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iiiir

i11

11N

1

UDR-TR-82-119

r^ A PRELIMINARY STUDY OFCRACK INITIATION AND GROWTH AT

STRESS CONCENTRATION SITES

W

By

D. S. DawickeJ. P. GallagherG. A. Hartman

A. M. Rajendian

University of DaytonResearch Institute

September 1982

Prepared For:

National Aeronautics and Space AdministrationLewis Research Center

Cleveland, Ohio

v o^Rf lg8?

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(NASA-CR-169358) A PRELi1LNAFY STILLY CF N81-3373bCRACK !"IITIATION AND GPUWA!3 AT STRESSCONC _N rRATION SITES Intec ka ' clinicalReport, 1 Jan. - 31 Aug. IJb2 (Dayton Univ., UnclasOhio.) 30 p HC A03 /MF A01 .CSCt 20K G3/39 35443

UNIVERSITY OF DAYTONDAYTON, OHIO

A

4

is

UDR-TR-82-119

t

A PRELIMINARY STUDY OFCRACK INITIATION AND GROWTH AT

STRESS CONCENTRATION SITES

By

D. S, Dawic3.eJ. P. GallagherG. A. Hartman

^m

A. M. Rajendran

University of Daytonm , Research Institute

September 1982

Prepared For: j

National Aeronautics and Space AdministrationLewis Research Center

Cleveland, OhioORO

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i^

i

aei x

x

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kt,

y

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ABSTRACT

A literature search was conducted to examine current crack

initiation and propagate= models For notches. The Dowling crack

initiation model and the E1 Haddad et al. crack propagation model

were chosen for additional study. Existing data was used to make

a preliminary evaluation of the crack propagation model. The

results indicate that fo g- the crack sizes in the test, the

elastic parameter K gave good correlation for the crack growth rate

data. Additional testing, directed specifically toward the

problem of small cracks initiating and propagating from notches

is necessary to make a full evaluation of these initiation and

propagation models.

r

FOREWORD

This t^.s First Interim Technical Report is submitted inaccordance with the NASA Gant NAG '3-246. It summarizes the

work accomplished during the period 1 January 1982 through

31 August 1982.

y The work is being performed for the National Aeronautics

and Space Administration. The work is being performed by theE ( Aerospace Mechanics Department of the University of Dayton

Research Institute, with Mr. George Hartman and Dr. J. P. Gallagher

acting as co-Principal Investigator. Technical support is being

provided by Dr. A. M. Rajendran and Mr. D. S. Dawicke.

I.

.a

f

r

i)

f

t

sTABLE OF CONTENTS

r

SECTION

,^. INTRODUCTION

2 LITERATURE REVIEW

t 2.1 STRAIN LIFE INITIATION MODEL

• 2.2 CRACK PROPAGATION MODELS

2.2.1 Elastic J-Integral _(K,)

2.2.2 Elastic-Plastic J-Integral

2.2.3 Strain Based Stress-IntensityFactor

3 APPLICATION OF EXISTING DATA TO CRACKPROPAGATION MODELS

3.1 ELASTIC J-INTEGRAL (K)

3.2 ELASTIC-PLASTIC J-INTEGRAL

3.3 STRAIN BASED STRESS-INTENSITY FACTOR

3.4 SUMMARY OF EXPERIME14TAL RESULTS

4 SUMMARY OF WORK ACCOMPLISHED AND WORKPLANNED FOR THE NEXT PERIOD

REFERENCES

R

IF

f

r

1

PAGE

1

2

2

3

36

7

8

8

13

16

20

21

22

OF Pot;i4a Quit a r

LIST OF IGURES

The Long and Short Crack Stress-IntensityFactor Solutions for a 1" DiametricallyCracked Hole.

Effect of Notch Severity on Dowing's CrackInitiation Cutoff Parameter.

Finite Element Mesh Representation of a 4"Wide Plate with a Crack Propagating from OneSide of a 1" Diameter Hole.

The Elastic Finite Element Solution and theBowie Approximation.

Crack Growth Rate Data Correlated with theElastic Stress-Intensity "'actor for industrialGrade Buss Bar Copper Subjected to a FatigueLoading with Stress Ratio (R) of 0:1.

Extent of Plastic Zone Along Expected Path ofCrack for a 1 " Diameter Hole in a 4" Wide Plate.

Extent of Plastic Enhancement of the Stress-Intensity Factor, as Calculated by the FiniteElement Solution.

Crack Growth Rate Data Correlated with thePlastically Enhanced Stress-Intensity Factorfrom the Finite Element Solution.

Fatigue Crack Growth Rate Behavior of CopperAlloy Using the Stress-intensity FactorProposed by El Haddad and the Elastic LongCrack Data.

Extent of Strain Hardening Experienced by theCopper and Finite Element Predictions of Strain.

PAGE

4

5

110

i

11

si12

14

it

15

17 t

18

19

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\7.

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^ }\

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LIST Of TABLES

TALE .

l CRACK GROWTH RATE TESTS FOR INDUSTRIAL GRADEBAR COPPER TnSTED AT A STRESS RATIO (R) OF 0.1

1

PAGE

§

SECTION I

INTRODOCTXON

The growth of cracks can be divided into the two categories

of initiation and propagation. In a notched structure, the

initiatipn and subsequently the propagation of small cracks are

governed by the stress field created by the notch. After the

crack grows beyond the influence oA4 the notch, crack propagation

behavior is governed by the nominal stress in the structure,.

A, literature search was performed to examine the state-of-

the-art crack propagation and initiation models. Thirteen recently

published studies [1-131 provide examples of crack initiation and

propagation models. The most promising of these models are:

the El Haddad et al. crack propagation model [1], the elastic-

plastic J-integral based crack propagation model described by

Ratwani et al. (2], and the Dowling crack initiation cut-off

model based on linear elastic fracture mechanics (3,4 t 5). The

work to date has included a close examination of the promising

crack propagation models, specifically by testing their ability

to describe existing data.

L

M,y

2

SECTION 2

LITERATURE REVIEW

w

A literature review was conducted to determine theexisting approaches taken to solve the problem of cracks ,initi-sting an;d propagating from notches. This reN°,iew was presentedat NASA-Lewis on 16 July 1982. The remainder of this sectionsummarizes the important ,ispects of the crack initiation an.,Ipropagation models that have been chosen for additional study.

2.1 STRAIN LIFE INITIAT,XON MODEL

A crack initiation model developed by Dowling (3,4,5)

predicts life to initiation for notch geometries using smooth

specimen strain life data. The severity of the notch is taken

into account by elevating the applied stress by the elastic

stress concentration factor (kt). The equivalent strain ampli-

tude for notched geometry is obtained from the cyclic stress

strain curve of the material.

Dowling found this method to be effective in predicting

crack initiation life if the initiation crack length remained

within the influence of the notch. Dowling defined an upper

bound on the initiation crack length which ensured the crack waswithin the influence of the notch. This upper bound was defined

to be the intersection of what he termed the short crack stress-

intensity factor (K s ) and the long crack stress-intensity factor

(KZ ) solutions described as a function of crack length. The

short crack stress-intensity factor was defined as a function of

the stress concentration factor (kt) of the notch and the crack

length measured from the edge of the notch (R), r

Ks = 1.12S k (1)

where 1.12 is the free surface correction factor for an edge

crack. The long crack stress-intensity factor assumed the notch

I

has no effect on the crack behavior,

K9 WS V7 (2)

where a is the sum of the crack length°(Z) and radius (r) of thenotch; S in Equations l and 2 x.s the applied stress.

TY{o long crack and short crack stress- intensity factor

solutions for a diametrically cracked 1.0 inch diameter hole in awide plate are shown in Figure 1. Analytically, the crack length(Rm) associated with the intersectioii of the long crack and shortcrack stress- intensity factors can be expressed in terms of notchradius (r) and stress concentration factor of the notch.

R r(3)M (1. 12 kt ) 2

The Dowling crack initiation cut-off parameter (R m) given byEquation 3 is shown in Figure 2 as a function of notch severity.

1

` As the notch severity increases the initiation crack length

decreases, i.e., the notch behaves like a crack. it can also be

noted that the initiation crack length is on the order of 10 to

20% of the notch radius.

Dowling also suggested that linear elastic fracture mecha-nics parameters could be utilized beyond the initiation cut-off.

2.2 CRACK PROPAGATION MODELS

2.2.1 Elastic J-Integral (K)

The linear elastic fracture mechanics (LEFM)parameter K, the stress-intensity factor f has been found to be

an effective basis for life analysis procedures when extensive

plasticity does not occur. Localized cyclic plasticity around

the root of a notch has been observed to accelerate the growth

rate of cracks propagating from notches. E1 Haddad et al. [1],

Lein et al. [11,12], and Newman [13] have shown that the LEFM

parameter K, correlates the crack growth rate data in a mannerzwhich would lead to unconservative life predictions for small

cracks propagating from plastically deforming notches under fully

reversed loading.

„ 3

EW

1

r

r N

4 ♦'y ^1

Q Vin.^

^.; StressNormalized

K1 R Str@ss—r

' Intensityk '^5 Factor

3^1

{ 9

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Y}i_. ,eke

OF POOR

"C,'

Ks R1,.12Kt ^

4}V M

KV (R+R)

a^

!

ai-w^

yw

0.0.00 .. 02 .04 .05 .08 .10k (in.)

Crack length measured from edge of notch.

Figure 1. The Long and Short Crack Stress-Intensity FactorSolutions for a 1 inch Diametrically Cracked Hole.

4

OF POOR QUAUTY

s

dr t ^^

a ^^

F2y

^0.

8.

i kt 7.

StressConcentration ^+Factor

5.

4.

3.

2.

1.

0.r 00 .05 .10 .15 .20 .25 .30R, m (in. )

Dowling's crack initiation cutoff parameter.

Figure 2. Effect of Notch Severity on Dowling's CrackInitiation Cutoff Parameter.

R

R

^ }}ppp^^ ^e

4, t

lie

5

The advantage of using the LEFM parameter K, is

that it is a material independent property of a specific geometry.

Handbook solutions exist for many simple geometries and a series

of relatively inexpensive linear finite element runs will generatesthe solution for more complex geometries. One of the more direct

methods tfor calculating the stress-,intmn.sity factor via finiteelement analysis involves using the path independent line integralderived by Rice [141, i.e., the J-integral, which is given by

i j (Wdy - T 3—x ds) (4)r

In the elastic case, the stress-intensity factor is proportionalto the square root of the J-integral according to the formula,

a

K = J

(5)

Y

E,

a

where E is the elastic. modulus. The LEFM parameter K provides a

starting point in the life analysis of cracks propagating from3

notches due to the availability of the solution ano its acceptance

in making linear elastic life predictions.

2.2.2 Elastic-Plastic J-integral

Ratwani et al. (2) suggested that the behavior of

cracks propagating from plastically deforming notches could be

de:,cribed using the J- ,ntegral calculated to account for the

plastic behavior. The J-integral proposed for this correlation

is calculated with Equation 4 based on finite element results for

elastic-plastic material properties. Actually, once Ratwani etal. calculated the value of the J--integral under elastic-plastic

conditions for a given loading and crack length, they converted

J to a quasi-stress intensity factor (Kp) using Equation 5, i.e.,

KpJE

(5)_

which was then used for calculating crack growth behavior in the

notch region.

6 OF POOR QUALITY

2.2.3 Strain Based Stress-Intensity Factoru

El Haddad et al. (1) proposed a different type of

' plasticity enhanced stress-intensity factor to account for the

behavior of cracks propagating from notches. This approach is

based upon a strain concentration factor (kE), which is the ratio

of local strain (e) as a function of distance (x) from the notch

to the nominal gross strain (e).

e (x)kE = kE {x) = e (7)

The stress-intensity factor (KEH ) proposed by E1 Haddad et al. is

givon by

KEH = S kE V-7rT (S)

where R is the crack length measured from the notch root and kE

is evaluated for x=k (See Equation 7).n

The strain concentration factor is evaluated by

a nonlinear finite element solution of the untracked,geometry.

The El Haddad et al. stress-intensity factor is equivalent to the

LEFM K for different levels of load when the material at the notch

root does not experience plastic deformation. For stress levels

which result in plastic deformation at the notch root, the strain

concentration factor will elevate the stress-intensity factor to

account for the accelerated growth rate of the crack. For crack

lengths outside the plastic cone of the notch, the stress-intensity

factor approaches that of the LEFM long crack stress-intensity

factor. El Haddad et al. presented results which indicate this

method correlates the crack growth rate data for cracks propa-

gating from several types of notches under several loading

conditions.

a^

7

isF.fCI:fI

1

ORIGINAL PAGE 19!OF POOR QUALITY

SECTION 3

APPLICATION OF EXISTING DATA TOCRACK PROPAGATION MODELS

A small data base of crack growth rate data [15] and a

series of finite element results [16] for industrial grade buss

bar copper were available for use in analyzing the crack propa-

gation models. Fatigue crack growth rate data had been collected

using 4.0 inch wide plates with center cracks (CCP) and withradially cracked holes . (RCH). Specimen types and test stress

levels are given in Table 1. Tensile tests and incremental step

tests were conducted to obtain the monotonic and cyclic stress-

strain curves for material modeling. The finite element based

study was conducted using the MAGNA code and the material model was

obtained from the tensile test results.

3.1 ELAST'I'C J-INTEGRAL (K)

A series of elastic finite element solutions were obtained

for a crack propagating from the RCH specimen. The LEFM para-

meter K, stress-intensity factor, for this geometry was calculated

from the J-integral results of the elastic finite element solu-

tion. Figure 3 describes the finite element grids, crack tip

elements, and a J-integral path for a 1 inch crack as measured

from the edge of the 1 inch diameter,hole in the 4 - inch wide plate.

The finite element solution for the stress- intensity factor isshown in Figure 4. The Bowie [17] solution for the stressintensity factor of a crack propagating from a 1 inch diameter

hole in an infinite plate has been added for comparison.The difference between the Bowie solution and the finite

element solution for K is due to the finite width of the plate.

The RCH crack growth rate data correlated with the finite

element determined K are shown in Figure 5, along with the crack

growth rate data for the CCP specimens. The fatigue cWk grc*th

rate obtained from these specimens were correlated with the

analytical elastic K solution with a width correction factor used

in A gTM standard E647 [18].

80

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V

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as

'^ G

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UniformApplied Load

^r

crack

Coarse Finite Element Mesh

Figure 3. Finite Element Mesh Representation of a 4 inchWide Plate with a Crack Propagating from OneSide of a 1 inch Diameter Hole.

10

Crack Tip J-IntegralPath

Fine Finite Element Mesh for Crack TipRegion

ElasticFiniteElementSolution

r

r.

2.5in

1 5

a*bStress

F,. NormalizedStress—Intensity 1. 0Factor

ORIGINAL. <<:^

OF POOR QUAUTV

0.0 2 4 .B .8 1.00. 0 .Crack Length

Figure 4. The Elastic Finite Element Solution and theBowie Approximation.

11

OF Pa

XX

e^

00

U*

0

00

40

Kmax(ksi)

Figure 5. Crack Growth Rate Data Correlat3d with theElastic Stress-Intensity Factor forIndustrial Grade Buss Bar Copper Subjectedgto a Fatigue Loading with Stress Ratio (R)

r of 0.1.i^;d

12

Y

BLn CCP 2 (25.0 ksi)

B o CCP 3 (21.3 ksi)

T RCH 1 (22.0 ksi)

4 X RCH 2' (22.0 ksi)

X RCH 3 (24.2 ksi)

• RCH 4 (24.2 ksi)

20e

3

13F I

4

ix

^th

e ;^

n^

r

K - S V71a Sec (W ) (9)

where a is the-half crack length and W is the plate width.

For this material, the correlation with the finite element

elastic K is very good. The accelerated crack growth rate for

small cracks near the notch reported by ethers was not observed.

The smallest crack lengths shown in Figure 5 are on the order oZ

0.1 inch which is small compared to the calculated size of the

plastic zone shown in Figure 6.

A study by Newnan [13] suggests that crack closure is the

cause for the accelerated crack growth rate experienced by small

cracks propagating from notches. The current data base seems to

agree with this conclusion. The previously reported accelerated

crack growth rate was seen in tests run with a stress ratio of

-1 (fully reversed loading). The existing test results were for

tests at a stress ratio of 0.1, thus crack closure may not be as

important in the cracking process. We are planning further

investigation in this area.

Newman reasoned that once closure has occurred, the load

needed to open a short crack is less than that needed to open a

long crack. This would result in a greater stress range rn which

damage is being done for the short crack, thus an increased crack

growth rate.

3.2 ELASTIC-PLASTIC J-INTEGRAI

Nonlinear finite element based studies were also conducted

for cracks propagating in the RCH specimens. Figure 7 summarizes

the results of the J-integral calculations in terms of the ratio

a of "plastically" corrected stress-intensity factor K (given by

Equation 6) to the elastic stress-intensity factor, for the two

applied stress levels utilized during the test."s

r#413

ORIGINAL P `^''-` IS

^aOF POOP? QUALITY

X

g,^,^.rLi^YJs

202000.

y

>

SO100©a.

n

°fl7II.

0.

ORIGINAL Pi =- (A'-I

pools QUALITY

R24,000 PSI

22,00' PSI RCH 1RCH 2.

Plastic Zone Size As Measured From Notch (In.)

rY

Figure 6. Extent of Plastic Zone Along Expected Path ofCrack for a i inch viameter Hole in a 4 inchWide plate.

a

14

ORIGINAL ^ 4' I

QF poon QUAU-i '

AppliedStress

5 2 k ' 2 ks ----------a

22.0 ksi

hotel 1 •`elastic

0.0.0 .1 .2 .3 .4 .5 .5 .7

zCrack Length (An,)

Figure 7. Extent of Plastic Enhancement of the Stress-Intensity Factor, as Calculated by the FiniteElement Solution.

P

^t 8

IR ^

15

2.

1.

.

The "plastically" corrected stress-intensity factors basad

on Equation 6 were used to correlate the crack growth rate data

from the RCH specimens as shown in •Figure 8, Finite element

solutions for the CCP specimens are not complete; thus, the crack

growth rate data for the CCP specimens were not included in the

figure. t`

3.3 STRAIN BASED STRESS-INTENSITY FACTORr

A nonlinear finite element solution for an uncracked RCH

specimen was used as an initial, estimate of the plastic strain

near the notch. The results of this program were used to calcu-

late the strain concentration factor (kr). This factor was used

in the plastically enhanced stress-intensity parameter (K EH).

KEH Was the^'i used to correlate the crack growth rate data for the

four RCH tests. The results of this correlation, along with the

elastic long crack (CCP) data are shown in Figure 9.^, 4

The plastically enhanced stress-intensity factor, as

described by El Haddad et al., failed to correlate the crack

growth rate data from the RCH specimens with the long crack data.

An additional test was conducted to determine the validity of

E the strains calculated by the finite element analysis. The test

specimen consisted of an uncracked RCH specimen with strain gages

attached to 1-he inside of the notch. The specimen was then cycled

a' fi at a stress level similar to that used in the fatigue tests. The

experimental results indicate that the material at the notch root

3 experienced a large decrease in plastic strain after the first

cycle due to redistribution of stresses. The plastic . strain

amplitude stabilized after approximately the tenth cycle. The

finite element solution agrees with the first cycle but differs

substantially with the tenth and succeeding cycles, as shown in-

Figure 10. The strain concentration factor for the uncracked

RCH specimen, as calculated b the finite element solution, doesP ynot give an accurate description of the strain along the expected

crack path. A cyclic nonlinear finite element analysis is

'a

16

W ,

R-0. I I

to RCH 1 (22. 0 ksi)

0 RCH 2 (22.0 ksi)

4—RCH 3 (24.2 ksi)

+ RCH 4 (24.2 ksi)

2— +

r*c4o

10,

46.

r

e,l

4

ORIGINAL PAGX^ Na

OF POOR QUALITY4

daTN

(in. /cyc.)1 0 4

ws, 103Q 40 5Q 60 70

NT

/JE I—n)(KSIY'

Figure 8. Crack Growth Rate Data Correlated with the PlasticallyEnhanced Stress-Intensity Factor from the Finite ElementSolution.

^x

a

t

qfe

F

1K

ti

d

U

S{4F[

URIGI 1 t"t»r __. i t,/

OF POOR QUALM Y

RCH SPECUMNS, RM0.1

c max "" 24.2 Ks

6' Q max " 24.2 Ksi r

c max " 22.0 Ksi4 a CCP R=0.1SP25

OENS^comax

° a MAX"'= 21.3 Ksi

r

i '71

Z V_ 0da ft.

00

4 = — ''

IN.( NCLE }

8 an "-

34 cm

°

IliV

V

ry

1

10 5.___ 2

4 6

10 1K

10EHmax

(Ksi in. }

" r

Figure 9. Fatigue Crack Growth Rate Behavior of the Copper

Alloy Using the Stress-Intensity Factor Proposedby El Haddad and the Elastic Long Crack Data.

18

I

r

s.

No ."IF fLL

Pi -

20000.

Applied 15000.Stress

(psi)

10000. —

5000. —

0. 1.000

OF POOR Q: T"i I

I 10th cycle

lot cycle

finite elementresults

.005 .015

Strain (in./in.)

FIgure 10. Fxtent of Redistribution of Stresses Experiencedby the Copper and Finite Elvment Predictions ofStrain.

4n tie ij

i

i

^ S

fR

Sf

0

19

20

planned and when the data is available it will be used to

obtain a better estimate of the plastic strain for the 'Zl Haddard

model.

3.4 SUMMARY OF EXPERIMENTAL RESULTS

TNe experimental results used thus far to evaluate the

crack propagation models show that the elastic stress -intensity

factor (K) correlates the crack growth rate data for the condi-

tions tested. The elastic-plastic d^integral based (Krj and the El

Haddad et al. plastically enhanced stress-intensity factor (K EH )

are strongly dependent upon they material model used in the finite

element analysis.

Additional testing is a.equired to determine the cond itions

under which the elastic K fails to correlate the data. Tests are

planned which will specifically address the problem of cracks

from ,. o These tests will crack growthpropagating ^.^.u^^ ► no,.cho^. w ,.rate data for small. cracks (less than 0.05 inch) near notches.

Tests will be run under fully reversed loading and under a stress

ratio of 0.1, to determine the efect of crack closure on the

small crack growth rate.

E

i

J

SECTION a

SUMMARY OF WORK ACCOMPLISHED AND WORKPLANNED FOR THE NEXT PERIOD

During the past six months, the work accomplished can be

summarized as follows:

A. A literature review of current crack initiation and

propagation model for notched structures was conducted.

B. A crack initiation model and several crack propagation

models were chosen for further study.

C. Analysis, using existing crack growth rate data and

finite element solutions, was performed to evaluate the crack

apropagation models.

D. The materials for additional testing has been acquired

and the machining of the specimens has begun.

Between September 1 and 31 December 1982, the following

work will be accomplished:

A. Baseline testing of the aluminum will be conducted.

B. Life predictions using the crack initiation and crack

propagation models will be made.

{ C. Fatigue tests will be conducted on the aluminum.

D. The crack initiation and crack propagation models will

be evaluated for the aluminum.

E. A plate of INCONEL 718 will be procured.

I

^l

21

REFERENCES

4 'J

1. E1 Haddad, M. J., Smith, K. N:, and "hopper, T. li., " AStrain based Intensity Factor Soll,tion for Short FatigueCracks Initiating from Notches," Fracture Mechanics, ASTMSTP 577, C. W. Smith, Ed., American Society for Testing andMatezials, 1979, pp. 274-289.

2. Ratwani, M. M., "Fatigue Crack Growth at Stress Concentra-tions Subject to Strains Beyond Elastic Range," J. ofAircraft, Vol. 18, No. 3, March 1981, pp. 206--212.

3. Dowling, N. E., "Fatigue at Notches and the Local Strainand Fracture Mechanics Approaches," Fracture Mechanics, ASTMSTP 677, C. W. Smith, Ed., American Society for Testing andMaterials, 1979, pp. 247-273.

4. Dowling, N. E. and W. K. Wilson, "Geometry and Size Require-ments for Fatigue Life Similitude Among Notched Members,"Advances in Fracture Research, D. Francois, et al., Ed.,Pergamon Press, 1980, pp. 581-588.

5. Dowling, N. E., "Notched Member Fatigue Life PredictionsCombining Crack Initiation and Propagation,"' Fati gue ofEngineering Materials and Structures, Vol. 2 1 pp. 129-138,1979.

6. Hanunouda, M. M. and Miller, K. J.., "Elastic-Plastic FractureMechanics Analysis of Notches," Elastic-Plastic Fracture,ASTM STP 668, American Society for Testing and Materials,1979, pp. 703-719.

7. Kurath, P., Socie, D. F., and Morrow, J., "A Nonarbitrary'atigue Crack Size Con4ept to Predict Total Fatigue Lives,"AFFDL-TR-79-3144, Air Force Flight Dynamics Laboratory,Wright--Patterson Air Force Base, Ohio, Dec. 1979.

8. Duggan. T. V., "Fatigue Integrity Assessment," Int. J. ofFatigue, April 1981, pp. 61-70

9. Wood, H. A. and Rudd, J. L., "Evaluation of Small Cracks inAirframe Structures," AGARD Publication, CESME, Turkey,April 1981.

10. Liu, A. F., "Crack Growth at a Preyielded Hole," Papersubmitted to J. of Engineering Mat'ls and Tech. forpublication 1981.

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11. Leas, B. N., Galliher, R. D., "Growth of Physically ShortCorner Cracks at Circular Notches," Battelle ColumbusLaboratories, September 1980.. .

12. Leas, B. N., "Fatigue Crack Propagation Through IynelastieGradient Fields," Battelle Columbus Laboratories, May 1981.

13. Newman, J. C., Jr., "A Nonlinear Fracture Mechanics Approachto+the Growth of Small Cracks," Presentation at the AGARDSpecialists Meeting on Behavior of Short Cracks in AirframeComponents, September 20-21, 1982. •

r

14. Rice, J. R., "A Path Independent Integral and the Approxi-mate Analysis of Strain Concentration by Notches and Cracks,"Transactions of the ASME, Journal of Applied Mechanics,Volume 34, 1968, pp. 379-386.

- 15. Unpublished Test Results, Air Force Contract No, F33615-78-C-5184, July 1982.

16. Unpublished Finite Element Results, Air Force Contract No.F33615-78-C-5184, July 1982.

17. Bowie, O. L., Journal of Mathematics and Physics, Vol. 35,' 1956, pp. 60-71.

18. "Standard Test Method for Constant-Lead-Amplitude FatigueCrack Growth Rates Above 10- 8 m/cyc," ASTM E647-81, pp. 765-783.

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