1-s2.0-014211239599737U-main

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TTERWORTH

I N E M A t q ~

Int. J, Fatigue Vol, 17, No. 5, pp. 365-373, 1995

Copyright © 1995 Elsevier Science Limited

Printed in Great Britain. All rights reserved

0142-1123/95/$10.00

D e t a i l e d e v a l u a t i o n o f m e t h o d s f o r e s t i m a t i o n

o f f a t i g u e p r o p e r t i e s

J u n - H y u b P a r k a n d J i - H o S o n g

Departmen t of Auto ma t ion and Design Engineering, Korea Advanc ed Inst itute of

Science and Technology, 207-43, C heongry angri-dong, Dongdaem un-gu, Seoul,

Korea

Received 7 Nov emb er 1994)

Using extensive experimental strain-life curve data on 116 steels, 16 aluminium alloys and six titanium

alloys, nearly all methods currently available for estimation of fatigue properties from simple tensile

data are discussed in detail. The four-point correlation method, the universal slopes method, Mitchell's

method, the modified universal slopes method, the uniform material law by Bgumel and Seeger. and

the modified four-point correlation method by Ong are evaluated in a quantitative manner by using

new criteria proposed in this work, along with conventional error criterion. The modified universal

slopes, Seeger's and Ong's methods give good life predictions. Among them, the modified universal

slopes method provides best results.

( Key wo rd s: f a t ig u e p ro p er ti e s; f a t ig u e l i f e p red ic t io n ; e s t im a t io n

method; simple

t en s i l e d a ta ; e v a lu a t io n cr i t er ia )

Fatigue analysis is very important in the design

of mechanical structures and components. Fatigue

properties o f materials are essential for fatigue analysis.

Fatigue properties such as stress-life

( S - N )

or

strain-life (e-N) curves are usually obtained by

performing fatigue tests. However, as fatigue testing

requires a lot of time and effort, there have been

many attempts to estimate fatigue properties from

simple tensile data.

Manson ~ first proposed two me thods , the four-point

correlation method and universal slopes method, to

estimate the strain-life curve composed of the plastic

and elastic lines on a log-log scale, using only

tensile data. Mitchell-~ has proposed another meth od,

particularly suitable for steels. Recently, B~iumel and

Seeger. 3 have proposed a new m ethod, the uniform

material law. In contrast, Ong 4 has proposed a modified

four-point correlation method.

The universal slopes method, in which the slopes of

the plastic and elastic lines are universalized as -0.6

and -0.12 respectively for all materials, has been

widely used for its simplicity and ease of application.

However, the method tends to produce overconserv-

ative estimates of life in the very high cycle life range.

To improve the original universal slopes method,

Muralidharan and Manson5 have proposed a new,

modified universal slopes method.

The assumption that the slope of the plastic

strain-life relationship (the plastic line) on a log-log

plot is almost constant for all materials is relatively

well accepted. However, the slope of the elastic

strain-life relationship (the elastic line) has been

frequently observed to vary over a relatively wide

range and to be material-dependent. Considering this

material dependence, Mitchell proposed a method to

estimate the slope of the elastic line from the tensile

strength of material.

The uniform material law proposed by Baumel and

Seeger. has been derived from a large amount of

fatigue data collected by them. This method may be

said to be a kind of universal slopes method, which

assigns di fferent slopes to una lloyed and low-alloy steels

and to aluminium and ti tanium alloys respectively. One

advantage of the method is that only the tensile

strength of the material is needed for estimation of

the strain-life curve, in contrast to other methods,

which also require the data of the reduction in area

or the fracture ductility of the material. Using the

fatigue data collected by them, they checked the

prediction capability of the uniform material law and

the modified universal slopes method. They found

that both methods showed larger deviations between

the predicted and experimental results for aluminium

and titan ium alloys and for high-alloy steels, compared

with unalloyed and low-alloy steels.

Quite recently, Ong 6 has evaluated three commonly

used methods, the four-point correlation method, the

original universal slopes method and Mitchell's method

(or the method proposed by Socie e t a l . ) , and found

that the four-point correlation method and the original

universal slopes meth od give satisfactory results, while

Mitchell's method gives inferior predictions. Moreover,

he proposed a new, modified four-point correlation

method to improve the original four-point correlation

method. However, the data used for the evaluation

were not actual experimental data but calculated values

from the fatigue properties of materials provided in

the

A S M M e t al s H a n d b o o k 7.

365

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3 6 6

Jun-Hyub Park and J i -Ho Song

T h o u g h m a n y u s e fu l e s ti m a t i o n m e t h o d s h a v e b e e n

p ro p o s e d a s d e s c r i b e d a b o v e , t h e r e i s l i t t l e w o rk t o

e v a l u a t e t h e m e x t e n s i v e l y , p a r t i c u l a r l y u s i n g a c t u a l

e x p e r i m e n t a l l y o b t a i n e d d a t a o n a v a r i e ty o f m a t e r ia l s .

T h e i m p o r t a n c e o f f a t i g u e d a t a h a s b e e n e m p h a s i z e d

r e c e n t l y i n se v e r a l r e m a r k a b l e f a t i g u e d a t a b o o k s ,

c o n t a i n i n g a l a rg e a m o u n t o f e x p e r i m e n t a l f a t i g u e d a t a

f o r a b r o a d r a n g e o f m a t e r i a l s , p u b l i s h e d i n J a p a n

a n d G e r m a n y .

In t h i s w o rk , u s i n g t h e a c t u a l f a t i g u e d a t a a v a i l a b l e

i n f a t i g u e d a t a b o o k s , t h e p r e d i c t i v e a c c u ra c y o f a l l

t h e e s t i m a t i o n m e t h o d s d e s c r i b e d a b o v e i s d i s c u s s e d .

I n g e n e r a l , t h e a c c u r a c y o f t h e e s t i m a t i o n m e t h o d s

h a s b e e n e v a l u a t e d i n t e r m s o f t h e e r r o r o r d e v i a t i o n

o f th e p r e d i c t e d v a l u e s f r o m t h e a c t u a l o n e s . H o w e v e r ,

i t s e e m s t h a t t h e e r ro r c r i t e r i o n o n i t s o w n i s n o t

a l w a y s s u f f i c i e n t f o r d e t a i l e d d i s c u s s i o n o f e s t i m a t i o n

m e t h o d s . S o m e a d d i t i o n a l c r i t e r i a a r e p r o p o s e d i n t h i s

w o r k t o e x a m i n e i n m o r e d e t a i l th e p r e d i c t i v e a c c u r a c y

o f e s t i m a t i o n m e t h o d s . A s t h e f a t i g u e d a t a u s e d

h e r e f o r d i s c u s s i o n o f e s t i m a t i o n m e t h o d s a r e a l m o s t

i n d e p e n d e n t f r o m t h e d a t a f r o m w h i c h t h e e s t i m a t i o n

m e t h o d s h a v e b e e n d e r i v e d , t h e c o n c l u s io n s o b t a i n e d

i n th i s w o r k m a y b e u n b i a s e d a n d h i g h l y r e l ia b l e .

M E T H O D S F O R E S T IM A T I O N O F F A T I G U E

P R O P E R T I E S F R O M T E N S I L E D A T A

T h e e s t i m a t i o n m e t h o d s t o b e d i s c u s s e d i n t h i s w o r k

a re b r i e f l y e x p l a i n e d b e l o w .

O r i g i n a l f o u r - p o i n t c o r r el a ti o n m e t h o d b y M a n s o n 1

T h e s t r a i n - l i f e r e l a t i o n i s u s u a ll y e x p r e ss e d a s t h e

s u m o f t h e e l a s t i c a n d p l a s t i c l i n e s o n a l o g - l o g s c a l e

as fo l lows :

A ~ = A ~ . e ~ A ~ . p = C ~ N } + C p N 'f (1 )

w h e r e A e , A ~ a n d A % a r e t h e t o t a l , e l a st ic a n d p l a s ti c

s t r a i n s r e s p e c t i v e l y , a n d N f i s t h e n u m b e r o f c y c l e s t o

f a i l u r e . I n t h e o r i g i n a l f o u r - p o i n t c o r r e l a t i o n m e t h o d ,

t h e c o e f f i c i e n t s C ~ , b , C p , c o f Eq u a t i o n (1 ) a r e

c a l c u l a t e d u s i n g t h e f o u r p o i n t s , P 1 - P 4 , l o c a t e d o n

t h e e l a s t i c a n d p l a s t i c l i n e s a s s h o w n i n F i g u r e 1 .

E v e r y p o i n t i s d e t e r m i n e d f r o m t e n s i l e d a t a : t h e t r u e

f r a c t u r e s t r e s s ~ r f , t h e t e n s i l e s t r e n g t h 0% a n d t h e

f r a c t u r e d u c t i l i t y e l . A e * i n t h e f i g u re i n d i c a t e s t h e

F i g u r e 1

- - + _ c

~a

i pl a%}cp.~

Ae~ = A ~

I ~ 1 p 4 ~ , ~ I ~

10 0 ..L 101 10 2 10 3 10 4 10 5

4

Num ber of cycles to failure, Nf

Four-point correlation method by Manson

v a l u e o f e l a s t i c s t r a i n a t N f = 10 4 c y c l e s o n t h e e l a s t i c

l ine .

As the t rue f rac tu re s t res s ~rf i s no t a lwa ys g iven in

t h e l i t e r a t u r e , M a n s o n ~ r e c o m m e n d e d t h e f o l l o w i n g

a p p r o x i m a t i o n s u g g e s t e d b y J . O ' B r i e n :

trf = ~r~(1 + El) (2)

O r i g i n a l u n iv e r s a l s l o p e s m e t h o d b y M a n s o n ~

In t h i s m e t h o d , t h e c o e f f i c i e n t s

C ~ , b , C p , c

o f

Eq u a t i o n (1 ) a r e a s g i v e n i n t h e fo l l o w i n g e q u a t i o n :

Ae = Ae~ + Aep = 3 . 5 E

N ( ° . '2 + o-° f.~N~ ,~ '

(3 )

M e t h o d b y M i t c h e l F

I n t h i s m e t h o d , t h e s t r a i n - l i f e r e l a t i o n i s e x p r e s s e d

u s i n g t h e s t r a in a m p l i t u d e A e / 2 a n d t h e n u m b e r o f

r e v e r s a l s 2 N f a s

A~ = Aee + Aep

2 2 2

: tyf (2N f)h + e~ (2N f) c

E

(4 )

w h e re ~ } i s t h e f a t i g u e s t r e n g t h c o e f f i c i e n t , b i s t h e

f a t i g u e s t r e n g t h e x p o n e n t , e } i s t h e f a t i g u e d u c t i l i t y

c o e f f i c i e n t a n d c i s t h e f a t i g u e d u c t i l i t y e x p o n e n t . Th e

e q u a t i o n is r e p r e s e n t e d g r a p h i c a ll y in

F i g u r e 2 . 2 N ,

in

t h e f i g u re i s t h e t r a n s i t i o n f a t i g u e l i f e a t w h i c h t h e

e l a s t i c a n d p l a s t i c l i n e s i n t e r s e c t . M i t c h e l l s u g g e s t e d

t h a t f o r s t e e l s w i t h h a r d n e s s b e l o w 5 0 0 B H N ,

~ ~ ~rf

= a u +

50 (ks i )

= ~ u + 3 4 5 (M P a )

(5)

o

-

1o

b =

log ( lO o ) - log ( lO 6 )

i log 2 (o '~ + 345)

= - ~ l o g = - a

~' \ trB / [ O-B J

(6 )

~5

" 0

e'~

F i g u r e 2

reversals

] A £ ¢ O f ~ b

t I

l0 0 2Nt 10 6 107

Num ber of reversals to failure,~d',lf

Representation of strain amplitude versus num ber of

to failure

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Detailed evaluation of methods for estimation of fat igue propert ies

3 6 7

6 ~ e f = I n 1 0 R A

w h e r e R A i s t h e p e r c e n t a g e r e d u c t i o n i n a r e a .

H e f u r t h e r r e c o m m e n d e d c o n s t r u c t i n g t h e p l a s t i c

l in e u s in g a n e m p i r i c a l r e p r e s e n t a t i o n o f t h e h a r d n e s s

v e r s u s t r a n s i t i o n f a t i g u e l i f e 2 N t , i n s t e a d o f u s i n g a

s p ec i fi c v a l u e o f c. H o w e v e r , a s t h e h a r d n e s s - t r a n s i t i o n

l i f e d a t a a r e n o t a l w a y s r e a d i l y a v a i l a b l e , a c o n s t a n t

v a l u e o f c = - 0 . 6 , w h i c h is r e g a r d e d a s a n a v e r a g e

v a l u e o f c f o r a l m o s t a l l m a t e r i a l s , m a y b e a s s i g n e d

t o t h e s l o p e o f t h e p l a s t i c l i n e , f o r c o n v e n i e n c e .

T h e r e s u l t i n g e q u a t i o n w i l l b e

A t E _ _ A e c - ~ A ~ p

9 ~ - 2 2

[ -2( o-B + 3 4 5

- ]

(orB + 3 4 5 ) -U ° g l ~

[

= E (2 N f ) 6

L

+ ~ , ( 2 N , ) " ~ ( 8 )

T h i s w il l b e h e r e a f t e r r e f e r r e d t o a s M i t c h e l l 's e q u a t i o n .

M o d i f ie d u n i v e r s a l s l o p e s m e t h o d b y M u r a l i d h a r a n

a n d M a n s o n 5

A s w a s b r ie f l y m e n t i o n e d i n t h e i n t r o d u c t i o n , t h i s

m e t h o d h a s b e e n p r o p o s e d t o i m p r o v e t h e o r i g i n a l

u n i v e r s a l s l o p e s m e t h o d , a n d e x p r e s s e s t h e s t r a i n - l i f e

r e l a t i o n a s

[ O.B \ 0 . 8 3 2

A e = 1 .1 7 [ / ~ ) U ~ ° '° 9

/ O " \ - ( I . 5 3

1 55 B

+ 0 . 02 6 6 ~ I - E ) N ? ° 5 6 ( 9 )

T h i s m e t h o d p r o v i d e s l o w e r u n i v e r s a l i z e d s l o p e s f o r

t h e e l a s t i c a n d p l a s t i c l i n e s t h a n t h e o r i g i n a l m e t h o d .

T h e crB/E t e r m i s n e w l y i n t r o d u c e d i n t o t h e c o e f f i c i en t

o f t h e p l a s t i c l i n e , i n d i c a t i n g t h a t t h e t e n s i l e s t r e n g t h

o f th e m a t e r i a l a l s o h a s a m a r k e d e f f e c t o n t h e l if e in

t h e l o w -c y c l e f a t i g u e .

U n i f o r m m a t e r i a l l a w b y B i i u m e l a n d S e e g e r 3

I n t h i s m e t h o d , t w o d i f f e r e n t e x p r e ss i o n s f o r s t r a in

v e r s u s l if e a r e u s e d f o r u n a l l o y e d a n d l o w - a l l o y s t e e ls

a n d f o r a l u m i n i u m a n d t i t a n i u m a l l o y s a s f o l l o w s .

F o r u n a l l o y e d a n d l o w - a l l o y s t e e l s :

A e - - A ~ e -{ - A E p

2 2 2

. . . . - E - \ ~ , , f ]

- b 0 . 5 9 t ~ 2 N f ) - ° 5 8

w h e r e

( l O )

O"B

- - < ~ 0 . 0 0 3 , 0 = 1

E

O'BE > 0 .0 03, 0 = 1 .375 - 1 2 5 .0 ~

F o r a l u m i n i u m a n d t i t a n i u m a l l o y s :

A e = Ae~ + A %

2 2 2

Th e s l o p e s o f t h e e l a s t i c a n d p l a s t i c l i n e s fo r s t e e l s ,

- 0 . 0 8 7 a n d - 0 . 5 8 , a r e v e r y c lo s e t o th e v a lu e s - 0 . 0 9

a n d - 0 . 5 6 i n t h e m o d i f i e d u n i v e r s a l s l o p e s m e t h o d .

H o w e v e r , t h e s l o p e o f th e p l a s ti c li n e f o r a l u m i n i u m

a n d t i t a n i u m a l l o y s , - 0 . 6 9 , i s r a t h e r s t e e p .

M o d i f i e d fo u r - p o i n t c o r r e la t io n m e t h o d b y O n g 4

T o i m p r o v e t h e o r i g i n a l f o u r - p o i n t c o r r e l a t i o n

m e t h o d , t h i s m e t h o d u s e s t h e f o u r p o i n t s P I ' - P 4 '

s h o w n i n F i g u r e 3 , i n s t e a d o f t h e f o u r p o i n t s P 1 - P 4

in F i g u r e 1 . Th e s t r a i n - l i f e r e l a t i o n i s g i v e n a s

Ae Ae~ + Aep

-- = - __

2 2 2

= u f (2 N f )b + ~ f (2 N f )c

E

w h e r e

b ~ { l o g [ t ( r \ ' " 8 ' -

= 0 1 6 t ]

(1 2 )

c = log 2~074 - log ( e , )

cr B 0 .81 t r l , -

Th e v a l u e o f u f i s n e e d e d a l s o in t h is m e t h o d a s i n

t h e o r i g in a l f o u r - p o i n t c o r r e l a t io n m e t h o d . O n g h a s

n o t e d t h a t t h e v a l u e o f (re c a n b e a p p r o x i m a t e d b y

E q u a t i o n ( 2 ) .

F A T I G U E D A T A F O R D I S C U S SI O N O F

E S T I M A T IO N M E T H O D S

T h e f a t i g u e d a t a u s e d a r e s t r a i n - l i f e d a t a o b t a i n e d

t h r o u g h f a t i g u e te s t i n g , w h i c h a r e c o n t a i n e d m a i n l y in

t h e d a t a b o o k s , t h e J S M E D a t a B o o k : F a t ig u e o f M e t a ls

IV 8 a n d M a t e r i a ls D a t a f o r C y c l ic L o a d i n g b y S e e g e r

a n d o t h e r s 3 '9 . I n p a r t i c u la r , w e h a v e u s e d t h e d a t a

o b t a i n e d u n d e r t h e s o - c a l l e d n o r m a l t e s t i n g c o n d i t i o n :

t h a t i s , o n p o l i s h e d s m o o t h o r h o u r g l a s s - t y p e s p e c i m e n s

t e s t e d i n a i r a t r o o m t e m p e r a t u r e u n d e r a x i a l l o a d i n g

.@ E

pi

10 10 106

Num ber of reversals to failure,2Nf

= 1 1 u t ~ - I z . ,z v f g c ' l O ' B [ ' 3 1 ~

--(I.095 q'- 0.3 5( 2N r) -° '69 (11 ) Figure 3 Modified four-point correlation me thod by Ong

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3 6 8

J u n H y u b P a r k a n d J i H o S o n g

at a s t r e s s r a t i o o f R ,~ ( = O m a x / O m i n ) = - - i o r a s t r a i n

r a ti o o f R~ ( = e m a Je r, i , ) = - - 1 . R u n - o u t d a ta a r e

e x c l u d e d . F u r t h e r , o n l y t h e d a t a o n m a t e r i a l s w i t h

b o t h d a t a o f t e n s i l e s t r e n g t h o -B a n d p e r c e n t r e d u c t i o n

i n ar e a R A ( o r f r a c t u r e d u c t i l i t y e l ) a r e u t i l i z e d .

T h e d a t a u s e d a r e l i s t e d i n Ta b l e 1 c l a s s i f i e d b y

d a ta s o u r c e s a n d m a t e r ia l s . T h e t o t a l n u m b e r o f

e - N

c u r v e s i s 3 1 5 a n d t h e t o t a l n u m b e r o f e x p e r i m e n t a l

s t r a i n - l i f e d a t a p o i n t s i s 2 7 1 7 o b t a i n e d o n 1 3 8 m a t e r i a ls .

F o r c o n v e n i e n c e , e a c h i n d i v id u a l s t r a i n - l i f e d a t u m i s

r e f e r r e d t o a e - N d a t a p o i n t , a n d a s e t o f ~ - N d a t a

p o i n t s f o r m i n g a n E - N c u r v e i s h e r e a f t e r r e f e r r e d t o

a ( E - N ) d a ta s e t . T o a p p l y s u c c e s s f u l ly t he e v a l u a t i o n

c r i t e r i a t o b e p r o p o s e d l a t e r , d a t a s e t s t h a t c o n s i s t o f

l e s s t h a n f o u r d a t a p o i n t s a r e n o t c o n s i d e r e d i n

t h e d i s c u s s i o n . W h e n n e e d e d , t h e v a l u e o f ~ rf i s

a p p r o x i m a t e d b y E q u a t i o n ( 2 ) .

C R I T E R I A F O R E V A L U A T I O N O F

E S T I M A T I O N M E T H O D S

A c r i te r i o n m o s t f r e q u e n t l y u s e d i s th e e r r o r c r i t e r io n ,

w h i c h e v a l u a t e s t h e p r e d i c t i v e a c c u r a c y o f a n e s t i m a t i o n

m e t h o d i n t e r m s o f t h e f r a c t i o n o f d a t a f a l l i n g w i t h i n

a s c a t te r b a n d o f a s p e c i f i e d f a c t o r s a s f o l l o w s :

N u m b e r o f d a t a fa l l in g w i t h i n

T h e e x p e r i m e n t a l v a l u e ~< s

T o t a l n u m b e r o f d a t a ( 1 3 )

T h e r e i s a p r o b l e m i n h e r e n t i n t h e c r i t e r i o n ,

a s i l l u s t r a t e d b e l o w . A s s u m e t h a t t h r e e d i f f e r e n t

e s t i m a t i o n m e t h o d s p r o v i d e t h r e e d i f f e r e n t l i f e p r e d i c -

t i o n s , a s s h o w n i n

F i g u r e 4

w h e r e N p a n d N e a r e t h e

p r e d i c t e d a n d e x p e r i m e n t a l l i v e s r e s p e c t i v e l y . T h e

v a l u e s c a l c u l a t e d b y E q u a t i o n ( 1 3 ) a r e a l l t h e s a m e

f o r t h e th r e e m e t h o d s . H o w e v e r , a s ca n b e e a s il y

f o u n d , m e t h o d C c a n h a r d l y e x p l a i n t h e e x p e r i m e n t a l

r e s u l t s a n d m e t h o d B m a y b e e x p e c t e d t o g i v e s u p e r i o r

p r e d i c t i o n s t o m e t h o d A f o r a w i d e r r a n g e o f l if e .

T h i s d r a w b a c k in t h e e r r o r c r i t er i o n m a y b e o v e r c o m e

b y c o n s i d e r i n g t h e l i n e a r r e l a t i o n s h i p b e t w e e n t h e

p r e d i c t e d a n d e x p e r i m e n t a l l i v e s i n t h e f o l l o w i n g

m a n n e r .

I f th e r e l a t io n s h i p b e t w e e n t h e p r e d i c t e d a n d e x p e r -

i m e n t a l l i v e s i s a p p r o x i m a t e d b y a l e a s t - s q u a r e s l i n e

as

l o g ( 2 N p )

= a + N o g ( 2 N , . ) ( 1 4 )

10

~ Method

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. i l i

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10

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Nf 109

F i g u r e 4 I l l u s t r a ti o n o f d r a w b a c k i n h e r e n t i n t h e e r r o r c r i t e r i o n

T a b l e 1 Data u s e d f o r c o m p a r i s o n o f e st i m a t i o n m e t h o d s

J S M E D a t a B o o k.

M a t e r i a l s u n i t F a t ig u e o f M e t a ls I V ~

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a l l o y s N u m b e r o f 100

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7/17/2019 1-s2.0-014211239599737U-main


Deta i led eva lua t ion o f methods fo r es t imat ion o f fa t igue p roper t ies

3 6 9

i t c a n b e s a i d t h a t t h e c l o s e r t h e v a l u e s o f i n t e r c e p t a

a n d s l o p e / 3 a r e t o 0 a n d 1 r e s p e c t i v e l y , t h e b e t t e r t h e

e s t i m a t i o n m e t h o d i s . A n d i f a i s r a t h e r g r e a t e r t h a n

z e r o , t h e c a s e w h e n / 3 i s m o d e r a t e l y s m a l l e r t h a n u n i t y

g i ve s b e t t e r p r e d i c t i o n s t h a n t h e c a s e w h e n / 3 i s g r e a t e r

t h a n u n i t y . F o r a r a t h e r l e s s t h a n z e r o , t h e c a s e

w h e n / 3 i s m o d e r a t e l y g r e a t e r t h a n u n i t y g i ve s b e t t e r

p r e d i c t io n s t h a n t h e c a s e w h e n / 3 i s s m a l le r t h a n u n i t y .

T h u s t h e v a l u e s o f a , / 3 a n d ( a + /3 ) c a n b e u t i li z e d

t o e v a l u a t e t h e p r e d i c t i v e a c c u r a c y o f a n e s t i m a t i o n

m e t h o d f r o m t h e v i e w p o i n t o f a g r e e m e n t b e t w e e n t h e

p r e d i c t e d a n d e x p e r i m e n t a l l i v e s . T h e d r a w b a c k a b o v e

d e s c r i b e d c a n b e o v e r c o m e b y e m p l o y i n g t h e v a l u e s

o f a , /3 a n d ( a + /3 ) a s a d d i t i o n a l c r i t e r i a .

I n u s i n g t h e a d d i t i o n a l c r i t e r i a , t h e fo l l o w i n g p o s s i-

b i li ty s h o u l d b e c o n s i d e r e d . A s s u m e t h a t F i g u r e 5

s h o w s t h e p r e d i c t i o n r e s u l t s o b t a i n e d b y a p p l y i n g tw o

e s t im a t i o n m e t h o d s , D a n d E , t o t w o ( e - N ) d a t a s et s

I a n d I I . T h a t i s , m e t h o d D g i ve s d i ff e r e n t l ea s t -

s q u a r e s l in e s f o r ( e - N ) d a t a s e t s [ a n d I I , w h i l e

p ro v i d i n g a l e a s t - s q u a re s l i n e o f a ~ 0 a n d / 3 ~ 1 fo r

t h e c o m b i n e d d a t a o f d a t a s e t s I a n d I I , a s s h o w n i n

Figure 5a . I n c o n t r a s t , m e t h o d E g i v es n e a r l y t h e

s a m e l e a s t - s q u a re s l i n e s o f a ~ 0 a n d /3 ~ - 1 fo r b o t h

d a t a s e t s I a n d I I , a n d a l s o f o r t h e c o m b i n e d d a t a o f

d a t a s e t s I a n d I I , a s s h o w n i n

F i g u r e 5 b .

L i m i t i n g

a t t e n t i o n t o o n l y t h e r e s u l t s o b t a i n e d f o r t h e c o m b i n e d

d a t a , m e t h o d s D a n d E a p p e a r t o b e e q u a l l y e x c e l l e n t

m e t h o d s . H o w e v e r , i t i s e v i d e n t f r o m t h e r e s u l t s

o b t a i n e d f o r i n d i v i d u a l ( E - N ) d a t a s e t s t h a t m e t h o d

E i s p r e f e r a b l e .

T h e c o r r e l a t i o n c o e f f i c ie n t b e t w e e n t h e p r e d i c t e d

a n d e x p e r i m e n t a l v a l u e s , r , w i l l a l s o b e o n e o t h e r

a d d i t i o n a l c r i t e ri o n . W h e n t w o d i f f e r e n t m e t h o d s

h a p p e n t o p ro v i d e i d e n t i c a l l e a s t - s q u a re s l i n e s , a n

e s t i m a t i o n m e t h o d t h a t g i v e s a h i g h e r r v a l u e i s

p r e f e r a b l e .

z ~

e~

- - t - - - f o r e N c u r v e l I I1 f f

- - m - - f o r e Nc ur ve l l l T , 1+I1

f o r c u r v e s 1 +

I l l I

i

i "2'

/

a) Method D

•f/I II•

/

" A "

. /

/

N , = 4

i

b ) M e t h o d E

E x p e r i m e n t a l l i fe , N f

F i g u r e 5 A p p l i c a t i o n o f l e a s t - s q u a r e s a n a l y si s t o t h e p r e d i c t e d d a t a

b y t w o d i f f e r e n t e s t i m a t i o n m e t h o d s

I n t h i s w o r k , t h e p r e d i c t i v e a c c u r a c y o f e s t i m a t i o n

m e t h o d s w i l l b e e v a l u a t e d i n a q u a n t i t a t i v e m a n n e r

b y u s i n g t h e e v a l u a t i o n m e a s u r e d e f i n e d b e l o w .

F r a c t io n o f d a t a w i t h i n a f a c t o r o f s , E j s )

A s n o t e d a b o v e , t h e e r r o r c r i t e r i o n c a n b e e x p r e s s e d

b y E q u a t i o n ( 1 3 ) , i . e . f r a c t i o n o f d a t a w i t h i n a f a c t o r

o f s , E f ( s ) . A v a l u e o f s o f 3 i s e m p l o y e d fo r l i f e

p r e d i c t i o n . T h e e v a l u a t i o n v a l u e Er(s = 3 ) i s g iven as

Ef(s = 3 ) (1 5 )

1

N p

N u m b e r o f d a t a f a l l i n g w i t h i n _ < - N r -<

3

N u m b e r o f to t a l d a t a

Th e c l o s e r E l (S ) is t o 1 , th e b e t t e r t h e p r e d i c t i o n .

A c c o r d i n g l y , a l l t h e e v a l u a t i o n m e a s u r e s t o b e h e r e

e m p l o y e d w i ll b e f o r m u l a t e d t o b e u n i t y fo r i d e a ll y

g o o d p r e d i c t i o n .

G o o d n e s s o f f i t b e tw e e n t h e p r ed i c te d a n d

exper imen ta l va lues , E .~

U s i n g t h e a d d i t i o n a l c r i t e r i a p r e s e n t e d a b o v e , t h e

g o o d n e s s o f f it is d e f i n e d f o r t h e c o m b i n e d d a t a o f a ll

( e - N ) d a t a s e ts a n d f o r i n d i v id u a l ( E - N ) d a t a s e t s.

s e p a r a t e l y a s

l - I O f t o t a l [ ) - c 1 - - l 1 - - / - ~ t o ta l I )

(E.),,,,,,1 =

4

( 1 - - [ 1 -

O ~ t o t a l - ~ [~ to ta I

[ ) + ( 1 - - 1 1 - - r ,, ,, ,[ ) ( 16 )

+ - - 4

1 / V

(E.)D~c, = N ,=~ (E~,),

N z.., [ 4

i = 1

~+ (1 - - 1] - - ~ ' - - ~ ' 1) + (1 - - [ I - - ' " l ) ] ( 1 7 )

w h e re t h e s u b s c r i p t s , t o t a l a n d i, r e f e r I o t h e c o m b i n e d

d a t a o f al l ( e - N ) d a t a s e ts a n d t h e it h ( e - N ) d a t a s e t

r e s p e c t iv e l y , a n d N is t h e n u m b e r o f ( e - N ) d a t a s e ts .

(E,0D s et o f Eq u a t i o n (1 7 ) r e p re s e n t s t h e g o o d n e s s o f

f i t f o r i n d i v i d u a l ( e -N ) d a t a s e t s .

C O M P A R I S O N O F E S T IM A T I O N M E T H O D S

L i f e p r e d i c t i o n s w e r e p e r f o r m e d w i t h t h e a f o r e m e n -

t i o n e d s i x e s t i m a t i o n m e t h o d s o n t h e f i v e m a t e r i a l

g ro u p s l i s t e d i n Table 1 . F igure 6 s h o w s a t y p i c a l

r e s u l t o f li f e p r e d i c t i o n s o b t a i n e d o n l o w -a l l o y s t e e l s ,

f o r w h ic h t h e l a r g e st a m o u n t o f d a t a i s o b t a i n e d a m o n g

t h e f i v e m a t e r i a l g ro u p s . I n t h e f i g u re , t h e p e r f e c t

c o r r e l a t i o n l i n e a n d a f a c t o r o f 3 s c a t t e r b a n d a r e

e x p re s s e d b y t h e d a s h e d a n d s o l i d l i n e s r e s p e c t i v e l y .

T h e e q u a t i o n a n d t h e v a l u e o f r in t h e f ig u r e r e p r e s e n t

t h e r e s u l t o f l e a s t - s q u a re a n a l y s i s a n d t h e c o r r e l a t i o n

c o e f f ic i e n t f o r t h e c o m b i n e d d a t a r e s p e c ti v e l y .

A s c a n b e f o u n d f r o m F i g u r e 6 a , t h e p r e d i c t e d d a t a

b y t h e f o u r - p o i n t c o r r e l a t i o n m e t h o d t e n d t o l e v e l o f f

i n t h e l o n g - l i f e r a n g e , r e s u l t i n g i n o v e r - c o n s e rv a t i v e

l i fe p r e d i c t i o n s . S u c h a t r e n d i s a l s o d i s c e rn i b l e i n t h e

r e s u l t f o r t h e o r i g i n a l u n i v e r s a l s l o p e s m e t h o d s h o w n

in Figure 6b . I n c o n t r a s t , t h e m o d i f i e d u n i v e r s a l s lo p e s

m e t h o d , t h e u n i f o r m m a t e r i a l l a w b y B ~ i u m e l a n d

7/17/2019 1-s2.0-014211239599737U-main


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109

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107

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10 101 102 103 104 105 106 107 10 109

Experimental eversals,2N

109

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107

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106

105

>

104

o

;.~ 103

102

101

b) OriginalUniversalSlopes

0 0

o o ~ 8 0

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r = 0.936

10 ~ ' ' ~ ~ l ° g ( 2 N=0.858+0.781 log(2N r

10 101 102 103 104 105 106 I07 10 109


f

109

1 0 8

107

Z

t"q

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105

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~ 104

;~ 103

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101

c) ModifiedUniversal

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r = 0.936

I0 ° ~ l o g ( 2 N=0.100+0.9531og(2Nr)

I0 ° l01 102 103 l04 105 106 107 108 109

Experimental eversals,2Nf

1 0 9

1 0 s

107

Z

t'-,I

106

~ 1¢

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~ 10

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d) Mitchell'sMethod

ee

m

engB ~ n o ~

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log(2Nv)=0.642+0.9271og(2Nr )

]0 ° l 01 10 2 10 3 10 4 10 5 10 6 10 7 l 0 8 10 9


f

1 0 9 1 0 9

e) Seeger'sMeth°d ~ ~ ~ If)Ong 'sMeth°d ~

108 108

mull •

107 ~k

106 , ~ 10 o |

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05

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lO r = 0.934 r = 0.935

10 ~ log(2Np)=0 .330+ 0.9161og(2 Nf) ll010 ~ I°g(2N p)=0"5 63+ 0"8621 °g(2N I)

l0 ° 101 102 103 104 105 106 107 l08 109 10 ° lO 1 102 103 104 105 106 107 l0 x 109

Experimental eversals,2Nf Experimental eversals,2N

f

Figure 6 Com parison of the predicted and expe rimen tal fa t igue lives for low-al loy s teels

7/17/2019 1-s2.0-014211239599737U-main


Detai led evaluat ion of methods for est imat ion of fa t igue propert ies 37 1

S e e g e r. a n d t h e m o d i f i e d f o u r -p o i n t c o r r e l a ti o n m e t h o d

b y O n g p r o v id e r e l a t i v e ly g o o d l i f e p r e d i c t i o n s . H o w -

e v e r , Mi t c h e l l ' s me th o d t e n d s t o g iv e c o n s id e r a b ly

n o n - c o n s e r v a t i v e p r e d i c t i o n s i n t h e l o n g e r - l i f e r a n g e .

A c lo s e i n s p e c t i o n o f Fi gure 6 r e v e a l s t h a t t h e

mo d i f i e d u n iv e r s a l s l o p e s me th o d t e n d s t o g iv e s l i g h t l y

c o n s e r v a t i v e p r e d i c t i o n s a t s h o r t e r l i v e s , b u t n o n -

c o n s e r v a t i v e p r e d i c t i o n s a t l o n g l i v e s . A s imi l a r t e n d -

e n c y c a n b e f o u n d f o r S e e g e r 's m e t h o d . T h e p r e d i c t e d

d a t a d u e t o O n g ' s m e t h o d a r e f o u n d o n a v e r a g e t o

f a l l s l i g h t l y a b o v e t h e p e r f e c t c o r r e l a t i o n l i n e .

T h e t r e n d o b s e r v e d o n l o w - a l l o y s t e el s w a s o b s e r v e d

c o n s i s t e n t l y o n o th e r ma te r i a l s . H o w e v e r , p a r t i c u l a r l y

o n a lu m in iu m a l l o y s , a f e w e x c e s s iv e ly n o n - c o n s e r v a -

t i v e p r e d i c t i o n s w e r e o b s e r v e d a t s h o r t l i v e s f o r a l l

t h e me th o d s e x c e p t Mi t c h e l l ' s .

U n l e s s o t h e r w i s e s t a t e d , F i g u r e 6 i s h e r e a f t e r

e mp lo y e d a s r e p r e s e n t a t i v e o f t h e r e s u l t s f o r a l l

ma t e r i a l s , f o r c o n v e n i e n c e .

Tab l e 2 s h o w s c o m p a r i s o n s o f t h e p r e d i c t i v e a c c u r a c y

o f e st i m a t i o n m e t h o d s o n t h e b a s is o f th e e v a l u a ti o n

valu__es de f ined in the pre v io us sec t ion . Th e va lue s of

E , E a n d E i n t h e t a b l e a r e g iv e n a s

3

= k _ 7: t - = E f ( s = 3 ) + ( E a ) t o t a l + ( E a ) D s e t

3 3

( 1 8 )

5

(E k ) i U i

E k - - j = l

- ~ - - - ( 1 9 )

ENj

J = l

E E k E l ( s = 3 ) + ~ +

: k = l = . . . . . t o t a l _ O s e t

3 3

( 2 0 )

w h e r e t h e s u b s c r ip t s j a n d k r e f e r t o t h e j t h ma te r i a l

a n d k th e v a lu a t i o n v a lu e r e s p e c t i v e ly , a n d N j i s t h e

n u m b e r o f

( e - N )

d a t a s e t s o f t h e j t h ma te r i a l .

T h e b o ld - f a c e d fi g u re s in t h e t a b l e r e p r e s e n t t h e

b e s t p r e d i c t i o n f o r e a c h e v a lu a t i o n i t e m. F i r s t , g iv in g

a t t e n t i o n t o t h e r e s u l t o n l o w - a l l o y s t e e l s a l o n g w i th

Figure 6 ,

t h e e v a lu a t i o n v a lu e o f f r a c t i o n o f d a t a

E f ( s

= 3 ) i s n e a r l y t h e s a me f o r t h e o r i g in a l a n d

mo d i f i e d u n iv e r s a l s l o p e s me th o d s , S e e g e r ' s a n d O n g ' s

m e t h o d s , a m o n g w h i c h O n g ' s m e t h o d g i v es t h e b e s t

p r e d i c t i o n s . W i th r e s p e c t t o t h e g o o d n e s s o f f it , t h e

mo d i f i e d u n iv e r s a l s l o p e s me th o d i s t h e b e s t f o r t h e

c o m b in e d d a t a , w h i l e S e e g e r ' s i s t h e b e s t f o r i n d iv id u a l

( E - N ) d a t a s e t s . I t i s i n t e r e s t in g t h a t t h e m e th o d g iv in g

th e b e s t r e s u l t d i ff e r s d e p e n d in g o n t h e e v a lu a t i o n i t e m.

C o n s id e r in g t h e e v a lu a t i o n v a lu e a v e r a g e d o v e r t h r e e

i t e m s , E , t h e m o d i f i e d u n iv e r sa l s l o p e s m e t h o d m a y

b e s a id t o b e t h e b e s t f o r l o w - a l l o y s t e e l s .

B a s e d o n t h e v a lu e o f E , t h e mo d i f i e d u n iv e r s a l

s l o p e s me th o d i s a l s o t h e b e s t f o r u n a l l o y e d a n d h ig h -

a l l o y s t e e l s . F o r u n a l l o y e d a n d h ig h - a l l o y s t e e l s ,

S e e g e r ' s me th o d a l s o g iv e s g o o d r e s u l t s , c o mp a r a b l e

t o t h e m o d i f i e d u n iv e r sa l s l o p e s m e t h o d . O n t h e o t h e r

h a n d , Mi t c h e l l ' s me th o d i s t h e b e s t f o r a l u min iu m

a l lo y , w h i l e O n g ' s i s t h e b e s t f o r t i t a n iu m.

T a b l e 2 C o m p a r i s o n o f e s t i m a t i o n m e t h o d s i n t e r m s o f e v a l u a t i o n v a lu e s

O r i g i n a l

M a t e r i a l F o u r - p o i n t u n i v e r s a l

g r o u p E v a l u e c o r r e l a t i o n s l o p e s

M o d i f i e d

u n i v e r s a l

s lopes

M i t c h e l l ' s

m e t h o d

S e e g e r ' s

m e t h o d

O n g ' s

m e t h o d

U n a l l o y e d E f ( s = 3) 0 . 570 0 . 769 0 . 8 2 4 0.738 0.805 0.811

steels (Ea)total 0.431 0.669 0.925 0.774 0.885 0.868

( E ~ ) D ~ 0 . 2 3 4 0 . 4 62 0 . 5 42 0 . 3 3 0 0 . 5 6 9 0 .4 9 l

E 0 . 412 0 . 634 0 . 764 0 . 614 0 . 753 0 . 723

Low-al loy

E f ( s

= 3) 0 . 548 0 . 750 0 . 764 0 . 532 0 . 768 0 . 793

s tee l s (E , )~o , , 0 . 229 0 . 444 0 . 747 0 . 528 0 . 654 0 . 562

(E~)Ds~, 0.224 0.503 0.621 0.514 0.632 0.614

0 . 334 0 . 566 0 . 711 0 . 524 0 . 684 0 . 656

H i g h - a l l o y

E r ( s

= 3) 0.663 0.865 0.867 0.60/) 0.846 0.865

s tee l s (E , ) , , , ~ 0 . 313 0 . 648 0 . 956 0 . 808 0 . 970 0 . 801

(E~)D~e, 0.4549 0.645 0.656 0 .605 0 .662 0.681

0 . 477 0 . 719 0 . 826 0 .671 0 . 826 0 . 782

A l u m i n i u m

E f ( s =

3) 0 . 567 0 . 8 0 0 0 . 796 0 . 721 0 . 738 0 . 713

a l loys (E~)~ , ,~ -0 . 0 43 0 . 586 0 . 720 0 . 907 0 . 877 0 . 723

(E,)Ds~, 0.437 0.666 0.704 O.734 0 . 7 3 5 0 . 712

0 . 320 0 . 684 0 . 740 0 . 787 0 . 783 0 . 716

T i t a n i u m

E f ( s =

3) 0 . 519 0 . 759 0 . 852 0 . 648 0 . 778 0 . 8 7 0

a l loys ( E , ) , , , ~ 0 . 331 0 . 558 0 . 892 0 . 847 0 . 746 0 . 9 2 1

( E , )D ~, 0 . 462 0 . 500 0 . 448 0 . 580 0 . 472 0 . 592

0 . 437 0 . 606 0 . 731 0 . 692 0 . 665 0 . 794

Al l E f ( s = 3) 0 . 576 0 . 780 0 .801 0 . 609 0 . 788

0 . 8 0 5

mate r i a l s = 0 . 271 0 . 556 0 . 834 0 . 688 0 . 798 0 . 708

(E~)~o,~J 0.303 0.539 0.613 0.514 0.62 9 0.609

( E~ )1 . . .

Tota l ~ 0 . 384 0 . 627 0 . 7 5 1 0.606 0.739 ( / .709

7/17/2019 1-s2.0-014211239599737U-main


37 Jun Hyub Park and Ji Ho Song

I n t h e e v a l u a t i o n a v e r a g e d o v e r a l l m a t e r i a l g r o u p s

a n d e x p r e s s e d b y t h e v a l u e o f E , a l m o s t t h e s a m e

r e s u l t s a r e o b t a i n e d a s d e s c r i b e d a b o v e o n l o w - a l l o y

s tee l s ,

I n t h e t o t a l e v a l u a t i o n r e p r e s e n t e d b y t h e v a l u e o f

~ , t h e m o d i f i e d u n i v e r s a l s l o p e s m e t h o d i s b e s t a n d

S e e g e r ' s m e t h o d i s n e x t b e s t , f o l l o w e d b y O n g ' s .

D I S C U S S I O N

F r o m t h e r e s u l t s i n t h e p r e c e d i n g s e c t i o n , t h e f e a t u r e s

o f e a c h e s t i m a t i o n m e t h o d c a n b e s u m m a r i z e d a s

fo l lows .

T h e p r e d i c t i v e a c c u r a c y o f b o t h t h e f o u r - p o i n t

c o r r e l a t i o n m e t h o d a n d t h e o r i g i n a l u n i v e r s a l s l o p e s

m e t h o d i s c o n s i d e r e d i n f e r i o r t o t h e o t h e r m e t h o d s ,

b e c a u s e t h e p r e d i c t i o n r e s u l t s f r o m b o t h m e t h o d s t e n d

t o l e v e l o f f in t h e l o n g - l i f e r a n g e , a s n o t e d i n Figure

6 . A s c a n b e f o u n d f r o m t h e v a l u e o f E f ( s = 3 ) in

Table 2 o n l y a b o u t 5 8 % o f t h e d a t a p r e d i c t e d b y t h e

o r i g i n a l f o u r - p o i n t c o r r e l a t i o n m e t h o d f a l l w i t h i n a

f a c t o r o f 3 s c a t t e r b a n d , i n d i c a ti n g th a t t h e m e t h o d

m a y n o t b e r e c o m m e n d e d .

T h e o r i g in a l u n i v e r sa l s l o p e s m e t h o d m a y b e s a t is f a c-

t o r y in t h e r e s p e c t t h a t a b o u t 8 0 % o f th e d a t a f a ll

w i t h i n a f a c t o r o f 3 s c a t te r b a n d , H o w e v e r , t h e g o o d n e s s

o f f it d u e t o t h _e m e t h o d i s i n f e r i o r , a s t h e v a l u e s o f

(E~)tot~l and (E~)Dset in Table 2 i n d i c a t e .

I t i s w o r t h n o t i n g t h a t t h e a b o v e t w o m e t h o d s t e n d

t o p ro v i d e s l i g h t l y n o n -c o n s e rv a t i v e p r e d i c t i o n s a t

s h o r t l i v es , b u t a r e e x c e s s i v e l y o v e rc o n s e r v a t i v e a t

l o n g l i v e s . T h i s f a c t h a s b e e n a l r e a d y p o i n t e d o u t b y

M a n s o n a n d M u r a l i d h a r a n 5 , p a r t i c u l a r l y f o r th e o r i gi n a l

u n i v e r s a l s l o p e s m e t h o d .

I n c o n t r a s t , t h e m o d i f i e d u n iv e r s a l s lo p e s m e t h o d ,

d e v e l o p e d t o o v e r c o m e t h i s d r a w b a c k o f t h e o r ig i n a l

u n i v e r s a l s l o p e s m e t h o d , t e n d s t o g i v e s l i g h t ly c o n s e rv a -

t i v e p r e d i c t i o n s a t s h o r t e r l i v es , b u t i s n o n -c o n s e rv a t i v e

a t l o n g l i v e s , a s a l r e a d y n o t e d i n Figure 6c. W h i l e t h e

f r a c t i o n o f d a t a w i t h i n a f a c t o r o f 3 s c a t t e rb a n d i s

8 0 % , a s c a n b e f o u n d i n Table 2 t h e g o o d n e s s o f f i t

i s r e m a r k a b l y i m p r o v e d i n t h e m o d i f i e d u n i v e r s a l

s l o p e s m e t h o d . A s t h e e q u a t i o n i n Figure 6c s h o w s ,

t h e v a l u e s o f i n t e r c e p t (~ = 0 .1 0 0 ) a n d s l o p e

( /3 = 0 .9 5 3 ) a r e v e ry c l o s e t o 0 a n d 1 r e s p e c t i v e l y .

Th i s r e s u l t i m p l i e s t h a t t h e m o d i f i e d u n i v e r s a l s l o p e s

m e t h o d g i v e s a c c u r a t e p r e d i c t i o n s .

A s c a n b e s e e n i n

Figure 6d

M i t c h e l l ' s m e t h o d a l s o

p ro v i d e s a v a l u e o f s l o p e , /3 = 0 .9 2 7 , r e l a t i v e l y c l o se

t o 1 , i n d i c a t i n g e x c e l l e n t d i r e c t c o r r e s p o n d e n c e

b e t w e e n t h e p r e d i c t e d a n d e x p e r i m e n t a l l i v e s . H o w -

e v e r , t h e v a l u e o f t h e i n t e r c e p t , a = 0 ,6 4 , is r e l a t i v e l y

h i g h , l e a d i n g t o n o n - c o n s e r v a t i v e p r e d i c t i o n s o v e r t h e

e n t i r e l if e r a n g e . C o n s e q u e n t l y , o n l y a b o u t 6 0 % o f

t h e d a t a f a l l w i t h i n a f a c t o r o f 3 s c a t t e rb a n d a s l i s t e d

in Table 2.

G o o d d i r e c t c o r r e s p o n d e n c e ( /3 = 0 . 91 4 ) b e t w e e n

t h e p r e d i c t e d a n d e x p e r i m e n t a l l i v e s c a n b e o b s e r v e d

a l so in Figure 6e f o r S e e g e r ' s m e t h o d . I n p a r t i c u l a r ,

t h e m e t h o d i s th e b e s t i n g o o d n e s s o f f it f o r i n d i v i d u a l

d a t a s e t s , a s c a n b e fo u n d f ro m t h e v a l u e o f (E~ )D ~ c t

in Table 2. T h e f r a c t i o n o f d a t a f a ll i ng w i t h in a f a c t o r

o f 3 s c a t t e r b a n d i s 7 9 % .

A s c a n b e s e e n i n Figure 6f O n g ' s m e t h o d g i v e s

t h e v a l u e o f s l o p e , /3 = 0 .8 6 2 , l e s s t h a n t h o s e fo r t h e

a b o v e t h r e e m e t h o d s , a n d a r e l a t i v e ly h ig h i n t e r c e p t

v a l u e , c~ = 0 , 5 6 3 . C o n s e q u e n t l y , O n g ' s m e t h o d i s

i n f e r i o r to b o t h t h e m o d i f i e d u n iv e r s a l s lo p e s m e t h o d

a n d S e e g e r ' s m e t h o d i n t h e g o o d n e s s o f f i t, a n d

t e n d s t o p r o v i d e n o n - c o n s e r v a t i v e p r e d i c t i o n s m o r e

f r e q u e n t l y , c o m p a r e d w i t h t h e m o d i f i e d u n iv e r s a l

s l o p e s m e t h o d . T h e f r a c t i o n o f d a t a w i t h i n a f a c t o r o f

3 s c a t t e r b a n d i s s l ig h t ly m o r e t h a n 8 0 % .

I t c a n b e n o t e d f ro m t h e a b o v e d i s c u s s i o n t h a t t h e

m o d i f i e d u n i v e r s al s lo p e s m e t h o d , S e e g e r ' s m e t h o d

a n d O n g ' s m e t h o d a l l g i v e r e a s o n a b l y g o o d l i f e

p r e d i c t io n s , w i t h a b o u t 8 0 % o f t h e d a t a f a l li n g w i t h in

a f a c t o r o f 3 s c a t t e rb a n d . B a s e d o n t h e g o o d n e s s o f

f i t , t h e m o d i f i e d u n i v e r s a l s l o p e s m e t h o d i s t h e b e s t

a n d S e e g e r ' s m e t h o d is th e n e x t b e s t . W h i l e n o t s h o w n

h e re e x p l i c i t l y , t h e m o d i f i e d u n i v e r s a l s l o p e s m e t h o d

h a s t h e f e w e s t n o n - c o n s e r v a t i v e d a t a f a l l i n g o u t s i d e

t h e u p p e r l i n e o f a f a c t o r o f 3 s c a t t e r b a n d , i n

c o m p a r i s o n w i t h o t h e r t w o m e t h o d s .

I n c o n c l u s i o n , t h e m o d i f i e d u n i v e r s a l sl o p e s m e t h o d

c a n b e r e c o m m e n d e d a s t h e b e s t e s t i m a t i o n m e t h o d

a t p r e s e n t . T h i s m e t h o d e s s e n t i a l l y r e q u i r e s t h e e l a s t i c

m o d u l u s E , t e n s i l e s t r e n g t h o -B a n d f r a c t u r e d u c t i l i ty

el, o f t h e m a t e r i a l . W h e n t h e f r a c t u r e d u c t i l i t y i s n o t

a v a i l a b le , S e e g e r ' s m e t h o d , w h i c h r e q u i re s o n l y t h e

e l a s t i c m o d u l u s a n d t e n s i l e s t r e n g t h , m a y b e u t i l i z e d

a s a n a l t e rn a t i v e t o o b t a i n s a t i s f a c t o ry r e s u l t s .

O n t h e o t h e r h a n d , O n g ' s m e t h o d , w h i l e b e i n g a

g o o d e s t i m a t i o n m e t h o d , r e q u i r e s t h e t r u e f r a c t u r e

s t res s o -t,, wh ic h i s no t a lw ays g iven , in ad d i t io n to the

a b o v e t h r e e m a t e r i a l p r o p e r t i e s .

A l t h o u g h n o t s h o w n h e r e , t h e f r a c t io n o f d a t a f a ll in g

w i t h i n a f a c t o r o f 2 s c a t t e r b a n d i s f o u n d t o b e a b o u t

6 0 % f o r a ll t h e a b o v e t h r e e m e t h o d s . T h e m a j o r

c o n c l u si o n s o b t a i n e d a b o v e f o r a f a c t o r o f 3 s c a t te r b a n d

a r e h a r d l y a f f e c t e d b y e m p l o y i n g a f a c t o r o f 2 .

I t h a s a l r e a d y b e e n n o t e d i n t h e i n t r o d u c t i o n t h a t ,

o n i t s o w n , t h e e r r o r c r i t e r i o n e x p r e s s e d b y E q u a t i o n

(1 5 ) i s n o t a l w a y s s u f f i c ie n t f o r d e t a i l e d e v a l u a t i o n o f

e s t i m a t i o n m e t h o d s , B a s e d o n t h e e r r o r c r i t e r i o n

a l o n e , t h e o r i g i n a l u n i v e r s a l s l o p e s m e t h o d m a y b e

s a i d t o b e o n e o f t h e b e s t e s t i m a t i o n m e t h o d s , a s t h e

f r a c t i o n o f d a t a f a l l i n g w i t h i n a f a c t o r o f 3 (o r 2 )

s c a t te r b a n d a m o u n t s t o 7 8 % ( o r 6 0 % ) , c o m p a r a b l e

t o t h e r e s u l t s d u e t o t h e a b o v e t h r e e m e t h o d s ,

H o w e v e r , t h e o r i g i n a l u n i v e r s a l s l o p e s m e t h o d i s i n

p ra c t i c e i n f e r i o r t o t h e m o d i f i e d u n i v e r s a l s l o p e s

m e t h o d a s r e p o r t e d b y M a n s o n a n d M u r a l i d h a r a n ~.

S u c h a f a c t c a n b e e a s i l y c l a r i f i e d i n a q u a n t i t a t i v e

m a n n e r b y u s in g t h e a d d i t i o n a l c r it e r ia p r o p o s e d h e r e ,

as Table 2 s h o w s .

C O N C L U S I O N

T h e p r e d i c t i v e a c c u r a c y o f v a r i o u s m e t h o d s d e v e l o p e d

t o e s t i m a t e f a t i g u e p ro p e r t i e s f ro m s i m p l e t e n s i l e d a t a

i s q u a n t i t a t i v e l y e v a l u a t e d b y u s i n g t h e a d d i t i o n a l

c r i t e r i a p ro p o s e d i n t h i s w o rk . w i t h t h e c o n v e n t i o n a l

e r r o r c r i t e r io n , T h e c o n c l u s i o n s o b t a i n e d a r e s u m m a r -

ized as fo l lows :

1 . N e w c r i t e r i a b a s e d o n a l e a s t - s q u a re s a n a l y s i s

b e t w e e n t h e p r e d i c t e d a n d e x p e r i m e n t a l f a t i g u e

l i v e s a r e p r o p o s e d t o e v a l u a t e e s t i m a t i o n m e t h o d s .

2 . T h e t h r e e m e t h o d s - t h e m o d i f i e d u n i v e r s a l s l o p e s ,

S e e g e r ' s a n d O n g ' s m e t h o d s - g i ve r e a s o n a b l y

g o o d l if e p r e d i c t io n s . A m o n g t h e m , t h e m o d i f i e d

u n i v e r s a l s l o p e s m e t h o d p ro v i d e s t h e b e s t r e s u l t s .

7/17/2019 1-s2.0-014211239599737U-main


D e t a i l e d e v a l u a t i o n o f m e t h o d s f o r e s t im a t io n o f f a t i g u e p r o p e r t i e s 3 7 3

3 . C o n s e q u e n t l y , t h e m o d i f i e d u n i v e r s al sl o p e s m e t h o d

c a n b e r e c o m m e n d e d a s t h e b e s t e s t i m a t i o n m e t h o d

a t p r e s e n t . W h e n t h e f r a c tu r e d u c t i l i t y r e q u i r e d b y

t h e m e t h o d i s n o t a v a i l a bl e , S e e g e r ' s m e t h o d m a y

b e u t i l i ze d a s a n a l t e r n a t iv e t o o b t a i n s a t i s f a c t o r y

r e s u l t s .

4 . T h e f r a c t i o n o f d a t a f a l l in g w i t h i n a f a c t o r o f 3 ( o r

2 ) s c a t t er b a n d i s f o u n d t o b e a b o u t 8 0 % ( o r 6 0 % )

f o r a l l t h e a b o v e t h r e e m e t h o d s .

R E F E R E N C E S

1 M an so n , S . S ,

Exp Mech

1965, 5, 193

2 M i t c h e l l , M . R . Fat igu e an d M i c r os t r u c t u r e ', A m e r i c an

S oc i e t y f or M e t a l s , M e t a l s P ac k , OH, 1979 , p . 385

3 B~iume l , A. Jr and Seeg er , T. ~Mater ials Dat a for Cycl ic

L o a d i n g , S u p p l e m e n t 1 '

E l se v i e r S c i e n c e P u b l i sh e r s , A m st e r -

d a m , 1990

4 O n g , J . H .

Int . J . Fatigue

1993, 15, 213

5 M u r a l i d h a r a n , U . a n d M a n s o n , S . S . J .

Eng . Mater . Techno l .

1988, 110, 55

6 O n g , 3 . H .

Int . J . Fatigue

1993, 15, 13

7 ' A S M M e t a l s H a n d b o o k ' , 9 t h e d n , A m e r i c a n S o c i e t y f o r

M e t a l s , M e t a l s Park , O H, 1978 , 680

8 " J SME D at a B ook : Fat i gu e o f M e t a l s ' , I V L o w C yc l e Fat i gu e

S t r e n gt h , Jap an S oc i e t y o f M e c h an i c a l E n g i n e e r s , T o k y o ,

1983

9 BOller, Ch r , J r

an d S e e ge r , T . ' M at e r i a l s D at a f or C yc l i c

L o a d i n g ' , V o l . l - 4 , E l se v i e r S c i e n c e P u b l i sh e r s , A m st e r d am ,

1987

1 0 W e t z e l , R . M . ( e d . ) ' F a t i g u e U n d e r C o m p l e x L o a d i n g :

A n a l y s e s a n d E x p e r i m e n t s ' , S o c i e t y o f A u t o m o t i v e E n g i n -

eer s , I nc . , 1975 p . 41

1-s2.0-014211239599737U-main

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