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UBC Theses and Dissertations

A comparison between two methods of fatigue lifetime predictions for random loads Jonk, Eric Frederick 1986

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A C O M P A R I S O N B E T W E E N TWO M E T H O D S O F F A T I G U E L I F E T I M E P R E D I C T I O N S F O R R A N D O M L O A D S B y E R I C F R E D E R I C K J O N K B . S c , T h e U n i v e r s i t y o f C a l g a r y , 1 9 8 2 A T H E S I S S U B M I T T E D I N P A R T I A L F U L F I L L M E N T T H E R E Q U I R E M E N T S F O R T H E D E G R E E O F MASTER OF APPLIED SCIENCE i n T H E F A C U L T Y O F G R A D U A T E S T U D I E S ( D E P A R T M E N T O F M E C H A N I C A L E N G I N E E R I N G , T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A ) We a c c e p t t h i s t h e s i s a s c o n f o r m i n g t o t h e r e q u i r e d s t a n d a r d T H E U N I V E R S I T Y O F B R I T I S H C O L U M B I A S e p t e m b e r , 1 9 8 6 (£> E r i c F r e d e r i c k J o n k , 1 9 8 6 In presenting this thesis in partial fulfilment of the requirements for an advanced degree at the University of British Columbia, I agree that the Library shall make it freely available for reference and study. I further agree that permission for extensive copying of this thesis for scholarly purposes may be granted by the head of my department or by his or her representatives. It is understood that copying or publication of this thesis for financial gain shall not be allowed without my written permission. Department of The University of British Columbia 1956 Main Mall Vancouver, Canada V6T 1Y3 Date Ar^<h l\. tftL DE-6(3/81) Abstract The purpose of t h i s thesis i s to compare the r e s u l t s obtained from two d i f f e r e n t methods to account f o r fatigue, the Root Mean Square (RMS) and the Histogram, to determine which method better represents r e a l i t y . The te s t procedure used subjected compact tension specimens to randomly selected block loads, then compared the actual l i f e t i m e s obtained by experiment to the l i f e t i m e s predicted by the methods. A s t a t i s t i c a l analysis was attempted to determine which method was superior. The r e s u l t s of the analysis suggest that the RMS model i s superior. However, no firm conclusions can be drawn, since the data obtained suggest that the Paris Law parameters used i n the analysis are possibly biased. ( i i ) Table of Contents page Abstract i i L i s t of Tables v L i s t of Figures v i L i s t of Symbols v i i Acknowledgements x Introduction 1 Theoretical Background 3 Experimental Procedure 21 Observations 29 Treatment of Data 32 Results 40 Discussion 47 Conclusion 50 References 51 Appendix I - Material Properties 53 - Y i e l d Strength, Ultimate Strength and 54 Young's Modulus - Procedure 54 - Analysis 56 - Results 56 - Determining Fracture Toughness 56 - Procedure 56 - Analysis 59 - Results 63 ( i i i ) Table of Contents - Threshold Stress Intensity - Procedure - Analysis - Results - The Paris Law Parameters - Procedure - Analysis - Results - Grain Structure - Procedure - Results Appendix II - Computer Programs - Fortran Fatigue Program - Program to Generate Random Load Sequence (iv) L i s t o f T a b l e s p a g e T a b l e I . O b s e r v a t i o n s 31 T a b l e I I T e s t 1 R e s u l t s 3 2 , 4 0 T a b l e I I I T e s t 1 I n d e p e n d e n c e T e s t 3 5 T a b l e I V N o r m a l M o d e l F i t T e s t f o r T e s t 1 3 7 , 4 4 T a b l e V T e s t 2 R e s u l t s 4 0 T a b l e V I T e s t 2 R e s u l t s ( u s i n g p u b l i s h e d P a r i s L a w p a r a a m e t e r s ) 41 T a b l e V I I R e s u l t s f r o m t h e S e n s i t i v e A n a l y s i s 42 T a b l e V I I I N o r m a l M o d e l F i t T e s t f o r T e s t 2 4 4 T a b l e I X D a t a I n d e p e n d e n c e T e s t R e s u l t s 4 5 T a b l e X R e s u l t s f r o m t h e S t a t i s t i c a l A n a l y s i s 46 T a b l e X I R e s u l t s F r o m t h e T e n s i o n T e s t s 5 8 T a b l e X I I R e s u l t s F r o m t h e F r a c t u r e T e s t s 64 ( v ) L i s t o f F i g u r e s page F i g u r e 1 E l l i p t i c a l Crack i n I n f i n i t e P l a t e 4 F i g u r e 2 Crack i n I n f i n i t e P l a t e 9 F i g u r e 3 The Three Loading Modes f o r a Crack 12 F i g u r e 4 A T y p i c a l P a r i s P l o t 15 F i g u r e 5 Sample Histogram 17 F i g u r e 6 Histogram Method 20 F i g u r e 7 Small Frame MTS 22 F i g u r e 8 Compact T e n s i o n Specimen 23 F i g u r e 9 S t r e s s I n t e n s i t y F a c t o r f o r a 25 Standard Specimen F i g u r e 10 A Loading D i s t r i b u t i o n 26 F i g u r e 11 C r e a t i o n of a Random Sequence of Loads 27 F i g u r e 12 Load D i s t r i b u t i o n s f o r Load Cases 30 One and Two F i g u r e 13 A Set o f O b s e r v a t i o n s (Test 1) 33 F i g u r e 14 T e n s i o n T e s t Specimen 55 F i g u r e 15 Load v s . Displacement Record 57 f o r a T e n s i o n T e s t F i g u r e 16 D e f i n i t i o n of Crack Length 60 F i g u r e 17 Load v s . Displacement Record 62 f o r a F r a c t u r e T e s t F i g u r e 18 Raw Data T r a n s f o r m a t i o n t o S t r e s s 68 I n t e n s i t y and Crack Growth Rate Data F i g u r e 19 P a r i s P l o t f o r A-36 S t e e l 70 F i g u r e 20 A-36 S t e e l G r a i n S t r u c t u r e 72 ( v i ) L i s t of Symbols a One-half or whole crack length or one-half of the major axis of an e l i p s e . a c,aj C r i t i c a l , i n i t i a l crack lengths. a T H Threshold crack length. A Paris Law parameter. A j I n i t i a l cross se c t i o n a l area. b One-half of the minor axis of an e l i p s e or a d i -mension on a compact t e s t specimen. c C r i t i c a l value. c,d,D Dimensions on a compact tension specimen. F N * F L N Normal, Log-Normal cumulative d i s t r i b u t i o n f r e -quency ( c d f ) . F, Function of compact tension specimen dimensions fo r determining a stress i n t e n s i t y f a c t o r . E Young's Modulus. h Dimension on compact tension specimen. H0,H, Hypothesis. i Complex number / -1' or a sample number. k Number of d i f f e r e n t loads. K Stress i n t e n s i t y f a c t o r . K j Mode I stress i n t e n s i t y f a c t o r . K i o K i c Plane s t r a i n , plane stress c r i t i c a l stress inten-s i t y f a c t o r . Kmax/Kmin Maximum, minimum stress i n t e n s i t y f a c t o r . K r m s Root Mean Square (RMS) stress i n t e n s i t y f a c t o r . K T H Threshold stress i n t e n s i t y f a c t o r . L j , L F I n i t i a l , f i n a l tension specimen lengths. ( v i i ) n P a r i s Law parameter. nj C y c l e s per pseudo-histogram or o b s e r v a t i o n s . nj Cumulative frequency. N Number of c y c l e s . Nj Number o f c y c l e s per l o a d . P A p p l i e d l o a d . P Q T e s t l o a d f o r d e t e r m i n i n g K or K . Pf F a t i g u e l o a d . P U,P V U l t i m a t e , y i e l d l o a d . r P o l a r c o o r d i n a t e . r p Radius of p l a s t i c zone. s Standard d e v i a t i o n . S x,Sy Standard d e v i a t i o n of a p a r t i c u l a r p o p u l a t i o n . t Value o b t a i n e d from a t - d i s t r i b u t i o n . u A random number. U e l a s t i c energy. W R e s i s t a n c e t o c r a c k growth. x,y R e c t a n g u l a r c o o r d i n a t e s or observed v a l u e . x,y Sample average. z Normal v a l u e or a complex number. Z Complex f u n c t i o n . a S i g n i f i c a n c e l e v e l f o r one-sided t e s t s and type I e r r o r s . 0 S i g n i f i c a n c e l e v e l f o r type I I e r r o r s . 7 S p e c i f i c s u r f a c e energy. 0 P o l a r c o o r d i n a t e . ( v i i i ) Mo. Mi Value f o r hypothesis H 0 / H i . p Crack t i p radius or sample c o r r e l a t i o n c o e f f i c i e n t . p c C r i t i c a l value of sample c o r r e l a t i o n . a Applied s t r e s s . orms R S A S s t r e s s . a t t p Stress at crack t i p . o0 Ultimate s t r e s s . o x,aY Stress i n the x-dire c t i o n , y - d i r e c t i o n or y i e l d s t r e s s . OyS Y i e l d stress r x y Shear stress i n x-y plane. >*i> Stress function. (ix) Acknowledgements I would l i k e to thank Dr. Nadeau, Ph.D. and my supervisor, Dr. Vaughan, Ph.D., for t h e i r h e l p f u l suggestions throughout t h i s work; a l l the technicians i n both the Mechanical and Metallurgy Engineering Departments of the University of B r i t i s h Columbia, who a s s i s t e d me i n a l l phases of the experimental work; Peter and John Somoya for t h e i r help i n the preparation of the manuscript; and f i n a l l y Jim Fraser, who spent a great deal of time proof reading t h i s t h e s i s . (x) 1 Introduction During the past f i f t y years, engineers have become increasingly aware of the problem of crack growth caused by fatigue. (Fatigue occurs when a structure or part wears out due to c y c l i c loading.) Regrettably, t h i s enlightenment has only arisen as a r e s u l t of the sudden f a i l u r e of some structures, examples of which include the American Liberty and T2-tankers during the Second World War[1] and the bridge at Point Pleasant, West V i r g i n i a i n 1 967[2].- From these and other s i m i l a r f a i l u r e s , engineers have been able to es t a b l i s h the p r i n c i p l e s of a new science c a l l e d fracture mechanics. The purpose of fracture mechanics i s to determine the e f f e c t that a crack has on the strength of a structure. This i s done by determining the crack's "stress i n t e n s i t y f a c t o r " , a measurement of stress i n t e n s i t y at a crack t i p . Stress i n t e n s i t y i s governed by the geometry of a structure and the load applied to i t . When an applied load reaches a c r i t i c a l l e v e l , c a l l e d the " c r i t i c a l load", the structure to which i t i s applied to w i l l f a i l . The value of stress i n t e n s i t y at the c r i t i c a l load i s known as the " c r i t i c a l stress i n t e n s i t y " and i t i s assumed to be a material constant. Most structures are subjected to some form of c y c l i c loading. This requires consideration of the e f f e c t s of fatigue, using a proven design procedure, the most common 2 of which i s the P a r i s Law e q u a t i o n [ 3 ] , which s t a t e s t h a t the r a t e of c r a c k growth i s d i r e c t l y p r o p o r t i o n a l to the change of s t r e s s i n t e n s i t y caused by c y c l i c l o a d i n g . The P a r i s Law has been confirmed e x p e r i m e n t a l l y f o r c o n s t a n t amplitude c y c l i c l o a d i n g , but s t i l l needs m o d i f i c a t i o n to take i n t o account the e f f e c t s of random amplitude l o a d i n g . One such m o d i f i c a t i o n c o n s i s t s of c a l c u l a t i n g the Root Mean Square (RMS) v a l u e of the c y c l i c l o a d s , which i s then used t o c a l c u l a t e s t r e s s i n t e n s i t y ( a l s o a RMS v a l u e ) a t the c r a c k t i p , which i n t u r n i s used i n the P a r i s Law. The method has been shown t o work w e l l [ 4 ] but i t i s not complete; the RMS method i n c o r r e c t l y assumes t h a t a l l loads cause c r a c k g r o w t h [ 5 ] . However, cr a c k growth does not occur i n some m a t e r i a l s (such as s t e e l ) i n the presence of s t r e s s e s below a l e v e l known as a " s t r e s s i n t e n s i t y t h r e s h o l d " . U s i n g t h i s knowledge, an a l t e r n a t i v e approach known as the Histogram method [ 6 ] has been d e v i s e d . T h i s paper w i l l t e s t the Histogram method t o determine i f i t can produce any s t a t i s t i c a l improvement over the standard RMS method. 1 3 T h e o r e t i c a l Background The development of fracture mechanics began i n the l a t e 1 9th and e a r l y 20th centuries when engineers noticed that many s o - c a l l e d b r i t t l e f a i l u r e s seemed to s t a r t at a crack. (A b r i t t l e f a i l u r e occurs when a structure or part f a i l s suddenly due to r a p i d crack growth.) Suspecting that such cracks could weaken structures, various attempts were made to analyse the e f f e c t s that they could have on s t r u c t u r a l strength. The f i r s t attempt i n c a l c u l a t i n g the stress around a crack t i p was made by I n g l i s [ 7 ] i n 1913. Representing the crack as an e l l i p s e , I n g l i s came up with the following-equation f o r the stress at a crack t i p (see f i g u r e 1). " t i p " 2 a ( a / p ) 1 / 2 1 where a.. i s the stress at the crack t i p t i p r a i s the applied stress a i s half the crack length p i s the crack t i p radius Obviously, the radius of the crack t i p must be known i n order to determine the stress around i t using t h i s equation, but that f i g u r e i s generally not a v a i l a b l e , so an assumption i s made. Since an assumption of a zero radius produces the useless r e s u l t of i n f i n i t e stress at a crack t i p , the usual F I G U R E I E l l i p t i c a l C r a c k i n I n f i n i t e Plate 5 assumption i s that the radius i s equal to the atomic spacing i n the subject material. This produces a f i n i t e answer, but one which i s equally useless, since i t y i e l d s a product stress which i s much higher than the ultimate strength of the material. G r i f f i t h [ 8 ] subsequently developed another approach, which used an energy method based on I n g l i s ' s r e s u l t s . G r i f f i t h ' s equation i s : dU _ 2 i T a 2 a 2 da E where U i s the e l a s t i c energy a i s h a l f the crack length a i s the a p p l i e d s t r e s s E i s Young's Modulus The e l a s t i c energy represents the amount of energy a v a i l a b l e to increase the length of a crack by amount "da". G r i f f i t h reasoned that where a crack i s produced, e l a s t i c energy had to be equal to the amount of energy r e s i s t i n g the growth of a crack dW/da. He further assumed that the energy r e s i s t i n g the growth of a crack d i d not change with crack length and that when the e l a s t i c energy release rate equalled the energy r e s i s t i n g the growth of a crack, a structure would break. G r i f f i t h reasoned that the c r i t i c a l energy r e s i s t i n g crack growth should be equal to the energy required to 6 create new surface, known as surface energy. dU = dW 4ay 3 da da where W i s t h e r e s i s t a n c e t o c r a c k growth Y i s t h e s p e c i f i c s u r f a c e energy Hence, 2 2 i r o a _ 4ay 4 5 a t which f a i l u r e o c c u r s Surface energy i s not an easy material property to measure, but i t can be measured for c e r t a i n materials l i k e glass. After measuring the surface energy of the glass used i n h is experiments, G r i f f i t h found that his assumptions about c r i t i c a l energy were reasonably accurate. Engineers f i n a l l y had a reasonable method of determin-ing when cracks i n a structure would lead to f a i l u r e but problems s t i l l remained. One shortcoming of the G r i f f i t h method was the d i f f i c u l t y of determining c r i t i c a l resistance to crack growth. In b r i t t l e materials, c r i t i c a l resistance i s equal to surface energy, but the l a t t e r property cannot w h i c h g i v e s o p = (2Ey/^a)li where a„ i s t h e s t r e s s 7 b e m e a s u r e d a c c u r a t e l y . R e s i s t a n c e i n d u c t i l e m a t e r i a l s c o n s i s t s n o t o n l y o f s u r f a c e e n e r g y b u t a l s o t h e a m o u n t o f e n e r g y r e q u i r e d t o f o r m a p l a s t i c z o n e i n f r o n t o f a c r a c k . T h i s i s u s u a l l y s o l a r g e t h a t s u r f a c e e n e r g y c a n b e n e g l e c t e d . T h e e a s i e s t w a y t o d e t e r m i n e r e s i s t a n c e t o c r a c k g r o w t h i s t o m e a s u r e i t i n d i r e c t l y b y m e a s u r i n g t h e s t r e s s r e q u i r e d t o b r e a k a s p e c i m e n w i t h a c e r t a i n c r a c k l e n g t h . A n o t h e r s h o r t c o m i n g o f t h e s o l u t i o n w a s t h a t i t w a s o n l y c o r r e c t f o r a c r a c k i n a n i n f i n i t e p l a t e . G r i f f i t h s o l v e d t h i s p r o b l e m i n h i s e x p e r i m e n t s b y u s i n g g l a s s b u l b s . ( T h e s u r f a c e o f a s p h e r e h a s n o e d g e s , j u s t l i k e a n i n f i n i t e p l a t e . ) S o l u t i o n s t o p r a c t i c a l p r o b l e m s w e r e n o t a v a i l a b l e u n t i l t h e w o r k o f W e s t e r g a a r d w a s p u b l i s h e d [ 9 ] . W e s t e r g a a r d ' s s o l u t i o n b e g i n s w i t h t h e a s s u m p t i o n t h a t a c r a c k c a n b e r e p r e s e n t e d a s e i t h e r a p l a n e s t r a i n o r p l a n e s t r e s s e l a s t i c i t y p r o b l e m . T h i s a s s u m p t i o n p e r m i t s t h e u s e o f A i r y ' s s t r e s s f u n c t i o n [ 1 0 ] : A24> = V 2 ( V2<|> ) = 0 6 2 2 2 where V = _ a _ 2 + 3 2 ax ay <t> i s a s t r e s s f u n c t i o n t h a t s a t i s f i e s t h e above e q u a t i o n and a l l b o u n d a r y c o n d i t i o n s W e s t e r g a a r d 1 s c o n t r i b u t i o n w a s t o d e f i n e t h e f o l l o w i n g f u n c t i o n t h a t s a t i s f i e s A i r y ' s f u n c t i o n . 8 $ = Re "Z + y l m "z 7 where dZ = Z, dZ = Z, d_Z _ Z' dz dz ~~ dz" Z(z} = Re Z + ilm Z 8 w i t h z = x + i y Subs t i t u t i n g Westergaard 1s function i n the s o l u t i o n f o r the equilibrium equations[11] gives: 2 a _ 3 it _ Re Z - y l m Z' 8y 2 2 a _ 3 • _ Re Z 4 y l m Z 1 9a-c Y ~ "3^2 ~ 2 x 3 Q> -yRe Z' J ax3y where a , a , x , a r e t h e s t r e s s e s i n t h e j£ y x y d i r e c t i o n s shown i n f i g u r e 2 For the problem shown i n the figu r e 2, the following equation f o r "Z" meets a l l the boundary conditions: Z _ a (z + a) ~ / z l z + 2a) 4 1 0 F I G U R E 2 C r a c k i n I n f i n i t e P l a t e 10 L e t t i n g the value of "a" be much larger than the value of "z", i . e . l e t t i n g the solut i o n only be v a l i d near a crack t i p , gives: Z = aa//Taz 11 Changing t h i s equation to polar coordinates gives: Z _ a/TS exp(-i9/2) 12 /2irr ' F i n a l l y , s u b s t i t u t i n g the above into equations 9 a - c , y i e l d s the solut i o n f o r the stresses around a crack t i p . a _ o/Ta c o s ( 6 / 2 ) [ l - sin(6/2)sin(36/2)] X ~ /2TF a _ a/ia c o s ( 6 / 2 ) [ l + sin(6/2)sin(36/2)] y - TT^r 1 3 a c T a/ia 1 sin ( 9/2 )cos ( 8/2 )cos ( 3 6/2 ) The solution can be considered to consist of two parts. The geometric part y i e l d s stress d i s t r i b u t i o n around a crack, including the s i n g u l a r i t y at the crack t i p . The second part consists of a sc a l i n g term, ajna* , a simple function of applied s t r e s s . This function i s defined as the "Mode I Stress Intensity Factor" and i s given the symbol K j . The d i f f e r e n t types of stress i n t e n s i t y modes are i l l u s -11 t r a t e d i n f i g u r e 3. Rewriting the crack t i p stress equations using the stres s i n t e n s i t y f a c t o r produces the equations: o _ K i c o s ( e / 2 ) [ l - sin(e/2)sin(3e/2)] x ~ /nrF a = K I c o s ( 8 / 2 ) [ l + sin(9/2)sin(39/2)] 1 4 a _ c *w - K I sin(9/2)cos(e/2)cos(36/2) I t i s t h i s stress i n t e n s i t y factor that i s relevant, because when i t reaches a c r i t i c a l value, a structure w i l l break. I t i s assumed that t h i s c r i t i c a l value w i l l be constant f o r any given material. The c r i t i c a l s tress inten-s i t y c r i t e r i o n i s s i m i l a r to the c r i t i c a l energy c r i t e r i o n G r i f f i t h proposed, but unlike G r i f f i t h ' s method, Westergaard's can be used to solve p r a c t i c a l crack prob-lems [12], Experimental r e s u l t s confirm the assumption of a c r i t i -c a l s tress i n t e n s i t y as long as a structure meets c e r t a i n thickness and plane s t r a i n requirements which assure that the p l a s t i c zone, which formed at the crack t i p , remains small compared to the crack length. Since the whole of Westergaard's analysis i s based on a l i n e a r e l a s t i c ap-proach, the p l a s t i c zone must be small compared to the dimensions used i n the analysis i f f r a c t u r e mechanics i s to F I G U R E 3 T h e T h r e e L o a d i n g M o d e s f o r a C r a c k ro 13 be v a l i d . If a structure does not meet these requirements, then each thickness of a material proposed f o r the structure has i t s own unique c r i t i c a l stress i n t e n s i t y . Westergaard 1s solut i o n i s useful i f f a i l u r e i s the r e s u l t of a single load. Most f a i l u r e s are not. Structures are usually subjected to c y c l i c loading and i n many cases cracks grow as a r e s u l t of fatigue. A design procedure f o r fatigue i s h e l p f u l and the one suggested by Paris[13] i s the most accepted method of determining the l i f e of a structure under c y c l i c loading. The l i f e i s defined as the number of cycles necessary to enlarge a crack from i t s i n i t i a l s ize to i t s c r i t i c a l s i z e . Paris suggested that cracks experiencing the same va r i a t i o n s of stress i n t e n s i t y w i l l grow at the same rate. This s o l u t i o n i s expressed as follows: da = f ( A K ) 15 dN where da i s the incremental crack growth dN i s the incremental l i f e i n cycles AK i s the v a r i a t i o n of stress i n t e n s i t y Generally i t i s i n v a l i d to assume that fatigue crack growth i s due to the v a r i a t i o n of stress i n t e n s i t y alone. There are other influences such as the maximum stress inten-s i t y experienced by a crack and the constraint on a crack. 14 Other fact o r s are relevant, but f o r two crack growth cases to be considered s i m i l a r , a l l the factors that a f f e c t crack growth must be s i m i l a r . In most cases, the procedure proposed by Paris i s accurate enough to be useful f o r p r a c t i c a l design. For the purposes of t h i s paper, the Paris r e l a t i o n s h i p i s assumed. A l l the tests hereinafter described were designed so that change i n stress i n t e n s i t y i s the dominant factor i n crack growth. Experimental data show that the function required to r e l a t e crack growth rate to the range of stress i n t e n s i t i e s i s a simple power function of the following form: da = A(AK) n 16 dN where 'A' and 'n' are material properties This equation i s known as the Paris Law. The Paris Law i s useful i n many p r a c t i c a l s i t u a t i o n s , but i s not v a l i d f o r stress i n t e n s i t y at e i t h e r end of the stress i n t e n s i t y range (see f i g u r e 4 ) . However, i f threshold stress i n t e n s i t y and c r i t i c a l stress i n t e n s i t y are used as l i m i t s on the range of the Paris Law, the error i n using i t i s small. The Paris law i n the above form can only be used to c a l c u l a t e the l i f e of a structure undergoing a constant v a r i a t i o n of loads. I t must be modified to c a l c u l a t e the 15 T H K I C Log(AK) FIGURE 4 A Typical Par is Plot 1 6 e f f e c t of random loads. One method i s to c a l c u l a t e the Root Mean Square (RMS) value of the loads applied to a structure and use the RMS value i n the Paris Law[14]. The consequent changes to the Paris Law are as follows: da A(AK ) n 17 = rms dN where AK _ / ZN,(AK,) rms = /. , l i i = l —\ 2 k EN i = l k i s the number of d i f f e r e n t loads N i s the number of c y c l e s f o r each load Barson and Rolfe[15] have shown that t h i s r e l a t i o n s h i p works w e l l . However, i t f a i l s to recognize that crack growth does not occur i n c e r t a i n materials i f the stress experienced by a structure i s less than a c r i t i c a l value known as the "stress i n t e n s i t y threshold". By including t h i s value i n a "Histogram Method", Vaughan[16] has developed a method for c a l c u l a t i n g fatigue crack growth that should be better than the RMS Method. A histogram i s a block representation of a load d i s t r i b u t i o n on a graph showing load vs. number of cycles that a load i s applied to a structure (see figure 5). The use of the histogram i n 17 Stress (ksi) 60 50 40 30 20 10 55 Stress (ksi) 60 50 AO 30^ 20 -10 20 Kilocycles 10 55 40 H1 10 20 30 Kilocycles H2 Stress (ksi) 60 50 40 30 20 10 55 40 25 10 20 30 40 Kilocycles H3 F I G U R E 5 Sample Histogram 18 c a l c u l a t i n g the l i f e of a structure i s shown as follows. F i r s t the threshold crack length i s calculated. I t i s the crack length that w i l l cause stress i n t e n s i t y at the crack t i p to be equal to the stress i n t e n s i t y threshold. Threshold crack lengths are required f o r each load l e v e l on the histogram. Next, for the largest stress l e v e l , c r i t i c a l crack s i z e i s calculated. C r i t i c a l crack s i z e i s reached when the stress i n t e n s i t y at a crack t i p i s equal to the c r i t i c a l stress i n t e n s i t y . If i t i s assumed that there i s i n i t i a l l y a crack of a ce r t a i n s i z e i n a structure and that crack i s smaller than the the smallest threshold crack length calculated, there can be no fatigue crack growth. I f the i n i t i a l crack i s larger than the c r i t i c a l crack s i z e , f a i l u r e of a structure can be expected almost immediately. I f a crack i s larger than some of the threshold crack lengths, but smaller than the c r i t i c a l crack s i z e , crack growth occurs. The l a t t e r case i s studied by constructing a sub-category of the histogram known as a pseudo-histogram (for example see histograms H1-3, figur e 5), which i s used to ca l c u l a t e the RMS stress i n t e n s i t y value used i n the Paris Law. This value i s calculated for currently e f f e c t i v e loads, which make up the pseudo-histogram, and i s used u n t i l the crack grows so that the next lowest load i n the main histogram has an e f f e c t on crack growth. Then an ad d i t i o n a l section of the histogram i s used to construct a new pseudo-h i s t o g r a m , w h i c h i s u s e d t o c a l c u l a t e a n e w RMS s t r e s s i n t e n s i t y f a c t o r . T h i s p r o c e d u r e i s c o n t i n u e d u n t i l t h e c r a c k r e a c h e s c r i t i c a l c r a c k l e n g t h . T h e s t r u c t u r e i s t h e n a s s u m e d t o f a i l a n d i t s l i f e t i m e i s c a l c u l a t e d f r o m t h e s u m o f a l l t h e p s e u d o - h i s t o g r a m l i f e t i m e s . 1 1 . S e e f i g u r e 6 f o r a n e x a m p l e o f h o w t h e H i s t o g r a m M e t h o d i s u s e d . 20 Given: K X = oVTra' K T } J : 6.0 ksi/ir? K I ( ^ 100. ksi/iTT a = .005 in, a . = 0 ksi mm. . . . a is obtained from histograms in figure 5 m a x 3 3 Paris Law: _da = 1.4A*10 " 9 (AK R M S ) 3 dn a r m s a T H aCR 2 n i ' d a 3 d N 4 d N * a 5 ( k s i ) ( i n ) ( i n ) ( k c ) ( i n ) ( e y e . ) ( e y e . ) H I 5 5 . 0 . 0 0 4 1 1 . 0 5 2 1 5 . 0 0 2 3 , 2 B 3 2 6 , 2 6 1 H 2 4 6 . 2 . 0 0 7 1 . 0 5 2 4 0 . 0 1 1 1 1 , 3 7 8 3 4 , 1 3 4 H 3 3 7 . 8 . 0 1 8 1 . 0 5 2 7 5 . 0 3 3 1 3 , 9 7 1 2 2 , 3 5 3 H 4 3 1 . 3 . 0 5 1 1 . 0 5 2 1 2 0 1 . 0 0 1 2 8 , 0 8 5 2 8 , 0 8 5 F o o t n o t e s 1. 2. 3 . 4 . 5 . N = 1 1 0 , 8 3 0 T h e i n i t i a l c r a c k l e n g t h 4 a ' i s g r e a t e r t h a n ' a T H f ° r H i s t o g r a m H I . 1 C r i t i c a l c r a c k l e n g t h i s t h e s a m e f o r a l l h i s t o g r a m s b e c a u s e t h e m a x i m u m l o a d c a n o c c u r a t a n y t i m e . d a = a T H 1 " a i o r a T H ( . i + l ) " f T H i °f *CR a T H N u m b e r o f c y c l e s e x p e r i e n c e d b y e a c n H i s t o g r a m . T o t a l a m o u n t o f c y c l e s e x p e r i e n c e d b y t h e s t r u c t u r e d u r i n g t h e t i m e t h a t h i s t o g r a m i s a p p l i e d ; N . B . n = 1 2 0 / Z n . F I G U R E 6 H i s t o g r a m M e t h o d 2 1 E x p e r i m e n t a l P r o c e d u r e T h i s p a p e r w i l l a t t e m p t t o d e t e r m i n e w h i c h m e t h o d , t h e RMS o r t h e H i s t o g r a m , m o r e a c c u r a t e l y r e p r e s e n t s c r a c k g r o w t h d u e t o f a t i g u e . T h e e x p e r i m e n t s w h i c h f o r m t h e b a s i s f o r t h e s t u d y r e q u i r e t h e s u b j e c t i o n o f a n u m b e r o f s p e c i m e n s t o r a n d o m l o a d s a n d t h e r e c o r d i n g o f t h e n u m b e r o f c y c l e s t h a t e a c h s p e c i m e n s u r v i v e s . T h i s n u m b e r i s c o m p a r e d t o t h e n u m b e r o f c y c l e s p r e d i c t e d b y e a c h m e t h o d f o r t h a t s p e c i m e n . U s i n g s t a t i s t i c s , a c o m p a r i s o n i s t h e n m a d e t o d e t e r m i n e w h i c h m e t h o d c o m e s c l o s e r t o t h e a c t u a l r e s u l t s . T h e d e t a i l s o f t h e p r o c e d u r e a r e a s f o l l o w s . S i n c e i t i s a c o m m o n m a t e r i a l u s e d f o r e n g i n e e r i n g p u r p o s e s , i t w a s d e c i d e d t h a t A S T M - 3 6 t y p e s t e e l w o u l d b e u s e d i n t h e e x p e r i m e n t s . O n c e i t w a s s e l e c t e d , v a r i o u s t e s t s w e r e c o n d u c t e d o n t h e m e t a l t o d e t e r m i n e i t s m a t e r i a l p r o p e r t i e s ( s e e A p p e n d i x I ) . T h e n e x t s t e p w a s t o s e l e c t t h e g e o m e t r y o f t h e t e s t s p e c i m e n ; t h e w r i t e r c h o s e a c o m p a c t t e n s i o n s p e c i m e n c o m m o n l y u s e d i n d e t e r m i n i n g f r a c t u r e t o u g h n e s s a n d o t h e r k i n d s o f f r a c t u r e e x p e r i m e n t s . I t s s i z e w a s m a d e a s l a r g e a s p o s s i b l e t o p r o d u c e a s a t i s f a c t o r y r a n g e f o r c r a c k g r o w t h . T h e a p p a r a t u s u s e d f o r t h e e x p e r i m e n t s w a s t h e s m a l l f r a m e M T S t e s t i n g m a c h i n e ( s e e f i g u r e 7 ) l o c a t e d i n t h e M e t a l l u r g i c a l E n g i n e e r i n g b u i l d i n g a t t h e U n i v e r s i t y o f B r i t i s h C o l u m b i a ( U B C ) . W i t h i n t h e l i m i t s i m p o s e d b y t h i s m a c h i n e , t h e f i n a l s p e c i m e n s i z e w a s d e t e r m i n e d a n d t h e 22 r e s u l t i s shown i n figure 8. F I G U R E 7 S m a l l F r a m e M T S The specimens were prepared as shown i n figure 8 and polished using a wet sanding process with the following g r i t s izes; 180,320,400,600. Polishing a specimen made i t easier to measure the crack length, as was necessary to do for determining the Paris Law parameters (see Appendix I ) . It also makes i t easier to measure the i n i t i a l crack length of about .7 inches that each specimen was precracked to. Precrack length was measured on both sides of the specimen before fatigue tests were begun. Since the experiment measures the number of cycles applied to a specimen from the 23 ALL DIMENSIONS IN INCHES NOTE 1 - Surfaces shall be perpendiculars, parallel as applicable to within .004 TIR. F I G U R E 8 C o m p a c t T e s t S p e c i m e n i n i t i a l c r a c k l e n g t h t o t h e c r i t i c a l c r a c k l e n g t h , t h e i n i t i a l c r a c k c r e a t e d b y p r e c r a c k i n g s h o u l d b e a f f e c t e d b y t h e f i r s t l o a d a p p l i e d e x p e r i m e n t a l l y t o t h e s p e c i m e n . T h e o n l y w a y t o e n s u r e t h a t t h i s i s s o i s t o r e q u i r e t h e p l a s t i c z o n e s i z e r e s u l t i n g f r o m t h e a p p l i e d l o a d t o b e g r e a t e r t h a n t h a t c r e a t e d b y t h e p r e c r a c k l o a d , t h u s t h e l o a d u s e d f o r p r e c r a c k i n g w a s l e s s t h a n t h e f i r s t l o a d a p p l i e d i n t h e t e s t . T h e n e x t p r o b l e m w a s t o c o n s t r u c t a r a n d o m s e q u e n c e o f b l o c k l o a d s . T h e f i r s t s t e p w a s t o s e l e c t a l o a d i n g d i s t r i b u t i o n t h a t w o u l d p r o d u c e a s i g n i f i c a n t d i f f e r e n c e b e t w e e n s p e c i m e n l i f e t i m e p r e d i c t i o n s m a d e b y t h e RMS a n d t h e H i s t o g r a m m e t h o d s . T h e l o a d i n g d i s t r i b u t i o n s e l e c t i o n w a s r e s t r i c t e d b y t i m e a n d o p e r a t i n g c o n s t r a i n t s o n b o t h t h e M T S m a c h i n e a n d i t s o p e r a t o r . V a r i o u s l o a d i n g d i s t r i b u t i o n s w e r e t r i e d , u s u a l l y b a s e d o n s o m e p o l y n o m i a l r e l a t i o n s h i p b e t w e e n s t r e s s a n d c y c l e s , u n t i l t w o l o a d i n g d i s t r i b u t i o n s m e t t h e c o n s t r a i n t s . T h i s w a s d o n e b y c o n s t r u c t i n g a h i s t o g r a m f o r e a c h d i s t r i b u t i o n f r o m w h i c h t h e l i f e o f t h e s p e c i m e n f o r b o t h t h e RMS a n d H i s t o g r a m m e t h o d s c o u l d b e c a l c u l a t e d a n d c o m p a r e d . A c o m p u t e r p r o g r a m w a s u s e d t o d e t e r m i n e t h e s e l i f e t i m e s ( s e e A p p e n d i x I I ) , s i n c e t h e s t r e s s i n t e n s i t y f u n c t i o n " K j 1 1 u s e d i s n o t a s i m p l e f u n c t i o n t o i n v e r t ( s e e f i g u r e 9 ) . T h e g e n e r a l p r o c e d u r e u s e d b y t h e c o m p u t e r p r o g r a m t o c a l c u l a t e t h e s e l i f e t i m e s i s t h e s a m e a s t h e e x a m p l e s h o w n i n t h e T h e o r e t i c a l B a c k g r o u n d , w i t h t h e 25 P For ,3 ^ g/b < .7 K, = oVa^(a/b,h/b,d/b) where cr = P/b F, = 29.6 - 185(a/b) • 655.7(a/b)2- 1017(a/b)3 • 638.9(a/b)4 F I G U R E 9 S t r e s s I n t e n s i t y F a c t o r f o r a S t a n d a r d S p e c i m e n [12] 26 m a t e r i a l p r o p e r t i e s , i n i t i a l c r a c k l e n g t h a n d a h i s t o g r a m o f t h e l o a d i n g d i s t r i b u t i o n b e i n g t h e i n p u t s t o t h e p r o g r a m . Load Case One Cycles 0 I 1 1 1 0 .45 1 2 3 3.15 Load (kips) F I G U R E I O A L o a d i n g D i s t r i b u t i o n O n c e a l o a d d i s t r i b u t i o n i s s e l e c t e d ( s e e f i g u r e 1 0 ) , a m e a n s m u s t b e d e v i s e d t o c r e a t e a r a n d o m s e q u e n c e o f l o a d s . T h e e a s i e s t w a y o f d o i n g t h i s i s t o c h o o s e a r a n d o m n u m b e r b e t w e e n 0 a n d 1 a n d t r a n s f o r m t h e r a n d o m n u m b e r i n t o t h e e q u i v a l e n t r a n d o m l o a d . T h e f r e q u e n c y w i t h w h i c h a n y l o a d o c c u r s c a n b e o b t a i n e d f r o m t h e l o a d i n g d i s t r i b u t i o n . T h e r e m a i n d e r o f t h e s e q u e n c e i s c r e a t e d i n t h e s a m e w a y w i t h t h e s e l e c t i o n o f m o r e r a n d o m l o a d s . ( S e e f i g u r e 11 f o r a n e x a m p l e o f t h e p r o c e d u r e . ) 27 Stress (ksij 3.0 2.5 2.01 1.5 1.0 .5 0' 0 50 100 ' Kilocycles Load Case One -A random number (0 < u £. 1) (Linear If u = .56 / i n - t a p p i n g < W .45 • u(3.15 - .45) = 2,0 ksi o00No. of cycles = 30 kc If u = .24 a m a x = 1-1 ksi o'oNo. of cycles = 39 kc 2 3 3.15 Load (kips) F I G U R E I I C r e a t i o n o f a R a n d o m S e q u e n c e o f L o a d s 2 8 I t w o u l d h a v e b e e n t e d i o u s t o c a l c u l a t e a r a n d o m l o a d s e q u e n c e o f 2 0 t o 3 0 l o a d s , s o a c o m p u t e r p r o g r a m ( s e e A p p e n d i x I I ) w a s w r i t t e n t o g e n e r a t e t h e s e q u e n c e s t o b e a p p l i e d t o a p r e p a r e d s p e c i m e n . T h e s a m e l o a d s e q u e n c e w a s r a n o n a n u m b e r o f t e s t s p e c i m e n s i n o r d e r t o o b s e r v e t h e r a n d o m v a r i a t i o n s o f t h e e x p e r i m e n t . W i t h m o r e t h a n o n e r e s u l t f o r e a c h s e q u e n c e , a s t a t i s t i c a l a n a l y s i s c o u l d b e a t t e m p t e d . T h e f i n a l s t e p o f c o u r s e i s t h e t e s t i t s e l f . T h e t e s t i s s i m p l y a m a t t e r o f a p p l y i n g l o a d s t o a s p e c i m e n f o r t h e r e q u i r e d n u m b e r o f c y c l e s p e r l o a d . T h e M T S w a s p r o g r a m m e d t o s h u t i t s e l f o f f w h e n t h e c r a c k o p e n i n g r e a c h e d a c r i t i c a l v a l u e , w h i c h w a s p r e d e t e r m i n e d b y v i s u a l i n s p e c t i o n a n d s e t s o t h a t a t e s t w o u l d e n d b e f o r e a n y g r e a t a m o u n t o f p l a s t i c w o r k w a s d o n e i n f a t i g u i n g t h e s p e c i m e n . 2 9 O b s e r v a t i o n s T h e l o a d d i s t r i b u t i o n s s e l e c t e d f o r t h i s w o r k a r e s h o w n i n f i g u r e 1 2 . E a c h l o a d d i s t r i b u t i o n w a s t e s t e d o n a g r o u p o f s p e c i m e n s ; k n o w n a s t e s t 1 f o r l o a d c a s e o n e a n d t e s t 2 f o r l o a d c a s e t w o . F I G U R E ( 2 L o a d D i s t r i b u t i o n s f o r L o a d C a s e s O n e a n d T w o CO o 31 T h e o b s e r v a t i o n s a r e s h o w n i n T a b l e I. T h e f i r s t c o l u m n s h o w s t h e s p e c i m e n n u m b e r , a n u m b e r s y s t e m u s e d t o k e e p t r a c k o f d a t a o n e a c h i n d i v i d u a l s p e c i m e n ; t h e s e c o n d s h o w s t h e t e s t g r o u p t o w h i c h t h e s p e c i m e n b e l o n g s ( i . e . t h e l o a d d i s t r i b u t i o n ) ; t h e t h i r d c o l u m n s h o w s t h e i n i t i a l c r a c k l e n g t h i n t h e s p e c i m e n a n d t h e f o u r t h s h o w s t h e n u m b e r o f c y c l e s a s p e c i m e n e n d u r e d t o t h e p o i n t o f f a i l u r e . Spec- Test I n i t i a l Cycles imen Crack (in) 10 1 . 710 399,780 11 1 . 701 420,040 12 1 . 700 391,810 13 1 . 704 401,290 15 1 .699 401,170 42 1 .698 422,000 43 1 .699 398,000 44 1 . 690 420,520 45 1 .695 399 ,000 46 1 .712 396,700 48 2 .697 681,820 49 2 . 706 751,010 50 2. . 722 749,000 51 2. .702 848,570 52 2. .691 737,930 53 2 .689 846,000 54 2 . 708 747,600 55 2 .691 785,390 56A 2 .716 594,640 56B 2 .690 741,600 57 1 . 707 281,500 58 1 .686 244,000 59 1 .695 367,000 60 1 .695 401,730 61 1 .715 405,800 TABLE I Observations 3 2 T r e a t m e n t o f D a t a O n c e t h e e x p e r i m e n t s w e r e c o m p l e t e d , a s t a t i s t i c a l a n a l y s i s o f t h e d a t a w a s p e r f o r m e d t o d e t e r m i n e w h i c h m e t h o d f i t t h e d a t a b e t t e r . F i r s t t h e e x p e c t e d l i f e t i m e o f e a c h s p e c i m e n w a s c a l c u l a t e d f o r b o t h t h e m e t h o d s u s i n g t h e s a m e c o m p u t e r p r o g r a m u s e d t o s e l e c t l o a d i n g d i s t r i b u t i o n s , t h e n d i v i d e d b y a c t u a l l i f e t i m e s r e c o r d e d i n t h e e x p e r i m e n t s , y i e l d i n g a s e t o f r a t i o s t o b e u s e d i n t h e a n a l y s i s . I f t h e m e t h o d u s e d t o p r e d i c t t h e l i f e t i m e o f a s p e c i m e n i s p e r f e c t l y a c c u r a t e , t h e r a t i o s h o u l d h a v e a v a l u e o f o n e ; t h e m e t h o d t h a t p r e d i c t s t h e l i f e o f a s p e c i m e n m o r e a c c u r a t e l y s h o u l d y i e l d a r a t i o c l o s e r t o o n e t h a n t h a t y i e l d e d b y t h e r i v a l m e t h o d . F i g u r e 1 3 s h o w s t h e s e e f f e c t s a n d T a b l e I I i l l u s t r a t e s t h e r e s u l t s o b t a i n e d f o r t h e f i r s t t e s t r u n , g r o u p 1 . Spec-imen No. Actual Cycles RMS Cycles Predicted Ratio RMS Histogram Cycles Predicted Ratio Histogram Actual Actual 10 399 ,780 265,970 .665 252,560 .632 11 420,040 280,600 .668 265,890 .633 12 391,810 282,250 . 720 267,390 .682 13 401,290 275,670 .687 261 ,390 .651 15 401,170 283,920 . 708 268,910 .670 42 422,000 285,580 .677 270 ,430 .641 43 398,000 283,920 .713 268,910 .676 44 420,520 299,160 . 711 282,800 .673 45 399,000 290 ,630 .728 275,030 .689 46 396,700 262,790 . 662 249,660 . 629 Mean .694 .658 Standard Deviation .025 .023 TABLE II Test 1 • Results Predicted Lifetimes (kc) Actual Lifetimes (kc) FIGURE 13 A S e t o f O b s e r v a t i o n s C T e s t I ) 34 T h e l a s t f i v e o b s e r v a t i o n s f o r t e s t 1 w e r e n o t u s e d i n a n y o f t h e f o l l o w i n g a n a l y s i s . T h e r e a s o n f o r t h i s i s t h a t a s t a t i s t i c a l t e s t , k n o w n a s a s a m p l e t e s t , s h o w s t h a t t h e d i s p e r s i o n a n d m e a n o f t h e s e f i v e o b s e r v a t i o n s i s d i f f e r e n t t h a n t h a t o f t h e f i r s t t e n o b s e r v a t i o n s . T h e s a m p l e t e s t u s e d w a s a r a n k s u m [ 1 8 ] , u s i n g s i g n i f i c a n c e l e v e l s ( t h e m e a n i n g o f w h i c h i s d i s c u s s e d b e l o w ) o f 0 . 0 5 a n d 0 . 1 . T h e r e a s o n f o r t h i s r e j e c t i o n i s u n k n o w n , b u t m a y b e a t t r i b u t e d t o s o m e i m p r o p e r c o n t r o l s e t t i n g s o n t h e M T S . T h e n e x t s t a t i s t i c a l t e s t p e r f o r m e d w a s d e s i g n e d t o d e t e r m i n e w h e t h e r t h e o b s e r v a t i o n s a r e i n d e p e n d e n t o f o n e a n o t h e r . T h e t e s t u s e d w a s t h e L a b e l t e s t [ 1 9 ] , w h i c h i s p e r f o r m e d b y c o n s i d e r i n g t h e s e q u e n c e o f r e s u l t s a s a v a r i a b l e a n d a p p l y i n g a t e s t o f i n d e p e n d e n c e t o t h e s e q u e n c e . T h i s p r o d u c e s a s a m p l e c o r r e l a t i o n c o e f f i c i e n t w h i c h i s u s e d t o a n a l y z e w h e t h e r t h e r e i s a r e l a t i o n s h i p b e t w e e n t h e s e t o f n u m b e r s . T h e L a b e l t e s t c a n o n l y " r e j e c t " o r " n o t r e j e c t " ( t h a t i s , s h o w t h a t t h e d a t a d o e s n o t c o n f l i c t w i t h ) t h e h y p o t h e s i s t h a t t h e d a t a i s r a n d o m . T h i s i s k n o w n a s a s i g n i f i c a n c e t y p e o f t e s t , s i n c e n o a l t e r n a t i v e h y p o t h e s i s c a n b e s e l e c t e d . F o r a l l s t a t i s t i c a l t e s t s , a s i g n i f i c a n c e l e v e l 1 1 a 11 m u s t b e s e l e c t e d . T h e h i g h e r t h e s i g n i f i c a n c e , t h e m o r e s i g n i f i c a n t i s t h e t e s t , w i t h v a l u e s o f . 0 1 , . 0 5 a n d .1 b e i n g t y p i c a l v a l u e s f o r " a ' ' . N o s a m p l e i s r e j e c t e d f o r a v a l u e f o r " a" e q u a l t o z e r o . T h e v a l u e o f "a", a l o n g w i t h 35 t h e s a m p l e s i z e , i s u s e d t o s e l e c t a c r i t i c a l v a l u e o f a p a r a m e t e r t h a t i s u s e d t o " r e j e c t " o r " n o t r e j e c t " a h y p o t h e s i s . F o r t h e L a b e l t e s t , t h i s i s t h e s a m p l e c o r r e l a t i o n c o e f f i c i e n t . I f i t i s l e s s t h a n t h e c r i t i c a l v a l u e , t h e n t h e h y p o t h e s i s t h a t t h e d a t a i s n o t r a n d o m c a n n o t b e r e j e c t e d . A n e x a m p l e o f a L a b e l t e s t i s s h o w n b e l o w f o r t h e d a t a f r o m g r o u p t e s t 1. Label Ratio ( y ) ( x ) RMS Actual 1 -665 2 .668 3 . 720 4 .687 5 .708 6 .677 7 . 713 8 . 711 9 .728 10 • .662 Label Ratio Mean Standard Deviation niabelMRatio) 5.5 3.03 38 .694 . 025 37 p = £xy - xyn (n - 1)S xS y .295 38.37 - 5.5(.69A)(10) (10 -1M3.03H.025) For a = .20 and n = 10: pe = .443 Hence. P < P < Therefore, the hypothesis that each reading i s independent of the previous r e s u l t s cannot be rejected. TABLE I I I Test 1 Independence Test T h e g r o u p o f r a t i o s w e r e t h e n f i t t e d t o t h e N o r m a l d i s t r i b u t i o n m o d e l s e l e c t e d f o r i t s e a s e o f u s e i n h y p o t h e s i s t e s t i n g . T h e K o l m o g o r o f f " g o o d n e s s - o f - f i t " t e s t [ 2 0 ] w a s c a r r i e d o u t t o s e e h o w w e l l t h e d i s t r i b u t i o n f i t t e d . T h e p u r p o s e o f t h i s t e s t i s t o f i n d t h e m a x i m u m a b s o l u t e d i f f e r e n c e b e t w e e n s a m p l e c u m u l a t i v e d i s t r i b u t i o n 36 a n d t h e c u m u l a t i v e d i s t r i b u t i o n p r e d i c t e d b y a d i s t r i b u t i o n m o d e l ( s e e T a b l e I V ) . T h i s t e s t i s a l s o a s i g n i f i c a n c e t e s t a n d a c r i t i c a l v a l u e m u s t b e s e l e c t e d f o r a g i v e n s i g n i f i c a n c e a n d s a m p l e s i z e . I f t h e l a r g e s t d i f f e r e n c e i s l e s s t h a n t h e c r i t i c a l v a l u e , t h e h y p o t h e s i s t h a t t h e d a t a f i t s a N o r m a l d i s t r i b u t i o n c a n n o t b e r e j e c t e d . T h e f i n a l a n a l y t i c a l p r o c e d u r e w a s a t e s t o f w h i c h f a t i g u e m e t h o d f i t t e d t h e d a t a b e t t e r . I t w a s p e r f o r m e d o n t h e h y p o t h e s i s t h a t t h e t w o d i s t r i b u t i o n s f r o m t h e r a t i o s r e s u l t i n g f r o m t h e RMS a n d H i s t o g r a m m e t h o d s c o u l d b e d i s t i n g u i s h e d f r o m o n e a n o t h e r , a s o p p o s e d t o t h e h y p o t h e s i s t h a t n o d i f f e r e n c e c o u l d b e d e t e c t e d . T h i s i s a h y p o t h e s i s t e s t a n d i t d i f f e r e d f r o m t h e s i g n i f i c a n c e t e s t s p r e v i o u s l y d e s c r i b e d i n t h a t t w o h y p o t h e s e s a r e p r o p o s e d . T h e c h a r a c t i s t i c b e h i n d t h e t e s t i s t h e a c c e p t a n c e o f o n e h y p o t h e s i s a n d t h e r e j e c t i o n o f t h e o t h e r . A g a i n , a c r i t i c a l v a l u e m u s t b e s e l e c t e d t o d e t e r m i n e w h i c h h y p o t h e s i s i s c o r r e c t . T h e c r i t i c a l v a l u e i s d e t e r m i n e d a c c o r d i n g t o s a m p l e s i z e a n d t h e p o s s i b i l i t y o f e r r o r i n d e t e r m i n i n g t h e r e s u l t . F o r t h e h y p o t h e s i s t e s t h o w e v e r , t h e r e a r e t w o p o s s i b l e t y p e s o f e r r o r . O n e i s c a l l e d t y p e I e r r o r a n d i t s p r o b a b i l i t y i s a s s i g n e d t h e s y m b o l " a " . T y p e I e r r o r o c c u r s w h e n t h e f i r s t h y p o t h e s i s s h o u l d b e a c c e p t e d b u t i s n o t . T h e s e c o n d t y p e o f e r r o r , t y p e I I , o c c u r s w h e n t h e f i r s t h y p o t h e s i s s h o u l d b e r e j e c t e d b u t i s n o t . T y p e I I e r r o r i s g i v e n t h e s y m b o l " / 3 " a n d i t , Class Observed Observed Cumulative Sample Normal Normal Absolute i Value Frequency Frequency Dis. Value c . d. f . Dif f erence ni 2nj Snj /n Zi F N \Znyn - F N | 1 . 6624 1 1 . 1 -1.270 .1020 .0020 2 . 6653 1 2 . 2 -1.555 .1240 .0760 3 . 6680 1 3 .3 -1.045 .1480 .1520 4 .6767 1 4 .4 - .696 .2432 .1568 5 .6870 1 5 .5 - .285 .3859 .1141 6 .7078 1 6 .6 :553 .7098 .1098 7 .7114 1 7 .7 . 695 .7564 .0564 8 .7134 1 8 .8 .774 .7806 .0194 9 .7204 1 9 .9 1.055 .8542 .04 58 10 .7284 1 10 1.0 1.377 .9158 .0842 TABLE IV Normal Model F i t Test for Test 1 u 38 l i k e t y p e I , s h o u l d b e a s s m a l l a s p o s s i b l e . F o r t y p e I e r r o r , t h e c r i t i c a l v a l u e i s c a l c u l a t e d o n s a m p l e s i z e a n d t h e d e s i r e d m a g n i t u d e o f t h e e r r o r . T y p e I I e r r o r d e p e n d s o n s a m p l e s i z e , m a g n i t u d e o f t y p e I e r r o r a n d t h e n a t u r e o f t h e h y p o t h e s i s . T y p e I I e r r o r s a r e d i f f i c u l t t o d e t e r m i n e a n d a r e c a l c u l a t e d o n l y f o r t h o s e t e s t s f o r w h i c h t h e d i a g r a m s h o w i n g t h e a m o u n t o f t y p e I I e r r o r i n r e f e r e n c e 21 a p p l i e s . An e x a m p l e o f a t w o - s i d e d h y p o t h e s i s t e s t f o r t h e d a t a o b t a i n e d i n t e s t 1 f o l l o w s : H y p o t h e s i s H 0 : P x - ' u = 0 H y p o t h e s i s H-j: \x - \i 0 x y L e t t y p e I e r r o r b e n o g r e a t e r t h a n 0 . 0 5 L e t t y p e I I e r r o r b e n o g r e a t e r t h a n 0 . 0 5 W i t h e v e r y v a l u e u s e d t o e s t i m a t e t h e m e a n "MX" h a v i n g a c o r r e s p o n d i n g v a l u e u s e d t o e s t i m a t e t h e m e a n 'Vy " , o b t a i n e d f r o m t h e s a m e t e s t p i e c e , t h e m e t h o d o f c o r r e l a t e d p a i r s c a n b e u s e d t o s e l e c t o n e o f t h e h y p o t h e s i s [ 2 2 ] . F o r t e s t 1 : F o r a = 0 . 0 5 , 0 = 0 . 0 5 , a n d n = 1 0 ( d . o . f . ; v = 9 ) F r o m r e f e r e n c e 2 1 : &/aa/T = 2 . 8 7 w h e r e " A " i s t h e c r i t i c a l v a l u e f o r t e s t s t a t i s t i c s . T e s t s t a t i s t i c : d = x - y = 0 . 0 3 6 4 3 aa = s d = 0 . 0 0 2 1 3 H e n c e , A = 0 . 0 0 9 a n d i s l e s s t h a n t h e t e s t s t a t i s t i c T h e r e f o r e , h y p o t h e s i s H Q i s r e j e c t e d i n f a v o r o f 3 9 h y p o t h e s i s H - |, t h a t i s ; u - u £ 0 40 R e s u l t s T h e f i r s t r e s u l t s ( s e e T a b l e s I I , V ) s h o w t h e r a t i o o b t a i n e d b y d i v i d i n g t h e e x p e c t e d l i f e t i m e b y t h e a c t u a l l i f e t i m e f o r e a c h o f t h e s p e c i m e n s . T h e m e a n a n d t h e s t a n d a r d d e v i a t i o n a r e a l s o g i v e n . Spec- A c t u a l RMS R a t i o Histogram R a t i o imen C y c l e s Cyc1e s RMS Cycles Histogram No. Predicted A c t u a l Predicted A c t u a l 10 399,730 265,970 .665 252 ,560 .632 11 420,040 280,600 .668 265,390 .633 12 391,810 282,250 . 720 267 ,390 .682 13 401,290 275,670 .687 251 ,390 .651 15 401,170 283,920 . 708 263,910 .670 42 422,000 285,530 .677 270 ,430 .641 43 398,000 283 ,920 .713 268 ,910 .676 44 420 ,520 299,160 . 711 282 .800 .673 45 399 ,000 290,630 . 728 275,030 .689 46 396 ,700 262,790 .662 249 ,660 . 629 Mean .694 .653 Standard D e v i a t i o n .025 .023 TABLE I I (repeated) T e s t 1 R e s u l t s Spec- A c t u a l RMS R a t i o Histogram R a t i o imen C y c l e s C y c l e s RMS Cycles Hi s t c c r r a T n No. Predicted A c t u a l Predicted A c t u a l 43 681,820 587,850 .862 530,370 .778 49 751,010 563 ,130 . 750 508,900 .678 50 749 ,000 521,230 . .696 472,070 .630 51 848,570 574 ,020 .676 518,350 .611 52 737 ,930 604,800 . 820 545 ,080 .739 53 846,000 610,530 .722 550 ,060 .650 54 747,600 557,760 .746 504,230 .674 55 785,390 604,800 .770 545,080 .694 56A 594 ,640 536,640 .902 485,740 .817 56B 741 ,600 607 ,660 . 819 547.570 .733 M e a n .776 .700 S t a n d a r d D e v i a t i o n .073 . 0 6 6 TABLE V T e s t 2 R e s u l t s 41 T h e r e s u l t s a r e f a r f r o m t h e i d e a l r a t i o o f o n e . T o d i s c o v e r w h y , a s e n s i t i v i t y a n a l y s i s w a s c o n d u c t e d t o s h o w t h e e f f e c t s t h a t c h a n g e s i n t h e i n p u t a n d m a t e r i a l p a r a m e t e r s h a v e o n t h e r e s u l t s . T h e r e s u l t s a r e s h o w n i n T a b l e V I I ( f o l l o w i n g p a g e ) a n d i t i s s u b m i t t e d t h a t t h e y d e m o n s t r a t e t h a t c h a n g e s i n t h e P a r i s L a w p a r a m e t e r s c a u s e t h e g r e a t e s t c h a n g e s i n t h e r e s u l t s . T o s e e h o w s u c h c h a n g e s w o u l d c h a n g e t h e p r e d i c t e d l i f e t i m e s , t h e d a t a f o r t e s t t w o w a s r e c a l c u l a t e d , b u t u s i n g P a r i s L a w p a r a m e t e r s o b t a i n e d f r o m l i t e r a t u r e [ 2 3 ] ( s e e T a b l e V I ) . Spec-imen No. Actual Cycles RMS Cycles Predicted Ratio RMS Histogram Cycles Predicted Ratio Histoqram Actual Actual . 48 681,820 763 ,850 1.120 695,990 1.021 49 751,010 732,410 .975 668,310 .890 50 749,000 746,260 .996 680,500 . 909 51 848,570 785,380 .926 714,960 . 843 52 737,930 679 ,040 .920 620 ,800 . 841 53 846,000 792,670 .937 721,380 . 853 54 747,600 725,570 . 971 662,280 . 886 55 785,390 785,380 1. 000 714,960 .910 56A 594,640 698,680 1.175 638,420 1. 074' 56B 741,600 789 ,020 1. 064 718,160 .968 Mean 1.008 .919 Standard Deviation .086 .C78 TABLE VI Test 2 Results (using published Paris Law parameters) U s i n g t h e p u b l i s h e d P a r i s L a w p a r a m e t e r s r e s u l t s i n r a t i o s t h a t a r e c l o s e r t o t h e i d e a l r a t i o o f o n e . T h o u g h t h e c o n c l u s i o n s a b o u t w h i c h m e t h o d p r o d u c e s b e t t e r l i f e t i m e p r e d i c t i o n s d o e s n o t c h a n g e f o r t h e s e v a l u e s ( s e e T a b l e X , n C y c l e s A % A % -3. 03 29.98 -1. 52 13.90 0. 0. 1.52 -12.28 3. 03 -23.04 P a r i s Law Parameter 4 n ' Log(A) C y c l e s A % A% 1. 01 30.00 0.51 18.18 0. 0. -0.51 - 7.14 -1. 01 -18.75 P a r i s Law Parameter , L o g ( A ) ' C y c l e s A% A % -1. 14 3.87 -0.57 1.92 0. 0. 0.57 -1. 89 1. 14 -3. 75 I n i t i a l Crack Length (aj ' C y c l e s A % A % -10. -2.03 - 5. -0.88 0. 0. 5. 0.69 10. 1.23 F r a c t u r e Toughtness *K1c' A% C v c l e s ~A% -16.67 2. 53 - 8.33 1. 06 0. 0. 8.33 -1.16 16.67 -2. 10 F r a c t u r e T h r e s h o l d 1 F o o t n o t e s : 1. A % = Changed Parameter - Base Parameter x Base Parameter Base Parameters: a; = 0.700" Kic = 40. k s i . y/ir?. Kn_| = 6.0 k s i . / i r T . n = 3.3 Log(A) = -9.9 TABLE V I I R e s u l t s from the Sensitivity Analysis 43 p a g e 4 6 ) , i t d o e s s h o w h o w c r i t i c a l i t i s t o u s e a c c u r a t e v a l u e s f o r t h e P a r i s L a w p a r a m e t e r s i n o r d e r t o o b t a i n v a l i d c o n c l u s i o n s a b o u t m e t h o d a c c u r a c y . T h e n e x t s e t o f t a b l e s ( p a g e 4 4 ) s h o w s g o o d n e s s - o f - f i t r e s u l t s f o r t h e N o r m a l d i s t r i b u t i o n s u s e d a t a s i g n i f i c a n c e l e v e l o f 0 . 2 0 . I t s h o u l d b e n o t e d t h a t t h e t e s t w a s b a s e d o n l y o n t h e r e s u l t s f o r o n e s e t o f P a r i s L a w p a r a m e t e r s . I t i s s u b m i t t e d t h a t c h a n g i n g t h e p a r a m e t e r s w i l l n o t c h a n g e t h e s h a p e o f t h e d i s t r i b u t i o n c u r v e , s i n c e s h a p e w o u l d d e p e n d m o r e o n t h e v a r i a t i o n o f t h e e x p e r i m e n t a l r e s u l t s . T a b l e I X ( p a g e 4 5 ) i l l u s t r a t e s t h a t t h e o b s e r v a t i o n s o b t a i n e d a r e i n d e p e n d e n t o f o n e a n o t h e r . T h e l a s t s e t o f r e s u l t s , ( t a b l e 1 9 , p a g e 4 6 ) r e v e a l s w h i c h m e t h o d p r o d u c e d b e t t e r r e s u l t s i n e a c h s e t o f e x p e r i m e n t s . D a t a i n t a b l e X s u m m a r i z e s t h e d i f f e r e n c e s o b s e r v e d b e t w e e n t h e h i s t o g r a m a n d RMS l i f e t i m e r a t i o s . I t a l s o c o n t a i n s t h e r e s u l t s o f t h e h y p o t h e s i s t e s t s ( s e e t r e a t m e n t o f d a t a ) t h a t w e r e c o n d u c t e d t o d e t e r m i n e w h e t h e r o r n o t t h e s e d i f f e r e n c e s a r e s i g n i f i c a n t . T h e d i f f e r e n c e s a r e s h o w n t o b e s i g n i f i c a n t , u s i n g l i m i t s o f 0 . 0 5 f o r t y p e I a n d I I e r r o r . F o r t e s t o n e a n d t w o t h i s m e a n s t h a t t h e m e t h o d t h a t p r o d u c e s a l i f e t i m e r a t i o c l o s e r t o t h e i d e a l r a t i o o f o n e i s m o r e a c c u r a t e . F o r t h e s e c a s e s , t h i s m e t h o d i s t h e RMS o n e . Class Observed Observed Cumulative Sample Normal Normal Absolute i Value Frequency Frequency Dis. Value c.d.f . D i f f erence \lnyn - F M | . X | ni Zn, Znj /n Z | F N 1 .6624 1 1 .1 -1.270 .1020 .0020 2 .6653 1 2 . 2 -1.555 .1240 .0760 3 .6680 1 3 .3 -1.045 .1480 .1520 4 .6767 1 4 .4 - .696 .2432 .1568 5 .6870 1 5 .5 - .285 .3859 .1141 6 .7078 1 6 .6 :553 .7098 .1098 7 .7114 1 7 .7 . 695 .7564 .0564 a .7134 1 8 .8 .774 .7806 .0194 9 .7204 1 9 .9 1.055 .8542 .04 58 10 .7284 1 10 1.0 1.377 .9158 .0842 TABLE IV (repeated) Normal Model F i t Testj for Test 1 Class Observed Observed Cumulative Sample Normal Normal Absolute i Value Frequency Frequency Dis. Value c.d.f. Difference 1V 1 X | n( In, Snj /n Z | F N | I ry/ n " F N | 1 . 67 65 1 1 . 1 -1.367 : .0858; .0142 : 2 . 6959. 1 2 . 2 -1.101 .13 5 5 .0 64 5 1 3 . 7217 1 3 .3 - .748 .2272 .072 8 4 .7461 1 4 .4 - .414 .3394 .0606 5 . 7498 1 5 .5 - .363 .3583 .1417 6 .7701 1 6 .6 - .008 .4960 .1040 7 . 8194 1 7 .7 .589 . .7221 .0221 8 . 8196 1 8 .8 .591 .7227 .0773 9 . 8622 1 9 .9 1.174 .8798 .0202 10 .9025 1 10 1.0 1.725 . 9591 .0409 TABLE VIII Normal Model F i t Test for Test 2 No. of Samples n S i g n i f i c a n t Level a Sample Correlation C o e f f i c i e n t p C r i t i c a l Correlation C o e f f i c i e n t P Is P<P ? Comments Test 1 10 .20 .295 .433 Yes H 0 cannot be rejected Test 2 10 .20 .292 .433 Yes H0 cannot be rejected H : That each reading i s independent of previous r e s u l t s . Table IX Data Independence Test Results O l Test Type I Error a Type II Error 3 n X o X l X o " X l S d C r i t i c a l Point M o _ 1 J l Is x , > d C r i t i c a l Point ? Comments 1 .05 .05 10 .694 .658 .036 .002 .009 Yes Accept H-j 2 .05 .05 10 .776 .701 .075 .007 .030 Yes Accept 2Pub .05 .05 10 1.008 .919 .089 .007 .030 Yes Accept Hypothesis: H^: MQ - = 0 H l : 0^ ' Vl = ° Notes: The variables with subscribt '0' represent the l i f e t i m e s for the RMS method. The variables with subscript '1' represent the l i f e t i m e s for the Histogram method. The variables with subscript 'd' represent the differences in the l i f e t i m e s between the two methods. Table X Results from the S t a t i s t i c a l Analysis O) 47 D i s c u s s i o n A b r i e f c o m m e n t s h o u l d b e m a d e o n t h e r e s u l t s o b t a i n e d f r o m t h e s i g n i f i c a n c e t e s t s t h a t w e r e c o n d u c t e d t o t e s t t h e q u a l i t y o f t h e e x p e r i m e n t . F o r e x a m p l e , t h e s i g n i f i c a n c e t e s t f o r i n d e p e n d e n c e o f o b s e r v a t i o n s s h o w s t h a t t h e e x p e r i m e n t h a s l i t t l e o r n o b i a s , s i n c e t h e o r d e r o f t h e d a t a a p p e a r s i n d e p e n d e n t f o r a r e a s o n a b l e g i v e n s i g n i f i c a n c e l e v e l . T h e s e l e c t i o n o f t h e N o r m a l d i s t r i b u t i o n a l s o a p p e a r s t o b e a g o o d c h o i c e , a s c o n f i r m e d b y t h e s i g n i f i c a n c e l e v e l u s e d f o r t h e K o l m o g o r o f f t e s t . T h e r e s u l t s f r o m t e s t s o f f a t i g u e m e t h o d s u p e r i o r i t y i n d i c a t e s t h a t t h e RMS m e t h o d i s s u p e r i o r f o r t h e l o a d d i s t r i b u t i o n s t e s t e d . H o w e v e r , t h e l i f e t i m e p r e d i c t i o n p r o d u c e d b y e i t h e r m e t h o d i s q u i t e i n c o n s i s t e n t w i t h a c t u a l l i f e t i m e s m e a s u r e d i n t h e e x p e r i m e n t . T h e s e n s i t i v i t y a n a l y s i s s u g g e s t s t h a t t h e m o s t l i k e l y s o u r c e o f e r r o r i s t h e e s t i m a t e o f t h e P a r i s L a w p a r a m e t e r s . O t h e r p o s s i b l e s o u r c e s o f e r r o r d o n o t h a v e s u c h a l a r g e e f f e c t o n t h e l i f e t i m e p r e d i c t i o n o f t h e s p e c i m e n . F o r e x a m p l e , T a b l e V I I ( p a g e 4 2 ) s h o w s t h a t c h a n g i n g t h e f r a c t u r e t o u g h n e s s o f t h e m a t e r i a l h a s l i t t l e e f f e c t o n t h e l i f e t i m e p r e d i c t i o n o f t h e s p e c i m e n . T h i s i s t o b e e x p e c t e d , s i n c e m o s t o f t h e l i f e t i m e o f t h e s p e c i m e n o c c u r s w h e n t h e c r a c k i s s h o r t a n d c r a c k p r o p a g a t i o n s p e e d i s l o w . T h a t m e a n s t h a t i n i t i a l c r a c k l e n g t h c a n h a v e a c o n s i d e r a b l e e f f e c t o n t h e l i f e o f a s p e c i m e n , a n d t h i s i s c o n f i r m e d b y 48 t h e s e n s i t i v i t y a n a l y s i s . T h e a c c u r a c y i n t h e m e a s u r e m e n t o f i n i t i a l c r a c k l e n g t h i n t h e e x p e r i m e n t i s h i g h , s o t h a t t h i s i s a n u n l i k e l y s o u r c e o f e r r o r . S e n s i t i v i t y a n a l y s i s a l s o s h o w s t h a t s m a l l c h a n g e s i n t h e s t r e s s i n t e n s i t y t h r e s h o l d h a v e l i t t l e e f f e c t o n t h e p r e d i c t e d l i f e o f a s p e c i m e n . I n s u m m a r y , t h e b i a s s e e n i n t h e r e s u l t s p r o b a b l y r e s u l t s f r o m e r r o r i n t h e e s t i m a t i o n o f t h e P a r i s L a w p a r a m e t e r s . T h e s o u r c e o f t h i s e r r o r i s p r o b a b l y e r r o r s i n c i d e n t a l t o d e t e r m i n i n g t h e c h a n g e i n c r a c k l e n g t h t h a t h a s o c c u r r e d i n a g i v e n n u m b e r o f c y c l e s . I n t h e p r o c e d u r e u s e d t o d e t e r m i n e t h e P a r i s L a w p a r a m e t e r s , c h a n g e s i n c r a c k l e n g t h " d a " w e r e q u i t e s m a l l , u s u a l l y a b o u t a h u n d r e d t h o f a n i n c h o r l e s s , b u t e r r o r i n t h e m e a s u r e m e n t o f c h a n g e i n c r a c k l e n g t h c o u l d b e a s h i g h a s t w o t h o u s a n d t h s o f a n i n c h , t h a t i s , a n e r r o r o f a s m u c h a s t w e n t y p e r c e n t . E r r o r o f t h i s m a g n i t u d e w o u l d p o s s i b l y p r o d u c e b i a s i n p a r a m e t e r e s t i m a t i o n t 2 4 ] . H o w e v e r , i f e n o u g h m e a s u r e m e n t s w e r e t a k e n o f t h e c r a c k g r o w t h r a t e , t h e e r r o r i n d e t e r m i n i n g " d a " c o u l d b e r e d u c e d . T h e b i a s o b s e r v e d i n t h e l i f e t i m e p r e d i c t i o n s m a d e t h e P a r i s L a w p a r a m e t e r s o b t a i n e d f o r t h i s e x p e r i m e n t , s u g g e s t t h a t t o o f e w m e a s u r e m e n t s w e r e t a k e n . T h e e r r o r i n t h e P a r i s L a w p a r a m e t e r s i s p r o b a b l y n o t g r e a t , s i n c e b y c h a n g i n g o n e o f t h e p a r a m e t e r s o b t a i n e d i n t h i s w o r k b y i t s 80% c o n f i d e n c e l i m i t , t h e p a r a m e t e r s m a t c h s o m e o f t h e r e s u l t s f o u n d i n t h e l i t e r a t u r e [ 2 5 ] . T h e r e s u l t 49 i s t h a t t h e n e w p r e d i c t e d l i f e t i m e s a r e m u c h c l o s e r t o t h e a c t u a l l i f e t i m e s m e a s u r e d . H o w e v e r , t h e RMS m e t h o d s t i l l g i v e s b e t t e r r e s u l t s . T h e r e s u l t s o b t a i n e d i n t h i s w o r k s h o w t h a t t h e RMS m e t h o d g i v e s a h i g h e r a c c u r a c y t h a n t h e H i s t o g r a m m e t h o d . T h i s r e s u l t c o n f l i c t s w i t h t h e b a s i c n o t i o n t h a t m e t h o d s t h a t i n c o r p o r a t e m o r e i n f l u e n t i a l p a r a m e t e r s t h a n s i m i l a r r i v a l m e t h o d s s h o u l d b e m o r e a c c u r a t e . F o r t h i s c a s e , t h e H i s t o g r a m m e t h o d i n c o r p r a t e s a " s t r e s s i n t e n s i t y t h r e s h o l d " , t h a t t h e RMS m e t h o d d o e s n o t . T h e n e t r e s u l t i s a RMS m e t h o d t h a t n e g l e c t s s t r e s s e s t h a t d o n o t c a u s e c r a c k g r o w t h . T h e r e f o r e , i t w o u l d b e e x p e c t e d t h a t t h e H i s t o g r a m m e t h o d w o u l d g i v e b e t t e r r e s u l t s . T h e f a c t t h a t i t d o e s n o t c a n b e e x p l a i n e d b y t h e f o l l o w i n g . T h e m o s t r e a s o n a b l e e x p l a i n a t i o n i s t h a t t h e P a r i s L a w p a r a m e t e r s u s e d a r e b i a s e d . I t h a s a l r e a d y b e e n s h o w n t h a t a s m a l l c h a n g e i n a P a r i s L a w p a r a m e t e r c a n p r o d u c e l a r g e v a r i a t i o n s i n t h e r e s u l t i n g l i f e t i m e p r e d i c t i o n s . I t a l s o h a s b e e n d e m o n s t r a t e d t h a t t h e l i f e t i m e p r e d i c t i o n s o b t a i n e d w e r e n o t h i g h l y a c c u r a t e . I t i s t h e r e f o r e s u b m i t t e d t h a t t h e l i f e t i m e p r e d i c t i o n s m a d e i n t h i s t h e s i s w o u l d p o s s i b l y s u p p o r t t h e H i s t o g r a m m e t h o d i f " c o r r e c t " v a l u e s f o r t h e P a r i s L a w p a r a m e t e r s w e r e u s e d . 50 C o n c l u s i o n T h e v a r i o u s s t a t i s t i c a l m e t h o d s u s e d t o t e s t t h e e x p e r i m e n t a l p r o c e d u r e s u g g e s t t h a t i t w a s s o u n d a n d t h a t i t s r e s u l t s s h o u l d b e a c c u r a t e . T h e l i f e t i m e s p r e d i c t e d b y b o t h m e t h o d s f o r a n y s p e c i m e n w e r e c o n s i d e r a b l y d i f f e r e n t f r o m t h e a c t u a l l i f e t i m e s m e a s u r e d , d u e t o s u s p e c t e d e r r o r i n t h e P a r i s L a w p a r a m e t e r s . T h e u s e o f P a r i s L a w p a r a m e t e r s p u b l i s h e d i n l i t e r a t u r e c o n f i r m e d t h i s s u s p i c i o n . T h e r e s u l t s s h o w t h a t t h e RMS m e t h o d i s s u p e r i o r t o t h e H i s t o g r a m m e t h o d . T h i s r e s u l t w a s n o t e x p e c t e d , s i n c e b y i n c l u d i n g a n e x t r a p a r a m e t e r , t h e " s t r e s s i n t e n s i t y t h r e s h o l d " , t h e H i s t o g r a m m e t h o d s h o u l d m o d e l r e a l i t y m o r e a c c u r a t e l y . T h e m o s t l i k e l y r e a s o n g i v e n f o r i t n o t d o i n g s o w a s t h a t t h e P a r i s L a w p a r a m e t e r s u s e d i n t h e l i f e t i m e p r e d i c t i o n s w e r e b i a s e d . 51 R e f e r e n c e s [ I ] B r o e k , D . , " E l e m e n t a r y E n g i n e e r i n g F r a c t u r e M e c h a n i c s " , M a r t i n i s N i j h o f f , ( 1 9 8 1 ) . [ 2 ] B a r s o n , J . M . a n d R o l f e , S . T . , " F r a c t u r e a n d F a t i g u e C o n t r o l i n S t r u c t u r e s " , P r e n t i c e - H a l l , ( 1 9 7 7 ) . [ 3 ] P a r i s , P . C . , " F a t i g u e - A n I n t e r d i s c i p l i n a r y A p p r o a c h " , P r o c . 1 0 t h S a g a m o r e C o n f . , S y r a c u s e U n i v . P r e s s , ( 1 9 6 4 ) p . 1 2 5 . [ 4 ] B a r s o n , J . M . a n d R o l f e , S . T . , o p . c i t . [ 5 ] V a u g h a n , H . , " F a t i g u e a n d F r a c t u r e o f S t r u c t u r e E l e m e n t s u n d e r R a n d o m L o a d s " , R o y a l I n s t , o f N a v a l A r c h i t e c t s , ( 1 9 8 3 ) p p . 2 0 9 - 2 2 0 . [6 ] s e e I b i d . [ 7 ] I n g l i s , C . E . , " S t r e s s e s i n a P l a t e d u e t o t h e P r e s e n c e o f C r a c k s a n d S h a r p C o r n e r s " , T r a n s . I n s t . N a v a l A r c h i t e c t s , 5 5 ( 1 9 1 3 ) p p . 2 1 9 - 2 4 1 . [ 8 ] G r i f f i t h , A . A . , " T h e P h e n o m e n a o f R u p t u r e a n d F l o w i n S o l i d s " , P h i l . T r a n s . R o y . S o c , L o n d o n , A 221 ( 1 9 2 1 ) p p . 1 6 3 - 1 9 7 . [ 9 ] W e s t e r g a a r d , H . M . , " B e a r i n g P r e s s u r e a n d C r a c k s " , J . A p p l . M e c h . , 61 ( 1 9 3 9 ) p p . A 4 9 - 5 3 . [ 1 0 ] T i m o s h e n k o , S . P . a n d G o o d i e r , J . N . , " T h e o r y o f E l a s t i c i t y " , M c G r a w - H i l l , ( 1 9 7 0 ) . [ I I ] s e e I b i d . [ 1 2 ] T a d a , H . , " T h e S t r e s s A n a l y s i s o f C r a c k s H a n d b o o k " , D e l R e s e a r c h C o r p o r a t i o n , ( 1 9 7 3 ) . [ 1 3 ] P a r i s , P . C , o p . c i t . [ 1 4 ] B a r s o n , J . M . , " F a t i g u e C r a c k G r o w t h U n d e r V a r i a b l e A m p l i t u d e L o a d i n g i n A S T M A 5 1 4 G r a d e B S t e e l " , A S T M 5 3 6 , A S T M ( 1 9 7 3 ) . [ 1 5 ] B a r s o n , J . M . a n d R o l f e , S . T . , o p . c i t . [ 1 6 ] V a u g h a n , H . , o p . c i t . [ 1 7 ] T a d a , H . , o p . c i t . 52 [ 1 8 ] B u r y , K . V . , " S t a t i s t i c a l M o d e l s i n A p p l i e d S c i e n c e " , J o h n W i l e y a n d S o n s , ( 1 9 7 5 ) . [ 1 9 ] s e e I b i d . [ 2 0 ] s e e I b i d . [ 2 1 ] P e a r s o n , E . S . a n d H a r t l e y , H . O . , " B i o m e t r i k a T a b l e s f o r S t a t i s t i c i a n s " , V o l . I , 3 r d E d t i o n , C a m b r i d g e U n i -v e r s i t y P r e s s , 1 9 6 6 . [ 2 2 ] L a p i n , L . L . , " S t a t i s t i c s : M e a n i n g a n d M e t h o d " , H a r c o u r t B r a c e J o v a n o v i c h , New Y o r k , 1 9 7 5 . [ 2 3 ] B a r s o n , J . M . a n d R o l f e , S . T . , o p . c i t . [ 2 4 ] W e i , R . P . , W e i , W. a n d M i l l e r , G . A . , " E f f e c t o f M e a s u r e m e n t P r e c i s i o n a n d D a t a P r o c e s s i n g P r o c e d u r e o n V a r i a b i l i t y i n F a t i g u e C r a c k G r o w t h R a t e " , J o u r n a l o f T e s t i n g a n d E v a l u a t i o n , J T E V A , V o l . 7 , N o . 2 , M a r c h 1 9 7 9 , p p . 9 0 - 9 5 . [ 2 5 ] B a r s o n , J . M . a n d R o l f e , S . T . , o p . c i t . [ 2 6 ] L e M a y , I . , " P r i n c i p l e s o f M e c h a n i c a l M e t a l l u r g y " , E l s e r v i e r , ( 1 9 8 1 ) . [ 2 7 ] B u r y , K . V . , o p . c i t . [ 2 8 ] A m e r i c a n S o c i e t y f o r T e s t i n g a n d M a t e r i a l s , " 1 9 7 8 A n n u a l B o o k o f A S T M S t a n d a r d s " , P a r t 1 0 , ( 1 9 7 8 ) . [ 2 9 ] B u r y , K . V . , o p . c i t . [ 3 0 ] B r o e k , D . , o p . c i t . [ 3 1 ] B a r s o n , J . M . a n d R o l f e , S . T . , o p . c i t . [ 3 2 ] M y e r s , R . H . , a n d W a l p o l e , R . E . , " P r o b a b i l i t y a n d S t a t i s t i c s f o r E n g i n e e r s a n d S c i e n t i s t s " , M a c M i l l a n , ( 1 9 7 8 ) . [ 3 3 ] B a r s o n , J . M . a n d R o l f e , S . T . , o p . c i t . [ 3 4 ] A m e r i c a n S o c i e t y o f M e t a l s , " M e t a l s H a n d b o o k " , V o l . 7 , (1 9 7 2 ) . 53 A p p e n d i x I_ M a t e r i a l P r o p e r t i e s I n o r d e r t o o b t a i n a n a c c u r a t e a n a l y s i s o f t h e d a t a p r e s e n t e d i n t h i s t h e s i s , a d e t a i l e d k n o w l e d g e o f t h e m a t e r i a l p r o p e r t i e s o f t h e m e t a l b e i n g t e s t e d w a s n e c e s s a r y . T h e s e h a d t o b e d e t e r m i n e d f r o m s e p e r a t e t e s t s r a t h e r t h a n u s i n g p u b l i s h e d v a l u e s , w h i c h a r e f o r a l a r g e n u m b e r o f s p e c i m e n s t a k e n f r o m m a n y d i f f e r e n t p l a t e s . T h e s p e c i m e n s u s e d i n t h e e x p e r i m e n t s c a m e f r o m o n e s m a l l s e c t i o n o f a l a r g e p l a t e , s o t h e m a t e r i a l p r o p e r t y v a l u e s o b t a i n e d a n d u s e d h e r e i n s h o u l d b e m o r e s p e c i f i c t h a n t h o s e f o u n d i n l i t e r a t u r e . T e s t r e s u l t s w e r e a l s o u s e f u l f o r c o m p a r i s o n w i t h p u b l i s h e d v a l u e s a s a c h e c k o n t h e t e c h n i q u e s u s e d b y t h e a u t h o r . T h e m a t e r i a l p r o p e r t i e s r e q u i r e d f o r t h e a n a l y s i s o f t h e e x p e r i m e n t a r e f r a c t u r e t o u g h n e s s , t h r e s h o l d s t r e s s i n t e n s i t y a n d t h e c o n s t a n t s " n " a n d " A " f o r t h e P a r i s L a w e q u a t i o n . O t h e r m a t e r i a l p r o p e r t i e s m e a s u r e d w e r e y i e l d s t r e n g t h , u l t i m a t e s t r e n g t h a n d Y o u n g ' s M o d u l u s . A m i c r o s t r u c t u r e a n a l y s i s w a s a l s o c a r r i e d o u t . D e t a i l s o f t h e m e a s u r e m e n t s a n d t h e r e s u l t s o b t a i n e d a r e d e s c r i b e d b e l o w . 54 Y i e l d S t r e n g t h , U l t i m a t e S t r e n g t h a n d Y o u n g 1 s M o d u l u s A t e n s i o n t e s t w a s p e r f o r m e d o n t h e t e s t s p e c i m e n s m a c h i n e d t o t h e s p e c i f i c a t i o n s s h o w n i n f i g u r e 14 t o d e t e r m i n e y i e l d s t r e n g t h , u l t i m a t e s t r e n g t h , a n d Y o u n g ' s M o d u l u s . T h e t e s t s w e r e p e r f o r m e d w i t h t h e T i n u s O l s e n m a t e r i a l s t e s t i n g m a c h i n e a t t h e D e p a r t m e n t o f M e c h a n i c a l E n g i n e e r i n g . P r o c e d u r e B e f o r e t e n s i o n t e s t s w e r e b e g a n , i t w a s n e c e s s a r y t o s e l e c t a s t r a i n r a t e t h a t w o u l d p r o d u c e c o n s i s t e n t r e s u l t s . I t i s k n o w n [ 2 6 ] t h a t a r a t e o f b e t w e e n . 0 1 / s a n d . 0 0 0 1 / s h a s l i t t l e o r n o e f f e c t o n t h e s t r e s s - s t r a i n c u r v e a t r o o m t e m p e r a t u r e . A c c o r d i n g l y , a r a t e o f a p p r o x i m a t e l y . 0 0 1 / s w a s s e l e c t e d f o r t h e s e t e s t s . E a c h s p e c i m e n w a s t h e n m e a s u r e d a n d l o a d e d i n t o t h e m a c h i n e . T e s t i n g w a s c a r r i e d o u t i n t w o s t a g e s . T h e f i r s t s t a g e c o n s i s t e d o f r e c o r d i n g t h e l o a d v s . t h e d i s p l a c e m e n t o f t h e s p e c i m e n u s i n g t h e T i n u s O l s e n m o d e l S - 1 0 0 0 - 2 A s t r a i n g a u g e , w h i c h i s a c c u r a t e t o . 0 0 0 1 i n c h , g i v i n g a n a c c u r a t e m e a s u r e o f Y o u n g ' s M o d u l u s . T h e T i n u s O l s e n w a s t h e n s t o p p e d a n d t h e s t r a i n g a u g e w a s r e m o v e d . T h e s e c o n d s t a g e c o n s i s t e d o f t h e s a m e r e c o r d i n g p r o c e d u r e b u t u t i l i z e d t h e T i n u s O l s e n t y p e D - 2 D e f l e c t o m e t e r , w h i c h m e a s u r e s t h e e n t i r e l o a d v s . t h e d i s p l a c e m e n t c u r v e . B o t h c u r v e s w e r e r e c o r d e d o n t h e s a m e s h e e t o f g r a p h p a p e r a n d w e r e u s e d t o d e t e r m i n e t h e v a r i o u s l o a d s a n d d i s p l a c e m e n t s r e q u i r e d t o FIGURE 14 Tension Test Specimen 5 6 c a l c u l a t e t h e m a t e r i a l p r o p e r t i e s . A n a l y s i s F o r a s i m p l e t e n s i o n t e s t , Y o u n g ' s m o d u l u s c a n b e d e t e r m i n e d q u i t e e a s i l y a s t h e s l o p e o f t h e l i n e s h o w n i n f i g u r e 1 5 . T h i s s i m p l e c a l c u l a t i o n i s s h o w n i n f i g u r e 1 5 , a s a r e t h e y i e l d a n d t h e u l t i m a t e l o a d s , t h e i n i t i a l a n d f i n a l d i m e n s i o n s o f t h e s p e c i m e n a n d t h e c a l c u l a t i o n s r e q u i r e d t o d e t e r m i n e t h e y i e l d a n d u l t i m a t e s t r e n g t h o f t h e m a t e r i a l . R e s u l t s T h e r e s u l t s f r o m s i x t e s t s a r e s u m m a r i z e d i n T a b l e X I V . A n a n a l y s i s w a s p e r f o r m e d o n t h e d a t a a s s u m i n g i t f i t t e d a L o g - N o r m a l d i s t r i b u t i o n [ 2 7 ] . T h e r e s u l t s a r e s u m m a r i z e d i n T a b l e X I V . D e t e r m i n i n g F r a c t u r e T o u g h n e s s T h e f r a c t u r e t o u g h n e s s o f t h e s t e e l u s e d f o r t h e e x p e r i m e n t s w a s d e t e r m i n e d u s i n g a s t a n d a r d c o m p a c t t e n s i o n s p e c i m e n ( s e e f i g u r e 8 ) . T h e t e s t m a c h i n e u s e d w a s t h e M a t e r i a l s T e s t i n g S y s t e m M T S 8 1 0 ( r e f e r r e d t o a s j u s t t h e M T S ) o p e r a t e d b y U . B . C . ' s D e p a r t m e n t o f M e t a l l u r g y E n g i n e e r i n g . P r o c e d u r e T h e t e s t p r o c e d u r e f o l l o w e d w a s t h a t d e s c r i b e d i n t h e A S T M E 3 9 9 - 7 2 s t a n d a r d " P l a n e - S t r a i n F r a c t u r e T o u g h n e s s o f M e t a l l i c M a t e r i a l s " [ 2 8 ] , u s i n g t h e M T S t o a p p l y t h e l o a d s Ultimate Stress V,/ = 22,500 lb./.3S in, =643 ksi. Slope = Young's Modulus 'E' '• = (13,000 lb.-0 lb.)/35 in2 (.0025"-0")/2" = 29.7 *103 ksi. Area of Specimen = .5"* .7" = .350 in2 .2 .4 .6 .8 1.0 1.2 U 1-6 Displacement (in.) L o a d v s . D i s p l a c e m e n t R e c o r d f o r a T e n s i o n T e s t Test. Speci-ment I n i t i a l Area A| (in**2) I n i t i a l Lenath U (in) F i n a l Length (in) Percent Elonga-tion Y i e l d Load Py (lb) Ulimate Load Pu (lb) Y i e l d Stress (ksi) Ulimate Stress (ksi) Young's Modulus (ksi) E A . 351 6. 7.60 21 % 13,000 22,200 37.0 63.2 N.M.1 B .350 6. 7.60 21 % 12,750 22,000 36.4 62.9 N.M. C .357 6. 7.65 22 % 15,000 22,900 42.0 64.1 31,700 D . 361 6. 7.65 22 % 13,800 22,400 38. 2 62. 0 31,600 E . 362 6. 7.50 20 % 15,000 22,800 41.4 61.9 28,900 F . 362 6. 7. 50 20 % 15,400 22,800 42. 5 63. 0 32,200 Expected Value: 39.6 62.9 31,100 Standard Deviation: 2.7 0.8 1,500 Footnotes: 1. Not Measured (N.M.) TABLE XI Results From the Tension Tests in 09 59 a n d a B a r s o n - T u r n e r I n s t r u m e n t s s e r i e s 8 0 0 0 D a t a C e n t e r s P r o g r a m V e r s i o n 4 t o r e c o r d t h e l o a d v s . c r a c k d i s p l a c e m e n t . C r a c k d i s p l a c e m e n t w a s m e a s u r e d w i t h a I n s t r o m C r a c k O p e n i n g 2 6 7 0 - 0 0 4 . O n e n o n - s t a n d a r d m o d i f i c a t i o n w a s m a d e , r e d u c i n g s p e c i m e n t h i c k n e s s b y a f a c t o r o f a b o u t o n e - h a l f t h e s t a n d a r d , s o t h a t s p e c i m e n s w o u l d h a v e a l a r g e r c r a c k g r o w t h r e g i o n f o r u s e i n t h e e x p e r i m e n t . T h e c h a n c e s o f t h e p l a n e s t r a i n c o n d i t i o n s o f t h e s t a n d a r d s b e i n g m e t w e r e s l i m , s o t h e s e c o n d i t i o n s w e r e i g n o r e d . T h u s , i n s t e a d o f m e a s u r i n g t h e p l a n e s t r a i n f r a c t u r e t o u g h n e s s , w h i c h i s a m a t e r i a l c o n s t a n t , t h e f r a c t u r e t o u g h n e s s v a l u e o b t a i n e d w a s a f u n c t i o n o f t h e t e s t s p e c i m e n g e o m e t r y , w h i c h r e m a i n e d t h e s a m e t h r o u g h o u t t h e e x p e r i m e n t , e x c e p t f o r a c h a n g e i n c r a c k l e n g t h d u r i n g t h e t e s t . T h e r e s u l t s o f t h e f r a c t u r e t o u g h n e s s t e s t s s u g g e s t t h a t t h e r e w a s l i t t l e c h a n g e i n f r a c t u r e t o u g h n e s s d u e t o d i f f e r e n t c r a c k l e n g t h s . A n a l y s i s T h e a n a l y s i s r e q u i r e d t o d e t e r m i n e t h e f r a c t u r e t o u g h n e s s o f t h e s p e c i m e n i s d e s c r i b e d c l e a r l y i n t h e A S T M s t a n d a r d s . H o w e v e r , a b r i e f d i s c u s s i o n o f t h e p r o c e d u r e f o l l o w s f o r c o n v e n i e n c e . T h e f i r s t s t e p i s t o d e t e r m i n e c r a c k l e n g t h b y m e a s u r i n g i t d i r e c t l y w i t h a v e r n i e r c a l i p e r a t t h r e e l o c a t i o n s a l o n g t h e c r a c k f r o n t . T h e s e l o c a t i o n s a r e t h e q u a r t e r , t h e h a l f a n d t h e t h r e e - q u a r t e r p o i n t s a l o n g t h e FIGURE 16 Definition of Crack Length 61 l e n g t h o f t h e c r a c k ' f r o n t ( s e e f i g u r e 1 6 ) . T h e a v e r a g e o f t h e s e c r a c k l e n g t h v a l u e s i s n o m i n a t e d t h e c r a c k l e n g t h f o r t h e s p e c i m e n . H o w e v e r , i f t h e l e n g t h o f t h e c r a c k d e v i a t e s m o r e t h a n f i v e p e r c e n t a l o n g t h e c r a c k f r o n t , a s c o m p a r e d t o t h e a v e r a g e l e n g t h o f t h e c r a c k , t h e s p e c i m e n c a n n o t b e u s e d t o d e t e r m i n e f r a c t u r e t o u g h n e s s . O n c e c r a c k l e n g t h h a s b e e n d e t e r m i n e d , t h e n e x t s t e p i s t o c a l c u l a t e t h e f r a c t u r e l o a d b y d e t e r m i n i n g t h e i n i t i a l s l o p e o f t h e l o a d v s . t h e c r a c k d i s p l a c e m e n t p l o t s h o w n i n f i g u r e 1 7 , d e t e r m i n i n g t h e 9 5 p e r c e n t s l o p e a n d d r a w i n g i t o n t h e p l o t . T h e f r a c t u r e l o a d o f t h e s p e c i m e n i s t h a t v a l u e a t t h e i n t e r s e c t i o n o f t h i s s l o p e a n d t h e l o a d v s . t h e c r a c k d i s p l a c e m e n t c u r v e . I f t h e f a t i g u e l o a d u s e d t o c r e a t e t h e i n i t i a l c r a c k i s l e s s t h a n o r e q u a l t o 6 0 p e r c e n t o f t h e f r a c t u r e l o a d , t h e n t h e f r a c t u r e t e s t m e e t s t h e l o a d c r i t e r i a o f t h e s t a n d a r d a n d i s a c c e p t a b l e . T h e l a s t s t e p i n t h e a n a l y s i s i s t o c a l c u l a t e f r a c t u r e t o u g h n e s s , w h i c h c a n b e d o n e b y e m p l o y i n g t h e e q u a t i o n s h o w n i n f i g u r e 9 . F o r t h e l o a d a n d c r a c k l e n g t h v a l u e s d e t e r m i n e d b y t h e a b o v e p r o c e d u r e : ( s e e p a g e 6 3 ) 62 Displacement (in.) FIGURE 17 Load vs. Displacement Record for a Fracture Test K j = a / a F 1 ( a / b ) F 1 ( a / b ) = 2 9 . 6 - 1 8 5 ( a / b ) + 6 5 5 . 7 ( a / b ) 2 - 1 0 1 7 ( a / b ) 3 + 6 3 8 . 9 ( a / b ) 4 f o r a = . 7 3 5 II b = 2 . II * a / b = . 3 6 8 = 1 1 . 3 f o r P = 4 . 2 5 k i p s ; t = . 5 1 0 o = P / b t = 4 . 2 5 k i p s / ( ( 2 " ) ( . 5 1 0 " ) ) = 4 . 1 6 7 k s i H e n c e , K J C = 4 . 1 6 7 k s i / . 7 3 5 " ' ( l l . 3 ) = 4 0 . 3 k s i / I n R e s u l t s T h e r e s u l t s f r o m t h e f i v e t e s t s a r e s u m m a r i z e d i n T a b l e X V . T h e a n a l y s i s w a s c a r r i e d o u t b y a s s u m i n g t h a t t h e d a t a f i t s a L o g - N o r m a l d i s t r i b u t i o n [ 2 9 ] . T h e r e s u l t s o f t h e a n a l y s i s a r e a l s o s h o w n i n T a b l e X V . T h r e s h o l d S t r e s s I n t e n s i t y T h r e s h o l d s t r e s s i n t e n s i t y i s c r i t i c a l i n t e s t i n g t h e H i s t o g r a m a n d t h e RMS m e t h o d s f o r p r e d i c t i n g s p e c i m e n l i f e ; u n f o r t u n a t e l y , i t i s a l s o o n e o f t h e h a r d e s t m a t e r i a l p r o p e r t i e s t o m e a s u r e . P r o c e d u r e T h e M T S t e s t i n g m a c h i n e w a s u s e d t o c a r r y o u t t h i s t e s t a s w e l l a n d w a s s e t t o t h e c o n s t a n t s t r o k e m o d e , w h i c h c a u s e s t h e M T S t o d e c r e a s e t h e l o a d a s t h e c r a c k g r o w s , s o t h a t t h e c r a c k d i s p l a c e m e n t ( s t r o k e ) o n t h e s p e c i m e n r e m a i n s c o n s t a n t . Test Crack Length (in) Ave. Crack Slope 95% Fracture Maximum Is Fracture No. 1 / 4 1 / 2 3 / 4 Crack Crite Slope Load Load Pi /Pf Tough-pt. pt. pt. Length -r ion (kips (kips Pf PF < .6 ? ness (on t l le spec: .men) (in) Met ? /in) /in) (kips) (kips) (ksiy in) 1 6 . 7 4 0 . 7 3 4 . 7 2 5 . 7 3 5 Yes 3 7 2 3 5 4 4 . 2 5 2 . 5 0 Yes 4 0 . 3 1 9 . 7 2 0 . 7 1 7 . 7 1 6 . 7 1 8 Yes 4 5 3 4 3 0 4 . 2 0 2 . 5 0 Yes 3 9 . 0 20 . 7 7 1 . 7 6 6 . 7 5 8 . 7 6 5 Yes 4 5 7 4 3 0 4 . 1 0 2 . 5 0 ok 4 0 . 3 34 . 9 7 2 . 9 6 7 . 9 6 1 . 9 6 7 Yes 2 6 0 2 4 7 3 . 1 0 1 . 6 8 Yes 3 9 . 7 3 5 1 . 0 0 9 1 . 0 2 0 1 . 0 1 4 1 . 0 1 4 Yes 2 5 4 2 4 1 2 . 8 5 1 . 6 8 Yes 3 9 . 1 Expected Value: 3 9 . 7 Standard Deviation: 0 . 6 TABLE XII Results From the Fracture Tests 65 S i n c e l o a d i s d e c r e a s e d a s c r a c k l e n g t h i n c r e a s e s , t h e r e i s a n i n s t a n t i n t h e g r o w t h o f a c r a c k w h e r e s t r e s s i n t e n s i t y , u p o n w h i c h t h e c r a c k t i p b e c o m e s e q u a l t o t h r e s h o l d s t r e s s i n t e n s i t y , a n d t h e c r a c k s t o p s g r o w i n g . B y r e c o r d i n g t h e l o a d a n d t h e l e n g t h o f t h e c r a c k , t h r e s h o l d s t r e s s i n t e n s i t y c a n b e d e t e r m i n e d . A n a c c u r a t e m e a n s o f m e a s u r i n g c r a c k l e n g t h c a n b e o b t a i n e d b y b r e a k i n g t h e s p e c i m e n i n h a l f w i t h a l a r g e f o r c e a n d m e a s u r i n g t h e f a t i g u e c r a c k u s i n g t h e s a m e t e c h n i q u e u s e d i n t h e f r a c t u r e t o u g h n e s s t e s t . U n f o r t u n a t e l y i n t h i s c a s e , t h i s p r o c e d u r e s i m p l y d i d n o t w o r k . T h e M T S w a s n o t s t a b l e e n o u g h t o k e e p t h e d i s p l a c e m e n t a t t h e c r a c k o p e n i n g c o n s t a n t a t t h e l o w l o a d s r e q u i r e d f o r t h e t e s t , s o t h e p r o c e d u r e w a s c h a n g e d t o t h e s i m p l e a l t e r n a t i v e o f m e a s u r i n g a v e r a g e g r a i n s i z e i n t h e c o m p o s i t i o n o f t h e m e t a l . A p i c t u r e o f t h e g r a i n s t r u c t u r e w a s m a d e a c c o r d i n g t o t h e i n s t r u c t i o n s l a i d o u t i n t h e s e c t i o n o f t h i s a p p e n d i x t r e a t i n g t h e s u b j e c t o f g r a i n s t r u c t u r e . A s e c t i o n o f t h e p i c t u r e w a s m a r k e d o f f a n d t h e n u m b e r o f g r a i n s w i t h i n t h e s e c t i o n w e r e c o u n t e d . T h e a v e r a g e a r e a o f e a c h g r a i n w a s c a l c u l a b l e o n c e t h e a r e a o f t h e s e c t i o n a n d t h e n u m b e r o f g r a i n s p e r s e c t i o n w e r e d e t e r m i n e d . I f i t i s a s s u m e d t h a t t h e a v e r a g e g r a i n i s r o u n d , a n a v e r a g e g r a i n r a d i u s c a n b e c a l c u l a t e d . A c c o r d i n g t o r e f e r e n c e 3 0 , t h e p l a s t i c z o n e h a s a d i a m e t e r o f t h e s i z e o f t h e a v e r a g e g r a i n w h e n t h e t h r e s h o l d s t r e s s i n t e n s i t y i s 6 6 r e a c h e d . A s s u m i n g t h e p l a s t i c z o n e i s a l s o r o u n d , t h r e s h o l d s t r e s s i n t e n s i t y m a y b e c a l c u l a t e d a s f o l l o w s . A n a l y s i s F r o m t h e p h o t o g r a p h i n f i g u r e 21: A v e r a g e d i a m e t e r o f g r a i n : .0067" 2 2 R a d i u s o f p l a s t i c z o n e : r = K T / u o ^ p r ys H e n c e , K T R = / r u o ^ = / . 00 67 " x TT X ( 39 . 6 k s i ) 2 > = 5.7 k s i / I n R e s u l t s T h e r e s u l t o f 5 . 7 k s i i n i s w i t h i n f i v e p e r c e n t o f t h e v a l u e p u b l i s h e d i n r e f e r e n c e 31 f o r m i l d s t e e l s o f t h e t y p e u s e d i n t h e e x p e r i m e n t s . T h e P a r i s L a w P a r a m e t e r s T h e P a r i s L a w s t a t e s t h a t t h e r a t e o f c r a c k g r o w t h i s d i r e c t l y p r o p o r t i o n a l t o t h e n t h p o w e r o f t h e s t r e s s i n t e n s i t y f a c t o r . T h i s m e a n s t h a t t w o p a r a m e t e r s h a v e t o b e d e t e r m i n e d i n o r d e r t o c a l c u l a t e t h e r a t e o f c r a c k g r o w t h . T h e p a r a m e t e r s a r e t h e p r o p o r t i o n a l f a c t o r " A " a n d t h e p o w e r f a c t o r "n" , a n d a r e c o n s t a n t s f o r a g i v e n m a t e r i a l . P r o c e d u r e T o d e t e r m i n e t h e p a r a m e t e r s , a s t a n d a r d t e s t s p e c i m e n w a s p r e c r a c k e d a n d t h e c r a c k l e n g t h w a s m e a s u r e d o n b o t h s i d e s o f t h e s p e c i m e n w i t h a n o p t i c a l c r a c k m e a s u r i n g d e v i c e m a d e b y G r e r t n e r S c i e n t i f i c C o r p o r a t i o n a n d h a v i n g a n a c c u r a c y o f a p p r o x i m a t e l y o n e t h o u s a n d t h o f a n i n c h . A f t e r t h e c r a c k h a d b e e n m e a s u r e d , t h e s p e c i m e n w a s p l a c e d b a c k 67 i n t o t h e M T S , t h e c r a c k m e a s u r i n g d e v i c e w a s s e t u p t o m e a s u r e r e l a t i v e c r a c k g r o w t h o n t h e s p e c i m e n a n d t h e s p e c i m e n w a s t h e n p l a c e d u n d e r a c o n s t a n t c y c l i c l o a d . W h e n c y c l i c l o a d i n g h a d b e g u n , a r e a d i n g w a s t a k e n f r o m t h e c r a c k m e a s u r i n g d e v i c e a n d t h e c y c l e c o u n t e r o n t h e M T S m a c h i n e . A f t e r s o m e t i m e , t h e c r a c k i n t h e s p e c i m e n w a s m e a s u r e d t o r e c o r d h o w m u c h i t h a d g r o w n , a n d t h e n u m b e r o f c y c l e s a p p l i e d t o t h e s p e c i m e n t o t h a t p o i n t w a s r e c o r d e d . T h e s e m e a s u r e m e n t s w e r e t a k e n i n t e r m i t t e n l y d u r i n g c r a c k g r o w t h , f r o m p r e c r a c k l e n g t h u n t i l t h e c r a c k w a s g r o w i n g i n a u n s t a b l e m a n n e r . A n a l y s i s T h e f i r s t t a s k i n d e t e r m i n i n g t h e P a r i s L a w p a r a m e t e r s i s t o t r a n s f o r m r a w d a t a i n t o s t r e s s i n t e n s i t y f a c t o r s a n d c r a c k g r o w t h r a t e v a l u e s . T h e l a t t e r i s t h e e a s i e s t o f t h e t w o v a l u e s t o c a l c u l a t e . T h e d i f f e r e n c e b e t w e e n t w o c r a c k l e n g t h s g i v e s t h e c h a n g e i n l e n g t h o f t h e c r a c k : " d a " . D i v i d i n g " d a " b y t h e n u m b e r o f c y c l e s r e c o r d e d t o g r o w t h e c r a c k b y " d a " p r o d u c e s t h e c r a c k g r o w t h r a t e " d a / d N " . E x a m p l e s o f t h e s e c a l c u l a t i o n s a r e s h o w n i n f i g u r e 1 8 . T o c a l c u l a t e t h e r e l a t e d s t r e s s i n t e n s i t y , t h e t o t a l c r a c k l e n g t h t o t h a t i n s t a n t m u s t b e d e t e r m i n e d f r o m t h e r e a d i n g s o b t a i n e d f r o m t h e c r a c k m e a s u r i n g d e v i c e . O n c e c r a c k l e n g t h i s k n o w n , t h e s t r e s s i n t e n s i t y f a c t o r c a n b e c a l c u l a t e d b y u s i n g t h e e q u a t i o n s h o w n i n f i g u r e 9 . E x a m p l e s o f t h e s e c a l c u l a t i o n s a r e a l s o s h o w n i n f i g u r e 1 8 . C R A C K N O . O F L E N G T H C Y C L E S R E A D I N G C O U N T E R 5 8 . 3 7 0 0 . 5 9 . 1 7 3 2 1 8 0 0 . 5 9 . 5 5 9 3 1 8 0 0 . 5 9 . 6 9 5 3 7 3 0 0 . 5 9 . 8 9 7 4 4 2 0 0 . 6 0 . 1 6 5 5 0 6 0 0 . 6 0 . 4 2 8 5 7 8 0 0 . 6 0 . 6 2 8 6 3 8 0 0 . 6 1 . 1 4 1 7 2 9 0 0 . 6 1 . 3 9 5 7 8 9 0 0 . 6 1 . 6 2 5 8 4 0 0 0 . 6 1 . 9 7 3 9 0 2 0 0 . 6 2 . 1 4 0 9 5 4 0 0 . 6 2 . 4 4 2 1 0 0 4 0 0 . 6 2 . 8 7 3 1 0 5 0 0 0 . 6 3 . 0 0 7 1 0 6 8 0 0 . 6 3 . 1 3 3 1 0 7 6 0 0 . 6 3 . 2 0 7 1 0 9 1 0 0 . 6 3 . 4 9 6 1 1 1 3 0 0 . 6 3 . 5 3 4 1 1 3 0 0 0 . 6 3 . 6 4 9 1 1 4 6 0 0 . 6 3 . 7 6 1 • 1 1 6 1 0 0 . • • Initial crack length 'a^= .636" Load 'P'=1.88 kips da = (59.173 mm - 58.370 mm)/(25.4 mm/in) = .032" dN =21,800 - 0 = 21,800 cycles da/dN = .032"/21,800 cycles = 1.450 *10"6in./cycle Loq(da/dN) = -5.8386 a = .636"* .032"/2 =-.652 inches From Figure b = 2" = P7c?[29.6 -185.5(a/b) • 655.7(a/b)2- 1017.(a/b)3 • 638.9(a/b)A] = 16.372 ksi/In LogtK^ = 1.2144 L 0 G ( K j ) L O G / d a ^ 1 . 2 1 4 4 - 5 . 8 3 8 6 1 . 2 2 5 4 - 5 . 8 1 8 2 1 . 2 3 0 3 - 6 . 0 1 1 7 1 . 2 3 3 6 - 5 . 9 3 8 3 1 . 2 3 8 1 - 5 . 7 8 2 9 1 . 2 4 3 3 - 5 . 8 4 2 2 1 . 2 4 7 9 - 5 . 8 8 2 0 1 . 2 5 5 1 - 5 . 6 5 3 8 1 . 2 6 2 9 - 5 . 7 7 8 1 1 . 2 6 7 9 - 5 . 7 5 0 7 1 . 2 7 3 9 - 5 . 6 5 5 6 1 . 2 7 9 3 - 5 . 8 9 8 1 1 . 2 8 4 2 - 5 . 6 2 3 8 1 . 2 9 2 0 - 5 . 4 3 3 1 1 . 2 9 8 1 - 5 . 5 3 3 0 1 . 3 0 0 9 - 5 . 2 0 7 6 1 . 3 0 3 1 - 5 . 7 1 1 7 1 . 3 0 7 0 - 5 . 2 8 6 4 1 . 3 1 2 3 - 5 . 5 4 8 3 1 . 3 1 4 8 - 5 . 5 3 1 7 1 . 3 1 8 2 - 5 . 4 8 3 3 * FIGURE 18 Raw Data Transformation to Stress Intensity & Crack Growth Rate Data O) 09 69 O n c e t h e s t r e s s i n t e n s i t y f a c t o r a n d c r a c k g r o w t h r a t e a r e k n o w n , t h e P a r i s L a w p a r a m e t e r s c a n b e d e t e r m i n e d b y p l o t t i n g t h e c r a c k g r o w t h r a t e a g a i n s t t h e s t r e s s i n t e n s i t y f a c t o r o n l o g p a p e r . A s t r a i g h t l i n e s h o u l d b e p r o d u c e d a n d b y u s i n g l i n e a r r e g r e s s i o n t e c h n i q u e s [ 3 2 ] , t h e P a r i s L a w p a r a m e t e r s " n " a n d " A " c a n b e d e t e r m i n e d . R e s u l t s A p l o t o f a l l t h e p o i n t s u s e d i n t h e l i n e a r r e g r e s s i o n a n d t h e r e s u l t s o f t h e a n a l y s i s a r e s h o w n i n f i g u r e 1 9 . G r a i n S t r u c t u r e I t w a s d e c i d e d t h a t t h e m e t a l u s e d f o r t h e s p e c i m e n s s h o u l d b e s u b j e c t e d t o a o p t i c a l g r a i n s t r u c t u r e a n a l y s i s t o d e t e r m i n e t h e r o l l i n g d i r e c t i o n o f t h e p l a t e , s i n c e t h e o r i e n t a t i o n o f a s p e c i m e n t o t h e r o l l i n g d i r e c t i o n w o u l d e f f e c t t h e v a l u e s o b t a i n e d f o r t h e m a t e r i a l p r o p e r t i e s . P r o c e d u r e A n a p p r o x i m a t e l y h a l f i n c h b y t h r e e q u a r t e r i n c h s e c t i o n o f s t e e l w a s c u t f r o m o n e s p e c i m e n u s e d f o r t h e t e s t s a n d p o l i s h e d o n o n e s i d e t o a o n e - m i c r o n s u r f a c e f i n i s h . A s e q u e n c e o f r o u g h p o l i s h i n g u s i n g w e t s a n d p a p e r o f 1 8 0 , 3 2 0 , 4 0 0 , a n d 6 0 0 g r i t s w a s f o l l o w e d b y f i n e p o l i s h i n g u s i n g d i a m o n d p o w d e r o f s i x a n d t h e n o n e - m i c r o n s i z e . A l l p o l i s h i n g w a s d o n e i n t h e M e t a l l u r g y p o l i s h i n g l a b u s i n g m o t o r i z e d p o l i s h i n g w h e e l s . A t e a c h s t a g e , t h e s p e c i m e n w a s p o l i s h e d i n a l t e r n a t e d i r e c t i o n s t o m i n i m i z e g r a i n FIGURE 19 Paris Plot for A-36 Steel 7 1 d i s t o r t i o n . T h e p o l i s h e d s u r f a c e w a s e t c h e d i n t w o p e r c e n t n i t a l , a s o l u t i o n o f t w o p e r c e n t n i t r i c a c i d i n a l c o h o l , f o r t e n s e c o n d s , e x a m i n e d a n d p h o t o g r a p h e d u s i n g a C a r l Z e i s s " U L T R A P H O T " C a m e r a M i c r o s c o p e . R e s u l t s T h e p i c t u r e i n f i g u r e 2 0 s h o w s t h e g r a i n s t r u c t u r e o f t h e s t e e l u s e d . A s i s o b v i o u s f r o m t h e p h o t o g r a p h , t h e r o l l i n g d i r e c t i o n o f t h e p l a t e , w h i c h w o u l d b e s h o w n b y t h e e n l o n g a t i o n o f t h e g r a i n s i n a p r e d o m i n a n t d i r e c t i o n , c a n n o t b e d e t e r m i n e d . T h i s d o e s n o t i n v a l i d a t e t h e e x p e r i m e n t s , s i n c e a s l o n g a s e a c h t e s t s p e c i m e n h a s t h e s a m e o r i e n t a t i o n t o t h e p l a t e a s a l l t h e o t h e r s , t h e r e s u l t s w i l l b e c o n s i s t e n t a n d v a l i d . A c o m p a r i s o n o f t h e p h o t o g r a p h o f t h e g r a i n s t r u c t u r e o f t h e s t e e l u s e d i n t h e e x p e r i m e n t a n d a p h o t o g r a p h o f t h e s a m e k i n d o f s t e e l t a k e n f r o m t h e M e t a l s H a n d b o o k [ 3 4 ] ( f i g u r e 2 0 ) , r e v e a l s t h a t t h e g r a i n s t r u c t u r e o f e a c h a r e s i m i l a r . 2% nital 26 5 * Grain Structure From A - 3 6 Steel Tested (1/2" Plate) Figure 20 A-36 Steel Grain Structure -4 A P P E N D I X I I C F O R T R A N F A T I G U E P R O G R A M C C N O . O F C Y C L E S T O F A I L U R E D U E T O C V A R I A B L E L O A D I N D C C I N P U T P A R A M E T E R S C C C I K = C R I T I C A L F R A C T U R E T O U G H N E S S C H T H = T H R E S H O L D F R A C T U R E T O U G H N E S S C A P = P A R I S L A W C O N S T A N T C A N = P A R I S LAW E X P O N E N T C I N O L = N O . O F L O A D S C A N I C = N 0 . O F C Y C L E S P E R L O A D C X P = L O A D P C A I = I N I T I A L C R A C K S I Z E C N O = N O . O F L O A D C A S E S C I M P L I C I T R E A L * 8 ( A - H , 0 - Y ) D I M E N S I O N A N I C C 3 0 ) , A N N ( 2 ) , X P ( 3 0 ) , A T ( 3 0 ) , A C ( 3 0 ) 5 F O R M A T ( 1 2 ) 7 F 0 R M A T ( F 7 . 0 , F 1 5 . 4 ) 10 F 0 R M A T ( F 1 5 . 4 ) C A L L A S S I G N (. 1 , ' F A T I N . D A T ' ) C A L L A S S I G N ( 2 , ' F A T O U T . D A T ' ) R E A D ( 1 , 5 ) NO W R I T E ( 2 , 1 5 0 0 ) N O DO 2 3 0 1 1 = 1 , N O R E A D ( 1 , 1 0 ) C 1 K R E A D d . l O H T H R E A D ( 1 , 1 0 ) A P R E A D ( 1 , 1 0 ) A N R E A D ( 1 , 1 0 ) A I R E A D ( 1 , 5 ) I N O L C C E C H O I N P U T D A T A C W R I T E ( 2 , 1 5 1 0 ) I I , C 1 K , H T H , A P , A N , A I , I N 0 L W R I T E ( 2 , 1 5 2 0 ) DO 2 0 1 = 1 , I N O L R E A D ( 1 , 7 ) A N I C ( I ) , X P ( I ) W R I T E f 2 , 1 5 3 0 ) I , A N I C ( I ) , X P ( I ) 20 C O N T I N U E C C C A L U L A T E C R I T I C A L C R A C K L E N G T H S C A = A I A T M I = 1 0 0 0 . DO 2 2 1 = 1 , I N O L 74 CALL CRAC(HTH,XP(I),A,AT(I)) IF(AT(I).LT.ATMI) ATMI=AT(I) CALL CRAC(C1K,XP(I),A,AC(I)) WRITE(5,21) AT(I),AC(I) 21 F0RMAT(2(1X,F7.4)) 22 CONTINUE IF(AT.LT.ATMI) GOTO 500 C C HISTOGRAM METHOD C 70 XN=0. J = 0 AT1=AI ATM=AT(1) ACM=AC(1) SC=ANIC(1) DO 85 1=2,INOL SC=SC+ANIC(I) IF(AT(I).GT.ATM) ATM=AT(I) IF(AC(I).LT.ACM) ACM=AC(I) 85 CONTINUE IF(ATM.GT.ACM) ATM=ACM AT2=ACM AT3=ACM 90 AL=0. TC=0. DO 100 1 = 1, INOL IF(AT(I).GT.ATI) GOTO 95 AL=AL+ANIC(I)*XP(I)** 2 TC=TC+ANIC(I) IF(AT(I).LE.ATI) GOTO 100 95 IF(AT(I).LT.AT2) AT2=AT(I) 100 CONTINUE AL=(AL/TC)**.5 IF((AT2.E0.ATM).AND.(J.E0.2)) AT2=ACM IF(ATI.EQ.AT2) GOTO 110 N=20 WRITE(5,105) ATI,AT2 105 FORMAT(2(IX,F8.4)) CALL CRAF(ATI,AT2,AL,XN1,N,AP,AN) XN=XN+XN1*(SC/TC) IF(AT2.EQ.ACM) GOTO 110 ATI=AT2 IF(AT2.EQ.ATM) J=2 AT2=ATM GOTO 90 110 WRITE(2,1110) XN C C RMS METHOD C XN=0. AT1=AI AL = 0. TC = 0. DO 115 I=1,IN0L AL=AL+AMIC(I)*XP(I)**2 115 TC=TC+ANIC(I) AL=(AL/TC)**.5 N=20 CALL CRAF(AT1,AT3,AL,XN1,N,AP,AN) XN=XN+XN1 WRITE(2,1120) XN GOTO 230 500 WRITE(2,1000) 1000 FORMAT(IX,//6X,'INITIAL CRACK LENGTH IS BELOW THE 1THRESHOLD'/IX,'CRACK LENGTH. THEREFORE, THE LIFE 20F THE SPECIMEN IS'/IX,'INFINITE FOR THE HISTO-3GRAM AND RMS METHODS.'//) 230 CONTINUE CALL CLOSE(l) CALL CLOSE(2) C 1110 FORMAT(IX,//6X,'THE LIFE OF THE SPECIMEN AS 1DETERMINED BY THE',/IX,'HISTOGRAM METHOD IS:' 2.D12.5,' CYCLES.'/) 1120 FORMAT(6X,'THE LIFE OF THE SPECIMEN AS DETERMINED 1BY THE',/IX,'RMS METHOD IS:',D12.5,' CYCLES.'/) 1500 FORMAT(IX,//11X,'RESULTS FROM FATIGUE PROGRAM', 1///6X,'THE NUMBER OF LOAD CASES IS: ',12,/) 1510 FORMAT(IX,/IX,'THE INPUT DATA FOR LOAD CASE ',12, 1' IS:',//6X,'CRITICAL FRACTURE TOUGHNESS:',D12.5, 2/6X,'THRESHOLD FRACTURE TOUGHNESS:',Dl2.5, 3/6X,'PARIS LAW CONSTANT:',D12.5,/6X,'PARIS LAW 4EXPONENT:',D12.5,/6X,'INITIAL CRACK SIZE:', 5D12.5,/6X'N0. OF LOADS: ',12,//) 1520 FORMAT(IX,'LOAD NO. OF',7X,'LOAD',/2X, l'NO. CYCLES',9X,'P',/) 1530 FORMAT(2X,I2,2D12.5) C STOP END C SUBROUTINE CRAC(FT,AL,AI,A) C C SUBROUTINE TO CALCULATE CRITICAL CRACK SIZE C GIVEN FRACTURE TOUGHNESS 'FT' AND LOAD 'AL' C IMPLICIT REAL*8(A-H,0-Y) CALL FRAC(Al,AL,FT1,IOS) IF(IOS.EQ.O) GOTO 5 WRITE(2,7) 7 F0RMATUX.//1X,'***WARNING*** Al IS OUT OF 1B0UNDS'//) 5 A1=AI 76 1 = 0 J = 0 B=FT1-FT A2=2.*AI 10 CALL FRAC(A2,AL,FT2,I0S) IF<(IOS.EQ.l)-AND.(I.EQ.l)) GOTO 32 IF(IOS.EQ.l) 1=1 C=FT2-FT 12 IF((B.GE.O.).AND.(C.LE.O. )) GOTO 32 IF((B.LT.O.).AND.(C.GT.O.)) GOTO 32 IF(J.EQ.l) GOTO 15 IF(J.E0.2) GOTO 20 Bl=DABS(B) Cl=DABS(C) IF(Bl.LT.Cl) GOTO 20 15 A1=A2 A2=2.*A2 B=C J=l GOTO 10 20 A2=A1 Al=.5*A1 C=B J=2 CALL FRAC(Al,AL,FT1,IOS) IF((IOS.EQ.l).AND.(I.EQ.l)) GOTO 32 IF(IOS.EQ.l) 1=1 B=FT1-FT GOTO 12 32 A=(Al+A2)/2. T=DABS((A-A2)/A) IF(T.LT.5.D-06) GOTO 40 CALL FRAC(A,AL,FT1,IOS) B=FT1-FT IFCB.LE.O.) A1=A IF(B.GE.O.) A2=A GOTO 32 40 RETURN END C SUBROUTINE CRAF(A,A2,AL,XN,N,AP,AN) C C SUBROUTINE TO CALCULATE THE NO. OF CYCLES 'DELTA C N' TO GROW CRACK FROM CRACK LENGTH A TO A2 C IMPLICIT REAL*8(A-H.0-Y) DIMENSION ANN(2) IM=1 3 CALL FRAC(A.AL,C1K,I0S) X0=1./(C1KAAAN) CALL FRAC(A2,AL,C1K,I0S) IF(IOS.EQ.l) GOTO 40 77 X0=X0+1./(C1K**AN> DO 25 K=IM,2 M=K*N X1 = 0. X2 = 0. A1=A H=(A2-A)/(2.*M) J=2*M-1 DO 20 1=1,J X=A1+I*H 2E1=FL0AT( I.) 12. 11=1/2 2E2=FL0AT(I1) IFCZE1.EQ.ZE2) GOTO 10 CALL FRAC(X,AL,C1K,I0S) X1=X1+1./(C1K**AN) GOTO 20 10 CALL FRAC(X,AL,C1K,I0S) X2=X2+1./(C1K**AN) 20 CONTINUE 25 ANN(K)=H*(X0+2.*X2+4.*X1> / ( 3 .*AP) T=DABS(ANN(2)-ANN(1))/ANN(2) IF(T.LT.3.125D-07) N=N/2+l WRITE(5,27) ANN(2> 27 F0RMAT(1X,F10.2) IF(T.LT.5.D-06) GOTO 30 ANN f1)=ANN(2) IM=2 N=2*N GOTO 3 30 XN=ANN(2) 40 RETURN END C SUBROUTINE FRAC(A,AL,C1K,IOS) C C SUBROUTINE TO CALCULATE FRCTURE TOUGHNESS Kl C GIVEN CRACK SIZE 'A', LOAD 'AL': 'OS' IS A LIMIT C FLAG C IMPLICIT REAL*8(A-H,O-Y) IOS = 0 IF((A.LT.0.6).OR.(A.GT.2.)) I0S=1 IF(A.LT.0.6) A=0.6 IF(A.GT.2.) A=2. AB=A/2. IF(AB.GT.0.7) GOTO 10 Fl=29.6-185.5*AB+655.7*AB**2-1017.*AB**3+638.9* 1AB**4 GOTO 20 10 F2^0.03651067*(AB-.7)+.6520468 IF(AB.EQ.l.) AB=0.9999D 00 F l = 2 r M 2 .+AB)*F2/((1 .-AB)**1.5*<AB>*. C1K=F1*ALMA**.5) / l . 01 RETURN END 79 C C P R O G R A M T O G E N E R A T E R A N D O M L O A D S E Q U E N C E C I M P L I C I T R E A L * 8 ( A - H , 0 - Y ) D I M E N S I O N F ( 3 0 ) W R I T E ( 5 , 1 0 0 ) R E A D ( 5 , 1 2 0 ) I I W R I T E ( 5 , 1 1 0 ) R E A D ( 5 , 1 2 0 ) 12 W R I T E ( 5 . 1 3 0 ) R E A D ( 5 , 1 2 5 ) I C L W R I T E ( 5 , 1 6 0 ) R E A D ( 5 , 1 7 0 ) B , S , S F DO 1 1 = 1 , 3 0 1 F ( I ) = 0 . A = l . C 1 = 1 . / ( 1 . - D E X P ( - ( B - A ) / S ) ) C A L L A S S I G N ( 1 , ' R A N . D A T ' ) DO 5 1 = 1 , I C L U = R A N ( I 1 , I 2 ) X = A - S * D L O G ( l . - U / C l ) DO 5 J = l , 2 9 I F ( ( X . G T . J ) . A N D . ( X . L E . ( J + 1 ) ) ) F ( J ) = F ( J ) + l . 5 C O N T I N U E DO 7 1 = 1 , 2 9 7 W R I T E ( 1 , 1 8 0 ) I , F ( I ) / F ( 2 9 ) 10 C A L L C L O S E ( l ) C 1 0 0 F O R M A T ( I X , ' 1 1 = ? ' ) 1 1 0 F O R M A T ( I X , ' 1 2 = ? ' ) 1 2 0 F O R M A T ( 1 5 ) 1 2 5 F O R M A T ( 1 7 ) 1 3 0 F O R M A T ( I X , ' I C L = ? ' ) 1 6 0 F O R M A T ( I X , ' B = ? , S = ? , S F = ? ' ) 1 7 0 F O R M A T ( E 1 2 . 5 , / E 1 2 . 5 , / E 1 2 . 5 ) 1 8 0 F O R M A T ( I X , 1 2 , 1 X , E 1 2 . 5 ) C S T O P E N D 

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