Multi Axial Fatigue Experiments-I Modelling

8/4/2019 Multi Axial Fatigue Experiments-I Modelling

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E L S E V I E R

Int . J . Fat igue Vol . 19, No. 3, pp. 201-207, 1997 199 7 E l s ev i e r S c i ence L imi ted . A l l r i gh t s re s e rved

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P I I : 0 1 4 2 -1 1 2 3 (9 6 )0 0 0 7 4 -6

M u l t ia x i a l f a t i g u e b e h a v i o u r o f u n i d i re c t io n a lp l i e s b a s e d o n u n i a x i a l f a t i g u ee x p e r i m e n t s m I. M o d e l l i n gM a h m o o d M . S h o k r i e h a n d L a r ry B . L e s s ar dDepartment of Mechanical Engineering McGill University, 817 Sherbrooke StreetWest, Montreal, Quebec, Canada H3A 2K6(Received 15 October 1995; revised 9 June 1996; accepted 23 September 1996)The fatigue behav iour of a unidirectional com posite lamina is examined from theoretical and expe rimentalviewpoints. The goal is to establish a technique to use experimental data from a unidirectional plyunder uniaxial fatigue to simulate the beha viour of that ply in multiaxial fatigue loading. Traditionalpolynomial failure cri teria for fatigue are of l imited use because fatigue strength is a function of numberof cycles and applied fatigue stress ratio. In practice, a problem such as a fatigue-loaded complexcom pos ite structure con taining stress concentrations, is subjected to varying stress ratios at diffe rentpoints, especially if material or geom etric nonlinearities are involved. H owe ver, fatigue testing u nder awide range o f stress ratios is t ime consum ing and expensive. The refore i t is important to establish atechnique to consider fatigue damage due to arbitrary stress ratio without having to perform excessiveamounts of testing. Here the theoretical basis of a new model, called the generalized residual materialproperty degradation model is explained in detail. C opyright 199 7 Elsevie r Science Limited. A llrights reserved( K e y w o r d s : c o m p o s i t e s , f a t i g u e , s t r e n g t h d e g r a d a t i o n , b i a x i a l; f a i lu r e c r i t e r i a , l if e , s t r e s s r a t i o )

I N T R O D U C T I O NA l t h o u g h t h e r e i s a n e x t e n s i v e a m o u n t o f r e s e a r c h o nb i a x i a l / m u l t i a x ia l f a t i g u e o f m e t a l s l , r e s e a r c h i n t h es a m e f i e ld o n c o m p o s i t e m a t e r i a l s 2-8 is l e s s c o m p l e t e .L i t e r a t u r e r e v i e w s o f m u l t i a x i a l a n d b i a x i a l f a t i g u el o a d in g o f c o m p o s i t e m a t e ri a ls h a v e b e e n p r e s e n t e d b yF o u n d 7, a n d C h e n a n d M a t t h e w s 8 , a n d t h e s e p a p e r ss t a te t h a t f u r t h e r r e s e a r c h i s n e e d e d . T h e i d e a o f u s i n gp o l y n o m i a l f a i l u r e c r i t e r i a f o r p r e d i c t i n g f a t i g u e f a i l u r eo f c o m p o s i t e l a m i n a te s h a s b e e n u s e d b y m a n yau th o r s 9 -2 , h o w ev e r , t h e ap p l i ca t i o n o f t h i s i d ea , d u eto ex p er imen ta l d i f f i cu l t i e s , i s l imi t ed t o sp ec i a l cases .A d e e p u n d e r s t an d i n g o f t h e b e h a v i o u r o f a c o m p o s i t el a m i n a u n d e r m 0 ] t i a x i a l f a t i g u e l o a d i n g , w i t h a r b i t r a r ys t r e s s r a t i o s , i s a k e y p o i n t t o s t u d y t h e b e h a v i o u r o fa c o m p l i c a t e d p r o b l e m .

I n t h is r e s e a r c h , a f t e r a r e v i e w o f d i f f e r e n t r e s i d u a ls t r e n g t h m o d e l s p r o p o s e d i n t h e l i t e r a t u r e , a s u i t a b l em o d e l t o s i m u l a t e t h e b e h a v i o u r o f a u n i d i r e c t i o n a ll a m i n a u n d e r a u n i a x i a l s t a t e o f s t r e s s i s s e l e c t e d . T h e na p r o c e d u r e t o f i n d t h e f a t i g u e l i f e o f a u n i d i r e c t i o n a ll a m i n a u n d e r a u n i a x i a l s t a t e o f s tr e s s , w i t h a n a r b i t r a r ys t r e s s r a t i o , i s e x p l a i n e d . F i n a l l y , b a s e d o n t h e p r e v i o u ss e c t i o n s , a m o d e l t o s i m u l a t e t h e b e h a v i o u r o f a u n i -d i r e c t i o n a l p l y u n d e r m u l t i a x i a l f a t i g u e l o a d i n g , w i t ha rb i t r a ry s t a t e o f s t r es s , an d s t r es s r a t i o i s e s t ab l i sh ed .

S T R E N G T H D E G R A D A T I O N U N D E R F A T I G U EL O A D I N GT h e r e a r e t w o m a j o r a p p r o a c h e s t o s i m u l a t e t h er e s i d u a l s t r e n g th o f l a m i n a t e d c o m p o s i t e s u n d e r u n i a x -i a l f a t i g u e l o ad in g 2 1, wh ich a r e ca l l ed t h e s t a t i s ti ca l( p r o b a b il i ty - b a s e d d a m a g e ) a n d m e c h a n i s t ic / p h e n o m -e n o l o g ic a l ( e m p h a s is o n d a m a g e m e c h a n i c s )a p p r o a ch e s . W o r k s o f H a p l in et al . 22"23 and Br ou tm ana n d S a h u 24 a r e t h e t w o e a r l i e s t e x a m p l e s o f s t a ti s ti c a la n d m e c h a n i s t i c a p p r o a c h e s , r e s p e c t i v e l y . I n v e s t i -g a t i o n s b y Ha h n an d Kim 25, Ch o u an d Cr o m an 26,27,W h i tn e y 28, S en d eck y j 29 , Ra d h ak r i sh n a n 3 , an d wo rk s o fY a n g a n d h i s c o - w o r k e r s 3~ -38 a r e s o m e e x a m p l e s i nt h e s t a t i s t i c a l c a t e g o r y . O n t h e o t h e r h a n d , R e i f s n i d e ra n d S t i n c h c o m b 39, R e i f s n i d e r 4 -42 , Ry d e r an d Cro ss -ma n 2~ , D an i e l an d Ch are wic z 4 3.4 4, Ro t em 45, Whit-w o r t h 4 6, S p e a r i n g a n d B e a u m o n t 4v, a n d H a r r i s a n d h i sc o - w o r k e r s 4 8 ' 4 9 p r e s e n t e d d i f f e r e n t m e c h a n i s t i c /p h e n o m e n o l o g i c a l m o d e l s . I n b o t h c a t e g o r i e s , s t a t i s t i c a la n d m e c h a n i s t i c / p h e n o m e n o l o g i c a l , n o c o m p r e h e n s i v es t u d y o f t h e b e h a v i o u r o f u n i d i r e c t i o n a l p l i e s u n d e rm u l t i a x i a l f a t i g u e l o a d i n g h a s b e e n c o n d u c t e d .Residual strength degradation models under uniaxialloading

C o n s i d e r a u n i d i r e c t i o n a l l a m i n a u n d e r a c o n s t a n tu n i a x i a l f a t i g u e l o a d i n g . U n d e r s t a t i c l o a d i n g , o r e q u i v -a l e n t l y a t n = 0 . 2 5 c y c l e s ( q u a r t e r o f a c y c l e ) i n f a t i g u e ,

201


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202 M .M . Shokr ieh a nd L .B . Lessa rd

R ( n ) Ir e s i d u a l s t r e n g t hR ( ) ~ ~ u r vk e ~ c a t a s t ro p hic

~ s ~ f a i l u r e p o in tc u ~ e , , , . . ~ ~ / n

0 . 0 . 2 5 N f nFi g ure 1 S t r e ng t h de gra da t i on o f a un i d i re c t iona l l a mi na unde r aconstant uniaxia l fa t igue loading (s trength vs log cyc les)

the st rength of the unidi rec t ional lamin a i s R(0)( F i g u r e 1 ) . By inc rea s ing t he number o f cyc l e s , unde ra constant appl ied st ress (o ' ) , fa t igue st rength ( R ( n ) )dec rea se s . F ina l l y , a f t e r a ce r t a in number o f cyc l e swhich i s c a l l ed number o f cyc l e s t o f a i l u re (Nf ) , t hemagn i tude o f t he s t reng th dec rea se s t o t he magn i tudeof t he app l i ed s t r e ss . At t h is po in t t he co mp os i t e l aminafa i l s ca tast rophica l ly . In F i g u r e 1 t h e residual s t rengthan d S - N curves a re shown in one g raph . As shown, i ti s c lear tha t for each sta te of s t ress the S - N curvepasse s t h rough the end po in t ( ca t a s t roph ic f a i l u re po in t )o f t he r e s idua l s t r eng th cu rve .The fa t i gue behav iour o f a compos i t e l amina va r i e sunder di f ferent s ta tes of s t ress . For instance , underhigh level s ta te of s t ress , the residual s t rength as afunc t ion o f number o f cyc l e s i s nea r ly cons t an t and i tdec rea se s d ra s t i c a l l y a t t he number o f cyc l e s t o f a i l u re( F i g u r e 2 ) . T h e s u d d e n d e a t h m o d e l 2 6 ' 2 7 has been usedto de sc r ibe t h i s behav iour . Howeve r , a t l ow l eve l s t a t eof s t ress the residual s t rength of the lamina , as afunc t ion o f number o f cyc l e s , degrades g radua l ly( F i g u r e 2 ) . T h e w e a r o u t m o d e l 23 has been used t op re sen t t h i s behav iour . These two mode l s r ep re sen tex t remes i n t he obse rva t i ons o f f a t i gue degrada t ionrepre sen t ing sudden and more g radua l behav iour , t husa re no t impl i ed t o be a comprehens ive l i s t o f f a i l u remodes . In p rac t i ce de s igne r s mus t dea l wi th a widerange o f s t a t e s o f s t r e ss , va ry ing f rom low to h igh ,t he re fo re a mode l t o p re sen t t he behav iour o f compos i t elaminates under a genera l s ta te of s t ress i s essent ia l .

Di f fe ren t mode l s have been p re sen t ed i n t he l i t e ra -R(n) ,~

R(O)%~ nO.

l o w l e v e l s t r e s s\ ut,,

= : : : : S S : : : : : S : : : : " i . .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. .. . ii iv0 . 2 5 N t [ N i l I

Fi gu re 2 S t r e ng th de gra da t ion unde r d i f f e re n t s ta te s o f s t re s s(s trength vs log cyc les)

t u re t o s imula t e t he r e s idua l s t r eng th o f compos i t elaminates under fa t igue loading. In the fol lowing acompar i son be tween d i f f e ren t mode l s i s made . I t shou ldbe men t ioned t ha t t he re a re numerous de t a i l s i n t he semode l s wh ich a re no t d i scussed he re . The re fo re i n t hefo l l owing , t he mode l s and equa t ions used by o the rau thor s a re on ly cons ide red f rom our po in t o f v i ew.Rea l i z ing t ha t va r ious no t a t i ons have been used bydifferent authors, for s impl ic i ty a uni f ied nota t ion hasbeen app l i ed he re i n o rde r t o p re sen t t he mode l s o fother authors in an informat ive manner . I t must bement ioned tha t in th is paper there i s no a t tempt tostudy the probabi l i s t ic fea tures of the residual s t rengthof compos i t e ma te r i a l s ; on ly t he mechan i s t i c cha rac t e r -i s t ics are considered here .T he s u d d e n d e a t h m o d e l i s very simple and st ra ight -fo rward . The s t r eng th o f the com pos i t e l amina i s con-stant unt i l the number of cycles to fa i lure (Nf) , wherethe composi te lamina fa i l s ca tast rophica l ly .In t he w e a r o u t m o d e l , which was i n i t i a l l y p re sen t edb y H a p l i n e t a l . 23, i t i s assumed tha t the residualst rength ( R ( n ) ) i s a mono ton i ca l l y dec rea s ing func t ionof number o f cyc l e s (n ) , and t he change o f t he r e s idua ls t r eng th i s approx ima ted by a power - l aw g rowth equ-ation.

dR(n) _ - A ( o ' ) / m [ R ( n ) ] m -1 ( 1 )d ni n wh ich A(o- ) i s a func t i on o f t he max imum cyc l i cst ress (o ' ) , and m is a constant . This model has beenused by man y authors 25-38 in prob abi l i s t ic an d mech an-ist ic models .By in t eg ra t i ng Equa t ion (1 ) f rom no to n l cyc l e s ,

R " ( n O = R m (n o) - A ( o ) ( n l - n o ) (2 )wi th no = 0 and nl = n , Eq uat ion (2) redu ces to:

R m ( n ) = R m (O ) - a ( o ' ) n (3 )where R(0) i s the sta t ic s t rength.By cons ide r ing t ha t a t t he num ber o f cyc l e s tofa i lure (Nf) , the residual s t rength ( R ( n ) ) i s equal to theappl ied st ress (o-) , Equ at ion (3) reduces to:

Rm(O) - a mR m ( n ) = R ' ( O ) n (4 )NfEquat ion (4) expresses the residual s t rength ( R ( n ) ) , asa func t i on o f s t a ti c s t r eng th (R(0)) , numb er o f cyc l e s(n ) , and number o f cyc l e s t o f a i l u re (N0 . Al so m i s acons t an t wh ich i s found expe r imen ta l l y . For d i f f e ren tsta tes of s t resses, m has di f ferent va lues, therefore toful ly charac ter ize a mater ia l , la rge number of exper i -men t s shou ld be pe r fo rmed . Th i s i s expens ive , t imeconsuming , and imposs ib l e i n p rac t i ce .For compar ing be tween d i f f e ren t mode l s p roposedby di f ferent authors, Equat ion (4) can be rewri t ten inthe fo l l owing fo rm,

R ' n ( n ) - R ' ( O ) n ( 5 )R m ( O ) - 0 a n - - N fand by t he fo l l owing a lgebra i c ope ra t i on ,

R m ( n ) - R m ( O ) - o "m + o a n n (6 )R m ( O ) - a " - - N fEquat ion (5) reduces to:


3/7

M u l t i a x i a l f a t i g u e b e h a v i o u r o f u n i d i r e c t i o n a l p l i e s - - I 203R m ( n ) - a m n- 1 - ( 7 )R m ( O ) - - a m N f

E q u a t i o n ( 7 ) i s a n o r m a l i z e d f o r m o f E q u a t i o n ( 4 )w h i c h c a n b e u s e d f o r th e p u r p o s e o f c o m p a r i n gd i f f e r e n t m o d e l s .B y u s i n g a u n i fi e d n o t a t io n a n d a p p l y i n g s i m i l a ra l g e b r a i c o p e r a t i o n s t o th e o t h e r m o d e l s p r o p o s e d i nt h e l it e ra t u re , a s u m m a r y o f t h e re s i d u a l s t r e n g th m o d -e l s i s p r e s e n t e d i n T a b l e 1 . A s h o w n i n T a b l e 1 , H a h na n d K i m 2 5, Y a n g e t a l . 3 1 - 3 8 , C h o u a n d C r o m a n 2 6'2 7,a n d S e n d e c k y j 29 a p p l i e d e x a c t l y t h e s a m e m o d e l p r o -p o s e d b y H a p l i n e t a l . 23 B r o u t m a n a n d S a h u 24 a s s u m e da 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 f a t i g u e s t r e n g t h a n dn u m b e r o f c y c l e s . T h i s a s s u m p t i o n i s n o t c o n s i s t e n tw i t h t h e e x p e r i m e n t a l e v i d e n c e o f s t re n g t h d e g r a d a t i o na t l o w l e v e l s t a t e o f s t r e s s n o r a t h i g h l e v e l s t a t e o fs t r e s s . I n a m o d e l p r e s e n t e d b y D a n i e l e t al . 43, a nu n d e f in e d f u n c t i o n o f n o r m a l i ze d n u m b e r o f c y c l e s " g "i s i n t r o d u c e d . T h e r e i s n o e f f o r t i n t h e i r p a p e r 43 t od e f i n e t h i s f u n c t i o n ( T a b l e 1 ) . I n t h e a l g e b r a i c e q u a t i o nw e w i l l u s e t o r e p r e s e n t a r e d u c e d f o r m o f t h e in t e g r ale q u a t i o n p r o p o s e d b y R e i f s n i d e r a n d S t i n c h c o m b 39 a n dR e i f s n i d e r 4 ~ 2 , ' K ' i s a c u r v e f i t ti n g p a r a m e t e r w h i c hm u s t b e f o u n d e x p e r i m e n t a l ly . H a r r i s e t a t . 4 8 ' 4 9p r e s e n t e d a n o r m a l i z e d e q u a t i o n c o n s i s t i n g o f t w oc u r v e f i t t i n g p a r a m e t e r s ( T a b l e 1 ) . T h e y s h o w e d t h a t' a ' a n d ' / 3 ' a r e s t r e s s i n d e p e n d e n t c u r v e f i t t i n g p a r a -m e t e r s w h i c h m u s t b e f o u n d e x p e r i m e n t a l l y . T h e ye m p h a s i z e d t h a t s t r e s s - in d e p e n d e n t m o d e l s , l i k e t h em o d e l p r o p o s e d b y F o n g s , w h i c h i s b a s e d o n t h ea s s u m p t i o n t h a t t h e f a t i g u e p r o c e s s i s c o n t r o l l e d b y as i n g l e p r i m a r y d a m a g e m e c h a n i s m i s n o t re a l is t ic . T h e yp o s t u l a t e d t h a t t h e i r m o d e l p e r m i t t e d t h e i n c o r p o r a t i o no f a ll m o d e s o f d a m a g e a c c u m u l a t i o n , f r o m w e a r o u tto s u d d e n d e a t h , b y t h e a d j u s t m e n t o f t h e c u r v e f i t ti n gp a r a m e t e r s a a n d / 3. I n th e i r s t u d i e s , t h e e q u i v a l e n tn u m b e r o f f a t ig u e c y c l e s f o r a s ta t ic l o a d i n g c o n d i t i o n

i s a s s u m e d t o b e 0 . 5 , h o w e v e r , b y c o n s i d e r i n g t h a t as t a t i c l o a d i n g i s a q u a r t e r o f a c y c l e , t h e e q u i v a l e n tn u m b e r o f c y c l e s s h o u ld b e c h a n g e d t o 0 .2 5 . B y u s i n gt h e n o r m a l i z a t io n t e c h n i q u e a l l d i f f e r e n t c u r v e s , f o rd i f f e r e n t s t a t e s o f s t r e s s , i n F i g u r e 2 c o l l a p s e t o as i n g l e c u r v e ( F i g u r e 3 ) .F o r u s e i n t h i s r e s e a r c h , t h e e q u a t i o n p r e s e n t e d b yH a r r i s e t a l . 4 8 ' 4 9 i s c h a n g e d t o t h e f o l l o w i n g f o r m :R ( n , o - ) : [ 1 ( l o g ( n ) - l o g( 0 .2 5 ) /t 3 1 1

- \ log~Nff) - l ~ g ( O ~ 5 - ) / J S ( 8 )(R( 0) - dr) + rB y h a v i n g s t a t ic s t r e n g t h ( R ( 0 ) ) , s t a te o f s t r e s s ( o ') ,n u m b e r o f c y c l e s t o f a i l u r e ( N f ) r e l a t e d t o t h e s t a t e o fs t r e s s , a n d t h e c u r v e f i t t i n g p a r a m e t e r s ( a a n d / 3 ) ,r e s i d u a l s t r e n g t h ( R ( n , o -) ), a s a f u n c t i o n o f n u m b e r o fc y c l e s ( n ) , a n d t h e s t a t e o f s t r e s s ( o - ) , i s f o u n d .S i n c e i n E q u a t i o n ( 8 ), a a n d / 3 a r e s tr e s s i n d e p e n d e n tp a r a m e t e r s , t h i s m o d e l i s c a l l e d t h e n o r m a l i z e d r e s i d u a ls t r e n g t h m o d e l . H o w e v e r n u m b e r o f c y c l e s t o f ai lu r e( N f ) i s a f u n c t i o n o f t h e s t a t e o f s t r e s s ( ~ r ) a n d t h e

1

0 1 l o 2 n - l o g . 2 5l o g N t - l o g . 2 5

F i g u r e 3 N o r m a l i z e d s t r e n g t h d e g r a d a t i o n c u rv e [ E q u a t i o n (8 ) ]

T a b l e 1 A s u m m a r y o f d i f f e r e n t s t r e n g t h d e g r a d a t i o n m o d e l sR e f e r e n c e s M o d e l s E x p l a n a t i o n sH a p l i n e t a l. 23 R " ( n ) - a m - 1 n ' m ' i s a c u r v e f i t t i n gH a h n a n d K i m 25 R m ( 0 ) - o ~ N t p a r a m e t e r , f o u n dY a n g e t a / . 3 L 3 8 e x p e r i m e n t a l l yC h o u a n d C r o m a n 2 6,27S e n d e c k y f 9B r o u t m a n a n d S a h u 2 4 R ( n ) - ~ _ 1 n L i n e a r s t r e n g t h d e g r a d a t i o n

R ( O ) - ~ N rDaniel etal.43'44 e ( n ) - o - ( N f ) f k ( ~ )R ( 0 ) - t r - g i s u n d e f i n e d f u n c t i o nR e i f s n i d e r a n d S t i n c h c o m b 39 R ( n ) - o - [ n ~ ' a c u r v e f i t t i n g p a r a m e t e r , f o u n dR e i f s n i d e r4 ~ 2 R ( 0 ) - t r 1 - N f ) e x p e r i m e n t a l l yH a r r i s e t a / ? 8,49 { R ( n ) - t r l " = ( l o g ( n ) - l o g ( 0 . 5 ) ] 0 ' a ' a n d ' /3 ' a r e t w o c u r v e f i tt i ng

\R(O)-o-] 1 - \ l o g ( N f ) - l o g ( 0 . 5 ) ] p a r a m e t e r s , f o u n d e x p e r i m e n t a l l yN o t e s : H a h n a n d K i m 25 u s e F=R(n),t=nY a n g et al. 23 u s e m=c,s=cr,N = N fC h o u a n d C r o m a n 2 6,2 7 u s e S=tr, N=NrS e n d e c k y j 2 9 u s e s t r a = tr , o ' f = R ( n ) , o - e = R ( 0 ), 1/S=m, C= ( n o t d e r i v e d b y h i m 2 9)

. nB r o u t m a n a n d S a h u 24 u s e t r ~ = R ( n ) , f = ~ , O ~ u S ~ s= ~ t~ = R ( 0 ) ( f o r o n e s t a t e o f s t r e s s )D a n i e l et aL 43,44 u s e f= R ( n ) , s= c r, N = N fR e i f s n i d e r a n d S t i n c h c o m b 39 u s e Sr(n)=R(n), S u = R ( 0 ) , S a = o , N=Nf, i=k ( u s e d i n t h e c r i t i c a l m o d e l )H a r r i s e t aL 48 '49 us e f= R ( n ) , s= o -, N = N e


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6/7

206 M.M. Shokrieh and L.B. Lessard( 0 " 2 2 ) a n d s h e a r ( 0 - 1 2 ) d i r e c t i o n s . T o p r e d i c t t h e m a t r i xf a i lu r e m o d e o f t h e u n i d i r e c ti o n a l p l y u n d e r t w o - d i m e n -s i o n a l p l a n e s t r e s s f a t i g u e l o a d i n g c o n d i t i o n s , H a s h i n -t y p e p o l y n o m i a l f a i l u r e c r i t e r i o n 11 c a n b e u s e d . I ts h o u l d b e n o t e d t h a t t h e d e n o m i n a t o r s o f th e f o l l o w i n ge q u a t i o n a r e n o t c o n s t an t s b u t f u n c t i o n s o f n u m b e r o fc y c l e s , s t a t e o f s t r e s s a n d s t r e s s r a t i o . T h i s t y p e o fe x p r e s s i o n h a s b e e n u s e d p r e v i o u s l y b y S i m s a n dB r o g d o n 9 a n d R o t e m et a L 1 3 - 1 5 a m o n g o t h e r s .

0 - 2 2 / 2 ( 0 - 1 2 tR 2 2 (~ ,K ) J "Jr- \R12(n,0- ,K) ~ = 1 ( 1 5 )w h e r eR z 2 ( n , 0 - , K ) = t r a n s v e r s e t e n s i l e f a t i g u e s t r e n g t hR12(n,0-,K) = i n - p l a n e s h e a r f a t i g u e s t r e n g t hB y u s i n g E q u a t i o n ( 9 ), t h e a s s o c i a te d f a t i g u e s t r e n g t hf o r e a c h s t a t e o f s t r e s s ( 0- 22 a n d 0 - ,2 ) a r e i n t h e f o l l o w -i n g f o r m : 1~22l o g ( n ) - 1 0 g (0 . 25 ) / ] a 2 2

R22(n ,o ',K ) = 1 - \ lo g(N f22) _ 10g (0 .25 )} ](R22(0) - 0-22) + 0-22 (16 )

a n d 1I ( l o g , l o g 0 2 , ) q1 2(n ,0 -,K ---- 1 - - l ~ 2 ~ _ _ ~ )(RI2(0 ) - 0-12) + o '12 (17 )

w h e r e e a c h u n i a x i al d i r e c t io n h a s a d i f f e r e n t s e t o fco ns tan ts c~ an d /3, i . e .a = , /3 22 = e x p e r i m e n t a l p a r a m e t e r s m e a s u r e d f r o mt r a n s v e r s e t e n s i l e f a t i g u e t e s t s ; a n dc q 2 , /3 12 = e x p e r i m e n t a l p a r a m e t e r s m e a s u r e d f r o ms h e a r f a t i g u e t e s t s .B y s u b s t i t u t i n g E q u a t i o n s ( 1 6 ) a n d ( 1 7 ) i n t o E q u -a t i o n ( 1 5 ) , t h e f o l l o w i n g e q u a t i o n i s d e r i v e d .

0-22{ l o g ( n ) - l o g( 0 .2 5 ) ~ t % ] ~ +1 - ] 0 " 2 2

0"12( log(n)- 1og(0.25) ~,21a35.- ] , 2 ( R 1 2 ( 0 ) - O'12) n- O.12)) = 1

(18)B y h a v i n g t h e s t a t e o f s t r e s s , s tr e s s r a t i o s , s t a ti cn u m e r i c a l p r o p e r t ie s , a n d b y u s i n g E q u a t i o n s ( 1 0 ) -( 1 4 ) , 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 f o r e a c h u n i a x i a ls t a t e o f s t r e s s ( N f2 2 a n d N f 1 2 ) a r e e x t r a c t e d f r o m t h ec u r v e s d e r i v e d b y E q u a t i o n s ( 1 3 ) a n d ( 1 4 ) , r e s p e c t i v e l y .T h e n b y s u b s t i t u t i n g a l l t h e p r e v i o u s i n f o r m a t i o n i n t oE q u a t i o n ( 1 8 ) , a n d a p p l y i n g a n o n l i n e a r i t e r a t i v e t e c h -n i q u e ( e. g. N e w t o n - R a p h s o n t e ch n i q ue ) , t h e n u m b e ro f c y c l e s t o f a i l u r e f o r t h e m u l t i a x i a l c a s e ( n = N ~ f) i sf o u n d . C l e a r l y , b y c o u p l i n g t h e s e t w o t e c h n i q u e s ,( n o r m a l i z e d r e s i d u a l s t r e n g t h m o d e l [ E q u a t i o n ( 9 ) ] , a n dn o r m a l i z e d f a t i g u e l i f e [ E q u a t i o n ( 1 0 ) - ( 1 4 ) ] , t h e s e v e r er e s t r i c t io n o f a p p l i c a t i o n o f f a t i g u e f a i l u r e c r i te r i a [ l i k eE q u a t i o n ( 1 5 ) ] t o a s p e c i a l s t r e s s r a t i o i s a v o i d e d .C o n s e q u e n t l y , b y d o i n g a li m i t e d n u m b e r o f s im p l eu n i d i r ec t i o n a l e x p e r i m e n t s , a n a l y s i s o f t h e b e h a v i o u r

o f u n i d ir e c t io n a l c o m p o s i t e l a m i n a t e s u n d e r m u l t ia x i a lf a t i g u e l o a d i n g i s p o s s i b l e .E q u a t i o n ( 1 8 ) i s a n e x a m p l e o f a v e r y s i m p l e c a s eo f b i a x i a l f a t i g u e l o a d i n g i n p l a n e s t r e ss . F o r a m u l t i a x -i a l c a s e s u i t a b l e e q u a t i o n s c a n b e d e r i v e d b y a s i m i l a rp r o c e d u r e . F o r i n s t a n c e i n t h r e e - d i m e n s i o n a l c a s e s t h es u i t a b l e f a i l u r e c r i t e r i o n f o r f i b e r b r e a k a g e c a n h a v et h e f o l l o w i n g f o r m ( a d a p t e d f r o m H a s h i n 1 1 ) :( R 0 - 1 1 ) 2 ( O - 1 2 12( O.13 )2, (n- ,, 0- ,K) / + \R1 2(n ,~,K ) / + \RI3 (n ,0- ,K ) / = 1( 1 9 )

w h e r eR 1 3 ( n , o ' , K ) = o u t o f p l a n e s h e a r f a t i g u e s t r e n g th .B y a p p l y i n g a s i m i l a r p r o c e d u r e t o t h a t w h i c h e n d e dw i t h E q u a t i o n ( 1 8 ) , a r e l e v a n t e q u a t i o n f o r t h i s c a s ec a n b e d e r i v e d . T h e n b y h a v i n g a l l n e c e s s a r y i n f o r -m a t i o n , t h e n u m b e r o f c y c l e s t o f a i lu r e f o r a m u l t i a x ia lc a s e c a n b e f o u n d b y a p p l y i n g a n u m e r i c a l i t e r a t i v et e c h n i q u e a s w a s p r o p o s e d f o r t h e p l a n e s t r e s s c a s e .U n d e r s t a n d i n g t h e b e h a v i o u r o f a u n id i r e c t io n a l p l y ,u n d e r m u l t i a x i a l s t a t e s o f s t r e s s a n d g e n e r a l s t r e s sr a ti o , h e l p s t o s i m u l a t e t h e f a t ig u e b e h a v i o u r o f ac o m p o s i t e l a m i n a t e u n d e r c o m p l i c a t e d s t a te s o f s t re s s .T h e c a p a b i l i ty o f th e generalized residual materialproperty degradation model t o s i m u l a t e t h e r e s i d u a ls t r e n g t h o f a u n i d i r e c t i o n a l p l y u n d e r a n a r b i t ra r yu n i a x i a l s t a t e o f s t r e s s a n d s t r e s s r a t i o c a n b e u s e d i nt h e a n a l y s is o f a c o m p l i c a t e d p r o b l e m . I n a c o m p l e xf a t i g u e - l o a d e d s t r u c t u r e , a l t h o u g h t h e e x t e r n a l f a t i g u el o a d r a t i o (Fmin/Fmax) m a y b e c o n s t a n t , t h e i n t e r n a ls t r e s s e s a n d s t r e s s r a t i o , d u e t o s t r e s s c o n c e n t r a t i o n s ,n o n - l i n e a r i t i e s a n d f a i l u r e , a t d i f f e r e n t p o i n t s a r e n o tc o n s t a n t . I n o t h e r w o r d s , e a c h p o i n t i s u n d e r a r e d i s t r i -b u t i n g s t a t e o f s t r e s s a n d v a r y i n g s t r e s s r a t io . T h e r e f o r ea f a i l u r e c r i t e r i o n , l i k e E q u a t i o n s ( 1 5 ) o r ( 1 9 ) c a n n o tb e u s e d d i r e c t l y t o f i n d t h e f a t i g u e l i f e , a n d a c y c l eb y c y c l e s t u d y o f t h e m a t e r i a l p r o p e r t y d e g r a d a t i o na n d r e d i s t r i b u t i o n o f t h e s t a t e o f s t r e s s e s a n d s t r e s sr a t i o s i s n e e d e d . F o r t h e p u r p o s e o f s i m u l a t i n g t h eb e h a v i o u r o f th e c o m p o s i t e l a m i n a t e u n d e r a c o m p l i -c a t e d f a t i g u e l o a d , t h e i d e a o f progressive damagemodeling m u s t b e m o d i f i e d t o fatigue progressive dam -age modeling, w h i c h h a s b e e n p u b l i s h e d e l s e w h e r e 57.C O N C L U S I O N SD u e t o e x p e n s i v e a n d t i m e c o n s u m i n g e x p e r im e n t sn e e d e d t o c h a r a c t e r iz e t h e m a t e r ia l p r o p e r t i e s o f ac o m p o s i t e l a m i n a i n a w i d e r a n g e o f th e s t a t e o f s t re s sa n d s t r e s s r a t i o s , th e a p p l i c a t i o n o f p o l y n o m i a l f a i l u r ec r i t e r i a i n f a t i g u e a n a l y s i s i s l i m i t e d t o c e r t a i n c a s e s .B y c o u p l i n g t h e n o r m a l i z e d r e s i d u a l s t r en g t h a n d t h en o r m a l i z e d f a t i g u e l i f e m o d e l s , a n e w m o d e l i s e s t a b -l i s h e d w h i c h i s c a l l e d t h e generalized residual materialproperty degradation model. B y u s i n g t h i s m o d e l t h en u m b e r o f e x p e r i m e n t s t o c h a r a c t e ri z e t h e m a t e r i a lp r o p e r t i e s o f a c o m p o s i t e l a m i n a c a n b e m i n i m i z e d ,a n d t h e s e v e r e r e s t r ic t i o n o f a p p l i c a t io n o f p o l y n o m i a lf a i l u r e c r i t e r i a t o c e r t a in s t a t e o f s t r e s s e s a n d s t r e s sr a t i o s i s a v o i d e d . A b i a x i a l l y l o a d e d f a t i g u e t e s t c a nb e m o d e l l e d a n d p r e d i c t e d b a s e d o n d a t a t a k e n f r o mo n l y u n i a x i a l f a t i g u e c h a r a c t e r i z a t i o n t e s t s . T h e e s t a b -l i s h e d m o d e l i s o n e o f th e i m p o r t a n t c o m p o n e n t s o ff a t i g u e p r o g r e s s i v e d a m a g e m o d e l l i n g t h a t i s c a p a b l eo f f a ti g u e a n a l y s i s o f c o m p l i c a t e d p r o b l e m s .


7/7

M ul t i ax i a l f a t i gue behav i our o f un i d i r ec t i ona l p l i es - - I 2 0 7ACKNOWLEDGEMENTThe support of the Structures and Materials Laboratoryof the National Research Council of Canada througha grant (31946-1-0008/01) is gratefully acknowledged.REFERENCES

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28 Whitney, J.M. 'Fatigue of Fibrous Composite Materials', ASTMSTP 723, American Society for Testing and Materials, Philadel-phia, 1981, pp. 133-15129 Sendeckyj, G.P. 'Test Methods and Design Allowable forFibrous Composites', ASTM STP 734 (Ed. C.C. Chamis) Amer-ican Society for Testing and Materials, Philadelphia, 1981,pp. 245-26030 Radhakrishnan, K. J. Composite Mater. 1984, 18, 21-3131 Yang, J.N. and Liu, M. D. J. Composite Mater. 1977, 11,176-202.32 Yang, J.N. 'Composite Materials: Testing and Design (FourthConference)', ASTM STP 617, American Society for Testingand Materials, Philadelphia, 1977, pp. 272-29533 Yang, J.N .J. Composite Mater. 1978, 12, 19-3934 Yang, J.N. and Jones, D. L. J. Composite Mater. 1978, 12,371-28935 Yang, J.N. and Sun, C.T.J . Composite Mater. 1980, 14, 168-17636 Yang, J.N., Miller, R.K. and Sun , C. T. J. Composite Mater.1980, 14, 82-9437 Yang, J.N. and Jones, D.L. 'Fatigue of Fibruous CompositeMaterials', ASTM STP 723 , American Society for Testing andMaterials, Philadelphia, 1981, pp. 2t 3-23238 Yang, J.N. and Du. S. J. Composite Mater. 1983, 17, 511-526.39 Reifsnider, K.L. and Stinchcomb, W.W. 'Composite Materials:Fatigue and Fracture', ASTM STP 907 (Ed. H.T. Hahn) Amer-ican Society for Testing and Materials, Philadelphia, 1986,pp. 298-31340 Reifsnider, K.L. Engng Fracture Mech. 1986, 25, 739-74941 Reifsnider, K.L. 'Fatigue of Composite Materials' (Ed . K.L.Reifsnider), Elsevier, Amsterdam, 1991 pp. 11-7742 Reifsnider, K.L. 'Composite Materials: Testing and Design(Tenth Volume)', ASTM STP 1120 (Ed. G.C. Crimes) AmericanSociety for Testing and Materials, Philadelphia, 1982, pp. 205 -22343 Daniel, I.M. and Charewicz, A. Engng Fracture Mech. 1986,25, 793-80844 Charewicz, A. and Daniel, I.M. 'Composite Materials: Fatigueand Fracture',ASTM STP 907 , (Ed. H. T. Hahn), AmericanSociety for Testing and Materials, Philadelphia, 1986, pp. 274-29745 Rotem, A. Engng Fracture Mech. 1986, 25, 819-82746 Whitworth, H.W .J. Composite Mater. 1987, 21, 2-1347 Spearing, S.M. and Beaumont. P.W.R. Composites Sci. Technol.1982, 44, 299-30748 Adam, T., Dickson, R.F., Femando, G, Harris, B. and Reiter, H.'The Fatigue Behaviour of Kevlar/Carbon Hybrid Composites',IMechE Conference Publications (Institute of MechanicalEngineers), Vol. 2, 1986, pp. 329-33549 Adam, T., Dickson, R.F., Jones, C.J., Reiter, H. and Harris, B.'Proceedings of the Institution of Mechanical Engineers, part c:Mechanical Engineering Science', Vol 200 n C3, 1986,pp. 155-16650 Fong, J.T . 'Damage in Composite Materials', ASTM STP 775(Ed. K.L. Reifsnider), American Society for Testing andMaterials, Philadelphia, 1982, pp. 243-26651 Boller, K.H. Modern Plastics, 1957, June , 163-29352 Adam, T., Fernando, G., Dickson, R.F., Reiter, H. and Ham s,B. Int. J. Fatigue, 1989, 11, 233-23753 Harris, B., Reiter, H., Adam, R., Dickson, R.F. and Fernando,

G. Composites 1990, 21, 232-24254 Gathercole, N., Adam, T., Ham s, B. and Reiter, H. 'Develop-ments in the Science and Technology of Composite Materials,ECCM 5th. European Conference on Composite Materials', Bor-deaux, France, 9-12 June, 1992, pp. 89-9455 Adam, T., Gathercole, N., Reiter, H. and Hams, B. Adv. Com-posites Lett. 1992, 1, 23-2656 Gathercole, N., Reiter, H., Adam, T. and Harris, B. Int. J.Fatigue 1994, 16, 523-53257 Shokrieh, M.M. and Lessard, L.B. "CANCOM Second Inter-national Composites Conference and Exhibition', Ottawa, Onta-rio, Canada, 1993, pp. 871-87758 Shokrieh, M.M. and Lessard, L.B. Int. J. Fatigue 1997, 19,209-217

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