- Email: [email protected]
Growth of surface cracks under constant and variable amplitude loading
Engineering Fracture Mechanics 71 (2004) 1725–1735 www.elsevier.com/locate/engfracmech
Growth of surface cracks under constant and variable amplitude loading F. Nilsson a
a,*
, T. Hansson b, T. M ansson
b
Department of Solid Mechanics, KTH, Royal Institute of Technology, SE-100 44 Stockholm, Sweden b Volvo Aero Corporation, SE-461 83 Trollh€attan, Sweden Received 17 March 2003; accepted 16 July 2003
Abstract Fatigue crack growth experiments were performed on surface cracked tensile specimens of Inconel 718 at 400 C. The loading was carried out at constant as well as at variable amplitude. The experimental results for the mean growth rate were compared with predictions based on data obtained from testing of compact tension specimens. Both nominal data as well as data corrected from measured crack closure were used in the predictions. The corrected data provided much better predictions than the nominal ones indicating that the level of crack closure during the testing of the surface cracked specimens was much lower than in compact tension specimens. 2003 Elsevier Ltd. All rights reserved. Keywords: Crack closure; Surface crack; Variable amplitude
1. Introduction The main goal of fatigue crack growth research is to ensure transferability between laboratory testing and crack growth in engineering structures and components. Fatigue crack growth laboratory experimentation is almost always performed on nominally two-dimensional specimens such as for instance the compact tension (CT) specimen. At this geometry a significant part of the crack front is influenced by the specimen side surfaces and the crack front state may here be far from plane deformation. In consequence the crack closure under positive values of the stress-intensity factor KI may be more pronounced than under more constrained conditions. Three-dimensional effects on crack closure are observed in numerous experimental investigations (e.g. [1–3]) and are also predicted by numerical analyses (cf. [4]). In practical applications it is on the other hand more common that the crack has a shape that causes more constrained conditions. Thus the growth behaviour may differ in a non-conservative way from that
*
Corresponding author. Tel.: +46-8790-7549; fax: +46-8411-2418. E-mail address: [email protected] (F. Nilsson).
0013-7944/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0013-7944(03)00242-X
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Nomenclature a r da dN dr dN K0 b a_ 0 u R KI KI;max KI;min DKI DKI;nom DKI;eff KI DK I;large DK I;small PD PDref PDmeas n rY
crack length crack radius crack growth rate average crack growth rate for surface cracks parameter in crack growth equation exponent in crack growth equation reference growth rate angular coordinate load ratio stress-intensity factor maximum value of the stress-intensity factor during a cycle minimum value of the stress-intensity factor during a cycle stress-intensity factor range nominal stress-intensity factor range effective stress-intensity factor range stress-intensity factor averaged over the crack front range of the stress-intensity factor averaged over the crack front for the large cycles range of the stress-intensity factor averaged over the crack front for the small cycles potential drop signal reference potential drop signal measured potential drop signal number of small cycles within a load block yield strength
obtained from testing of CT specimens. In view of this it is surprising that so few investigations have been devoted to predictions of surface cracksÕ growth. Yngvesson and Nilsson [5] consider the growth of elliptic surface cracks under non-symmetric loading. The authors find that shapes are well predicted, but the prediction of the growth rate is non-conservative using crack growth data from CT specimens. No data corrected for crack closure were however available for the analysis in [5], so the use of such data could not be studied. James and de los Rios [6] study growth of very short surface cracks, typically in length 20–80 lm. Their observations do not seem entirely applicable to the growth of macroscopic cracks and clearly the crack closure effects may be different. One assumption that ought to result in conservative predictions is that no crack closure occurs at positive KI -values. Furthermore it is nowadays widely believed that crack closure is the main cause of the history effects that occur under variable amplitude loading. If constraint effects preclude crack closure it is therefore to be expected that history effects should be less prominent provided that data based on effective stress-intensity factor ranges are used for the prediction. The object of the present study was to conduct experiments with surface cracks in tensile specimens subjected to both constant amplitude and variable amplitude loading. The experimentally observed growth rates were then compared with predictions based on constant amplitude data obtained in a previous investigation of the same material [7] and also data from the present study.
F. Nilsson et al. / Engineering Fracture Mechanics 71 (2004) 1725–1735
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2. Experimental set-up and details In the present investigation the growth of surface cracks in tensile specimens (see Fig. 1) of a nickel base alloy (Inconel 718) under constant amplitude and variable amplitude loading at the elevated temperature 400 C was studied. The specimens (cross-section 4.27 mm · 10.16 mm) were equipped with a spark machined semi-circular starter notch with an initial radius of about 0.075 mm and a width of about 0.075 mm. Pre-cracking to crack depths of about 0.2 mm was needed to ensure that the stress field at the crack front was not influenced by the stress concentration from the notch. The semi-circular shape was kept reasonably well throughout the ensuing propagation. The fatigue crack growth experiments as described above were conducted on a gas turbine disk material, Inconel 718. The yield strength of this material during monotonically increasing load at 400 C was 1101 MPa, while the yield strength for cyclic loading was 850 MPa. All specimens were cut from turbine disks in such a way that the crack plane coincided with the x–z-plane, where x is the radial direction and z the thickness direction of the disk. The instantaneous size of the crack was monitored by a direct current (DC) potential drop technique. A constant current of 10 A was applied through the specimens. Thin wires with a diameter of 0.05 mm of a Ni-base alloy were spot-welded to each side of the starter crack. These wires measured the drop in potential over the crack, PDmeas . Another set of wires was spot-welded in a location of the gauge section where the stress field was not influenced by the crack. These wires were used to measure a reference potential drop signal, PDref . The potential drop signal used as a measure of the crack depth was defined as
r
4.27 mm
10.16 mm
Fig. 1. Sketch of specimen.
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PD ¼
PDmeas : PDref
ð1Þ
The ratio PDmeas =PDref was calculated in order to eliminate any influence of temperature on the potential drop signal. By this technique the crack radius could be measured at very short cycle intervals. This together with the scatter in the crack position determination necessitated a smoothing scheme when evaluating the growth rates. A straight line was fitted by the least-squares procedure to a number of crack sizes around the point where the rate was to be determined. The growth rate was then evaluated simply as the slope of this line. The number of points used in the evaluation scheme was chosen so that the necessary stability and resolution were obtained. In order to detect crack closure, records of the PD-value as a function load were taken periodically on full loading blocks. This technique is earlier described by Andersson et al. [8].
3. Scope of investigation and assumptions The testing of the surface cracked specimens was performed both under constant amplitude tension and variable amplitude testing as follows. The constant amplitude testing was performed at R ¼ 0:05. In these cases the maximum nominal stress was held constant through the experiment. Thus the stress-intensity factor range increased with crack size. The variable amplitude testing was performed for the two types of load blocks shown in Fig. 2. Type 1 is an idealised service cycle of a typical gas turbine, while type 2 is a modification of the single overload profile. For these types of sequences the range pair procedure and rainflow counting procedure give the result that each block contains one large cycle and n small cycles. Again the maximum nominal stress during each load block was held constant such that the block load as measured in stress-intensity factor was scaled up with the crack size.
σ0 σ 0, max 0.5 σ 0, max
Load block type 1 n
1
… n = number of small cycles
0.05 σ 0, max time
σ0
σ 0, max
Load block type 2 n = number of small cycles
0.55 σ 0, max … 0.05 σ 0, max
1
n time
Fig. 2. Block loading types.
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Three variants of the two types of blocks were tested. These are designated (1 þ n) where n ¼ 8, 16, 32. The load range for the large cycle is denoted DK I;large and the size of the small cycle DK I;small was around half of the maximum stress-intensity factor. The load ratio R was about 0.05 for the large cycle and for the small cycles in the type 2 sequences, while it was about 0.5 for the small cycles of the type 1 sequences. In order to study the transferability, the experimentally measured growth rates were compared with predictions based on nominal data (i.e., not corrected for crack closure) obtained in a previous study by M ansson et al. [7] and effective growth data obtained by new experiments on CT specimens. Thus, the crack growth rate was assumed to be described by a curve fit of the following type. bðRÞ da DKI ¼ a_ 0 ; dN K0 ðRÞ
ð2Þ
where DKI ¼ KI;max KI;min is the stress-intensity factor range and R ¼ KI;min =KI;max is the load ratio. Here a_ 0 is an arbitrary reference crack growth rate introduced for dimensional reasons. This rate was here chosen to 109 m/cycle. K0 ðRÞ and bðRÞ are material parameters. This way of formulating the growth equation has the advantage over the conventional practice that the quantities appearing have easily interpretable units. Numerous experiments show that if the crack growth test is performed according to the usual procedure, i.e. by keeping R constant and under decreasing DKI , the parameters in Eq. (1) depend on the load ratio R. It is suggested by Elber [9] that this dependence is caused by the crack surfaces closing at positive KI . This closure effect is also held responsible for sequence effects. A commonly used procedure is to monitor the closure level by compliance measurements and define an effective stress-intensity factor. This procedure is not unambiguous as is documented in many studies. An alternative procedure is to arrange the loading strategy in the test so that KI;min is always larger than a possible closure level. It is been suggested and verified by several studies (cf. [10]) that the so-called constant KI;max method can be used to obtain growth data supposedly free from crack closure and thus that b and K0 become independent of R. In such a procedure the maximum stress-intensity factor KI;max is kept constant during the test, but the minimum stress-intensity factor KI;min is successively increased. By such a test strategy crack closure does not seem to affect the growth as is evident from several investigations (for instance [10,11]). The method was employed in the present investigation by use of standard CT-50 specimens with a thickness of 12.55 mm and resulted in the data shown in Fig. 3. In this figure the regression lines for data at the stress ratios R ¼ 0:05 and 0.5 taken from M ansson et al. [7] are also shown. These data are nominal, i.e. not corrected for any closure effects. Numerical values of the parameters in the growth equations for the different regression lines are given in Table 1. In the case of a surface crack the stress-intensity factor is not constant but varies along the crack front and the question of which value to insert in the growth equations arises. Nilsson [12] suggests a scheme based on a least-squares procedure for growth rates. This scheme results in average stress-intensity factor values corresponding to the parameters chosen to describe the crack evolution. In the present case the only parameter is the crack radius r. Under such circumstances the procedure in [12] reduces to a simple average of the crack growth rate evaluated along the crack front and the stress-intensity factor can be averaged according to KI ¼
2 p
Z
p=2
1=b b KI ðuÞ du :
ð3Þ
0
The symbol u is an angular coordinate for the position along the quarter circular crack front. The stressintensity factor values obtained by Raju and Newman [13] were utilised in the evaluation of Eq. (3). The mean growth rate for the semi-circular crack is denoted dr=dN . This rate was evaluated according to Eq. (4) for a loading block in three different ways and compared with the experimental results.
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da –1 ------- ⁄ [ mm ⋅ cycle ] dN
10-4
effective data R = 0.5
R=0.05
10-5
10
5
20
∆K I ⁄ [ MPa m ] Fig. 3. Effective and nominal (R ¼ 0:05 and 0.5) crack growth data.
Table 1 Crack growth law parameters Nominal stress-intensity factor range R ¼ 0:05 Nominal stress-intensity factor range R ¼ 0:5 Effective stress-intensity factor range
dr a_ 0 ¼ dN n þ 1
DK I;large K0 ðR1 Þ
bðR1 Þ
DK I;small þn K0 ðR2 Þ
bðR2 Þ ! :
b
K0 /[MPa m1=2 ]
3.588 2.544 2.9029
6.99 3.54 3.87
ð4Þ
The first type of predictions was based on nominal data. Here the appropriate values of K0 and b were used for different values of R. The second type of predictions was based on the effective data and thus no dependence on R was assumed. If the data are indeed obtained from experiments with no crack closure, this method of evaluation should yield upper bound values of the growth rate. The third type of predictions was based on a technique that has proven successful for variable amplitude loading of plane specimens (cf. [14,11]). It is assumed that the closure level is solely determined by the large amplitude cycles and equal to the corresponding constant amplitude closure level. The closure level for the actual geometry at constant amplitude loading is obtained by comparing the actual stress-intensity factor with the one that would cause the same growth rate for closure free conditions. The so calculated closure level is then used to calculate DK I;eff and insert the so obtained values into Eq. (4). Growth rates according to this procedure are in the following referred to as reduced data. Assuming a closure level corresponding to the large cycles during the entire load block should provide an overestimate and thus yield lower bound values for the growth rates. This evaluation was only performed for the type 2 loading blocks. For the type 1 loading blocks the small cycles at R ¼ 0:5 should essentially be free from crack closure and thus the reduced data evaluation should coincide with the evaluation using nominal data.
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4. Results The results from the experiments as well as those predicted using either nominal data or effective data are plotted in Figs. 4–6. In the figures the mean growth rate dr=dN is shown as a function of the crack radius r. In Fig. 4 the results from two constant amplitude tests are shown. Fig. 5 contains the results from loading with load block 1 and Fig. 6 those obtained with loading according to load block 2. The overall pattern was similar for all the experiments. Predictions of the growth rate using effective data were closer to the experimental results, while the use of nominal data led to non-conservative results by up to two orders of magnitude. For the two constant amplitude tests the ratio between the predicted and the experimentally observed one is relatively constant and equals about 1.8 for both experiments in Fig. 4 for a crack radius r < 1:5 mm. This ratio of rates corresponds to an effective stress-intensity factor range of 0:82DKI;nom . Taking the applied R-value into account this means that the closure level was 0:23KI;max . For comparison the same type of estimate but for the curve denoted nominal load (i.e. equivalent to the uncorrected results from the CT testing) led to a closure level of 0:70KI;max . The result for the surface crack is somewhat lower than the level predicted in finite element analysis of a comparable geometry by Skinner and Daniewicz [15]. These authors report an opening level of about 0:3KI;max . They point out that the effects of hardening are important and the present material may exhibit different hardening characteristics than those assumed in [15]. The presently estimated closure level was on the other hand in fair agreement with experimental results on 7075-T6 aluminium obtained by Putra and Schijve [16] through fractography observations of striation patterns.
-3
dr –1 ------- ⁄ [ mm ⋅ cycle ] dN
10
effective data -4
10
experimental data
-5
10
nominal data -6
dr –1 ------- ⁄ [ mm ⋅ cycle ] dN
10
0
10-3
0.5
1
1.5 2 r ⁄ [ mm ]
2.5
3
effective data experimental data
-4
10
nominal data
-5
10
-6
10
0
0.5
1
1.5 2 r ⁄ [ mm ]
2.5
3
Fig. 4. Comparison between predictions and experimental results. The crack growth rate as a function of the crack radius at constant amplitude loading.
F. Nilsson et al. / Engineering Fracture Mechanics 71 (2004) 1725–1735
dr –1 ------- ⁄ [ mm ⋅ cycle ] dN
dr –1 ------- ⁄ [ mm ⋅ cycle ] dN
1732
effective data
experimental data
10-5
nominal data -6
10
0
0.5
1
1.5 2 r ⁄ [ mm ]
2.5
3
effective data
-4
10
n = 16 experimental data
-5
10
nominal data -6
10
–1 dr ------- ⁄ [ mm ⋅ cycle ] dN
n=8
-4
10
0
0.5
1
1.5 2 r ⁄ [ mm ]
2.5
3
-4
10
effective data
n = 32
nominal data experimental data
-5
10
-6
10
0
0.5
1
1.5 r ⁄ [ mm ]
2
2.5
3
Fig. 5. Comparison between predictions and experimental results. The crack growth rate as a function of the crack radius at type 1 block loading.
The results from the type 2 tests (Fig. 6) exhibited a behaviour similar to that of the constant amplitude tests. In all three cases the experimentally observed rates were bracketed by the reduced data and the effective data. For short crack lengths the predictions using reduced data fell near the experimental values, while for longer crack lengths the effective data provided better agreement. The results from type 1 tests (Fig. 5) showed almost no indications of crack closure except for the case n ¼ 32. In fact for n equal to 8 and 16 the effective data provided very good predictions for crack radii up to about 1.5 mm. The difference between the effective data and the nominal ones for R ¼ 0:5 is much smaller than that in Figs. 4 and 6. However it should be remembered that nominal data from CT specimens obtained for R ¼ 0:5 need not necessarily be completely free from crack closure. As can be seen from Fig. 3 there is a certain difference between these nominal data and the effective at the growth typical for the present experiments. The recorded loops for the PD-value as a function of load did not indicate any significant crack closure. This was a further indication that the levels of crack closure were comparatively low during the present experiments. In all cases the experimentally observed rates approached and in cases even exceeded the predictions by effective data for r larger than about 2 mm. This could possibly have been due to deviations from the linear elastic fracture mechanics regime.
dr –1 ------- ⁄ [ mm ⋅ cycle ] dN
F. Nilsson et al. / Engineering Fracture Mechanics 71 (2004) 1725–1735 -4
n=8
10
effective data -5
experimental data
10
reduced data -6
nominal data
10
0
0.5
1
–1 dr ------- ⁄ [ mm ⋅ cycle ] dN
-4
1.5 2 r ⁄ [ mm ]
2.5
3
n = 16
10
effective data -5
10
experimental data reduced data
-6
nominal data
10
0 –1 dr ------- ⁄ [ mm ⋅ cycle ] dN
1733
0.5
1
1.5 r ⁄ [ mm ]
2
2.5
3
-4
10
n = 32 effective data
-5
10
experimental data reduced data
-6
10
nominal data 0
0.5
1
1.5 r ⁄ [ mm ]
2
2.5
3
Fig. 6. Comparison between predictions and experimental results. The crack growth rate as a function of the crack radius at type 2 block loading.
5. Discussion and conclusions Since the absolute dimensions of the specimen were rather small, the limits of linear elastic fracture mechanics (LEFM) are of interest. The commonly used ASTM-criterion [17] for fatigue crack growth requires that any significant dimension d should satisfy the following relation. dP
4 2 ðKI =rY Þ : p
ð5Þ
Here rY is the yield strength. This requirement led together with the maximum applied stress level to that LEFM could be applied if r is smaller than about 2.6 mm. Thus it could be expected that in the latter parts of the growth process deviations from LEFM occurred gradually. It is likely that non-linear effects of this kind should be more pronounced for the type 1 load blocks where maximum stress is reached in every cycle, while in the type 2 loading the small cycles are far from the maximum stress. This might explain the tendency to non-conservatism during the latter part of the process for the type 1 loading.
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The results of the present study clearly show that closure effects can be very dependent of the crack geometry. It may thus be dangerous to use data where the closure effects are unknown to predict crack growth processes in other geometries. The constant KI;max method seems to be realistic alternative to produce data free from closure by a relatively simple technique. The present study indicates that conservative predictions can be obtained by use of such data provided LEFM conditions are satisfied. The third type of evaluation did not provide such good an agreement between predictions and experiments as in the previous studies [11,14]. Still the predictions using this method underestimated the growth rates as expected. Thus it is concluded that using effective data according to the first type of evaluation and reduced data according to the third type of evaluation is a engineering alternative to provide bounds for the growth rates under variable amplitude loading. Acknowledgements The present work was part of a larger project supported by KME, which is a consortium formed between the Swedish Energy Authority and the energy-oriented industry in Sweden. This support is gratefully acknowledged.
References [1] Dawicke DS, Grandt AF, Newman Jr JC. Three-dimensional crack closure behavior. Engng Fract Mech 1990;36:111– 21. [2] Shuter DM, Geary W. The influence of specimen thickness on fatigue crack growth retardation following an overload. Int J Fatigue 1995;17:111–9. [3] Marci G, Packman PF. The effects of the plastic wake zone on the conditions for fatigue crack propagation. Int J Fatigue 1980;16:33–53. [4] Solanki K, Daniewicz SR, Newman Jr JC. Finite element analysis of plasticity-induced fatigue crack closure: an overview. Engng Fract Mech, in press. [5] Yngvesson M, Nilsson F. Fatigue crack growth of surface cracks under non-symmetric loading. Engng Fract Mech 1999;63:375– 93. [6] James MN, de los Rios ER. Variable amplitude loading of small fatigue cracks in 6261-T6 aluminium alloy. Engng Fract Mech 1996;19:413–26. [7] M ansson T, Skantz J, Nilsson F. High temperature fatigue crack growth in two metals under constant and variable amplitude loading. Int J Fatigue 2002;24:1159–68. [8] Anderssson M, Persson C, Melin S, Hansson T. Numerical examination of potential drop technique for crack closure measurements. In: Blom AF, editor. Proceedings of the Eighth International Fatigue Congress, 3–7 June 2002, Stockholm, EMAS 2002, vol. 4. p. 2301–8. [9] Elber W. Fatigue crack closure under cyclic tension. Engng Fract Mech 1970;2:15–30. [10] Hertzberg RW, Herman WA, Clark T, Jaccard R. Simulation of short crack and other low closure conditions utilizing constant Kmax DK decreasing fatigue crack growth procedures. In: Larsen JM, Allen JE, editors. Small-crack test methods. ASTM STP, 1149. West Conshohocken, PA, USA: American Society of Testing and Materials; 1992. p. 197–229. € [11] M ansson T, Oberg H, Nilsson F. Closure effects on fatigue crack growth rates at constant and variable amplitude loading. Engng Fract Mech, in press. [12] Nilsson F. A consistent few parameter crack growth description procedure. Int J Fract 1992;54:35–44. [13] Raju IS, Newman Jr JC. Stress-intensity factors for internal and external cracks in cylindrical vessels. J Pressure Vessel Tech 1972;104:293–8. [14] Socie DF. Prediction of fatigue crack growth in notched members under variable amplitude loading histories. Engng Fract Mech 1997;9:849–65. [15] Skinner JD, Daniewicz SR. Simulation of plasticity-induced fatigue crack closure in part-through cracked geometries using finite element analyses. Engng Fract Mech 2002;69:1–11.
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[16] Putra IS, Schijve J. Crack opening stress measurements of surface cracks in 7075-T6 aluminium alloy plate specimen through electron fractography. Fatigue Fract Mater Struct 1992;15:323–38. [17] ASTM E-647. Standard test method for measurement of fatigue crack growth rates, ASTM Book of Standards, American Society for Testing and Materials; West Conshohoken, PA.
Growth of surface cracks under constant and variable amplitude loading F. Nilsson a
a,*
, T. Hansson b, T. M ansson
b
Department of Solid Mechanics, KTH, Royal Institute of Technology, SE-100 44 Stockholm, Sweden b Volvo Aero Corporation, SE-461 83 Trollh€attan, Sweden Received 17 March 2003; accepted 16 July 2003
Abstract Fatigue crack growth experiments were performed on surface cracked tensile specimens of Inconel 718 at 400 C. The loading was carried out at constant as well as at variable amplitude. The experimental results for the mean growth rate were compared with predictions based on data obtained from testing of compact tension specimens. Both nominal data as well as data corrected from measured crack closure were used in the predictions. The corrected data provided much better predictions than the nominal ones indicating that the level of crack closure during the testing of the surface cracked specimens was much lower than in compact tension specimens. 2003 Elsevier Ltd. All rights reserved. Keywords: Crack closure; Surface crack; Variable amplitude
1. Introduction The main goal of fatigue crack growth research is to ensure transferability between laboratory testing and crack growth in engineering structures and components. Fatigue crack growth laboratory experimentation is almost always performed on nominally two-dimensional specimens such as for instance the compact tension (CT) specimen. At this geometry a significant part of the crack front is influenced by the specimen side surfaces and the crack front state may here be far from plane deformation. In consequence the crack closure under positive values of the stress-intensity factor KI may be more pronounced than under more constrained conditions. Three-dimensional effects on crack closure are observed in numerous experimental investigations (e.g. [1–3]) and are also predicted by numerical analyses (cf. [4]). In practical applications it is on the other hand more common that the crack has a shape that causes more constrained conditions. Thus the growth behaviour may differ in a non-conservative way from that
*
Corresponding author. Tel.: +46-8790-7549; fax: +46-8411-2418. E-mail address: [email protected] (F. Nilsson).
0013-7944/$ - see front matter 2003 Elsevier Ltd. All rights reserved. doi:10.1016/S0013-7944(03)00242-X
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Nomenclature a r da dN dr dN K0 b a_ 0 u R KI KI;max KI;min DKI DKI;nom DKI;eff KI DK I;large DK I;small PD PDref PDmeas n rY
crack length crack radius crack growth rate average crack growth rate for surface cracks parameter in crack growth equation exponent in crack growth equation reference growth rate angular coordinate load ratio stress-intensity factor maximum value of the stress-intensity factor during a cycle minimum value of the stress-intensity factor during a cycle stress-intensity factor range nominal stress-intensity factor range effective stress-intensity factor range stress-intensity factor averaged over the crack front range of the stress-intensity factor averaged over the crack front for the large cycles range of the stress-intensity factor averaged over the crack front for the small cycles potential drop signal reference potential drop signal measured potential drop signal number of small cycles within a load block yield strength
obtained from testing of CT specimens. In view of this it is surprising that so few investigations have been devoted to predictions of surface cracksÕ growth. Yngvesson and Nilsson [5] consider the growth of elliptic surface cracks under non-symmetric loading. The authors find that shapes are well predicted, but the prediction of the growth rate is non-conservative using crack growth data from CT specimens. No data corrected for crack closure were however available for the analysis in [5], so the use of such data could not be studied. James and de los Rios [6] study growth of very short surface cracks, typically in length 20–80 lm. Their observations do not seem entirely applicable to the growth of macroscopic cracks and clearly the crack closure effects may be different. One assumption that ought to result in conservative predictions is that no crack closure occurs at positive KI -values. Furthermore it is nowadays widely believed that crack closure is the main cause of the history effects that occur under variable amplitude loading. If constraint effects preclude crack closure it is therefore to be expected that history effects should be less prominent provided that data based on effective stress-intensity factor ranges are used for the prediction. The object of the present study was to conduct experiments with surface cracks in tensile specimens subjected to both constant amplitude and variable amplitude loading. The experimentally observed growth rates were then compared with predictions based on constant amplitude data obtained in a previous investigation of the same material [7] and also data from the present study.
F. Nilsson et al. / Engineering Fracture Mechanics 71 (2004) 1725–1735
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2. Experimental set-up and details In the present investigation the growth of surface cracks in tensile specimens (see Fig. 1) of a nickel base alloy (Inconel 718) under constant amplitude and variable amplitude loading at the elevated temperature 400 C was studied. The specimens (cross-section 4.27 mm · 10.16 mm) were equipped with a spark machined semi-circular starter notch with an initial radius of about 0.075 mm and a width of about 0.075 mm. Pre-cracking to crack depths of about 0.2 mm was needed to ensure that the stress field at the crack front was not influenced by the stress concentration from the notch. The semi-circular shape was kept reasonably well throughout the ensuing propagation. The fatigue crack growth experiments as described above were conducted on a gas turbine disk material, Inconel 718. The yield strength of this material during monotonically increasing load at 400 C was 1101 MPa, while the yield strength for cyclic loading was 850 MPa. All specimens were cut from turbine disks in such a way that the crack plane coincided with the x–z-plane, where x is the radial direction and z the thickness direction of the disk. The instantaneous size of the crack was monitored by a direct current (DC) potential drop technique. A constant current of 10 A was applied through the specimens. Thin wires with a diameter of 0.05 mm of a Ni-base alloy were spot-welded to each side of the starter crack. These wires measured the drop in potential over the crack, PDmeas . Another set of wires was spot-welded in a location of the gauge section where the stress field was not influenced by the crack. These wires were used to measure a reference potential drop signal, PDref . The potential drop signal used as a measure of the crack depth was defined as
r
4.27 mm
10.16 mm
Fig. 1. Sketch of specimen.
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PD ¼
PDmeas : PDref
ð1Þ
The ratio PDmeas =PDref was calculated in order to eliminate any influence of temperature on the potential drop signal. By this technique the crack radius could be measured at very short cycle intervals. This together with the scatter in the crack position determination necessitated a smoothing scheme when evaluating the growth rates. A straight line was fitted by the least-squares procedure to a number of crack sizes around the point where the rate was to be determined. The growth rate was then evaluated simply as the slope of this line. The number of points used in the evaluation scheme was chosen so that the necessary stability and resolution were obtained. In order to detect crack closure, records of the PD-value as a function load were taken periodically on full loading blocks. This technique is earlier described by Andersson et al. [8].
3. Scope of investigation and assumptions The testing of the surface cracked specimens was performed both under constant amplitude tension and variable amplitude testing as follows. The constant amplitude testing was performed at R ¼ 0:05. In these cases the maximum nominal stress was held constant through the experiment. Thus the stress-intensity factor range increased with crack size. The variable amplitude testing was performed for the two types of load blocks shown in Fig. 2. Type 1 is an idealised service cycle of a typical gas turbine, while type 2 is a modification of the single overload profile. For these types of sequences the range pair procedure and rainflow counting procedure give the result that each block contains one large cycle and n small cycles. Again the maximum nominal stress during each load block was held constant such that the block load as measured in stress-intensity factor was scaled up with the crack size.
σ0 σ 0, max 0.5 σ 0, max
Load block type 1 n
1
… n = number of small cycles
0.05 σ 0, max time
σ0
σ 0, max
Load block type 2 n = number of small cycles
0.55 σ 0, max … 0.05 σ 0, max
1
n time
Fig. 2. Block loading types.
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Three variants of the two types of blocks were tested. These are designated (1 þ n) where n ¼ 8, 16, 32. The load range for the large cycle is denoted DK I;large and the size of the small cycle DK I;small was around half of the maximum stress-intensity factor. The load ratio R was about 0.05 for the large cycle and for the small cycles in the type 2 sequences, while it was about 0.5 for the small cycles of the type 1 sequences. In order to study the transferability, the experimentally measured growth rates were compared with predictions based on nominal data (i.e., not corrected for crack closure) obtained in a previous study by M ansson et al. [7] and effective growth data obtained by new experiments on CT specimens. Thus, the crack growth rate was assumed to be described by a curve fit of the following type. bðRÞ da DKI ¼ a_ 0 ; dN K0 ðRÞ
ð2Þ
where DKI ¼ KI;max KI;min is the stress-intensity factor range and R ¼ KI;min =KI;max is the load ratio. Here a_ 0 is an arbitrary reference crack growth rate introduced for dimensional reasons. This rate was here chosen to 109 m/cycle. K0 ðRÞ and bðRÞ are material parameters. This way of formulating the growth equation has the advantage over the conventional practice that the quantities appearing have easily interpretable units. Numerous experiments show that if the crack growth test is performed according to the usual procedure, i.e. by keeping R constant and under decreasing DKI , the parameters in Eq. (1) depend on the load ratio R. It is suggested by Elber [9] that this dependence is caused by the crack surfaces closing at positive KI . This closure effect is also held responsible for sequence effects. A commonly used procedure is to monitor the closure level by compliance measurements and define an effective stress-intensity factor. This procedure is not unambiguous as is documented in many studies. An alternative procedure is to arrange the loading strategy in the test so that KI;min is always larger than a possible closure level. It is been suggested and verified by several studies (cf. [10]) that the so-called constant KI;max method can be used to obtain growth data supposedly free from crack closure and thus that b and K0 become independent of R. In such a procedure the maximum stress-intensity factor KI;max is kept constant during the test, but the minimum stress-intensity factor KI;min is successively increased. By such a test strategy crack closure does not seem to affect the growth as is evident from several investigations (for instance [10,11]). The method was employed in the present investigation by use of standard CT-50 specimens with a thickness of 12.55 mm and resulted in the data shown in Fig. 3. In this figure the regression lines for data at the stress ratios R ¼ 0:05 and 0.5 taken from M ansson et al. [7] are also shown. These data are nominal, i.e. not corrected for any closure effects. Numerical values of the parameters in the growth equations for the different regression lines are given in Table 1. In the case of a surface crack the stress-intensity factor is not constant but varies along the crack front and the question of which value to insert in the growth equations arises. Nilsson [12] suggests a scheme based on a least-squares procedure for growth rates. This scheme results in average stress-intensity factor values corresponding to the parameters chosen to describe the crack evolution. In the present case the only parameter is the crack radius r. Under such circumstances the procedure in [12] reduces to a simple average of the crack growth rate evaluated along the crack front and the stress-intensity factor can be averaged according to KI ¼
2 p
Z
p=2
1=b b KI ðuÞ du :
ð3Þ
0
The symbol u is an angular coordinate for the position along the quarter circular crack front. The stressintensity factor values obtained by Raju and Newman [13] were utilised in the evaluation of Eq. (3). The mean growth rate for the semi-circular crack is denoted dr=dN . This rate was evaluated according to Eq. (4) for a loading block in three different ways and compared with the experimental results.
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da –1 ------- ⁄ [ mm ⋅ cycle ] dN
10-4
effective data R = 0.5
R=0.05
10-5
10
5
20
∆K I ⁄ [ MPa m ] Fig. 3. Effective and nominal (R ¼ 0:05 and 0.5) crack growth data.
Table 1 Crack growth law parameters Nominal stress-intensity factor range R ¼ 0:05 Nominal stress-intensity factor range R ¼ 0:5 Effective stress-intensity factor range
dr a_ 0 ¼ dN n þ 1
DK I;large K0 ðR1 Þ
bðR1 Þ
DK I;small þn K0 ðR2 Þ
bðR2 Þ ! :
b
K0 /[MPa m1=2 ]
3.588 2.544 2.9029
6.99 3.54 3.87
ð4Þ
The first type of predictions was based on nominal data. Here the appropriate values of K0 and b were used for different values of R. The second type of predictions was based on the effective data and thus no dependence on R was assumed. If the data are indeed obtained from experiments with no crack closure, this method of evaluation should yield upper bound values of the growth rate. The third type of predictions was based on a technique that has proven successful for variable amplitude loading of plane specimens (cf. [14,11]). It is assumed that the closure level is solely determined by the large amplitude cycles and equal to the corresponding constant amplitude closure level. The closure level for the actual geometry at constant amplitude loading is obtained by comparing the actual stress-intensity factor with the one that would cause the same growth rate for closure free conditions. The so calculated closure level is then used to calculate DK I;eff and insert the so obtained values into Eq. (4). Growth rates according to this procedure are in the following referred to as reduced data. Assuming a closure level corresponding to the large cycles during the entire load block should provide an overestimate and thus yield lower bound values for the growth rates. This evaluation was only performed for the type 2 loading blocks. For the type 1 loading blocks the small cycles at R ¼ 0:5 should essentially be free from crack closure and thus the reduced data evaluation should coincide with the evaluation using nominal data.
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4. Results The results from the experiments as well as those predicted using either nominal data or effective data are plotted in Figs. 4–6. In the figures the mean growth rate dr=dN is shown as a function of the crack radius r. In Fig. 4 the results from two constant amplitude tests are shown. Fig. 5 contains the results from loading with load block 1 and Fig. 6 those obtained with loading according to load block 2. The overall pattern was similar for all the experiments. Predictions of the growth rate using effective data were closer to the experimental results, while the use of nominal data led to non-conservative results by up to two orders of magnitude. For the two constant amplitude tests the ratio between the predicted and the experimentally observed one is relatively constant and equals about 1.8 for both experiments in Fig. 4 for a crack radius r < 1:5 mm. This ratio of rates corresponds to an effective stress-intensity factor range of 0:82DKI;nom . Taking the applied R-value into account this means that the closure level was 0:23KI;max . For comparison the same type of estimate but for the curve denoted nominal load (i.e. equivalent to the uncorrected results from the CT testing) led to a closure level of 0:70KI;max . The result for the surface crack is somewhat lower than the level predicted in finite element analysis of a comparable geometry by Skinner and Daniewicz [15]. These authors report an opening level of about 0:3KI;max . They point out that the effects of hardening are important and the present material may exhibit different hardening characteristics than those assumed in [15]. The presently estimated closure level was on the other hand in fair agreement with experimental results on 7075-T6 aluminium obtained by Putra and Schijve [16] through fractography observations of striation patterns.
-3
dr –1 ------- ⁄ [ mm ⋅ cycle ] dN
10
effective data -4
10
experimental data
-5
10
nominal data -6
dr –1 ------- ⁄ [ mm ⋅ cycle ] dN
10
0
10-3
0.5
1
1.5 2 r ⁄ [ mm ]
2.5
3
effective data experimental data
-4
10
nominal data
-5
10
-6
10
0
0.5
1
1.5 2 r ⁄ [ mm ]
2.5
3
Fig. 4. Comparison between predictions and experimental results. The crack growth rate as a function of the crack radius at constant amplitude loading.
F. Nilsson et al. / Engineering Fracture Mechanics 71 (2004) 1725–1735
dr –1 ------- ⁄ [ mm ⋅ cycle ] dN
dr –1 ------- ⁄ [ mm ⋅ cycle ] dN
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effective data
experimental data
10-5
nominal data -6
10
0
0.5
1
1.5 2 r ⁄ [ mm ]
2.5
3
effective data
-4
10
n = 16 experimental data
-5
10
nominal data -6
10
–1 dr ------- ⁄ [ mm ⋅ cycle ] dN
n=8
-4
10
0
0.5
1
1.5 2 r ⁄ [ mm ]
2.5
3
-4
10
effective data
n = 32
nominal data experimental data
-5
10
-6
10
0
0.5
1
1.5 r ⁄ [ mm ]
2
2.5
3
Fig. 5. Comparison between predictions and experimental results. The crack growth rate as a function of the crack radius at type 1 block loading.
The results from the type 2 tests (Fig. 6) exhibited a behaviour similar to that of the constant amplitude tests. In all three cases the experimentally observed rates were bracketed by the reduced data and the effective data. For short crack lengths the predictions using reduced data fell near the experimental values, while for longer crack lengths the effective data provided better agreement. The results from type 1 tests (Fig. 5) showed almost no indications of crack closure except for the case n ¼ 32. In fact for n equal to 8 and 16 the effective data provided very good predictions for crack radii up to about 1.5 mm. The difference between the effective data and the nominal ones for R ¼ 0:5 is much smaller than that in Figs. 4 and 6. However it should be remembered that nominal data from CT specimens obtained for R ¼ 0:5 need not necessarily be completely free from crack closure. As can be seen from Fig. 3 there is a certain difference between these nominal data and the effective at the growth typical for the present experiments. The recorded loops for the PD-value as a function of load did not indicate any significant crack closure. This was a further indication that the levels of crack closure were comparatively low during the present experiments. In all cases the experimentally observed rates approached and in cases even exceeded the predictions by effective data for r larger than about 2 mm. This could possibly have been due to deviations from the linear elastic fracture mechanics regime.
dr –1 ------- ⁄ [ mm ⋅ cycle ] dN
F. Nilsson et al. / Engineering Fracture Mechanics 71 (2004) 1725–1735 -4
n=8
10
effective data -5
experimental data
10
reduced data -6
nominal data
10
0
0.5
1
–1 dr ------- ⁄ [ mm ⋅ cycle ] dN
-4
1.5 2 r ⁄ [ mm ]
2.5
3
n = 16
10
effective data -5
10
experimental data reduced data
-6
nominal data
10
0 –1 dr ------- ⁄ [ mm ⋅ cycle ] dN
1733
0.5
1
1.5 r ⁄ [ mm ]
2
2.5
3
-4
10
n = 32 effective data
-5
10
experimental data reduced data
-6
10
nominal data 0
0.5
1
1.5 r ⁄ [ mm ]
2
2.5
3
Fig. 6. Comparison between predictions and experimental results. The crack growth rate as a function of the crack radius at type 2 block loading.
5. Discussion and conclusions Since the absolute dimensions of the specimen were rather small, the limits of linear elastic fracture mechanics (LEFM) are of interest. The commonly used ASTM-criterion [17] for fatigue crack growth requires that any significant dimension d should satisfy the following relation. dP
4 2 ðKI =rY Þ : p
ð5Þ
Here rY is the yield strength. This requirement led together with the maximum applied stress level to that LEFM could be applied if r is smaller than about 2.6 mm. Thus it could be expected that in the latter parts of the growth process deviations from LEFM occurred gradually. It is likely that non-linear effects of this kind should be more pronounced for the type 1 load blocks where maximum stress is reached in every cycle, while in the type 2 loading the small cycles are far from the maximum stress. This might explain the tendency to non-conservatism during the latter part of the process for the type 1 loading.
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The results of the present study clearly show that closure effects can be very dependent of the crack geometry. It may thus be dangerous to use data where the closure effects are unknown to predict crack growth processes in other geometries. The constant KI;max method seems to be realistic alternative to produce data free from closure by a relatively simple technique. The present study indicates that conservative predictions can be obtained by use of such data provided LEFM conditions are satisfied. The third type of evaluation did not provide such good an agreement between predictions and experiments as in the previous studies [11,14]. Still the predictions using this method underestimated the growth rates as expected. Thus it is concluded that using effective data according to the first type of evaluation and reduced data according to the third type of evaluation is a engineering alternative to provide bounds for the growth rates under variable amplitude loading. Acknowledgements The present work was part of a larger project supported by KME, which is a consortium formed between the Swedish Energy Authority and the energy-oriented industry in Sweden. This support is gratefully acknowledged.
References [1] Dawicke DS, Grandt AF, Newman Jr JC. Three-dimensional crack closure behavior. Engng Fract Mech 1990;36:111– 21. [2] Shuter DM, Geary W. The influence of specimen thickness on fatigue crack growth retardation following an overload. Int J Fatigue 1995;17:111–9. [3] Marci G, Packman PF. The effects of the plastic wake zone on the conditions for fatigue crack propagation. Int J Fatigue 1980;16:33–53. [4] Solanki K, Daniewicz SR, Newman Jr JC. Finite element analysis of plasticity-induced fatigue crack closure: an overview. Engng Fract Mech, in press. [5] Yngvesson M, Nilsson F. Fatigue crack growth of surface cracks under non-symmetric loading. Engng Fract Mech 1999;63:375– 93. [6] James MN, de los Rios ER. Variable amplitude loading of small fatigue cracks in 6261-T6 aluminium alloy. Engng Fract Mech 1996;19:413–26. [7] M ansson T, Skantz J, Nilsson F. High temperature fatigue crack growth in two metals under constant and variable amplitude loading. Int J Fatigue 2002;24:1159–68. [8] Anderssson M, Persson C, Melin S, Hansson T. Numerical examination of potential drop technique for crack closure measurements. In: Blom AF, editor. Proceedings of the Eighth International Fatigue Congress, 3–7 June 2002, Stockholm, EMAS 2002, vol. 4. p. 2301–8. [9] Elber W. Fatigue crack closure under cyclic tension. Engng Fract Mech 1970;2:15–30. [10] Hertzberg RW, Herman WA, Clark T, Jaccard R. Simulation of short crack and other low closure conditions utilizing constant Kmax DK decreasing fatigue crack growth procedures. In: Larsen JM, Allen JE, editors. Small-crack test methods. ASTM STP, 1149. West Conshohocken, PA, USA: American Society of Testing and Materials; 1992. p. 197–229. € [11] M ansson T, Oberg H, Nilsson F. Closure effects on fatigue crack growth rates at constant and variable amplitude loading. Engng Fract Mech, in press. [12] Nilsson F. A consistent few parameter crack growth description procedure. Int J Fract 1992;54:35–44. [13] Raju IS, Newman Jr JC. Stress-intensity factors for internal and external cracks in cylindrical vessels. J Pressure Vessel Tech 1972;104:293–8. [14] Socie DF. Prediction of fatigue crack growth in notched members under variable amplitude loading histories. Engng Fract Mech 1997;9:849–65. [15] Skinner JD, Daniewicz SR. Simulation of plasticity-induced fatigue crack closure in part-through cracked geometries using finite element analyses. Engng Fract Mech 2002;69:1–11.
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[16] Putra IS, Schijve J. Crack opening stress measurements of surface cracks in 7075-T6 aluminium alloy plate specimen through electron fractography. Fatigue Fract Mater Struct 1992;15:323–38. [17] ASTM E-647. Standard test method for measurement of fatigue crack growth rates, ASTM Book of Standards, American Society for Testing and Materials; West Conshohoken, PA.