AFWAL-TR-82-4040o VOLUME II(N)

"STATISTICS OF CRACK GROWTH INI ENGINE MATERIALS

Volume II Spectrum Loadingand Advanced Techniques

C. 0. Annie, Jr.

Pratt A Whitney Aircraft

West Palm Beach, Florida 33402

Q. C. SallvarFlorida Atlantic UniversityBoca Raton, Florida 3431

J. N. YangThe George Washington UniversityWashington, D.C. 20012

February 1984

Final Report for Period 15 January 1981 - 30 November 1982

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STATISTICS OF CRACK GROWTH IN ENGINE Final for PeriodMATERIALS 1___Jan_81_- __30_Nov_82

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B. J. Schwartz G. C. Salivar F33615-80C.5189C. G. Annis, Jr. J. N. Yang

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Statistical Fatigue Crack Growth Spectrum LoadingEngine Materials Lognormal DistributionElevated TemperatureFracture Mechanics

110. A88TRACT (Ccelhntir an revere aide It noooee.r and IddeidIl by bMock namberw)

)A method is developed capable oincorporating the scatter observed in crack growth data intocomponent residual life analyse., s ch an used in damage-tolerant and retirement-for-cause concepts.Two fracture mechanics-based at tiotical models for fatigue crack growth damage accumulation inengine materials are proposed. Ti are based on synergistic fracture mechanics models: a hyperbolicmine crack growth rate function'~eeoe "yPrt' ' jhte Arrft" a linear crack growthrate function of the Paris type.DDI FOAmRM 1473 EDITION or I No 6 s O*ST

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Test results of IN100 at various temperatures, loLding frequencies and stress ratios were compiledand analyzed statistically. The statistical distributions of crack growth rate, propagation life to reachany given crack size, and crack sise at any service life, are derived. Correlation between the IN100 testresults and the two statistical models is very good.

Test results for IN100 under block type spectrum loading under various test conditions were usedwith a lognormal statistical model to predict statistical distributions of life to reach any given cracksize, and of the crack size at any given service life. Good correlation is shown.

Available fatigue crack growth rate data for Ti 6-2-4-6 and Waspaloy under various test conditionsare also analyzed statistically. The lognormal statistical model, using either the Paris or hyperbolicsine crack growth rate function, gives good correlation with extrapolated test results. When applied toa collection of 7475-.7361 aluminum fastener hole specimen data, the Paris relation does not givereasonable correlation.

Finally, a statistical theory, based on the concept of fracture mechanics and random processes, isproposed for the analysis of fatigue crack propagation. Examples are presented for the case where thecrack grwth rate is governed by a power law. Parameters are estimated from experimental data from

7475-T7351 aluminum fastener hole specimens subjected to an aircraft bomber load spectrum.

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PREFACE

An investigation of the statistical nature of fatigue-crack growth rate data, resulting in theincorporation of the observed scatter into component residual life analyses, was conducted fromFebruary 1981 through January 1983, The authors wish to acknowledge the assistance of severalPratt & Whitney Aircrafte.mployees who made significant contributions to the program. Theseinclude Messrs. M. L. Poormon, E. H. Hindle III, C. K. Kraft, G. Scott and D. W. Ogden whoaided in the verification testing, and T. Watkins, Jr. who aided in data analysis.

This work was performed under Materials Laboratory Contract F33615-80-C-5189,"Statistics of Crack Growth in Engine Materials," The program wao conducted by theMechanics of Materials and Structures section of Pratt & Whitney Aircraft, GovernmentProduct. Division, under the cognizance of Dr. R. C, Donath, AFWAL/MLLN. Mr. B. J.Schwartz and Dr. G, C, Salivar were the responsible engineers and Mr. C. G. Annim, Jr. was theprogram manager reporting to Mr. M. C. VanWanderham, Manager, Mechanics of Materialsand Structures. Professor J. N, Yang, of The George Washington University, Washington, DC,was the principal investigator.

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iii

TABLE OF CONTENTS

Section Page

1 INTRIODUCTION ............................................. ................. 1

11 FATIGUE CRACK GROWTH UNDER SPECTRUM LOADING CON-D IT IO N S ................................................................. .............. 3

1. Theoretical Model ................. ................... 32. Experimental Program ..................................................... 73. Correlation Between Statisticel Model and Experimental Test

R esults ......................................................................... 10

III LOGNORMAL STATISTICAL MODEL APPLIED TO THE PARIS

EQUATION FOR IN100 ........................................................... 14

1. Correlation With IN100 Test Results ................................ 21

IV ANALYSIS OF TI 6-2-4-6 AND WASPALOY DATA ..................... 33

V STATISTICAL ANALYSIS OF A HOMOGENEOUS DATA SET ...... 60

VI ADVANCED STATISTICAL MODEL FOR FATIGUE CRACKGROWTH ........................................................ ............. 65

1. Model for X(t) ............................................ 662. Approximation of 1 (t) by a Markov Random Process ........... 683. Time to Reach a Given Crack Size ................................... 704. Power-Law Crack Propagation .......................................... 725. Estimation of Model Parameters ....................................... 74

VII CONCLUSIONS ...................................................................... 83

1, Spectrum Loading .......................................................... 832. Related Developments ..................................................... 83

REFERENCES ........................................................................ 84

APPEN D IX ............................................................................ 86

UIAIK-aft n

v

LIST OF ILLUSTRATIONS

Figure Page

1 Representation of One Cycle Block ............................................. 4

2 'y Percentiles of Crack Length Versus Number of Cycles ................. 6

a Laboratory Spectrum Test Loading Conditions for IN 100............... -7

4 Verification Test Results for IN100 .............................................. 9

5 Predicted Crack Size Versus Cycles Behavior Under Spectrum Loading 10

6 Distribution of Cycles to Reach 0.8 Inch, for the IN100 Mission Veri-fication T est ........................................................................... 11

7 Distribution of Cycles to Reach 1.4 Inches for the IN100 MissionV erification T est ....................................................................... 11

8 Probability of Crack Exceedance at 24,000 Cycles for IN100 MissionVerification Testing ...................... .......... ...... ......................... 12

9 Probability of Crack Exceedance at 35,000 Cycles for IN100 MissionVerification Testing ................................................................. 12

10 Data for Crack Growth Rate Versus Stress Intensity Range for TestCondition No. 1 .......................... ................. 16

11 Normal Probability Plot for Z (Test Condition No. 3) ..................... 17

12 Percentiles of Log Crack Growth Rate as a Function of Log StressIntensity Range for Test Condition No. 1 .................................... 18

13 Distribution of Crack Size an Function of Load Cycles Based on Sta-tistical Model; (a) Test Condition No. 1, (b) Test Condition No. 2 .... 20

14 Distribution of Cycles To Reach Given Crack Size for Test Cor itionNo. 1; (a) a = 1.0 Inch, and (b) a - 2.0 Inches ............................ 22

15 Crack Exceedance Curve After 25,000 Cycles for Test Condition No. 1 23

16 Distribution of Cycles To Reach Given Crack Size for Test ConditionNo. 2; (a) a = 0.8 Inch, and (b) a = 2.0 Inches ............................ 24

17 Crack Exceedance Curve after 10,000 Cycles for Test Condition No. 2 25

18 Distribution of Cycles To Reach Given Crack Size for Test ConditionNo. 3; (a) a = 0.76 Inch, and (b) a - 1.7 Inches ........................... 26

19 Crack Exceedance Curve after 40,000 Cycles for Test Condition No. 3 27

S. L'v.

LIST OF ILLUSTRATIONS (Continued)

Figure Page

20 Distribution of Cycles to Reach Given Crack Size for Test ConditionNo. 4; (a) a = 0.8 Inch, and (b) a = 2,0 Inchep ............................. 28

21 Crack Exceedance Curve after 10,000 Cycles for Test Condition No. 4 29

22 Distribution of Cycles to Reach Given Crack Size for Test ConditionNo. 5; (a) a - 0.8 Inch, and (b) a = 2.0 Inches ....................... 30

23 Crack Exceedance Curve after 8,000 Cycles for Test Condition No. 5 31

24 Extrapolated Test Results for a Versus n; (a) Test Condition No. 1and (b) Test Condition No. 2 .................................................... 32

25 Crack Growth Rate Model for Waspaloy at 12000 F, 20 Hz, and R =0.05 ............................................................. 35

26 Crack Growth Rate Model for Titanium 6-2.4-6 at Room Temperature,20 Hz, and R = 0.1 ........................................................... ..... 36

27 Distribution Curve and Crack Exceedance Curve for Waspaloy for TestCondition No. 1; (a) Distribution of Cycles to Reach Given Crack Size,and (b) Crack Exceedance Curve After 50,000 Cycles ...................... 39

28 Distribution Curve and Crack Exceedance Curve for Waspaloy for TestCondition No. 2; (a) Distribution of Cycles to Reach a Given CrackSize, and (b) Crack Exceedance Curve After 10,000 Cycles ............... 40

29 Distribution Curve and Crack Exceedance Curve for Waspaloy for TestCbndition No. 3; (a) Distribution of Cycles to Reach Given Crack Size,and (b) Crack Exceedance Curve After 50,000 Cycles ...................... 41

30 Distribution Curve and Crack Exceedance Curve for Titanium for TestCondition No. 1, Using the Hyperbolic Sine Model; (a) Distribution ofCycles to Reach Given Crack Size, and (b) Crack Exceedrnce CurveAfter 30,000 Cycles .................................................................. 42

31 Distribution Curve and Crack Exceedance Curve for Titanium for TestCondition No. 2, Using the Hyperbolic Sine Model; (a) Distribution ofCycles to Reach Given Crack Size, and (b) Crack Exceedance CurveAfter 20,000 Cycles ................................................................. 43

32 Distribution Curve and Crack Exceedance Curve for Titanium for TestCondition No. 3, Using the Hyperbolic Sine Model; (a) Distribution ofCycles to Reach Given Crack Size, and (b) Crack Exceedance CurveA fter 5,000 Cycles ..................................................................... 44

vii

?I

LIST OF ILLUSTRATIONS (Continued)

FA re Page

33 Distribution of Cycles to Reach Given Crack Size for Titanium forTest Condition No. 4, Using the Hyperbolic Sine Model; (a) Distribu-tion of Cycles to Reach Given Crack Size, and (b) Crack ExceedanceCurve After 5,000 Cycles ........................................................... 45

34 Distribution Curve and Crack Exceedance Curve for Titanium for TestCondition No. 5, Using the Hyperbolic Sine Model; (a) Distribution ofCycles to Reach Given Crack Size, and (b) Crack Exceedance CurveAfter 100,000 Cycles ................................................................. 46

35 Titanium Crack Growth Rate Spocimen at Test Condition No. 1 ....... 48

36 Distribution Curve and Crack Excesdance Curve for Titanium for TestCondition No. 1, Without Specimen No. 1582; (a) Distribution of Cy-cles to Reach Given Crack Size, and (b) Crack Exceedance CurveAfter 30,000 Cycles .................................................................. 49

37 Distribution Curve and Crack Exceedance Curve for Waspaloy for TestCondition No. 1, Using the Paris Model; (a) Distribution of Cycles toReach Given Crack Size, and (b) Crack Exceedance Curve After 50,000Cycles .................................................................................... 52

38 Distribution Curve and Crack Exceedance Curve for Waspaloy for TestCondition No. 2, Using the Paris Model; (a) Distribution of Cycles toReach Given Crack Size, and (L) Crack Exceedancc Curve After 10,000Cycles ................................................................................... . 53

39 Distribution Curve and Crack Exceedance Curve for Waspaloy for TestCondition No. 3, Using the Paris Model; (a) Distribution of Cycles toReach Given Crack Size, and (b) Crack Exceedance Curve After 50,000Cycles ................................................................................... 54

40 Distribution Curve and Crack Exceedance Curve for Titanium for TestCondition No. 1, Using th-.e Paris Model; (a) Distribution of Cycles toReach Given Crack Size, and (b) Crack Exceedance Curve After 30,000Cycles ................................................................................... 55

41 Distribution Curve and Crack Exceedance Curve for Titanium for TestCondition No. 2, Using the Paris Model; (a) Distribution of Cycles toReach Given Crack Size, and (b) Crack Exceedance Curve After 20,000Cycles .................................................................................... 56

42 Distribution of Cycles to Reach Given Crack Size for Titanium forTest Condition No. 3, Using the Paris Model; (i) Distribution ofCycles to Reach Given Crack Size, and (b) Crack Exceedance CurveAfter 5,000 Cycles .................................................................... 57

viii

', . .. • ... .• .r• ; o • • ,.• ... . .

LIST OF ILLUSTRATIONS (Continued)

Figure Page

43 Distribution of Cycles to Reach Given Crack Size for Titanium forTest Condition No. 4, Using the Paris Model; (a) Distribution ofCycles to Reach Given Crack Size, and (b) Crack Exceedance CurveAfter 5,000 Cycles ................................................................... 58

44 Distribution Curve and Crack Exceedance Curve for Titanium for TestCondition No. 5, Using the Paris Model; (a) Distribution of Cycles toReach Given Crack Size, and (b) Crack Exceedance Curve After100,000 Cycles ......................................................................... 59

45 Homogeneous Data Set of Virkler, et al. with P.. = 23.35 kn and R= 0 .2 ...................................................................................... 60

46 Lognormal Statistical Model Prediction of Virkler, et al. Data Set ..... 61

47 Distribution Function of Number of Cycles to Reach 21mm Half-CrackLength ................................................................................... 62

48 Distribution Function of Number of Cycles to Reach 49.8mm Half-

Crack Length ................................................. 63

49 Crack Exceedance Curves at 150,000 Cycles .................................. 64

50 Actual Crack Propagation Time-Histories of Some WPB FastenerH oles ..................................................................................... 66

51 Typical Sample Function of Random Process X(t) .......................... 67

52 Autocorrelation Function of Random Process Y(t) .......................... 69

53 Regression Analysis for the Estimation of Model Parameters for WPBFastener H oles ......................................................................... 7A

54 Theoretical Mean and Standard Deviation of Random Time to ReachVarious Crack Sizes Computed for Different A Values for WPB Fas-tener H oles ............................................................... .............. 77

55 Mean and Standard Deviation of Random Time to Reach VariousCrack Sizes Computed Directly from Some Actual Time-Histories ofW PB Fastener Holes ................................................................ 77

56 Simulated Sample Functions of Crack Propagation Time-Histories forW PB Fastener Holes ................................................................ 78

57 Comparison Between Weibull-Type Approximation for the Distributionsof Random Time tr% Reach a Given Crack Size and Actual Test Re-suits for WPB Fastener Holes .................................................... 79

ix

qn.............................. • ............ I

LIST OF ILLUSTRATIONS (Continued)

ftur Page

58 Actual Crack Propagation Time-Histories of Some XWPB FastenerH oles ..................................................................................... 80

59 Regression Analysis for the Estimation of Model Parameters forXW PB Fastener Holes .............................................................. 80

60 Theoretical Mean and Standard Deviation of Random Time to ReachVarious Crack Sizes Computed for Different A Values for XWPB Fas-tener H oles ............................................................................. 81

61 Mean and Standard Deviation of Random Time to Reach VariousCrack Sizes Computed Directly from Some Actual Time-Histories ofXW PB Fastener Holes .............................................................. 81

62 Comparison Between Weibull-Type Approximation for the Distributionof Random Time to Reach a Given Crack Size and Actual SampleDistribution of XWPB Fastener Holes ........................................ 82

63 Simulated Sample Functions of Crack Propagation Time-History forXW PB Fastener Holes ............................................................... 82

x

1 71

LIST OF TABLES

Table Page

1 Verification Test Results for IN100 .............................................. 8

2 Parameter Values C2, C. and C4 as Well as Standard Deviation a . ..... 10

3 Maximum Likelihood Estimate of b, Q, Standard Deviation ay andCoefficient of Variation of da/dn ....... ................................. 16

4 Assumed Homogeneous Test Environments for Test Specimens 19

5 Specimen Geometry and Maximum Load for Each Test Specimen of"Ti 6-2-4-6 ................................... ...................... 33

6 Specimen Geometry and Maximum Load for Each Test Specimen ofW aspaloy ................................................................................. 34

7 Maximum Likelihood Estimate of C2 , C3 , C4, Standard Deviation p =

a, and Coefficient of Variation, V, of da/dn for 'ri 6-2-4-6 (HyperbolicSine F unction) ......................................................................... 34

8 Maximum Likelihood Estimate of C2, Ca, C, Standard Deviation aya. and Coefficient of Variation, V, of da/dn for Waspaloy (HyperbolicSine Function) ........................ .............................................. 34

9 Parameters C,, Ca, and C4 for Each Test Specimen of Titanium WithC, = 0.7, Hyperbolic Sine Crack Growth Rate Function ................... 37

10 Parameters C,, Cp, and C4 for Each Test Specimen of Waspaloy WithC, = 05, Hyperbolic Sine Crack Growth Rate Function ................ 37

11 Assumed Homogeneous Test Environments for Each Test Conditionfor W aspaloy ........................................................................... 38

it12 Assumed Homogeneous Test Environments for Each Test Condition

for T itanium ........................................................................... 47

l.. 13 Maximum Likelihood Estimate of b, Log Q and Standard Deviation a.a- y for W aspaloy (Paris Function) .............................................. 50

14 Maximum Likelihood Estimate of b, Log Q and Standard Deviation a,- ay for Titanium (Paris Function) ............................................. 50

15 Parameters b and Log Q for Each Test Specimen of Waspaloy, (ParisFunction) .................................................... 50

16 Parameters b and Log Q for Each Test Specimen of Titanium (ParisF unction) ................................................................................ 51

17 Maximum Likelihood Estimate of C2, Ca, C4, Standard Deviation a =

(r. and Coefficient of Variation, V, of da/dn; C1, 0.5 ...................... 61

xi

8ECTION I

INTRODUCTION

Engine components have traditionally been designed using a crack initiation criterion.This approach has been very successful from a safety standpoint, but the conservatism inherentin this method has resulted in poor utilization of the intrinsic life of the component. Thedevelopment of high temperature fracture mechanics has permitted basing residual life analyseson a crack propagation criterion. Using this approach, lives from a specifed defect size can becalculated and combined with periodic inspection to determine component retirement.

Propagation analyses classically employed in residual life predictions are deterministicallybased. They typically account for materials scatter by the incorporation of a safety factor. Amore rigorous treatment of materials scatter is necessary to permit maximum utilization ofcomponent life.

The objective of this program was to develop a methodology capable of incorporating thescatter observed in crack growth data into component residual life analyses. Such a methodologyis desired for use in damage-tolerant and retirement-for-cause (RFC) concepts. This objectiveincluded: identification of the distribution functions which best describe crack growth rate(da/dn) behavior; characterization of fatigue-crack progagation (FCP) controlling parameters asto their effects on crack growth rate variability; examination of the correlations between thisvariability and propagation life distributions; and development of a generic methodologyapplicable to all engine materials, although particular program emphasis was placed on IN100,Waspaloy, and Ti 6-2-4-6.

Two fr-. ire mechanics-based statistical models for fatigue crack growth damageaccumulation in engine materials were proposed and investigated (Reference 1). These modelswere based on hyperbolic sine crack growth rate functions developed by Pratt & WhitneyAircraft (Reference 2). Test results of IN100 (a superalloy used in the F100 engine) at variouselevated temperatures, loading frequencies, stress ratios, etc., were compiled and analyzedstatistically. The statistical distributions of (1) crack growth rate; (2) propagation life to reachany given crack size; and (3) crack size at any service life, were derived and reported. It wasdemonstrated that the correlation between the IN100 test results and the two statistical modelswas very good.

In this report, the lognormal statistical model (Reference 1) is extended to the case ofspectrum loading. Test results for IN100 under block type loading are generated and used toevaluate the capability of this model for predicting the statistical distributions of (1) the life toreach any given crack size; and (2) the crack size at any given service life. A good correlation isshown between the model and the test results.

The lognormal statistical model is also applied to a power law (linear) crack growth ratefunction (i.e., Paris equation). Test results for IN100 under various conditions used to evaluatethe hyperbolic sine function (Reference 1) are also used to eval,.iate the Paris function. Thismodel is only applicable within a certain range of stress intensity and care should be exercised inits application. The Paris model proves to be mathematically simple for practical engineeringapplications.

Available fatigue crack growth rate data for Ti 6-2-4-6 and Waspaloy under various testconditions are also analyzed statistically. The lognormal statistical model, along with either theParis or hyperbolic sine crack growth rate function, is shown to corre!ate well with the testresults.

The lognormal model, using both the hyperbolic sine and the Paris equations, is applied to

the homogeneous data set of Virkler, et al. (References 3 and 4). It is shown that the Parisrelation does not provide a reasonable representation of this data set. Finally, a new statisticaltheory is proposed for the analysis of fatigue crack propagation. This theory is based on theconcept of fracture mechanics and random processes. Exawples are presented to demonstratethe application of the new theory using available test results.

2

SECTION II

FAT;GUE CRACK GROWTH UNDER SPECTRUM LOADING CONDITIONS

Two statistical fracture mechanics-based models for predicting the fatigue crack propaga-tion life or engine materials under any single test condition have been proposed and investigatedin Reference 1. These models are based on the hyperbolic sine crack growth rate functiondeveloped by Pratt & Whitney Aircraft. It has been demonstrated that the correlations betweenthe proposed statistical models and the test results are very good.

A single test condition is defined by a single constant value of each of the followingparameters: temperature T, loading frequency Y, stress ratio R and maximum load P,.. Serviceloading spectra to engine components, however, consist of variable histories of temperature T,frequency Y, loading magnitude R and P.., and holding time Tb. The objectives of this sectionare: (1) to extend a statistical model proposed in Reference 1 for any single test condition to becapable of predicting the fatigue crack propagation of engine materials under spectrum loadings,(2) to conduct experimental tests using IN100 compact tension specimens subject to loadingspectra in order to generate statistically meaningful data, and (3) to correlate the test resultswith the proposed statistical model for verification purposes.

1. Theoretical Model/

rhe lognormal crack growth rate model proposed in Reference 1 for a single test conditionis expressed as

Y - Clsinh[Cg(X+CA)] + C4 + Z (1)

in which C1 is a material constant, C., C, and C4 are functions of test conditions (T, P, R) andda

Y -log- , X--logK (2)

where AK is the stress intensity range. In Equation 1, Z is a normal random variable with zeromean and standard deviation o, that also depends on the test conditions (see Reference 1).

In order to extend the statistical model given by Equation 1 to account for the crackpropagation under spectrum loadings, the composition of loading spectra will be described asfollows: A loading spectrum is assume I to consist of repeated identical cycle blocks (or missionsor duty cycles). Each cycle block (or duty cycle) is composed of m different segments (or m testconditions). Each segment or test condition is defined by constant values of temperature T,loading frequency v, stress ratio R, maximum load P." and numbers of load cycles n. Therefore,the jth segment in one cycle block (or duty cycle) can be denoted by (T1, Yj, Rý, Pj, nj) whereasone cycle block consists of (Tj, Pj, Rj, Pica, nj) for j =1,2,...,m, as schematically shown in Figure 1.

For the jth test condition, or segnient, in one cycle block, it follows from Equation 1 that

-iYj - C inh[C2j(X+C2 j)I + C4j + Zj ; j-1,2,...,m (3)

in which Z is a normal random variable with zero mean and standard deviation a. Values of C•,C31, C41 and a. for different test conditions are given in Reference 1.

3

T4

T2 TImo IIT I I I I

Cycles, n

FD 25079

Figure 1. Representation of One Cycle Block

Since Zj is a statistical variable, a specimen having a faster crack growth rate will preserveit over the entire crack propagation life in the jth test condition, and the variability of Zj isexclusively contributed by the material variability (loading is deterministic) as described inReference 1. Hence, under spectrum loadings Zj for jfi,2,3,,..,m are completely correlated notonly in one cycle block but also over the entire crack propagation life. Consequently, Equation 5can be integrated segment-by-segment not ordy in one cycle block but also over the entirepropagation life to obtain the crack length a(n) as a function of the number of load cycles, n (orcycle blocks).

For simplicity, the crack propagation in the first cycle block is illustrated by the following.Let zj, be the y percentile of the normal random variable Z, associated with the jth segment.

oy% - P[Zj>zh] - 1 - 6(z1/C.-j)

in which 4 ( ) is the standardized normal distribution function and a, is the standard deviationof Zj. The determination of r was described in Reference 1.

The y percentile, z,, of Zj can be obtained from Equation 4 as

zjY - J-l-' (1 - y %) for jl,2,...,m

where - () is the inverse standardized normal distribution function.

4

T *• •''- ... ................. I *' " ............ ,- ,.... ,•';.'• •',,•:•,• ,.•= ••'•' Ir"'-'-•'•'v•.•l•i•' -,, :. ..

The y percentile of the crack growth rate for the jth segment, denoted by Y,, follows fromEquation 3 as

Yh - CIsinh[C~j(X + Caj)] + C4J + zh for j-1,2,...,m (6)Then, Equation 6 can be integrated segment-by-segment to yield the y percentile of the cracklength a7 (n) versus the number of load cycles n as follows:

a,(n,) - a, + E A+",

,a(n, + n2) - aY (n) + Aa(7) k-I (7)

a, n,. a, nn + ÷ Aa.J-L iI- k-I

in which a. is the initial crack size, a7 (n1) is the y percentile crack length after the first segmentthat consists of n, load cycles, and Aal.k is the increment of the crack length during the kth loadcycle in the first segment which is computed numerically using Equation 6 for j-I as follows:

Y1, - C s, inh [C gi(X + C o.)] + C41 + z j, (8)

In Equation 8, z, is determined from Equation 5 with j-1. Other notations appearing inEquation 7 are self-exp;anatory for segment by segment numerical integrations.

Repeating numerical integrations similar to Equation 7 and using appropriate initial cracklengths for each segment and cycle block, we obtain one -y percentile for the crack length, a T (n),as a function of load cycles n. Furthermore, by varying values of 'Y, we obtain the distribution ofthe crack length as a function of load cycles as shown in Figure 2. It should be noted that thenumerical integration to generate a set of a7 (n) is deterministic, straightforward, and verysimple.

The approach described above recognizes the difference of the statistical dispersion of thecrack growth rate, da/dn, for each segment (or test condition). However, since the spectrumloading is idealized by repeated identical cycle blocks, it may be reasonable to average thestatistical dispersion of the crack growth rate for each segment over one cycle block and toapproximate the statistical dispersion of the crack growth rate for one cycle block by such anaverage value. Such an approximate approach is described in the following:

The coefficient of variation of the crack growth rate Vj in the jth segment is related to Y,S~ through

Vj -[e(1h10)' 1129(9)

* Then, the coefficient of variation, V, of the crack growth rate for one cycle block isapproximated by the weighted average of Vj associated with all segments in one cycle block, i~e,,

V • Vj nj/I njJ.1 J-1 (10)

5

and Equation 3 is approximated by

Yj - C, sinh [Cj(X+C3 j)] + C~j + Z

in which Z is a normal random variable with zero mean and standard deviation a, which isrelated to V (Equation 10) through

- f +T /nio (12)

1% 10% 25% 50% 75% 90% 95%

a;

Cycles, n

FD 25MU

Figure 2. -y Percentiles of Crack Length Versus Number of Cycles

Therefore, the statistical dispersion of the crack growth rate over the entire loadingspectrum is characterized, in approximation, by the random variable Z.

Applying the same procedures described previously, with all Z (j=1,2,...,m) being replacedby Z, one dan obtain the distribution of the craok length as a function of load cycles shown inFigure 2.

To account for the scatter of crack propagation life in the prediction of engine components,two distributions are of practical importance: (1) the distribution FN(.1)(n) of the propagationlife N(a1) to reach any given crack length a, (including the critical crack length), and (2) thedistribution, F..()(x), of the crack length a(n) after any specific number, n, of load cycles, In fact,the distribution of the crack length as a function of load cycles shown in Figure 2 contains all theinformation mentioned above. For instance, by drawing a horizontal line through any specificcrack length a, in Figure 2, one obtains the distribution, FN(s,)(n), of the number of load cyclesto reach that crack length. Likewise, the distribution, F.1 .)(x), of the crack length a(n) after agiven number, n, of load cycles can be obtained by drawing a vertical line through n in Figure 2.

A computer program based on the above theoretical model (Equations 6-8) has beenestablished to predict the crack growth damage accumulation for any given loading spectrum.

0.. :

No attempt was made in this program to account for retardation or acceleration effects,although the statistical model developed can be modified to take these effects into account.

2. Experimental Program

An experimental test program has been carried out in order to verify the capability of theproposed statistical model in predicting the crack propagation behavior of engine materialsunder spectrum loading, The specimens used were ASTM compact tension specimens with athickness (B) of 0,5 inches (12.7 mm) and width (W) of 2.5 inches (63,5 mm). The initial flawsize (ao) is 0,5 inches (12.7 mm) and the final flaw size (as) is 2.0 inch (60,8 mm).

The spectrum loading employed wrn composed of repeated block loading (or cycle block)consisting of four test conditions (segmsxt. as shown in Figure 3. Each segment was applied for1,000 cycles with the next test condition in esccession. The entire block was repeated after testcondition number 4. This procedure was contin aed until specimen fracture occurred.

Test Test Test TestCondition Condition Condition Condition

1 ,,, 2 _ _ 3 _ • 4

Pmax

Load, -P

Start Over

Cycles, n

Test Condition 1 1200 0 F, 10 cpm, R - 0,1 1,000 CyclesTest Condition 2 13500F, 10 opm, R = 0.1 1,000 CyclesTest Condition 3 1200 0 F, 10 cpm, R A 0,5 '1,000 CyclesTest Condldon 4 1000 0 F, 10 cpm, R - 0,1 1,000 Cycles

FD 247364

Figure 3. Laboratory Spectrum Te'st Loadi,ig Conditiona for IN100

7

In one block loading, the maximum load, Pm, for each segment was fixed at 2.4 kips(10.7 kn) with all positive stress ratios, R, in order to eliminate the need for consideration ofretardation or acceleration effects due to overloads or underloads. The results of twelve tests aresummarized in Table 1, and displayed in Figure 4. A complete listing of these tests is shown inthe Appendix. The three solid curves shown in Figure 4 are the theoretical predictions that willbe described later.

TABLE 1. VERIFICATION TEST RESULTS FORIN100

FinalCrack Cycles to

Specimen Length Failure Cycles atNo. ao N 1.4 inches

1. 2396 1.74 100,000 97,5002. 2W97 1.68 65,000 62,0003. 2398 1.68 68,000 65,8004. 2399 1.43 73,000 72,8005. 240 1.84 70,000 68,5006, 2401 1.48 97,000 96,5007. 2402 1.63 68,000 82,0008, 2403 1.63 75,000 73.3009, 2404 1.59 115,000 113,60010. 2408 1.41 81,000 81,00011. 2406 1.53 81,000 80,00012, 2407 1.45 85,000 84,500

Failure: Mean Life (N) - 82,900Standard Deviation - 14,800Moan Life at 1.4 in. - 81,500Standard Deviation - 15,000

For theoretical predictions the parameter values C2, Caj and C,,, as well as the standarddeviation a4, of Z, under each test condition (or segment) in one block loading, Figure 3, havebeen estimated using the method of maximum likelihood in Reference 1. The results are shownin Table 2 along with the total number of test specimens used. The parameter C, is a materialconstant which is equal to 0.5 for IN100, Note that the parameter values presented in Table 2were estimated using compiled IN100 crack growth rate data, which wore generated over a longperiod of time for different purposes (Reference 1). The total number of test specimens undereach test condition is very small. Even with such a small sample size, the specimen dimensions,the initial and final flaw sizes, and the maximum cyclic load for each test specimen are alldifferent. Thus, the data base used for estimating the parameter values is highly nonhomoge-neous; however, it is representative of the expected data base found in industry (seeReference 1).

Using the statistical model described in the previous section, Equation 7, and Table 1, weconstruct various 7 percentiles for the crack size, a (n), as a function of the number of loadcycles n. The results are shown in Figure 5 as solid curves and the numerical w lues arepresented in the Appendix. For instance, the curve associated with 'Yl0 indicates that theprobability is 10% that a specimen will have a crack size growing faster than that denoted bythe curve. These solid curves are the predicted distribution of crack gruwth damageaccumulation based on the statistical model. The crack propagation associated with y=5, 50 and95 percentiles are also shown in Figure 4 as solid curves for the purpose of comparison with testresults.

i8

IWD

co z

r - 6 6 ý -*ukD 0 lasgj

tam

TABLE 2. PARAMETER VALUES C,2 C3 AND C4 AS WELL ASSTANDARD DEVIATION,

Test Number ofSegment Condition C, CO C' as Specimens

1 5 3,8982 -1.5376 -3,9341 0.1026 92 2 4.9323 -1.4073 -3.9895 0.1692 43 3 4.3098 -1.3032 -4.4450 0.1240 44 1 3,8033 -1.5239 -4.3563 0.1673 5

9A -'5% 10% 2.% 50% 76% 90% 05%2,0

1.4

S1,40

0.41

0''

0 20,000 40,000 60,000 60,000 i00,000 120.000 140,000 160,000cycift. n FD 251640

Figure 5. Predicted Crack Size Versus Cycles Behavior Under Spectrum Loading

3. Correlation Between Btatlsticosl Model and Experimental Test Results

As observed from Figure 4, the statistical model predicts faster average crack growthdamage accumulation and slightly larger statistical dispersion. Thus, from the analysis anddesign standpoint, the statistical model is conservative. The correlation between the test resultsand theoretical prediction appears to be very reasonable.

The theoretical predictions for the statistical distribution of the number of load cycles toreach crack sizes 0.8 and 1.4 inches (20.3 and 35.6 mm) are presented as solid curves in Figures 6and 7, respectively. Also plotted in these figures as circles are the experimental test results fortwelve specimens. Furthermore, predictions for the probability of crack exceedance, which isequal to one minus the distribution function of the crack size, at 24,000, and 35,000 load cyclesare displayad ia Figures 8 and 9 as solid curves, respectively. Also shown in these figures ascircles are the corresponding experimental test results. Again, Figures 6 to 9 indicate that thestatistical model is slightly conservative, in the sense that the model predicts shorter crackpropagation life and larger statistical dispersion. The correlation between the theoretical modeland the test results is very reasonable.

10

S. ... ... ... ......u- -

1.00

0.8 0

Ho .6- 0:

0.4

0

0.2 00 a -0.8 In.

F 010 4 8 12 Is

Cyleo, n- X 104

FO 20MI

Figure 6. Distribution of Cycles to Reach 0.8 Inch, for the INIO0 Mission VerificationTeat

1.0

0.89

0.71-

0.4-

I,4-0

0.3 --

0.2 0-

0.1L a,•• ••- - 1.4 In.

0 4 8 12 16

Cycles. n x x 104

FU 26MB

Figure 7, Distribution of Cycles to Reach 1.4 Inches for the INIO0 Mission VerificationTest

S~11 ,

I .

1.0 -(

0.4 n -24,000 Cycles

' 0.2.

00

0.5 0,6 0.7 0.8 0.9 1.0

Crack Size - In.

Figure 8. Probability of Crack Exceedance at 24,000 Cycles for INlO0 Mission

Verification Testing

1.0-

I:I0.8-

0 n - 35,000 Cycles

'~0.4 0

0050.2 0

00 1

0.6 0.6 0.7 0.8 0.9 1,0

Crack Size, a - In. Fl) 268007

Figure 9. Probability of Crack Exceedance at 35.,000 Cycles for INIO0 MissionVerification Testing

12

I I L J -- -- " ' ... . I " ili I . - -- . -

Since the initial crack size is deterministic, which is equal to 0,5 inch (12.7 mm), thedeviation between the theoretical predictions and the test results is expected to increase as thenumber of load cycles increases. This is clearly demonstrated by the crack exceedance curvesshown in Figures 8 and 9. Furthermore, the statistical dispersion of the crack size increases withrespect to the number of load cycles as exemplified by these figures, Hence, the maximumdiscrepancy occurs when the random number of load cycles reaches a 1.4 inch (35.6 mm) crack,that is very close to the critical crack size (see Figure 4). It is observed from Figure 7 that thediscrepancy in predicting the average number of load cycles to reach a 1.4 inch (35.6 mm) crackis approximately 13%.

Some of this conservatism may be dae to retardation at the crack tip caused by the changein yield stress as a function of temperature. The size of the plastic zone is inversely proportionalto the square of the yield stress. At the 13500F test condition, the lower yield stress will cause alarger plastic zone which will retard the crack when the temperature is decreased to 1200'F.This will also occur from the 1200*F to the 1000*F temperature change. The initial retardationof the crack growth rate in each test condition would result in a longer test life than thepredicted life, as the computer program (theoretical model) does not address retardation effects.

From the statistical standpoint, the main reason for a slight discrepancy comes from thecompiled data base as described in the following.

The theoretical prediction is computed based on the parameter values C, C, Can and.for j = 1, 2, 3, 4, These parameters values were estimated using a compiled data set ýor each testcondition which was generated over a long period of time for different purposes. As a result, thespecimen preparations, heat treatments and processes of materials, test machines, measure-ments and environments, etc. will not be identical to those specimens tested under the presentprogram. The number of test specimens compiled for the data base for each test condition is alsovery small (see Table 1). Likewise, the data base is highly inhomogeneous, indicating that thespecimen dimension, initial and final flaw sizes, and the maximum applied load are all differentfor each test specimen. Consequently, it may be reasonable to expect some inconsistenciesbetween the highly inhomogeneous data base, from which the theoretical model parameters arecalibrated, and the homogeneous test results generated in the present program. Nevertheless,the worst discrepancy is only 13% on the conservative side, which is quite reasonable inpractice.

In practical situations, however, the data base is usually not plentiful, highly nonhomoge-neous, and compiled over a long period of time under different test environments. Nevertheless,such a homogeneous and limited data base can be used to calibrate the model parameters, and

_-* the present statistical model is capable of predicting the crack propagation behavior underspectrum loading.

One important aspect of the fracture mechanics approach is that available crack growthrate data from inhomogeneous data bases can be pooled together to estimate the crack growthrate parameters, although the type of specimen, the specimen dimension, initial and final flawt lengths, and the maximum cyclic load for each test specimen may he different in the datia base.In contrast to other statistical models (e.g., References 14, 15, 18-21), the statistical model

proposed herein is based on fracture mechanics, thus it possesses the same important advantagepertinent to the fracture mechanics approach

Finally, the present statistical model does not address the rethrdation or accelerationeffects. Further research is needed in this regard in order to provide a better predictioncapability necessary for retirement-for-cause analysis of engine components tinder serviceloading spectra.

13

...............................

SECTION III

LOGNORMAL STATISTICAL MODEL APPLIED TO THE PARIS EQUATION FOR IN100

The lognormal crack growth rate model has been applied to data for IN100 at various testconditions using the hyperbolic sine (SINH) function (Reference 1). In this section, thelognormal model is developed for the Paris crack growth rate function. Emphasis is on thesimplicity of this model for practical application. The distributions of propagation life to reachany specific crack size and the distribution of crack size at any service life are derived. Thecorrelation between the extrapolated test results for IN100 and the statistical model is shown tobe reasonable.

The Paris crack growth rate equation is given by

d a - Q [ ,&K ] hd (13)

in which a = crack size after n load cycles, AK = stress intensity range, Q and b are functions oftemperature T, loading frequency Y, stress ratio R and others.

To account for the statistical variability of the crack growth rate da(n)/dn, thedeterministic crack growth rate function (Equation 13) is randomized as follows

- X(AK)QIAK]b (14)

in which X(AK) is a non-negative random function taking values around unity, and introducedto reflect the statistical nature of the crack growth rate.

Taking the logarithm on both sides of Equation 14, one obtains

Y - bU + q, + Z(AK)

in which

I log -- , U -- logAK (16)

q logQ, Z(AK) - logX(AK)

Whereas X(AK) is a random function, the solution for the statistical distribution of thecrack growth damage accumulation is rather complicated (e.g., Reference 5). In this connection,two extreme cases of the random function X(AK) should be mentioned. At one extreme, X(AK)is completely independent at any two values of the stress intensity range, referred to as thewhite noise process. Based on the central limit theorem, it can be shown that the statisticalvariability of the crack size or the fatigue life, after integrating lUquation 14, is the smallestwithin the class of random functions. Hence, it is unconservative for engineering analyses andapplications. Also, the mathematical solution for the white noise random process model isdifficult. At another extreme, the random function X(AK) is totally correlated at any two valuesof the stress intensity range, indicating that X(AK) is a random variable, ie., X(AK) = X. Forthe case of random variable X, the statistical dispersion of the fatigue crack growth damage

14

accumulaition is the largest in the class of random functions. Consequently, the random variablemodel is conservative for the prediction of the crack propagation life and for practicalapplications.

Because of mathematical simplicity in analysis and the conservative nature in crackpropagation prediction, we shall investigate the random variable model, i.e., X(AK) = X is a

positive random variable taking values around unity. It is assumed that X follows the lognormaldistribution with a median equal to unity. Tests for goodness of fit for the lognormaldistribution will be performed later,

it follows from Equation 16 that Z = log X is a normal random variable with zero mean. Byvirtue of Equation 15 the log crack growth rate, Y = log(da/dn), is a normal random variablewith mean value pand standard deviation a given by

AY bU + q(17)

ay-- a,(18)

in which it is obvious that the mean value pyis a function of AK, and the standard deviation a. isa constant.

The parameters b and q (or Q), as well as the standard deviation a, (or a), given inEquations 15 through 18 car be estimated from the test results of the crack growth rate versusthe stress intensity range using Equation 15 and the method of maximum likelihood estimate.Since Y and Z are normal random variables and Equation 3 is linear, the method of maximumlikelihood is identical to the linear regression analysis and the method of least squares.

Teat results from compact tension specimens, for log crack growth rate Y = log (da/dn)versus the log stress intensity range U = log AK are presented in Figure 10 as discrete points forfive IN100 specimens in the test condition No. 1 (T = 1000F, v = 10 cpm, R = 0.1). The crackgrowth rate data shown in Figure 8 have been used to estimate b, Q and a. using the method ofmaximum likelihood. The r "suits are presented in the first row of Table 3 and plotted in Figure10 as a solid straight line for ay = 0. The maximum likelihood estimates of b, Q and 0,determined from the test results of Y versus U using ASTM CT specimens, are shown in Table 3for different test conditions (v,R,T).

The power law for the crack growth rate (Equation 14) holds only in the central region ofAK. In the regions where AK is either too low or too high, the crack growth rate appears tobehave asymptotically. Hence several data points for the crack growth rate have already beencensored in Figure 10. Also, the crack growth rate data seem to scatter around the solid straightline indicating the validity of the Paris function.

To show the validity of the assumption that Z follows the normal distribution, samplevalues of Z, aenoted by zy, are computed from test rsults of the log crack growth rate Y = logda/dn versus log stress intensity range U = log AK, denoted by (yj,uj), using Equation 15

z) = yj - buj - q for j -1,2,...,n (19)

in which b and q have been estimated by the method of maximum likelihood and n is the totalnumber of test data.

15

-2

-3 00

OBo-4o 0

00

[ ~ ~-6 0

Test Condition I

0.8 1.1 1.4 1.7 2.0 2.3

log (delta kc)

Figure 10. Data for Crack Growth Rate Versus Stress Intensity Range for TestCondition No. I

TABLE 3, MAXIMUM LIKELIHOOD ESTIMATE OF b,Q, STANDARD DEVIATION v ANDCOEFFICIENT OF VARIATION OF da/dn

Test Coef of No. ofCondition* b Q a-d, Vawiation Data

1 2.0617 5.016X10-1 0.1516 36.0% 2512 2.9216 7,409X10'- 0.1455 34.5% 1473 2.8439 7.141X10'- 0.1038 24.3% 1294 2.3850 1.870X1IO- 0,1961 45,2% 1855 2,3180 3.141X10-s 0.1131 26.5% 338

Average 33.3%

*Test Conditions

1. T - t000F, P - 10 epm, R - 0.12. T - 1350F, P - 10 cpm, R - 0.13. T - 12001F,v " 10 cpm, R- 0,54. T - 1200*F, j - 20 cpm, R - 0.055. T, 1200'F, - 10 cpm, R - 0,.

Sample data of Z (j=1, 2, .... n) for test condition No, 3 are plotted on normal probabilitypaper in Figure 11 as circles along with a straight lile repre;enting the estimated normaldistribution for Z (with zero mean and standaid deviation a, determined previously, see Table3). A linear scale is used in Figure 11 in which the sample data z, P.,e arranged in fn ascendingorder, ie., z, :5 z2 : za... :s; z., and the ordinate (orresponding to zj is given by 4-t'0/(u4 1)] with

16

,-i K ) being the inversed standardized normal distribution function. It is observed from Figure

11 that the sample values of Z are scattered around the straight line without a nonlinear trend,indicating that the normal distribution is acceptable.

20

00

-2

'ift

0/

0/ Test Condlton 3-3 .. ..

-1 -0.5 0 0.5 1

Figure 11. Normal Probability Plot for Z (Test Condition No. 3)

The Kolmogorov-Smirnov test for goodness of fit was performed to determine the observedK-S statistic D.. The normal distribution was found to be acceptable at least at a 10% level ofsignificance.

Since Z - log X and Y - log da/dn are normal random varableb, the crack growth rate G =da/dn follows the lognormal distribution. The coefficient of variation, V, of da/dn is re!ated tothe standard deviation, o,, a., through

V - [e(VY'n"'° - 11/2( (20)

The coefficients of variation, V, for the crack growth rate, G f da/dn, for five test conditions arealso presented in Table 3.

The distribution of Z, and Y = log[da/dn], are both normal with the same standarddeviation, ayl fi.%, that has been determined previously. Let zY be the y percentile of Z, i.e.,

'Y% • P[Z>z,] - 1 -- (z'/a,)(21)

or inversely,

S• 4@-l(1 - y%) (22)

17

They percentile of the log crack growth rate Y, denoted by yy (AK,b,q), follows from Equation 3as

y.(AK,b,q) - bU + q + z, (23)

in which z. is given by Equation 22. Note that y7 (AK,b,q) is a function of b and q which in turndepend on test conditions (p, R, and T).

Therefore, by varying the value of y, one obtains the distribution of the log crack growthrate in terms of percentiles, The results are shown in Figure 12 for test condition No. 1 (P =10 cpm, R = 0.1 and T = 1000°F). As an example, the crack growth rate path associated with ',= 10 indicates that the probability is 10% that a specimen will have a growth rate faster thanthat shown by the curve.

-2

-3

10

-4C 25

75

-6 90

95 Test Condition 1

-7 1 - I 1 110.8 1.1 1.4 1.7 2.0 2.3

Iog(dalta k)Fr) 25fl34

Figure 12. Percentiles of Log Crack Growth Rate as a Function of Log Stress IntensityRange for Test Condition No. I

The ,y percentile of the random variable X, denoted by Xr, is computed from Equation 4 as

X - (10)' 24(24)

in which z,, is given by Equation 10.

The 7 percentile of the crack size after n load cycles, denoted by a,, is obtained bysubstituting Equation 24 into Equation 4

da•

(25)

18

- - , - -- - - - . ....-

in which the stress intensity range AK7 for the ASTM CT specimen is expressed as

K APAK' -- -Bfial]S(26)

where B = specimen thickness, W - specimen width, AP = applied load range and

f[a7 ] (I (l-a)/ (0.866+4.64a-13,32a 2 +14.72ac-5.6a4) (27)

a -

(28)

Thus, Equation 25 can be integrated numerically over particular limits to obtain a set ofcrack lengths, a (n), versus the number of load cycles, n, for different -y percentiles. It should bementioned that the numerical integration using Equation 26 for each y percentile of the cracklength, a(n), is deterministic and straightforward.

Test environments, including the initial crack size so, the final crack size ap the maximumload P.., and the width W and thickness B of the ASTM CT specimen, are assumed for eachtest condition as shown in Table 4. The necessity to assume homogeneous test environments inTable 4 for the correlation study has been described previously (Reference 1). IntegratingEquation 25 and using Tables 3 and 4, one obtains five sets of crack length, a,, as a function ofthe number of load cycles, n, for different y percentiles. Only the results for test conditionsNo. 1 and No. 2 are displayed in Figure 13.

TABLE 4. ASSUMED HOMOGENEOUS TEST ENVIRON-MENTS FOR TEST SPECIMENS

Tut a. at W Bcondition (in.) (in.) on) (in.) (hip..)

1 0.5 2.0 2.502 0.250 1.4002 0.5 2.0 2.503 0.501 2.6003 0.5 1.8 2.000 0.500 2.6004 0.5 2.0 2.506 0.500 3.0005 0.5 2.0 2.497 0.501 3.633

S- initial crack sizet, final crack sianW - specimin widthB - specimen thicknessP. - maximum load

In the life prediction of engine components, two statistical distributions are of practicalimportance; (1) the distribution function, FN(. , of the number of cycles, N(a1 ), to. reach anygiven crack length, a, and (2) the distribution function, F.(x), of the crack length a after anynumber of load cycles n. Since Figure 13 represents the distribution of the crack length as afunction load cycles n, it contains all the information needed to determine the distributionsmentioned above. For instance, by drawing a horizontal line in Figure 13 through a crack lengthof interest, the distribution for the number of cycles to reach that crack length is obtained.Likewise drawing a vertical line in Figure 13 through a given load cycle n, one obtains thedistribution of the crack length after n load cycles. The complement, F:(x), of the distributionfunction, F,(x), of the crack length, i.e., F:(x) - 1 - F,(x), is the probability that the cracklength after n cycles will exceed a certain value x. Hence, the plot of F:(x) is referred to as thecrack exceedance curve.

19

S]1

1.9 %10% 25% 50% 75% 90% 95%

1.7

~1.1=0 1.3

0.9

0.7 Test Condition I

0.50 2 4 a 8 10 12 14

Cycles, n X- 104

(a)

10% 25% 50% 75% 90% 05%1.7

1.5

A 1.3 -

S0,9

0.7Test Condition 2

0.5 1 1 10 5 10 15 20 25 30 35 40 45 50

Cycles, n - X 103

(b)FO 2605

Figure 13. Distribution of Crack Size as Function of Load Cycles Based on StatisticalModel; (a) Test Condition No. 1, (b) Test Condition No. 2

20

i I..

For instance, the distribution function, FN(-,), for the number of load cycles to reach the"crack lengths a, = 1.0 and 2,0 inches is obtained from Figure 13 and displayed in Figure 14 as asolid curve for test condition No. 1. For the same condition, the crack exceedance curve F:(x)after n = 25,000 cycles is shown in Figure 15 as a solid curve.

In a similar manner, the distribution functions for the number of load cycles to reachcertain crack sizes and the exceedance curves after certain number of load cycles for testconditions No, 2 to No. 5 are presented in Figures 16 through 23 as solid curves. Note that thesolid curves depicted in Figures 14 through 23 are obtained based on the statistical model thatutilizes only the crack growth rate data, e.g., Figure 10, for estimating the model parameters b, Qand ay = ea as shown in Table 3.

1. Correlation with INlOb Test Results

A qualitative correlation study of the IN100 results is carried out in the following manner.

Test results of the crack growth rate for each specimen are best fitted by the hyperbolicsine crack growth rate function proposed by Pratt & Whitney Aircraft (Reference 2).

Y - C0 sinh [Cg(U + C2)] + C& (29)

in which Y and U are given in Equation 16, C1 is a material constant, and Cp, C. and C4 areconstants depending on the test condition (Y, R, T). Equation 29 involving four parameters issupposed to provide a better fit to the crack growth rate data.

Given the crack growth rate data, the least squares best fit procedures to estimate values of02, C, and C4 for each specimen were described in Reference 2 and the results of C2, C. and C4 foreach specimen in five test conditions were presented in Reference 1. Then, Equations 29 and 16can be integrated to yield the crack size, a, as a function of load cycles, n. It has been shown thatthe crack growth damage accumulation a thus reproduced by integrating Equations 29 and 16correlates very well with the test results for each specimen (Reference 1).

In order to correlate the nonhomogeneous test results with the predictions based on thestatistical model proposed previously, homogeneous test environments have been assumed foreach test condition, as shown in Table 4. The maximum loads Pm.. given in Table 4 are chosen inorder to avoid excessive extrapolation far into the region of AK in which actual test results donot exist.

By integrating Equations 16 and 29 over the homogeneous test environments, withappropriate values of C1, C2, C1 and C4 for each specimen (see Reference 1), one obtains five setsof the homogeneous crack size, a, as a function of load cycles n. These homogeneous data sets arereferred to as the extrapolated test results, since they are not the results obtained directly fromexperimental tests. The extrapolated test results for the crack size a versus load cycles n for thefive test conditions are given in Reference 1 and only the results for test conditions No. 1 andNo. 2 are depicted in Figure 24. A comparison between Figures 13 and 24 indicates that thecorrelation between the theoretical model and the extrapolated test results is reasonable.

21

1.0

0.8

0.6

0.4 Test Condition I

a - 1.0 In.

-- Theoretical Model

0.20 Test Results

00 4 8 12 16 20

Cycles, n - X 104(a)

1.0

0.8

S0.0

0.4 Toot Condition 1

a - 2.0 in.- Theoretical Model

0.2

0 Teat Results

0 4 L 12 18 20

Cycles, n - X 104(b)

Figure 14. Distribution of Cycles To Reach Given Crack Size for Test Condition No. 1;(a) a 1.0 Inch, and (b) a - 2.0 Inches

22

1.0

t0.4

0.2

.n -25,0 Cycles

0,5 0.6 0,7 0.8 0.9 1.0 1.1 1.2Crock Size, a - In.

Figure 115. Crack Exceedance Curve After 25,000 CycLe8 for Test Condition No. I

From the extrapolated test results for the crack size a versus the number of load cycles n,one obtainis (1) extrapolated teat data for the number of load cycles to reach any given crack sizea, by drawing a horizontal line through a,, and (2) extrapolated test data for the crack size afterany number of load cycles n, by drawing a vertical line through n-.

The distribution function is constructed from the extrapolated test data obtained above byarranging them in an ascending order. The ordinate of the ith data is given by i/(m+1) where mis the total number of data points. For instance, one obtains five data points from Figure 22 bydrawing a horizontal line through a, - 1.0 inch. These five data points, for the number of cyclesto reach a crack size of aI = 1 inch, are ranked in an ascending order where the ordinate of theith data point is equal to i/6 (i=1, 2, ..., 5). These results are shown in Figure 14(a) by circles. Inlike manner, the distributions of the extrapolated test results for the number of load cycles toreach other crack sizes, as well as their creck exceedances after some numbers of cycles havebeen constructed and shown in Figures 14 through 23 as circles. It is observed from Figures 14through 23 that the correlation between the statistical model (solid curves) and the extrapolatedtest results (circles) is very reasonable,

23

1.0

0.8

j0.0So.4 i

0o.4 )Test Condition 2

a - 0.8 In.Theoretical Model

0.2

0 Test Results

00 1 2 3 4

Cycles, n - x 104

(a)

0.81.01

o0.6

a - 2.0 In.

""-. Theoretical Model

0.2 00 Test Results

00 1 2 3 4 6 a

Cycles, n - X 104(b)

Figure 16. Distribution of Cycles To Reach Given Crack Size for Test Condition No. 2;(a) a = 0.8 Inch, and (b) a = 2.0 Inches

24

1.0

,o.a - 0

U10.6-

Test Condition 2

.- Theoretical Model0.4 0 0 Test Reaults

0.2

0,0 1 -, I . .....0.5 0.6 0.7 0.8 O.9 1.0 1.1 1.2

Crack Size, a - in.

Figure t7, Crack Exceedance Curve after 10,000 Cycles for Test. Condition No. 2

2i

' 25

2 • ~a.,. .. . . -itt. ... f. th k a.1bg..4~ -

1.0

0.8

0.4 Teat Condition 3a 0.76 In.

-Theoretical Model

0.2

0 Extrapolated Teat Results

0 ___________

2 4 6 8 10 12Cycles, n - X 104

(a)

0.60

0.4 0

~ 04 Teat Condition 3

a - 1.7 In.-Theoretical Model

0.20 Test Results

02 4 6 8 10 12

Cycles, n - X 10'(b)

Figure 18. Distribution of Cycles To Reach Given Crack Size for Teat Condition No. 3;(a) a 0.76 Inch, and (b) a - 1.7 Inches

26

1.0

0.8

1 0.60i o., \Test Condition 3

Theoretical Model

0.4 0 Tet Condite

- •n - 40,000 Cycles

0.2

0.8 0.6 0.7 0.8 0.9 1.0 1.1 1.2

Crack Sim, a - In.

Figure 19. Crack Exceedance Curve after 40,000 Cycles for Test Condition No. 3

27

1.00

0.8

0.6

S0.40.4 Test Condition 4

a - 0.8 in.- Theoretical Model

0.20.2 0 Test Results

0 2 4 6 8 10

Cyles, n - X 104(a)

1.0

m 00.8

- 0

0

a - 2.0 In."" Theoretical Model

0,2

0 Test Results

00 2 4 6 8 10

Cycles, n - X 104

(b)

Figure 20, Distribution of Cycles to Reach Given Crack Size for Test Condition No. 4;(a) a = 0.8 Inch, and (b) a = 2.0 Inches

28

•1•' .L

1.0

0

0.8

0

0.6 0

Test Condition 4

0.40 Theoretical Model: 040 T-st Results-3, (n 10,000 Cycles

0.2

0

0.6 0.6 0.7 0.0 0.9 1.0 111 1.2

Crack Size, a - In. Omos

Figure 21. Crack Exceedance Curve after 10,000 Cycles for Test Condition No. 4

2

29

SiI

1.0 -, •

0.8

S0.6

10.4 Test Condition 5

Theoretical Model

0.2 (0 Test Results

010 6 16 24 32 40

Cycles, n - x 103

(a)1.0.

- 0

018 0

0

4 0.6 0

Test ConditIon 5

. 4- 2.0 In.

- Theoreticall Model

C2 00 Test Reouits

'~010 8 16 24 32 40

Cycles, n - X ios(b)

?Pgure 22. Distribution of Cycles to Reach Given Crack Size for Test Condition No. 5;(a) a 0.8 tnch, and (b) a = 2.0 Inches

30

0

8 0.8 0

0

0.6- 0

Test Condition 5-Theorstiosil Model

OA 0 Extrapolated Test Results

0.2

0.00.5 0.6 0.7 0.8 0. 1.0 1.1 1.2

Crack Size, a - rin.

Figure 23. Crach Exceedance Curve after 8,000 Cycles for Test Condition No, 5

31

1.9-

1.7

1.5

* 1.3

0.9

0.70,5o 2 4 0 8 10 12 14

Cyolee, n - x 104

(a)

1.7

1.3I 1,3-

0.7 --

0.5

10 15 20 25 30 35 40 45 50Cyol.., n --x 104

(b)FO 25•,

Figure 24. El.xtropolated Test Results for a Versus it; (a) Test Condition No. 1 and (b)Test Condition No, 2

32

r

SECTION IV

ANALYSIS OF TI 6-2-4.6 AND WASPALOY DATA

Available Ti 6-2-4-6 and Waspaloy crack propagation test results from constant amplitudecyclic loading with various test conditions were compiled. The specimen dimensions, initialcrack size, final crack size, and maximum load for each specimen in each test condition areshown in Tables 5 and 6. The crack growth rate data for all specimens in each test condition arealso pooled together for analysis purposes. The pooled data for Waspaloy and Ti 6-2-4-6 fromtest condition Number 1 are shown therein, respectively.

TABLE 5. SPECIMEN GEOMETRY AND MAXIMUM LOADFOR EACH TEST SPECIMEN OF Ti 6-2-4-6

Test Specimen a. a R W P..Condition -No.. (in,) _..in.) (in. Cin.) (i•.t.

T - Room Temperature 1536 0.8138 10724 0,249 2.496 005480v - 20 Hz 1508 0,8130 1.6130 0.2970 2.5007 0.7200R - 0.1 1582 0,380 2,2080 0.3690 2.0085 1,0640

1682 0.7360 1.0620 0.5004 1.4970 0,6821663 017680 110590 0.499 1.49 0,6810

1676 0,3990 1,1780 0,5020 1.910 0.95801677 0,4040 141900 0.3830 1.9987 0,9851680 0,4730 1,1164 0,3735 120955 1.00601972 0,1919 0,6025 0.900 0.99 1.19601535 0.7835 1,7618 0.2485 204999 4.27

2T -Room Tmperature 03 0.3850 017973 0.0970 2,0000 0,624v- 10 epth 0448 0,8629 1,7005 0,2520 2.2551 0,568R - 0.1 1484 1.3789 13577 0.1250 2.4951 1,037

010M 0,6784 1,1716 0,5004 109929 1,1821886 0.5733 1.1711 0.5005 2.0060 1,1381892 0.5149 1.4487 0.3890 2.4993 1,2962103 0.5577 1,2964 0.3760 2"003 1,301

S

T - Reto Temperature 1744 0.0828 0.2261 0.2935 0.996 6.611V - 10 eHz 1741 0.0766 0.1982 0.2900 0.996 7.157R - 0,0 1742 0,0887 0489 0,2970 0.950 6.815

1672 0.116 0.31 0,9050 0 .1997 4.2734T - 800'F 0358 0.597 1,7926 0,2750 2.2520 U,24v - 10 opra 0359 0.6697 1.5878 0.2750 2.2556 0.626R - 0.1 0409 0.3690 0,8359 0.1260 2.0000 4,644

0410 0.3674 0.6907 0.1205 2.0000 4.8911245 0.6453 0.7775 0,5010 1.0030 0.365

1881 0.3068 1.4964 0.5240 1.9959 1.7631883 0,3679 1,4914 0,5040 2.0045 1,577

T - 400OF 0413 0,6325 2,06M0 0,2500 2,5000 0.3330S-20 Hz 0415 0.3741 0,8276 0,1260 2.0000 2,4500

R - 0,1 0416 0.9426 2,0489 0.2500 2.5W02 0.26901678 0.4260 0.9300 0,3440} 1.9972 1.1150IBM6 0.6128 1.4754 0.5015 1.9986 1.1210

i3

TABLE 6. SPECIMEN GEOMETRY AND MAXIMUM LOADFOR EACH TEST SPECIMEN OF WASPALOY

Test Specimen a. a/ B W P.Condition No. (in.) (in.) (in.) (in.) (kip,)

T , 1200F 0738 1.0706 1L7666 0.432 2,502 1.621u- 20 Ha 1464 0,.60 2.0820 0.500 2.500 1.240R - 0.06 1470 0,86510 1.7460 0.491 2,491 1.224

1304 1.0830 1.6670 0.500 2.09 2.263

2T - 1200*F 1004 0.0703 0.3326 0.301 0.998 11.613v - 10 epm 1301 1.1072 1.7094 0,601 2,519 1.769R - 0.05 1013 0.9688 1.8553 0,132 2.484 1.261

3T - 800*F 1002 0,8339 1.7598 0.499 2,606 2.574V- 10 epm 1003 0.0635 0,3047 0.303 0.997 13,316R - 0.05 1018 1.01U4 1.89" 0.298 2.5W 1.417

Using the hyperbolic sine crack growth rate function (Equation 29), and the lognormalstatistical model (Reference 1), we obtain the maximum likelihood estimates for C. , C3, C4 andthe standard deviation o a . These results are presented in Tables 7 and 8. The crack growthrates based on the values of CI, C2, C. and C4 given in these tables, for Z - 0, are shown as solidcurves in Figures 25 and 26 (test condition Number 1).

TABLE 7. MAXIMUM LIKELIHOOD ESTIMATE OF C2 , CS, C4,STANDARD DEVIATION a = a AND COEFFICIENTOF VARIATION, V, OF da/d'n FOR Ti 6-2-4-6 (HYPER-BOLIC SINE FUNCTION)

No, ofTest Data

Condition C, C, q, C4 -fr, V Points

1 0.7 4.68 -1.0898 -6.4878 0.0672 15.6% 2692 0.7 4.5715 -1.1088 -5.378 0.1156 27.1% 179a 0.7 3.0949 -0.9651 -6,7946 0.1174 27,5% 504 0.7 2.1954 -0.9086 -5,4958 0.0561 13.0% 2515 0.7 3.5301 -1.0641 -6.7081 0.0939 21.9% 163

TABLE 8. MAXIMUM LIKELIHOOD ESTIMATE OF C2, CS,C4 , STANDARD DEVIATION o, = or AND COEFFI-CIENT OF VARIATION, V4 OF da/dn FORWASPALOY (HYPERBOLIC SINE FUNCTION)

No, ofTest Data

Condition C,• I C, G.y-O, V Points

1 0.5 3,5484 -1,4054 -5,0269 0.0783 18.2% 1302 0.5 4.0154 -1,3694 -4.8774 0.0470 10.9% 603 0.5 4,3817 -1.4941 -4.9847 0.0554 12.8% 67

34

AK, MPa m

10- ,_2 6 10 2 6 100 2 5 1000 7APe7N1N Symbol

,o, - iiiiiiiiiiiiiliiiiiil '''liii 'iIII ' ,o.7AN0736 a]- 10-1 7AN1484 C

* 7AN14707AN1304 +

10-3 ./ 1-~... ... ... ... .. .. .................. .. ...........

10--

10-4

I 10-3

_,• ,10 -'6 _ . ........................... 1 .. ........ .......... ..........................

_ __ 10-4

1065

1 2 5 10 2 5 100 2 5 1000

AK, kl V./1•

Figure 25. Crack Growth Rate Model for Waspaloy at 12000F, 20 Hz, and R 0.05

35 6

!4

AK, MPa %/M-

1- 2 5 10 2 5 100 2 5 1000

SWe No. Symbol

10-' 7AN1535 V37AN1536 07AN1580 A

- ANISO2 +7AN16S2 X

1-3........................................................ 7ANI 163 !

7AN1676 +'7AN1677 x7AN1680 Z

10-1 IAN1972 Y

10-3

;i~~~~~1- 106...... . .

10-1

101*-6

%omit 10-6

2 5 10 2 5 100 2 5 1000AK, kal v~-

Figure 26. Crack Growth Rate Model for Titanium 6-2-4-6 at Room Temperature, 20 Hz, and R 0.1

36

Crack growth rate data for each specimen are beet-fitted by the hypcrbolic sine function(Equation 3), to determine the parameter values C0, C3 and C4. The results of the parametervalues for each specimen are shown in Tables 9 and 10.

TABLE 9. PARAMETERS C2, C0, AND C4 FOR EACH TESTSPECIMEN OF TITANIUM WITH C, = 0.7, HYPER-BOLIC SINE CRACK GROWTH RATE FUNCTION

Test Typo ofCondition Specimen No, CO Vs C4 R specimen

1 - 15.98 4.692 -1.0205 -5.735 0.1 MCI'

1580 4.1696 -1.2051 -5.1217 0,1 MCT1582 5.0229 -1.3336 -4.6700 0.1 MCT1662 4,4731 -1.0217 -5.6851 0.1 MCT1683 4,75a7 -1.0399 -5.6814 0.1 MCT1676 4.1695 -0.9868 -56.824 0.1 MCT1677 4,2736 -0,9858 -5,8294 0.1 MCT1680 4,1694 -1,2464 -4.9161 0.1 MCT1972 3,7766 - 1.0662 --5.5379 0.1 PItT1535 4,8335 -1.0630 -5,6762 0,1 MCT

2 - 0423 4.6117 -1,1744 -5,2811 0.1 ON0448 4.4517 -1.0333 -5.6816 0.1 MCT1484 4.8831 -1,4252 -'44558 0,1 MNOT18 5.2785 -1,1319 -5,2332 0.1 MCT1886 4,0642 -0,8937 -6,1048 0.1 MCT1892 4.6282 -1.0629 -5,6214 0,1 MCT2103 5.1135 -0.9827 -5.7010 0.1 MC('

3 - 1744 5.4560 -1.3166 -4.6476 0.0 ON1732 3.1613 -0.8448 -5.6793 0.0 ON1741 5.1208 -1.2970 -4.7369 0.0 ON1542 4.7786 -1.4152 -4,3071 0.0 ON

4 - 0558 2.1554 -0,8647 -5,5219 0.1 MOT0369 1.6211 -0.4380 -8,1660 0.1 MCT0409 3.0045 -1.7454 -3,7829 0.1 CN0410 3.3857 -1,4531 -4.4011 0,1 CN1245 2.2045 -0.8054 -5,619 0.1 MCT1881 2.7011 -1,0981 -5,1852 0.1 MCT1883 2.3800 -0.9634 -5,4166 0.1 MCT

8 -- 0413 3.5875 -1.1146 -5,5747 0.I MCT0415 3.4828 -1.2919 -5.1373 0.1 CN0416 3.8776 -0.8233 -6,4855 0.1 MCT1678 3.7288 -1.2226 --5,2324 0.1 MCT1886 3,590A -1.1537 -5.3697 0.1 MCT

TABLX 10. PARAMETERS C2, C3, AND C4 FOR EACH TESTSPECIMEN OF WASPALOY WITH C1 - 0.5, HYPER-BOLIC SINE CRACK GROWTH RATE FUNCTION

Teat Type o/Condition Specimen No, Cr Cl C4 R Specimen

1 0736 3.9744 -1.5278 -4.8472 0,05 MCT1464 3.4968 -1.4725 -4.8366 0.05 MCT1470 4.4457 -1.2349 -5.8545 0,05 MCT1304 3.8747 -1.5438 -4,7752 00.1 MOT

2 1004 4,9009 -1.4710 -4.6875 0,05 CN1301 3.7554 -1.2698 -5.1172 0.05 MCT1013 3,8140 -1.4985 -4.5702 0.1 MCT

3 1002 4,4664 -1.408 -4.8281 0.05 MCT1003 4.9438 -1.5226 --4.9012 0.05 CN1.018 3.827 -1.3911 -5.2284 0.05 MCI'

37

In order to perform correlation studies, homogeneous test environments should beassumed, since each test specimen has different geometry, initial crack size, final crack size, andmaximum load P,., as shown in Tables 5 and 6. Homogeneous test environments assumed forWaspaloy are shown in Table 11.

TABLE 11. ASSUMED HOMOGENEOUS TEST ENVIRON-MENTS FOR EACH TEST CONDITION FORWASPALOY

Tesat a at B W P,.. Type ofCondition (in,) ((n,),_ 0n)ki(...) p .___) Specimen

1 0.6 2.0 0. 2.5 1.6 MCT2 0.5 2.0 0.5 2.5 6.2 MCT8 0,5 2.0 0.6 2A 4,0 MCT

The distribution of the crack growth damage accumulation, a, as a function of load cycles,n, based on the lognormal statistical model can be obtained using Tables 8 and 11. Then, thedistribution functions for (1) the number of load cycles to reach any specific crack size, and (2)the crack exceedance probability at any given number of load cycles, can easily be determined(Reference 1). The distribution functions for the number of load cycles to reach crack sizes 0,875and 2.0 inches are displayed as solid curves in Figures 27(a), 28(a), and 29(a), for the three testconditions, Likewise, the probabilities of crack exceedance at n - 50,000, 10,000 and 50,000cycles, for tho three test conditions, are shown as solid curves in Figures 27(b), 28(b), and 29(b).

With the aid of the best-fitted parameter values for C2, Cp, and C4, given in Table 10, thecrack growth damage accumulation, a, for each individual specimen, is obtained by integratingthe crack growth rate equation and using the assumed homogeneous test environments given inTahle 11., The distribution of the number of load cycles to reach any specific crack size and thecrack exceedance probability at any givon number of load cycles is then established. Theseextrk,)olated results are presented as circles and triangles in Figures 27 through 29, ThehonogeiLeows test environment in Table 11,, is assumed to avoid excessive extrapolation into theregion of 4K for which test data do not exist. This may be difficult to achieve because the crackgrowth rate data for each specimen do not cover the same region of AK. Therefore, theextrapolation cannot be avoided for some specimens, This problem becomes more serious for theTi 6,-2-4-6 data and will be discussed later.

It is observed from Figures 27 through 29 that the correlation between the lognormalstatistical model and the extrapolated test results is very reasonable for Waspaloy,

In a similar manner, homogeneous test environments are assumed for TI 6-2-4-6. Based onthe lognormal statistical model, the distribution functions foe' the number of cycles to reach twodifferent crack sizes are shown as solid curves in Figures 30(a) through 34(a) while the crackexceedance curves are shown in Figures 30(b) through 34(b). The corresponding extrapolatedtest results are shown in the same figures by circles and triangles. It is observed that except fortest condition number 5, the correlation is less than satisfactory,

38

1.0 Test Condition 1Waipaloy

a 0.875 In./ ) - 3.6a-0.6 - Hyperbolic Sine Model

-0.7

0.6 oCIS ,-2.0 in.

0.3

0.1

2 3 4 5 6 7 8 9 10 11 12 13, .,es, n -- )' .104

1.0 Test Condition 1

CIO WmapaloyPm- " 3.6

0I 8 0 Hyperbolic Sine Modeln , 50,000 Cycles

0.7

a0,e

10.50.40

0.3

0.2-

0.1

00.7 0,8 6.9 1.0 1.1 1.2 1.3

Crack S1m, a - In.(b)

Figure 27, Distribution Curve and Cnrk Ezceedance Curve for Waspoloy for TestCondition No. 1; (a) Distribution of Cycles to Reach Given Crack Size, and(b) Crack Exceedance Curve After 50,000 Cycles

39

WOW......I

1.0

0.9 - Tost Condition 2

0.8 WaspaloyPm" 5.2

"0.7 - Hyperbolic Sine Model

0,6 - a 0875 a " 2.0

0,5 -

0.4

"0.3& 0.2

0.1

5 7 9 11 13 15 17 19 21 23 25

Cyoles, n - x 103,,(a) I'D 21164

1,0 -

0.9-Test Condition 2

0,8 - WasepaoyPm,, -5.2

0.7 Hyperbolic Sine Modeln - 10,000 Cycles

0.60,600,4

00

0.10.1

0.60 0.65 0.70 0.75 0.80 0,85Crao Size, a - In.

(b) FD asi M~

" Figure 28. Distribution Curve and Crack Exceedance Curve for Waspaloy for TestCondition No. 2; ('a) Di.'stribution of Cycles to Reach a Given Crack Size,and (b) Crack Exceedance Curve After 10,000 Cycles

40

1.0 STest Condition 31,0E •Wapaloy

0.9 Pmp - 4.0Hyperbolic Sine Model

0.8 &

Oe a- 0.875 In. - 2.0 in.

0.5

0,430.3

0.1

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

cycles, a - x O0,(a)

1.00.9 Test Condition 3

0.8 Waspawoy- 4.0

0.7 -Hyperbolic Sine Modeln - 50,000 Cycles

0.60.50.4

0.3

0.2

0,65 0.75 0.85 0.95

Crack Size, 9 - in.

(b) F0 2C0iob

Figure 29. Distribution Curve and Crack Exceedance Curm: for Waspaloy for Test

Condition No. 3; (a) Distribution of Cycles to Reach. Guen Crack Size, and

(b) Crack Exceedance Curve After 50,000 Cycles

41

/ I i ..... .... . .. ........ .. ... . I I I R ... ... ....... .... . .. . .

0.9 -024.2 X 10 27.2 X10

0.7 0 1.70 in.

0.6 -a 0,725 In.0.5

I%=P • Test Condition 1S0.4 A Titanium

Pmax - 1.40.3 0 Hyperbolic Sine Model

0,2 - 0 Original Data

0.1 L I t

6 7 8 0 10 11 12 13 14 15 16 17 18

Cyoles, n - x 10t

0.7 '4

0 .6

"60.4 ,- Test Condition 1 (i ~Titanium0,3 -- P..x -1,4

Hyperbolic Sine Model0.2 n - 30,000 CyclesOriginal Data 00.1 - 0

0.40 0.42 0.44 0.46 0.48 0.50 0.52

Crack Size, a - in.(b) F0r Memo

Figure .90. Distribution Curve and Crack. Exceedance Curve for Titanium /or TestCondition No. 1, Using the Hyperbolic Sine Model; (a) Distribution ofCycles to Reach Given Crack. Size, and (b) Crack Exceedance Curve After.30,000 Cycles

42

10.710 =--,•./

0.9 -- ,

0.9 - 0.725 In.17

Sa - 1.70 In,

0.6

0o.4-

o 0.3 -- Test Condition 2STitanium0.2 -- Pma~x - 2.4

0.1 -Hyperbolic $Ina Model

01 2 3 4 5 6 7 8 9 10

Scycles, n -x 104

(a)FO P25020

0.7 Test Condition 2Titanium0.6 Prmx "- 2.4

Hyparbolic Sine Model0.5 - 0 n = 20,000 Cyclee

oS 0.4-

1 0.3-

0.2 -0.1 --

o , I ,I• = ...

0.45 0.50 0.55 0.60 0.A6 0.70 0.75 0.80Crack Size, a - in,

(b) FO 256007

Figure 31. Distribution Curve and Crack. Rxceedance Curve for Titanium for TestCondition No. 2, Using the Hyperbolic Sine Model; (a) Distribution ofCycles to Reach Given Crack Size, and (b) Crack Et.ceedance Curve After20,000 Cycles

43)

1.0°- ,0.16

0.8

0.7 a 0.16 In. - 0.4 in.

0.6

04

0.3 - Test Condition 302 -Titanium

P,,, 7.933

U.1 -Hyperbolic Sine Model

0 _.10 18 20 25

Cycles, n - X 103

1.0

08 ... Test Condition 30.8 0 Titanium

P1 - 7.933Hyperbolic Siem Model

*0.6 o E N - 5,000 Cycles

S 0.4 -Q

1 0.2

00 10 12 14 16 18 20

Crocf Size, a - In. x 10-

(b)

Figure 32. Distribution Curve and Crack Exceedance Curve for Titanium for TestCondition No. 3, Using the Hyperbolic Sine Model; (a) Distribution ofCycles to Reach Given Crack Size, and (b) Crack Exceedance Curve After5,000 Cycles

44

1.0 wo0.9

0.8

0.7 -a 0.7265 In.

S0.6 Test Condition 4 a 1.7 In.

0.5 -TitaniumP. - 4.64

0.4 Hyperbolic Sine Model / /

0.3 I

0.20 -0 2 4 6 8 10 12

(a) - Cycles, n -- 1 103

(a)

1.00.9 - , Tet Cundi.ion 4

0.9 Titanium0. °"-" %, -4.nm0.8 I Hyperbolic Sine Model

0.7 N - 5,000 Cycles

: 0 ." 0 )0.5- 0

? 0.4

1 00.20.1 0

0~0.5 0.6 0,7 0.8

Crack Size, a - In.

.'igure 33, Distribut•mn of Cycles to Rc•u'h Oiven C-ack Size for Titanium for TestCondition No, 4, Using the ffyperb.li, Sine Model; (a) Dintribution ofCycles to Reach Given Crack Size, and (b) Crack Exceedance Curve After5,000 Cyclos

45

1.0

0.9

0,8- 0 725 In.0.8

0O.5Teat Condition 5

0.4 - Titanium(ED) Pmax - 1.115o.0.3 Hyperbolic Sine Model

0.2

0L

10 15 20 25 30 35 40 45

Cycles, n -X 104

(a) PO 2Uo01

Test Condition 50.9 Titanium

Pnm - 1.115e - Hyperbollo Sine Model

07 - 100,000 Cycles

0.0S0.5

S0.4 -

0.1 -

0.44 0.48 0,52 0.56 0.80 0.64

Crack Size, a - In.DO 256O12(b)

Figure 34. Distribution Curve and Crack Exceedance Curve for Titanium for TestCondition No. 5, Using the Hyperbolic Sine Model; (a) Dibtribution ofCycles to Reach Given Crack Size, and (b) Crack Exceedance Curve After100,000 Cycles

46

A careful examination of the data indicates that the ext.apolation for some specimens isexcessive, due to the following reasons: (1) The crack growth rate data for each specimen coverdifferent ranges of AK, and therefore, it is impossible to assume a homogeneous testenvironment without excessive extrapolations for some specimens, and (2) The hyperbolic sinefunction has two asymptotes, and the extrapolation is very sensitive in the region of AK wherean asymptote occurs. In test condition Number 1, for instance, the crack growth rate data of onespecimen (Number 7AN1592) covers only a very small range of AK. Consequently, the best-fitted hyperbolic sine function results when the left asymptote falls into the region ofextrapolation as shown in Figure 35. Thetefore, the propagation life is too long as indicated inFigure 30 by numerical values. Consequently, it seems reasonable to censor this specimen. Withsuch a specimen being censored, the results are shown in Figures 36(a) and (b). It is observedthat the correlation is very reasonable. Similar situations exist for test conditions Numbers 2through 4.

The Paris crack growth model was also used to analyze statistically the Waspaloy andTitanium data.

From the pooled crack growth rate data for each teat condition, the method of maximumlikelihood was employed to estimate the parameters b and log Q as well as the standarddeviation a. -- v. The results are presented in Tables 13 and 14, respectively, for Waspaloy andTitanium. Furthermore, the crack growth rate data for each test specimen were best-fitted bythe Paris function to determine the corresponding b and log Q values. The results for eachspecimen are shown in Tables 15 and 16, respectively.

Following the same procedures described in Section III and using the assumed homoge-neous test environmenta given in Tables 11 and 12, the correlation between the statistical modeland the extrapolated test results was obtained. The correlation results for the distribution ofrandom number of load cycles to reach any given crack size and the crack exceedance curve aredepicted in Figures 37 through 44. It is observed that the correlation is vwry good for Waspaloydata sets whereas a considerable improvement in correlation has been achieved for Titaniumdata sets.

TABLE 12. ASSUMED HOMOGENEOUS TEST ENVIRON-MENTS FOR EACH TEST CONDITION FORTITANIUM

Test ao at B W P., Type ofCondition (in,) (ira.) --Lin.) (in.) (hips) specimen

1 0.4 1.7 0.3830 1.9987 1.40 MCT2 0.4 1.7 0.3890 2.4943 2.40 MCT3 0,08 0.4 0,3000 1.0000 7.90 CN4 0.4 1.7 0.5000 2.0000 4.64 MCT5 0,4 1.7 0.3440 1.9972 1.12 MCT

47

7I I I I . .. ... . " "-r

AK, MP& V'M

,,1-' 2 6 10 2 6 100 2 6 1000

7AN1692 +10-1

I - 10"~I

10-3• ............. ........ ...... ... .............. ................ .. ........................

10-s -0-2

/ i0.-s

10-4 ............................ .... ............ .................. .........................

10-3

SAK, kal \/,

F •igure 85. Titanium Crach Growth Rate Spe',imern at Test Condition No. 1!

'~ ~~~~ ~ ~~ 10-5 ..... '•. . ... . • , ;, ," i, .•.i•'... ....... ........ t,•• •r... . .. ,... ..... '..........

48 .--

10 - ..... ...... ...... ......... ........ ....... .....

1,0

0.9 a - 0,725 In. -

•0.8 Ga -1.700 Ini.a0.7

0.6

S0.4

0.3 "-. Test Condition I0 Titanium0.2 - Pro - 1,4

0,1 Hyperbolic Sine Model

6 6 7 8 9 10 11 12 13 14 15 16

Cycles, n - X 104(e) I-D 995013

(a) 1 AI

1.0.-

0,9J 0,8--

0,7

0,6

S0.5 Test Condition 1

1 0.4 TitaniumPMOX " 1.4

0.3 F Hyperbolic Sine Modeln - 30,000 Cycles

0.2 -

0.1- 00o wmim I , I L I0,40 0,42 0,44 0,46 0,48 0.50

Crack Size, a - In,FD 285014(b)

Figure 36, Distribution Curve and Crack Exceedonce Curve for Titanium for Test

Condition No. 1, Without Specimen No. 1582; (a) Distribution of Cycles toReach Given Crack Size, and (b) Crack Exceedance Curve After 30,000Cycles

49

•-i , .. . ... I :,

TABLE 13. MAXIMUM LIKELIHOODESTIMATE OF b, LOG QAND STANDARD DEVIA-TION a, = c. FOR WASPA-LOY (PARIgFUNCTION)

Test Condition b log Q 0, - a.1 2.1841 -8.0979 0,08262 2.1429 -7.8101 0,04753 2.4193 -.8,8940 0,0876

TABLE 14. MAXIMUM LIKELIHOODESTIMATE OF b, LOG QAND STANDARD DEVIA-TION o, " . FOR TITANI-UM (PARIS VUNCTION)

Teat Condition b log Q 0.- ay

1 3,6062 -9,4166 0,07972 3,240 -9.2847 0,11803 2,8691 -8,4737 0,12734 2,1210 -7,8039 0.09265 3.1094 -9,0305 0.0767

TAB'LE 1.. PARAMETERS b AND LOG QFOR EACH TEST SPECIMEN OFWASPALOY, (PARIS FUNCTION)

Test Condition I

.- S , oirn Ito, b lot Q0736 2.2388 -8,26421464 2,2937 -8,21061470 2,171 -8.46111304 2.0887 -7,9979

Teas Condition 2

Specimen No. b log Q

100o 2,5790 --8.47921301 2.1821 •-7.88261013 2.1302 -7,7809

Teat Condition 3

Spocimen No, b log Q

1002 2.4759 -8,66851003 2,8664 -9,23611018 2.1978 -8.2906

60

TABLE 16. PARAMETERS b AND LOG QFOR EACH TEST SPECIMEN OFTITANIUM (PARIS FUNCTION)

Teat Condition I

Spcimen No. b lo Q1535 3.8179 -9.7058153 3,989 -9.40061580 3.5938 -9.40741582 4.02•8 -10.0165162 3.6577 -9.420D1663 3.8253 -9,0413

1676 8.5012 -9.27971677 34t373 -9.4230l•0 a.760c -9,5302

172$ 2.Ta 2 -8.7004

Teat Condition 2

Specime N o. b log Q0423 3.3759 -9.2429

0444 8.189 -90,0721484 5.0674 -11.2012

is" 4.0291 -9.79M21886 3.4m$0 -9.2101

12 3.69220 -9,43"1704 4.8748 -10.0253

Te#t Condition 9

Specimen No. b Wo Q

1744 4.5104 -70.8830402 2.0T72 -8.65951741 8.4522 -95W481742 4A.9 -11.021

Te•t Condition 4

Specimen No, b tog Q

0451 1.9648 -7.21903K9 1.96119 -7,.8631

0415 2.1940 -8.55980410 2.4115 -7.91871248 1.7779 -7.1221101I 2,.74% -7.W062

188I 2.2311 -7.6245

Test Conditiont 6

S041. 5 -0.2172S041 2,.635 -8.5358, i0418 IgND -8.i)137i:. I'1678 2.9,Mb~l -8,.mu

lw 2 88 .7W.3 -. 6,4"

a

0.8 0.875 in.

ja -, 80.20 In.i .O in,

0.60,15

0.4 Test CondItion 1Waspeloy

0 P.3 - /m,. - 3.6&0,32m• Paris Model

0.2 -

0.1

2 4 6 6 7 8 9 10 11 12 13 14Cycles, n X 104

(a) FO 25 •60

Test Condition 10.8 W•spwloy

SPMA - 3,60.7 -Paris Model

0.6 n 50,000 Cyolam

S0.5

0,4

0.3

0.2 -

0.1

.65 .75 .85 .95 1.05 1.15 1,26

Crack Size, a - In.

(b) F) 266010

Figure 37. Distribution Curve and Craock Exceedanve Curve for Waspaloy for TestCondition No. 1, Using the Paris Model; (a) Distribution of Cycles to ReachGiven Crack Size, and (b) Crack Exceedance Curve After 50,000 Cycles

52

1.0

i0.8,-0*0.70,7 b.. a - 0.875 In.

0,5 - a2.0 In. >

j0.4 -2 Test Condition 2

.0.3 Waspaloy0.2 Pmax - 5.2

0.2 •Paris Model0,1-

yoliea, n - X 103

(a)

1.0

0.9

0,le8 Test Condition 2

0,7 WaspaloyPmax - 8.2

0.6 Paris Modeln 10,000 Cyoles

0.6

0.3

0,1

a6 .64 .68 .72 .76 .UO .84

Criok Sis, a - in.

(b) Fo 265016

Figure 38. Distribution Curve and Crack Exceedance Curve for Waspuloy for TestCondition No. 2, Using the Paris Model; (a) Distributiun of Cycles to ReachGiven Crack Size, and (b) Crack Exceedance Curve After I0,000 Cycles

1.0

0.8 -

i)•• i0.875In0.5

0.6 --- • - 2.0 In,

"t ,4 •Test CondItion 3013 Waspaloy

0.3 -,Pmax -. 4,0

0.2 Paris Model

0.1

013 4 5 8 7 8 9 10 11 12 13 14

Cycle, n - X 104

(a) .10 MOOl

1.0

0,9

0,8

0.7 - Test Condlllon 3• Waspaloy

0.8 - Pmax - 4.0Parrs Model

0.5 - n - 50,000 Cyo~es

0.4 -

0.1

.aa 171 .7a ,81 88 , .. 6

Crack Size, a - In,

"(b) FO 255020

Figure 39, Distributioit Cu~rve and Crack Exceedance Curve for Waspaloy for TestCondition No. 3, Using the Paris Model; (a) Distribution of Cycles to ReachGiven Crac. Size, and (b) Crack Exceedance Curve After 50,000 Cycles

-4

1.0oa- 0.725 In,

0.9

0.8 -

07a- 1.70 In.0.6-05 6

o.4 Test Condition ITitanium0.4 - Pmax - 1.4

0.2 -Paris Model

K 0.1 Q L5 6 7 8 9 10 11 12 13 14 15 16 17 18

Cycle., n -) X 104

(a) po 265021

1.0 -

0.9 -S0.8 •Test Condition 3

0,7 0D TitaniumC) Pmax - 1.4

0,6 Paris Modeln 30,000 Cycles

0.6 j.

* 0.4-

0.3

0.2 -0'

01 0

,40 .42 .44 .46 .48 .50 .52

Crack 81ze, a - in.(b) ro .-.

Pigure 40. Distribution Curve and Crack Exceedance Curve for Titanium for TestCondition No. 1, Using the Paris Model; (a) Distribution of Cycles to ReuchGiven Crack Size, and (b) Crack Exceedance Curve After 30,000 Cycles

55

S.. . . . . ... . .. ... ... .. . .. . . . . . ... . . .' .,."I

1.0

0.9 a 0.725 In..-

0.8 G0.7

0.4

•.~~ IL ,7 n

0.4 00,L

0.3 Test Condition 2Titanium

0.21- Pmax - 2.4Paris Model

1 2 3 4 5 6 7 8 9 10 11

Cycles, n - X 104

(a) FO 255023

1.0

0.9 Teat Condition 2a Titanium

0.8 Pmax - 2.40.7 Pairls Modfil

- n .- 20,000 Cyolas

0o6

~0.5 00,4

0.3

S0.2 -

0,1 - )I . I I - I.44 .5 .se .62 .68 .74

Crack Size, a - In.(b) I'D 255024

Figure 41, Distribution Curve and Crack Exceedance Curve for Titonium for TestCondition No. 2, Using the Paris Model; (a) Distribution of Cycles to ReachGiven Crack Size, and (b) Crack Exceedance Curve After 20,000 Cycles

56

0.9

0.8"

0.7 a 0.16n. - 0.4 n.

00-

0.4 0 Test Condition 3

03 Titanium0.3 -P,.. - 7.933

0,2 Pads Model

02

0 10 20 30

Cycles, n - X 103

(a)

1.0

*0.8 0Test Condition 3

TitaniumPm,, - 7.933

0.6 - Paris ModelN - 5,000 Cycles

" 0.4- 0

I 0.2

0.

0.10 0.12 0.14 0.16 0.18 0.20

Crack Size a, - In.

(b)

"Figure .12. Dis8tribution of Cycles to Reach Given Crack Size for Titanium for TestCondition No. 3, Using the Paris Model; ý'v) Distribution of Cycles to ReachGiven Crack Size, and (b) Crack Exceedonce Curve After 5,000 Cycles

57

' ... .. .. .. L, : ... • • ' . - .... . . •' *... , .;i;. .& -t • • • a • - - .' ..• * .•w ,• •-

A

1.0

0.8- / L

06 - 0.725 In.0.6 -

a - 1.7 In.

0.4

Test Condition 4Titanium

0.2 - Pn, - 4.64Paris Model

0100 10 20 30Cycles, n - X '103

0.9-

0.8 - Test Condition 4

0.7 TitaniumPmaw " 4.64

0.6 - Paris Model

0.5 N - 5,000 Cycles

S0.4-

j 0.3 -

0.1 -

010.5 0,6 0.7 0.8

Crack Size, a - In.

(b)

Figure 43. Distribution of Cycles to Reach Given Crack Size for Titanium for TestCondition No. 4, Using the Paris Model; (a) Distribution of Cycles to ReachGiven Crack Size, and (b) Crack ExcOedance Curve After 5,000 Cycles

58

wo~5

'1110.6 -1.70 In.0.5

0.4 Test Condition 5

0.3 Titanium~~oPma .- a 1.701In

t0. -- Pmaris ModelI 04TatCnDto0.1mx- .1

00

L ,10 2,0 30 40

SCycles, n -- X 104

(a) FO ,02

Test Condition 5

0.9 Titanium0(• Pmax - 1.115

S0.6 Parli Modeln `0100,000 Cyoea

0.1

: 0.6

-0.4 ---'i. i 0.3

0.2

.40 .44 .48 .52 .56 .60Crack Sie, a - In.

(b) 20 50

Figure 44. Distribution Curve and Crack Exceedance Curve for Titanium foi TestCondition No. 5, Uasng the Paris Model; (a) Distribution of Cycles to ReachGiven Crack Size, and (b) Crack Exceedance Curve After 100,000 Cycles

Ih

SECTION V

STATISTICAL ANALYSIS OF A HOMOGENEOUS DATA SET

A homogeneous crack propagation data set for 2024-T3 aluminum canter-cracked

specimens under constant amplitude cyclic loading generated in References 3 and 4 is analyzed.

The lognormal statistical model, developed under the present program, and an advanced model

currently being developed under another program, have been used. The results of such analyses

reveal the importance of both the selection of the crack growth rate function and the data

analysis technique used for deriving the crack growth rate data from the experimentcl

measurements.

Sixty-four sample functions for the crack growth damage accumulation, a, versus the

number of load cycles, n, are shown in Figure 45, The initial half-crack length is 9 mm and the

final half-crack length is 49.8 mm. The crack growth rate data, da/dn, versus the stress intensity

range, AK, are derived from Figure 45 using the seven-point polynomial method and the secant

method. Based on the hyperbolic sine crack growth rate function, (Equation 1), the method of

maximum likelihood is employed to estimate the parameters C2, Cj, C4 and the standard

deviation cy, = a. (see Reference 1). The results are shown in Table 17. It is observed from Table

17 that the statistical dispersion of the crack growth rate, da/dn, is smaller when the seven-point

polynomial method is applied.

49.8

41.5

EE 33.2

S24.9

'16.6

8.3

00 8 16 24 32 40

Cycles, n - X 104

Figure 45. Homogeneous Data Set of Virkler, et al. with P,. = 23.35 kn and R 0.2

Using the lognormal statistical model and integrating the crack growth rate equation

(Equation 1), for different y percentiles, one obtains the distribution of the half-crack length as

a function of load cycles n. The results one shown in Figure 46, in which the seven-point

polynomial method has been used. Note Lat the stress intensity range AK for the Wcr

specimen is given by

AP 7fraAK= -]Wa--sec (Ora/2W) (30)

60

TABLE 17. MAXIMUM LIKELIHOOD ESTIMATE OF C2, Cap C4,STANDARD DEVIATION - a, AND COEFFICIENT OFVARIATION, V, OF da/dn; C,- 0.5

Analys isTechnique C' CaC YYY €, V

7-Point 3.4477 -1.3902 -4.5348 0.08235 19.18%Polynomial

Secantmethod 5.0148 -1.1336 -5,0733 0.09589 22,35%

5%1 %25%60%75%90% 98%49.8 -45•

4011 35I.I

~30

265

20

15

0 4 G 12 16 20 24 28 32 36 40Cycles, n - X 104 F 26031

Figure 46. Lognormal Statistical Model Prediction of Virkler, et al. Data Set

A comparison between Figures 45 and 46 indicates that the lognormal statistical modelcorrelates well with the tost results for the 50% crack propagation life. However, the statisticaldispersion of the crack growth damage accumulation based on the model is larger than theexperimental test results.

The dlistribution functions for the number of cycles to reach half-crack lengths of 21 mmand 49.8 mm are presented in Figures 47 and 48, respectively. In these figures, the solid curvemid the dashed curve represent the results based on the lognormal statistical model. The formeris obtained using the 7-point polynomial method, whereas the latter is obtained using the secantmethod. The experimental test results, Figure 45, are shown by circles in Figures 47 and 48.

61

v . . . ... , . . .. ,... .. .

1.0 '

0.9r

0.8 IJI

0.7 =a1 - 21 mm

-- Lognormal Model"0.6 7 Point Polynomial

Method"" 0.15. Lognormal Model

A Secant Method..4- ........ Adv Model

7 Point PolynomialMethod

0.3 ------ Adv ModelSecant Method

0.2 / 0 Test Results

0.1

0.0 10 12 14 16 18 20 22 24 26 28 30

Cycles, n - X 104

Figure 47, Distribution Function of Number of Cycles to Reach 21mm Half-CrackLength

The probabilities of crack exceedance at n = 150,000 cycles are shown in Figure 49 as asolid curve, a dashed curve, and circles which have the same meaning as those in Figures 47 and48.

As observed from Figures 47 through 49, the correlation betwoen the lognormal statisticalmodel and the test results is very good for the 50% crack propagation life. However, the modolpredicts a larger statistical dispersion for crack growth damage accumulation than the actualtest data which results in a conservative prediction for the crack propagation life. Furthermore,the seven-point polynomial method appears to be superior to the nccant method.

The observation that the lognormal statistical model results in a larger statisticaldispersion for the crack growth damage accumulation was expected as described in Reference 1.This stems from the assumption that the crack growth rate is completely correlated and hence Zis a random variable. At another extreme, wheii the crack growth rate is assumed to becompletely uncorrelated at any two different values of log AK, oile obtains a white noise process.The white noise process model results in the smallest statistical dispersion for the crackpropagation life (Reference 1) as evidenced by the results presented in References 3 and 4.However, for the white noise model, the prediction for the crack propagation life isunconservative.

62

W1014"

1.0

0.90.8 ,* r

0.7

0a - 49.8 mm

0.5 - Lognormal Model7 Point PolynomialMethod

0.4..... Lognormal ModelSecant Method

0. 3 - ....... Adv Model7 Point Polynomial

0.2 " - Method-----. Adv Model

/ / /Secant Methodo Test Results

0.0 ,0

18 20 22 24 26 28 30 32 34 36 38

Cycles, n - X 104

Figure 48, Distribution Function of Number of Cycles to Reach 49.8mm Half-CrackLength

The actual test results in Figure 45 indicate a definite correlation for the crack growth rateat different values of log AK. Therefore, a statistical model taking into account such acorrelation has been explored in Section VI, which is referred to as the advanced statisticalmodel. A complete development for the analytical solution of such a model has beenaccompl~sbed under the sponsorship of another program (Reference 28). It cem be shown that

the statistical dispersion of the crack propagation life decreases as the correlation for the crackgrowth rate reduces.

Based on the advanced model with the correlation parameter A = 40,000 (see Section VI),the distribution functions for the number of cycles to reach half-crack lengths 21 mm and49.8 mm are indicated by dotted and dash-dot curves in Figures 47 and 48. The dotted curverepresents the results of the advanced model using the seven-point polynomial method, whilethe dash-dot curve denotes the results using the secant method. Similarly, the crack exceedancecurve at n - 150,000 cycles is shown in Figure 49 by dotted curve and dash-dot curve having thesame meaning as those in Figures 47 and 48.

An indicated from Figures 47 through 49, the advanced model with the method of seven-point polynomial correlates well with the experimental test data (circles). For the advancedmodel, the 7-point polynomial method is definitely superior to the secant method.

63

1.0 0

0.0 n 150,000 Oycles0.- - - Lognormal Model

7 Point Polynomial

0.8 MethodLognormal ModelSecant Method

0.7 \ .......... Adv Model7 Point Polynomial

j06 MethodAdv ModelSecant Method

0.5 0 Test Results

S0.4

* " -4 .0.3 -4\

0.1 -''%,,

0.0 L.-, " ""'12 14 16 18 20 22 24 26 28

Half Crack Length, mm

Figure 49, Crack Exceedance Curves at 150,000 Cycles

In conclusion, the Paris crack growth rate function, Equation 13, has been used inconjunction with both the lognormal and advanced statistical models. The correlation betweenthe test data and the prediction based on the Paris crack growth rate funtion is extremely pooreven for the 50% crack propagation life, Therefore, it is not worthwhile to present the results.

The results of analyses for the homogeneous data set presented above indicate two salientfeatures associated with the prediction of fatigue crack propagation. For a particular type ofspecimen under a particular type of fatigue loading, one should select a crack growth rate

function which can best describe the crack growth rate behavior. For the homogeneous data Betanalyzed, the hyperbolic sine function fit the crack growth rate data very well, whereas the Parisfunction did riot. Using the method of maximum likelihood, the goodness-of-fit can be judgedfrom the standard deviation ff = a,. A crack growth rate function has a better fit if thecorresponding standard deviation a. is smaller.

Data for the crack size, a, versus the number of load cyci L3, n, are measured directly fromexperiments. Then, the crack growth rate data are derived from a versus n measurements.Various analysis techniques for obtaining the crack growth rate data have been propocud in theliterature, such as tho secant method, the method of seven-point polynomial, etc. Unfol.tunately,the accuracy for the statistical prediction of fatigue crack propagation depends on the datareduction methodology employed. Further research is needed to identify a best technique forthe reduction of the crack growth rate data as well as the measurements of a versus n.

64

SECTION VI

ADVANCED STATISTICAL MODEL FOR FATIGUE CRACK GROWTH

Several mathematical models have been proposed for the prediction of crack growthdamage accumulation for structures under dynamic loads based on the principles of fracturemechanics (References 6 and 7). These models have the general form of

da-da- Q(K,AK,s,a,R)

(31)

where a(t) - crack size at time t, Q - a non-negative function, K = stress intensity factor, AK =

stress intensity range, s ý stress amplitude, R =stress ratio. For instance, the well-known Parismodel is given by

da-d•- Q,(AK)b

(32)

AK , a(a) As V (33)

where As = stress range, a( ) = a function depending on the specimen and crack geometries.However, even in a well-controlled laboratory environment, results obtained from crack growthexperiments under either a constant-amplitude cyclic loading or a given spectrum loadingusually exhibit considerable statistical variability, This is illustrated in Figure 50, which showsthe actual crack growth records of some fastener hole specimens subjected to the excitation of aspecific load spectrum in a laboratory. It is, therefore, not surprising to find that statisticalanalyses have been applied quite frequently to such problems in recent years (References 6through 22).

If we restrict our attention to a laboratory setting so that the loading time-variation isdeterministic, then the mathematical model, Equation (1), can be "randomized" as follows-

driS- Q(K,AK,s,i,R) X(t) (34)

where the additional factor X(t) is a non-negative random process, and ii(t) is a random processrepresenting the crack length at time t. It is of interest to note that Virkler, Hillberry and Goel(References 3 and 4) have undertaken simulation studies of crack propagation which amount toamuming X(t) in Equation (34) to be totally uncorrelated at any two different times. At theother extreme, Yang (References 1, 22, 17) has replaced the random process X(t) in Equation(34) with a random variable, which is equivalent to the case where X(t) is totally correlated at alltimes. It was pointed out in Reference 1 that a totaily uncorrelated X(t) would lead to thesmallest statistical dispersion and a totally correlated X(t) to the greatest statistical dispersionfor the time at which a given crack size is reached. A nwore realistic modeling of fatigue crackgrowth should lie momewhere between the two extremes. Therefore, the ability to account for anarbitrary correlation in X(t) is a major consideration In the present section.

tC

0.04

0,03

S0.02

0.01

0 2 4 6 8 10 12 14 16 18 20

Flight Time, t X- ) 103FD 255038

Figure 50. Actual Crack Propagation Time-Histories of Some WPB Fastener Holes

From the ztandpoint of fatigue design and schedule maintenance (References 11, 14, 15),

two types of statistical information are of interest: the distribution of the random crack size at

any given time, and the distribution of the random time to reach a given crack size. As shown in

Reference 1, these two problems are interrelated. Attention will be focused on the statistical

properties of the latter problem. In particular, analytical solutions will be given for the

statistical moments of the random time to reach any crack size, given the knowledge of an initial

size. A procedure to estimate the parameters in a power-law mathematical model also will be

presented, using the fractographical results (Reference 23) of some 7475-T7051 aluminum

fastener hole specimens subjected to the excitation of a bomber load spectrum.

1. Model for X(t)

We shall model X(t) as a random pulse train (see Reference 24), i.e.,

X(t) -& Zkw(t,rk) (:g.)k-1

"

where N(t) - a homogeneous Poisson counting process, giving the total number of pulses that

arrive within the time interval (-co, t); rk = the arrival time of the kth pulse; Zk = the random

amplitude of the kth pulse, and1, ~ ~0 < t-i S (6

w(ti) = w(t.r) =(36)

0, otherwise

66

We further assume that Zk for different k are independent, identically distributed randomvariables, with a common probability distribution Z,

A typical sample function• of X(t) is shown in Figure 51.

7 1 r2 r3 r4Time, t

FD 2a503r

Figure 51. Typical Sample Function of Random ProcesA X(t)

The 3tatistical properties of X(t) can be described by its cumulant (or semi-invariant)functions. The ruth cumulant function im given by:

K.JX(t1 ),.,.,X(t.)I - E[Zp],fM " min w(t, - ) .,,w(t. - r)dr

in which E[ ] denotes an ensemble average, X is the average arrival rate of the Poisson process,and min( ) indicates the smallest of the parenthesized quantities. In particular, the firstcumulant is the mean function, and the second cumulant is the covariance function. These arefound to be:

- EIX(t)] - E-H[ZlXf w(t-r)dr

SIZ]iX fw(u)du - EJZIXA (38)

67

:1'! --...' .. .. " • •,, . .- - . . .... .. . .. .... .. ...... ...... ... . .. i

and

Cov[X(t,), X(t 2)] E[Z 2pX f'1 w(t, - T)w(t 2 - r)dT

"E[Z'pX f7 w(u)w(t 2 t1 + u)du

I 2 o9(l -At-t/), I - , <

_"0' t A.- a,>• ,,(39)

in which# = E [Z2] XA.

2. Approximation of ! (t) by a Markov Random Proceso

We now re-write Equation (84) as follows:

•r • -- Q(0)i0 + Ylt)(40)

where the dependence of Q on K, AK, a and R has been suppressed for simplicity. Clearly, Y(t)is a random process with a zero mean and the correlation function of Y(t) is the same as thecovariance function of X(t); namely,

S2•(1- u/A), H < A,y EI") - E1Y(t) Y(t+T)] 2( (41)

O, otherwise

A sketch of this correlation function is shown in Figure 52. If the correlation time of Y(t) is shortcompared with the characteristic time of N (t), then i (t) is close to a diffusive Markov process(Reference 25) which is governed by an Ito's stochastic differential equation (Reference 26):

di - m(l,t)dt + u(C,t)dB(t) (42)

where in is called the drift coefficient, Y the diffusion coefficient, and B(t) is a unit Brownianmotion process (also called the Wiener's process), which has the property that dB(t1 ) and dB(t.)are Independent for t, 0 tg.

Tho correlation time of Y(t) may be defined as follows:

- f ,Rvv(r)l dr-; car JR-lyT)l d-r (43)

Substitution of Equation (41) into Equation (43) results in ro = A/3.

68

4-. . .. . -, -l

2JII

R ,(t)

2f#

FD L55039

Figure 52, Autocorrelation Function of Random Process Y(t)

Strictly speaking, the W (t) process in Equation (42) is ani approximation of that inEquation (40), and they could be represented more clearly by two different symbols; however,the same symbol will be used in this report for both processes as long as no confusion will result.

Stratonovich (Reference 25) has given the required formulas to compute the drift anddiffusion coefficients from the original physical equation when the Markov approximation isjustified. In the case of Equation (40)

m -QA + f Q H t t+r d

QY + Q ,,,( Q-(A + #A Qa (44)

2 f , Q' J Y(t)Y(t + -)Idr 2Q2#4k" '" (45)

where io > A. These equations imply that Q and •iQ/i'a vary slowly within the integrationinterval to justify their being taken outside the integrals. Thus, the integrals account basicallyfor the contribution towards the drift and diffus;1on due to the correlation between the past andthe present "excitations" Y(t). This contribution is lumped at the present, when m and a areused in Equation (42). The replacement of Equation (40) by Equation (42) amounts tosubstituting Y(t) by a white noise. Theoretically, the substitution introduces a small errorassociated with a small probability for da to become negative. This error is negligible so long asthe drift dominates the diffusion, which should be verified in each practical case. Stratonovich'sformulas are applicable to other types of correlation functions for Y(t) as long as Q and dQ/davary slowly within an interval of r where such a correlation is not negligibly small. In this case,the lower limit of integration can even be extended to - oo. Stratonovich's method is known asthe stochastic averaging method, originally prop'sed on a physical ground, but later provedrigorously by Khasminskii (Reference 27) and gwven a rgorous mathematical interpretation,

69

The transition probability density qa (a, t I no, to) of the Markov process K (t) is aconditional probab-Aity density which describes the distribution of1 (t) under the condition thatthe initial crack size is A (to) = ao at an earlier time t0. It is governed by the following Fokker-Planck equation (Reference 26):q4 +.- ai -Q(q)- o

-t a [ q - a (46)

or by the adjoint of Equation (46):q, 1A [Q(+0, Q -A q+ Q-{-)J~ + Q~/iA-~- (q "- 0

a (47)

subject to the condition,

q.(a,t0 I ao,t,) - h(a-a0)

(48)

In Equation (46) Q is treated as a function of "a" which is a sample ,alue of the crack size R (t),whereas in Equation (47) it is treated as a function of ao. These equations are also known as theKolmogorov's forward and backward equations, respectively,

3. Time to Reach a Given Crack Size

We now focus our attention on the random time when the crack size A (t) reaches a specificvalue a,. This event may be considered as the passage for the first time across an absorbingboundary, Introduce g(a, t I ao, to) such that

g(a, t aIn. t,,)da - Prob Ia : U(t) _- a + da I 0 n Z(T) < ad, to -5 T < tJ,

so -< a < a,, (49)

Although g is similar to q.a it is not a probability density since its integration on a from a0 to a,

is generally smaller than one. In fact, g describes only the sample functions which do not reachthe absorbing boundary al before time t. However, g also satisfies the Kolniogorov backwardequation, Equation (47).

The integration of g yields

CT(t) -. f.",, 04a t I n,,, t,) da

Prob survival in (t,,t) I (tu) na (50)

Here the dependence of G on ao and to has been suppressed. The G(t) function also satisfies theKolmogorov backward equation, namely,

G 2

G+Q(M+U - )- G + 0 (U)G

70

subject to the following conditions:

G(to) - 1 (52)

G(oo) - 0 (53)

Condition (53) indicates that sooner or later fatigue failure will occur. Letting r - t - to, weobtain

) .'G + ± Q 2P ALG -0

Tr - 08 0 86a (54)

Now C(t 0) - G(t) - 1 - G(t) is the probability thaqt first passage (reaching the crack size a,)

occurs prior to t, This is the distribution function of the random first passage time T; i.e.,

Foy(r) - 1 - G(to + -) (55)

The probability density of T follows froLr1 a differentiation of Equation (55);

+ 7) (56)

The average first pasage time is

E[T - - fr ý- dr - f:G(to+ 7)dd ()

In obtaining the second part of Equation (57) use has been made of tho condition (53).

Integrating Equation (54) and using Fquations (56) and (67), we obtain an equation for theaverage first passage time:

+ Q(A+ A - , E[T] + Q da& E[T] - 0(58)

The above second order equation requires two boundary conditions, one of which is clearly

E[T] - 0, ifa 0 - a, (59)

We shall assume that another condition is

-a-E[T] - 0, if ao- 0 S(60)

Equations for higher order moments can also be derived from Equation (54). Note that

E[Tm ] - - d -- nfi -G(t, + O)dr (61)

S71

Multiplying Equation (54) by T' and integrating on T, we obtain

(n+ 1)E[T+J + Q + O q-l] EITr + _C 0+(62)

subject to the conditions:

E[T" + 11 . 0, if , - a, (63)

(63)

-Ia- 4 ~EITI1+ 1] -O0 ifa 0o- 0 (64)

It is clear that Equation (64) which inchldes Equation (60) as a special case implies that a 0 e isa reflective boundary. Equation (62) can be used recursively to obtain higher moments from thelower order moments. Equation (62) reduces to Equation (58) when n = 0.

Note that Equation (62) is first order in dE[Ta" ]/daO which can be solved readily. Let QQof(N) where Qo has the unit of length/time and f(.) is a dimensionless function. We obtain

d +!. da0 •m oE[T " +] " - exP - - f

T e (65)

where

A - 0 #A -Q (66)

in which the common practice of using the same symbol for the integration variable and limit ofintegration has been adopted for convenience. This solution satisfies condition (64), Equation(65) implies that d/d% E[T7+1] is nonpositive or E[TO"+] is non-increasing when the initial cracksize N is increased, which is physically reasonable.

The (n+ 1)th moment of T follows from integration of Equation (65):

""[Tn + 11 - - E[TM + ']dao (67)

which satisfies condition (63).

4. Power-Law Crack Propagation

We now consider a special form for the Q function which has been proposed by severalauthors (References 13 through 16):

Q Qo,(/a .) (68)

72

L..

. - • ' •- , - , .. . _•. . . • - -....-... ' ,. . .. . . ' , * , . . . . - - • ' * ". . .. . ' • •

aid in the case of backward equations, we have Q = Q0(ao/ag)b, In this case, the integration in

Equations (66) and (67) can be carried out in closed form. Specifically

dao

-•o Qnu, b -i(69)

where u ao/a.. To compute E[T], we require Jo. From Equation (66) tind with n = 0,

1 [exp k u I.b Hkl-b)], b - 1

Jo -

U k b- I(70)

adswhere k and H is the Heaviside function, i.e.,

H(i-b) 1, b < (1/ 0, b > 1 (71)

Equations (69) and (70) are substituted into the expression for dElTI/dao which is thenintegrated to obtain E[T]; the results are

(1i-u'-b) 4- H(I-b) ( -)[exp(-

EITi- exp( - k _b) b

(72)

a,: - -Q - - 2 n u , b - 1

To compute E[TI] we require

- , f EiTjd[exP(-- f .3)]

73

Substituting Equation (72) into Equation (73), we obtain

01k U - b X b)

+ H(1-b)) [ (___,

- (u- + 1)- b 1 (74)

04s2

it- U ( 1 nu), b 1 (75)

In obtaining Equation (75), we have used the formula

f: ua.n u du - - fuadu d'J-v

- (a+1) -2x+l[(a+1)Qnx-1] (76)

The second part of Equation (76) tollowa from changing the order of integration.

Substitute Equations (74) and (75), respectively, into the expression for dE[T']/da, andintegrate1

•• - ( + 7-)(1---uI -•H90lE[VI + 1 ) ~(I_ U1- b)(- u 1-b)

2ak 1 I - )+U - b)+k

Inotann Euton (78-ehaeue the'- f-=ormula:" ,.-•-)

-e-'-p (1+ 2 + --L-exp

b I (77)

041

l-aT (78)

In obtaining Equation (78) we have used the formula:

Th fx- R inu du - - fx-'du f'-dv - - -2-nx))u (79)

The second part of Equation (79) follows from an interchange of the order of integration.

5. *stlmetion of Model Parameters

The estimation of the parameters of a mathematical model will be illustrated by use of twoexamples. Refer again to Figure 50 which shows the crack propagation time-histories of dome

74

t _________

specimens selected from the fractographical data of 7475-T7351 aluminum fastener holesa',ailable in Reference 23. The data set is identified as WPB, indicating that the specimens weredrilled with Winslow Spacematric machines (W), with a proper drilling technique (P), andsubjected to a given B-1 bomber load spectrum (B). The data set has been censored to include 9only those specimens having fatigue crack growth through the crack length interval from 0.004inch to 0.04 inch (corner crack). This censoring procedure is necessary in order to ndrmalize thedata set to zero life at 0.004 inch, thus obtaining a homogeneous crack growth data base. Theresulting data set consists of 16 specimens. Within a small crack size range, the power-lawpropagation, Equation (68), was shown to be valid (References 13 through 17).

The parameters of our mathematical model, specialized to a power-law crack propagationthat fits the behavior of the above WPB specimens, have been estimated using the experimentaldata. We began by taking logarithms on the two sides of Equation (68), with Q - Qo(a/a,)b toyield

da(t)

log dat b log a(t) + log(Q 0/a•) + log X(t)(8o)

Variations of log d /dt versus log K were plotted for the 16 specimens in the WPB data set asshown in Figure 53. Implicit in the mathematical model is the assumption that all points wouldfall along a single straight line if the random element log X(t) were not present. Since X(t) hasbeen assumed to be a stationary random process, the mean and the standard deviation of logX(t) are constants. A linear regression analysis was then carried out to estimate the slope, theintercept, and the standard deviation of the random element log X(t). The slope is equal to b,and the intercept is log (Q0/ ab),

The results obtained from the 16 specimensi are

b - 0.92971,

Q0/a. - 1.1051 X 10"4,

•gX(t) =0,087635

The linea~r regression analysis imples that each point in Figure 53 has been treated as anindependent sampling of a Gaussian random variable. Thus, estimates of the mean and varianceof X(t) itself can be computed from those of log X(t), using the log-normal to normal conversionformulas; namely,

ft- lp P Lp x(t)X 10]j -- 1 -]

S(82)

Application of Equations (81) and (82) resulted in

p - 1.0206, 0,021643

75

-4

-5

-7

-8i i, J- -2 -

log a

Figure 53. Regression Analysis for the Estimation of Model Parameters for WPBFastener Holes

The rationale behind these conversion rules is the use of linear regression to obtain the meanand standard deviation for log X(t). It does not necessarily require that X(t) must be log-normal. In fact, when the crack size 1(t) is treated as a diffusive Markov process in anapproximate sense, the "excitation" process X(t) is also effectively replaced by a constant meanA plus a Gaussian white noise. Of course, this replacement excitation process cannot be log-normal.

Having the values of b, Qo, )A and 0, the mean and mean-square values of the time T toreach a given crack size a, can be computed using Equations (72) and (78) for different values ofk = ajs!(#4Q0), or equivalently for different values of A. Figure 54 shows the results of suchcomputations, in terms of E[T] and F[T] j± aT, where aT - (E[T'] - E[T] 1)112 is the standarddeviation of T. The mean values E[T] are practically uninfluenced by the choice of A since thelast two terms in Equation (72) are several orders of magnitude smaller than the first term.However, the statistical dispersion UT increases as A increases. This A--oT relationship agreeswith our earlier observation concerning the results of Yang (References 1, 22), and of Virkler,Hillberry and Goel (References 3 and 4), the former case being equivalent to A --+ co and thelatter case to A --* 0. Estimates of the mean E[T] and standard deviation, UT, of T were alsoobtained directly from the data of the 16 specimens, and the results plotted in Figure 55.Comparison between Figures 54 and 55 shows that A 8,000 is a reasonable choice for thisparticular mathematical model and data set.

76

0.040

0.032 1.A 5002.A 1,000

. - 3. A - 5,0000.024 4. A- 8,000"0.024 - 5. A - 10,000

0.016

0.008

0.0I I0 2 4 6 a 10 12 14 118 18

Flight Time, t - hr X 103

Figure 54. Theoretical Mean and Standard Deviation of Random Time to ReachVarious Crack Sizes Computed for Different A Values for WPB FastenerHoles

0,040

0.032

0.024

0.016

S0.008

0.0 2 4 6 8 10 12 14 16 18Flight Time, t - hr x 103

Figure 55, Mean and Standard Deviation of Random Time to Reach Various CrackSises Computed Directly from Some Actual Time-Histories of WPB FastenerHoles

77

S-

Some computer simulated sample functions using the present mathematical modeldescribed in Equations (34) and (35) are shown in Figure 56. These have been obtained using"parameters O&X- 1.0206, vXr f 0.043286, b = 0.92971, Q0/ah - 1.1051 X 10-4, A = 8,000 and anaverage Poisson pulse rate of X f 0.1. Their general characteristics are remarkably similar to theactual records in Figure 50.

0.04 ,

0.03 ..

IdI.

0.02 " . ' *

0.01

o.0

0 2 4 6 a 10 12 14 16 18 20

Flight Time, t - hr X 103

Figure 56. Simulated Sample Functions of Crcick Propagation Time-Histories for WPBFastener Holes

It has been suggested that the distribution of the time T to reach a given crack size a. canbe approximated by a two-parameter Weibull distribution (Reference 1),

FT(t) - PITSt] I - i-exp[-(t/0o)"fO (83)

in which a0 and f0 are, of course, related to the values of E[T] and av. Approximate distributionfunctions of T thus computed are shown in Figure 57 as solid curves for a. = 0.01, 0.02 and 0.04inch, respectively. Also displayed in Figure 57 as circles, triangles and rectangles are thecorresponding distributions of the test results obtained from Figure 50. It is observed fromFigure 57 that on the basis of the approximation, Equation (83), the test results correlate verywell with the present statistical fatigue crack propagation model.

For additional comparison, the above procedure has been applied to another set of data, tobe referred to as XWPB where X signifies a 15% load transfer in the fasteners and WPB has thesame meaning as before. This second data set also has been censored to include only those witha crack length growth from 0.004 to 0.07 inch, resulting in a total of 22 specimens. The crack

propagation time-histories of these specimens are shown in Figure 58. The linear regressionanalysis for the estimation of model parameters, illustrated in Figure 59, resulted in

b = 0.985, Q0/a, - 2.4414 x 10 -, ,x(t) 0.12896

'78

1.0- O0 ", ,(JA 0

0 a

0.2- 0• 0

_ 0 , 0

01A

0 A aa

0.0

0 2 4 I 8 10 12 14 16 18 20Flight Time, i - hr X 103

Figure 57, Comparison Between Weibul-Type Approximation for the Distributions ofRandom Time to Reach a Given Crack Size and Actual Test Results forWPB Fastener Holes

An application of Equations (81) and (82) yielded

1.0451, 6, -0.0503

The computed E[T] ± aT curves for a number of A values are shown in Figure 60. When theyare compared with the specimen mean and specimen standard deviation, shown in Figure 61, thebest agreement is obtained for A - 8000. Approximate distributions of the random times toa, - 0,008, 0.025 and 0.07, based on a Weibull form, Equation (83), are shown in Figure 62, alongwith the sample distributions. Computer generated simnlations using A - 8000 and X = 0.1 areshown in Figure 63, Again, excellent theoretical end experimental correlations are seen in thissecond example.

79

0.07

0.04 'I"

0.01

08 12 16 20

Time, t - hr x 10'

Figure 58, Actual Crack Propagation Time-Hiatories of Some XWPB Fastener Holes

-4

-7

S-3 -2

log a

Figure 59. Regression Analysis for the Estimation of Model Parameters for XWPB

Fastener Holes

80

0.0700.064-

.1. 4- 500

2. A- 1,000 33. A- 5,000 4

0.048 4. A - 8,000

S0.032

0.016

0.00 2 4 8 8 10 12 14

Flight Time, t - hr X 103

Figure 60, Theoretical Mean and Standard Deviation of Random Time to ReachVarious Crack Sizes Computed for Different A Values for XWPB FastenerHoles

0.070

0.084

0.048

0.032

0.01 -

0 2 4 a 8 10 1U 14

Flight Time, t - hr x 103

Figure 61. Mean and Standard Deviation of Random Time to Reach Various CrackSizes Computed Directly frow, Sonw Actual Time-Histories of XWPBFastener Holes

81

1.0- 0

0.8[A

-0.4

0A0 A 0

0.2 A

000o

0.00 2 4 e 8 10 12 14 16

Flight Time, t - hr X 103

Figure 62. Comparison Between Weibull-Type Approximation for the Distribution ofRandom Time to Reach a Given Crack Size and Actual Sample Distributionof XWPF Fastener Holes

0.07

0,04 -.. .

.004

0.02 -§J• j:• r 2

0.01

0- 6 8 10 12 14 is

Flight Time, t - hr X 103

Figure 63. Simulated Sample Functions of Crack Propcgation Time-History for XWPBFastener Hales

82

I ,

SECTION VII

CONCLUSIONS

1. Spectrum Loading

A fracture mechanics-based statistical model developed in Reference 1 for the predictionof fatigue crack growth behavior under any single test condition has been extended to the case ofspectrum loading. The distributions of propagation life to reach any given crack size and thecrack size at any specific number of load cycles were obtained. A specific block loading waschosen and applied to IN100 laboratory test specimens to obtain statistically meaningful crackpropagation test data for the verification of the statistical model. The same block loading wasused for the theoretical prediction of the crack propagation behavior of IN100 baaed on themodel, Comparison of the theoretical results with the verification test data indicated areasonable correlation. The model provided conservative predictions of both the mean behaviorand the variability of the data. Thus, the proposed fracture mechanics-based statistical model isquite practical, since it requires only a small nonhomogeneous data base for predicting crackpropagation under spectrum loading. Crack growth retardation or acceleration effects due to theapplied spectrum (temperature, loading) must be addressed in further work on this model.However, the theoretical model is judged appropriate for practical applications due to itssimplicity and conservative predictions.

2. Related Developments

A statistical model based on the Paris crack growth rate function for fatigue crack growthin IN100 was investigated and applied to test results at various temperatures, loadingfrequencies, and stress ratios, Again, the distributions of life to reach any specific crack size andthe crack size at any specific number of cycles were obtained. Homogeneous test environmentswere assumed to avoid excessive extrapolation into the region where crack growth data did notexist, The life integration was then based on the best-fitted crack growth rate parameters. Thisapproach was necessary in order to obtain the homogeneous data sets, referred to as theextrapolated test results, for the correlation study,

The correlation between the statistical model and the extrapolated test results wasreasonable, Care should be taken in the application of this model as the Paris crack growth ratefunction is applicable only to a certain region of the stress intensity range. This model ismathematically simple and practical for engineering applications.

Closed form solutions have been obtained for the statistical moments of the random timewhen a dominant crack reaches a given size for a rather general class of crack growthmechanisms. The key to this success was the approximation of the random crack size by adiffusive Markov process. Theoretically, the approximation introduced an error associated withthe possibility for the crack propagation rate to be negative at times. However, in terms ofstatistical properties, the error was negligible if the tendency for drift dominates the tendencyfor diffusion. The application of the proposed theory to two real examples seemed tosubstantiate our contention. The theoretical results have been shown to correlate well with theexperimental results when the parameters for the mathematical model were obtained from alinear regression procedure, Since the calculation of a few key parameters using such aprocedure dose not require a very large data base, the proposed theory is quite practical in viewof the limited experimental results available at the present time.

83

REFERENCES

1. Yang, J. N., G. C. ,alivar, and C. G. Annis, Jr., "Statistics of Crack Growth in EngineMaterials - Vol. 1: Constant Amplitude Fatigue Crack Growth at Elevated Tempera.tures," Air Force Wright Aeronautical Laboratories Technical Report, AFWAL-TR-82-4040, June 1982,

2. Larsen, J. M., B. J. Schwartz, and C. G. Annis, Jr., "Cumulative Damage FractureMechanics Under Engine Spectra," Technical Report AFML-TR-79-4159, Air ForceMateriils Laboratory, WPAFB, Jan. 1080.

3. Virkler, D. A., B. M. Hillberry and P. K. Goel, The Statistical Nature of Fatigue CrackPropagation, Journal of Engineering Materials and Technology, ASME, 101, 148-152(1979).

4. Virkler, D. A.. B. M. Hillberry and P. K. Goel, The Statistical Nature of Fatigue CrackPropagation, Air Force Flight Dynamics Laboratory, Technical Report AFFDL-TR.78-43,WPAFB (1978).

5. Lin, Y. K., and J. N. Yang, "On Statistical Moments of Fatigue Crack Propagation," paperto appear in Jouznal of Engineoring Fracture Mechanics, August 1983.

6. MUler, M. S., and J. P. Gallagher, An Analysis of Several Fatigue Crack Growth Rate(FCGR) Descriptions, in Fatigue Crack Growth Measurement and Data Analysis, ASTM-STP 738, 1981, pp. 205-251.

7. HIoeppner, D. W., and W. ER Krupp, Prediction of Component Life by Application ofFatigue Crack Growth Knowledge, Engineering Fracture Mechanics, 6,47-70 (1974).

8. Swanson, S. R., F. Cicci and W. Hoppe, Crack Propagation in Clad 7079-T6 AluminumAlloy Sheet Under Constant and Random Amplitude Fatigue Loading, Fatigue CrackPropagation, ASTM..STP415, 312-362 (1967).

9. Smith, S, H., Random Loading Fatigue Crack Growth Behavior of Some Aluminum andTitanium Alloys, Structural Fatigue in Aircraft, ASTM-STP404 (1966).

10. Yang, J. N., Statistics of Random Loading Relevant to the Fatigue, Journal Engr. Mech.Div., ASCE, 100, No. EM3, 469-475 (1974).

11. Yang, J. N., and W. H. Trapp, Reliability Analysis of Aircraft Structures Under RandomLoading and Periodic Inspection, AIAA Journal, 12, 1623-1630 (1974).

12. Wood, H. A., R. M. Engle, J. P. Gallagher and ,J. M. Potter, Current Practice on EstimatingCrack Growth Damage Accumulation With Specific Appl~cation to Structural Safety,Durability and Reliability, U. S. Aii Force Flight Dynamics Lab., AFFDL-TR-75-32,WPAFB (1976).

13. Yang, J. N., and S. D. Manning, Distribution of Equivalent Initial Flaw Size, 1980Proceedings of Annual Reliability and Maintainability Symposium, 112-120 (1980).

84

14. Yang, J. N., Statistical Eatimation of Economic Life for Aircraft Structures, Proc.AIAA/ASME/ASCE/AHS SDM Conf., St. Louis, 240-248, (1979), Journal of Aircraft,AIMA, Vol. 17, No. 7, 528-535 (1980).

15. Yang, J. N., S. D. Manning and W. R. Garver, Durability Methods Development, VolumeV - Durability Analysis Methodology Development, Air Force Flight Dynamics Laborato-ry, AFFDL-TR-3113, Vol. 5 (1979).

15. Rudd, J. L., J. N. Yang, S. D. Manning and W. R. Garver, Durability Design Requirementsand Analysis for Metallic Airframe, in Design of Fatigue and Fracture ResistantStructures, ASTM-STP 761, 1982, pp. 133-151.

17. Yang, J. N., Statistical Crack Growth in Durability and Damage Tolerant Analyses, Proc.AIAA/ASME/ASCE/AHS 22nd Structures, Structural Dynamics, and Materials Confer-ence, 38.44 (1980),

18. Birnbaum, Z. W., and S. C. Saunders, A New Family of Life Distribution, Journal of

Applied Probability, 319-327 (1969).

19. Birnbaum, Z. W., and S. C. Saunders, Estimation for a Family of Life Distributions withApplications to Fatigue, Journal of Applied Probability, 328-347 (1969).

20. Bogdanoff, J., A New Cumulative Damage Model -- Part 4, Journal ofApplied Mechanics,ASME, 47, 40-44 (1980).

21. Kozin, F., and J. Bogdanoff, A Critical Analysis of Some Probabilistic Models of FatigueCrack Growth, Journal of Engineering Fracture Mechanics, 14, 59-89 (1981).

22. Yang, J. N., G. C. Salivar, Statistical Modeling of Fatigue-Crack Growth in a Nickel-BasedSuperalloy, to appear in Journal of Engineering Fracture Mechanics, August 1983.

23. Norohna, P. J., et al., Fastener Hole Quality, Vol I, Air Force Flight Dynamics Lab.,AFFDL-TR-78-206, WPAFB (1978).

24. Lin, Y. K., Probabilistic Theory of Structural Dynamics, McGraw-Hill Book Co. (1967).Reprinted by R. E. Krieger Publishing Co. (1976).

25. Stratonovich, R. L., Topics in the Theory of Random Noise, Vol. II, translated by R. A.Silverman, Gordon and Breach, New York (1967).

26. Arnold, L., Stochastic Differential Equations: Theory and Applications, John Wiley andSons, Inc., New York (1974).

27. Khaaminskii, R. Z., A Limit Thooren, for the Solution of Differential Equations withRandom Right Hand Sides, Theory of Probability and Application, 11, 390-405 (1966).

28. Lin, Y. K., and J. N. Yang, A Stochastic Theory of Fatigue Crack Propagation, AIAAPaper No. 83-0978, Proc. AIAA/ASME/ASCE/AHS 24th SDM Conference, 2-4 May 1983,Lake Tahoe.

85

.= sc.

APPENDIX

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44 1 rlr4 0-4 I.PI4 H 4 w"N N eiCN4Nm m m mMm W

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NIte" Q~r .wmiLAcM 02-OH4c"Ml4tLh-cNCv'o04ONJMeIA

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