From NASGRO to Fractals: Representing Crack Growth in Metals ⇑ R

International Journal of Fatigue 82 (2016) 540–549

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International Journal of Fatigue

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From NASGRO to fractals: Representing crack growth in metals ⇑ R. Jones a, F. Chen a, S. Pitt a, M. Paggi b, , A. Carpinteri c a Centre of Expertise in Structural Mechanics, Department of Mechanical and Aerospace Engineering, Monash University, Clayton, Victoria 3800, Australia b IMT Institute for Advanced Studies Lucca, Piazza San Francesco 19, 55100 Lucca, Italy c Department of Structural, Geotechnical and Building Engineering, Politecnico di Torino, C.so Duca degli Abruzzi 24, 10129 Torino, Italy article info abstract

Article history: This paper presents the results of an extensive experimental analysis of the fractal properties of fatigue Received 9 June 2015 crack rough surfaces. The analysis of the power-spectral density functions of profilometric traces shows a Received in revised form 7 September 2015 predominance of the box fractal dimension D = 1.2. This result leads to a particularization of the fatigue Accepted 10 September 2015 crack growth equation based on fractality proposed by the last two authors which is very close to the Available online 25 September 2015 generalized Frost–Dugdale equation proposed by the first three authors. The two approaches, albeit based on different initial modelling assumptions, are both very effective in predicting the crack growth rate of Keywords: short cracks. Fatigue crack growth models Ó 2015 Elsevier Ltd. All rights reserved. Fractality Fatigue experiments Profilometric analysis Roughness

1. Introduction used to represent the growth of short cracks if closure effects are set to near zero and the threshold is set to a very small value. The breakdown of physical similitude in fatigue crack growth In the present contribution, an extensive experimental analysis has been pioneeringly revealed in the early work of Barenblatt of the fractal properties of fatigue crack rough surfaces is and Botvina [1]. Since then, several attempts have been made in presented. order to understand the impact of this incomplete similarity on Given the fact proven in the previous publications that the gen- the Paris and Wöhler representations of fatigue. Experimental con- eralized fatigue crack growth equation based on fractality [8] and firmation of size-scale effects has been reported by Ritchie [2] and the generalized Frost–Dugdale equation [14] are both viable generalized theories of fatigue based on dimensional analysis and methods to model and simulate the anomalous fatigue behaviour incomplete similarity have been proposed [3–6] to interpret this of short cracks, in this study we attempt at applying the two anomalous phenomenon. Using these concepts, a dependency of approaches to a set of experimental data whose fatigue curves the Paris’ law on the grain size has also been put into evidence are known and from which it is possible to compute the profile by Plekhov et al. [7]. fractal dimension, a key parameter in the fractal model. Analysis Supported by these dimensional analysis considerations, a new of the power-spectral density functions of profilometric traces fatigue crack growth theory based on fractality of crack profiles has shows a predominance of the fractal dimension D = 1.2. This result been developed [3,6,8–12] to interpret the (experimentally leads to a particularization of the fatigue crack growth equation evidenced) anomalous crack growth rate of short cracks and based on fractality which is very close to the generalized crack-size effects on the fatigue threshold, facts that were not fully Frost–Dugdale equation, thus explaining why the two approaches, explained by previous theories. At the same time and indepen- albeit based on different initial modelling assumptions, are both dently, Jones and coworkers [13–17] have shown that the anoma- very effective in predicting the crack growth rate of short cracks. lous crack growth rate of short cracks can also be captured using a Finally, by analogy with the generalized Frost–Dugdale equation, generalization of the pioneering Frost–Dugdale equation [18].It a generalization of the fractal approach in case of crack closure is has also been shown [19] that the NASGRO crack growth equation proposed. with the constants determined form tests on long cracks can be

2. Crack growth equations

⇑ Corresponding author. Tel.: +39 0583 4326 604; fax: +39 0583 4326 565. The recent state-of-the-art review of fatigue crack growth and E-mail address: [email protected] (M. Paggi). damage tolerance [19] revealed that the growth of both long and http://dx.doi.org/10.1016/j.ijfatigue.2015.09.009 0142-1123/Ó 2015 Elsevier Ltd. All rights reserved. R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549 541 short cracks could be represented by the NASGRO crack growth equation [20], viz: = ¼ D ðmpÞðD D Þp=ð = Þq ð Þ da dN D Keff Keff Keff;thr 1 Kmax A 1 where D is a constant, DKeff is an effective stress intensity factor and the terms DKthr and A are best interpreted as parameters chosen so as to ﬁt the measured da/dN versus DK data. The term DKeff is gen- erally expressed in the form

DKeff ¼ Kmax Kop ð2Þ where Kop is deﬁned as the value of the stress intensity factor at which the crack ﬁrst opens. A range of formulae for determining

Kop have been suggested in the related literature. However, the most 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 stress intensity factor Kop decays to Kmin as the crack length reduces. One way to achieve this is via the McEvily et al. [22] formulation:

ka DKop ¼ð1 e ÞDKopl ð3Þ

D where DKopl is the long crack value of DKop (=Kop Kmin) and k is a Fig. 1. Comparison of the various da/dN versus K test data for AA7050-T7451, material dependent constant. from Jones et al. [25]. Jones [19] and Jones et al. [23] have shown that crack growth in a large cross-section of aerospace and rail materials can be cap- In this case if the crack has a fractal dimension D (1 < D < 2), tured by Eq. (1) with m = p and q = p/2 and that in such cases p then the near crack-tip stress field takes the form [27–29] often takes a value that is approximately 2. Jones et al. [24] have r / ðD2Þ=2 ð Þ also shown that this approach applies to Mode I, Mode II and r 6 Mixed Mode I/II disbonding in adhesively bonded structures. so that the nature of the near tip stress field is fundamentally chan- On the other hand Jones et al. [13–17] have shown that the ged and nop longer has the classical linear elastic fracture mechanics crack growth can also be captured using the Generalized Frost– (LEFM) 1/ r singularity. Dugdale equation, viz: The crack tip stress field only equates to the classical linear elas- c= Þ c da=dN ¼ C að1 2 ðDjÞ ðda=dNÞ ð4Þ tic fracture mechanics (LEFM) solution when the crack profile is a 0 line in Euclidean space and thus has a fractal dimension D =1.In where Dj is a crack driving force all other cases the near tip stress field differs from the classical linear ð Þ elastic fracture mechanics solution. In such cases, Spagnoli [3] and Dj ¼ DKpðK Þ 1 p ð5Þ max then Paggi and Carpinteri [8] have shown that crack growth depends Here p, c, which in most instances is approximately equal to 3, and both on the stress intensity factor K and the crack length viz: * C are constants, and the term (da/dN) reflects both the fatigue ð Þð þ = Þ 0 da=dN ¼ C =Da 1 D 1 m 2 ðDKÞm ð7Þ threshold and the nature of the notch (defect/discontinuity) from which cracking initiates. where m and C* are constants. This law is formally identical to the It was also shown that, in numerous instances, crack growth can classical Paris law except that the coefficient multiplying the be captured equally well by using Eqs. (1) or (4). Figs. 1–5 illustrate stress-intensity factor range is now dependent on the crack length this by presenting crack growth data for 7050-T7451 and the MC a, whereas in the Paris law [30,31] it was assumed to be a material and MB wheel steels replotted as per Eqs. (1) and (4). This phe- constant. At this stage it should be noted that Bouchaud [32] has nomenon is also seen by comparing the crack growth representa- shown that for small cracks the fractal dimension D is slightly vary- tions presented in Jones et al. [25], which presents a crack ing and is about 1.2 in several natural examples. This finding was growth data for a range of materials expressed as per Eq. (1), and substantiated by Mandelbrot [33]. by Jones et al. [17], which presents crack growth data for a range Frost and Dugdale [34] were the first to note that for many of materials expressed as per Eq. (4). materials the crack growth rate is often (approximately) proportional to the cube of the stress amplitude (Dr)3. This has been con- 3. Fatigue crack growth of fractal cracks 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 as Having established that the Generalized Frost–Dugdale equa- a result is now adopted in the Royal Australian Air Force (RAAF) tion and the Nasgro equation can effectively be used to describe Structural Assessment Manual, Main [39], the RAAF P3C (Orion) the phenomenon of fatigue crack growth over a wide range of crack repair assessment manual, Duthie and Matricciani [40] and Ayling sizes, let us now examine the fractal representation of cracks. Let et al. [41] and the Association of State Highway and Officials [42]. us consider a simple test such that the crack grows in pure Mode In addition to the assumption of cubic stress dependency, if we I, i.e. straight ahead, as per Fig. 6. can also experimentally validate that D in Eq. (7) is approximately In this case the near tip stress field is proportional to r1/2, equal to 1.2, then Eq. (7) yields = where r is the radial distance from the crack tip. da=dN ¼ Ca 1 2ðDKÞ3 ð8Þ Now consider the case when the crack surface is rough and roughness is modelled as a fractal, as motivated by micromechan- which is a subset of the Generalized Frost–Dugdale equation, viz: ical considerations and molecular dynamics simulations, see Figs. 7 = ¼ 1=2½D pð Þð1pÞ3 ð = Þ ð Þ and 8. da dN C a K Kmax da dN 0 9 542 R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549

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

Short crack Test 2 R = 0.7 Short crack Test 2 R =0.1 1.0E-05 Linear fit DSTO CT R = 0.1 DSTO CT R = 0.8 DSTO CT R = 0.5 1.0E-06 DSTO MT R = 0.2 Short crack Test 3 R = 0.1 Short crack Test 3 R = 0.7 Short crack Test 4 R = 0.5 1.0E-07

1.0E-08 da/dN (m/cycle) y = 6.1E-12x 2.01 1.0E-09 y = 3.5e-10 x R = 0.980

1.0E-10 10 100 1000 10000 100000 1000000 10000000 ((ΔK/(1−R)0.2)3/a0.5 (MPa3 m)

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

2.0E-04 Fig. 5. Nasgro representation of crack growth in MC and MB wheel steels, from MC test1 Jones et al. [17]. MC test 2

MC test 3 1.5E-04 MB test 1 da/dN (mm/cycle) MB test 2

MB test 3 1.0E-04

y = 1.26E-09x -1.37E-05 R² = 9.97E-01

5.0E-05

0.0E+00 0 50000 100000 150000 200000 ΔK3/a1/2 (MPa3 metres)

Fig. 4. Generalised Frost–Dugdale representation of crack growth in MC and MB wheel steels, from Jones et al. [17]. Fig. 6. A classical Mode I fatigue crack growth. R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549 543

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.

Noting in fact that Kmax = DK/(1 R), where R = Kmin/Kmax is the Radhakrishnan [48]). Therefore, the more general expression in loading ratio, Eq. (9) can be recast as follows: Eq. (10) would appear to be an extension of the fractal crack growth law to account for mean stress effects and for cases show- = ¼ 1=2D 3=ð Þ3ð1pÞ ð = Þ ð Þ da dN C a K 1 R da dN 0 10 ing a threshold term. The validity of the assumption that D can be approximated by 1.2 for fatigue rough surfaces is investigated in Eq. (10) matches Eq. (8) exactly if p = 1 and if the term (da/dN)0 is vanishing. However, in principle, a power-law dependency from the next section according to a proﬁlometric analysis. the dimensionless number (1 R) is admissible from dimensional analysis considerations, as theoretically demonstrated in Ciavarella 4. Fractal dimension measurements et al. [4] and conﬁrmed by several semi-empirical fatigue crack growth models (see e.g. Roberts and Erdogan [43], Klesnil and Jones et al. [16] presented the results of crack growth from Lukas [44], Hojo et al. [45], Liu and Chen [46], Walker [47], and small near micrometer size initial defects in 7050-T7451, which 544 R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549

Table 1 is used in both the F/A-18 Classic Hornet and the Super Hornet. The Fractal dimension results for 7050-T7451 aluminium alloys. experimental setup, which is described in more detail in the Point Specimen number Appendix A, found that crack growth in this material could be TD ET D2 E described by Eq. (4) with p = 0.2. The failure surfaces of four specimens, designations TD, ET, D2 1 1.182 1.200 1.425a 1.246 2 1.220 1.229 1.280 1.222 and E, were examined and the variation of the fractal dimension 3 1.299 1.267 1.243 1.255 was measured as described in the Appendix A. A total of twenty- 4 1.122 1.194 1.199 1.270 five equally distributed points on each surface were examined. 5 1.264 1.317 1.167 1.260 The size of each of the locations examined was approximately a 6 1.329 1.295 1.144 1.222 ranging from 1.2 by 0.9 mm and the results for one such location 7 1.162 1.294 1.120 1.263 8 1.236 1.335 1.218 1.354 are shown in Fig. 9. The resulting values of D obtained from a study 9 1.107 1.197 1.174 1.286 of the profilometric traces are given in Table 1. 10 1.245 1.326a 1.163 1.319 Away from the regions where the failure surface transitioned to 11 1.299 1.295 1.257 1.304 rapid crack growth, the value of D was approximately constant and 12 1.227 1.332 1.129 1.254 13 1.248 1.292 1.183 1.320a varied from approximately 1.17 to 1.3. The mean values of D for 14 1.173 1.315 1.183 1.305 specimens TD, ET, D2 and E were 1.23, 1.26, 1.19 and 1.29 which 15 1.140 1.240 1.277 1.272 yields a mean value for all of the specimens of approximately 16 1.159 1.263 1.150 1.305 1.24. As such this example supports Bouchard’s hypothesis that 17 1.287 1.296 1.123 1.263 the fatigue crack growth surfaces have a fractal dimension of 18 1.333a 1.180 1.159 1.281 19 1.490a 1.194 1.101 1.306a approximately 2.2, i.e., their profiles have D = 1.2. 20 1.176 1.219 1.174 1.308 Fractal dimension measurements were also performed on an in- 21 1.246 1.314a 1.190 1.314 service fatigue crack in a Queensland Rail (now Aurizon) wheel, 22 1.299 1.218 1.167 1.280 (see Fig. 10). The resultant fractal dimensions are given in Table 2. 23 1.190 1.178 1.128 1.257 24 1.184 1.310a 1.140 1.334a Here it should be noted that locations 6–10 are associated with 25 1.252 1.256 1.029 1.373a rapid crack growth whilst locations 1–5 are associated with slow Ave value of D 1.23 1.26 1.18 1.29 fatigue crack growth. Again, these results further confirm that the fractal dimension of profiles extracted from fatigue crack a These values are associated with the transition to rapid crack growth. growth surfaces is approximately equal to 1.2.

Rail Wheel Proﬁlometer Locaons

8 10

11 6 7 12 2 4 3 1 5

Fig. 10. The failed surface and the associated measurement locations. R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549 545

Fig. 11. Compact specimen test geometry.

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

Fractal dimension, D 123456a 78910b 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 1.26 Y 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.

10 9 8 7 6 5 y 4 3 2 1 X

6 2 5 4 3 Fig. 12. Locations of the fractal measurements. 1

Fig. 13. A fatigue failure in a BHP bolt. Table 3 Fractal dimension D at various locations on the surface.

Fractal dimension Table 4 1234567a 8910b Fractal dimension associated with the BHP bolt. X 1.28 1.21 1.24 1.29 1.24 1.14 1.35 1.35 1.21 1.28 Y 1.28 1.32 1.26 1.29 1.29 1.30 1.29 1.23 1.29 1.22 Fractal dimension 123456 a From this point onwards crack growth was very rapid. b Surface too rough to obtain an accurate reading. X 1.232 1.239 1.205 1.168 1.258 1.218 Y 1.241 1.193 1.163 1.161 1.225 1.225

The fractal dimension associated with laboratory fatigue crack tests on an ASTM standard compact tension (CT) MC wheel steel As such, the value associated with the lab tests and with the in- specimens [14] was also measured, see Fig. 11. The locations at service cracking are very close. which the fractal dimension were measured are shown in Fig. 12 To further evaluate the nature of the fractal dimension associ- and the resultant values are given in Table 3 where we see a fractal ated with in-service cracking, tests were performed on a fatigue dimension in the direction of crack growth of approximately 1.28. failure in a bolt from a BHP mining machine, see Fig. 13. Here 546 R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549

Squat

4 5 6 x 3 1 2

Fig. 14. The ﬁrst rail squat fatigue surface.

5 4 6

1 2 3

Fig. 15. The second rail squat fatigue surface.

Table 5 Table 6 Fractal dimension associated with the ﬁrst rail squat. Fractal dimension associated with rail squat number 2.

Fractal dimension, D Fractal dimension, D 123456 1234a 56 X 1.250 1.277 1.17 1.200 1.226 1.190 X 1.18 1.227 1.215 1.284 1.168 1.171 Y 1.260 1.272 1.18 1.292 1.255 1.160 Y 1.19 1.230 1.243 1.334 1.187 1.234

In each case the size of the region investigated was 228 lm by 300 lm. a At this point we are seeing very rapid crack growth. we again see a near uniform fractal dimension with a mean value of approximately 1.2, see Table 4. Measurements were also made We thus see that all of the specimens examined have revealed a on squats in two different head hardened rail specimens, see fractal dimension in the direction of crack growth that was in the Figs. 14 and 15. The associated fractal dimension measurements range between 1.17 and 1.29. Here it is important to pinpoint that are given in Tables 5 and 6. with a fractal dimension D = 1.2 and m = 3, the various growth laws 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], – PSI mode (Phase-shifting interferometry) – This mode enables Paggi and Carpinteri [8] and Carpinteri and Paggi [5,6,9,10] the measurement of smooth surfaces. coincide. – VSI mode (Vertical scanning interferometry) – This mode enables the measurement of rough surfaces. 5. Conclusions The VSI mode was used in this study due to the rough nature of The anomalous behaviour of short-cracks in fatigue, which vio- the various fatigue fracture surfaces. In this mode the range of lates the assumption of physical similitude behind the Paris’ law, is heights that the profiler can accurately measure is approximately crucial for the design of safe aerospace and railways applications, 2 mm. The range in depth that the profiler can accurately measure with well ascertain lifetimes. So far, two independent approaches in this mode is approximately 5 nm to 2 mm (see the manuals in the literature, namely the generalized Frost–Dugdale equation Veeco [51] and Wyko [52] for more details). and the generalized fractal approach, have shown to be able to sat- In VSI mode the value of fractal dimension is obtained by means isfactorily predict crack fatigue growth of short cracks. To under- of the power spectrum method which is introduced as Power Spec- stand why the two methods lead to very similar results, although tral Density (PSD) in the WYKO surface profiler system. It uses being based on very different assumptions, an experimental cam- Fourier decomposition of the measured surface profile into its paign has been undertaken by analyzing the fractal dimension of component spatial frequencies f. The PSD function calculates the fatigue rough surfaces at failure for a wide range of materials power spectral densities for each horizontal (X) or vertical (Y) line where short-crack effects were reported in previous publications. in the data and then averages all X and Y profiles. The range of The findings presented in this paper show that rough profiles spatial frequencies measured is limited by the objective field of extracted from fatigue crack growth surfaces have a fractal dimen- view and spatial sampling. The orientation of a surface with non- sion approximately equal to 1.2. Since for the MC and 7050-T7451 random features also affects the average X and Y PSD plots. These materials the exponent m is approximately equal to 3, we found average PSD data is also used to calculate fractal box dimension. that the Generalized Frost–Dugdale equation and the fractal based The average PSD is modelled as crack growth equation are linked and provide very close predic- B PSDðf Þ¼A=f ðA:1Þ tions of fatigue crack growth rates. Moreover, from a detailed analysis of the equations, an exten- where A, B are constants and f is the spatial frequency. The program sion of the current fractal crack growth equation to account for R uses a linear least-squares fit to determine A and B, viz: ratio effects is suggested, viz, ex: logðPSDðf ÞÞ ¼ logðAÞB logðf ÞðA:2Þ = ¼ ð1DÞð1þm=2Þ½D pð Þð1pÞm ð = Þ ð Þ da dN C a K Kmax da dN 0 11 The interval of frequencies used in the above equation is We also see that this formulation and the Nasgro equation are defined by the ISO standard as: often both capable of representing crack growth in a range of mate- 1=D < f < 1=C ðA:3Þ rials, see Jones et al. [17] and Jones et al. [23]. For sub-micrometer cracks, Bouchaud revealed a fractal box dimension of approxi- where 1/D is the program’s low frequency cut-off option and 1/C is mately 1.5, in line with multi-fractal properties of cracks put for- the program’s high frequency cut-off option. The fractal box dimen- ward in Carpinteri [50]. In this case, Eq. (11) reduces to sion D is then calculated as

ð Þ m D ¼ð5 BÞ=2 ðA:4Þ = ¼ 1=2ð1þm=2Þ½D pð Þ 1 p ð = Þ ð Þ da dN C a K Kmax da dN 0 12 Further details are given in the Wyko [52] Vision32 users man- which is a particularization of the generalized fatigue crack growth ual and are discussed also in Zavarise et al. [53]. In the present model based on fractal concepts proposed in Carpinteri and Paggi study, each proﬁlometric trace was taken over the same physical [11] for the growth of short cracks, where any possible fractal distance 0.9–1.2 mm. The array size used was 736 480. dimension between 1 and 2 was allowed. Regarding the short crack test specimens used in this Although Eq. (12) appears to be a straightforward generaliza- study, they were made from 7050-T7451 which is high strength tion of the fractal approach by analogy with the generalized aluminium alloy with a Young modulus of approximately Frost–Dugdale equation, further research is deemed to be required 71,000 MPa and its Poisson’s ratio is 0.33. The specimens had a to provide a general theoretical framework to account for the load- ‘‘dog bone” geometry as shown in Fig. A1 with the surface etched ing ratio effects in the fractal modelling of fatigue crack growth. to create small pits from which fatigue cracks could initiate. The specimens were 11 mm thick and 42 mm wide, see Jones et al. Acknowledgement [49] for more details. To assist with quantitative fractography the specimens were tested under repeated marker block loading, see MP and AC would like to thank the Italian Ministry of Education, Fig. A2. Each block consisted of a number (C) cycles of constant University and Research for his support to the project FIRB 2010 amplitude loading with a maximum stress and stress ratio RI fol- Future in Research ‘‘Structural mechanics models for renewable lowed by M constant amplitude load cycles with the same maxi- energy applications” (RBFR107AKG).

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

The measurement of fractal dimension is carried out by a VEECO NT1100 Microscope in conjunction with the associated WYKO Vision32 computer program, which (together) are denoted as the WYKO surface profiler system. The WYKO surface profiler system is a non-contact optical profiler that offers two modes for measuring a wide range of surface topologies. These two approaches are briefly explained as follows: Fig. A1. The Geometry of the dog-bone test specimens. 548 R. Jones et al. / International Journal of Fatigue 82 (2016) 540–549

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