unidirectional_frcs.pdf

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N . & M . N o . 3 7 8 2

P R O C U R E M E N T E X E C U T I V E , M I N I S T R Y O F D E F E N C EAERONAUTICAL RESEARCH COUNCIL

REPORTS AND MEMORANDA

O n t h e E la s ti c M o d u l i o f U n id irec t io n a l F ib reR e i n f o r c ed C o m p o s i te s

B y E . H . MANSFIELD, F . R . S .Structures Dept., R.A.E., Farnborough

LONDON: HER MAJESTY'S STATIONERY OFFICE1976

2.30 NET


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O n t h e E l a s t ic M o d u l i o f U n i d i r e c t io n a l F i b r eR e i n f o r c e d C o m p o s i t e s

By E. H. MANSFIELD,F.R.S.Structures Dept., R.A.E., Farnborough

Repo r t s and M em oran da N o . 3 7 8 2 *Ju l y , 1974

SummaryA simple analytical technique is presented for obtaining lower bounds to the effective moduli of unidirec-tional fibre-reinforced composites with any repetitive array of fibres. The analysis depends upon a direct

meth od involving stress-equilibrium and strain-continuity considerations, and thus it differs from the type ofanalysis due to Hashin and Hill (and others) involving variational methods. The composite is envisaged asbeing sliced along all 1,2 planes so tha t condit ions of plane stress exist throughout and the stiffness propertiesof each vanishingly thin slice are readily determined in terms of the constituent properties of the fibre andmatrix. The effective elastic m0duli are obtained by suitable integration of such slice properties over arepeating fibre/matrix pattern. Comparison with known accurate values of the longitudinal shear modulusshows that the technique underestimates the modulus by factors of about 1.1 for the square array and 1.2 forthe hexagonal array, depend ing on the fibre volume fraction and f ibre/matrix stiffness ratio. By judicious useof such correction factors it should be possible to estimate the longitudinal shear modulus over a wide range ofdesign parameters to within about 5 per cent; for the square and hexagonal arrays Symm's accurate but limitedvalues have been augmented by new results which have an accuracy of 0.3 per cent. New results are alsopresented for the transverse modulus.

* Replaces R.A.E. Technical Report 74106 -- A.R.C. 35 647


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LIST OF CONTENTS1. Introduction2. Method of Analysis

2.1. The Longitudinal Shear Modulus of Unidirectional FRC2.2. The Longitudinal and Transverse Moduli of Unidirectional FRC2.3. Evaluation of ( S * ( z ) )

3. Comparisons with Known Solutions3.1. The Hexagonal Arra y3.2. The Square Array

4. Extension of Known Accurate Values for G * 2 / G m

5. The Transverse Modulus Ratio E * / E m6. Examples7. ConclusionsAcknowledgmentList of SymbolsReferences

Tables 1 fo 3Illustrations - - Figures 1 to 7Detachable Abstract Cards


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1. IntroductionUnidirectional fibre- reinforced composite (FRC) may be used in a structural context for carrying unidirec-

tional loads, as in the manne r of a strut or reinforcing stringer, or as one of a number of laminates in a built-upsheet for ca rrying multi-directional loads. In either case there is the problem of load transfer into the membervia end fittings. A knowledge of the effective longitudinal, transverse and shear moduli of the member is anessential preliminary to the efficient design of such fittings.In the multi-laminate sheet it is customary to have at least three distinct alignments for the fibres so that theeffective moduli are domina ted by the stiffness of the fibres rather than the matrix, and are adequately andsimply given by 'nett ing analysis'.1 However, in a unidirectional FRC member the transverse and shear moduliare primarily dependent on the stiffness of the matrix rather than the fibre. Theoretical and numericaltechniques 2-7 are available for predicting these moduli assuming that the fibres are spaced in regular arrays anda limited number of accurate numerical results for hexagonal and square arrays are given by Symm.6 There arealso techniques available for deriving upper and lower bounds to the effective moduli for what are calledrandom arrays of fibres.8

Here we present a technique for obtaining lower bounds to the effective moduli in unidirectional FRC withan y speci f ied repet i tive array. Comparisons with Symm's accurate results for the longitudinal shear modu lus ofFRC with hexagonal and square arrays show that the technique under -estimates the true value by about 20 percent for the hexagonal array and 10 per cent for the square array depending on the fibre volume fraction v andthe f ibre/matr ix stiffness ratio. In comparison with Hashin's8 best lower bound' the present technique is moreaccurate only at high values of v , namely v > 0.85 for the hexagonal array and v > 0-60 for the square array.Such an improvement, however limited, is to be expected because Hashin's model requires the presence offibres with ever-decreasing diameter, which cannot adequately represent a composite with fibres of constantdiameter--particularly near the limits of maximum fibre packing. But the main advantages of the presentmethod are (i) its simplicity, (ii) its generality , and (iii) the fact that all known accurate values of the effectivemoduli are under- estimated by amounts which vary only slightly with v and the f ibre/matrix stiffness ratio,thus enabling the meth od to augment the range of accurate results for hexagonal and square arrays. Finally,(iv) the method enables results for the effective longitudinal shear modulus to be used to derive results for theeffective transverse modulus of unidirectional FRC.

2. Me thod o f Analys is (see Fig. 1 for notation)The method of analysis is approximate but simple. It is based on the following closely related facts. First,

under an applied longitudinal shear stress r*2, the dominant stresses in the matr ix and fibre are the longitudinalshear stresses r~"~ and ~2 in the plane of the applied shear stress: the longitudinal shear stresses in a normalplane, r~'~ and ~3, are significantly less. Second, under an applied transverse stress o-* the normal stresses inthe matrix and fibre, o-~ and o'~ are small in comparison with the transverse stresses o-~ and o-~.The present me thod ignores these normal shear and direct stresses by adopting a slicing technique in whichthe composite is envisaged as being sliced along all 1, 2 planes (like a pack of cards), thus ensuring conditions ofplane stress throughout. It is to be noted that this technique introduces an artificial degree of flexibility nto thecomposite so that the resulting values of the composite moduli are lower limits to the true values. The degree towhich this technique under-estimates the true moduli is determined by comparison with some known exactsolutions and appropriate compensating factors are introduced.

The slicing technique greatly simplifies the theoretical determination of the longitudinal shear andtransverse moduli of unidirectional FRC with any repetitive array of fibres. In such an FR C the cross-sectionmay be covered by contiguous and equal rectangles containing identical fibre patterns. The sides of therectangles may cut through individual fibres and the rectangles themselves may be staggered, as shown in Fig.3. The effective elastic moduli of such an F RC are the same as for the individual strips of rectangular sectionwhose faces are displaced by fixed amounts. The effective elastic moduli of an individual strip are determinedby integrat ion (through the height h) of the effective moduli of the constituen t slices of width w bounded byvanishingly close planes at z, z + 8z.The matrix is assumed to be isotropic with shear modulus G m, Young's modulus E " and Poisson's ratio v" .The fibre material is assumed to have a longitudinal shear modulus G s and orthotropic direct stiffnesscharacteristics with longitudinal and transverse Young 's moduli E{ and E~ and corresponding Poisson's ratiosV~l and v{2. (For numerical purposes we later take v{2 = v m as this considerably simplifies the analysis withoutsignificantly affecting results.) Average or effective values of the moduli and stresses for the composite are


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d e n o t e d b y a n a st e r is k . T h e o v e r a l l v o l u m e f r a c t i o n s a r e d e n o t e d b y vm a n d v ( v , , + v = 1 ) , w h i l e th e v o l u m ef r a c t i o n s a p p r o p r i a t e t o a s li c e a t z a r e d e n o t e d b y v m ( z ) a n d v t ( z) . W e a d o p t t h e n o t a t i o n ( f ( z ) } to i n d i c a te t h ea v e r a g e v a l u e o f f ( z ) t h r o u g h t h e h e i g h t h o f t h e r e p e a t i n g r e c t a n g l e , i . e.

a n d w e n o t e i n p a ss i n g t h a t , f o r e x a m p l e ,

l f o/ ( z ) ) = g f ( z ) d z(v t (z)> = yr.

F i g . 2 s h o ws a v e ra g e a n d l o c a l s t r e s s e s i n a t y p i c a l s l i c e .2 .1 . The Longi tudinal Shear Modulus o f Unidirecf ionaa FRC

T h e s t r e s s - s t r a i n r e l a t i o n s i n s h e a r f o r t h e m a t r i x a n d f i b r e m a t e r i a l a r e

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

v ~ = r ~ % l G " , }v { ~ = r { 2 1 G r , (1 )

r*2(z) = r~'z(z) = Z{z(Z). (2)T h e a v e r a g e s h e a r s t r a in i n a t y p ic a l l a m i n a a t z i s g i v e n b y

v *2 = v . , ( z ) w % ( z ) + v r ( z )v { 2 ( z ) , (3 )a n d a c c o r d i n g l y t h e e f f e c t i v e l o n g i t u d i n a l s h e a r m o d u l u s i s g i v e n b y

G * 2 ( z ) = ~ * ~ ( z ) l v T 2{ v m ( z ) + ~ i ( z ) ~ - '= \ G m G I ] , ( 4 )a n d i t i s c o n v e n i e n t t o r e - w r i t e t h i s in t h e f o r m

G * 2 ( z ) / G " = S * (z ), s a y= { 1 - ( 1 - n ) v s ( z ) } - 1 ( 5 )

w h e r e r ) = G " / G I .F i n a l l y , b y i n t e g r a t i n g t h r o u g h t h e h e i g h t h o f th e r e p e a t i n g r e c t a n g l e w e o b t a i n

G * 2 / G " = ( S * ( z ) ) . ( 6)T h e a b o v e a n a ly s is c a n r e a d i l y b e e x t e n d e d t o i n c lu d e a n F R C c o n t a i n i n g f ib r e s w i t h d i f fe r i n g m o d u l i . A n

e x a m p l e is g iv e n i n S e c t i o n 6 .2 . 2 . The Lo ng i tud ina l a nd Tra nsv ers e M o duf i o f U ni d i rect i o na l F R C

T h e d i r e c t s t r e s s - s t r a i n r e l a t i o n s in t h e m a t r i x a n d f i b r e m a t e r i a l a r e g i v e n b ye 7 = ( , ~ 7 - v%-~ ) l E%

m m rr l m rnE 2 = ( O " 2 - - / " 0 " 1 ) / E ,e { = o - { / E { - * t * (7)l t2 i 0 " 2 / E 2 ,~ = 4 / e r 2 - * , ," 1 2 O ' l / E 1 ,

4


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w h e r e , f o r e x a m p l e , / ' { 2 i s t h e r a t i o ( t r a n s v e r s e c o n t r a c t i o n o f f i b r e ) / ( l o n g i t u d i n a l e x t e n s i o n ) d u e t o al o n g i t u d in a l s t re s s. F o r a n e l a s ti c m a t e r i a l - - a s t h e f i br e s a r e a s s u m e d t o b e - - i t f o l lo w s f ro m t h e R e c i p r o c a lT h e o r e m t h at

f f f f~'El~E2 =/'12/E1. (8)E q u i l i b r i u m o f a v e r a g e d i r e c t s t re s s e s w i t h c o m p o n e n t s t r e s s e s i n a t y p i c a l s l ic e gi v e s

a n d o * ( z ) = v , .( z ) 0 . ? ( z ) + v t ( z ) 0 .{ ( z )J. * (z ) = 0 .7 ( z ) = ~ ( z ) , (9 )

w h i l e t h e a v e r a g e d i r e c t s tr a in s a r e r e l a t e d t o t h e c o m p o n e n t s t ra i n s b y

la n d e* ( z ) = v , ,, ( z) e2 ( z ) + v t ( z ) J z ( z ) . JE q u a t i o n s ( 7 ) t o ( 1 0 ) e n a b l e u s t o r e l a t e t h e a v e r a g e d i r e c t s t r a i n s t o t h e a v e r a g e d i r e c t s t r e s se s i n a ty p i c a ls l ice :

w h e r e

E * (z ) = 0 . * ( z ) / E ~ ( z ) - / ' * , ( z ) 0 . * ( z ) / E * ( z ) ,e * ( z ) = o - * ( z ) / E * ( z ) - / ' * 2 ( z ) o ' * ( z ) / E * ( z ) ,

/ ' * , ( z ) / E * ( z ) = / ' * 2 ( z ) / E * ( z ) .( I 1 )

F i n a l l y , f o r t h e c o m p o s i t e a s a w h o l e , w e a r e n o w i n a p o s i t i o n t o r e l a t e t h e a v e r a g e d i r e c t s t ra i n s t o t h e a v e r a g ed i r e c t s t r e s se s :

w h e r e

- - 0 .1 ~ E l - - / ' 2 1 0 " 2 / E z ,E ~ - - ca * * * *- - 0 " 2 / E 2 - / ' 1 2 0 . 1 / E l ,

ca__ * ca/ ' 2 1 / E 2 - / ' 1 2 / E 1 .(12)

N o w i n d e t e r m i n i n g E * a n d E * i t i s n o t , in g e n e r a l, s i m p l y a m a t t e r o f i n t e g r a ti n g E * ( z ) a n d E * ( z ) t h r o u g ht h e h e i g h t h b e c a u s e w e m u s t e n s u r e t h a t both t h e a v e r a g e d i r e c t s t r a i n s s * ( z ) a n d e * ( z ) a r e t h e s a m e f o r a lls l ic e s. T h i s p r o b l e m d i d n o t a r i s e f o r t h e c a s e o f s h e a r b e c a u s e y*2(z ) w a s t h e o n l y n o n - z e r o s t ra i n . H o w e v e r ,f o r th e m o d u l u s E * a s im p l e i n t e g r a ti o n o f E * ( z ) i s va l id i f

/ " = / ' { 2 = / ' , s a y . ( 1 3 )

T h i s i s b e c a u s e u n d e r a n a p p l i e d s t r e s s 0 . * ( z ) t h e a v e r a g e s t r a i n s a r e n o w s u c h t h a te * ( z ) = - / ' e * ( z ) ( 1 4 )

s o t h a t e q u a l i t y o f l o n g i t u d i n a l s t r a i n s i n a ll s l ic e s n e c e s s a r i ly i m p l i e s e q u a l i t y o f t r a n s v e r s e s t r a i n s a n d , i np a r t i c u l a r ,,,1 "2 = / ' . ( 1 5 )

W i t h t h is s i m p l if y in g a s s u m p t i o n i t m a y b e c o n f i r m e d t h a tE * ( z ) = v m ( z ) E m + v t ( z ) E {

w h i c h m a y l e g i t im a t e l y b e i n t e g r a te d t o r e c o v e r t h e w e l l - k n o w n r e su l t:E * , , i= v inE +v t E1 . ( 1 6 )


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T h e s i m p l e s t w a y t o d e r i v e E * is v i a t h e p l a n e s t r a i n m o d u l u s

m a k i n g u s e o f t h e r e l a t io n

w h i c h y i e l d sF , * = E * / (1 * *- I ] 1 2 1 / 2 1 ) ~

(17)

(18)

/ 1 v 2 \ - IE * = i - ~ 2 + - ~ l ) , (19)i n v i r t u e o f e q u a t i o n ( 1 5 ) a n d t h e l a s t o f e q u a t i o n ( 1 2 ) .

N o w u n d e r c o n d i t i o n s o f p l a n e s t r a i n ( e * = 0 ) it m a y r e a d i l y b e v e r i f i e d t h a t

/ ~ * , , [ v ~ ' ( z )+ v t ( z ) V ' }w h e r e / ~ " = E r a ~ ( 1 - u z ) , (20)

~ = e~ / ( l - ~ rb ' 1 2 ~ 2 1 ) .T h i s r e s u lt h a s m u c h i n c o m m o n w i t h e q u a t i o n ( 4) f o r t h e s h e a r m o d u l u s a n d i t i s c o n v e n i e n t t o e x p r e s s i t i n li k ef o r m :

~ * ( z ) /~ . m = S * ( z )= { 1 - (1 - n ' ) v t ( z ) } -1 ( 2 1 )

w h e r e ~ ' = / ~ m / / ~ t z .F i n a l ly , b y i n t e g r a t i n g th r o u g h t h e h e i g h t h o f t h e r e p e a t i n g r e c t a n g l e w e o b t a i n

E * / E m = (S*(z ) ) , (22)w h i c h i s f o r m a l l y i d e n t i c a l t o e q u a t i o n ( 6) a p a r t f r o m t h e r e p l a c e m e n t o f "q b y " q' .F r o m a n u m e r i c a l s t a n d p o i n t t h e d e t e r m i n a t i o n o f G *2 ,/~ 2" , a n d h e n c e E * , t h u s r e q u i r e s t h e e v a l u a t i o n o ft h e s a m e f u n c t i o n , n a m e l y ( S * ( z ) ) , b u t w i t h d i f f e r i n g v a l u e s o f t h e c o n s t a n t s ~ , ~ ' .

2 . 3 . Eva lua t ion o f ( S * ( z ) )T h e n u m e r i c a l v a l u e o f ( S * ( z ) ) f o r a n y g iv e n F R C a r r a y m a y , o f c o ur s e , b e d e t e r m i n e d f r o m S * ( z ) b y s i m p l e

g r a p h i c a l i n t e g r a t i o n . A n a l y t ic a l o r c o m p u t e r - p r o g r a m m e d i n t e g r a t i o n r e q u ir e s a f u n c ti o n a l e x p r e s s i o n f o rv t ( z ) a n d h e n c e e x p r e s s i o n s f o r t h e c h o r d l e n g t h c . ( z ) o f a f i b r e c r o s s - s e c t i o n c u t b y a s e c t i o n a t z . Su c he x p r e s s i o n s f o r f i b r e s o f c i r c u l a r , b u t n o t n e c e s s a r i l y e q u a l , c r o s s - s e c t i o n a r e g i v e n b e l o w .

T h e e q u a t i o n f o r a c i r c l e o f r a d i u s r . w h o s e c e n t r e i s a t y . , z . i s g i v e n b y(y _ y ,)Z + (z - z , ) 2 = rZ,, (23)

a n d a c c o r d i n g l y a t a ty p i c a l s e c t i o n z ,y - y , = +{rZ~ (z -z,)2}~-'

a n d h e n c ec . ( z ) 2 { r ] - ( z z- z . ) } 2 .


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N o t e t h a t t h i s e x p r e s si o n , a n d h e n c e ( S * ( z ) ) , i s i n d e p e n d e n t o f t h e v a l u e o f y , ; t h is is a d i r e c t c o n s e q u e n c e o f- - Z n ) ~ rn ,h e i n h e r e n t a p p r o x i m a t i o n i n v o l v e d i n t h e ' sl ic i ng t e c h n i q u e ' . I t i s a l so t a c i tl y a s s u m e d t h a t ( z 2 2

o t h e r w i s e t h e r e i s n o i n t e r s e c t i o n a n d c . ( z ) i s z e r o . T h i s p o s s i b i l it y c a n b e c a t e r e d f o r s i mp l y b y w r i t i n gc . ( z ) = 2 ~ { r Z . - ( z - z . ) 2 '-}2 , (24)

w h e r e ~ s t a n d s f o r ' t h e re a l p a r t o f ' .T h e w i d t h o f th e r e p e a t i n g r e c t a n g u l a r e l e m e n t is w a n d t h e v o l u m e f r a c t io n v i ( z ) i s t h u s g i v e n b y

v r ( z ) =lx c . ( z )W n

= 2 E ~ { r ~ - ( z - z . ) 2 } , ( 2 5 )W n

w h e r e t h e s u m m a t i o n e x t e n d s o v e r a ll e f f e c t i v e v a l u e s o f n ; e f f e c t i v e , b e c a u s e a n y s e g m e n t s o f c i rc l es c u t b y t h es i de s o f th e r e c t a n g l e a t y = 0 , w c a n b e c o m b i n e d t o f o r m a c o m p l e t e s e g m e n t f o r p u r p o s e s o f c a l c u la t i o n .

3 . C o m p a r i s o n s w i t h K n o w n S o l u t io n sW e n o w t u r n o u t a t t e n t io n t o c o m p a r i s o n s w i t h S y m m ' s6 a c c u r a t e v a l u e s o f G * 2 / G " f o r h e x a g o n a l a n ds q u a r e a r r a y s o f f i b r e s o f e q u a l c i r c u l a r c r o s s - s e c t i o n . I n t h i s c o n n e c t i o n w e n o t e t h a t f o r v < 0 . 4 ( w h i c h i s

g e n e r a l l y o u t s i d e t h e p r a c t i c a l r a n g e o f i n t e r e s t ) t h e f i b r e s a r e s u f f i c i e n t l y s p a r s e f o r t h e i n t e r a c t i o n e f f e c tb e t w e e n f i br e s to b e s o s m a l l t h a t t h e d i s t i n c t io n b e t w e e n G *2 f o r t h e h e x a g o n a l a n d s q u a r e a r r a y s i s n e g l ig i b lea n d b o t h a r e a d e q u a t e l y g i v e n b y H a s h i n ' s 8 s i m p le m o d e l a n d f o r m u l a :

O*2 _ (1 + v t ) + ~( 1 - v )G " ( 1 - v t) + r t ( l + v r)" (26)

3 . 1 . T h e H e x a g o n a l A r r a yA s u i t a b le r e c t a n g u l a r r e p e a t i n g e l e m e n t f o r t h e h e x a g o n a l a r r a y i s i d e n t i fi e d b y t h e c o r n e r p o i n t s O A B C in

t h e y , z p l a n e i n F ig . 4 , w h e r ew = f ib re p i t ch ]

a n d t ( 2 7 )h = --~- w.

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

r = { 4 - Jw ~ 2 - ~ v r J " (28)[ I t is c o n v e n i e n t a t t h i s p o i n t t o t a k e w = 1 , s o t h a t r , h , z a r e e x p r e s s e d a s p r o p o r t i o n s o f t h e f i b r e p i t c h . ] T h ec i rc l es c e n t r e d o n O a n d A c a n b e c o m b i n e d t o f o r m a s i n g le e f f e c ti v e c i rc l e, a s p re v i o u s l y d e s c r i b e d , s o t h a te q u a t i o n ( 25 ) b e c o m e s

v f ( z ) = 2 ~ [ ( r 2 - z 2 ) + { r z - ( z - h)Z}k].w

(29)

W h e n r ~< h , w h i c h c o r r e s p o n d s t o t h e r a n g e

vr ~


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n o s in g l e z - s e c t i o n c u t s b o t h u p p e r a n d l o w e r c i r cl e s , a n d t h e t w o t e r m s i n s q u a r e b r a c k e t s i n e q u a t i o n ( 2 9 ) d on o t e x i s t s i m u l t a n e o u s l y . T h e e x p r e s s i o n f o r v r ( z ) a c c o r d i n g l y s p li ts u p i n t o t h e f o l l o w i n g si m p l e c o m p o n e n t s :

2 2 ~v t ( z ) = - - ( r - z )~,W

= 0= 2 { r2 - - (Z - h )2 } ~

W

o v e r t h e r a n g e 0 < z < r , /

o v e r t h e r a n g e r < z < (h - r ) ,o v e r t h e r a n g e (h - r ) < z < h .

(31)

T h e c o r r e s p o n d i n g v a l u e s o f S * ( z ) , g i v e n b y e q u a t i o n ( 5) , le n d t h e m s e l v e s t o c l o s e d f o r m i n t e g r a ti o n t o y ie l dG * 2 1 2 a 1 , - - / - T r + 4c =

( I + P @ /- - t a n - 1 \ 1 - - - ~ 1 / '

w h e r e a = , - 7 - */and ~ = A (1 - ~) .

( 3 2 )

F o r v a l u e s o f v f g r e a t e r t h a n t h a t g i v e n b y e q u a t i o n ( 3 0 ) , G * 2 / G m h a s b e e n o b t a i n e d b y a s im p l e c o m p u t e rp r o g r a m .T h e f a c t o rs K b y w h i ch t h e p r e s e n t v a l ue s o f G * 2 / G " m u s t b e m u l t i p li e d t o b r i n g t h e m i n to a g r e e m e n t

w i t h S y m m ' s a c c u r a t e v a l u e s a r e s h o w n i n F i g. 5 . It is s e e n t h a t K v a r i e s w i t h v a n d G r / G '~ in a s l igh t ands m o o t h m a n n e r e v e r y w h e r e . T h e r e l a t i v e l y l a r g e m a g n i t u d e o f K is a r e f l e c t io n o f t h e f a c t t h a t t h e s l ic i ngt e c h n i q u e , w h i c h i g n o r e s t h e s h e a r s tr e s s e s % > c a n n o t a d e q u a t e l y r e p r e s e n t t h o s e l i n e s o f s h e a r w h i c h , i n t h eh e x a g o n a l a r r a y , z i g z a g b e t w e e n a d j a c e n t r o w s o f fi b re s . M a r k e d l y s m a l l e r v a l u e s o f K a r e t o b e e x p e c t e d f o rt h e s q u a r e a r r a y w h e r e , f r o m s y m m e t r y , t h e r e a r e n o l in e s o f s h e a r w h ic h c r o ss f r o m r o w t o r ow .

3 . 2 . T h e S q u a r e A r r a yT h e a n a ly s is f o r th e s q u a r e a r r a y h a s m u c h i n c o m m o n w i th t h a t f o r t h e h e x a g o n a l a r r a y a n d a c c o r d i n g ly t h e

d e t a i l s w i ll b e o m i t t e d . S u f fi c e i t t o s a y t h a t t h e f o l l o w i n g c l o s e d f o r m e x p r e s s i o n e x i s t s f o r a ll v a l u e s o f v i :

G m = ( 1 - T / ) t 2 ~ t a n - l ~ l - p / J '/ 4w h e r e K = ~ -~ v f] (33 )

an d p = K(1 - 77).T h e f a c t o r s K b y w h i ch t h e s e v a l u e s m u s t b e m u l t i pl i e d t o b r in g t h e m i n to a g r e e m e n t w i t h S y m m ' s a c c u r a t ev a l u e s a r e s h o w n i n F i g . 6 .

4 . E x t en s i o n of K n o w n A c c u r a t e V a l u e s o r G ~ 2 / G mI n t h is s e c t io n u s e i s m a d e o f t h e p r e s e n t t e c h n i q u e a n d t h e s m o o t h l y v a r y i n g n a t u r e o f t h e c o r r e c t i o n f a c t o r s

K t o e x t e n d S y m m ' s 6 r e s u l ts , w h i c h c o r r e s p o n d t o t h e c i r c l e s i n F ig s . 5 a n d 6 , t o t h e f o l l o w i n g a d d i t i o n a l v a l u e so f v f: 0 . 4 5 , 0 . 5 0 , 0 - 6 0 , 0 - 6 5 , 0 . 7 5 . T h e m e t h o d p e r m i t s a n a c c u r a c y o f a b o u t 0 . 3 p e r c e n t .

W h i l e t h e v a l u e s o f G * a / G " g i v e n b y S y m m h a v e b e e n d e s c r i b e d a s a c c u r a t e , i t is o n l y f a i r t o p o i n t o u t , a sS y m m h a s d o n e , t h a t t h e c o l l o c a ti o n p r o c e d u r e h e a d o p t e d d i d n o t c o n v e r g e f o r c e r t a in v a l u e s o f v a n dG t / G ' '. F o r t h e h e x a g o n a l a r r a y n o n c o n v e r g e n c e o c c u r r e d a t v = 0 . 9 f o r G r / G ' ' = 1 2 0 , ~ , w h i l e f o r t h es q u a r e a r ra y n o n c o n v e r g e n c e o c c u r r e d at v t = 0 . 7 8 f o r G r / G m = 1 2 , 2 0 , 1 2 0 , ~ . T h e v a l u e o f G * z / G "~d e r i v e d f o r t h e l a st c o m b i n a t i o n c i t e d is c l e a r ly in e r r o r a s i t is less ( b y a b o u t 1 .1 p e r c e n t ) t h a n t h e p r e s e n tl o w e r l i m i t . T h e o t h e r v a l u e s t e n d t o s h o w i r r e g u l a r v a r i a t i o n s i n t h e f a c t o r s K . B e c a u s e o f t h e r e s u l t i n gu n c e r t a i n t y t h e s e v a lu e s a r e q u e r i e d i n T a b l e 1 a n d o m i t t e d a l t o g e t h e r f r o m T a b l e 2 .


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5 . T h e T r an s ve r se M o d u l u s R a t i o E ~ / E mT h e t r a n s v e r s e m o d u l u s E * i s g i v e n b y e q u a t i o n ( 1 9 ) , i n w h i c h E * i s g i v e n b y e q u a t i o n ( 1 6 ). T h u s w e f i n d

E ~ [. ( /~2* /E ,'~) ~- 1 - v f + v f ( E ~ / E m) (34)w h e r e a l o w e r l i m i t t o / ~ * i s g i v e n b y e q u a t i o n ( 2 2) .N o w w e h a v e s e e n i n S e c t i o n 2 .2 t h a t t h e s l ic i n g t e c h n i q u e y i e l d s a l o w e r l i m i t f o r t h e p l a n e s t r a i n m o d u l u sr a t i o / ~ . / / ~ m w h i c h is f o r m a l l y i d e n t ic a l to e q u a t i o n ( 6) f o r t h e l o n g i t u d i n a l s h e a r m o d u l u s r a t i o G*2/G m.A c c u r a t e v a l u e s f o r G*2/G m a r e a l s o k n o w n a n d i t is t e mp t i n g t o a s s u m e t h a t , f o r g i v e n v a l u e s o f r /, - ,/ ', t h el o w e r l i m i t s f o r G*2/G" a n d / ~ , / / ~ m u n d e r - e s t i m a t e t h e t r u e v a l u e s b y t h e s a m e a m o u n t s . W i t h t h i sa s s u m p t i o n w e c a n u s e T a b l e 1 t o d e t e r m i n e ' c o r r e c t e d ' v a l u e s o f E* / E ~ a n d h e n c e ( p r e s u m a b l y ) m o r ea c c u r a t e v a l u e s f o r E* /E m. T h i s h a s b e e n d o n e i n T a b l e 3 f o r t h e h e x a g o n a l a r r a y , m a k i n g t h e f u r t h e ra s s u m p t i o n t h a t

v = 0 " 2 5 , ( 35 )a n d

E~ / E m =/~r2//~m, (36)

s o t h a t t h e f i b r e s a r e e f f e c t i v e l y i s o t r o p i c . Fo r t h e p a r t i c u l a r c a s e i n w h i c h Er2/E m = 2 0 , C h e n a n d C h e n g 3o b t a i n e d n u m e r i c a l r e s u l ts f o r t h e h e x a g o n a l a r r a y w h i c h a r e b a s e d o n a c l as s ic a l e l a s ti c i ty s o lu t i o n . T h e i rd e r i v e d v a r i a t i o n o f E * w i t h v i s i n e x c e l l e n t a g r e e m e n t w i t h v a lu e s o b t a i n e d f r o m T a b l e 3 , a l t h o u g h t h e i rg r a p h ic a l m e t h o d o f p r e s e n t a t i o n d o e s n o t p e r m i t m u c h a c c u ra c y .A s f o r t h e i n f l u e n c e o f fi b re a n i s o t r o p y , i t is c le a r t h a t t h e t r a n s v e r s e m o d u l u s o f FRC d e p e n d s m o r e o n t h et r a n s v e r s e m o d u l u s o f t h e f ib r e s E ~ t h a n o n t h e m o r e r e a d i l y m e a s u r e d l o n g i t u d i n a l m o d u l u s E l . T h e u s e o fT a b l e 3 r e q u i r e s a k n o w l e d g e o f EY2 ( o r mo r e p r e c i se l y / ~ I2 / / ~m) a n d t h e d e r i v e d v a l u e s o f E * d i f f e r l i tt l e f r o mt h o s e a p p r o p r i a t e t o a n i s o t r o p i c f ib r e s w i t h v a l u e s o f E l w h i c h d o n o t s a ti s f y e q u a t i o n ( 36 ). A n e x a m p l e i sc o n s i d e r e d i n S e c t io n 6 .

6 . E x a m p l e sT h e f o l l o w i n g e x a m p l e s i l l u st r a t e a c c u r a t e m e t h o d s o f i n t e r p o l a t i o n i n t h e t a b l e s o f e la s t ic m o d u l i r a ti o s , t h e

i n f l u e n c e o f f ib r e a n i s o t r o p y a n d t h e u s e o f t h e s l i ci n g t e c h n i q u e t o i n v e s ti g a t e a n F R C c o n t a i n i n g t w o d i f f e r e n tt y p e s o f f i b r e .( 1 ) D e t e r m i n e G*2/G m f o r a n F R C i n w h i c h v = 0 .6 5 a n d Gr/G '' ( = 1 / ~ /) = 5 0 .

E q u a t i o n ( 26 ) s u g g e st s th a t t h e f o l l o w i n g g r o u p o f t e r m s

h e r e a f t e r d e n o t e d b y ~b( r/) , w i l l v a r y s m o o t h l y w i t h "0. A s i mp l e c a l c u l a t i o n f r o m T a b l e 1 s h o w s t h a t a s r /a s s u m e s s u c c e s s i v e v a l u e s o f 1 / 6 , 1 / 1 2 , 1 / 2 0 , 1 / 1 2 0 , 0 , th e f u n c t i o n qb (~ ) e q u a l s 1 . 0 3 8 , 1 . 0 2 4 , 1 . 0 2 0 , 1 . 0 1 4 ,1 . 0 1 4 r e s p e c t i v e l y . A p l o t o f q b( ~/) a g a i n s t r / i n d i c a t e s t h a t q ) ( 1 / 5 0 ) = 1 . 0 1 6 , a n d h e n c e

G~*2 _ 1. 01 6~1 - - ~ v J 5 0

= 4 . 3 8 .( 2) D e t e r m i n e t h e i n f l u e n c e o f f i b re a n i s o t r o p y o n th e t r a n s v e r s e m o d u l u s o f a n F R C i n w h i c h

v = 0 . 6 5 , / ~ / / ~ m = 2 0 , u = 0 . 2 5 .


11/20

I f E t l / E " = / ~ t 2 / / ~ " t h e t r a n s v e r s e m o d u l u s i s g i v e n b y T a b l e 3 :E * / E ~ = 4 . 0 7 .

H o w e v e r , i f E t / E " = k U ~ / f f , , , , s a y , w h e r e k i s a m e a s u r e o f t h e f i b r e a n i s o t r o p y , i t m a y b e s h o w n f r o me q u a t i o n ( 3 4 ) t h a t a s k i n c r e a s e s f r o m 1 t o ~ t h e r a t i o E * / E " i n c r e a s e s f r o m 4 . 0 7 t o 4 . 1 4 , a n i n c r e a s e o f le s st h a n 2 p e r c e n t .( 3 ) ' D e t e r m i n e G * 2 / G " f o r a n F R C c o n t a i n i n g f ib r e s w i th d i f f e ri n g m o d u l i a n d v o l u m e f r a c t io n s , i d e n t if i e db y s u f f ic e s 1 a n d 2 , s p e c i f i e d b y

a n d

v ,, = 0 . 3 51vrt = ~vt2 ,

G~I ' / G " = 6G f 2 / G m = 50.

( s o t h a t vr~ + vt2 = 0 . 6 5 ) ,

T h e f i b r e s a r e o f t h e s a me s i z e a n d i n a n h e x a g o n a l a r r a y a s s h o w n i n F i g . 7 .A p p l i c a t i o n o f t h e s l i ci n g t e c h n i q u e o v e r t h e r e p e a t i n g r e c t a n g l e s h o w n d o t t e d i n F i g . 7 sh o w s t h a t o v e r t h er a n g e 0 < z < h t h e F R C h a s a l o n g i tu d i n a l s h e a r m o d u l u s a p p r o p r i a t e t o t h a t d e t e r m i n e d i n e x a m p l e (1 ):

m[ G , z / G ]o


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7 . Co n c l u s i o n sA simple analytical technique has been presented for obtaining lower bounds to the effective elastic moduliof unidirectional fibre reinforced composite (FRC) with any repetitive array of fibres. The unidirectional FRCis envisaged as being sliced along all 1,2 planes so that conditions of plane stress exist throughout and thestiffness properties of each vanishingly thin slice are readily determined in terms of the constituen t propertiesof the fibre and matrix. The effective elastic moduli of the unidirectional FRC are obtained by suitableintegration of such slice properties over a repeating fibre/ma trix pattern. The lower bound for the transversemodulus is shown to depend on a similar function of the section properties as does the longitudinal shearmodulus. Comparison with known accurate values of the longitudinal shear modulus shows that the technique

under-est imates the modulus by factors of about 1.1 for the square array and 1-2 for the hexagonal array,depending on the fibre volume fraction and fibre/matr ix stiffness ratio. By judicious use of such correctionfactors it should be possible to estimate the longitudinal shear modulus over a wide range of design parametersto within about 5 per cent; for the square and hexagonal arrays Symm's accurate but limited values have beenaugmented by new results which have an accuracy of 0.3 per cent. New results are also presented for thetransverse modulus.Acknowledgment

The author is grateful to Miss Doreen Best for the computation.

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c . ( z )EP~GhKr .S * ( z )12W

X , y , Zy , , , Z .

L I ST O F S Y M B O L S

C h o r d l e n g t h o f f i b re c u t b y s e c t i o n a t zY o u n g ' s m o d u l u sP l a n e s t r a in m o d u l u s , s e e e q u a t i o n s ( 1 7 ) , ( 2 0)S h e a r m o d u l u sH e i g h t o f r e p e a t i n g r e c t a n g l e i n f ib r e a r r a yC o r r e c t i o n f a c t o r s f o r lo n g i tu d i n a l s h e a r m o d u l u sR a d i u s o f n t h f i br eD e f i n e d b y e q u a t i o n ( 5) o r ( 2 1 )V o l u m e f r a c t i o nW i d t h o f r e p e a t i n g r e c t a n g l e i n f ib r e a r r a yC a r t e s i a n c o o r d i n a t e s , 0 x i n f i b r e d i r e c t i o n , 0 y , 0 z p a r a l l e l t o w , hC o o r d i n a t e s o f c e n t r e o f n t h f ib r e o f c i r c u la r s e c t io n

3 ' S h e a r s t r a i ne D i r e c t s t r a i n ( ~ ) I n t r o d u c e d i n S e c t i o n 6r l, r f G " i G I , F , m l E ,~ r e s p e c t i v e l yv P o i s s o n ' s r a t i oo - D i r e c t s t r e s sr S h e a r s t r e s s1 f o. . . > - . . . d zS u f f ic e s o r i n d i c e s m , f r e f e r t o m a t r i x o r f i b r eS u f f ic e s l , 2 , 3 r e f e r t o c a r t e s i a n a x e s x , y, z r e s p e c t i v e l yA s t e r i s k * r e f e r s t o a v e r a g e o r e f f e c t i v e v a l u e s

i 2


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No. Author(s)R E F E R E N C E S

Title, etc.

1 H . L . C o x . . . . . . . . . . . . . . . . . . . . . . . .

2 C . H . C h e n . . . . . . . . . . . . . . . . . . . . . . .

3 C . H . C h e n a n d S . C h e n g . . . . . . . . . . .

4 M . D . H e a t on . . . . . . . . . . . . . . . . . . . . .

5 G . P i ck e tt . . . . . . . . . . . . . . . . . . . . . . . .

6 G . T . S y m m . . . . . . . . . . . . . . . . . . . . . .

7 G . P . S e n d e c k y j . . . . . . . . . . . . . . . . . . .

8 Z . H a s h i n a n d B . W . R o s e n . . . . . . . . .

E l a s t i c it y a n d s t r e n g t h o f p a p e r a n d o t h e r f i b ro u s m a t e r i a l s .Brit. J. Appl. Phys., 3 , 7 2 ( 1 9 5 2 )F i b r e - r e i n f o r c e d c o m p o s i t e s u n d e r l o n g i t u d i n a l s h e a r l o a d i n g .J. Appl. Mech., 3 7 , 1 9 8 ( 1 9 7 0 ) .M e c h a n i c a l p r o p e r t i e s o f f ib r e - r e i n f o r c e d c o m p o s i t e s.J. C ompo site M aterials, 1 , 3 0 ( 1 9 6 7 ) .A c a l c u l a t i o n o f t h e e l a s t i c c o n s t a n t s o f a u n i d i r e c t i o n a l f ib r e -r e i n f o r c e d c o m p o s i t e .Brit. J. Appl. Phys. Se t 2 , 1 , 1 0 3 9 ( 1 9 6 8 ) .E l a s t i c m o d u l i o f f i b r e - r e i n f o r c e d p l a s t i c c o m p o s i t e s i n :Fu nda me ntal aspects of fibre-reinforced plastic composites.( S c h w a r t z & S c h w a r t z , e d s .) I n t e r s c i e n c e P u b l is h e r s , N e w Y o r k( 1 9 6 8 ) .T h e l o n g i t u d i n a l s h e a r m o d u l u s o f a

c o m p o s i t e .J. C omposite M aterials, 4 , 4 2 6 ( 1 9 7 0 )u n i d i r e c t i o n a l f i b r o u s

L o n g i t u d i n a l s h e a r d e f o r m a t i o n o f c o m p o s i t e s - - e f f e c ti v e s h e a rm o d u l u s .J. Co mposite Ma terials, 4, 500 ( 1 9 7 0 ) .T h e e l a s ti c m o d u l i o f f ib r e - r e i n f o r c e d m a t e r i a ls .J. Appl. Mech., 3 1 , 2 2 3 ( 1 9 6 4 ) .

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TABLE 1V a l ues o f G*2/Gm fo r Hex a g o n a l A rra y

v t G t / G ~ = 6 12 20 120 oo

0-40 1.80 2"02 2.14 2-30 2"330.45 1.95 2.23 2"37 2.59 2.640.50 2.11 2.47 2-65 2"94 3-000"55 2-30 2"75 2-99 3"37 3.460-60 2"50 3'08 3"39 3.91 4"030-65 2.74 3"47 3"89 4.61 4"780.70 3"02 3"95 4-53 5"55 5"810.75 3"34 4.57 5"38 6"93 7.350-80 3.73 5.39 6"60 9.18 9"960.85 4-22 6.58 8"54 13.70 15.700.90 4.87 8.63 12.80 35"3? 56.2?

TABLE 2V a l ues o f G*2/Gm o r Squa re A rra y

v t G Q G m = 6 12 20 120 oo0.40 1-81 2.03 2.15 2-31 2.350.45 1.96 2.24 2.39 2.62 2.670.50 2.13 2.50 2.70 3-00 3.080.55 2.33 2.81 3-08 3.51 3.610.60 2-56 3.19 3.56 4-19 4.350.65 2.83 3.69 4.23 5.20 5.460.70 3.17 4.38 5.21 6.93 7.430.75 3-62 5.47 7.00 11.20 12.80

TABLE 3V a l ues o [ E * / E m o r Hex a g o n a l A rra y (v = 0 . 2 5 )

E ~ / E m =6 12 20 120 eo

0.40 1.84 2.10 2.25 2.45 2.490.45 2.00 2-32 2.49 2.76 2.820.50 2.16 2.57 2.78 3.12 3.200.55 2.35 2.86 3.13 3.58 3.690.60 2.56 3.20 3.55 4.15 4.300-65 2.80 3.60 4.07 4.90 5.100-70 3.09 4.08 4.73 5.88 6.200.75 3.40 4.71 5.62 7.36 7~840-80 3.80 5.56 6.85 9.71 10.600.85 4-27 6.73 8.77 14.50 16.800.90 4.90 8.75 13-00 36.9? 60.0?

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Repe af, n~

S

r e c t a n g l e w x n / - ' 5 "

~ . . ~ , " . . 6 o : . _ " o o o _

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Y

FIG. 1 . D ia g r a m show ing no ta t i on

i ~ ~ ! ! ~ % i i i i i l i ~ i ! i i l l ' ! ! i i i : : : i l ! ] i i ! : i : i : : : : : : : ! ! i i ! ! : i : i :: : : ; : : : i ! ! : i : i : : : : : : :: : : : : i i : : :: : : :

" " : ' : ' : ' :' : ' : ' " ' " " " ' " " " " " " " " " " " : ' : ': ' : ' : ' : ': ' : ' : ; : ': ' : ' : ' " : ' " : ' ~ ' i ' " : ': " " " " " ' " " " " " " " ' " " ' " " " ' : ' : ' : ': " " : ' : ' ~ ' ~ m : : ' " " " ' " " " ' " " ' ' " " ' " " ' " ' " " ' " "

~ : ~ : ~ : ! : ~ : ! : ~ i ~ i : ~ : ? : i : i: i : i: i : i: ! : ~ : i : :i ~ : : ~ : i : ! :: : ! : ! :i : i: : ~ : ! : ~ : ~ : i : ~ i ~ i : i ~ : ~ : i : ~ : i ~ ? i : ! : i: ! ~ ! : i : : :i : i: ~ : ? : i ~ i

~ > a , * ( z )- - - - a , m ( z )- - - o r / (z )

FIG. 2 . S t resse s ac t ing on thin s l ice

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Z ,

19

v

YFIG. 3 . Gen era l r epea t ing r ec t ang le

F IG. 4 . Rep ea t ing r ec t ang le fo r hexagona l a r r ay

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1"4 o 0f 120 /

~--'~"m 2 0 /G 1 2 I - " - -1"3

K

1"3

i .

I'1

I-0 I I I I0 " 4 0 " 5 0 " 6 0 - 7 0 " 8 0 . 9vFIG. 5 . C or re c t i on fa c t o r s fo r he xa go na l a r ra y

K

l 'i z o ~ .f /2,o G--- /I - IO

I .O5\ \

I I I.0 0 " 4 O - S 0 - 6 0 - 7 0 - 8v

FIG. 6 . C or re c t i on fa c to r s fo r squa re a r ra y

17


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D u

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F I G . 7 . F i b r e s w i t h d i f f er i n g m o d u l i in h e x a g o n a l a r r a y

Pr inted in En gland for H er M ajes ty ' s Sta t ionery Office by J . W . Arrow smi th Ltd . , Br is to l BS3 2NTDd. 289031 K5 2/76

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G o v e r n m e n t B o o k s h o p s49 High Holborn, London WCIV 6HB13a Castle Street, Edinburgh EH2 3AR

41 The Hayes. CardiffCF1 IJWBrazennose Street, Manchester M60 8AS

Southey House, Wine Street, Bristol BSI 2BQ258 Broad Street, Birmingham B1 2HE80 Chichester Street, Belfast BTI 4JY

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