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Experimental and calculation challenges of crack propagation in notches – From initiation to end of lifetime
International Journal of Fatigue 103 (2017) 395–404
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Experimental and calculation challenges of crack propagation in notches – From initiation to end of lifetime Stefan Kolitsch a,b,⇑, Hans-Peter Gänser a, Reinhard Pippan b a b
Materials Center Leoben Forschung GmbH, Roseggerstrabe 17, A-8700 Leoben, Austria Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Jahnstrabe 12, A-8700 Leoben, Austria
a r t i c l e
i n f o
Article history: Received 22 December 2016 Received in revised form 17 May 2017 Accepted 14 June 2017 Available online 17 June 2017 Keywords: Notch Fatigue Short crack growth Semi-elliptical crack Direct current potential drop method
a b s t r a c t Notches reduce significantly the lifetime of cyclically loaded components due to their stress concentration and early crack initiation. In this work the initiation and crack growth behaviour of cracks in notches has been studied with the direct current potential drop technique. Due to the notch geometry and semielliptical shape of the starting crack, significant differences between the crack length determination by the Johnson equation and the real crack length have been observed. By means of numerical simulations for single and multiple crack initiation sites, a calibration function for determining the crack length from the potential drop has been developed. The calibration function results in a good agreement between the estimated and measured changes of the potential drop and permits a quite accurate estimation of the crack initiation life time. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The estimation of the lifetime of notched components is an important technical issue. The lifetime can be distinguished into a crack initiation and a crack growth period. In order to specify inspection intervals, the information about the initiation and crack growth periods would be an essential advantage. The investigation of crack initiation by microscopy is a difficult and ambitious task. For this reason, the present contribution is devoted to investigating the measurability of crack initiation and early crack growth behaviour in notched specimens by means of the direct current potential drop (DCPD) method. The influence of notches on the fatigue limit and crack propagation have been studied since years. An interesting work on this topic, considering circumferential notched bars in torsion and tension is presented by Tanaka [1]. In this study the crack initiation and end of lifetime is determined for different notch radii using the DCPD method for the measurement of the crack length. The results indicate, the higher the notch radius is the lower is the lifetime and the lower is the crack propagation in the first stage. Berto [2] reanalyzed the results using the strain energy density method of Lazzarin and Zambardi [3] and provides a good prediction of the experimental results of Tanaka.
⇑ Corresponding author at: Materials Center Leoben Forschung GmbH, Roseggerstrabe 17, A-8700 Leoben, Austria. E-mail address: [email protected] (S. Kolitsch). http://dx.doi.org/10.1016/j.ijfatigue.2017.06.016 0142-1123/Ó 2017 Elsevier Ltd. All rights reserved.
The consequences of the notch concentration on the endurance limit and Woehler curves is presented by Atzori [4], where approaches of the general notch mechanic and fracture mechanics are compared. Based on the suggestions of Glinka [5], Atzori [6] provides analytical equations of the stress distribution of U- and V-shaped notches, considering bending and tension loads. The DCPD method is commonly used to determine the initiation and propagation of cracks. Nevertheless, the change of the electrical potential depends on the measuring position, the specimen and notch geometry and calibrations based on experimental results [7,8] or using finite element simulations [9] are recommended. 2. Fatigue experiments of notched specimens To determine the crack initiation and growth depending on the number of cycles, SENB (single edge notched bending) specimens with two different notch geometries are tested at two different load levels, each in a four-point bending device, shown in Fig. 1. All specimens were subjected to cyclic loading with a sinusoidal frequency of 10 Hz in a computer-controlled electro-servo hydraulic tension/compression machine. The specimens are applied with a constant load with a load ratio R = 0.1. By using the direct current potential drop (DCPD) method and the formulation of Johnson [10] and Schwalbe [11] assuming a straight through-thickness crack, the crack length can be computed. The tested material is a commonly used pearlitic steel with a tensile strength of 1070 MPa. The height of the specimens is W = 20 mm, the width B = 0.15 W and the length L = 5.5 W.
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Nomenclature a aDCPD af
total crack length [mm] measured crack length by DCPD method [mm] crack length at fracture [mm] ak elastic stress concentration factor at the notch root [–] B specimen width [mm] c crack width of a semi-elliptical crack [mm] C1 . . . C7 constants to describe the potential depending on the crack geometry and specimen [–] Da crack extension [mm] Dastart initial crack length [mm] DaJohnson crack extension of a straight crack front as calculated by the Johnson formula [mm] Dasemi-ellipse crack extension of a semi-elliptical crack front [mm] Dastraight crack extension of a straight crack front [mm] p DK stress intensity factor range [MPa m] DKth threshold value of the stress intensity factor range p [MPa m] DKth,eff intrinsic (effective) threshold value of the stress intenp sity factor range [MPa m] DU electric potential drop [–] DYA,C alternative geometry factor for point A or C [–] F(R,Da) NASGRO crack growth rate factor [–]
Due to a different notch width 2q, the measuring distance 2y for the electric potential drop DU varies depending on the notch radius (Fig. 1). In (Fig. 2a) the mild notch with a depth t = 0.2 W, a notch radius q = 0.05 W and a measuring distance 2y = 0.375 W and in (Fig. 1b) the sharp notch with a depth t = 0.2 W, a notch radius of q = 0.01W and measuring distance 2y = 0.15 W mm are displayed. The stress concentration estimated with Eq. (1) [13] is 5.0 for the mild notch and 9.9 for the sharp notch.
sffiffiffiffi t ak ¼ 1 þ 2 :
q
ð1Þ
In Fig. 3 the experimental results for the crack length over the number of cycles are displayed for the different notch geometries.
m p q R
constant in the NASGRO crack growth equation [–] constant in the NASGRO crack growth equation [–] variable for the initial crack length [–] load ratio [–] q notch radius [mm] rmax maximum applied stress [MPa] rF flow stress [MPa] t notch depth [mm] W specimen height [mm] x characterization variable for the transition from the Newman/Raju approach to c/W = 0.9 [–] YA,C enhanced geometry factor for a semi-elliptical crack shape [–] YA,C (c/B = 0.9) geometry factor for point A or C for the ratio c/ B = 0.9, determined from the FE simulations [–] YA,C-NR geometry factor for a semi-elliptical crack shape by Newman and Raju [–] Ystraight geometry factor for a straight (through-thickness) crack front by Tada [–] YN (Da) geometry factor for a crack emanating from a notch [–]
It can be seen that in general the crack growth period for a given notch geometry and applied stress is similar. The crack initiation varies within the same notch geometry and applied load, supposedly depending on the initiation site. Judging from these results – obtained by assuming a straight through-thickness crack, the initiation period forms a major part of the total lifetime. In order to verify the underlying assumptions, some further experiments were conducted where testing was stopped at different crack lengths and the specimens were broken up subsequently. Fractography showed that the fatigue cracks initiated at clearly localized spots and exhibited initially a semi-elliptical crack shape (Fig. 4). As it will be shown in Section 3, the DCPD method grossly underestimates the crack length when assuming a straight crack front, and thereby underestimates the early crack propagation behaviour and overestimates the macroscopic crack initiation time. Consequently, for obtaining more accurate crack growth curves and macro-crack initiation times, the evolution of the crack shape in the course of the experiment has to be taken into account; the following Section is devoted to this task.
3. The shape of the crack and deviation from the Johnson approach
Fig. 1. Fatigue test bench: SENB specimen in a four point bending device [12].
To estimate the short crack growth behaviour, starting with a semi-elliptical crack shape, several experiments are conducted and stopped at crack lengths Da af. The analyzed fracture surfaces of selected experiments are displayed in Fig. 4, where the dimensions of an approximated semi-elliptical crack of depth Da and width 2c are shown. The experiments are stopped at different pre-defined lengths of a fictitious straight through-thickness crack as estimated by the DCPD method and the Johnson equation [10]; here, for (Fig. 4a) 0.1 mm, (Fig 4 b) 0.25 mm and (Fig. 4c) 0.2 mm. In comparison to the analyzed real crack shape, the estimated fictitious straight crack length from the Johnson equation is markedly smaller than the actual depth of the semi-elliptical crack. The influence of different crack shapes and the resulting measured potential drop DU was already investigated for different configurations by Riemelmoser et al. [15]. Extending this work,
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Fig. 2. Schematic representation of the notch geometries and distances of the measuring points of the DCPD method. (a) mild notch and (b) sharp notch.
Fig. 3. Crack growth curves between initiation and failure: (a) q = 0.05W, t = 0.2W and Drnotch = 1540 MPa; (b) q = 0.05W, t = 0.2W and Drnotch = 1260 MPa; (c) q = 0.01W, t = 0.02W and Drnotch = 1085 MPa; (d) q = 0.01W, t = 0.02W and Drnotch = 945 MPa [14].
several finite element (FE) simulations were performed to determine the potential drop DU for different configurations of semi-elliptical cracks with respect to the specimen dimensions, where the focus was set on different ratios Da/c and c/B up to
1.0 to estimate the transition from the semi-elliptical to the straight (through-thickness) crack front. In addition, the difference between the crack growth of a single and a double semi-elliptical crack is worked out.
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Fig. 4. Light microscopic fracture surface analysis of interrupted experiments and measured crack length for (a) DaJohnson = 0.1 mm, (b) DaJohnson = 0.25 mm and (c) DaJohnson = 0.2 mm, the fractographs clearly illustrate the very large underestimation of the crack length by the Johnson approximation.
Fig. 5. Schematic sections in the crack plane for the determination of the normalized potential drop DU/U0 in the DCPD method: (a) single, (b) double ellipse.
In Fig. 5 the schematic cross sections for determination of the normalized potential drop DU/U0 is displayed. The geometry configurations were done for different crack lengths for ratios inbetween 0.3 Da/c 1.0, 0 c/B 1.0 and 0.1 t/W 0.3. In Fig. 6 the results for the normalized potential drop DU/U0 as a function of the crack length Da are exemplarily displayed for the notch geometries from the experiments (t/W = 0.2 and q/ W = 0.05 resp. q/W = 0.01); (a) for the mild notch and a single crack, (b) for the mild notch and a double crack, (c) for the sharp notch and a single crack and (d) for the sharp notch and a double crack. The different marks represent different Da/c ratios and, for comparison, the black circles show the results for the straight through-thickness crack. Additionally the solution from the Johnson equation is displayed by the dashed black lines. The different combinations in Fig. 6 clearly show that the fictitious straight crack length clearly underestimates the actual crack depth, see also Fig. 4. Even with the straight crack front a difference between FE results and Johnson’s prediction is recognizable because the latter is only valid for an ideally sharp crack and does not account for the notch radius and the distance of the notch flanks.
For designing a prediction of the potential DU/U0, using a linear regression model was carried out [16]. The equations for different ratios (Da/c, a/W, c/B, and t/W) can then be calculated by Eqs. ((2)--(4)) for the semi-elliptical crack and Eqs. ((3)--(5)) for the straight crack front by using the parameters shown in Table 1. The potential drop across the semi-elliptical crack (or the two semi-elliptical cracks, respectively) is
2 DU c t a t ¼ 1 þ C1 q þ C2 q þ C3 q þ C4 q U0 B W W W a 2 c 3 a 3 þ C5 q þ C6 q þ C7 q ; W B W where
q¼
a t W W
ð2Þ
ð3Þ
and a describes the added notch and crack length consisting of the fatigue crack extension Da and the initial notch depth t,
a ¼ Da þ t:
ð4Þ
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Fig. 6. Comparison of the normalized potential drop for different crack and specimen geometries with a straight crack front and the solution derived from Johnson equation. Exemplarily shown for different ratios Da/c and t/W = 0.2 (a) for single crack and (b) double crack, both for q/W = 0.05; (c) for single and (d) for double crack, both for q/ W = 0.01.
Table 1 Parameters for prediction of the potential drop for a single and double semi-elliptical and a straight crack front. Parameter
C1 C2 C3 C4 C5 C6 C7
Mild notch (q/W = 0.05)
Sharp notch (q/W = 0.01)
1 Ellipse
2 Ellipses
Straight
1 Ellipse
2 Ellipses
Straight
0.53 5.06 3.85 4.58 0 0.48 6.14
0.49 7.72 6.53 6.12 0 3.15 9.62
0.92 13.90 17.21 11.63 25.14 16.51 –
2.07 24.46 27.94 49.18 86.88 0.80 76.26
3.11 30.79 32.52 61.19 87.82 6.10 61.00
8.19 59.83 20.47 88.65 32.62 23.00 –
The potential drop across the straight through-thickness crack is
2 DU t a t ¼ 1 þ C1 q þ C2 q þ C3 q þ C4 q U0 W W W a 2 a 3 þ C5 q þ C6 q : W W
ð5Þ
The range of validity of Eqs ((2) and (5)) is 0.1 a/W 0.7, 0.1 t/W 0.3, 0 c/B 1. In Fig. 7 the comparison of the FE results (horizontal axis) and the analytical prediction by Eqs ((2)--(5)) (vertical axis) is displayed. Here the blue marks represent the prediction for the mild notch and the green marks the sharp notch in (a) for a single and a double semi-elliptical crack front and in (b) for the straight crack front. Furthermore, the full black line represents the exact and the dashed and dotted lines the 2.5% in (a) and 10% in (b) over- resp. under- prediction of the FE results. The comparison between the approximation and the FE results of the potentials in Fig. 7 shows an acceptable residual error of less
than 2.5% for the single and double semi-elliptical cracks and 10% for the straight through-thickness crack. To validate this approach, the new prediction is compared with the experiments depicted in Fig. 4. The depth of the semi-elliptical crack is computed from the potential drop via Eq. (2) using the Da/ c ratio from the micrograph, and then compared with the actual crack length determined from the fracture surface (Fig. 8). The predicted crack lengths using Eq. (1) are acceptably close to the actual crack lengths determined from the micrographs. Hence, Eqs. ((2)--(4)) will be used for the following considerations. (See Table 2) 4. Prediction of the crack growth behaviour starting from a notch After initiation, the fatigue crack growth can be calculated by commonly used fracture mechanics approaches. Here, the crack growth is described by a modified NASGRO equation from Maierhofer et al. [12], including the short crack growth behaviour [17] and the material parameters from [14]. This equation considers
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Fig. 7. Comparison of the analytical prediction of the potential drop with the FE results and the 2.5%/10% over- resp. under-prediction scatter lines, in (a) for the semi elliptical crack shape and (b) the straight (through thickness) crack.
Fig. 8. Comparison of crack lengths as estimated by the Johnson formula and by Eq. (2) for a semi-elliptical crack with Da/c from the micrograph Fig. 4.
Table 2 Comparison of crack lengths as estimated by the Johnson formula, by Eq. (2) and as determined from experiment. sample
DaJohnson [mm]
Dasemi-ellipse [mm]
Da from experiment [mm]
(Fig. 3a) (Fig.3b) (Fig. 3c)
0.10 0.25 0.20
0.84 1.03 0.87
0.778 1.072 0.890
8 2 !2:5 30:4 9 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi < = t þ D a 2:5 5 1 Y N ðaÞ ¼ Y A;C 1 þ 4ðak 1Þ þ : ; Da
the crack growth for physically short cracks depending on the stress ratio R and the crack extension Da,
da ¼ C FðR; DaÞ DK mp ðDK DK th ðR; DaÞÞp : dN
ð6Þ
To calculate the crack growth, a forward integration with a semi-elliptical starting crack shape and an initial crack length depending on the effective threshold DKth,eff is suggested,
Dastart ¼
DK th;eff 2 1 Y N ðDaÞ2 : Dr p
Due to the notch stress concentration, the calculation of the crack growth starting from a notch is different compared to a crack starting from a smooth surface. Depending on the notch depth t and radius q, the geometry factor can be calculated by the approach from Neuber [18] for a straight crack. For a semielliptical shape, the equation has to be modified as follows:
ð7Þ
ð8Þ
Here, YA,C denotes the geometry factor for the semi-elliptical crack front at points A and C, respectively (cf. Fig. 9). Newman and Raju (NR) [19–21] provide an empirical function for the geometry factor of a semi-elliptical crack. Nevertheless their function is restricted to a maximum ratio of c/B = 0.5. Therefore, an enhanced solution for 0.5 c/B 0.9 was developed by the authors [22]. Following this approach, for describing the geometry factor for the range 0 c/B 0.9, the equation proposed by NR for 0 c/ B 0.5 is combined with the new enhanced function for 0.5 < c/ B < 0.9. This is done by a combination of a linear approximation
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The enhanced function for the geometry factor for the range of 0.5 < c/B < 0.9 is developed by FE simulations in combination with a statistical analysis. Following this, the higher-order correction term can then be calculated for point A with Eq. (13) and for point C with Eq. (14).
2 2 Da Da 0:02 þ ðx2 1Þ 0:1 c W c 2 0:04 þ ðx2 1Þ B 2 2 Da Da DY C ¼ ðx2 1Þ 0:07 þ ðx2 1Þ 0:33 c W
DY A ¼ ðx2 1Þ
Fig. 9. Schematically explanation for point A and C at the elliptical crack front.
and a higher-order correction term DYA,C for points A and C, respectively,
Y A;C ¼ Y A;CNR
1x 1þx þ Y 0A;Cjc=B¼0:9 þ DY A;C ; 2 2
ð9Þ
where the auxiliary variable
x¼5
c 7 B 2
ð10Þ
varies in the range 1 < x < 1 for 0.5 < c/B < 0.9. YA,C-NR and YA,C|c/B = 0.9 is determined by a statistical analysis from FE results (Eqs. (11) and (12)) [22].
2 Da Da Da 1:13 þ 0:37 c W c 2 3 Da Da þ 2:91 2:88 W W 2 2 3 3 Da Da Da Da 1:79 þ 1:81 c W c W
YAjc=B¼0:9 ¼ 1:23 0:81
2 Da Da Da 0:28 3:07 c W c 2 3 3 Da Da Da þ 2:54 þ 1:3 1:34 W c W 2 2 Da Da 0:65 c W
ð11Þ
YCjc=B¼0:9 ¼ 0:12 þ 2:77
ð12Þ
ð13Þ ð14Þ
As the stress concentration leads to plastic deformation and thereby to residual stresses in front the notch, the local stress ratio in front of the notch will deviate from the applied stress ratio [14]. In summary, the crack growth and the evaluation of the crack shape can be predicted with the modified NASGRO equation (Eq. (6)), the enhanced geometry factor (Eqs. (9)--(14)) in combination with the stress concentration in front of the notch (Eqs. (1) and (8)) and the varying local stress ratio from [14] by means of a forward integration using a given initial crack length astart and a given initial Da/c ratio. The enhanced geometry function is valid up to c/B = 0.9. Assuming that the crack growth from c/B = 0.9 to c/B = 1.0 and the subsequent transformation into a straight crack front will take comparatively few cycles, this phase is approximated as follows: the geometry function is extrapolated up to c/B = 1.0; directly afterwards, the crack is assumed to exhibit a straight crack front. The geometry factor of a straight through-thickness crack can then be calculated by an existing solution from Tada [23]. Nevertheless, this assumption does not consider a three dimensional effect when the crack width intersects the free surface (c/B = 1.0). Investigations on this topic for the short crack growth are provided by Pook [24–26] and delivers a detail understanding on those corner points singularity. In this contribution, finite element simulations are conducted to describe the mixed mode loading, in combination with the strain energy density method. The analysis enables the estimation of the crack front intersection angle and stress intensity at the ratio c/B = 1.0 and is depending on the Poisson ratio and the applied load. The results of this estimation provide a high accuracy compared in the transition regime for c/B > 0.9 compared to the FE results.
Fig. 10. Crack growth and evolution of the Da/c ratio for single and double semi-elliptical cracks and corresponding fictitious crack lengths from the Johnson equation for a single and double semi-elliptical crack front and a straight through-thickness crack; for Da/c = 0.6, t/W = 0.2, q/W = 0.05 and Drnotch = 1260 MPa.
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In addition to the information of the crack shape and the specimen dimensions, the potential drop DU/U0 can then be calculated by Eqs. ((2)--(5)) and converted into the fictitious straight throughthickness crack length computed from the DCPD measurements using the Johnson equation. In Fig. 10 the crack growth of point A and the Da/c ratio (vertical right axis) is displayed over the number of cycles for the case Da/cstart = 0.6, t/W = 0.2, q/W = 0.05 and
Drnotch = 1260 MPa; additionally, the growth of a single semielliptical crack (black) is compared with that of a double semi-elliptical crack (blue). The full lines represent the calculated crack growth and the dotted lines the crack lengths as estimated from the DCPD measurements by Eqs. ((1)--(3)). Furthermore the growth of the semi-elliptical crack is compared with the resulting growth of a straight through-thickness crack (full red line).
Fig. 11. Comparison of the crack growth behaviour for different initial crack sizes 10 Dastart 100 lm exemplary shown for Da/c = 0.6, t/W = 0.2, q/W = 0.05 and Drnotch = 1260 MPa and assuming a single semi-elliptical crack front.
Fig. 12. Crack growth curves (full lines: experiment, dashed lines: single semi-elliptical crack, dotted lines: double semi-elliptical crack): (a) q = 0.05W, t = 0.2W and Drnotch = 1540 MPa; (b) q = 0.05W, t = 0.2W and Drnotch = 1260 MPa; (c) q = 0.01W, t = 0.2W and Drnotch = 1085 MPa; (d) q = 0.01W, t = 0.2W and Drnotch = 945 MPa.
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Starting at Da/c = 0.6, extracted from (Fig. 4a), the Da/c ratio increases rapidly nearly to 0.9 and then decreases due to the higher stress intensity factor at point C. At a crack length Da 1.3 mm, the crack width 2c reaches the specimen width and the crack is assumed to continue as a straight through-thickness crack. Compared to the single ellipse, the crack growth of the double semi-elliptical crack is about 6 times faster and the transition point from the semi-elliptical shape to the straight crack front occurs at a crack length Da of approximately 0.8 mm. The straight crack grows much faster than the semi-elliptical crack due to a higher stress intensity factor at the crack tip. In comparison to the actual crack dimensions, the estimates from the DCPD measurements using the Johnson equation (dashed lines) underestimate the actual crack growth behaviour. As mentioned, the differences in total lifetime depend markedly on the failure initiation site. Assuming that failure initiation occurs at a microstructural flaw of a certain size, this behaviour can be modelled by assuming different initial crack lengths corresponding to different microstructural flaw sizes. In Fig. 11 the crack growth is exemplarily calculated for Da/c = 0.6, t/W = 0.2, q/W = 0.05 and Drnotch = 1260 MPa and different initial crack lengths Dastart from 10 lm up to 100 lm. The crack growth curves show clearly that the lifetime can be about two times higher if the initial flaw is about 10 times higher. This explains well the scatter in the curves from Fig. 3. Starting from a given initial crack length and shape, the crack growth is calculated for different notch geometries and various applied stresses and displayed in Fig. 12. The colored lines represent the experimental data from the DCPD measurements and the Johnson equation; the black dashed and dotted lines describe the growth of a single resp. double semi-elliptical converted to the length of a fictitious straight through-thickness crack equal to the Johnson approach. Due to the fact that the number of cycles at initiation depends on the microstructural flaw size, the experimentally determined crack growth curves where shifted such that they have the same number of cycles at fracture. This provides an easier comparison with the analytical prediction of the crack growth behaviour. The prediction with a single semi-elliptical crack growth fits acceptably to the experimentally determined crack growth curves, whereas the double semi-elliptical crack overestimates the crack growth. The remaining differences between the predicted and the measured crack growth are quite small and may be explained by deviations from the idealized semi-elliptical shape and an eventual initiation of several cracks with different shapes.
5. Conclusion The determination of the crack initiation period from DCPD measurements by using the Johnson equation assuming a straight through-thickness crack leads insufficient accuracy, giving a gross overestimate of the crack initiation period. By investigating individual fracture surfaces, it was found out that the crack initiates with a semi-elliptical shape at microstructural flaws, which has to be taken into account for improved accuracy of the calculation results. For analytical prediction of the fatigue crack growth, a forward integration method with an enhanced geometry factor is suggested and presented in this work. The improved analytical calculation method gives results that fit acceptably to the experimental data. The higher the stress ratio, the earlier occurs the transition from the semi-elliptical to the straight crack front. It has been shown that (i) the number of cycles at crack initiation highly depends on the size of the individual microstructural flaw and that (ii) the crack growth period dominates the total lifetime.
Acknowledgements Financial support by the Austrian Federal Government (in particular from Bundesministerium für Verkehr, Innovation und Technologie and Bundesministerium für Wirtschaft, Familie und Jugend) represented by Österreichische Forschungsförderungsge sellschaft mbH and the Styrian and the Tyrolean Provincial Government, represented by Steirische Wirtschaftsförderungsgesellschaft mbH and Standortagentur Tirol, within the framework of the COMET Funding Programme is gratefully acknowledged. Appendix A Below, the equations for the NR solution [19–21] with / = 0° for point A and / = 90° for point C are summarized:
Y A;CNR ¼ rb H
pffiffiffiffi a a c Q F ; ; ;/ W c B
a1;65 c
Q ¼ 1 þ 1:464
ðA2Þ
a 2 a 4 f/ fW g þ M3 F ¼ M1 þ M2 W W M1 ¼ 1:13 0:09 M2 ¼ 0:54
M2 ¼ 0:5
f/ ¼
a c
0:89 0:2 þ ac
ðA5Þ
1:0 a24 a þ 14 1:0 0:65 þ c c
rffiffiffiffiffiffi!#0:5 a f w ¼ sec 2B W
pc
p
a a p ¼ 0:2 þ þ 0:6 c W H1 ¼ 1 0:34 H 2 ¼ 1 þ G1
ðA6Þ
ðA7Þ
ðA8Þ
a 2 2 ð1 sin /Þ g ¼ 1 þ 0:1 þ 0:35 W H ¼ H1 þ ðH2 H1 Þ ðsin /Þ
ðA3Þ ðA4Þ
2 0:25 a 2 cos2 / þ sin / c "
ðA1Þ
a a a 0:11 W c W
ðA9Þ ðA10Þ ðA11Þ ðA12Þ
a 2 a þ G2 W W
ðA13Þ
a c
ðA14Þ
G1 ¼ 1:22 0:12 G2 ¼ 0:55 1:05
a0:75 a1:5 þ 0:47 c c
ðA15Þ
References [1] Tanaka K. Crack initiation and propagation in torsional fatigue of circumferentially notched steel bars. Int J Fatigue 2014;58:114–25. [2] Berto F, Campagnolo A, Meneghetti G, Tanaka K. Averaged strain energy density-based synthesis of crack initiation life in notched steel bars under torsional fatigue. Frattura ed Integrità Strutturale 2016;38:215–23.
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[3] Lazzarin P, Zambardi R. A finite-volume-energy based approach to predict the static and fatigue behavior of components with sharp V-shaped notches. Int J Fract 2001;112:275–98. [4] Atzori B, Meneghetti G, Ricotta M. Fatigue and Notch Mechanics. Fatigue and Notch Mechanics. In: Boukharouba T, Pluvinage G, Azouaoui K, editors. Applied Mechanics, Behavior of Materials, and Engineering Systems. Lecture Notes in Mechanical Engineering. Springer; 2016. p. 9–23. [5] Glinka G. Calculation of inelastic notch-tip strain-stress histories under cyclic loading. Eng Fract Mech 1985;22:839–54. [6] Atzori B, Filippi S, Lazzarin P, Berto F. Stress distributions in notched structural components under pure bending and combined traction and bending. Fat Fract Eng Mater Struct 2004;28:13–23. [7] Clark G, Knott JF. Measurement of fatigue cracks in notched specimens by means of theoretical electrical potential calibrations. J Mech Phys Solids 1975;23:265–76. [8] Hicks MA, Pickard AC. A comparison of theoretical and experimental methods of calibrating the electrical potential drop technique for crack length determination. Internat J Frat 1982;20:91–101. [9] Gandossi L, Summers SA, Taylor NG, Hurst RC, Hulm BJ, Parker JD. The potential drop method for monitoring crack growth in real components subjected to combined fatigue and creep conditions: application of FE techniques for deriving calibration curves. Int J Press Vessels Pip 2001;78:881–91. [10] Johnson HH. Calibrating the electric potential method for studying slow crack growth. Mater Res Stand 1965;5. [11] Schwalbe KH, Hellman D. Application of the electrical potential method to crack length measurements using Johnson’s formula. J Test Eval 1981;9. [12] Maierhofer J, Pippan R, Gänser H-P. Modified NASGRO equation for physically short cracks. Internat J Fat 2014;59:200–7. [13] Radaj D, Vormwald M. Advanced Ermüdungsfestigkeit Grundlagen für Ingenieure. 3rd edition. Springer-Verlag; 2013.
[14] Kolitsch S, Gänser H-P, Pippan R. Determination of crack initiation and crack growth stress-life curves by fracture mechanics experiments and statistical analysis. Proced Struct Integ 2016;2:3026–39. [15] Riemelmoser FO, Pippan R, Kolednik O. The influence of irregularities in the crack shape on the crack extension measurement by means of the directcurrent potential drop Method. J Test Eval 1999;27:42–6. [16] Marques de Sá JP. Applied Statistics, Using SPSS, STATISTICA, MATLAB and R. Springer; 2007. [17] Tabernig B, Pippan R. Determination of the length dependence of the threshold for fatigue crack propagation. Eng Fract Mech 2002;69:899–907. [18] Neuber H. Die halbelliptische Kerbe mit Riß als Beispiel zur Korrelation von Mikro-und Makrospannungskonzentrationen. Ingenieur-Archiv 1977;46:389–99. [19] Newman JC, Raju IS. Analysis of surface cracks in finite plates under tension or bending loads. NASA Report TP-1578 1979. [20] Newman JC. An empirical stress intensity factor equation for the surface crack. Eng Fract Mech 1981;15:185–92. [21] Newman JC. A crack opening stress equation for fatigue crack growth. Int J Fract 1984;24:131–5. [22] Kolitsch S, Gänser H-P, Pippan R. Approximate stress intensity factor solutions for semi-elliptical cracks with large a/t and c/W under tension and bending. Theoret Appl Fract Mech 2017. under Review. [23] Murakami Y. Stress Intensity Factors Handbook. 3rd edition. Pergamon; 1986. [24] Pook LP. On fatigue. Int J Fatigue 1995;17:5–13. [25] Pook LP. A 50-year retrospective review of three-dimensional effects at cracks and sharp notches. Fatigue Fract Eng Mater Struct 2013. [26] Pook LP, Campagnolo A, Berto F, Lazzarin P. Coupled fracture mode of a cracked plate under anti-plane loading. Eng Fract Mech 2015;134:391–403.
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Experimental and calculation challenges of crack propagation in notches – From initiation to end of lifetime Stefan Kolitsch a,b,⇑, Hans-Peter Gänser a, Reinhard Pippan b a b
Materials Center Leoben Forschung GmbH, Roseggerstrabe 17, A-8700 Leoben, Austria Erich Schmid Institute of Materials Science, Austrian Academy of Sciences, Jahnstrabe 12, A-8700 Leoben, Austria
a r t i c l e
i n f o
Article history: Received 22 December 2016 Received in revised form 17 May 2017 Accepted 14 June 2017 Available online 17 June 2017 Keywords: Notch Fatigue Short crack growth Semi-elliptical crack Direct current potential drop method
a b s t r a c t Notches reduce significantly the lifetime of cyclically loaded components due to their stress concentration and early crack initiation. In this work the initiation and crack growth behaviour of cracks in notches has been studied with the direct current potential drop technique. Due to the notch geometry and semielliptical shape of the starting crack, significant differences between the crack length determination by the Johnson equation and the real crack length have been observed. By means of numerical simulations for single and multiple crack initiation sites, a calibration function for determining the crack length from the potential drop has been developed. The calibration function results in a good agreement between the estimated and measured changes of the potential drop and permits a quite accurate estimation of the crack initiation life time. Ó 2017 Elsevier Ltd. All rights reserved.
1. Introduction The estimation of the lifetime of notched components is an important technical issue. The lifetime can be distinguished into a crack initiation and a crack growth period. In order to specify inspection intervals, the information about the initiation and crack growth periods would be an essential advantage. The investigation of crack initiation by microscopy is a difficult and ambitious task. For this reason, the present contribution is devoted to investigating the measurability of crack initiation and early crack growth behaviour in notched specimens by means of the direct current potential drop (DCPD) method. The influence of notches on the fatigue limit and crack propagation have been studied since years. An interesting work on this topic, considering circumferential notched bars in torsion and tension is presented by Tanaka [1]. In this study the crack initiation and end of lifetime is determined for different notch radii using the DCPD method for the measurement of the crack length. The results indicate, the higher the notch radius is the lower is the lifetime and the lower is the crack propagation in the first stage. Berto [2] reanalyzed the results using the strain energy density method of Lazzarin and Zambardi [3] and provides a good prediction of the experimental results of Tanaka.
⇑ Corresponding author at: Materials Center Leoben Forschung GmbH, Roseggerstrabe 17, A-8700 Leoben, Austria. E-mail address: [email protected] (S. Kolitsch). http://dx.doi.org/10.1016/j.ijfatigue.2017.06.016 0142-1123/Ó 2017 Elsevier Ltd. All rights reserved.
The consequences of the notch concentration on the endurance limit and Woehler curves is presented by Atzori [4], where approaches of the general notch mechanic and fracture mechanics are compared. Based on the suggestions of Glinka [5], Atzori [6] provides analytical equations of the stress distribution of U- and V-shaped notches, considering bending and tension loads. The DCPD method is commonly used to determine the initiation and propagation of cracks. Nevertheless, the change of the electrical potential depends on the measuring position, the specimen and notch geometry and calibrations based on experimental results [7,8] or using finite element simulations [9] are recommended. 2. Fatigue experiments of notched specimens To determine the crack initiation and growth depending on the number of cycles, SENB (single edge notched bending) specimens with two different notch geometries are tested at two different load levels, each in a four-point bending device, shown in Fig. 1. All specimens were subjected to cyclic loading with a sinusoidal frequency of 10 Hz in a computer-controlled electro-servo hydraulic tension/compression machine. The specimens are applied with a constant load with a load ratio R = 0.1. By using the direct current potential drop (DCPD) method and the formulation of Johnson [10] and Schwalbe [11] assuming a straight through-thickness crack, the crack length can be computed. The tested material is a commonly used pearlitic steel with a tensile strength of 1070 MPa. The height of the specimens is W = 20 mm, the width B = 0.15 W and the length L = 5.5 W.
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Nomenclature a aDCPD af
total crack length [mm] measured crack length by DCPD method [mm] crack length at fracture [mm] ak elastic stress concentration factor at the notch root [–] B specimen width [mm] c crack width of a semi-elliptical crack [mm] C1 . . . C7 constants to describe the potential depending on the crack geometry and specimen [–] Da crack extension [mm] Dastart initial crack length [mm] DaJohnson crack extension of a straight crack front as calculated by the Johnson formula [mm] Dasemi-ellipse crack extension of a semi-elliptical crack front [mm] Dastraight crack extension of a straight crack front [mm] p DK stress intensity factor range [MPa m] DKth threshold value of the stress intensity factor range p [MPa m] DKth,eff intrinsic (effective) threshold value of the stress intenp sity factor range [MPa m] DU electric potential drop [–] DYA,C alternative geometry factor for point A or C [–] F(R,Da) NASGRO crack growth rate factor [–]
Due to a different notch width 2q, the measuring distance 2y for the electric potential drop DU varies depending on the notch radius (Fig. 1). In (Fig. 2a) the mild notch with a depth t = 0.2 W, a notch radius q = 0.05 W and a measuring distance 2y = 0.375 W and in (Fig. 1b) the sharp notch with a depth t = 0.2 W, a notch radius of q = 0.01W and measuring distance 2y = 0.15 W mm are displayed. The stress concentration estimated with Eq. (1) [13] is 5.0 for the mild notch and 9.9 for the sharp notch.
sffiffiffiffi t ak ¼ 1 þ 2 :
q
ð1Þ
In Fig. 3 the experimental results for the crack length over the number of cycles are displayed for the different notch geometries.
m p q R
constant in the NASGRO crack growth equation [–] constant in the NASGRO crack growth equation [–] variable for the initial crack length [–] load ratio [–] q notch radius [mm] rmax maximum applied stress [MPa] rF flow stress [MPa] t notch depth [mm] W specimen height [mm] x characterization variable for the transition from the Newman/Raju approach to c/W = 0.9 [–] YA,C enhanced geometry factor for a semi-elliptical crack shape [–] YA,C (c/B = 0.9) geometry factor for point A or C for the ratio c/ B = 0.9, determined from the FE simulations [–] YA,C-NR geometry factor for a semi-elliptical crack shape by Newman and Raju [–] Ystraight geometry factor for a straight (through-thickness) crack front by Tada [–] YN (Da) geometry factor for a crack emanating from a notch [–]
It can be seen that in general the crack growth period for a given notch geometry and applied stress is similar. The crack initiation varies within the same notch geometry and applied load, supposedly depending on the initiation site. Judging from these results – obtained by assuming a straight through-thickness crack, the initiation period forms a major part of the total lifetime. In order to verify the underlying assumptions, some further experiments were conducted where testing was stopped at different crack lengths and the specimens were broken up subsequently. Fractography showed that the fatigue cracks initiated at clearly localized spots and exhibited initially a semi-elliptical crack shape (Fig. 4). As it will be shown in Section 3, the DCPD method grossly underestimates the crack length when assuming a straight crack front, and thereby underestimates the early crack propagation behaviour and overestimates the macroscopic crack initiation time. Consequently, for obtaining more accurate crack growth curves and macro-crack initiation times, the evolution of the crack shape in the course of the experiment has to be taken into account; the following Section is devoted to this task.
3. The shape of the crack and deviation from the Johnson approach
Fig. 1. Fatigue test bench: SENB specimen in a four point bending device [12].
To estimate the short crack growth behaviour, starting with a semi-elliptical crack shape, several experiments are conducted and stopped at crack lengths Da af. The analyzed fracture surfaces of selected experiments are displayed in Fig. 4, where the dimensions of an approximated semi-elliptical crack of depth Da and width 2c are shown. The experiments are stopped at different pre-defined lengths of a fictitious straight through-thickness crack as estimated by the DCPD method and the Johnson equation [10]; here, for (Fig. 4a) 0.1 mm, (Fig 4 b) 0.25 mm and (Fig. 4c) 0.2 mm. In comparison to the analyzed real crack shape, the estimated fictitious straight crack length from the Johnson equation is markedly smaller than the actual depth of the semi-elliptical crack. The influence of different crack shapes and the resulting measured potential drop DU was already investigated for different configurations by Riemelmoser et al. [15]. Extending this work,
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Fig. 2. Schematic representation of the notch geometries and distances of the measuring points of the DCPD method. (a) mild notch and (b) sharp notch.
Fig. 3. Crack growth curves between initiation and failure: (a) q = 0.05W, t = 0.2W and Drnotch = 1540 MPa; (b) q = 0.05W, t = 0.2W and Drnotch = 1260 MPa; (c) q = 0.01W, t = 0.02W and Drnotch = 1085 MPa; (d) q = 0.01W, t = 0.02W and Drnotch = 945 MPa [14].
several finite element (FE) simulations were performed to determine the potential drop DU for different configurations of semi-elliptical cracks with respect to the specimen dimensions, where the focus was set on different ratios Da/c and c/B up to
1.0 to estimate the transition from the semi-elliptical to the straight (through-thickness) crack front. In addition, the difference between the crack growth of a single and a double semi-elliptical crack is worked out.
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Fig. 4. Light microscopic fracture surface analysis of interrupted experiments and measured crack length for (a) DaJohnson = 0.1 mm, (b) DaJohnson = 0.25 mm and (c) DaJohnson = 0.2 mm, the fractographs clearly illustrate the very large underestimation of the crack length by the Johnson approximation.
Fig. 5. Schematic sections in the crack plane for the determination of the normalized potential drop DU/U0 in the DCPD method: (a) single, (b) double ellipse.
In Fig. 5 the schematic cross sections for determination of the normalized potential drop DU/U0 is displayed. The geometry configurations were done for different crack lengths for ratios inbetween 0.3 Da/c 1.0, 0 c/B 1.0 and 0.1 t/W 0.3. In Fig. 6 the results for the normalized potential drop DU/U0 as a function of the crack length Da are exemplarily displayed for the notch geometries from the experiments (t/W = 0.2 and q/ W = 0.05 resp. q/W = 0.01); (a) for the mild notch and a single crack, (b) for the mild notch and a double crack, (c) for the sharp notch and a single crack and (d) for the sharp notch and a double crack. The different marks represent different Da/c ratios and, for comparison, the black circles show the results for the straight through-thickness crack. Additionally the solution from the Johnson equation is displayed by the dashed black lines. The different combinations in Fig. 6 clearly show that the fictitious straight crack length clearly underestimates the actual crack depth, see also Fig. 4. Even with the straight crack front a difference between FE results and Johnson’s prediction is recognizable because the latter is only valid for an ideally sharp crack and does not account for the notch radius and the distance of the notch flanks.
For designing a prediction of the potential DU/U0, using a linear regression model was carried out [16]. The equations for different ratios (Da/c, a/W, c/B, and t/W) can then be calculated by Eqs. ((2)--(4)) for the semi-elliptical crack and Eqs. ((3)--(5)) for the straight crack front by using the parameters shown in Table 1. The potential drop across the semi-elliptical crack (or the two semi-elliptical cracks, respectively) is
2 DU c t a t ¼ 1 þ C1 q þ C2 q þ C3 q þ C4 q U0 B W W W a 2 c 3 a 3 þ C5 q þ C6 q þ C7 q ; W B W where
q¼
a t W W
ð2Þ
ð3Þ
and a describes the added notch and crack length consisting of the fatigue crack extension Da and the initial notch depth t,
a ¼ Da þ t:
ð4Þ
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Fig. 6. Comparison of the normalized potential drop for different crack and specimen geometries with a straight crack front and the solution derived from Johnson equation. Exemplarily shown for different ratios Da/c and t/W = 0.2 (a) for single crack and (b) double crack, both for q/W = 0.05; (c) for single and (d) for double crack, both for q/ W = 0.01.
Table 1 Parameters for prediction of the potential drop for a single and double semi-elliptical and a straight crack front. Parameter
C1 C2 C3 C4 C5 C6 C7
Mild notch (q/W = 0.05)
Sharp notch (q/W = 0.01)
1 Ellipse
2 Ellipses
Straight
1 Ellipse
2 Ellipses
Straight
0.53 5.06 3.85 4.58 0 0.48 6.14
0.49 7.72 6.53 6.12 0 3.15 9.62
0.92 13.90 17.21 11.63 25.14 16.51 –
2.07 24.46 27.94 49.18 86.88 0.80 76.26
3.11 30.79 32.52 61.19 87.82 6.10 61.00
8.19 59.83 20.47 88.65 32.62 23.00 –
The potential drop across the straight through-thickness crack is
2 DU t a t ¼ 1 þ C1 q þ C2 q þ C3 q þ C4 q U0 W W W a 2 a 3 þ C5 q þ C6 q : W W
ð5Þ
The range of validity of Eqs ((2) and (5)) is 0.1 a/W 0.7, 0.1 t/W 0.3, 0 c/B 1. In Fig. 7 the comparison of the FE results (horizontal axis) and the analytical prediction by Eqs ((2)--(5)) (vertical axis) is displayed. Here the blue marks represent the prediction for the mild notch and the green marks the sharp notch in (a) for a single and a double semi-elliptical crack front and in (b) for the straight crack front. Furthermore, the full black line represents the exact and the dashed and dotted lines the 2.5% in (a) and 10% in (b) over- resp. under- prediction of the FE results. The comparison between the approximation and the FE results of the potentials in Fig. 7 shows an acceptable residual error of less
than 2.5% for the single and double semi-elliptical cracks and 10% for the straight through-thickness crack. To validate this approach, the new prediction is compared with the experiments depicted in Fig. 4. The depth of the semi-elliptical crack is computed from the potential drop via Eq. (2) using the Da/ c ratio from the micrograph, and then compared with the actual crack length determined from the fracture surface (Fig. 8). The predicted crack lengths using Eq. (1) are acceptably close to the actual crack lengths determined from the micrographs. Hence, Eqs. ((2)--(4)) will be used for the following considerations. (See Table 2) 4. Prediction of the crack growth behaviour starting from a notch After initiation, the fatigue crack growth can be calculated by commonly used fracture mechanics approaches. Here, the crack growth is described by a modified NASGRO equation from Maierhofer et al. [12], including the short crack growth behaviour [17] and the material parameters from [14]. This equation considers
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Fig. 7. Comparison of the analytical prediction of the potential drop with the FE results and the 2.5%/10% over- resp. under-prediction scatter lines, in (a) for the semi elliptical crack shape and (b) the straight (through thickness) crack.
Fig. 8. Comparison of crack lengths as estimated by the Johnson formula and by Eq. (2) for a semi-elliptical crack with Da/c from the micrograph Fig. 4.
Table 2 Comparison of crack lengths as estimated by the Johnson formula, by Eq. (2) and as determined from experiment. sample
DaJohnson [mm]
Dasemi-ellipse [mm]
Da from experiment [mm]
(Fig. 3a) (Fig.3b) (Fig. 3c)
0.10 0.25 0.20
0.84 1.03 0.87
0.778 1.072 0.890
8 2 !2:5 30:4 9 rffiffiffiffiffiffiffiffiffiffiffiffiffiffi < = t þ D a 2:5 5 1 Y N ðaÞ ¼ Y A;C 1 þ 4ðak 1Þ þ : ; Da
the crack growth for physically short cracks depending on the stress ratio R and the crack extension Da,
da ¼ C FðR; DaÞ DK mp ðDK DK th ðR; DaÞÞp : dN
ð6Þ
To calculate the crack growth, a forward integration with a semi-elliptical starting crack shape and an initial crack length depending on the effective threshold DKth,eff is suggested,
Dastart ¼
DK th;eff 2 1 Y N ðDaÞ2 : Dr p
Due to the notch stress concentration, the calculation of the crack growth starting from a notch is different compared to a crack starting from a smooth surface. Depending on the notch depth t and radius q, the geometry factor can be calculated by the approach from Neuber [18] for a straight crack. For a semielliptical shape, the equation has to be modified as follows:
ð7Þ
ð8Þ
Here, YA,C denotes the geometry factor for the semi-elliptical crack front at points A and C, respectively (cf. Fig. 9). Newman and Raju (NR) [19–21] provide an empirical function for the geometry factor of a semi-elliptical crack. Nevertheless their function is restricted to a maximum ratio of c/B = 0.5. Therefore, an enhanced solution for 0.5 c/B 0.9 was developed by the authors [22]. Following this approach, for describing the geometry factor for the range 0 c/B 0.9, the equation proposed by NR for 0 c/ B 0.5 is combined with the new enhanced function for 0.5 < c/ B < 0.9. This is done by a combination of a linear approximation
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The enhanced function for the geometry factor for the range of 0.5 < c/B < 0.9 is developed by FE simulations in combination with a statistical analysis. Following this, the higher-order correction term can then be calculated for point A with Eq. (13) and for point C with Eq. (14).
2 2 Da Da 0:02 þ ðx2 1Þ 0:1 c W c 2 0:04 þ ðx2 1Þ B 2 2 Da Da DY C ¼ ðx2 1Þ 0:07 þ ðx2 1Þ 0:33 c W
DY A ¼ ðx2 1Þ
Fig. 9. Schematically explanation for point A and C at the elliptical crack front.
and a higher-order correction term DYA,C for points A and C, respectively,
Y A;C ¼ Y A;CNR
1x 1þx þ Y 0A;Cjc=B¼0:9 þ DY A;C ; 2 2
ð9Þ
where the auxiliary variable
x¼5
c 7 B 2
ð10Þ
varies in the range 1 < x < 1 for 0.5 < c/B < 0.9. YA,C-NR and YA,C|c/B = 0.9 is determined by a statistical analysis from FE results (Eqs. (11) and (12)) [22].
2 Da Da Da 1:13 þ 0:37 c W c 2 3 Da Da þ 2:91 2:88 W W 2 2 3 3 Da Da Da Da 1:79 þ 1:81 c W c W
YAjc=B¼0:9 ¼ 1:23 0:81
2 Da Da Da 0:28 3:07 c W c 2 3 3 Da Da Da þ 2:54 þ 1:3 1:34 W c W 2 2 Da Da 0:65 c W
ð11Þ
YCjc=B¼0:9 ¼ 0:12 þ 2:77
ð12Þ
ð13Þ ð14Þ
As the stress concentration leads to plastic deformation and thereby to residual stresses in front the notch, the local stress ratio in front of the notch will deviate from the applied stress ratio [14]. In summary, the crack growth and the evaluation of the crack shape can be predicted with the modified NASGRO equation (Eq. (6)), the enhanced geometry factor (Eqs. (9)--(14)) in combination with the stress concentration in front of the notch (Eqs. (1) and (8)) and the varying local stress ratio from [14] by means of a forward integration using a given initial crack length astart and a given initial Da/c ratio. The enhanced geometry function is valid up to c/B = 0.9. Assuming that the crack growth from c/B = 0.9 to c/B = 1.0 and the subsequent transformation into a straight crack front will take comparatively few cycles, this phase is approximated as follows: the geometry function is extrapolated up to c/B = 1.0; directly afterwards, the crack is assumed to exhibit a straight crack front. The geometry factor of a straight through-thickness crack can then be calculated by an existing solution from Tada [23]. Nevertheless, this assumption does not consider a three dimensional effect when the crack width intersects the free surface (c/B = 1.0). Investigations on this topic for the short crack growth are provided by Pook [24–26] and delivers a detail understanding on those corner points singularity. In this contribution, finite element simulations are conducted to describe the mixed mode loading, in combination with the strain energy density method. The analysis enables the estimation of the crack front intersection angle and stress intensity at the ratio c/B = 1.0 and is depending on the Poisson ratio and the applied load. The results of this estimation provide a high accuracy compared in the transition regime for c/B > 0.9 compared to the FE results.
Fig. 10. Crack growth and evolution of the Da/c ratio for single and double semi-elliptical cracks and corresponding fictitious crack lengths from the Johnson equation for a single and double semi-elliptical crack front and a straight through-thickness crack; for Da/c = 0.6, t/W = 0.2, q/W = 0.05 and Drnotch = 1260 MPa.
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In addition to the information of the crack shape and the specimen dimensions, the potential drop DU/U0 can then be calculated by Eqs. ((2)--(5)) and converted into the fictitious straight throughthickness crack length computed from the DCPD measurements using the Johnson equation. In Fig. 10 the crack growth of point A and the Da/c ratio (vertical right axis) is displayed over the number of cycles for the case Da/cstart = 0.6, t/W = 0.2, q/W = 0.05 and
Drnotch = 1260 MPa; additionally, the growth of a single semielliptical crack (black) is compared with that of a double semi-elliptical crack (blue). The full lines represent the calculated crack growth and the dotted lines the crack lengths as estimated from the DCPD measurements by Eqs. ((1)--(3)). Furthermore the growth of the semi-elliptical crack is compared with the resulting growth of a straight through-thickness crack (full red line).
Fig. 11. Comparison of the crack growth behaviour for different initial crack sizes 10 Dastart 100 lm exemplary shown for Da/c = 0.6, t/W = 0.2, q/W = 0.05 and Drnotch = 1260 MPa and assuming a single semi-elliptical crack front.
Fig. 12. Crack growth curves (full lines: experiment, dashed lines: single semi-elliptical crack, dotted lines: double semi-elliptical crack): (a) q = 0.05W, t = 0.2W and Drnotch = 1540 MPa; (b) q = 0.05W, t = 0.2W and Drnotch = 1260 MPa; (c) q = 0.01W, t = 0.2W and Drnotch = 1085 MPa; (d) q = 0.01W, t = 0.2W and Drnotch = 945 MPa.
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Starting at Da/c = 0.6, extracted from (Fig. 4a), the Da/c ratio increases rapidly nearly to 0.9 and then decreases due to the higher stress intensity factor at point C. At a crack length Da 1.3 mm, the crack width 2c reaches the specimen width and the crack is assumed to continue as a straight through-thickness crack. Compared to the single ellipse, the crack growth of the double semi-elliptical crack is about 6 times faster and the transition point from the semi-elliptical shape to the straight crack front occurs at a crack length Da of approximately 0.8 mm. The straight crack grows much faster than the semi-elliptical crack due to a higher stress intensity factor at the crack tip. In comparison to the actual crack dimensions, the estimates from the DCPD measurements using the Johnson equation (dashed lines) underestimate the actual crack growth behaviour. As mentioned, the differences in total lifetime depend markedly on the failure initiation site. Assuming that failure initiation occurs at a microstructural flaw of a certain size, this behaviour can be modelled by assuming different initial crack lengths corresponding to different microstructural flaw sizes. In Fig. 11 the crack growth is exemplarily calculated for Da/c = 0.6, t/W = 0.2, q/W = 0.05 and Drnotch = 1260 MPa and different initial crack lengths Dastart from 10 lm up to 100 lm. The crack growth curves show clearly that the lifetime can be about two times higher if the initial flaw is about 10 times higher. This explains well the scatter in the curves from Fig. 3. Starting from a given initial crack length and shape, the crack growth is calculated for different notch geometries and various applied stresses and displayed in Fig. 12. The colored lines represent the experimental data from the DCPD measurements and the Johnson equation; the black dashed and dotted lines describe the growth of a single resp. double semi-elliptical converted to the length of a fictitious straight through-thickness crack equal to the Johnson approach. Due to the fact that the number of cycles at initiation depends on the microstructural flaw size, the experimentally determined crack growth curves where shifted such that they have the same number of cycles at fracture. This provides an easier comparison with the analytical prediction of the crack growth behaviour. The prediction with a single semi-elliptical crack growth fits acceptably to the experimentally determined crack growth curves, whereas the double semi-elliptical crack overestimates the crack growth. The remaining differences between the predicted and the measured crack growth are quite small and may be explained by deviations from the idealized semi-elliptical shape and an eventual initiation of several cracks with different shapes.
5. Conclusion The determination of the crack initiation period from DCPD measurements by using the Johnson equation assuming a straight through-thickness crack leads insufficient accuracy, giving a gross overestimate of the crack initiation period. By investigating individual fracture surfaces, it was found out that the crack initiates with a semi-elliptical shape at microstructural flaws, which has to be taken into account for improved accuracy of the calculation results. For analytical prediction of the fatigue crack growth, a forward integration method with an enhanced geometry factor is suggested and presented in this work. The improved analytical calculation method gives results that fit acceptably to the experimental data. The higher the stress ratio, the earlier occurs the transition from the semi-elliptical to the straight crack front. It has been shown that (i) the number of cycles at crack initiation highly depends on the size of the individual microstructural flaw and that (ii) the crack growth period dominates the total lifetime.
Acknowledgements Financial support by the Austrian Federal Government (in particular from Bundesministerium für Verkehr, Innovation und Technologie and Bundesministerium für Wirtschaft, Familie und Jugend) represented by Österreichische Forschungsförderungsge sellschaft mbH and the Styrian and the Tyrolean Provincial Government, represented by Steirische Wirtschaftsförderungsgesellschaft mbH and Standortagentur Tirol, within the framework of the COMET Funding Programme is gratefully acknowledged. Appendix A Below, the equations for the NR solution [19–21] with / = 0° for point A and / = 90° for point C are summarized:
Y A;CNR ¼ rb H
pffiffiffiffi a a c Q F ; ; ;/ W c B
a1;65 c
Q ¼ 1 þ 1:464
ðA2Þ
a 2 a 4 f/ fW g þ M3 F ¼ M1 þ M2 W W M1 ¼ 1:13 0:09 M2 ¼ 0:54
M2 ¼ 0:5
f/ ¼
a c
0:89 0:2 þ ac
ðA5Þ
1:0 a24 a þ 14 1:0 0:65 þ c c
rffiffiffiffiffiffi!#0:5 a f w ¼ sec 2B W
pc
p
a a p ¼ 0:2 þ þ 0:6 c W H1 ¼ 1 0:34 H 2 ¼ 1 þ G1
ðA6Þ
ðA7Þ
ðA8Þ
a 2 2 ð1 sin /Þ g ¼ 1 þ 0:1 þ 0:35 W H ¼ H1 þ ðH2 H1 Þ ðsin /Þ
ðA3Þ ðA4Þ
2 0:25 a 2 cos2 / þ sin / c "
ðA1Þ
a a a 0:11 W c W
ðA9Þ ðA10Þ ðA11Þ ðA12Þ
a 2 a þ G2 W W
ðA13Þ
a c
ðA14Þ
G1 ¼ 1:22 0:12 G2 ¼ 0:55 1:05
a0:75 a1:5 þ 0:47 c c
ðA15Þ
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