AN EVALUATION OF SEVERAL RETARDATION MODELS FOR CRACK GROWTH PREDICTION UNDER SPECTRUM LOADING

Transcription

1 AN EVALUATION OF SEVERAL RETARDATION MODELS FOR CRACK GROWTH PREDICTION UNDER SPECTRUM LOADING A. Brot and C. Matias Engineering Division Israel Aircraft Industries Ben-Gurion Airport, Israel INTRODUCTION It is well established that, under spectrum loading, load-interactions occur which generally retard the rate of crack growth. This effect can be very significant for certain types of loading spectra, the life increase due to retardation may be a factor ranging from 2 to 5. Unfortunately, it is very difficult to predict a priori the extent of retardation that can be expected for a specific combination of alloy, loading spectrum, stress level and crack configuration. Several retardation models have been proposed in the past 30 years, including the Wheeler, Willenborg, Generalized Willenborg, Closure and GRF Models. It has been found that these semi-empirical models have limited value in predicting crack growth behavior, since their dominating parameters must be calibrated for the specific alloy, loading spectrum, stress level and crack configuration. Since this calibration process can be performed only after crack growth testing, these methods are not very useful in predicting, during the design process, the expected crack growth life. The present study includes four widely used aluminum alloys, which have been tested under five distinct spectrum types. The test coupons include two crack configurations: CCT and open hole. The test results are evaluated using several state-of-the-art load-interaction models including the Strip-Yield Model (2 versions), the Generalized Willenborg Model and the Modified Generalized Willenborg Model. In all cases, the emphasis is on using the above retardation models with a minimal need for calibration. THE TEST PROGRAM A coupon test program was performed to establish retardation properties of four widely used aluminum alloys: 7075-T7351, 7050-T7451, 7475-T7351 and 2024-T351 under five distinct spectrum types. Two crack configurations were tested: (a) Center cracked tension (CCT) coupon, with a width of 3.15 inches and a thickness of 0.25 inch. The coupon was manufactured from a plate with the grain direction running along the loading axis. A 79 inch, through-the-thickness, EDM produced flaw was precracked, under constant-amplitude loading, to a total crack length (2a) of inches. Presented to the 2002 USAF Aircraft Structural Integrity Program Conference, Savannah, Georgia, 10 December

2 (b) Open hole coupon, with identical dimensions as the CCT coupon, having a centrally located, inch diameter open hole. A 39 inch, through-the-thickness, EDM produced flaw was placed at one side of the hole, and it was precracked to a crack length of 51 inches. For each alloy, all the coupons were manufactured from the same material batches. Straingages were bonded to all the coupons in order to insure correct alignment in the testing machine. Several coupons, from each alloy, were tested under constant amplitude loading in order to verify that they conform to published data. The coupons were tested under five distinct spectrum types, as is shown in Figure 1: (a) Lateral gust loading spectrum: This spectrum simulates lateral gust loading which affects the vertical tail of an aircraft. It is composed of seven levels, all in totally reversed loading (R = -1). The cycles are ordered randomly having approximately 13 cycles per five flights. (b) Ground loading spectrum: This spectrum simulates landing impact and taxi loads. The landing impact portion is composed of three levels, all with R = 0. One landing impact per flight is selected randomly. The taxi portion is composed of 21 levels having various values of R. The taxi cycles are ordered randomly with approximately 17 cycles per flight. (It should be noted that the landing impact cycle dominates the spectrum, relative to the taxi cycles, as can be seen from Figure 1.) (c) Fighter aircraft maneuver spectrum: This spectrum simulates the wing-root loading of a fighter aircraft performing maneuvers. The spectrum is composed of seven levels of loading, with a mean value of R = 6. The cycles are ordered randomly having approximately 28 cycles per flight. (d) Wing gust and maneuver spectrum: This spectrum simulates the gust and maneuver loading of the wing of a transport aircraft. The wing is loaded from zero load to a loadfactor of, once per flight, and on this is superimposed fifteen levels of gust and maneuver loading, having a mean R ratio of 0.6. The gust and maneuver loads are ordered randomly with approximately 16.5 cycles per flight. (e) Pressurized fuselage gust and maneuver spectrum: This spectrum simulates the gust and maneuver loading of a pressurized fuselage of a transport aircraft. The fuselage is pressurized once per flight, and on this is superimposed fifteen levels of gust and maneuver loading, having a mean stress-ratio of R = The gust and maneuver loads are ordered randomly with approximately 16.5 cycles per flight. These spectra were selected since they are representative of actual aircraft spectra and because they cover the stress-ratio range from R = -1 to R = Testing was performed in a flight-by-flight manner, with the cycles randomized within each flight. A repeating block of 2000 randomized flights was utilized for all the spectra except for the fighter aircraft maneuver spectrum which used a repeating block of 500 randomized flights. 2

3 stress, ksi Lateral gust loading spectrum (5 flights) cycles stress, ksi Ground spectrum (typical flight) cycles stress, ksi Fighter Aircraft maneuver spectrum (typical flight) cycles stress, ksi Wing gust and maneuver spectrum (typical flight) cycles stress, ksi Pressurized fuselage gust and maneuver spectrum (typical flight) cycles Figure 1 Spectrum Types Used for Testing and Analysis 3

4 Crack growth was monitored, during each test, by means of crack-propagation gages. Typical results are shown in Figures 2 6. Table 1 summarizes the scope of the test program from the standpoint of alloys, spectrum type and crack configuration. As is seen from Table 1, not every combination of alloy, spectrum and crack configuration was tested, but a total of 43 coupons were tested. In most cases, two coupons were tested for each combination. The correlation between the two test results was reasonably close, in most cases, as is shown in Figures 2 6. Table 1 Scope of the Test Program (Number of coupons tested by spectrum type, alloy and coupon configuration) Material Alloy Coupon Type Wing Gust and Maneuver Pressurized Fuselage Gust and Maneuver Lateral Gust Ground Fighter Aircraft Maneuver AL7075 T7351 CCT Open Hole AL7475 T7351 CCT Open Hole 2 AL7050 T7451 CCT Open Hole AL2024 T351 CCT Open Hole 2 1 TOTAL

5 RETARDATION MODELS It has been observed that crack growth under spectrum loading is characterized by a nonlinear load-interaction phenomenon. This means that the crack growth rate is usually considerably slower than predicted by da/dn vs. K data for the individual loads. This retardation effect is explained by the compressive residual stresses resulting from the plastic zones introduced at the crack tip when a load peak is encountered. This retardation phenomenon can be seen in Figures 2 6 by observing the great variation between the test data (shown as points) and the unretarded analytical solution (dashed lines). Several semi-empirical load-interaction (retardation) models have been proposed in the past 30 years, including the Wheeler [1], Willenborg [2], Generalized Willenborg [3], Closure [4] and GRF [5] Models. It has been found that these models have limited value in predicting crack growth behavior, since their dominating parameters must be calibrated for the specific alloy, loading spectrum, stress level and crack configuration. Since this calibration process can be performed only after crack growth testing, these methods are not very useful in predicting, during the design process, the expected crack growth life. These models generally assume that a tensile overload produces a plastic zone at the crack tip. When the overload is removed, compressive residual stresses develop around the crack tip. This compressive stress field affects the rate of crack growth, until the crack has left the residual stress field. The Generalized Willenborg (GW) Model [3] defines an effective stress-ratio, R eff, which is a function of the actual stress-ratio and the maximum stress-intensity for the overload cycle. A single parameter, R so, defined as the overload shut-off value, is used to calibrate the model to spectrum test results. This model does not account for additional retardation introduced by multiple overloads, nor reduced retardation resulting from underloads. The Modified Generalized Willenborg (MGW) Model [6] takes into account the reduction of retardation effects due to underloads. In this model, similar to the GW Model, a single parameter Φ o is used to calibrate the model to spectrum test results. The Strip-Yield Model is a mechanical model based on the assumption that a growing fatigue crack will grow through the residual strength field. The plastic deformation left in the wake of the crack will contribute to the interaction effect, and will explain such phenomena as stresslevel dependence, retardation and acceleration. It is strongly based on crack closure concepts first introduced by Elber [4] and the Dugdale crack-opening model [7]. The Strip-Yield Model uses a constraint factor, α, to account for plane-stress or plane-strain behavior. There exist two variations of the Strip-Yield Model: The constant constraint-loss option (SY-N), developed by NASA [8, 9], assumes that α is constant along the plastic zone but its value depends on the state-of-stress that changes from plane-strain to plane-stress as the crack grows. 5

6 The second variation of the Strip-Yield Model is called the variable constraint-loss option (SY-E), which was developed by ESA and NLR [10]. In this model, α varies along the plastic zone according to a parabolic expression. In addition, the state-of-stress is calculated by a different expression than that used by the SY-N Model. NASGRO (version 4) includes (among others) the Generalized Willenborg (GW) Model, the Modified Generalized Willenborg (MGW) Model, the Constant Constraint-Loss Strip-Yield (SY-N) Model and the Variable Constraint-Loss Strip-Yield (SY-E) Model. These four models were used to attempt to predict the crack growth behavior of the coupon tests under the five spectrum types. The following ground-rules were used to perform the predictions: (a) The material databases built into NASGRO were used with no attempt to tweak the parameters in order to improve the results. (b) The input parameters needed for the GW and MGW models were determined by calibrating the CCT coupon results (for each material alloy) under the lateral gust loading spectrum. In this way, it was felt, that the predictions would simulate the behavior used by a typical NASGRO user who has none, or a limited amount of test data. The analysis was performed in a flight-by-flight manner, with the cycles randomized within each flight. A repeating block of 2000 randomized flights was utilized for all the spectra except for the fighter aircraft maneuver spectrum which used a repeating block of 500 randomized flights. Another paper comparing the MGW and SY-N models to spectrum test data, for three types of spectra, was recently published [11]. However, the study included tweaking the models to optimize results as well as modifying certain features of the Strip-Yield Model. As a result, the results of the two studies are not directly comparable. COMPARISON OF PREDICTED LIVES vs. EXPERIMENTAL RESULTS Representative Results: A great variety of results were found in this study. Figures 2 6 illustrate some typical results. Figure 2 shows the crack growth of 7050-T7451 open hole coupons under the lateral gust loading spectrum. The test results were compared to an unretarded analysis and analyses using the GW, MGW and SY-N retardation models. (Material data was not available for this alloy to run the SY-E model.) All three models gave reasonable, but slightly unconservative, results. Figure 3 describes the crack growth of 2024-T351 CCT coupons under the fighter aircraft maneuver spectrum. The test results were compared to all four models. The results show good correlation with both strip-yield models and the MGW model, and poorer, but conservative correlation to the GW model. 6

7 Crack length, inches Test results- specimen 1 Test results- specimen 2 no retardation SY-N GW (Rso=3.6) MGW (Phi0=0.61) Flights Figure 2 - Crack growth of 7050-T7451 Open Hole Coupons Under the Lateral Gust Loading Spectrum Test results - specimen 1 Test results - specimen 2 no retardation SY-N SY-E GW (Rso=4.96) MGW (Phi0=0.391) Crack length, inches Flights Figure 3 - Crack Growth of 2024-T351 CCT Coupons Under the Fighter Aircraft Maneuver Spectrum 7

8 In Figure 4, the crack growth of 7075-T7351 CCT coupons under the wing gust and maneuver spectrum is shown. Here, the results are very variable, with the SY-E model showing the best correlation and the SY-N showing the worst correlation. The GW and MGW models were intermediate in correlation. Figure 5 illustrates the crack growth of 7075-T7351 open hole coupons under the lateral gust loading spectrum. In this case, all four retardation models greatly overestimated the degree of retardation, with the SY-E model having the poorest correlation to the test data. 1.3 Crack length, inches Test results - specimen 1 Test results - specimen 2 no retardation SY-N SY-E GW (Rso=4.28) MGW (Phi0=0.488) Flights Figure 4 - Crack Growth of 7075-T7351 CCT Coupons Under the Wing Gust and Maneuver Spectrum Crack length, inches Test results - specimen 1 Test results - specimen 2 no retardation SY-N SY-E GW (Rso=4.28) MGW (Phi0=0.488) Flights Figure 5 - Crack Growth of 7075-T7351 Open Hole Coupons Under the Lateral Gust Loading Spectrum 8

9 Crack length, inches Test results - soecimen 1 Test results - specimen 2 no retardation SY-N SY-E GW (Rso=4.96) MGW (Phi0=0.391) Flights Figure 6 - Crack Growth of 2024-T351 Open Hole Coupons Under the Pressurized Fuselage, Gust and Maneuver Spectrum Figure 6 describes the crack growth of 2024-T351 open hole coupons under the pressurized fuselage, gust and maneuver spectrum. This time, all four models greatly underestimated the degree of retardation. Although these are only five selected results of the study, they are representative of the great variations that were encountered. The challenge remains to examine all the results and to attempt to find the patterns of the behavior. Global Retardation: The concept of global retardation [5] allows us to examine the degree of retardation that is present in a system composed of a structural configuration, material alloy and loading spectrum. For analysis, the global retardation factor (GRF) is defined as the ratio of the calculated crack growth life (with retardation effects included) to the calculated unretarded crack growth life. For test results, the global retardation factor is defined as the ratio of the measured crack growth life to the calculated unretarded crack growth life. Figure 7 presents the mean global retardation factor as a function of spectrum type, for both analysis and testing. The spectra are arranged in Figure 7 by increasing level of mean stress-ratio, R. Figure 7 indicates that the degree of retardation generally decreases with increasing stressratio. For example, test results under the lateral spectrum (R = -1) had a mean GRF of 3.00 while test results under the pressurized fuselage spectrum (R = 0.83) had a mean GRF of only (An exception to this rule is the fighter aircraft spectrum (R = 6) which had an unusually high GRF of 3.27, due to the aggressive nature of this spectrum.) 9

10 By comparing the GRF for the calculated and test results, it becomes clear that the SY-N model can reasonably predict crack growth for spectra having stress-ratios in the range of -1 to 0, but is unable to accurately predict the degree of retardation for spectra having positive stressratios. This is especially evident for the wing gust spectrum (R = 0.60) tests, which resulted in a mean GRF of 1.84 while the SY-N model predicted a GRF of only GRF Crack GRF Growth Life / Calculated Unretarded Life Average Calculated Retarded Life / Calculated Unretarded Life (SY-N model) Average Test Growth Life / Calculated Unretarded Life Figure 7 Global Retardation Factor (GRF) for the Various Spectrum Types Lateral gust loading Ground loading Fighter aircraft maneuver Type of Spectrum Wing gust and maneuver loading Pressurized fuselage gust and maneuver loading Retardation Model Performance: All the predicted crack growth lives were compared to the measured results and the resulting ratios of lives were compiled for each of the four retardation models that were studied. Statistical analyses of the results were performed, under the assumption that the ratio of calculated life to measured life will fit a log-normal distribution. Under this assumption, mean values and standard deviations were calculated for each retardation model. These results are summarized in Figure 8. The results of Figure 8 show all the retardation models, on the average, correlated reasonably well with the range of test data. However, it is clear that the SY-N model gave the least variation. If we consider the log-mean value ± two standard deviations as a measure of the variation, the SY-N model gave predictions that ranged from 0.43 to 1.75 times the measured crack growth lives. The other three retardation models had much greater variations, as is shown in Figure 8. (It should be noted that, due to a lack of SY-E material properties for two of the alloys, the sample size for the SY-E evaluation was considerably smaller than those of the other retardation models.) 10

11 Calculated Life Life / Test Growth Life Mean + 2 SD Mean + SD Mean Mean - SD Mean - 2 SD SY-N SY-E GW MGW Retardation Model Figure 8 Retardation Model Performance Effect of Spectrum Type: As was previously stated, and as was shown in Figure 7, the five different spectrum types that were tested had various degrees of retardation and the retardation models had different degrees of success in their predictions, for the various spectrum types. Since the SY-N model gave the best performance, the effect of spectrum type was studied statistically using the results of the SY-N predictions. Again it was assumed that the ratio of calculated life to measured life fits a log-normal distribution. Under this assumption, mean values and standard deviations were calculated for the data corresponding to each spectrum type. These results are summarized in Figure 9. Figure 9 reveals several interesting results. The lateral gust spectrum, while having a mean calculated result of 9 times the mean test result, displays an extraordinary degree of variation under the log-mean value ± two standard deviations criterion. The reasons for this large variation are not evident. Figure 9 also indicates that, under the wing gust spectrum (R = 0.6), little variation was noted, but the mean calculated result was only 0.63 times the mean test result. The reason that the SY-N retardation model consistently underestimated the degree of retardation for this spectrum type is not evident. Effect of Material Alloy: This time, the data was statistically analyzed according to the alloy, as is shown in Figure 10. The log-normal distribution was again used to compare the ratio of calculated lives to test lives. Again, an interesting, but unexplainable result was noted. The log-mean results for the 7075-T7351 and 2024-T351 alloys were found to be very close to, but the variation was quite large. For the 7475-T7351 and 7050-T7451 alloys, the mean calculated life was found to 11

12 be less than 0.8 times the mean measured life, but the variation was much smaller. These differences were not consistent with constant amplitude test results that were performed on representative coupons of the various alloys. 3.0 Mean + 2 SD Calculated Calculated Life / Life Test Growth Life Mean + SD Mean Mean - SD Mean - 2 SD Lateral gust loading Ground loading Fighter aircraft maneuver Wing gust and maneuver loading Pressurized fuselage gust and maneuver loading Spectrum Type Figure 9 Statistical Variation Per Spectrum Type (SY-N Model) Calculated Calculated Life Life / Test Growth Mean + 2 SD Mean + SD Mean Mean - SD Mean - 2 SD AL7075 T7351 AL7475 T7351 AL7050 T7451 AL2024 T351 Alloy Figure 10 Statistical Variation Per Material Alloy (SY-N Model) 12

13 Calculated Calculated Life / Life Test Growth Life Mean + 2 SD Mean + SD Mean Mean - SD Mean - 2 SD CCT Open Hole Coupon Type Figure 11 - Statistical Variation Per Coupon Type (SY-N Model) Effect of Coupon Type: As was stated earlier, two types of coupons were tested: center-cracked tension (CCT) and open hole coupons. Table 1 summarizes which tests were performed with each coupon configuration. The data was separated according to configuration and was statistically analyzed, as is shown in Figure 11. The log-normal distribution was again used to compare the ratio of calculated lives to test lives. The results show that the open hole coupon type had a mean calculated result of 0.94 times the mean test result, but the variations were large. Under the logmean value ± two standard deviations criterion, the calculated results could vary between 0.38 and 2.34 times the test results. On the other hand, the CCT coupon type had a mean calculated result of only 0.82 times the mean test result, but the variations were much smaller. Under the log-mean value ± two standard deviations criterion, the calculated results would vary between 0.50 and 1.36 times the test results. It is theorized, but not proven, that the reason for the large variation of open hole coupon results is due to variations in the initial crack size, which nominally was 51 inch. The CCT coupon, starting with a much larger initial crack, (2a = inch) would be much less sensitive to variations in initial crack length. As a result of this statistical analysis, it was concluded that the results of the CCT type coupon were more reliable. 13

14 SUMMARY AND CONCLUSIONS 1. Predictions of the four retardation models gave large variations compared to crack growth measurements, over the entire range of testing. 2. The NASA Strip-Yield (SY-N) Model correlated reasonably well with the test data and had the least variation, over the range of testing, compared to the other models. 3. All the retardation models correlated reasonably well with test data under spectrum loading having mean stress-ratios ranging from -1 to 0, but performed much poorer under spectra having a positive stress-ratio (R > 0). 4. Significant differences were found in predictions for CCT coupons compared to predictions for open hole coupons. The CCT coupons seem to give more reliable results. REFERENCES 1. Wheeler, O. E., Spectrum Loading and Crack Growth, J. Basic Eng., Trans. ASME, Vol. D94, No. 1, Willenborg, J., R. M. Engle, and H. A. Wood, A Crack Growth Retardation Model Using an Effective Stress Concept, AFFDL TM-71-1-FBR, Wright Patterson Air Force Laboratory, Gallagher, J. P., A Generalized Development of Yield Zone Models, AFFDL-TM FBR, Wright Patterson Air Force Laboratory, Elber, W., The Significance of Fatigue Crack Closure, Damage Tolerance of Aircraft Structures, ASTM STP 486, Brot, A., GRF A Simple Method of Estimating Retardation Effects in Crack Growth, Proceedings of the Fatigue 90 Conference, Honolulu, Hawaii, NASGRO Reference Manual, (version 4.02), NASA Johnson Space Center and Southwest Research Institute, Dugdale, D. S., Yielding of Steel Shafts Containing Slits, J. of Mechanics and Physics of Solids, Vol. 8, Newman, J. C., Jr., A Crack-Closure Model for Predicting Fatigue Crack Growth under Aircraft Spectrum Loading, Methods and Models for Predicting Fatigue Crack Growth Under Random Loading, ASTM STP 748, Newman, J.C., Jr., "FASTRAN II - A Fatigue Crack Growth Structural Analysis Program," NASA-TM , NASA Langley Research Center, Hampton, Virginia,

15 10. de Koning, A. U., and Liefting, G., Analysis of Crack Opening Behavior by Application of a Discretized Strip Yield Model, Mechanics of Fatigue Crack Closure, ASTM STP 982, McClung, R. C., McMaster, F. J., and Feiger, J. H., Comparisons of Analytical Crack Closure Models and Experimental Results Under Flight Spectrum Loading, Fatigue Testing and Analysis Under Variable Amplitude Loading, ASTM STP 1439,