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Validation of stress intensity factors of diametrically opposed corner cracks in a hole
International Journal of Fatigue 31 (2009) 712–718
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Validation of stress intensity factors of diametrically opposed corner cracks in a hole S.A. Fawaz *, D. Hill Department of Engineering Mechanics, Center for Aircraft Structural Life Extension (CAStLE), United States Air Force Academy, 2354 Fairchild Hall, Suite 6L-155, CO 80840, USA
a r t i c l e
i n f o
Article history: Received 21 September 2007 Received in revised form 27 February 2008 Accepted 8 March 2008 Available online 16 March 2008 Keywords: Stress intensity Striation spacing Validation Unsymmetric corner cracks
a b s t r a c t Population of the world’s largest database of stress intensity factor (K) solutions began in 2002 with the calculation of 5.6 million K solutions for diametrically opposed unsymmetric corner cracks at a straight shank hole in a finite width sheet subject to remote tension, remote bending, and bearing loading. Previous work to validate these K solutions was in the form of fatigue life predictions and crack shape development. The current work attempts to build on the previous validation efforts with the addition of comparing the calculated K solutions with K solutions obtained from carefully controlled laboratory experiments. The latter are obtained via fatigue striation measurements at high magnification, up to 40,000, using a scanning electron microscope and crack growth rate data, in terms of da/dN vs. DK at the same test condition. The results show the numerical K solutions are within 20% of the experimentally derived K’s at discrete locations along the crack front. The relatively large error is due to the discontinuous crack extension process of the crack front. Moreover, the entire crack front does not instantaneously extend uniformly in a self similar fashion. The crack extends stepwise over discrete portions of the crack front. Possibly averaging the striation spacing over a specified arc length of the crack front would ameliorate the discontinuous nature of crack propagation resulting in better correlation between the numerical and experimental results. As a result of the current work, we have shown time consuming striation spacing measurements at high magnification are not required to validate K solutions. The best method for such validation efforts is using the fatigue life, crack history, and crack shape which can be obtained at 1/ 10th the cost of obtaining striation spacing measurements. Published by Elsevier Ltd.
1. Introduction Fatigue-related problems have cost the military and commercial aircraft operators’ loss of aircraft, expensive repair actions, and an ever increasing inspection burden. The uncertainty associated with fleet management activities has many sources, but can be separated into two broad categories; uncertainty related to aircraft structural life prediction and uncertainty related to maintenance actions. This paper only addresses the former with specific focus to provide stress intensity factor, K, solutions with known and controlled accuracy. Although damage tolerance analysis, DTA, methods have been in places for decades, much room for improvement is still possible. For example, the case of two unsymmetric corner cracks at a fastener hole, see Fig. 1 being modeled as symmetric corner cracks. In this figure, rbypass = ro – rbrg where ro is the remote applied stress; rbrg is the bearing stress calculated by strength of materials methods (rbrg = P/Dt). Specifically, the rbypass is the membrane stress remaining in the plate after load transfer through a rivet or bolt. In addition, h is measured from the top of hole at the hole centerline. Newman
* Corresponding author. Tel.: +1 7193336266; fax: +1 7193332944. E-mail address: [email protected] (S.A. Fawaz). 0142-1123/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.ijfatigue.2008.03.015
and Shivakumar found using Pcos2 h instead of just P gave better correlation of measured and calculated stress concentration factors of a pin-loaded hole [1]. Neither in the laboratory nor in service do two corner cracks at a fastener hole propagate in a symmetric fashion. Calculation of these new K’s was previously published in [2] with limited validation in [3]. The validation effort is extended in this work and presented below. 2. Background The most common method for assessing the accuracy of K solutions is in terms of predicting the fatigue life of a laboratory test coupon. In a controlled experiment, the material and mechanical properties, loading magnitude and sequence, initial and final crack size are known. Calculating the number of load cycles to propagate the crack from the initial size to final size is rather straightforward. The results of this process can be seen in Fig. 2 where a comparison would be made between the final measured and predicted crack lengths. Further assessment of the accuracy of the K solution is made by comparing the crack history, crack length in time, of the measured and predicted results. A third method is available for assessing the accuracy of K solutions which is to directly compare an experimentally derived K with the calculated K.
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σ bypass
4
σ bending
Test 2, 7075-T651 w=49.27 mm, t =6.33 mm, r =3.25 m m c 1 =1.58 mm, c 2 =2.68 mm
Pcos2θ
2h c2
2r c1 2b
σ bending
Crack Length (mm)
3.5
Measured LeftCrack AFGROW Right Crack Measured RightCrack
3
2.5
2
σo a2
AFGROW Left Crack
a1
1.5 87.000
89.500
92.000 94.500 K cycles
Fig. 1. Diametrically opposed unsymmetric corner cracks subject to tension, bending, and bearing loading.
97.000
4
The details of the experimental program were previously reported by Fawaz, Andersson, and Newman and only briefly discussed here. The test program was designed to provide an array of crack data; specifically, crack shape (crack depth to crack length, a/c) and size (crack depth to sheet thickness, a/t) at various times during the fatigue life. The crack data would then be used to experimentally verify the newly calculated K’s. The specimens were quite simple, 50 mm wide, 200 mm high, 6.35 mm thick sheet with a 6.35 mm diameter centrally located straight shank hole. The specimen length and width were chosen to eliminate the finite height effect as reported by Raju and Newman [5] and Tada, Paris, and Irwin[6], respectively. All specimens were made from 7075T651 plate and tested using a blocked fatigue spectrum on a closed loop servo-hydraulic fatigue machine at 70 MPa and 10 Hz. Electro-discharge machined (EDM) notches were used to better control the starter notch geometry. The spectrum, shown in Fig. 4 was designed to create marker bands, groups of fatigue striations, on the fracture surface. One pass through the spectrum contains 8170 cycles. In situ crack growth measurement was not used as nearly the entire crack history was reconstructed post-test using a Nikon SMZ645 Stereoscope and/or Jeol JSM-6480LV scanning electron microscope to detect the marker bands. This technique was successfully demonstrated by Pelloux, Warren, and O’Grady [7] and in [2] and [8]. When the part through crack grew threw the specimen thickness, determined by visual inspection, the test was halted in order to reduce the chance the fracture surfaces would come into contact during cycling and damage the marker bands. 3.1. Fatigue Crack Growth Characterization of 7075-T651 All specimens used in this effort were machined from a single piece of 7075-T651 aluminum. A compact tension, C(T), specimen was used for fatigue crack growth characterization. The specimens were cyclically loaded in a servo-hydraulic test frame. A clip gauge was used to measure the crack opening displacement which could then be used with a compliance equation to determine crack length. All test procedures and data analysis followed ASTM standard E647. Three specimens were pre-cracked with constant
Crack Length (mm)
3. Experiments
3.5
AFGROW Left Crack Measured LeftCrack AFGROW RightCrack Measured Right Crack
3
2.5
2
1.5 62.000
64.500
67.000
69.500 K cycles
72.000
74.500
4 Tes t 4,
70 75-T651
3.8 w=25.37 mm, t =6.38 mm, r =1.61 m m c 1 =2.38 mm, c 2 =0 mm
3.6 Crack Length (mm)
The basic crack scenario considered for K validation is shown in Fig. 3. To calculate a K value from a fatigue test, the procedure developed by James and Anderson is used [4]. The test protocol and procedure for calculating empirical K’s are discussed in the following sections.
Tes t 3, 7075- T651 w=49. 17 mm, t =6.36 mm, r =3.23 m m c 1 =2.76 mm, c 2 =1.75 mm
3.4
AFGROW LeftCrack Measured Left Crack
3.2 3 2.8 2.6 2.4 2.2 2 47.000 49.500 52.000 54.500 57.000 59.500 62.000 64.500 67.000 K cycles
Fig. 2. Predicted and experimental crack history of diametrically opposed, unsymmetric corner cracks at a centrally located hole in a finite width plate subject to remote tension5.
t D W Fig. 3. Cracking scenarios investigated.
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Fig. 4. Diagram of marker-band spectrum.
tain the constant K gradient. The tests with R of 0.05 and 0.5 were tested in constant amplitude fatigue running at 10 Hz. The data from the three tests were then analyzed and plotted in terms of da/dN vs. DK, Fig. 5.
1.E-04 R = 0.05 R = 0.1 R = 0.5
1.E-05
3.2. K Validation specimens
da/dN (m/cycle)
1.E-06
1.E-07
1.E-08
1.E-09
1.E-10 1
10
100
Δ K (MPa√m) Fig. 5. Crack growth rate curves for 7075-T651 sheet used in the test program.
amplitude sinusoidal cycles. The load was determined so that when the pre-crack length was approximately 0.1 inch from the p starter notch a Kmax of 243 MPa mm would be felt at the crack tip. The specimens were pre-cracked with an R (minimum stress/ maximum stress) of 0.1 and frequency of 10 Hz. After the specimens were pre-cracked, each specimen was tested at a different R, 0.05, 0.1, and 0.5. For the test with R = 0.1, a constant K gradient (10 mm1) test was performed. A clip gage was used to measure crack length with the MTS software controlling the load to main-
The dimensions of the specimens investigated here are given in Table 1. Since all specimens were cut from the same sheet of material, the nominal thickness is approximately the same for all specimens at 6.35 mm. All specimens had smooth machined sides and were machined to a tolerance of ±0.0762 mm. The hole was drilled in one pass using a standard drill bit. After the specimens were machined, diametrically opposed and perpendicular to the applied load elliptical electric discharge machine (EDM) notches were introduced on the edge of the hole to facilitate crack nucleation and asymmetric crack growth. Again a closed loop servo-hydraulic test frame was used to test the specimens. The frame was controlled by a TestStar IImTM controller with the TestStarTM software package. The specimens were gripped using MTS hydraulic grips and the test protocol is given below. Each specimen was tested using the marker-band spectrum, Fig. 4, with a maximum spectrum stress of 69 MPa. The spectrum begins with a constant amplitude block at 100% of the full load for 2000 cycles. After the 2000 cycle block the load drops to 80% of the full load for 100 cycles and then completes a quick 10 cycles back at 100% of the full load. The 100 cycles at 80% leave a dark distinguishable band on the fracture surface that can easily be seen under a microscope. This 100 cycle 10 cycles process is repeated 10 times therefore making the first marker-band set with 10 dark marker-bands. After a set of 10 marker-bands is created, the spectrum runs for another 2000 cycles at 100% load. When this block is complete a set of 4 marker cycles are applied just as before. Then 2000 more cycles are ran and another set of 6 marker cycles are made. Fig. 6 shows an image taken with (a) an optical microscope and (b) a scanning electron microscope (SEM). The striation spacing can be seen leading up to the dark marker-bands with 10 striations in between each band. When both cracks transition from corner cracks to through cracks the tests were stopped after the next full pass and before the set of ten marker-bands started. At this point the specimens were statically over-loaded until final failure occurred. By following
Table 1 K validation specimen dimensions Specimen
Width (mm)
Thickness (mm)
Hole diameter (mm)
Initial crack dimensions Left c-tip (mm)
Left a-tip (mm)
Right c-tip (mm)
Right a-tip (mm)
Test Test Test Test
49.250 49.276 49.225 25.374
6.325 6.35 6.35 6.35
6.35 6.35 6.35 3.175
1.862 1.214 1.029 1.225
3.102 1.277 3.801 1.282
1.274 .955 1.343 None
1.241 3.103 1.230 None
1 2 3 4
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Fig. 6. Micrograph of marker bands on specimen test 3 at 5500.
Table 2 Stress levels, life results, and number of cracks formed from the K validation specimens Specimen
Life (Passes)
Total cycles
Cracks
Valid crack configurations between (Pass/Markerband)
Test Test Test Test
10 15+ 12 11+
81,700 122,850 98,040 90,716
2 2 2 1
No valid combinations (12/4) (9/6) (8/6)
1 2 3 4
this procedure, the integrity of the markers bands was preserved. This also made it easier to find the last marker-band which would be a set of 6. The two halves were then placed in a desiccator awaiting fractographic analysis. Table 2 shows the relevant test results. In this table, columns two and three are the number of passes through the marker spectrum (8170 cycles per pass), and corresponding total number of fatigue cycles, respectively. The specimens either one crack on each side of the hole or just one crack on one side of the hole as indicated in column four. Columns five and six gives the range over which crack growth predictions could be made. For example in test 2, the prediction started at the crack size of a group of 6 marker bands in pass 11 and finished at a group of 4 marker bands in pass 12. The crack configuration is valid if the a1/c1, a2/ c2, a1/t, and a2/t ratio is within the analysis parameter space of the calculated K solutions. The results of test 1 did not contain any valid combination even though 30 marker cycles were applied (10 passes of the spectrum and 3 marker cycles per pass) thus indicating the challenge in collecting experimental data for validation efforts of this type. 3.3. Fractographic analysis Fractographic analysis was conducted to determine the final crack length, flaw shape development, and local crack growth rates. A Nikon Stereo Microscope equipped with Quadracheck and a motorized stage was used for all measurements. Magnification up to 1000 was used. In addition, due to the submicron resolution of the stage, accurate measurements outside the field of view of the lens could be made. Since each side of the hole had a unique crack, each crack was analyzed and investigated separately. The origin was defined as the intersection of the hole bore and sheet surface were the notch had been machined originally. The mapping of the marker-bands began at the static to fatigue fracture transition, Fig. 7. Following the procedure described above, a set of six marker-bands should be found at the edge of this transition point. By starting at this set of marker-bands and working towards the origin, each set of six, four, or ten marker-bands could be found in order and traced back to the nucleation site. As each set of marker-bands were found they were mapped out by finding 8–10
Points No valid combinations (11/6) (8/4) (4/10)
0 3 5 12
Fig. 7. Right side fracture surface of test 3.
points along the arc of the marker-band which is a record of the crack front at a given cycle count. These points could then be plotted as shown in Fig. 8. The first number of each dataset name indicates the number of passes through the marker spectrum and the second number indicates what type of marker band (10, 4, or 6) was measured. The ‘‘Points” in Fig. 8 are the a- and c-crack tips used for crack growth predictions. The crack on the opposite side of the hole was analyzed in the same manner. In the open literature, there is not a K solution for a part through crack on one side of a hole and a through crack on the opposing side; therefore, when this scenario occurred, which was quite frequently, this crack data could not be used for K validation. To remove the boundary layer effect, non-elliptical crack shape where the crack front intersects a free surface, a computer program was written in C++ to read from a text file the eight to ten points from the marker-band mapping and calculate a best fit ellipse using linear regression. From this new best fit ellipse, the maximum crack depth and crack length were used as the a- and cdimensions for the corner crack modeled in AFGROW. The crack
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Notch
6
12-6 12-4 12-10
5
11-6
Crack Depth (mm)
11-4 11-10
4
10-6 10-4 10-10
3
9-6 9-4 9-10 2
8-6 8-4 Points
1
0 0
1
2
3
4
5
6
Crack Length (mm) Fig. 8. Reconstructed marker-bands on the right side of test 3.
45
James Anderson
40 35
% Error
30 25 20 15 10 5 0 0
5
10
15
20
25
30
Crack length (mm) Fig. 10. Crack length vs. % error of James–Anderson data.
Fig. 9. Measuring of striation spacing on test 3.
histories are shown in Fig. 2 with the prediction and measured values starting/stopping at the particular pass and marker type given in Table 2. Similarly, the measured and predicted flaw shapes are given in Fig. 11. The fit a and c values from the linear regression program were then entered into AFGROW as a right and left corner crack on a specimen with the same width, thickness and hole diameter of the original test coupon. The crack growth rates determined during the material characterization part of this effort were used via the AFGROW material property table lookup. Any batch, lot, or manufacturer material variability is removed by this approach. The test spectrum, Fig. 4, was also used for the predictions; thus, even though the load sequence effects were shown to be negligible in reference [7] if any effects did exist, they would be captured in the prediction. Using AFGROW’s multi-point advanced model for this cracking scenario yields DK values at 11 points along the crack front
for a given cycle count. Thus, a point to point comparison of analytical and experimental DK values is possible. The empirical K’s are calculated from a striation spacing measurements, see Fig. 9 and the crack growth rate curve. The striation measurements were made at the end of the block of 2000 cycles just before the marker bands. This location was chosen because it was found that the striation spacing in between the marker bands experienced small load sequence effects. After 2000 cycles, the crack growth rate would not be affected by any prior load history, for example the 80% cycles. Numerous, 10–20 in most cases, striations were measured and then that distance was divided by the exact number of striations to give an accurate local da/dN. With an accurate da/dN calculation and the da/dN vs. DK curve that was created for this specific lot of material, an empirical DK could then be determined. 4. Discussion The discussion below is separated by the three validation efforts undertaken in this effort; namely, validation in terms of fatigue life predictions, flaw shape predictions, and empirical versus analytical stress intensity factor solutions.
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4.1. Fatigue life predictions The fatigue life predictions using the new K solutions, shown in Fig. 2, have a maximum error of 7% over the rather short interval of crack growth. Unfortunately the cracks grow rather quickly even at the low stress level of 69 MPa and thus the data collected is not extensive. The new K solutions over-estimate K which results in a shorter fatigue life than measured. 4.2. Flaw shape predictions The flaw shape predictions shown in Fig. 11 have a maximum error of 18% which occurs at the very last measurement of the sin-
gle crack case, test 4. Except for this one dataset, the correlation between measured and predicted (using the new K solutions) flaw shapes is acceptable. The new K solutions over-estimate K which results in a larger crack size all along the crack front than what was measured. 4.3. Comparison of empirical and analytical stress intensity factor solutions James and Anderson, in the late 1960’s, found empirical K solutions for crack geometries for which no analytical solution existed.6 Their work consisted of testing three thick walled pressure vessels. They then compared the results by plotting crack length vs. K. The
Fig. 11. Flaw shape development.
20%
10%
Test 2 Test 3 Test 4
10%
0% -10%
Error
0%
Error
Test 2 Test 3 Test 4
-10%
-20% -30%
-20%
-40%
-30% -40%
-50% 0
0.5
1
1.5
2
2.5
3
3.5
4
-60%
0
1
c-tip Crack Length (mm)
2
3
4
c-tip Crack Length (mm) Fig. 12. Crack length vs. % error comparisons.
5
6
7
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correlation of their results shows an error or variation of results between 11% and 39%. By plotting this error with respect to crack length, Fig. 10, the trend shows that at short and long cracks the error is the largest with this method and less variation around crack lengths of 12 mm. Since the crack lengths of interest in this study were below 5 mm in length, the current results are consistent in terms of magnitude of error as in [5]. In Fig. 12, a correlation between crack depth (a-tip) and length (c-tip) and error is not apparent. For all specimens tested, the error between the empirical and numerical DK’s ranged from 50% to 15%. Several sources of error are possible. With regard to testing procedures, specifically the uncertainty in the applied loading; the load command and response signal was recorded, and the maximum difference throughout each test was less than ±45 N resulting in a 0.2% error in DK at the peak stress of 69 MPa. As evident from the fractographic analysis, the EDM notch is not part-elliptical and this would contribute to the error in the predictions if the analysis used initial flaw dimensions of the EDM notch. However, the specimen was cycled for several passes through the spectrum to fully develop the part-elliptical crack front with the shape being verified by post-test fractography. The initial flaw dimensions were taken from the first fully developed part-elliptical crack front. A conclusion from reference [3] was that a part-elliptical EDM notch does not always result in a part-elliptical crack shape. AFGROW uses the parametric angle of an ellipse, u, to calculate the location of the point on the crack front for multi point analysis. The experimental measurements are also recorded in terms of u and the da/dN (and thus K) for nearly the same u is compared. Interpolation error in u was negligible. A comparison between using the as measured a- and c-tip dimensions (where a and c are the largest dimension in the sheet thickness and width directions, respectively) and those calculated from the elliptical linear regression shows no improvement in the correlation between the measured and calculated K values. The regression procedure changed the c-tip crack length on average 17 lm micron with a range of 112 to 235 lm where a negative change indicates that the measured crack is shorter than the crack length after linear regression. The average change in DK is p 0.0641 MPa m again signifying that the DK for the measured crack is less. The average difference between K’s calculated using as measured or linear regression data is 0.2%. A couple of sources of uncertainty in the striation measurement are possible but are more difficult to quantify. However, using digital images of the fracture surface from the SEM, an estimate has been made. The first possible source of uncertainty is not measuring from the same point between striations. To help eliminate this problem, several striations, usually greater than 10, were measured. Due to image resolution at high magnification and engineering judgment in selecting the measurement location, this type of measurement error is estimated to at 5 and 10 nm. Again, averaging over 10 striations results in an error of about 0.5–1 nm. The greater part of the striation measurement uncertainty is due to local variation in the crack growth process. Single striations were measured at several locations along the crack front near the free surface. A variation of 100–300 nm was quite common. Thus, local variations in DK were between 20% and 30% for a given c-tip crack length. As the crack length increased, the local DK variation increased as well which is evident in the increasing DK error with crack length in Fig. 12. With regard to material variability, ASTM E647 gives the following guidance. ‘‘. . . an overall measure of variability in da/dN versus DK is available from results of an inter-laboratory test program in which
14 laboratories participated. These data, obtained on highly homogeneous 10 Ni steel, showed the reproducibility in da/dN within a laboratory to average ±27% and range from ±13% to ±50%, depending on laboratory; the repeatability between laboratories was ±32%. Values cited are standard errors based on ±2 residual standard deviations about the mean response determined from regression analysis. . .Because a highly homogeneous material was employed in this program, the cited variabilities in da/dN are believed to have arisen primarily from random crack size measurement errors.” The extent of material variability is exacerbated at the a-crack tip where da/dN versus DK is obtained from tests conducted in the material long-transverse orientation, but crack growth is in the short-transverse direction. In this study, the largest errors between empirical and analytical K’s was indeed at the a-crack tip. The points of comparison for the empirical and analytical K’s are at discrete locations along the crack front. However, fatigue crack propagation is an inherently discontinuous phenomenon. The process is not smooth, but occurs in a stair-step fashion. This effect could be mitigated by developing some sort of K averaging methodology, but that is contrary to making point-wise comparisons of the empirical and analytical K’s. 5. Conclusion and recommendations Unsymmetric corner cracks at a hole subject to tension, bending, and bearing loading have been calculated and implemented in the USAF life prediction code AFGROW. These new K solutions were compared to experimental results in terms of fatigue life and flaw shape predictions. The maximum error in the former was 7% and 18% for the latter. Furthermore, empirically derived K’s were compared to the newly calculated K’s near the a- and ccrack tips. For this K comparison, the error was between 15% and 50% which is similar to what previous authors have found using this technique. Of the six different sources that could possibly affect the K comparison, local DK variation was the source of greatest error in the empirical vs. analytical results. The local DK variation is inherent in the fatigue process; thus, comparing analytical and empirical DK’s is not a recommended validation procedure. Although correlating fatigue life and flaw shape predictions are a more coarse validation technique, these two methods are preferred and recommended for K validation studies. References [1] Newman JC, Shivakumar KN. Stress concentrations for straight-shank and countersunk holes in plates. Journal of Applied Mechanics 1995;62:248–9. [2] Fawaz SA, Andersson Börje. Accurate stress intensity factor solutions for unsymmetric corner cracks at a hole. In: Proceedings of the fourth joint DoD/ FAA/NASA conference on aging aircraft, St. Louis, MO; 15–18 May 2000. [3] Fawaz SA, Andersson Börje, Newman JC Jr. Experimental verification of stress intensity factor solutions for corner cracks at a hole subject to general loading. In: Proceedings of the 22nd symposium of the international committee on aeronautical fatigue, Lucerne, CH, EMAS; 7–9 May 2003. [4] James LA, Anderson WE. A simple experimental procedure for stress intensity factor calibration. Engineering Fracture Mechanics 1969;1:565–8. [5] Raju IS, Newman JC Jr. Three-dimensional finite-element analysis of finitethickness fracture specimens. Technical Note NASA TN D 8414; May 1977. [6] Tada, Hiroshi, Paris Paul C, Irwin George R. The stress analysis of cracks handbook. 2nd ed. Paris productions incorporated and del research corporation, St. Louis; 1985. [7] Pelloux R, Warren A, O’Grady J. Fractographic analysis of nucleation and growth of fatigue cracks at rivet holes. In: Atluri SN, Sampath SG, Tong P, editors. Structural integrity of aging airplanes. Springer series in computational mechanics. Berlin: Springer-Verlag; 1991. [8] Fawaz SA, Schijve J, de Koning AU. Fatigue crack growth in riveted lap-splice joints. In: Proceedings of the 19th symposium of the international committee on aeronautical fatigue, 16-20 June 1997, Edinburgh, Scotland, UK: EMAS/ SoMat Systems International Ltd; 1997.
Contents lists available at ScienceDirect
International Journal of Fatigue journal homepage: www.elsevier.com/locate/ijfatigue
Validation of stress intensity factors of diametrically opposed corner cracks in a hole S.A. Fawaz *, D. Hill Department of Engineering Mechanics, Center for Aircraft Structural Life Extension (CAStLE), United States Air Force Academy, 2354 Fairchild Hall, Suite 6L-155, CO 80840, USA
a r t i c l e
i n f o
Article history: Received 21 September 2007 Received in revised form 27 February 2008 Accepted 8 March 2008 Available online 16 March 2008 Keywords: Stress intensity Striation spacing Validation Unsymmetric corner cracks
a b s t r a c t Population of the world’s largest database of stress intensity factor (K) solutions began in 2002 with the calculation of 5.6 million K solutions for diametrically opposed unsymmetric corner cracks at a straight shank hole in a finite width sheet subject to remote tension, remote bending, and bearing loading. Previous work to validate these K solutions was in the form of fatigue life predictions and crack shape development. The current work attempts to build on the previous validation efforts with the addition of comparing the calculated K solutions with K solutions obtained from carefully controlled laboratory experiments. The latter are obtained via fatigue striation measurements at high magnification, up to 40,000, using a scanning electron microscope and crack growth rate data, in terms of da/dN vs. DK at the same test condition. The results show the numerical K solutions are within 20% of the experimentally derived K’s at discrete locations along the crack front. The relatively large error is due to the discontinuous crack extension process of the crack front. Moreover, the entire crack front does not instantaneously extend uniformly in a self similar fashion. The crack extends stepwise over discrete portions of the crack front. Possibly averaging the striation spacing over a specified arc length of the crack front would ameliorate the discontinuous nature of crack propagation resulting in better correlation between the numerical and experimental results. As a result of the current work, we have shown time consuming striation spacing measurements at high magnification are not required to validate K solutions. The best method for such validation efforts is using the fatigue life, crack history, and crack shape which can be obtained at 1/ 10th the cost of obtaining striation spacing measurements. Published by Elsevier Ltd.
1. Introduction Fatigue-related problems have cost the military and commercial aircraft operators’ loss of aircraft, expensive repair actions, and an ever increasing inspection burden. The uncertainty associated with fleet management activities has many sources, but can be separated into two broad categories; uncertainty related to aircraft structural life prediction and uncertainty related to maintenance actions. This paper only addresses the former with specific focus to provide stress intensity factor, K, solutions with known and controlled accuracy. Although damage tolerance analysis, DTA, methods have been in places for decades, much room for improvement is still possible. For example, the case of two unsymmetric corner cracks at a fastener hole, see Fig. 1 being modeled as symmetric corner cracks. In this figure, rbypass = ro – rbrg where ro is the remote applied stress; rbrg is the bearing stress calculated by strength of materials methods (rbrg = P/Dt). Specifically, the rbypass is the membrane stress remaining in the plate after load transfer through a rivet or bolt. In addition, h is measured from the top of hole at the hole centerline. Newman
* Corresponding author. Tel.: +1 7193336266; fax: +1 7193332944. E-mail address: [email protected] (S.A. Fawaz). 0142-1123/$ - see front matter Published by Elsevier Ltd. doi:10.1016/j.ijfatigue.2008.03.015
and Shivakumar found using Pcos2 h instead of just P gave better correlation of measured and calculated stress concentration factors of a pin-loaded hole [1]. Neither in the laboratory nor in service do two corner cracks at a fastener hole propagate in a symmetric fashion. Calculation of these new K’s was previously published in [2] with limited validation in [3]. The validation effort is extended in this work and presented below. 2. Background The most common method for assessing the accuracy of K solutions is in terms of predicting the fatigue life of a laboratory test coupon. In a controlled experiment, the material and mechanical properties, loading magnitude and sequence, initial and final crack size are known. Calculating the number of load cycles to propagate the crack from the initial size to final size is rather straightforward. The results of this process can be seen in Fig. 2 where a comparison would be made between the final measured and predicted crack lengths. Further assessment of the accuracy of the K solution is made by comparing the crack history, crack length in time, of the measured and predicted results. A third method is available for assessing the accuracy of K solutions which is to directly compare an experimentally derived K with the calculated K.
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σ bypass
4
σ bending
Test 2, 7075-T651 w=49.27 mm, t =6.33 mm, r =3.25 m m c 1 =1.58 mm, c 2 =2.68 mm
Pcos2θ
2h c2
2r c1 2b
σ bending
Crack Length (mm)
3.5
Measured LeftCrack AFGROW Right Crack Measured RightCrack
3
2.5
2
σo a2
AFGROW Left Crack
a1
1.5 87.000
89.500
92.000 94.500 K cycles
Fig. 1. Diametrically opposed unsymmetric corner cracks subject to tension, bending, and bearing loading.
97.000
4
The details of the experimental program were previously reported by Fawaz, Andersson, and Newman and only briefly discussed here. The test program was designed to provide an array of crack data; specifically, crack shape (crack depth to crack length, a/c) and size (crack depth to sheet thickness, a/t) at various times during the fatigue life. The crack data would then be used to experimentally verify the newly calculated K’s. The specimens were quite simple, 50 mm wide, 200 mm high, 6.35 mm thick sheet with a 6.35 mm diameter centrally located straight shank hole. The specimen length and width were chosen to eliminate the finite height effect as reported by Raju and Newman [5] and Tada, Paris, and Irwin[6], respectively. All specimens were made from 7075T651 plate and tested using a blocked fatigue spectrum on a closed loop servo-hydraulic fatigue machine at 70 MPa and 10 Hz. Electro-discharge machined (EDM) notches were used to better control the starter notch geometry. The spectrum, shown in Fig. 4 was designed to create marker bands, groups of fatigue striations, on the fracture surface. One pass through the spectrum contains 8170 cycles. In situ crack growth measurement was not used as nearly the entire crack history was reconstructed post-test using a Nikon SMZ645 Stereoscope and/or Jeol JSM-6480LV scanning electron microscope to detect the marker bands. This technique was successfully demonstrated by Pelloux, Warren, and O’Grady [7] and in [2] and [8]. When the part through crack grew threw the specimen thickness, determined by visual inspection, the test was halted in order to reduce the chance the fracture surfaces would come into contact during cycling and damage the marker bands. 3.1. Fatigue Crack Growth Characterization of 7075-T651 All specimens used in this effort were machined from a single piece of 7075-T651 aluminum. A compact tension, C(T), specimen was used for fatigue crack growth characterization. The specimens were cyclically loaded in a servo-hydraulic test frame. A clip gauge was used to measure the crack opening displacement which could then be used with a compliance equation to determine crack length. All test procedures and data analysis followed ASTM standard E647. Three specimens were pre-cracked with constant
Crack Length (mm)
3. Experiments
3.5
AFGROW Left Crack Measured LeftCrack AFGROW RightCrack Measured Right Crack
3
2.5
2
1.5 62.000
64.500
67.000
69.500 K cycles
72.000
74.500
4 Tes t 4,
70 75-T651
3.8 w=25.37 mm, t =6.38 mm, r =1.61 m m c 1 =2.38 mm, c 2 =0 mm
3.6 Crack Length (mm)
The basic crack scenario considered for K validation is shown in Fig. 3. To calculate a K value from a fatigue test, the procedure developed by James and Anderson is used [4]. The test protocol and procedure for calculating empirical K’s are discussed in the following sections.
Tes t 3, 7075- T651 w=49. 17 mm, t =6.36 mm, r =3.23 m m c 1 =2.76 mm, c 2 =1.75 mm
3.4
AFGROW LeftCrack Measured Left Crack
3.2 3 2.8 2.6 2.4 2.2 2 47.000 49.500 52.000 54.500 57.000 59.500 62.000 64.500 67.000 K cycles
Fig. 2. Predicted and experimental crack history of diametrically opposed, unsymmetric corner cracks at a centrally located hole in a finite width plate subject to remote tension5.
t D W Fig. 3. Cracking scenarios investigated.
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Fig. 4. Diagram of marker-band spectrum.
tain the constant K gradient. The tests with R of 0.05 and 0.5 were tested in constant amplitude fatigue running at 10 Hz. The data from the three tests were then analyzed and plotted in terms of da/dN vs. DK, Fig. 5.
1.E-04 R = 0.05 R = 0.1 R = 0.5
1.E-05
3.2. K Validation specimens
da/dN (m/cycle)
1.E-06
1.E-07
1.E-08
1.E-09
1.E-10 1
10
100
Δ K (MPa√m) Fig. 5. Crack growth rate curves for 7075-T651 sheet used in the test program.
amplitude sinusoidal cycles. The load was determined so that when the pre-crack length was approximately 0.1 inch from the p starter notch a Kmax of 243 MPa mm would be felt at the crack tip. The specimens were pre-cracked with an R (minimum stress/ maximum stress) of 0.1 and frequency of 10 Hz. After the specimens were pre-cracked, each specimen was tested at a different R, 0.05, 0.1, and 0.5. For the test with R = 0.1, a constant K gradient (10 mm1) test was performed. A clip gage was used to measure crack length with the MTS software controlling the load to main-
The dimensions of the specimens investigated here are given in Table 1. Since all specimens were cut from the same sheet of material, the nominal thickness is approximately the same for all specimens at 6.35 mm. All specimens had smooth machined sides and were machined to a tolerance of ±0.0762 mm. The hole was drilled in one pass using a standard drill bit. After the specimens were machined, diametrically opposed and perpendicular to the applied load elliptical electric discharge machine (EDM) notches were introduced on the edge of the hole to facilitate crack nucleation and asymmetric crack growth. Again a closed loop servo-hydraulic test frame was used to test the specimens. The frame was controlled by a TestStar IImTM controller with the TestStarTM software package. The specimens were gripped using MTS hydraulic grips and the test protocol is given below. Each specimen was tested using the marker-band spectrum, Fig. 4, with a maximum spectrum stress of 69 MPa. The spectrum begins with a constant amplitude block at 100% of the full load for 2000 cycles. After the 2000 cycle block the load drops to 80% of the full load for 100 cycles and then completes a quick 10 cycles back at 100% of the full load. The 100 cycles at 80% leave a dark distinguishable band on the fracture surface that can easily be seen under a microscope. This 100 cycle 10 cycles process is repeated 10 times therefore making the first marker-band set with 10 dark marker-bands. After a set of 10 marker-bands is created, the spectrum runs for another 2000 cycles at 100% load. When this block is complete a set of 4 marker cycles are applied just as before. Then 2000 more cycles are ran and another set of 6 marker cycles are made. Fig. 6 shows an image taken with (a) an optical microscope and (b) a scanning electron microscope (SEM). The striation spacing can be seen leading up to the dark marker-bands with 10 striations in between each band. When both cracks transition from corner cracks to through cracks the tests were stopped after the next full pass and before the set of ten marker-bands started. At this point the specimens were statically over-loaded until final failure occurred. By following
Table 1 K validation specimen dimensions Specimen
Width (mm)
Thickness (mm)
Hole diameter (mm)
Initial crack dimensions Left c-tip (mm)
Left a-tip (mm)
Right c-tip (mm)
Right a-tip (mm)
Test Test Test Test
49.250 49.276 49.225 25.374
6.325 6.35 6.35 6.35
6.35 6.35 6.35 3.175
1.862 1.214 1.029 1.225
3.102 1.277 3.801 1.282
1.274 .955 1.343 None
1.241 3.103 1.230 None
1 2 3 4
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Fig. 6. Micrograph of marker bands on specimen test 3 at 5500.
Table 2 Stress levels, life results, and number of cracks formed from the K validation specimens Specimen
Life (Passes)
Total cycles
Cracks
Valid crack configurations between (Pass/Markerband)
Test Test Test Test
10 15+ 12 11+
81,700 122,850 98,040 90,716
2 2 2 1
No valid combinations (12/4) (9/6) (8/6)
1 2 3 4
this procedure, the integrity of the markers bands was preserved. This also made it easier to find the last marker-band which would be a set of 6. The two halves were then placed in a desiccator awaiting fractographic analysis. Table 2 shows the relevant test results. In this table, columns two and three are the number of passes through the marker spectrum (8170 cycles per pass), and corresponding total number of fatigue cycles, respectively. The specimens either one crack on each side of the hole or just one crack on one side of the hole as indicated in column four. Columns five and six gives the range over which crack growth predictions could be made. For example in test 2, the prediction started at the crack size of a group of 6 marker bands in pass 11 and finished at a group of 4 marker bands in pass 12. The crack configuration is valid if the a1/c1, a2/ c2, a1/t, and a2/t ratio is within the analysis parameter space of the calculated K solutions. The results of test 1 did not contain any valid combination even though 30 marker cycles were applied (10 passes of the spectrum and 3 marker cycles per pass) thus indicating the challenge in collecting experimental data for validation efforts of this type. 3.3. Fractographic analysis Fractographic analysis was conducted to determine the final crack length, flaw shape development, and local crack growth rates. A Nikon Stereo Microscope equipped with Quadracheck and a motorized stage was used for all measurements. Magnification up to 1000 was used. In addition, due to the submicron resolution of the stage, accurate measurements outside the field of view of the lens could be made. Since each side of the hole had a unique crack, each crack was analyzed and investigated separately. The origin was defined as the intersection of the hole bore and sheet surface were the notch had been machined originally. The mapping of the marker-bands began at the static to fatigue fracture transition, Fig. 7. Following the procedure described above, a set of six marker-bands should be found at the edge of this transition point. By starting at this set of marker-bands and working towards the origin, each set of six, four, or ten marker-bands could be found in order and traced back to the nucleation site. As each set of marker-bands were found they were mapped out by finding 8–10
Points No valid combinations (11/6) (8/4) (4/10)
0 3 5 12
Fig. 7. Right side fracture surface of test 3.
points along the arc of the marker-band which is a record of the crack front at a given cycle count. These points could then be plotted as shown in Fig. 8. The first number of each dataset name indicates the number of passes through the marker spectrum and the second number indicates what type of marker band (10, 4, or 6) was measured. The ‘‘Points” in Fig. 8 are the a- and c-crack tips used for crack growth predictions. The crack on the opposite side of the hole was analyzed in the same manner. In the open literature, there is not a K solution for a part through crack on one side of a hole and a through crack on the opposing side; therefore, when this scenario occurred, which was quite frequently, this crack data could not be used for K validation. To remove the boundary layer effect, non-elliptical crack shape where the crack front intersects a free surface, a computer program was written in C++ to read from a text file the eight to ten points from the marker-band mapping and calculate a best fit ellipse using linear regression. From this new best fit ellipse, the maximum crack depth and crack length were used as the a- and cdimensions for the corner crack modeled in AFGROW. The crack
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Notch
6
12-6 12-4 12-10
5
11-6
Crack Depth (mm)
11-4 11-10
4
10-6 10-4 10-10
3
9-6 9-4 9-10 2
8-6 8-4 Points
1
0 0
1
2
3
4
5
6
Crack Length (mm) Fig. 8. Reconstructed marker-bands on the right side of test 3.
45
James Anderson
40 35
% Error
30 25 20 15 10 5 0 0
5
10
15
20
25
30
Crack length (mm) Fig. 10. Crack length vs. % error of James–Anderson data.
Fig. 9. Measuring of striation spacing on test 3.
histories are shown in Fig. 2 with the prediction and measured values starting/stopping at the particular pass and marker type given in Table 2. Similarly, the measured and predicted flaw shapes are given in Fig. 11. The fit a and c values from the linear regression program were then entered into AFGROW as a right and left corner crack on a specimen with the same width, thickness and hole diameter of the original test coupon. The crack growth rates determined during the material characterization part of this effort were used via the AFGROW material property table lookup. Any batch, lot, or manufacturer material variability is removed by this approach. The test spectrum, Fig. 4, was also used for the predictions; thus, even though the load sequence effects were shown to be negligible in reference [7] if any effects did exist, they would be captured in the prediction. Using AFGROW’s multi-point advanced model for this cracking scenario yields DK values at 11 points along the crack front
for a given cycle count. Thus, a point to point comparison of analytical and experimental DK values is possible. The empirical K’s are calculated from a striation spacing measurements, see Fig. 9 and the crack growth rate curve. The striation measurements were made at the end of the block of 2000 cycles just before the marker bands. This location was chosen because it was found that the striation spacing in between the marker bands experienced small load sequence effects. After 2000 cycles, the crack growth rate would not be affected by any prior load history, for example the 80% cycles. Numerous, 10–20 in most cases, striations were measured and then that distance was divided by the exact number of striations to give an accurate local da/dN. With an accurate da/dN calculation and the da/dN vs. DK curve that was created for this specific lot of material, an empirical DK could then be determined. 4. Discussion The discussion below is separated by the three validation efforts undertaken in this effort; namely, validation in terms of fatigue life predictions, flaw shape predictions, and empirical versus analytical stress intensity factor solutions.
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4.1. Fatigue life predictions The fatigue life predictions using the new K solutions, shown in Fig. 2, have a maximum error of 7% over the rather short interval of crack growth. Unfortunately the cracks grow rather quickly even at the low stress level of 69 MPa and thus the data collected is not extensive. The new K solutions over-estimate K which results in a shorter fatigue life than measured. 4.2. Flaw shape predictions The flaw shape predictions shown in Fig. 11 have a maximum error of 18% which occurs at the very last measurement of the sin-
gle crack case, test 4. Except for this one dataset, the correlation between measured and predicted (using the new K solutions) flaw shapes is acceptable. The new K solutions over-estimate K which results in a larger crack size all along the crack front than what was measured. 4.3. Comparison of empirical and analytical stress intensity factor solutions James and Anderson, in the late 1960’s, found empirical K solutions for crack geometries for which no analytical solution existed.6 Their work consisted of testing three thick walled pressure vessels. They then compared the results by plotting crack length vs. K. The
Fig. 11. Flaw shape development.
20%
10%
Test 2 Test 3 Test 4
10%
0% -10%
Error
0%
Error
Test 2 Test 3 Test 4
-10%
-20% -30%
-20%
-40%
-30% -40%
-50% 0
0.5
1
1.5
2
2.5
3
3.5
4
-60%
0
1
c-tip Crack Length (mm)
2
3
4
c-tip Crack Length (mm) Fig. 12. Crack length vs. % error comparisons.
5
6
7
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correlation of their results shows an error or variation of results between 11% and 39%. By plotting this error with respect to crack length, Fig. 10, the trend shows that at short and long cracks the error is the largest with this method and less variation around crack lengths of 12 mm. Since the crack lengths of interest in this study were below 5 mm in length, the current results are consistent in terms of magnitude of error as in [5]. In Fig. 12, a correlation between crack depth (a-tip) and length (c-tip) and error is not apparent. For all specimens tested, the error between the empirical and numerical DK’s ranged from 50% to 15%. Several sources of error are possible. With regard to testing procedures, specifically the uncertainty in the applied loading; the load command and response signal was recorded, and the maximum difference throughout each test was less than ±45 N resulting in a 0.2% error in DK at the peak stress of 69 MPa. As evident from the fractographic analysis, the EDM notch is not part-elliptical and this would contribute to the error in the predictions if the analysis used initial flaw dimensions of the EDM notch. However, the specimen was cycled for several passes through the spectrum to fully develop the part-elliptical crack front with the shape being verified by post-test fractography. The initial flaw dimensions were taken from the first fully developed part-elliptical crack front. A conclusion from reference [3] was that a part-elliptical EDM notch does not always result in a part-elliptical crack shape. AFGROW uses the parametric angle of an ellipse, u, to calculate the location of the point on the crack front for multi point analysis. The experimental measurements are also recorded in terms of u and the da/dN (and thus K) for nearly the same u is compared. Interpolation error in u was negligible. A comparison between using the as measured a- and c-tip dimensions (where a and c are the largest dimension in the sheet thickness and width directions, respectively) and those calculated from the elliptical linear regression shows no improvement in the correlation between the measured and calculated K values. The regression procedure changed the c-tip crack length on average 17 lm micron with a range of 112 to 235 lm where a negative change indicates that the measured crack is shorter than the crack length after linear regression. The average change in DK is p 0.0641 MPa m again signifying that the DK for the measured crack is less. The average difference between K’s calculated using as measured or linear regression data is 0.2%. A couple of sources of uncertainty in the striation measurement are possible but are more difficult to quantify. However, using digital images of the fracture surface from the SEM, an estimate has been made. The first possible source of uncertainty is not measuring from the same point between striations. To help eliminate this problem, several striations, usually greater than 10, were measured. Due to image resolution at high magnification and engineering judgment in selecting the measurement location, this type of measurement error is estimated to at 5 and 10 nm. Again, averaging over 10 striations results in an error of about 0.5–1 nm. The greater part of the striation measurement uncertainty is due to local variation in the crack growth process. Single striations were measured at several locations along the crack front near the free surface. A variation of 100–300 nm was quite common. Thus, local variations in DK were between 20% and 30% for a given c-tip crack length. As the crack length increased, the local DK variation increased as well which is evident in the increasing DK error with crack length in Fig. 12. With regard to material variability, ASTM E647 gives the following guidance. ‘‘. . . an overall measure of variability in da/dN versus DK is available from results of an inter-laboratory test program in which
14 laboratories participated. These data, obtained on highly homogeneous 10 Ni steel, showed the reproducibility in da/dN within a laboratory to average ±27% and range from ±13% to ±50%, depending on laboratory; the repeatability between laboratories was ±32%. Values cited are standard errors based on ±2 residual standard deviations about the mean response determined from regression analysis. . .Because a highly homogeneous material was employed in this program, the cited variabilities in da/dN are believed to have arisen primarily from random crack size measurement errors.” The extent of material variability is exacerbated at the a-crack tip where da/dN versus DK is obtained from tests conducted in the material long-transverse orientation, but crack growth is in the short-transverse direction. In this study, the largest errors between empirical and analytical K’s was indeed at the a-crack tip. The points of comparison for the empirical and analytical K’s are at discrete locations along the crack front. However, fatigue crack propagation is an inherently discontinuous phenomenon. The process is not smooth, but occurs in a stair-step fashion. This effect could be mitigated by developing some sort of K averaging methodology, but that is contrary to making point-wise comparisons of the empirical and analytical K’s. 5. Conclusion and recommendations Unsymmetric corner cracks at a hole subject to tension, bending, and bearing loading have been calculated and implemented in the USAF life prediction code AFGROW. These new K solutions were compared to experimental results in terms of fatigue life and flaw shape predictions. The maximum error in the former was 7% and 18% for the latter. Furthermore, empirically derived K’s were compared to the newly calculated K’s near the a- and ccrack tips. For this K comparison, the error was between 15% and 50% which is similar to what previous authors have found using this technique. Of the six different sources that could possibly affect the K comparison, local DK variation was the source of greatest error in the empirical vs. analytical results. The local DK variation is inherent in the fatigue process; thus, comparing analytical and empirical DK’s is not a recommended validation procedure. Although correlating fatigue life and flaw shape predictions are a more coarse validation technique, these two methods are preferred and recommended for K validation studies. References [1] Newman JC, Shivakumar KN. Stress concentrations for straight-shank and countersunk holes in plates. Journal of Applied Mechanics 1995;62:248–9. [2] Fawaz SA, Andersson Börje. Accurate stress intensity factor solutions for unsymmetric corner cracks at a hole. In: Proceedings of the fourth joint DoD/ FAA/NASA conference on aging aircraft, St. Louis, MO; 15–18 May 2000. [3] Fawaz SA, Andersson Börje, Newman JC Jr. Experimental verification of stress intensity factor solutions for corner cracks at a hole subject to general loading. In: Proceedings of the 22nd symposium of the international committee on aeronautical fatigue, Lucerne, CH, EMAS; 7–9 May 2003. [4] James LA, Anderson WE. A simple experimental procedure for stress intensity factor calibration. Engineering Fracture Mechanics 1969;1:565–8. [5] Raju IS, Newman JC Jr. Three-dimensional finite-element analysis of finitethickness fracture specimens. Technical Note NASA TN D 8414; May 1977. [6] Tada, Hiroshi, Paris Paul C, Irwin George R. The stress analysis of cracks handbook. 2nd ed. Paris productions incorporated and del research corporation, St. Louis; 1985. [7] Pelloux R, Warren A, O’Grady J. Fractographic analysis of nucleation and growth of fatigue cracks at rivet holes. In: Atluri SN, Sampath SG, Tong P, editors. Structural integrity of aging airplanes. Springer series in computational mechanics. Berlin: Springer-Verlag; 1991. [8] Fawaz SA, Schijve J, de Koning AU. Fatigue crack growth in riveted lap-splice joints. In: Proceedings of the 19th symposium of the international committee on aeronautical fatigue, 16-20 June 1997, Edinburgh, Scotland, UK: EMAS/ SoMat Systems International Ltd; 1997.