Fatigue crack propagation under block loading analysed in terms of equivalent loading concepts

Engineer&tg Fracrure Mechanics

001%75’44/!32 S5.00 + 0.00 0 1992 Pergamon Pm8 Ltd.

Vol. 42, No. 1, pp. 59-71, 1992

FVintcdin Great Britain.

FATIGUE LOADING

CRACK PROPAGATION UNDER BLOCK ANALYSED IN TERMS OF EQUIVALENT LOADING CONCEPTS

N. RANGANATHAN,?

M. BENGUEDIAB,j

C. NICOLAS,t

G. HENAFFt

and J. PETIT?

tLaboratoire

de M&canique et de Physique des MatMaux, URA CNRS no. 863, ENSMA, Poitiers, France fInstitut National de 1’Enseignement Sup&ieure. en Electronique, Sidi-Bel-AbWs 22000, Algeria Abstract-The results of fatigue crack growth tests under a block loading consisting of four mean levels are analyzed in terms of equivalent constant amplitude loading concepts. The different models used to determine the equivalent loadings are based on the root mean square (RMS) method, a modified method based on Paris’ law, Elber’s method based on crack closure concepts and a newly proposed energy based method. The results obtained show that the RMS analysis fails under the studied test conditions and leads to highly unconservative life estimates. The modifications brought into Paris’ method based on a crack closure analysis lead to very good life estimations. Elber’s method also gives good results except for one of the spectra studied. Finally, the experimental energy based method gives excellent results for the studied test conditions.

INTRODUCTION THE FATIGUE crack growth (FCG) behavior under variable amplitude loading is a complex process and is influenced by various load interaction effects such as retardation due to occasional peak loads [l], acceleration due to under loads [2] or a combination of both. In some loading conditions changes in mean load levels can lead to additional load interaction effects [3]. Different models exist in the literature to predict FCG behavior under such conditions and the validity and limitations of some well known models have been discussed in a recent review paper [4]. Another approach to the understanding of FCG under variable amplitude (VA) loading is through the determination of an equivalent constant amplitude loading which should lead to the same crack growth as does the VA loading. Some of the techniques used to achieve this end are the following. The root mean square (RMS) method In this method the equivalent constant amplitude (ECA) loading is determined from a root mean squared analysis of the initial load spectrum [5]. In terms of stress intensity factors, the ECA loading is determined by:

where N is the number of cycles in the base loading. The number of equivalent cycles, NW, is simply the total number of cycles in the spectrum, i.e.: % = N.

(lb)

The root mean method on Paris’ law In this method, assuming that Paris’ law [6] can be used to describe FCG under CA conditions, the ECA is defined as: A~

,;, (AW” =

N

(= 59

‘lrn @a) 1

N. RANGANATHAN

60

et al.

where m is the Paris law exponent given by the well known relation under CA conditions,

where da/dN is the crack growth rate at a given AK level, and C is an experimentally defined constant. N_ for this method is the same as that for the RMS method. The ECA loading by the Paris method is the same as that given by the RMS method for materials having m = 2 (certain steels) [A. Elber’s method

This method is based on the crack closure concept developed by Elber [8]. Assuming that FCG under CA conditions in the mode I type of loading can occur only when the crack is fully open, the ECA for short and stationary spectra [9] is defined by respecting the following conditions: (i) the maximum load is the same for the ECA loading and base spectrum; (ii) the crack opening load, Pop, is the same for the ECA and base loading. Nq is next defined by:

where PM, is the maximum load level for cycle i in the base spectrum Pop is the crack opening level

; is p”” = OP

efff;tiv
M,

level

op

and P,,,, is the minimum load for the cycle number i in the base spectrum P,,,, is the maximum load for the spectrum

n’ is the exponent relating da/dN and the effective stress intensity level, &, conditions

under CA

&-=C’(Ak;,)“‘. In this paper, FCG behavior under simplified block loadings (BL) consisting of four mean levels is discussed in terms of the above mentioned ECA loading methods and an energy based concept is developed in our laboratory. EXPERIMENTAL

DETAILS

This study was conducted on the high strength aluminum alloy 2024 T351. The nominal composition, mechanical properties and full experimental details can be found elsewhere [lo]. Briefly, CA tests were carried out at five R ratios to determine the evolution of du/dN in terms of AK level. The crack growth rate covered is in the range: lo-* c du/dN < lo-‘m/cycle. The BL loading considered here is shown in Fig. 1 and the number of cycles at each level is given in Table 1. This type of loading was derived from a typical transport aircraft wing lower surface loading [lo]. The block A represents the most probable Ground Air Ground Cycle. Blocks B, C and D simulate load spectra with increasing gust load severity, represented by increasing number of cycles at the highest mean level. Block E is a modified loading representing a particular flight configuration. All the tests were carried out under computer control at 20 Hz in ambient air and at selected crack lengths, the evolution of the crack mouth opening displacement S (measured by a clip gage)

Fatigue crack propagation under block loading

61

1 block

F-

Fig. 1. Type of loading studied.

Table 1. Load levels and number of cycles at different levels for the spectra studied

A :

Step 1 Pti, = 80 P_ = 150 R = 0.53 nl 1 10

D E

10 46

Type

sptctnml

Step 2 Pmin= 160 P-=392 A = 0.41

Step 3 Pti, = 323 P-=600 R = 0.54

Step4 Pd.= 138 P_ = 323 R = 0.4

n2

03

n4

1 10

1 10

1 2

1: 31

: 2

and the differential displacement 6’ with respect to the load P were recorded on an XY plotter at a frequency of 0.2 Hz. 6’ is defined by: 6’=6

-UP

(4)

where OLis the specimen compliance at a particular crack length. The measurements were carried out during one cycle for CA tests and during one block for BL tests. Typical S vs P and 6’ vs P diagrams for CA and BL conditions are given in Fig. 2a-c. The following parameters were measured from these diagrams. CA tests (1) The crack opening load Pop, at the beginning of the horizontal portion of the 6’ vs P diagrams [ 111. (2) The energy dissipated per cycle, Q, which is the area enclosed by the 6’ vs P diagrams after suitable corrections [ 121.

Q env. = a 1 to 10

KS.’\

/a-p

Fig. 2a. 6 VI P and 6’ vs P diagrams, d&nition of Popand Q.

Fig. 2b. S’ vs P diagram at low K_

for BL tests.

62

N. RANGANATHAN

et al.

Fig. 2c. 6’ vs P diagram at high kT_ for BL tests.

BL tests (1) The crack opening level Pop (Fig. 2b). (2) The energy dissipated per block, Q,,. Two cases arise. As shown Fig. 2b, at low K values, the change in compliance, as shown by the slope of the near horizontal portion in the 6’ vs P diagrams, is negligible from the beginning to the end of the block. In this case Qtot is defined by:

Qtot = C Qi+ Qmv

(54

where Qi is the energy dissipated during one cycle and QenYis the energy determined from the envelope of all the cycles [13]. This amounts to a rainflow analysis for the particular loading [14]. The second case is shown in Fig. 2c. In this case the compliance change is significant during a block and in this case, Q,,, is defined as [13]:

Qtot = C Qi*

Fig. 3a. Evolution of da/m

with respect to AK.

Wd

Fatigue crack propagation under block ioading

63

Fig. 3b. Evolution of da/dN with respect to i&.

EXPERIMENTAL RESULTS

The evolution of da/dN with respect to AK and L for five different R ratios is given in Fig. 3a and b. AK and L are determined according to ASTM standards [15]. The effect of R ratio is usually analyzed in terms of crack closure [8] and coupled environmental effects [16]. The relationship between da/dN and measured A& values is shown in Fig. 4. AK, is determined by AK&= &, - Kq,3

(6)

being associated to the measured Poplevel. In this figure near threshold crack growth behavior for this material is added [16],while results at high growth rates (da/dN > 10m6m/cycle) have been omitted. This diagram shows that the relationship between da/dN and A& can be expressed in terms of the relation (3b) irrespective of the R value, but with the definition of two different slopes, with a transition at about da/dN = IO-* m/cycle. Thus the constants of Elber’s relation (3b) are different on either side of this transition. The values of the constants C’ and n’ determined are given in Table 2. l&,

Fig. 4. Relationship between da/dN and effective AK level.

N. ~NGANA~N

64

et al.

Table 2. Constants C’ and n’ of Elber’s relation Range considcrKi da iii?

<

IO-* m/cycle

2 >RF

m/cycle

C’

‘n’

3.83 x lo-‘O

2.50

1.51 x 10-10

4.0

,

1 8

(Q

K,

80

,.

Fig. 5. ~lationship

86

~~~a~~

between the efficiency factor U, and &,,_.

It should be remarked that for R = 0.7 no closure was detected in the studied AK range, therefore G = AK for this R value. The evolution of the crack load was analyzed in terms of the efficiency factor, U,, defined by PI:

uE=L\K:' AG’T

(W

The evolution U, with respect to K,,, is given in Fig. 5. This figure shows that Ua depends on K_ and R and the following empirical relationship was determined: W,=A+B&,,+CR.

00

The values of A, B and C in different K,,,,, ranges determined by regression techniques are given in Table 3. These constants do not cover near threshold values where a strong decrease of & is observed [16]. It is seen here that the relations determined are quite different from that proposed by Elber 181. CRACK GROWTH

FOR BL TESTS

The crack growth rates for BL tests, expressed in terms of crack advance per block (Aaiblock) with respect to K_, are shown in Fig. 6. It is seen here that the crack growth rate increases with spectrum severity for a given Ila, value, i.e.: As/block for BL D > for BL C > for BL B > for BL A. The block E gives similar results to block C but with slightly lower growth rates at low K_ values. Table 3 Range of L

(MPa&)

6 17

A 0.03 0.03 0.03

3 0.047 0.047 x & 0.80

C 0.36 0.36 0.36

65

Fatigue crack propagation under block loading

d

_

.Iii 6

7

6

I

LO

K max4lPa JG

Fig. 6. Crack growth rate evolution for BL tests.

The FCG behavior under BL loading conditions also shows the existence of two regions of different slopes. The transition, called TBL, between these two behaviors takes place at a & of about 16 MPa& for the different blocks. The crack growth life measured for a crack growth from Q = 24 mm to 52 mm is given in Table 4. The lives are divided into two ranges on either side of the transition defined above, which corresponds to a crack length of 30 mm as all the tests were done with the same maximum load. It is observed here that the crack growth lives in both the ranges decrease with spectrum severity for blocks A, B, C and D. Block E gives a higher life than block C in range 1 while the lives for these two blocks are the same in range 2. The results of crack closure measurements are shown in Fig. 7 in terms of the efficiency factor U, . The closure level, KOr, is found to be approximately constant during one block. U, is hence determined with respect to K,,,,, of the block using the following relation: KW - J&p “=

(8)

Kmm-k&

where K_ and lEain refer to the maximum and minimum loads in a block. The evolution & with respect to K,,,, is shown in Fig. 7. It can be noted here that U,is almost constant in the studied range and independent of the block type for the test conditions. This value of U,is about the same as that observed under CA conditions at R = 0.01 for K_ > 17 MPa&. This behavior can be related to the hypothesis of Schijve [17], who proposed that for short and stationary spectra, the crack opening level should be equal to that observed under CA conditions under maximum and minimum loads of the spectra.

Table 4. Crack growth life under BL conditions Life in number of blocks Block type

Range a = 24-30

Range u = 30-52

Total

A B C D E

39 590 12000 5750 3200 7850

35 530 11900 3850 2300 3850

75 120 23 900 9600 5600 11600

N. RANGANATHAN

I-

mA OB

Y

9

Yii

et al.

oc *O 0 ‘0’ we Oco$od . .

~opdO.*m~* ,

.

.o

.

0’0

0 0

-0

Y

P J

2

Fig. 7. Evolution of f& for BL tests.

DISCUSSION The discussion is divided into three parts. Firstly, modifications suggested to the Paris method for the determination of ECA loading are presented. Secondly, the salient features of the energy based method are presented. Finally, the results obtained by the different techniques of ECA determination are discussed. A modljied equivalent constant amplitude loading based on Paris’ method

As seen from Fig. 3a, there is no unique relationship between da/dN and AK for the studied material in the explored AK range. It is hence difficult to apply directly the relations (2a) and (2b) for an ECA analysis. Hence this method is modified to take into account the relations independent of R ratio determined in terms of the crack closure analysis. An ECA loading is proposed respecting the following conditions. (1) The K_ value is the same for ECA and BL loadings. This condition is the same as that proposed in Elber’s method [9]. (2) The equivalent R ratio for ECA loading is the mean R ratio for the base loading, i.e.: 2 Ri h=%*

(9a)

(3) A& for the ECA loading is now determined from the following relation:

(9b) where n’ is the average slope of 4 determined in this study. This approximation is proposed for the following reasons. If one considers the existence of two different slopes, two solutions are possible for (AKeff)nldepending upon the crack growth rate. To avoid this ambiguity, the slope of 4 which covers the range lo-’ < da/dN < lo-‘m/cycle is also extended to lower growth rates. Having determined (AK&),, the equivalent J&,,, value is obtained by solving the appropriate relation (7b), by substituting &, and the & value. Nep for this method is the total number of cycles in the base spectrum. An energy based method for the &termination of ECA loading It has previously been shown that for CA conditions the relationship between the crack growth rate and the energy dissipated per cycle, Q, follows distinct laws depending upon the micromechanism of crack growth for the studied material [12, 13, 18,191.

Fatigue crack propagation under block loading

67

Fig. 8. Crack growth rate evolution for BL tests with respect to energy dissipated.

At low growth rates, when the crack advances by a step-by-step mechanism, a power law relationship is obtained: da dN

=

A'QQ.

The experimentally determined values of A' and n, are 2.22 x lo5 and 3.8 respectively. At high growth rates as the crack grows by a striation mechanism a linear relationship obtained between da/dN and Q, i.e.:

is

da

==BQ.

In the case of the studied BL conditions the relationship between the crack advance per block and the global energy dissipated per block, Qtot ,shows a similar evolution as that described above for CA conditions, as shown in Fig. 8. In this figure the linear relationship obtained for CA conditions [eq. (lob)] is also projected for comparison. It is seen here that at high growth rates a linear relationship, which happens to be within a narrow scatter band with respect to that obtained by eq. (lob), is again observed, this time for BL conditions and independent of the spectrum type. At low growth rates, individual spectra can be distinguished and the limited number of experimental points suggests a power law relation between Au/block and Q.,. A critical fractographic analysis [13] shows that this change of behavior can be associated with a similar change of crack growth mechanism observed under CA conditions. It was shown that at low growth rates the crack advances by a step-by-step mechanism while it advances during each block at high growth rates. At the same time, it was observed that the spatial distribution of significant fractographic features is similar to that obtained under CA conditions at R = 0.01 at low growth rates for all the studied spectra. At high growth rates, except for spectrum A for which the fracture appearance is still similar to that obtained for CA conditions at R = 0.01, the fracture appearance becomes similar to that obtained at R = 0.54 (CA conditions) for all the other spectra. The significant details of this quantitative analysis of fracture appearance are presented elsewhere [lo, 131. Based on these observations ECA loadings were determined using the following relations, In all the studied K ranges for block A and for &,, < T,, for the other spectra, the number of equivalent cycles, Ncq, at R = 0.0 1, is given by:

Table 5. ECA loadings for RMS, Elber and energy methods Method RMS RMS with rainflow Elber Energy

Crack length range (mm)

PmucW daN

PWW daN

N,

24-52 24-52 24-33 33-52 24-32 30-52

416 423 600 600 600 600

6 6 6 324

3.96 3.07 7 18

Q0.01 is the energy dissipated at a given K,, value at R = 0.01 for CA tests and the crack advance is given by: Aa,, = &,A’QY!-o.ol

(1 lb)

where A’ and nl are given in relation (lOa). For K_ > TBL for spectra B, C, D and E: lot Ncrl=- Q Q0.54 where Qo.%is the energy dissipated at R = 0.54 for CA tests at a given K,, value and Aa, = NWBQ0.s

(12a)

(W

where B = 1.4 x 10T4J. Estimation of equivalent load cycles and crack growth lives The equivalent CA cycles for the studied spectra using the four methods presented here are given in Tables 5 and 6 for block B. For the RMS and Paris methods two analyses were made for spectra A and B, one with a rainflow analysis 1141and one without. For other spectra the rainflow analysis did not lead to any sibilant difference. For Elber’s method the crack opening level was considered to be constant and equal to the value given in Fig. 7 for all the spectra. It can be noted here that, except for the RMS method, the determined ECA loading depends upon the crack length range. For the modified Paris method changes in ECA loadings are clearly evident from reading Table 6. These changes can be associated with changes in estimated crack opening levels for the base loading based on relations (7a) and (7b). The life estimation was carried out by calculating incremental crack advance per block by block conditions up to the appropriate using the CA crack growth relations determined under CA crack length. The results are given in Table 7. The relative error in percentage is also calculated by:

Estimated life - Measured life x loo Measured life > ( In the case of Paris’ method, the ECA loading was supposed to evolve linearly within the considered crack length range. For example, for block B, ECA loading was supposed to vary linearly from = 490 daN as the crack length varied from 37.5 to 52 mm. A similar P m@W)= 469 daN to PmpxCeqj evolution was considered for Pmi, as well. Error % =

Table 6, ECA loadings from modified Paris method Crack length range (mm) Paris

Paris with rainflow

P,(sj daN

Pktq, daN

24 37.4 37.5 52

500 488 415 446

238 233 I98 212

24 37.4 37.5 52

520 522 469 490

240 241 217 226

NW :: 32 32 32 :; 32

Fatigue crack pupation

under block loading

69

Table 7 Error %

Life in number of blocks

B

C

D

E

Range 2

Total

Range 1

Range 2

Measured RMS RMS c RF Paris Paris + RF Elber Energy

39 586 638 600 26 21 100 209600 73 200 115500 48 250

35 534 388 600 141000 173 100 27 300 45 100 20 700

75 120 I 027 200 403 100 382 700 100500 160600 68 950

1513.2 562.1 429.5 84.9 191.8 21.9

993.6 296.8 387.1 -23.2 26.9 -41.7

1267.4 436.6 409.5 33.8 113.8 -8.2

Measured RMS RMS + RF Paris Paris + RF Elber Energy

12000 50900 45 200 22 500 17600 22 200 13 370

11900 30000 26 200 15300 9400 8600 12 790

23900 80900 71400 37 800 27000 30 800 26 160

324.2 276.7 87.5 46.7 85.0 11.4

152.1 120.2 28.6 -21.0 -27.7 7.5

238.5 198.7 58.2 13.0 28.9 9.5

Measured RMS Paris Elber Energy

5750 9500 9300 7000 6240

38.50 5200

9600 14700 12900 9700 10 210

65.2 61.7 21.7 8.5

35.1 -6.5 -29.9 3.1

53.1 34.4 1.0 6.4

Measured RMS Paris Efber Energy

3200 4500 3800 4200 3020

2300 2400 1600 2130

5500 6900 5800 5800 5150

40.6 18.8 31.3 -5.6

- 1::: - 30.4 -7.4

-:::

Measured RMS Paris Elber Energy

7850 17000 6700 7700 6240

3850 10 300 4300 2900 3840

11700 27 300 llooo 10600 10800

116.6 - 14.6 -1.9 -20.5

167.5 11.7 -24.7 -0.3

133.3 -6.0 -9.4 - 13.8

Block A

Range 1

Total

:z 3970

25.5 5.5

By examining Table 7, the following observations can be made with respect to the different methods. RMS method This method ove~stimates the life for all the spectra studied. The relative error is unrealistically large, especially for spectra A and B, The best life estimation is obtained for spectrum D. Modijied Paris method This method gives reasonably good estimations for all the studied spectra, the relative error being within -6 < RE < 34%. In the case of spectra A and B, the rainflow technique leads to a better life estimation. However, it appears that these results based on total life estimation have to be treated with caution. A careful examination of the results shows that this method in all of the cases (except spectrum E) overestimates the life in range 1 while it underestimates the life in range 2. The relative errors in the two ranges seem to compensate for each other. The overestimation of life in range 1 can be associated to the fact that in the BCA determination, we considered a unique slope of 4 even at low growth rates. By examining Fig. 4, it can be noticed that the extension of the crack growth law, obtained for da/dN > lo-* m/cycle, for lower growth rates underestimates the da/dN values in this range. Elber method This technique leads to very good life ~timation for all the spectra studied, except spectrum A. The relative error is within -9.4 < RE < 29% for spectra B, C, D and E, while a value of 114% is obtained for spectrmn 1. Since this method depends on an accurate determination of the crack opening load, any errors in its measurements for spectrum A may be one of the reasons for further error.

70

N. ~NGANA~AN

et of.

Energy based method This method leads to reasonably good life estimations for all the spectra studied. The relative error falls in the range - 13.8 < RE < 6.4%. No systematic deviations are observed on each of the crack length ranges. However, it should be remarked for spectrum A that the error in range 1, which is positive, seems to be compensated by a negative error in range 2. GENERAL

DISCUSSION

It is thus confirmed that the RMS method leads to an overestimation of life under the spectra studied. This can be att~buted to the facts that the slopeof the curve relating da/dlv and AK, is not equal to 2 and that the loads in the spectra do not follow a Gaussian distribution [I. The improvements brought into the Paris method based on crack closure measurements seem to give good results, especially coupled with a rainflow analysis. Elber’s method leads to very good results except for one case. This method relies on an accurate determination of the crack opening level which was done by experimental means in the present study. The energy based method also seems to be very attractive as good life estimates are obtained for all the cases studied. This method depends upon the relationship between the crack growth and energy dissipated and on an equivalent R ratio determined from a quantitative fractographic examination. However, this method is purely experimental and at the present stage cannot be considered to be a predictive method. This can be corrected for by estimating the hysteretic energy by numerical methods. It has been suggested that one of the tests for a model is that it should predict trends in life evolution associated with modification of the distribution of the loads in a spectrum [4]. In this respect the modified Paris method and the energy based one lead to the best results. However, the present tests relate to simple block loading where load interaction effects are minimal [IO]. It is suggested that such analyses should be extended to other spectra with predominant load interaction effects and other materials to extend the critical examination of the different models. CONCLUSIONS (1) Four different techniques for covalent constant amplitude destination have been examined on simple block level tests on a high strength aluminum alloy. (2) The RMS method systematically overestimates the total life for the studied test conditions. (3) Improvements brought into the Paris method based on crack closure measurements give encouraging results. (4) Elber’s method also gives good results except for the shortest spectrum studied. (5) A new method based on hysteretic energy measurements is presented which appears to be one of the best techniques for the studied test conditions. Acknowledgements-This work was part of a research program between out laboratory and the Aerospatiale, Toulouse. The authors have great pleasure in acknowledging the technical support of M. B. Lachaud. The ECA analysis was carried out as part of their research project by M. R. Faganello and MS L. Guedj, graduate students at the Ecole Nationale Supirieure de Mecanique et d’Aerotechnique, 1989, and their participation in this work is also appreciated.

REFERENCES [I] N. Ranganathan et al., ASTM STf loaS, 374 (1990). [2] R. I. Stephens et al., ASTM STP 595, 27 (1976). [3] K. Schulte et at., ASTM STP 1049, 347 (1990). [4] R. J. H. Wanhill and J. Schijve, in Fatigue Crack GrowthUnder Yariabfe Amp&u& Loading @ii&d by J. Petit et al.), p. 326. Ebevier, London (1988). [S] R. Hudson, et at., ASTM STP 748, 41 (1981). [6] P. C. Paris, The fracture mechanics approach to fatigue, in Fatigue, An bJteFdi.Wiphbary Approach,Proc. 10th Segumore ConjI Syracuse University Press, Syracuse, NY (1964). [7] A. Bignonnet et al., in Fatigue Crack Growth Under Variable AmplitudeLoading (Edited by J. Petit et al.), p. 372. Elsevier, London (1988).

Fatigue crack propagation under block loading

71

[8] W. Elber, ASTM STP 486, 230 (1971). [9] W. Elber, ASTM STP 595, 236 (1976). [lo] M. Renguediab et al., in Fatigue Crack Growth Under Variable Amplitu& Loading (Edited by J. petit et al.), p. 309. Elsevier, London (1988). [l l] M. Kikukawa et al., f. Mater. Sci. 26, 64 (1977). 1121N. Ranganathan et al., in Strength of Metais and Alloys (Edited by H. J. Mcqueen et al.), Vol. 2, p. 1267. Pergamon Press, Oxford (1987). [ 131 N. Ranganthan, Proc. Canadian Mechanical Engineering Forum, University of Toronto, Canada, Vol. 2. p. 295 (1990). [14] M. Matsuishi and T. Endo, Fatigue of Metais Subjected to Varying Stress. Japan Society of Me&a&al Engineers, Jukvoka, Japan (1968). [ 151 ASTM Test Method for Measurement of Fatigue Crack Growth Rates (E647-86a), American Society for Testing and Materials, Philadelphia, PA. [ 161 J. Petit, in Fatigue Crack Growth Threshofd Concepts (Edited by D. L. Davidson and S. Suresh), p. 3. The Metallurgical Society, AIME Publishers (1984). [I7] J. Schijve, ASTM STP 700, 3 (1980). [18] N. Ranganathan et aZ., ASTM STP 948,424 (1987). (191 N. Ranganathan et al., Scripta MetaN. 21, 1045 (1987). (Received 28 March 1991)