimension in the direction of crack growth that was in therange between 1.17 and 1.29. Here it is important to pinpoint thatwith a fractal dimension D = 1.2 andm = 3, the various growth laws

Fig. A1. The Geometry of the dog-bone test specimens.

R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549 547

presented in works by Jones et al. [14,15], Jones and Forth [49],Paggi and Carpinteri [8] and Carpinteri and Paggi [5,6,9,10]coincide.

5. Conclusions

The anomalous behaviour of short-cracks in fatigue, which vio-lates the assumption of physical similitude behind the Paris’ law, iscrucial for the design of safe aerospace and railways applications,with well ascertain lifetimes. So far, two independent approachesin the literature, namely the generalized Frost–Dugdale equationand the generalized fractal approach, have shown to be able to sat-isfactorily predict crack fatigue growth of short cracks. To under-stand why the two methods lead to very similar results, althoughbeing based on very different assumptions, an experimental cam-paign has been undertaken by analyzing the fractal dimension offatigue rough surfaces at failure for a wide range of materialswhere short-crack effects were reported in previous publications.

The findings presented in this paper show that rough profilesextracted from fatigue crack growth surfaces have a fractal dimen-sion approximately equal to 1.2. Since for the MC and 7050-T7451materials the exponent m is approximately equal to 3, we foundthat the Generalized Frost–Dugdale equation and the fractal basedcrack growth equation are linked and provide very close predic-tions of fatigue crack growth rates.

Moreover, from a detailed analysis of the equations, an exten-sion of the current fractal crack growth equation to account for Rratio effects is suggested, viz, ex:

da=dN ¼ C�að1�DÞð1þm=2Þ½DKpðKmaxÞð1�pÞ�m � ðda=dNÞ0 ð11ÞWe also see that this formulation and the Nasgro equation are

often both capable of representing crack growth in a range of mate-rials, see Jones et al. [17] and Jones et al. [23]. For sub-micrometercracks, Bouchaud revealed a fractal box dimension of approxi-mately 1.5, in line with multi-fractal properties of cracks put for-ward in Carpinteri [50]. In this case, Eq. (11) reduces to

da=dN ¼ C�a�1=2ð1þm=2Þ½DKpðKmaxÞð1�pÞ�m � ðda=dNÞ0 ð12Þwhich is a particularization of the generalized fatigue crack growthmodel based on fractal concepts proposed in Carpinteri and Paggi[11] for the growth of short cracks, where any possible fractaldimension between 1 and 2 was allowed.

Although Eq. (12) appears to be a straightforward generaliza-tion of the fractal approach by analogy with the generalizedFrost–Dugdale equation, further research is deemed to be requiredto provide a general theoretical framework to account for the load-ing ratio effects in the fractal modelling of fatigue crack growth.

Acknowledgement

MP and AC would like to thank the Italian Ministry of Education,University and Research for his support to the project FIRB 2010Future in Research ‘‘Structural mechanics models for renewableenergy applications” (RBFR107AKG).

Appendix A. details of the experimental tests and method forthe computation of the fractal dimension

The measurement of fractal dimension is carried out by aVEECO NT1100 Microscope in conjunction with the associatedWYKO Vision32 computer program, which (together) are denotedas the WYKO surface profiler system. The WYKO surface profilersystem is a non-contact optical profiler that offers two modes formeasuring a wide range of surface topologies. These twoapproaches are briefly explained as follows:

– PSI mode (Phase-shifting interferometry) – This mode enablesthe measurement of smooth surfaces.

– VSI mode (Vertical scanning interferometry) – This modeenables the measurement of rough surfaces.

The VSI mode was used in this study due to the rough nature ofthe various fatigue fracture surfaces. In this mode the range ofheights that the profiler can accurately measure is approximately2 mm. The range in depth that the profiler can accurately measurein this mode is approximately 5 nm to 2 mm (see the manualsVeeco [51] and Wyko [52] for more details).

In VSI mode the value of fractal dimension is obtained by meansof the power spectrummethod which is introduced as Power Spec-tral Density (PSD) in the WYKO surface profiler system. It usesFourier decomposition of the measured surface profile into itscomponent spatial frequencies f. The PSD function calculates thepower spectral densities for each horizontal (X) or vertical (Y) linein the data and then averages all X and Y profiles. The range ofspatial frequencies measured is limited by the objective field ofview and spatial sampling. The orientation of a surface with non-random features also affects the average X and Y PSD plots. Theseaverage PSD data is also used to calculate fractal box dimension.The average PSD is modelled as

PSDðf Þ ¼ A=f B ðA:1Þwhere A, B are constants and f is the spatial frequency. The programuses a linear least-squares fit to determine A and B, viz:

logðPSDðf ÞÞ ¼ logðAÞ � B logðf Þ ðA:2ÞThe interval of frequencies used in the above equation is

defined by the ISO standard as:

1=D < f < 1=C ðA:3Þwhere 1/D is the program’s low frequency cut-off option and 1/C isthe program’s high frequency cut-off option. The fractal box dimen-sion D is then calculated as

D ¼ ð5� BÞ=2 ðA:4ÞFurther details are given in the Wyko [52] Vision32 users man-

ual and are discussed also in Zavarise et al. [53]. In the presentstudy, each profilometric trace was taken over the same physicaldistance 0.9–1.2 mm. The array size used was 736 � 480.

Regarding the short crack test specimens used in thisstudy, they were made from 7050-T7451 which is high strengthaluminium alloy with a Young modulus of approximately71,000 MPa and its Poisson’s ratio is 0.33. The specimens had a‘‘dog bone” geometry as shown in Fig. A1 with the surface etchedto create small pits from which fatigue cracks could initiate. Thespecimens were 11 mm thick and 42 mm wide, see Jones et al.[49] for more details. To assist with quantitative fractography thespecimens were tested under repeated marker block loading, seeFig. A2. Each block consisted of a number (C) cycles of constantamplitude loading with a maximum stress and stress ratio RI fol-lowed by M constant amplitude load cycles with the same maxi-

Fig. A2. Schematics of cycles for marker band creation.

Table A1Stress information, in MPa, for Specimen TD.

Specimen TD rmax (Mpa) rmin (Mpa) Dr (Mpa) Stress ratio R No. cycles

Section C 250 25 225a 0.1 5000Section M 250 175 75 0.7 1500

Table A2Stress information, in MPa, for Specimen ET.

Specimen ET rmax rmin Dr Stress ratio R No. cycles

Section C 225 22.5 202.5 0.1 5000Section M 225 157.5 67.5 0.7 3000

Table A3Stress information, in MPa, for Specimen D2.

Specimen D2 rmax rmin Dr Stress ratio R No. cycles

Section C 225 �67.5 292.5 �0.3 5000Section M 225 112.5 112.5 0.5 3000

Table A4Stress information, in MPa, for Specimen E.

Specimen E rmax rmin Dr Stress ratio R No. cycles

Section C 200 �60 260 �0.3 7000Section M 200 100 100 0.5 7000

548 R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549

mum stress as per the C cycles and a stress ratio R2. The stresses inthe working section experienced by specimens TD, ET, D2 and E aregiven in Tables A1–A4.

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