Life Prediction in Composites

8/11/2019 Life Prediction in Composites

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ELSEVIER

Int. J. Pres. Ves. Piping 59(1994) 131-140 1994 Elsevier S c i e n c e i m i t e d

Printed in Northern Ireland. All rights reserved0308-0161/94/ 07.00

at igue l i fe predic t ion in composi tes

R J H u s t o n

Divis ion o f Aeronau t i ca l Sys tems Techno logy, CSIR, P re to r ia , Repub l i c o f Sou th Af r i ca

Because of the relatively large numb er of possible failure mechanisms in fibrereinforced composite materials, the prediction of fatigue life in a c ompone nt isnot a simple process. Several mathematical and statistical models have beenproposed, but the experimental evidence to support them is limited so far. Inthis paper, an attempt has been made to fit experimental results to two of themore promising models. Results obtained from repeated tension fatigue testson unidir ectional carbon fibre reinforced epoxy were used to test residualstrength and residual stiffness models. Further fatigue tests were carried outunder spectrum loading so that the results could be correlated with thecumulative damage predicted by the residual strength model.

I N T R O D U C T I O N

D u r i n g t h e l a s t t w o d e c a d e s o r s o , t h e u s e o fc o m p o s i t e m a t e r i a l s i n t h e p l a c e o f m e t a l s i nm a n y i n d u s t r i e s h a s b e c o m e w i d e s p r e a d , n o tl e a s t i n t h e a i r c r a f t in d u s t r y. I n g e n e r a l , ac o m p o s i t e m a t e r i a l c a n b e d e f i n e d a s a m a t e r i a lt h a t c o n si st s o f tw o o r m o r e d i s ti n c t c o m p o n e n t s .T h e c o m p o s i t e m a t e r i a l s u s e d t o m a n u f a c t u r ea i r c r a f t s t r u c t u r e s h a v e u s u a l l y c o m p r i s e d c o n -t i n u o u s f i b r e s e m b e d d e d i n a p o l y m e r i c m a t r i x .T h e f i b r e s p r o v i d e t h e s t r e n g t h a n d t h e s t i f f n e s sa n d t h e m a t r i x h o l d s t h e f i b r e s t o g e t h e r.C o n s i d e r a b l e w e i g h t s a v i n g s c a n b e a c h i e v e d b yu s i n g c o m p o s i t e m a t e r i a l s i n a i r c r a f t b e c a u s e o ft h e i r h i g h e r s p e c i f i c s t r e n g t h s a n d s p e c i f i cs t i f f n e s s e s c o m p a r e d w i t h a l u m i n i u m a l l o y s .

S i n ce t h e d a y s o f t h e C o m e t , f a t ig u e h a s b e e na s u b j e c t o f c o n s i d e r a b l e i n t e r e s t t o t h e a i r c r a f ti n d u s t r y. T h e i n c r e a s e d u s e o f c o m p o s i t em a t e r i a l s h a s e m p h a s i s e d t h e f a c t t h a t t h e i rf a t i g u e b e h a v i o u r i s m o r e c o m p l e x t h a n t h ef a t i g u e b e h a v i o u r o f m e t a l s . I n a m e t a l , f a t i g u ed a m a g e e v e n t u a l l y a p p e a r s a s a c l e a rl y d e f in e dc r a c k , t h e i n it ia t io n a n d p r o p a g a t i o n o f w h i c hc a n b e p r e d i c t e d b y f r a c t u r e m e c h a n i c s a n a l y s i s .I n a c o m p o s i t e m a t e r i a l , f a ti g u e d a m a g e c a n t a k et h e f o r m o f a n y o r a l l o f t h e f o l l o w i n g :d e l a m i n a t i o n , m a t r i x c r a c k i n g , f i b r e f a i l u r e ,m a t r i x c r a z i n g , f i b r e / m a t r i x d e b o n d i n g a n d v o i dg r o w t h . I t i s d e p e n d e n t o n v a r i a b l e s a s s o c i a t e d

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

T h e S - N c u r v e a p p e a r s s t i l l t o b e t h e m o s tp o p u l a r m e t h o d o f c h a r a c t e r i s i n g t h e f a t i g u eb e h a v i o u r o f c o m p o s i t e m a t e r i a ls . S e v e r a l e m -p i r i c a l e q u a t i o n s e x i s t f o r d e s c r i b i n g S - N c u r v e s .M o s t a r e b a s e d o n t h e c l a s s i c a l p o w e r l a w t h a tg ives a s t ra igh t l ine in a log- log p lo t o f the fa t igued a t a , b u t s o m e p o s s e ss r e f i n e m e n t s t o a c c o u n t f o rm e a n s t r e s s o r t h e r a t i o o f m a x i m u m s t r e s s t om i n i m u m s t re s s. O n e o f t h e s i m p l e r e q u a t i o n su s e d t o d e s c r i b e t h e f a t i g u e b e h a v i o u r o f s e v e r a lc o m p o s i t e m a t e r i a l s i s a s f o ll o w s : ~

Oa = O, -- b log N 1)

131

w h e r e o a is t h e m a x i m u m a p p l i e d s t r e s s , o u i s t h es ta t i c s t reng th , b i s a cons tan t and N i s then u m b e r o f c y c l e s t o f a i l u r e .

O t h e r t h e o r i e s h a v e b e e n f o r m u l a t e d f o r t h ec h a r a c t e r i s a t i o n o f t h e f a t i g u e b e h a v i o u r o fc o m p o s i t e m a t e r i a l s. T h e r e a r e e s s e n t ia l ly t h r e et y p e s : t h e o r i e s b a s e d o n t h e d e g r a d a t i o n o fr e s i d u a l s t r e n g t h , t h e o r i e s b a s e d o n c h a n g e s i nm o d u l u s a n d t h e o r i e s b a s e d o n a c t u a l d a m a g em e c h a n i s m s .

M o s t o f t h e l i f e p r e d i c t i o n m e t h o d s f o rp o l y m e r i c c o m p o s i t e m a t e r i a ls c u r r e n t l y i nf a v o u r a r e b a s e d o n r e s i d u a l st r e n g t h d e g r a d a t i o ne . g . R e f s 2 - 5 ) .

T h e s e h a v e b e e n d e v e l o p e d o n t h e b a s i s o f


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132 R J H u s t o n

t h r e e a s s u m p t i o n s :

1 ) the s t a t i s t i ca l va r i ab i l i ty o f th e s t a t i cs t r e n g t h o f t h e m a t e r i a l c an b e d e s c r i b e db y a t w o - p a r a m e t e r We i b u l l d i s t r i b u t i o n ;

2 ) t h e r e s i d u a l s t r e n g t h a f t e r N c y c l e s o ff a t i g u e l o a d i n g c a n b e r e l a t e d t o t h e s t a t i cs t r e n g t h b y a d e t e r m i n i s t i c e q u a t i o n ; a n d

3 ) w h e n t h e r es i d u a l s t re n g t h d e c r e a s e s t o th em a x i m u m a p p l i e d s t r e s s , f a t i g u e f a i l u r eO c c u r s

R e d u c t i o n i n s ti ff n es s c a u s e d b y f a t i g u e l o a d i n gh a s b e e n r e p o r t e d b y v a r i o u s o b s e r v e r s , 6-11 a n d i th a s b e e n r e c e n t l y r e v i e w e d b y R e i f s n i d e r. 12T h e o r i e s f o r f a t i g u e b a s e d o n t h e r e d u c t i o n i ns t i f f n e s s h a v e o n e s i g n i f i c a n t a d v a n t a g e o v e r t h er e s i d u a l s t r e n g t h t h e o r i e s , n a m e l y t h a t t h er e m a i n i n g l i fe c a n b e a s s e s s e d b y n o n - d e s t r u c t i v e

m e a n s .A s e c o n d a d v a n t a g e is th a t l e s s t e s t i n g n e e d s t o

b e c o n d u c t e d . A t y p i c a l r e d u c t i o n i n s t i f f n e s sm o d e l ~1 r e l at e s t h e d e g r a d a t i o n o f m o d u l u s t o t h ef r a c t i o n o f l i f e e x p e n d e d a t a g i v e n s t r e s sa m p l i t u d e , a s s u m i n g t h a t t h e r e s i d u a l s ti f fn e s sd e c re a s e s m o n o t o n i c a l l y a s th e n u m b e r o f l o a dc y c l e s i n c r e a s e s .

F a t i g u e t h e o r i e s b a s e d o n d a m a g e m e c h a n i s m sm o d e l t h e i n t r i n s i c d e f e c t s i n t h e m a t r i x a s s m a l lc r a c k s p a r a l l e l t o t h e f i b r e s , a n d t h e p r o p a g a t i o no f t h e s e c ra c k s a r e p r e d i c t e d b y l i n e a r f r a c t u r e

m e c h a n i c s a n a l y s i s . U n f o r t u n a t e l y, p r e d i c t i o n o ff a t i g u e f a i l u r e i s d i f f i c u l t b e c a u s e s o p h i s t i c a t e dm a t h e m a t i c a l t e c h n i q u e s a r e n e e d e d t o c a l c u l a t et h e f o r c e s r e q u i r e d t o p r o p a g a t e t h e c r a c k s a n dt h e d a m a g e a c c u m u l a t i o n p r o c e s s is s im u l a t e du s i n g M o n t e C a r l o m e t h o d s . t 3

I n t h i s p a p e r a n a t t e m p t i s m a d e t o c o r r e l a t et h e f a t ig u e t h e o r i e s o f S e n d e c k y j 5 a n dW h i t w o r t h ~ w i t h t h e e x p e r i m e n t a l r e s u l t s o b -t a i n e d f r o m u n i d i r e c t i o n a l c a r b o n f i b r e r e i n -f o r c e d e p o x y.

E X P E R IM E N T L P R O C E D U R E

Compos i te mate r ia l t es ted and i t s p rocess ingcondi t ions

T h e c o m p o s i t e m a t e r i a l s y s t em u s e d t o t e st t h er e s i d u a l s t r e n g t h a n d r e s i d u a l s t i f f n e s s m o d e l sw a s a u n i d i r e c t i o n a l c a r b o n f ib r e r e i n f o r c e de p o x y p r e p r e g s y s t e m , Vi c o t e x 9 1 3 C - T S , m a n u -f a c t u r e d b y B r o c h i e r S A , a n a f f i l i a t e o f

C i b a - G e i g y. T h e r e s i n m a t r i x w a s a m o d i f i e de p o x y w i th g o o d r e si s ta n c e t o t e m p e r a t u r e a n dh u m i d i t y, a n d t h e c a r b o n f i b r e s w e r e o f t h eTo r a y c a T 3 0 0 B t y p e .

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

t e s t s p e c i m e n s c o u l d b e p r e p a r e d . T h i s s c h e d u l ec o n s is t ed o f h e a t in g t h e u n c u r e d l a m i n a t e u n d e rv a c u u m t o 9 0 C a t a r a t e o f a b o u t 3 C p e r m i n u t ea n d t h e n h o l d i n g th e t e m p e r a t u r e a t 9 0 C fo r3 0 m i n . A p r e s s u r e o f 7 b a r w a s g r a d u a l l y a p p l i e da n d t h e v a c u u m v e n t e d w h e n t h e p r e s s u r er e a c h e d 1 .4 b a r. T h e t e m p e r a t u r e w a s i n c r e a se dt o 1 2 0 C , a g ai n a t a b o u t 3 C p e r m i n u t e , a n dh e l d t h e r e f o r 6 0 m i n . T h e l a m i n a t e w a s t h e nc o o l e d t o 9 0 C w h e n t h e p r e s s u r e w a s r e l e a s e d ,a n d i t w a s r e m o v e d f r o m t h e a u t o c la v e w h e n t h et e m p e r a t u r e h a d d r o p p e d t o b e l o w 6 0 C .

P r e p a r at io n o f s p e c i m e n s

T h e s a m e t y p e o f s p e c i m e n w a s u s e d i n t h et e n s i l e a n d f a t i g u e t e s t s . I t w a s a f l a t ,p a r a l l e l - s i d e d s p e c i m e n a p p r o x i m a t e l y 2 2 0 m ml o n g , 2 0 m m w i d e a n d 1 m m t h ic k . A b a n d s a ww a s u s e d t o c u t t h e s p e c i m e n s o u t o f t h e c u r e dl a m i n a t e s , a n d t h e y w e r e g r o u n d p a r a l l e l w i t h t h ea i d o f a s u r f a c e g r i n d e r. To f a c i l i t a t e l o a dt r a n s f e r in t h e g r i p s o f th e t e s t m a c h i n e ,a l u m i n i u m t a b s a b o u t 2 m m t h i ck w e r e b o n d e d

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

Te s t p r o g r a m m e

I n t h e f ir st s ta g e o f t h e t e s t p r o g r a m m e , t e n s i l et e s t s w e r e c o n d u c t e d o n 3 0 s p e c i m e n s t o p l a c e a nu p p e r l i m i t o n t h e s t r e s s l ev e l s t o b e a p p l i e d i nt h e s u b s e q u e n t f a t i g u e t e s t s a n d t o a s s e s s t h ev a r ia b i li ty i n t h e p r o p e r t i e s o f t h e m a t e r i a l . T h et e n s i l e t e s t s w e r e c o n d u c t e d i n a c c o r d a n c e w i t ht h e r ec o m m e n d a t i o n s o f A S T M D 3 0 3 9, a n d ine a c h t e s t t h e t e n s i l e s t r e n g t h a n d t h e t e n s i l em o d u l u s w e r e m e a s u r e d .

F o l l o w i n g t h e t e n s i le t e s t s , f a t i g u e t e s t s w e r ec a r r ie d o u t t o d e r i v e S - N c u r v e s fo r t h e m a t e r i a l .T h e s e t e s t s w e r e c a r r i e d o u t u n d e r l o a d c o n t r o li n r e p e a t e d t e n s i o n w i t h t h e s t re s s r a t i o R ) e q u a lt o 0 . 1. T h e f r e q u e n c y w a s 1 0 H z . F i v e r e p l i c a t es p e c i m e n s w e r e t e s t e d a t e a c h o f s e v e n s t re s sl e v el s t o g e n e r a t e t h e S - N c u r v e s . F i v e t en s i l es t r e n g t h r e s u l t s , s e l e c t e d r a n d o m l y f r o m t h et e n s i l e t e s t d a t a , w e r e a l s o i n c l u d e d . T h e f a i l u r ec r i t e r i o n w a s c o n s i d e r e d t o b e s p e c i m e n f r a c t u r e .


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Fa t ig u e l if e p r ed i c t i on i n compo s i t e s 133

Fatigue tests were continued until approximately2 million cycles, at which point the specimenswere considered to be run-outs. Tensile testswere conducted on each run-out to measureresidual strength.

At two stress levels, 1700 and 1800MPa,

further fatigue tests were carried out to assess thevariability of fatigue life. Twenty-five specimenswere tested at both stress levels to identify aprobable distribution.

In the next stage of the programme, a series ofload spectrum tests was performed to comparecumulative damage predictions with experiment alresults. A load spe ctrum base d on F AL ST AF F lawas devised and 10 specimens tested. The testswere continued either to failure or were stoppedafter 5 million cycles, and then residual tensilestrength was measured. As before, the tests were

carried out un der load control at a freq uency of10 Hz and with a stress ratio R) o f 0.1.

Finally, to assess the residual modulus model,a few more fatigue tests were performed. Atvarious stages in each of these tests, the statictensile modulus was measured. To do this, thefatigue tests were interru pted periodically so thatan extensomet er could be attached to thespecimen and the measurements made.

T e s t e q u i p m e n t

All the fatigue tests were conducted using aSchenck Hydropuls machine of 250 kN capacity.Tensile modulus measurements were made usingan HBM DD1 strain transducer connected to anHBM DWS73 amplifier and digital monitor.

T E N S I L E T E S T R E S U L T S

The results of the preliminary tensile tests aresummarised in Table 1. The tensile strength data

The most popular method of characterising thefatigue behaviour of composite materials is stillthe generation of S-N curves. To attach

statistical value to these curves, it has usuallybeen necessary to test large numbers of replicatespecimens at different stress levels to obtain thedistribution of fatigue lives. Examples of this areshown in Figs 3 and 4, where Weibulldistributions have been fitted to fatigue livesobtained at maximum stresses of 1700 and1800 MPa.

The probabilistic life prediction model ofSendeckyj 5 does not requir e a large data base.Static, fatigue and residual strengths are allconverted to equivalent static strengths and

pooled to produce a data base large enough forstatistical analysis. A two-parameter Weibulldistribution is fitted to the residual strength data.By assuming that the shape tr) and scale fl)parameters of the fatigue life and static strengthdistributions are functionally related to eachother and to the constants C and S in the classicalfatigue power law equatio n

o u tr a = C N s 2)

Tensile Tensilestrength, S~1 modulu s, EI~

MPa) GPa)

Mean 2 180 135Standard deviation 197 5.89Coefficient of variation 0.090 0-004No. of specimens 30 30Maximum value 2 526 148Minimum value 1 733 122Distribution Weibull - -B-Basis value 1 800 --A-Basis value 1 700 --

were analysed in the manner recommended inthe Military Handbook 17B, and a two-parameter Weibull distribution was found to bethe most appropriate, in prefere nce to thenormal, log-normal, and non-parametric dis-tributions. As can be seen from the Weibull plot

in Fig. 1, the linearity of the results isquestionable, as borne out by the correlationcoefficient of -0 .92 . The distribution of thetensile moduli is shown in Fig. 2. The data fit theNormal distribution well.

GENERATION OF FATIGUE CURVES

Ta b l e 1 . Te n s i l e s t r e n g t h a n d t e n s i le m o d u l u s r e s u l t s fo r u n i d i r e c t io n a lVi c o t e x 9 1 3 C T S c a r b o n f ib r e r e in f o r c e d e p o x y


4/10

O

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J~o

0L

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oZ

0.8

0 . 6

0.4

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-0 .4

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I I I I I I I I I I

3.20 3 .24 3 .28 3 .32 3 .36

Log Tensile strength, MPa)

Fig. 1 Weibu ll distribu tion of tensile strengths.

[]I

3 . 4 0

t h e f o l l o w i n g e x p r e s s i o n f o r f a t i g u e l if e c a n b eo b t a i n e d :

e re = [ a ; s + C ( N - 1 ) o J s ] s 3)

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

d a t a i s a n i t e r a t i v e p r o c e s s . F r o m t h e l o g o v e r s u slog N p lo t s ho wn in F ig . 5 , in i t i a l va lues o f Ca n d S a r e e s t i m a t e d . T h e f a t i g u e d a t a a r ec o n v e r t e d t o e q u i v a l e n t s t a t i c d a t a b y u s i n g t h e s ev a l ue s i n e q n 3 ) . A t w o - p a r a m e t e r We i bu Udis t r ibu t ion is then f i t t ed to th e cr~ da ta ,p r e f e r a b ly b y u s i n g t h e m a x i m u m l i k e li h o o d

m e t h o d , a n d t h e v a l u e s o f t h e s h a p e a n d s c a l ep a r a m e t e r s a r e s t o r e d . N e w v a l u e s f o r C a n d Sa r e c h o s e n a n d t h e p r o c e d u r e i s r e p e a t e d u n t i lt h e m a x i m u m v a l u e o f t h e s h a p e p a r a m e t e r i so b t a i n e d . T h i s v a l u e , a n d t h e c o r r e s p o n d i n g s c a le ,C a n d S v a l u e s , p r o v i d e t h e b e s t f i t o f t h e f a t i g u ed a t a . F o r t h e f a t i g u e t e s t s r e p o r t e d h e r e , t h ev a l u e s o f C a n d S w e r e f o u n d t o b e 0 - 0 0 0 7 5 a n d0 . 05 7 6 , r e s p e c t i v e l y. T h e s e g a v e r i s e t o a s h a p ep a r a m e t e r o f 1 3 . 2 4 a n d a s c a l e p a r a m e t e r o f 2 2 3 5f o r t h e d i s t r i b u t i o n o f t h e e q u i v a l e n t s t a t i cs t reng ths . This d i s t r ibu t ion i s shown in F ig . 6 .F r o m t h i s d i s t r i b u t io n a n d e q n 3 ) , S - N c u r v e s

3 . 0

2.0

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-1 .0 []

-2 .0

-3 .0

134 R . J. H u s t o n

I1 2 0

I I I I I t I I I I I1 2 4 1 2 8 1 3 2 1 3 6 1 4 0 1 4 4

T e n s i l e m o d u l u s , E l 1 + ( G P a )

Fi g 2. Normal distribution of tensile moduli.

I J [4 8


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a t igue i f e predic t ion in composi tes 135

0.6

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0 4

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- 0 . 8

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L o g C y c l e s t o f ai l u re )

Fig 3 W e i b u i l d i s t r i b u t i o n o f fa t i g u e l i v e s a t m a x i m u m s t r e s s o f1700 MP a

c a n b e c o n s t r u c t e d f o r v a r i o u s p r o b a b i l i t i e s o fs u r v iv a l . E x a m p l e s o f t h e s e a r e s h o w n i n F i g . 7 .

T h e S e n d e c k y j m o d e l i s a c o n v e n i e n t m e t h o do f d e s c r ib i n g f a t i g u e b e h a v i o u r b u t is d e p e n d e n to n t h e v a li d i ty o f t h e th r e e a s s u m p t i o n s u p o nwh ich i t i s based : 5

1 ) t h e S - N b e h a v i o u r c a n b e d e s c r i b e d b y ad e t e r m i n i s t i c e q u a t i o n ;

2 ) t h e s t a ti c s tr e n g t h s a r e u n i q u e l y r e l a t e d t o

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

3 ) t h e s ta t ic s t r e n g th d a t a c a n b e d e s c r i b e d b ya t w o - p a r a m e t e r We i b u l l d i s t r i b u t i o n .

F o r t h e f a t i g u e t e s t s p r e s e n t e d h e r e , t h e f i r s ta s s u m p t i o n i s p r o b a b l y v a l i d , s i n c e a n e m p i r i c a le q u a t i o n c a n b e f i t t e d t o t h e d a t a . T h e s e c o n da s s u m p t i o n c o u l d b e i n v a l i d a t e d i f d if f e r e n tf a i l u r e m o d e s w e r e p o s s i b l e , b u t t h i s i s u n l i k e l yin th i s inves t iga t ion as the s t r e s s d i s t r ibu t ionr e s u l t i n g f r o m t e n s i l e l o a d s a p p l i e d i n t h e

l o n g i t u d i n a l d i r e c t i o n t o a u n i d i r e c t i o n a l l yr e i n f o r c e d s p e c i m e n i s n o t c o m p l e x . T h e l i n e a r i t y

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Pig 4 W e i b u l l d i s t r i b u t i o n o f fa t i g u e l i v e s a t m a x i m u m s t r e s s o f1800 MP a


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o f t h e d a t a i n F i g . 6 p r o v i d e s a n i n d i c a t i o n o f t h ev a l i d i t y o f t h e t h i r d a s s u m p t i o n . T h e c o r r e l a t i o nc o e f f i c ie n t i s - 0 . 9 6 . T h i s s u g g e s t s th a t t h eWeibu l l d i s t r ibu t ion f i t s the equ iva len t s t a t i cs t r eng th da ta qu i t e we l l .

F AT I G U E L IF E P R E D I C T I O N U N D E RS P E C T R U M L O A D I N G

I f o n e r e a r r a n g e s e q n ( 3 ) t o

o r = [ o ~ s - C ( N - 1 ) o ~ , s ] s (4)

a n d i f o n e m a k e s t h e a s s u m p t i o n t h a t t h ise q u a t i o n c a n g e n e r a t e a v a l u e f o r r e s i d u a ls t r e n g th a f t e r e v e r y c y c l e , S e n d e c k y j s m o d e l c a nb e a p p l i e d t o s p e c t r u m l o a d i n g .

T h e f a t i g u e l o a d s p e c t r u m u s e d i n t h i si n v e s t i g a t i o n w a s b a s e d o n FA L S TA F F, b u t i tw a s s i m p l i f i e d a n d a f a c t o r w a s u s e d o n t h e l o a d st o e n s u r e t h a t s o m e s p e c i m e n s f a i l e d a f t e r n o tt o o m a n y c y c l e s . T h e s p e c t r u m i s g i v e n i n Ta b l e2

A n e x a m p l e o f h o w t h e r e s i d u a l s t r e n g t hm o d e l c a n b e a p p l i e d t o p r e d i c t t h e f a t i g u e l i f e o fa s p e c i m e n u n d e r s p e c t r u m l o a d i n g i s s h o w n i n

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atiguelife predi tion in omp osites 137

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Fig. 7. S- N curves for 0.9, 0.5 and 0.1 probab ilities of survival.

Ta b l e 2 F a t i g u e s p e c t r u m

Maximum stress MPa) Numb er of cycles

1 480 8 1001 200 50 000

990 225 0001 310 23 0001 800 5001 630 1 9001 100 150 000

850 1 000 000

Ta b l e 3 . F o r a p r o b a b i l i t y o f su r v i v a l o f 0 - 9 t h ee q u i v a l e n t te n s il e s tr e n g t h i s 1 9 4 5 M P a . T h er e s id u a l s t r e n g t h d e c r e a s e s p r o g r e s s iv e l y t h r o u g he a c h s ta g e o f e a c h b l o c k . T h e v a l u e a f t e r e a c hs t a g e i s g i v e n i n Ta b l e 3 . W h e n t h e r e s i d u a ls t r e n g t h h a s d e c r e a s e d t o a v a l u e b e l o w t h ea p p l i e d s tr e s s f a i l u r e o c c u r s . T h i s is p r e d i c t e d t o

h a p p e n a f t e r 4 5 7 6 6 0 0 c y c l e s w h e n t h e m a x i -m u m a p p l i e d s t r e s s i s 1 8 0 0 M P a i n t h e f o u r t hb lock .

S o m e e x p e r i m e n t a l r e s u l t s a r e p r e s e n t e d i nTa b l e 4 f o r c o m p a r i s o n w i t h t h e r e s i d u a l s t r e n g t hm o d e l p r e d i c ti o n . A s c a n b e s e e n t w o s p e c i m e n sh a d f a t i g u e l iv e s s h o r t e r t h a n t h a t p r e d i c t e d o n ec o n s i d e r a b l y s o w h i l e e ig h t s p e c i m e n s s u r v iv e d 5m i l l i o n c y c l e s . S p e c i m e n 4 f a i l e d d u r i n g t h e1 80 0 M P a s t a g e o f th e t h i r d b lo c k a n d S p e c i m e n7 f a i l e d d u r i n g t h e 1 63 0 M P a s t a g e o f t h e f ir s tb l o c k . T h e r e s i d u a l t e n s i l e s t r e n g t h s w e r em e a s u r e d f o r t h e o t h e r s p e c i m e n s a n d q u i t e aw i d e v a r i a ti o n w a s o b s e rv e d . N a t u r a l l y a ll w e r eh i g h e r t h a n t h e h i g h e s t m a x i m u m a p p l i e d s t r e s so f t h e f a t i g u e s p e c t r u m .

T h e p r e d i c t e d f a t i g u e l i f e w a s f o r a p r o b a b i l i t yo f s u rv i va l o f 0 . 9 s o o n e w o u l d e x p e c t o n e

Tab le 3 Fa t igue l i fe p red ic t ion : r e s idua l s t r eng th m od e l

Maximum stress, Oa

MPa)

Number of cycles, N Residual strength after N cycles, or

MPa)1st block 2nd block 3rd block 4th block

1 480 8 1001 200 50 000

990 225 0001 310 23 0001 800 5001 630 1 9001 100 115 000

850 1 000 000

.1 939 1 905 1 857 1 7721 938 1 904 1 855 1 7681 938 1 903 1 855 1 7691 936 1 901 1 850 1 7571 923 1 883 1 822 Fai lur e1 914 1 870 1 798 - -1 913 1 869 1 797 - -1 913 1 869 1 797 - -

Equiva lent tensile stre ngth = 1 945 MPa.Num ber of cycles to failure = 4 576 600.


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138 R . J . H u s t o n

Table 4 . Fa t igue spec trum tes t r e su lt s

Spec imen Fat igue l i fe Residu al s t rengthcyc les ) MPa)

1 -- 2 3222 - - 20983 - - 2 094

4 3 153 350 - -5 - - 2 4506 - - 19087 308 390 - -8 - - 18349 - - 2166

10 - - 1 9 7 1

s p e c i m e n i n t e n t o h a v e a s h o r t e r t h a n p r e d i c t e dl i f e . W h e n t h e t e s t s w e r e c a r r i e d o u t , t w os p e c i m e n s o u t o f t h e t e n h a d l iv e s s h o r t e r t h a np r e d i c t e d . C o n s i d e r i n g t h e sm a l l s a m p l e , t h e

c o r r e l a t i o n i s a d e q u a t e .

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

T h e r e s i d u a l s ti f fn e s s m o d e l o f W h i t w o r t h i s ap h e n o m e n o l o g i c a l m o d e l a n d i s l i m i t e d t os p e c i m e n s s u b j e c t e d t o c o n s t a n t a m p l i t u d ef a t i g u e l o a d i n g . T h e m o d e l r e q u i r e s t h e c a l c u l a -t io n o f t h r e e p a r a m e t e r s f r o m e x p e r i m e n t a l d a t a ,a n d i t i s a s s u m e d t h a t r e s i d u a l s t i f f n e s s d e c r e a s e sm o n o t o n i c a l l y a s t h e n u m b e r o f c y c le s in c r e a se s .

T h e b a s is o f t h e m o d e l is a n e q u a t i o n t h a ta s s u m e s t h a t t h e r a t e o f t h e r e s i d u a l m o d u l u sr e d u c t i o n i s i n v e r s e l y p r o p o r t i o n a l t o a p o w e r o ft h e r e s i d u a l m o d u l u s i t s e l f :

d E ( N ' ) - D f f ( E ( 0 ) , o ,)- - - 5 )

d N x E X - I ( N )

w h e r e E ( N ) i s t h e r e s i d u a l m o d u l u s a f t e r Nf a t i g u e ~ y c l e s, N ' i s t h e r a t i o o f t h e a p p l i e d c y c l e st o t h e f a t i g u e l i fe , E ( 0 ) i s t h e i n i t i a l m o d u l u s , xa n d D a r e c o n s t a n t s a n d f ( E ( 0 ) , O a) i s a f u n c t i o no f t h e i n i ti a l m o d u l u s a n d t h e s t r e s s r a n g e , O a.

I n t e g r a t i o n o f e q n ( 5 ) f r o m 0 t o N ' :

E ~ ( N ) = E ~ (O ) - D U ( E ( O ) , o , ) N (6 )

a n d t h i s r e p r e s e n t s t h e r e s i d u a l m o d u l u s a f t e r ac e r t a i n p o r t i o n o f t h e f a t i g u e l i fe h a s b e e nc o m p l e t e d .

I f it is a s s u m e d t h a t f i E ( 0 ) , o a ) is a l i n e a rf u n c t i o n o f t h e s t if f n es s c h a n g e o v e r t h e c y c li c l if e

f ( E ( 0 ) , o ~ ) = k A E ( N ) (7 )

a n d i f t h e s t r e s s - s t r a i n r e s p o n s e is l i n e a r t o

f a i l u re , t h e n a t f a i l u r e

E ( N ) = E f a n d a a ~ E f e f (8)

w h e r e E f is t h e r e d u c e d s t if f ne s s a t f a i lu r e a n d e fi s a c o n s t a n t r e l a t e d t o t h e s t r a i n a t f a i l u r e .E q u a t i o n ( 6 ) b e c o m e s

E x ( N ) = E x ( O ) - O k X [ E ( O ) - O a / ef ]X N

= E x ( o ) - H I E ( 0 ) - B ] x N (9)

w h e r e H = D k xand B = aa /e f .A s c a n b e s e e n i n e q n ( 9 ) , t h r e e p a r a m e t e r s

( x , H a n d B ) c h a r a c t e r i s e t h e s t if f n es s r e d u c t i o na n d a r e d e t e r m i n e d e x p e r i m e n t a l l y, x a n d H a r ei n d e p e n d e n t o f t h e t e s t v a r ia b l e s a n d B i sd e p e n d e n t o n t h e m a x i m u m a p p l i e d st re s s. M o r ed e t a i ls o f t h e m o d e l c a n b e f o u n d i n R e f s 11 a n d15.

To d e t e r m i n e t h e c o n s t a n t s x , H a n d B , f a t i g u e

t e s ts w e r e c a r r i e d o u t o n f i v e s p e c i m e n s . D u r i n ge a c h t e s t t h e s t a t i c m o d u l u s w a s m e a s u r e d o ns e v e r a l o c c a s i o n s . T h e r e d u c t i o n i n m o d u l u s f o ra ll fi v e s p e c i m e n s i s p l o t t e d a g a i n s t c y c l e s i n F ig .8 . T h e m o d u l u s a f t e r N c y c l e s is g i v e n a s af r a c t i o n o f t h e i n it ia l m o d u l u s a n d t h e n u m b e r o fc y c l e s is g i v e n a s a f r a c t i o n o f t h e f a t i g u e l if e .T h e m a x i m u m s tr e ss w a s 1 8 7 5 M P a a n d t h ef a t i g u e l iv e s v a r i e d f r o m 2 5 2 0 c y c l e s t o 3 4 4 3 1 0c y c l e s . F i g u r e 8 s h o w s t h a t t h e r e d u c t i o n i nm o d u l u s , t h o u g h s l i g h t , i s a l m o s t l i n e a r u p t o af r a c t i o n a l li fe o f a b o u t 0 - 75 , a f t e r w h i c h t h ed e c r e a s e is a l it tl e m o r e r a p i d . H o w e v e r , t h em a x i m u m o b s e r v e d d r o p w a s o n l y 6 , f r o m 1 35t o a b o u t 1 27 G P a .

T h e v a l u e s o f t h e c o n s t a n t s x, H a n d B w e r ef o u n d t o b e 1 - 1 5 , 1 .7 4 a n d 1 2 7 , r e s p e c t i v e l y .

T h e c u r v e g e n e r a t e d b y e q n ( 9 ) a p p e a r s t o f itt h e e x p e r i m e n t a l r e s u l t s q u i t e w e l l . T h e m e a s -u r e d r e d u c t i o n i n s ti f fn e s s w a s r a t h e r s m a l l a n dp e r h a p s u n i d i r e c ti o n a l c a r b o n r e i n f o r c e d e p o x yw a s n o t t h e c o r r e c t m a t e r i a l t o t e s t t h e m o d e l .H o w e v e r , o n e c a n c l e a r l y s e e t h e n e c e s s i t y f o ra c c u r a t e m o d u l u s m e a s u r e m e n t i f t h e m o d e l i s t ob e a p p l i e d s u c c e s s f u l l y.

C O N C L U S I O N S

T h e r e s id u a l s tr e n g t h m o d e l o f S e n d e c k y j a n dt h e r e s i d u a l st if f n es s m o d e l o f W h i t w o r t h h a v eb e e n u s e d t o d e s c r i b e t h e r e p e a t e d t e n s i o nf a t ig u e p r o p e r t i e s o f u n i d i r e c t i o n a l c a r b o n f i b rer e i n f o r c e d e p o x y. B o t h m o d e l s f i tt e d t h ee x p e r i m e n t a l r e s u l t s q u i t e w e l l .


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atigue life prediction in composites 139

1.0

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F r a c t i o n a l l if e , NF i g 8. Normal i sed s ta tic modulus as a func t ion of the f rac t ion of fa t igue l i fe a t a maximu m s t ress of 1875 MPa.

T h e r e s i d u a l s t r e n g t h m o d e l w a s a l s o u s e d t op r e d i c t f a t i g u e l i f e u n d e r s p e c t r u m l o a d i n g . T h ec o r r e l a t i o n b e t w e e n t h e p r e d i c t i o n a n d t h ee x p e r i m e n t a l o b s e r v a t i o n s w a s s a t i s f a c t o r y.

I f t h e a s s u m p t i o n s o n w h i c h t h e r e s i d u a ls t r e n g t h m o d e l i s b a s e d a r e s h o w n t o b e v a l i dt h e n t h e m o d e l c a n b e u s e d w i t h s o m ec o n f i d e n c e . T h e p a r a m e t e r s C a n d S a r e p r o b a b l yd e p e n d e n t o n f a c t o r s l i k e l a m i n a t e o r i e n t a t i o ne n v i r o n m e n t a l c o n d i t i o n s a n d R - r a t i o s o t h em o d e l w o u l d n e e d t o b e m o d i f i e d t o a c c o m m o d -a t e t h e s e . H o w e v e r w o r k c a r r i e d o u t a tN o r t h r o p h a s i n d i c a t e d t h a t i f s t r a i n i s u s e d t od e f i n e f a t i g u e l o a d i n s t e a d o f st r e s s l e ssm o d i f i c a t i o n i s r e q u i r e d .6

T h e r e s i d u a l s t i f f n e s s m o d e l a l s o p r o v i d e d ar e a s o n a b l e f i t t o t h e e x p e r i m e n t a l r e s u l t so b t a i n e d . I t s h o u l d p r o v e t o b e a u s e f u l t o o l i nm o n i t o r i n g f a t i g u e li f e a s lo n g a s t h e m o d u l u s i sm e a s u r e d a c c u r a t e l y. I t s m a i n d i s a d v a n t a g e i st h a t i t c a n b e a p p l i e d o n l y t o c o n s t a n t a m p l i t u d ef a t i g u e l o a d i n g s .

R E F E R E N E S

1. Mandel l , J . F. , Huan g, D. D. Mc Garry, F. J . ,Tens i le fa t igue per formance of g lass f ib re domina tedcomposites.Comp. Technol. Rev.,3 (1981) 96-102.

2 . Halp in , J . C . , Je r ina , K. L . Johnson , T. A. ,Charac te r iza t ion of compos i tes for the purpose ofrel iabil i ty evaluation. InAnalysis of Test Methods for

High M odulus Fibres and Com posites(ASTM STP 521).AST M, Phi lade lphia , PA, 1973 , 5 -64 .

3 . Yang, J . N. , Re l iab il i ty pred ic tion for compos i tes underperiodic proof tests in service. InProceedings of 4thConference on Composite M aterials: Testing and Design(ASTM STP 617) . ASTM , Phi lade lphia , PA, 1977 ,272-95 .

4 . Whi tney, J . M. , Res idua l s t rength degrada t ion modelfor compet ing fa i lu re modes . InLong Term Behaviourof Composites(ASTM STP 813) . ASTM, Phi lade lphia ,

PA, 1983, 483-97 .5 . Sendecky j , G. P . , F i t ting models to compos i te materia l sfat igue data. InTest Methods and D esign Allowables forFibrous Composites(ASTM STP 734). ASTM, Phila-delphia, PA, 1981, 245-60.

6. Com pon eschi , E. T. St inchcom b, W. W ., St iffnessreduc t ion as an ind ica tor o f damage in graphi te /epoxylaminates. InComposite Materials: T esting and DesignSixth Conference)(ASTM STP 787) . AST M, Phi lade l -

ph ia , PA, 225-46 .7. Schulte, K. , St iffness reduction and development of

longitudinal cracks during fat igue loading of compositelaminates. In Me chanical Characterization of LoadBearing Fibre Composite Laminates,e d . A . H . C a r d o n

G. Verchery. E lsev ie r Appl ied Sc ience Publ i shers ,London, 1985 , pp . 36-54 .

8 . Yang, J . N. , Jones , D. L . , Yang, S . H. Meskin i, A. ,A s ti ffness degrada t ion m odel for g raphi te /epoxylaminates. In P r o ce e di ng s o f A I A A / A S M E /A SC E /A H S 29th Structures, Structural Dynamics andMaterials Conference,Williamsburg, Virginia, 1988,958-66 .

9. Ro tem , A ., St iffness chang e of a graphite ep oxyliminate under reverse fat igue loading.J. Comp.Technol. Res.,U (1989) 59-6 4.

10. Naeem , M. , Fa t igue damage-com pl iance re la tionsh ipf o r G R P. Composites, 20(1989) 589-92.

11. Wh itworth, H . A . , M odell ing sti ffness red uction o fgraphi te /epoxy compos i te lamina tes .J. Comp. Mater.,21 (1987) 362-72.


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140 R. J . Hus ton

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