A critique on fatigue crack growth life estimation methodologies

Int J Fatigue 14 No 1 (1992) pp 30-34

A critique on fatigue crack growth life estimation methodologies T. A k y i i r e k and O . G . Bilir

In this study, three fatigue crack growth life estimation methodologies are reviewed and sample calculations are made using these methodologies. Comparison of the results with respect to the methodologies are made. Three computer codes which represent these methodologies, CRACKS IV, FASTRAN and FATIGUE are selected for the analyses. The estimations are also correlated to the test results found in the literature. FALSTAFF spectra are used in the analyses. Key words: crack growth; methodologies

There are many fatigue crack growth life estimation methodologies used by engineers and scientists, but there is no unified methodology commonly agreed upon. Many round-robin crack growth analyses have been made in the past, but none has resulted in a unified procedure.l-3 In this study, sample calculations are made using the three estimation methodologies. Three computer codes which represent these methodologies, CRACKS IV, 4 FASTRAN 5 and FATIGUE, 6 are selected for the analyses. Results are compared with respect to the methodologies and the test data found in the literature. FALSTAFF / spectra are used in the analyses.

Methodologies Many factors are dominant in the estimation of fatigue crack growth life under service loading, such as the linear crack growth equation, load interaction model, damage accumulation scheme, cycle counting, stress intensity factor formulation, etc. Different combinations of these factors result in different methodologies.

Fatigue crack growth rate relationship

= C[(1 - J~)m-'AK]~

0 ~
R=R

0~R

R = R:,t

~Rg, t

(1)

where C and n are the fatigue crack growth rate constants, M(" is the stress intensity factor range, R is the stress ratio, m is the stress ratio layer collapsing factor, and R2,t is the cut-off value of the stress ratio above which no further stress ratio layering is shown in a d a / d N versus M( plot. The threshold stress intensity range (M(TH) is not accounted for in the modified Walker equation. If the applied M( is less than MfTH, then Equation (1) is not applicable and no crack growth results. The values of MfTH are assumed to vary with respect to R in the following manner:

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AR) AK°H

(2)

where A is an empirical constant determined from constant amplitude data with various stress ratios, and AK°H is the threshold AK for R = 0.

Load interaction models The Wheeler, the Willenborg, the analytical crack closure and the Willenborg-Chang models were used in the analyses. The first two models were selected to represent the computer code CRACKS IV, and the last two represented FASTRAN and FATIGUE, respectively. For comparative purposes, estimations were also made using the non-interactive assumptions for each case.

The non-interactive mode/ The model is based on the hypothesis that each toad cycle extends the crack by the same degree as that produced in the case of constant amplitude loading. The method is generally conservative. In certain cases, as for narrow-band Gaussian sequences, the results are fairly accurate. 9

The Wheeler mode/

The selected crack growth rate equations for the analyses are tabular da/dN-AK data for CRACKS IV and FASTRAN, and the modified Walker equation for FATIGUE. The modified Walker equation can be expressed as followsS:

dN

AKTH = (1 -

To take into account the interaction effect due to overload, Wheeler 1° suggested the application of a corrective factor to the crack growth rate, which is obtained by a K-rate relationship (Paris, Forman, Walker or da/dN-AK data, etc). The corrective factor is given as:

CP=

(a o + rpi rpo -

Cp = 1

ai

)m forai+rpi
(3)

f o r ai + rpl >i go + rpo

(4)

where ai is the crack dimension when the ith load cycle is applied, rpi is the size of the plastic zone produced by the load in question, rpo is that of the plastic zone generated by the overload applied to the crack dimension ao, m is an empirical constant (shaping exponent) which is determined by means of the fit between specific experimental data and the results from the model. The basic shortcoming of the Wheeler model is the empirical nature of the constant m, which is highly dependent on the kind of material, on the loading spectrum and probably on the geometry.

0142-1123/92/010030-05 © 1992 Butterworth-Heinemann Ltd Int J Fatigue January 1992

The Willenborg model Willenborg 11 assumes that gmax, i actually occurring at the current crack length ai, will be effectively reduced by an amount K~¢d, given by: gred = g m . . . . . q

--

gm,x,

(5)

i

where Km. . . . q is the value of stress intensity factor required to grow the crack out of the overload plastic zone. The residual compressive stresses introduced by an overload reduce the effective stress at the crack tip. The magnitude of the residual stress is given by: SR = gmax , req

Kmax , i

(6)

Ke--f{n = Kmin - (I~ Kmax. OL

1

ZOL

= 1 - K'm/Kr~,x, og

Rso- 1

(16)

where AKTH

Keffax,t. = Kmax,i __ gred = 2Kma~, I __ Km . . . .

(7)

q

Kefm[n,i = Kmin,i -- Kred + Kmin,i + Kmax, i - Km . . . . q ( g ) If either K~!n or both K~x and K~!, are smaller than zero they are set at zero. The cycle ratio R ¢u becomes Re.

_ K m ~ . a - K~od _ K m ~ . , , +

Kmax,i -- Kr~d

Km=.~- K m . . . . q

2Km,x,i -- Km . . . . .

The crack closure model The crack growth equation proposed by Elber states that the growth rate is a power function of the effective stress intensity factor range only. Reference 12 shows that the power law is inadequate at low growth rates. The power law was modified by Newman 13 to the equation given below:

(10)

2

where Ago = C3 (1 - C 4 S m i . / S m a x ) = C 3 ( 1 - C 4 R ) AKef f = C (Srnax

-

-

So) ~

(11) (12)

The constants Ct to Cs are determined to best fit experimental data for constant amplitude loading. So is the crack opening stress to be calculated from the analytical crack closure model. The effective threshold stress intensity factor range, AKo, is determined from the threshold stress intensity factor range, /~TTH, as AKo -

(1 - S o / S i n , x )

(1 - R)

~¢I'TH

(13)

The Willenborg-Chang model is a combination of the generalized Willenborg model TM and the Chang acceleration scheme, ls It uses the modified Walker equation s as the baseline crack growth equation. The generalized Willenborg model can be expressed in the following form:

Int J Fatigue January 1992

1

K~{ n = A K

(18)

Reff_ K~!n K~x

(19)

Note that AKeffhas the same value as AK. Thus the generalized Willenborg model predicts the crack growth retardation by reducing the effective stress ratio below that remotely applied. The crack growth retardation model is expressed in terms of the effective stress intensity factor range and the effective stress ratio; that is, da d N = C[(1

-

Reff) ra-I AKeff] n

0 ~ R elf < Rc+ut

R elf = R elf

0 ~< R elf > Rc+t

R 'ei~ = R c+t

(20)

If the effective stress ratio is negative (R eu < 0) the Chang acceleration scheme in conjunction with the generalized Willenborg model is used to account for the compressive load acceleration effect: da dN

-

C {[1 + (ReU)2]qK~k}"

(21)

The value of acceleration index, q, is determined using the following relationship: q = [In (~/)/ln (1 + R2)]/n

R< 0

(22)

where ~/ is the ratio of the crack growth rate of a specific stress ratio to its R = 0 counterpart obtained from test data.

Damage accumulation schemes

The Willenborg-Chang model

K°--~L=Km=-* gm=.OC

The overload shut-off ratio (Rso) is the value of Km=, OL /Km= above which there is no crack growth. For spectrum loading, the effective stress intensity factor range and effective stress ratio are expressed in terms of the maximum and minimum effective stress intensity factors as follows: AKe. = K ~ -

Both AK,ff and R elf can be calculated and then da/dN can be calculated.

da dN - C, A K ~

(17)

(9)

q

( oy 1 - \AK~ff/

(15)

Where Km~,, OL is the stress intensity factor corresponding to the maximum stress of the overload cycle, Aa is the incremental growth following overload, and ZOL is the overload retardation zone size. The multiplier q) in the above equations is determined as follows:

KTH - 1 - R

Hence, the effective stress intensity is given by:

Km.x

Zo~

Km=

(14)

The Runge-Kutta integration technique, 16 Vroman's linear approximation method s and the linear approximation method t3 were selected as the damage accumulation schemes, which represent the computer codes CRACKS IV, FATIGUE and FASTRAN, respectively.

Cycle counting techniques Cycle counting is used to summarize the (often lengthy) irregular load versus time histories by providing the number of times cycles of various sizes occur. The definition of a

31

cycle varies with the method of cycle counting. These practices c o v e r the procedures used to obtain cycle counts by various methods, including level crossing counting, peak counting, simple-range counting, range-pair counting (or rainflow counting). 17 Estimations

for sample

cases

In this study, a comparison is made of crack growth lives of centre-cracked panels as estimated by the crack propagation programs CRACKS IV, FATIGUE and FASTRAN, under FALSTAFF and truncated-FALSTAFF spectra. The estimations are also correlated with the test results found in the literature. The Wheeler model and the Willenborg model using CRACKS IV, the Willenborg-Chang model using FATIGUE and the analytical crack closure model using FASTRAN were used in the estimations. Experimental fatigue crack growth data under FALSTAFF and truncated-FALSTAFF loading spectra were provided from Reference 7. The following sections describe the analytical predictions and correlation results. Analytical predictions Predictions were done in two steps for each computer code.

Step I: Creation of data files Data files were created for each material and for different stress histories. Since the stress ratio, R, for FALSTAFF loads is mostly positive and the available data sets are for R = 0.3 and - 1 , the data set for which stress ratio is equal to 0.3 was used in the calculations. 1 Material property data are summarized in Table 1.

Step 2: Estimation of crack propagation lives Estimations were made using the data files created in Step 1. FALSTAFF generation: the standard FALSTAFF sequence of 200 flights generated using the data provided in Reference

7 was used in the analyses. Since each flight starts with a negative-sloped load excursion from level 8 to level 6, the final level was taken fom the end of each flight and added to the beginning. In this way each flight began and ended with a positive-sloped excursion. This procedure was applied to the studies with the CRACKS IV and FATIGUE programs. FASTRAN has an input option for the FALSTAFF spectrum loads. The FALSTAFF sequence is generated within the program, however the above procedure could not be applied to the FALSTAFF generation by FASTRAN, and therefore each flight began and ended with a negative-sloped excursion. Analytical predictions and crack g r o w t h data correlations Analytical predictions by the fatigue crack growth estimation methodologies were correlated to the crack growth test data in order to verify the efficiencies of these methodologies. To assess the prediction accuracy of each method, the ratio of the predicted crack growth life to the test crack growth life, Npred/Nt.... was calculated for each test case. A summary of estimations and correlation to test results is given in Table 2, to allow comparison of the estimations made using the computer codes. For FATIGUE, the predictions with q = 1.0 for L93 and 7475-T7651, and those with q = 0.8 for BS2S99 were used in comparison (Chang 16 proposed that q = 1.0 for 2219-T851 aluminium alloy and q = 0.8 for Inco 718 steel). For FASTRAN, the analytical estimations with c~ = 1.73 were used for the comparison given in Table 2 (because the stress state was neither plane stress, et = 1, nor plane strain, c~ = 2.5, but between these extreme cases). Estimations with the truncated-FALSTAFF spectrum could not be made using FASTRAN since some modifications were required in the program, For comparison, crack growth curves for a sample case are shown in Fig. 1. Considering that the normal scatter in fatigue crack growth rates may range from a factor of two

Table 1. Material properties Material

Wheeler and Willenborg

Crack c l o s u r e

Willenborg-Chang

BSL93 (AI a l l o y )

Tabular da/dN-AK R+ut = 1.00 K~ = 60

Tabular da/dN-AK KF = 60 m = 0.0

C = 0.6132 x 10 -9 n = 3.7839 m = 0.5

q = 1.0 AK~H = 2.385

7475-T7651 (AI a l l o y )

BS2S99

(Steel)

32

Sy=398

NS=6

P l a n e stress m = 1.6 ( W h e e l e r e x p o n e n t )

Sy -- 398 e( = 1.73 13 = 1.0

Tabular da/dN-AK Rc+ut = 1.00 K~ = 70

Tabular da/dN-AK KF = 70 m = 0.0

C = 0.7168 x 10 -11 n = 5.9652 m = 0.5

Sy=448

NS = 6

Plane stress m = 1.6 ( W h e e l e r e x p o n e n t )

Sy = 448

q = 1.0 AK-~H = 2.385

e = 1.73 13 = 1.0

Rso = 2.5 Kc = 64.22

Tabular da/dN-AK KF = 136

C = 0.1037 x 10 -11 n = 4.7312 m = 0.5 q=0.8 /:?so = 2.5 Kc = 124.77

Tabular da/dN-AK R+cut = 1.00

K~ = 136

m = 0.0

S y = 1267 P l a n e stress m = 1.6 ( W h e e l e r e x p o n e n t )

NS=6 Sy = 1267 o~ = 1.73 13= 1.0

/:?so = 2.5 Kc = 55.04

Int J Fatigue January 1992

Table 2. Estimations and correlation to test results

Analytical predictions (Npred/Ntest) CRACKS IV Gross stress (Level-32; MPa)

Willenborg

Wheeler (m = 1.6)

FASTRAN ((~ = 1.73)

FATIGUE

BSL93

67 77 100 77 truncated FALSTAFF

1.66 2.29 2.08 1.09

1.58 2.04 1.80 1.22

0.97 1.07 0.76 -

1.10 1.22 0.84 1.35

7475-T7651

63 102 77 truncated FALSTAFF

1.32 0.65 0.93

1.28 0.53 0.96

1.02 0.34 -

1.27 0.22 0.84

BS2S99

240

1.24

1.04

0.35

0.91

Average Standard

(Npred/Ntost) Deviation

1.41 0.53

1.31 0.46

0.86 0.25

0.97 0.34

Material

to four under identical loading conditions, these results are quite good. All the computer codes underestimated the crack growth life at high spectrum stresses. This result may be attributed to the baseline data. Underestimation appears to be due to the high slope of the da/dN v e r s u s M ¢ curve at high fiat¢ values. This is probably because if the slope is steeper the crack will grow through an overload zone more quickly and so the effect of retardation will be reduced.

(iii)

Conclusions In this study, three different methodologies used for estimation of fatigue crack growth life are reviewed and sample calculations are made using these methodologies. Comparisons of the results with respect to the methodologies are made. Three computer codes which represent these methodologies, CRACKS IV, FATIGUE and FASTRAN, are selected for the analyses. The estimations are also correlated to the test data. From the results of the predictions, the following conclusions are reached: (i)

(ii)

(iv)

(v)

Considering that the normal scatter in fatigue crack growth rates may range from a factor of two to four 50

/

(vi)

/ 30

//'

////,/ u

20I"1 Crack closure (~= 1.73] - - - Test results (L-S} Test results (U-B) Wheeler (Cp: 1.6) • Willenborg • Willenborg-Chang

0

10

I

0

(vii)

50

I 100

I 150

I 200

I 250

I 300

I

350

N(FALSTAFF blocks) Fig. 1 Analytical crack growth estimations ( a - N diagram for BSL93, spectrum stress = 67 MPa)

Int J Fatigue January 1992

under identical loading conditions, it can be concluded from the predictions that all three methodologies, CRACKS IV, FATIGUE and FASTRAN, provide very good analytical predictions on crack growth lives under fighter spectrum loadings. Predictions for the very high spectrum loads usually result in underestimation of crack growth life, especially if the acceleration phenomenon is taken into consideration by the load interaction model. This is valid for all the methodologies. The order of the computer codes with respect to increasing computation time is FASTRAN, FATIGUE and CRACKS IV. If the spectrum loading is in the random cycle-bycycle format, it is essential to process the random spectrum through a cycle counting technique before crack growth analysis. However, predictions with the original spectra yield better results than those with the counted spectra for the analyses without load interaction. All methodologies, except FASTRAN, cannot be used for short cracks, due to the fact that linear elastic fracture mechanics might not be applicable to characterize the crack growth behaviour for very small crack sizes. The analytical crack closure model is the only one among the models mentioned that considers the crack growth acceleration observed for short cracks below the classical threshold stress intensity factor. Results of the non-interactive estimations are within acceptable limits. This result may be due to the random character of FALSTAFF load sequences: acceleration and retardation effects might have cancelled each other out. It can be concluded that FATIGUE is the best of the three methodologies, considering the efficiencies of the estimations.

References 1.

Perrett, B. 'TTCP-HTP3 collaborative exercise-validation of crack growth models (calculation by c o m p a r i s o n with crack growth test results' ('I-I'CP, 1984)

33

2.

3.

4.

5.

6.

7.

8.

9. 10. 11.

34

'An assessment of the accuracy of various fatigue crack growth modelling techniques. Part h methods and results' The Technical Cooperation Program (TTCP), Technical Pane/HTP-3 ('r-I'CP, 1989) Chang, J.B. 'Round-robin crack growth predictions on center-cracked tension specimens under random spectrum loading' ASTM STP 748 (American Society for Testing and Materials, 1981) pp 3-40 Kacprzynski, J.J. 'CRACKS IV--USAF crack propagation program, NRC modification no. 1' NAE Report LTR-ST1260 (National Research Council of Canada, 1981) Newman, J.C. 'Instruction for use of FASTRAN, a program for the fatigue crack growth analysis of various crack configurations' (NASA, 1984) AkyQrek, T. 'Damage tolerance life estimation of critical parts of a fighter aircraft' Ph.D.Thesis (Middle East Technical University, 1991) Aicher, W. et al 'FALSTAFF, description of a fighter aircraft loading standard for fatigue evaluation' (Joint publication of F&W, LBF, IABG and NLR, 1976) Szamossi, M. 'Crack propagation analyses by Vroman's model, program EFFGRO' NA-72-94 (Rockwell International, California, 1972) Salvetti, A., Cavallini, G. and Frediani, A. Fracture Mechanics, AGARD-AG-292, (NATO, 1983) pp 133-178 Wheeler, O.E. 'Spectrum loading and crack growth', J Basic Engng, Trans ASME, Ser D 94 1 (1972) pp 181-186 Willenborg, J.D., Engle, R.M. and Wood, H.A. 'A crack growth model using effective stress concept' AFFDL-TM71-1-FBR (Air Force Flight Dynamics Laboratory, 1971)

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Hardrath, H.F., Newman, J.C., Elber, W. end Poe, C.C. 'Recent developments in analysis of crack propagation and fracture of practical materials' in Perione, N. (Ed) Fracture Mechanics, University of Virginia (1978) Newman, J.C. 'A crack closure model for predicting crack growth under spectrum loading', ASTM STP 748 (American Society for Testing and Materials, 1981) pp 53-84 Gallagher, J.P. 'A generalized development of yield zone models' AFFDL-TM-FBR-74-28 (Air Force Flight Dynamics Laboratory, Ohio, 1974) Chang, J,B., Szamossi, M. and Liu, K.W. 'Random spectrum fatigue crack life predictions with or without considering load interactions' ASTM STP 748 (American Society for Testing and Materials, 1981) pp 115-132 Chang, J.B., Hiyama, R.M. and Szamossi, M. 'Improved methods for predicting spectrum loading effects' AFVVALTR-81-3092 1 ( 1981 ) 'Standard practices for cycle counting in fatigue analysis' ASTM EI049-85, Annual Book of ASTM Standards, 03.01 (American Society for Testing and Materials, 1988) pp 764-772

Authors T. Akyiirek is with the Ministry of Defence (MSB), Department of R&D (ARGE), Ankara, Turkey. O.G. Billr is with the Middle East Technical University (ODTii), Mechanical Engineer's Department, Ankara, Turkey. Received 21 April 1991; accepted in revised form 28 August 1991.

Int J Fatigue J a n u a r y 1992