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CRACK GROWTH PREDICTIONS USING SEVERAL RETARDATION MODELS

A. Brot and C. MatiasEngineering Division

Israel Aircraft Industries

Ben-Gurion Airport, IsraelE-Mail: abrot@iai.co.il

ABSTRACT

Under spectrum loading, load-interactions occur. The crack growth rate usually becomes

considerably slower than predicted by da/dN vs. K data. For certain types of loading spectra, the

life increase due to retardation may be a factor ranging from 2 to 5. The existing retardation

models are semi-empirical and have limited value in predicting crack growth behavior. Thepresent study includes four widely used aluminum alloys, which have been tested under five

distinct spectrum types. The test results are evaluated using several state-of-the-art load-

interaction models including the Strip-Yield Model (2 versions), the Generalized WillenborgModel and the Modified Generalized Willenborg Model. The predicted crack growth lives

were compared to the measured results. All the retardation models, on the average, correlated

reasonably well with the range of test data. However, the Strip-Yield Model (NASA) gave theleast variation. The study indicates that the extent of retardation generally decreases as the

spectrum that is being implemented has a higher mean stress-ratio. The retardation models, can

reasonably predict crack growth for spectra having stress-ratios in the range of -1 to 0, and have a

lower degree of accuracy to predict the extent of retardation for spectra having positive stress-ratios.

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 crackconfiguration.

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 thesesemi-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 aftercrack growthtesting, 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

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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 theModified 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 usedaluminum 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 runningalong the loading axis. A 0.079 inch, through-the-thickness, EDM produced flaw was

precracked, under constant-amplitude loading, to a total crack length (2a) of 0.236

inches.(b) Open hole coupon, with identical dimensions as the CCT coupon, having a centrally

located, 0.315 inch diameter open hole. A 0.039 inch, through-the-thickness, EDM

produced flaw was placed at one side of the hole, and it was precracked to a crack

length of 0.051 inches.

For each alloy, all the coupons were manufactured from the same material batches. Strain-

gages 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 reversedloading (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 variousvalues 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 = 0.06. The cycles are ordered randomly havingapproximately 28 cycles per flight.

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(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 load-factor of 1.0, 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 andmaneuver loading, having a mean stress-ratio of R = 0.83. The gust and maneuver loads

are ordered randomly with approximately 16.5 cycles per flight.

Wing gust and maneuver spectrum

(typical flight )

0

2

4

6

8

10

12

14

16

18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

cycles

stress,

ksi

Pressurized fuselage gust and

maneuver spectrum (typical flight )

0

2

4

6

8

10

12

14

16

18

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16

cycles

stress,

ksi

Fighter Aircraft maneuver spectrum

(typical flight )

0

2

4

6

8

10

12

14

16

18

1 4 7 10 13 16 19 22 25 28

cycles

stress,

ks

i

Lateral gust loading spectrum

(5 flights)

-15

-10

-5

0

5

10

15

1 2 3 4 5 6 7 8 9 10 11 12 13

cycles

stress,

ksi

Ground spectrum

(typical flight )

0

2

4

6

8

10

12

14

16

18

20

22

1 2 3 4 5 6 7 8 9 1 0 11 12 13 14 15 16 17 18

cycles

stress,

ks

i

Figure 1 Spectrum Types Used for Testing and Analysis

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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 = 0.83.

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.

Crack growth was monitored, during each test, by means of crack-propagation gages. Typical

results are shown in Figures 26. 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 everycombination 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 test point. The correlation between

the two test results was reasonably close, in most cases, as is shown in Figures 26.

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

GustGround

Fighter

Aircraft

Maneuver

CCT 2 2 1AL7075

T7351

Open Hole 2 2 2

CCT 2 2 1AL7475

T7351

Open Hole 2

CCT 2 2 2 2 2AL7050

T7451

Open Hole 2 2 2

CCT 2 2 2AL2024

T351

Open Hole 2 1

TOTAL 12 12 10 4 5

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RETARDATION MODELS

It has been observed that crack growth under spectrum loading is characterized by a nonlinearload-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 isexplained 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

26 by observing the great variation between the test data (data 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] andGRF [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 beperformed only aftercrack 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, Reff, which is a

function of the actual stress-ratio and the maximum stress-intensity for the overload cycle. Asingle parameter, Rso, 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

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 fatiguecrack 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 stress-leveldependence, retardation and acceleration. It is strongly based on crack closure concepts firstintroduced by Elber [4] and the Dugdale crack-opening model [7]. The Strip-Yield Model uses a

constraint factor, , 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 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.

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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, varies along the plastic

zone according to a parabolic expression. In addition, the state-of-stress is calculated by adifferent 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 fivespectrum types. The following ground-rules were used to perform the predictions:

(a) The material databases built into NASGRO were used with no attempt to tweakthe 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 exceptfor 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 tooptimize 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 26 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 usingthe 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 conservativecorrelation to the GW model.

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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

1.4

0 40000 80000 120000 160000 200000 240000 280000

Flights

Cracklength,

inches

Test results- specimen 1

Test results- specimen 2

no retardation

SY-N

GW (Rso=3.6)MGW (Phi0=0.61)

Figure 2 - Crack growth of 7050-T7451 Open Hole Coupons Under the Lateral Gust

Loading Spectrum

0.0

0.2

0.4

0.6

0.8

1.0

1.2

1.4

1.6

1.8

2.0

2.2

2.4

0 1000 2000 3000 4000 5000 6000 7000 8000 9000

Flights

Cracklength,

inches

Test results - specimen 1Test results - specimen 2no retardation

SY-NSY-EGW (Rso=4.96)MGW (Phi0=0.391)

Figure 3 - Crack Growth of 2024-T351 CCT Coupons Under the Fighter Aircraft

Maneuver Spectrum

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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 showingthe 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 gustloading 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.

0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.0

1.1

1.2

1.3

0 2000 4000 6000 8000 10000 12000 14000 16000 18000

Flights

Cracklen

gth,

inches

Test results - specimen 1Test results - specimen 2no retardationSY-NSY-EGW (Rso=4.28)MGW (Phi0=0.488)

Figure 4 - Crack Growth of 7075-T7351 CCT Coupons Under the Wing Gust and

Maneuver Spectrum

0.0

0.2

0.4

0.6

0.8

1.0

1.2

0 40000 80000 120000 160000 200000 240000 280000 320000

Flights

C

racklength,

inches

Test results - specimen 1

Test results - specimen 2

no retardation

SY-N

SY-EGW (Rso=4.28)

MGW (Phi0=0.488)

Figure 5 - Crack Growth of 7075-T7351 Open Hole Coupons Under the Lateral Gust

Loading Spectrum

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0.0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1.01.1

1.2

1.3

1.4

0 3000 6000 9000 12000 15000 18000 21000 24000 27000 30000 33000 36000 39000Flights

Cracklength,

inches

Test results - soecimen 1

Test results - specimen 2

no retardation

SY-NSY-E

GW (Rso=4.96)

MGW (Phi0=0.391)

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 pressurizedfuselage, 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 attemptto find the patterns of the behavior.

Global Retardation:

The concept of global retardation [5] allows us to examine the degree of retardation that ispresent 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. Fortest 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 inFigure 7 by increasing level ofmean stress-ratio, R.

Figure 7 indicates that the degree of retardation generally decreases with increasing stress-

ratio. For example, test results under the lateral spectrum (R = -1) had a mean GRF of 3.00while test results under the pressurized fuselage spectrum (R = 0.83) had a mean GRF of only

1.45. (An exception to this rule is the fighter aircraft spectrum (R = 0.06) which had an

unusually high GRF of 3.27, due to the aggressive nature of this spectrum.)

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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 stress-

ratios. 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 1.15.

3.069

1.988

2.617

1.154

1.281

3.002

2.118

3.273

1.835

1.448

1.0

10.0

Lateral gust loading Ground loading Fighter aircraft

maneuver

Wing gust and

maneuver loading

Pressurized fuselage

gust and maneuver

loading

Type of Spectrum

GRF

Average Calculated Retarded Life / Calculated Unretarded Life (SY-N model )

Average Test Growth Life / Calculated Unretarded Life

GRFCrackGrowthLife/CalculatedUnretardedLife

Figure 7 Global Retardation Factor (GRF) for the Various Spectrum Types

Retardation Model Performance:

All the predicted crack growth lives were compared to the measured results and the resultingratios 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 standarddeviations 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 reasonablywell 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, theSY-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 samplesize for the SY-E evaluation was considerably smaller than those of the other retardation models.)

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0.0

0.5

1.0

1.5

2.0

2.5

3.0

3.5

SY-N SY-E GW MGW

Retardation Model

CalculatedLife

Mean + 2 SD

Mean + SD

Mean

Mean - SD

Mean - 2 SD

CalculatedLife/TestG

rowthLife

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 bestperformance, 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 thedata 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 1.09 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 wasnoted, 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 1.0, but the variation was quitelarge. For the 7475-T7351 and 7050-T7451 alloys, the mean calculated life was found to be less

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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 representativecoupons of the various alloys.

0.0

0.5

1.0

1.5

2.0

2.5

3.0

Lateral gust

loading

Ground loading Fighter aircraft

maneuver

Wing gust and

maneuver loading

Pressurized

fuselage gust and

maneuver loading

Spectrum Type

CalculatedLife

Mean + 2 SD

Mean + SD

Mean

Mean - SD

Mean - 2 SD

C

alculatedLife/TestGrowthLife

Figure 9 Statistical Variation Per Spectrum Type (SY-N Model)

0.0

0.5

1.0

1.5

2.0

2.5

AL7075 T7351 AL7475 T7351 AL7050 T7451 AL2024 T351

Alloy

CalculatedLife

Mean+ 2 SD

Mean+ SDMean

Mean- SD

Mean- 2 SD

CalculatedLife/TestGrowth

Figure 10 Statistical Variation Per Material Alloy (SY-N Model)

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0.0

0.5

1.0

1.5

2.0

2.5

CCT Open Hole

Coupon Type

CalculatedLife

Mean + 2 SD

Mean + SD

Mean

Mean - SD

Mean - 2 SD

CalculatedLife/TestGrowthLife

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) andopen 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 ofcalculated 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 log-

mean 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 0.051 inch. The CCTcoupon, starting with a much larger initial crack, (2a = 0.236 inch) would be much less sensitiveto 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.

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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 underspectra 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, 1972.

2. Willenborg, J., R. M. Engle, and H. A. Wood, A Crack Growth Retardation Model Usingan Effective Stress Concept, AFFDL TM-71-1-FBR, Wright Patterson Air Force

Laboratory, 1971.

3. Gallagher, J. P., A Generalized Development of Yield Zone Models, AFFDL-TM-74-

28-FBR, Wright Patterson Air Force Laboratory, 1974.

4. Elber, W., The Significance of Fatigue Crack Closure, Damage Tolerance of Aircraft

Structures, ASTM STP 486, 1971.

5. Brot, A., GRF A Simple Method of Estimating Retardation Effects in Crack Growth,

Proceedings of the Fatigue 90 Conference, Honolulu, Hawaii, 1990.

6. NASGRO Reference Manual, (version 4.02),NASA Johnson Space Center and Southwest

Research Institute, 2002.

7. Dugdale, D. S., Yielding of Steel Shafts Containing Slits, J. of Mechanics and Physics

of Solids, Vol. 8, 1960.

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 GrowthUnder Random Loading, ASTM STP 748, 1981.

9. Newman, J.C., Jr., "FASTRAN II - A Fatigue Crack Growth Structural Analysis

Program,"NASA-TM-104159, NASA Langley Research Center, Hampton, Virginia, 1992.

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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, ASTMSTP 982, 1988.

11. McClung, R. C., McMaster, F. J., and Feiger, J. H., Comparisons of Analytical CrackClosure Models and Experimental Results Under Flight Spectrum Loading, Fatigue

Testing and Analysis Under Variable Amplitude Loading, ASTM STP 1439, 2002.