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M i c r o m e c h a n i c s o f c r a ck b r i d g in g i n fi b r e - r e in f o r c e d c o n c r e t eV. C. LIAdvanced C ivil Engineering Ma terials Research Lab oratory, Departm ent of Civil and Environm ental Engineering,University of Michigan, Ann Arbor, MI 48109-2125, USAH. STANG, H. KRENCHELDepartment of Structural Engineering, Technical University of Denm ark, DK 2800, L yngby, Denm arkThe s tress-crack width relationship has been determined experim ental ly for concretesreinforced with two typ es o f f ibres , s teel and polypropylene, of various f ibre volume fract ions.A micromechanics-based theoretical mod el is prop osed which captures the essential fea tur es o fthe s tress-cra ck width relat ionships at small crack widths (less than 0.3 mm).Microm echanisms accounted for include the bridging act ions due to aggregates and f ibres ,Co ok- Go rdo n interface debonding and fibr e pre-stress. The fibr e bridging action involvesinterfilce slip-dependent friction as well as snubbing fric tion fo r fibres bridging a t inclinedangles. Theoretical predictions based on independent para me tric inputs compa re fav ou rab lywith experimental measurements of the s tress- crack width relat ionship. Findings in thisresearch provide confidence in the use of the propo sed mo del fo r m aterials engineeringtargeted at prescribed structural performance.

1 . I N T R O D U C T I O NC r a c k w i d t h c o n t r o l is o f p r i m a r y i m p o r t a n c e i n m a n yr e i n f o r c e d c o n c r e t e s t r u c t u r e s , s i n c e i t i s b e l i e v e d t h a tt h e r e is a c lo s e re l a t io n s h i p b e t w e e n m e a n o r m a x i m u mc r a c k w i d t h s a n d t h e d u r a b i l i t y o f t h e s t r u c t u r e .F u r t h e r m o r e , w h e n t h e c o n c r e t e s t r u c t u r e i n q u e s t i o na c t s a s a c o n t a i n m e n t v e s s e l f o r f l u i d s , l e a k a g e m u s t b em i n i m i z e d o r m i g h t n o t b e t o l e r a t e d a t a l l . C o n s e q u e n t l y ,i n th e s e r v i c e a b i li t y l i m i t s t a t e a m e a n o r m a x i m u m c r a c kw i d t h l e s s t h a n a b o u t 0 . 1 - 0 . 3 m m i s u s u a l l y p r e s c r i b e d .

I t h a s b e e n f o u n d [ I ] t h a t t h e r e i s a c l o s e r e l a t i o n s h i pb e t w e e n t h e t en s il e lo a d c a p a c i t y o f t h e c o n c r e t e c r a c k s( t h e s o - c a l l e d s t r e s s - c r a c k w i d t h r e l a t i o n s h i p ) a n d t h ec r a c k w i d t h s f o r a g i v e n l o a d i n g o n a g i v e n c o n c r e t es t r u c t u r e . F u r t h e r m o r e , t h e u l t i m a t e l o a d - c a r r y i n g c a -p a c i t y o f a c r a c k e d u n r e i n f o r c e d c o n c r e t e s t r u c t u r e w a sf o u n d t o d e p e n d o n t h e t e n s i l e l o a d c a p a c i t y a c r o s s t h ecracks , e spec ia l ly a t smal l c rac k width s (e .g . [2] ) . Fo rt h e s e r e a s o n s t h e s t r e s s - c r a c k w i d t h r e l a t i o n s h i p f o rs m a l l c r a c k w i d t h s ( l es s t h a n a b o u t 0 . 1 - 0 . 3 m m ) i s o ff u n d a m e n t a l i m p o r t a n c e i n c o n c r e t e s t r u c t u r e d e s i g n i nb o t h s e r v i c e a b i li t y a n d u l t i m a t e l i m i t st a t e . F o r e x a m p l e ,c o n t r o l o f t h e s t re s s - c r a c k w i d t h r e l a t i o n s h i p c a n r e d u c et h e a m o u n t o f c o n v e n t io n a l r e i n f o r ce m e n t a n d t h e r ef o r er e d u c e l a b o u r c o s t s f o r a g i v e n a c c e p t a b l e c r a c k w i d t h[33.I n p l a i n c o n c r e t e , c r a c k b r i d g i n g i s p r o v i d e d b ya g g r e g a t e l o c k i n g a c t i o n . A l t h o u g h t h e d e t a i l e d p h y s i c sh a s n o t b e e n fu l ly u n d e r s t o o d i n q u a n t i t a t i v e t e rm s ,a g g r e g a t e b r i d g i n g a c t i o n c a n b e w e l l r e p r e s e n t e da n a l y t i c a l l y [ 3 ] . W h e n f i b r e s a r e a d d e d t o c o n c r e t e , a na d d i t i o n a l b r i d g i n g a c t i o n i s b r o u g h t i n t o e f fe c t, s u p e r -i m p o s i n g o n t h e a g g r e g a t e b r i d g i n g e ff ec t. T h e c o m b i n e d0025-5432/93 ~ RILE M

b r i d g i n g a c t io n i n a f i b r e -r e i n f o r c e d c o n c r e t e ( F R C ) c a nb e v e r y b e n e f i ci a l t o i n c r e a s e t h e s t r es s c a r r i e d a c r o s s t h ec r a ck . I n a d d i t i o n , f i b re s h a v e b e e n a d d e d t o c o n c r e t e t oc r e a t e p s e u d o - s t r a i n - h a r d e n i n g [ 4 , 5 ] . T h e l o a d t r a n s f e rc a p a c i t y a c r o s s a m a t r i x c r a c k i s c r i t i c a l i n d e t e r m i n i n gt h e a m o u n t o f fi b re s n e c e s sa r y i n a c h i e v i n g p s e u d o - s t r a i n -h a r d e n i n g .I n t h i s p a p e r , t h e p o s t - c r a c k i n g s t r e s s - c r a c k w i d t hr e l a t io n s h i p ( a r o f t h e c o m p o s i t e is s t u d i e d f r o m am i c r o m e c h a n i c s p o i n t o f v ie w , p a r t i c u l a r l y w i t h r e f e r e n c et o f i b r e b r i d g i n g a c t i o n c o m b i n e d w i t h t h e w e l l - k n o w nC o o k - G o r d o n d e b o n d i n g e f f ec t [ 6 ] . I n a d d i t io n , f ib r ep r e - s tr e s si n g a n d s l i p - d e p e n d e n t i n t e r f a ci a l b o n d s t r e n g t ha r e a c c o u n t e d f o r i n t h e m o d e l . A n a l y t i c a l r e s u l t s a r eu s e d t o p r e d i c t s t r e s s - c r a c k w i d t h r e l a t io n s h i p s f o r s t e elf i b re a n d p o l y p r o p y l e n e f i b r e r e i n f o r c e d c o n c r e t e s , f o rv a r i o u s fi b r e v o l u m e fr a c t io n s . C o m p a r i s o n s b e t w e e nm o d e l p r e d ic t io n s a n d e x p e r i m e n t a l ly m e a s u r e d a c - wr e l a ti o n s c o n f i r m m a n y o f t h e s a l ie n t f e a t u r e s i n t h e a c - wc u r v e s a s s o c i a te d w i t h t h e v a r i o u s m i e r o m e c h a n i s m s . Ac o m p a n i o n p a p e r [ 7- ] w il l d e s c r i b e t h e u s e o f t h e p r o p o s e da n a l y t i c a l m o d e l i n d e si g n a n d s t r u c t u r a l a p p l i c a t i o n s o ff i b r e - r e i n f o r c e d c o n c r e t e .

2 . C R A C K B R I D G I N G M O D E L2. 1 Aggregate br idg ing ac t ionT h e p o s t - p e a k t e n s i o n - s o f t e n i n g o f c o n c r e t e h a s b e e nm o d e l l e d a s t h e c o a l e s c e n c e o f r e g u l a r l y s p a c e d m i c r o -c r a c k s [ 8 - 1 0 ] . A l t h o u g h t h e s e m o d e l s a r e c o n v e n i e n t t ou s e, th e l a ck o f c o r r e s p o n d e n c e b e t w e e n t h e r e a l p h y s i c a ls t r u c t u r e o f c o n c r e t e a n d t h a t o f t h e m o d e l m a k e s i td i ff ic u l t t o d e t e r m i n e p h y s i c a l p a r a m e t e r s . F o r l a c k o f a

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I O0M o d e l f o r a g g r e g a t e b r i d g i n g

b0.80 ~, ,r= 5 . 0 M Po

cb ~ a u- , . M Po0 6 0 ~ a

oo0 . 4 0 o ~o - o ~ , ~ - - ' ~ ~ ~

~ ~ ~ li .o o0 . 2 0 - O o o ~ ~ ~P l a i n c o n c r e t e

0 . 0 0 . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . . , . . . . . . . . . . . . . . . . . . .O . O O E + O 2 . 0 0 E - 5 4,00E-5 6 . 0 0 E - 5 8 . 0 0 E - 5

w ( m )Fig. 1 Post-crack G-w relat ion for unreinforced concrete(maximum aggregate s ize = 8 mm ) shown to gether with afitted curve based on Equation 1. Experimental data includeboth the normal-s t rength concrete used in the polypropyleneFR C series (a~ = 3.4 M Pa) a nd the high-strength con crete(cr~ = 5.0 M Pa) used in the steel FR C series.

g o o d p h y s i c a l m o d e l , w e s h a l l a d o p t h e r e a n e m p i r i c a lm o d e l p r o p o s e d b y S t a n g [ 1 ] a n d w h i c h f it s a w i d e ra n g eo f e x p e r i m e n t a l d a t a e x t r e m e l y w el l. I n t h i s m o d e l t h ea g g r e g a t e b r i d g i n g s t r e s s o- i s e x p r e s s e d a s a f u n c t i o n o ft h e c r a c k o p e n i n g w :

uO"m~ r ~ - ( 1 )1 + ( W / W o )w h e r e a~m s th e m a x i m u m b r i d g i n g s t re s s d u e t o a g g r e g a t ea c t i o n a t w = 0 . T h e p a r a m e t e r p d e s c r ib e s t h e s h a p e o ft h e s o f t e n i n g p r o c e s s w i t h i n c r e a s i n g c r a c k o p e n i n g , a n dh a s b e e n d e t e r m i n e d t o b e c lo s e to u n i t y f o r m o s t c o n c r e t et e st e d t o d a t e . T h e p a r a m e t e r w o c o r r e s p o n d s t o t h e c r a c ko p e n i n g w h e n t h e s tr e ss h a s d r o p p e d t o h a l f o f c ry .E q u a t i o n 1 is s h o w n i n F i g . 1 w i t h p = 1 .2 a n dwo = 0 .0 1 5 r a m . T w o c o n c r e t e t y p e s w i t h a m a x i m u ma g g r e g a t e s i ze o f 8 m m w e r e u s e d i n t h e f i b r e - r e i n f o r c e dc o n c r e te s . T h e o n e r e i n f o r c e d w it h p o l y p r o p y l e n e f i b re sis a n o r m a l - s t r e n g t h c o n c r e t e w i t h ~,~ = 3 .4 M P a , w h e r e a st h e o n e r e i n f o r c e d w i t h s t e e l f i b r e s i s a h i g h - s t r e n g t hc o n c r e t e w i t h a ~ = 5 .0 M P a . E q u a t i o n 1 i s s h o w n t od e s c r i b e t h e t w o c o n c r e t e s v e r y w e l l , a t l e a s t f o r t h er a n g e o f w i n d i c a t e d ( w < 0 . 0 7 r a m ) .

2 . 2 F i b r e b r id g i n g a c t i o nB a s e d o n t h e c o n c e p t o f d e b o n d i n g a g a i n s t a f r i c ti o n a ls t r e n g t h o f r a n d o n t h e c o n c e p t o f a n i n c l in e d f i b rea c t i n g a s a f l e x i b le r o p e p a s s i n g o v e r a f r i c t i o n a l p u l l e ya g a i n s t a s n u b b i n g c o e f f i c i e n t f [ 1 1 ] , L i [ 1 2 ] d e r i v e d t h ef i b r e b r i d g i n g s t r e s s b y i n t e g r a t i n g t h e i n d i v i d u a lc o n t r i b u t i o n o f f ib r es l o c a t e d a t v a r i o u s c e n t r o i d a ld i s t a n c e s ( z ) f r o m t h e m a t r i x c r a c k a n d a t v a r i o u so r i e n t a t i o n s ( ~b ) r e l a t i v e t o t h e t e n s il e l o a d i n g d i r e c t i o n .F o r a c o m p o s i t e w i t h f i b r e v o l u m e f r a c t i o n Vf o f f i b r eso f le n g t h L f a n d d i a m e t e r d r , t h e b r i d g i n g s t re s s o f m a y

b e e x p r e s s e d a s a f u n c t i o n o f c r a c k i n g o p e n i n g 3 :4 V f F ~ 1 2 ~ l L r , ' 2 ) c ~ 1 6 2= - - P ( 6 ) p (q S ) p (z ) dz dq5 (2)O ' f ( 6 ) ~ d ? d 4 b = O d z=O

w h e r e P ( 6 ) i s t h e b r i d g i n g f o r c e e x e r t e d b y a s i n g l e f ib r ew i t h a n e m b e d d e d l e n g t h ( t h e s h o r t e r o f t h e t w o o n e i t h e rs i d e o f t h e m a t r i x c r a c k ) o f l a t a n o r i e n t a t i o n a n g l e ~ :P ( 5 ) = ~ [ (1 + r / ) E f d ~ 5 ] v 2 e i *

P ( 6 ) = x z l d f ( e 3 -_ 1 ~ ) o ) e S 4 ,

P ( 6 ) = 0

f o r b < 3 o ( 3 a)

fo r 3o < 3

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4 8 8 L i , S t a n g a n d K r e n c h e l

A c o n v e n i e n t p o l y n o m i a l f o r m h a s b e e n e m p l o y e dp r e v i o u s ly a n d a p p e a r s t o d e s c ri b e t h e e x p e r i m e n t a l d a t af o r f i b r e p u l l - o u t r e a s o n a b l y w e l l :

r ( 3 ' ) = r o + a t 3 ' + a 2 ( 3 ' ) 2 (7 )T h e c o n s t a n t s " co , a l a n d a 2 m u s t b e e x p e r i m e n t a l l yd e t e r m i n e d f o r e a c h s p e c if ic c o m b i n a t i o n o f f ib r e t y p e ,m a t r i x , a n d p r o c e s s i n g c o n d i t i o n s . B e c a u s e t h e l o c a ls l ip p a g e p r i o r t o c o m p l e t e d e b o n d i n g is t y p ic a l ly v e r ys m a l l , i t w o u l d b e u s e f u l t o s i m p l i f y E q u a t i o n 7 b ye q u a t i n g 3 t o 6 ' , s o t h a t t h e l o c a l s l i p p a g e i s a s s u m e d t ob e e q u a l t o t h e f i b r e e n d s l i p p a g e a f t e r c o m p l e t ed e b o n d i n g . T h u s E q u a t i o n 7 s h o u l d b e r e w r i t t e n a s

"c(5) = Zo fo r 6 < 5" ( 8 )"C(3 ) = "CO + a 1 3 + a 2 2 f o r 3 > 6 "

W e n o w c o n s i d e r t h e a d d i t i o n a l b r i d g i n g c o m p l i a n c ea s a re s ul t o f t h e C o o k - G o r d o n e ff ec t. C o o k a n d G o r d o n[ 6 ] p r e d i c t e d t h a t a c r a c k o f f i n it e r o o t r a d i u s i n a n e l a s ti cs o l i d u n d e r r e m o t e t e n s i l e l o a d w i l l c r e a t e a c r a c k t i ps t re s s fi e ld w i th a c r a c k - p l a n e - p a r a l l e l t e n s i le c o m p o n e n tw h i c h r e a c h e s a m a x i m u m a t a d i s t an c e o f th e r a d i u s o ft h e c r a c k t i p . T h u s a m a t r i x c r a c k a p p r o a c h i n g a ni s o l a t e d f i b r e c a n c a u s e i n t e r f a c e d e b o n d i n g b e f o r e t h e

(a )

?w = 3 + ~ c g

(b )Fig. 2 The C oo k- G or do n effect (a) induces f ibr e-m atrixseparation due to the tensile stress in the horizontal directionassociated with the elastic crack tip field of the appr oac hin gmatrix crack, and (b) leads to an addi t ional crack openingr du e to elastic stretching of the fibre segmen t ~ in additi onto that 3 associated with interface frictional debonding.

c r a c k t i p r e a c h e s t h e f i b r e - m a t r i x i n t er f a c e if t h ei n t e r f a c i a l s t r e n g t h i s a d e q u a t e l y w e a k . T h i s C o o k -G o r d o n e f f e c t is i ll u s t r a t e d s c h e m a t i c a l l y i n F i g . 2 .I n t e r f a c i a l d e b o n d i n g h a s b e e n i d e n t i f i e d b y S E Mo b s e r v a t i o n s i n f i b r e - r e i n f o r c e d c e r a m i c s t o a c t a s l o c a le v e n t s p r e c u r s o r y t o f i rs t c r a c k i n g [ 16 ]. D i r e c t o b s e r v a t i o n[ 1 7, 18 ] o f t h e C o o k - G o r d o n e ff ec t b y m e a n s o f in situS E M , i n a s te e l f ib r e - r e i n f o r c e d c e m e n t p a s t e , s u g g e s t st h a t t h e d e b o n d z o n e c a n e x t e n d f o r 2 0 0 g m t o o v e r1 00 0 l am , a n d is l ik e ly t o d e p e n d o n t h e i n h o m o g e n e i t yo f t h e i n t e rf a c i a l m i c r o s t r u c t u r e . A s a c o n s e q u e n c e , w ee x p e c t e d a n a d d i t i o n a l d i s p l a c e m e n t 6 cg r e l a te d t o t h ee l a s ti c s t r e t c h i n g o f t h e d e b o n d e d l e n g t h ~ , h e r e a f t e rl ab e ll ed a s t h e C o o k - G o r d o n p a r a m e t e r . F o r a s in g lef i b r e , t h e a d d i t i o n a l d i s p l a c e m e n t m a y b e e s t i m a t e d a sfo l lows :

4 ~ P3cg - - 7zdgEf (9 )

T h i s P - 3 r e l a t i o n s h i p f o r a s i n g le f i b r e m a y b e s u b s t i t u t e di n t o E q u a t i o n 2 t o o b t a i n

4c~5~g - crf (1 0)VfEfs o t h a t t h e t o t a l c r a c k w i d t h w is a p p r o x i m a t e l y g i v e n a s

w = 3 + 3r (11)w i t h 5 r e l a t e d t o a t b y E q u a t i o n s 4 a n d 6 , a n d 6 c g g i v e nb y E q u a t i o n 1 0 a b o v e . T h i s p r o c e d u r e a l l o w s u s t oc a l c u l a t e t h e f i b r e b r i d g i n g s t r e s s a f i n d i r e c t l y a s af u n c t i o n o f t h e t o t a l c r a c k w i d t h w .

F i n a l ly , it s h o u l d b e r e c o g n i z e d t h a t a t t h e f o r m a t i o no f a m a t r i x c r a c k , a n d p r i o r t o a n y c r a c k o p e n i n g , t h ef i b r e i s a l r e a d y i n a s t r e s s e d s t a t e . T h e l e v e l o fp r e - st r e ss i n g t r~ m a y b e e s t im a t e d f r o m t h e l o a d - s h a r i n gb e t w e e n f i b r e a n d m a t r i x a t t h e m a t r i x c r a c k i n g s t r a i n .T h i s l e a d s t o [ 1 9 ]

0a p s = r l o ? h e r n u E f V f (12)w h e r e e mu i s t h e m a t r i x c r a c k i n g s t r a i n , t y p i c a l l y a r o u n d0 .0 2~ o fo r c e m e n t i t i o u s m a t e r i a l s , a n d q o a n d q l a r eo r i e n t a t i o n a n d l e n g t h e f f i c i e n c y f a c t o r s , r e sp e c t i v e l y .F o r a t h r e e -d i m e n s i o n al u n i f o r m r a n d o m d i s tr i b u ti o n o ff i b re s , C o x [ 2 0 ] f o u n d t h a t rt0 = 1 / 6 f o r u n c o n s t r a i n e dd e f o r m a t i o n , a n d K r e n c h e l [ 2 1 ] f o u n d t h a t q 0 = 1 / 5 f o rt h e c a se w h e n d e f o r m a t i o n is c o n s t r a i n e d t o t h e d i r e c t i o no f a p p l i e d t e n s il e lo a d i n g . F o r t h e p r e s e n t p u r p o s e , s i n cew e i n te n d t o u s e E q u a t i o n 1 2 f o r t h e p o s t - c r a c k c o n t r i -b u t i o n o f f i b r es , a n d s i n c e i n t h i s s t a g e a l l b r i d g i n gf i b r e s e g m e n t s c r o s s i n g t h e m a t r i x c r a c k w i l l b er e o r i e n t a t e d a l o n g t h e l o a d d i r e c t i o n , i t w o u l d s e e ma p p r o p r i a t e t o e m p l o y r /0 = I . F o r t h e e f f i c i en c y f a c t o rr e l a t e d t o fi n i te f i b r e l e n g t h , b a s e d o n t h e a s s u m p t i o n o fa n e l a s ti c b o n d , L a w s [ 2 2 ] f o u n d t h a t ~h m a y b e e x p r e s s e di n t h e f o l l o w i n g f o r m :

emu Ef dfrh = 1 (13)4Lr ' c o

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M a t e r i a l s a n d S t r u c t u r e s 4 8 9

Fibre pres t ress ing may be expec ted to be reduced by theinterfacial debonding process, and the prestress shouldbe complete ly el iminated w hen the f ibre is fully deb ond ed.This occurs when b has reached 3*. As a f irst approxi-mation, we assume that the prestress level diminisheslinear ly with crack opening, so that

0 *ap~(w) = ap~(W - w)/w* for w < w* (14)aps(W) = 0 for w > w*where

W* = 3" + 6r = ao) (15)In a f ibre-reinforced concrete , and especial ly for a smallcrack op ening , the aggregate br idging effect and the f ibrebridging effect may be assumed to opera te simultaneously.Wh en the prestressing effect is included, the totalcomposite br idging stress at(w) is then given by

a~(w) = a~(w) + af(w) + aps(w) (16)with aa(w) as given by Equa t io n 1 and a f (w) and pps(W)as just descr ibed. Eq uatio n 16 is used to pred ictcompo si te br idging s t ress as a func t ion of c rack op eningfor two families of FRCs to be descr ibed below.

3. E X P E R I M E N T A L D E T E R M I N A T I O N O FS T R E S S - C R A C K W I D T H R E L A T I O N S H I P

In order to de te rmine the s t ress-c rack w idth re la t ionshipexper imenta l ly , deformat ion-co nt rol led tens i le test s werecondu c ted on n otched spec imens wi th th ickness 40 mm,width 50 mm and he ight 55 mm. To e l imina te the pre -s t ress ing inevi tably in t roduced in the spec imens whenusing convent iona l gr ips , for improved a l ignment , andfor maximum stif fness, a special specimen f ixture wasdeve loped. This f ix ture cons is t s of two inte rchangeablesteel blocks on to which the specimens were glued. Thes tee l b locks were both f ixed in advance to the f rame a ndthe c rosshead of a 100 kN 6025 Ins t ron tes t ing mach ineequipped for c losed- loop tes ting.

The deformat ion was measured us ing two s tandardInstro n exten som eters ( type 2620-602) with 12.5 mmgauge length mo unte d across each of the two 10 mm deepand 3 mm wide notches . The tes t was co nduc ted asa prescr ibed deformat ion- ra te tes t us ing the averages igna l f rom the extensometer as f eedback. An ini t ia lcross head rate of 1.25 x 10 .3 mm m in - 1 was used. Afterthe peak was reached the rate was slowly increased to6.25 x 10 -2 m m m in -1.

The ex per imenta l se t -up i s shown in F ig . 3 . Technica lde ta il s on tes t ing procedure and type of adhes ion usedcan be found e lsewhere [14] .

The raw da ta cons is ted of t ime , load, d isplacemen treading f rom each extensometer , and average displace-ment . Data were recorded by the tes t ing machine andt ransfe r red to a PC. The c rack opening was ext rac tedf rom the to ta l average displacement ac ross the c rack bysubt rac t ing the to ta l e las t ic and ine las t ic deformat ionouts ide the c rack. This was done by assuming tha t the

Fig. 3 Set-up for the uniaxial tension test of FRC.

mater ia l outs ide the c rack unloaded l inear ly wi th ast if fness corresponding to the ini t ial st if fness observed inuniaxial tension (see e .g. [23,24]) .

The s t ress-c rack width re la t ionship was de te rminedexper imen ta l ly on two se ts of FR C mater ial s : poly-propylene f ibres in normal- s t rength concre te and s tee lf ibres in high-stren gth concrete . Details of recipes, mixingprocedures and cur ing can be found e lsewhere for thepolyp ropy lene f ibre FRC [25] and the steel f ibre FRC[26] . The max imum aggrega te s ize was 8 mm in bothconcretes.

4. C O M P A R I S O N B E T W E E N M O D E LP R E D I C T I O N A N D E X P E R I M E N T A L D A TA

4.1 Independently measured or deduced parametersThe aggrega te br idging pa rameter s p = 1 .2 and wo =0.015 mm have been obta ined by f i tt ing Equa t io n 1 toexper imen ta l aa--W curves obta in ed wi th the two concre tematr ices, as shown in Fig. 1. Th e ult im ate agg regatebr idging s t ress a~ , has been obta ined di rec t ly f rom thepeak uniaxia l tens ile load of these matr ices . F or thenorm al- s t rength a~ = 3 .4 MPa and for the high-s t rength

u 5 .0 MP a. The same norm al- s t rength con-oncre te am =cre te mat r ix has been used in a l l the polypropylene FRCsamples , and the same high-s t rength concre te mat r ix hasbeen used in a l l the s tee l FRC ones . We make theassumpt ion tha t the aggrega te br idging parameter s a re

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4 9 0 L i , S t a n g a n d K r e n c h e l

Table I Fibre and interfacial parametric values used as model inputFibre Fibre param eters Interracial parameterstypes

Ef L f df 1:0 a t a2 f(GPa) ( ram) ( ram) (MPa ) (M Pa ram - i ) (M Pa ram - 1) (mm)Steel 210 25 0.4 4.2 -4 .0 1.0 0.75 6.0Po lypr op ylen e 11.9 12 0.048 0.8 0 0 0 0.72

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

T h e o n l y o t h e r c o n c r e t e m a t r i x p a r a m e t e r n e e d e d i nt h e c a l c u l a t i o n o f f ib r e p r e s tr e s s i n g l e v e l ( E q u a t i o n t 2 )i s th e f a i l u r e s t r a in e m u, w h i c h w e a g a i n a s s u m e n o t t ob e a f f e c t e d b y t h e p r e s e n c e o f t h e l o w v o l u m e o f f i b r e su s e d i n p r e s e n t s e r ie s o f t e st s. A s s u m i n g a l i n e a r r e s p o n s eup to fa i lure , we use he re

uO" m

e mu - - (17)E m

w h e r e E m h a s b e e n d e t e r m i n e d f r o m t h e i n i t i a l s l o p e o ft h e te n s i l e s t r e s s - s t r a i n c u r v e . F o r b o t h m a t r i c e sE m "~" 3 0 G P a , s o t h a t e ,mu f o r t h e n o r m a l - s t r e n g t hc o n c r e t e i s 0.0 13 ') /o a n d t h a t f o r t h e h i g h - s t r e n g t h c o n c r e t eis 0.017%.

A l l f i b r e p a r a m e t e r s E l , L f a n d d~ a r e m e a s u r a b l e w i t hg o o d a c c u r a c y . V a l u e s f o r t h e s e p a r a m e t e r s f o r t h e s t e e lf i b r es a n d p o l y p r o p y l e n e f i b re s u s e d c a n b e f o u n d i nT a b l e 1. W e n o t e h e r e t h a t t h e p o l y p r o p y l e n e fi b re u s e dh a s a r e c t a n g u l a r c r o s s - s e c t i o n , a n d a n e f f e c t iv e f i b r ed i a m e t e r h a s b e e n e m p l o y e d :

2 b tdf - (18)b + t

w h e r e b a n d t a r e t h e w i d t h a n d t h i c k n e s s o f th e f i b r e.E q u a t i o n 1 8 h a s ta k e n i n t o a c c o u n t t h e c h a n g e i n s u r f ac ec o n t a c t a r e a p e r f i br e a n d t h e c h a n g e i n t h e n u m b e r o ff i b r e s d u e t o t h e r e c t a n g u l a r c r o s s - s e c t i o n f o r a g i v e nf i b r e v o l u m e f r a c t i o n .T h e i n t e r r a c i a l p a r a m e t e r s r e q u i r e d i n t h e m o d e lc o n s i s t o f th e i n t e r f a c ia l b o n d s t r e n g t h v a s e x p r e s s e d i nE q u a t i o n 7 , t h e sn u b b i n g c o e f fi c ie n t f , a n d t h e C o o k -G o r d o n p a r a m e t e r ~ . W e d i s c u ss e a c h o f th e s e p a r a m e t e r sin the fo l lowing.

T h e s l i p - d e p e n d e n t b o n d s t r e n g t h h a s b e e n m e a s u r e db y G l a v i n d [ 1 5 ] f o r s te e l f i b r e b y m e a n s o f p u l l - o u t t e s ts .B o t h t h e f i b r e a n d t h e c o n c r e t e m a t r i x w e r e s i m i l a r t ot h o s e u s e d i n th e p r e s e n t e x p e r i m e n t . T h e r e s u l ts r e p o r t e d[ l 5 ] s h o w q u i t e la r g e v a r i a ti o n s i n t h e b o n d p a r a m e t e r sf o r r e la t i v e l y s m a l l v a r i a t i o n s i n m a t r i x c o m p o s i t i o n . T h ev a l u e s u s e d h e r e , a s s h o w n i n T a b l e 1 , a r e o f t h e s a m eo r d e r o f m a g n i t u d e a s th e p a r a m e t e r s r e p o r t e d p r e v i o u s l y[ 1 5 ] , T h e r e a r e n o s i m i l a r m e a s u r e m e n t s a v a i l a b l e f o rt h e p o l y p r o p y l e n e f ib r es , a l t h o u g h s li p h a r d e n i n g h a sb e e n p r e v i o u s l y r e p o r t e d [ 1 3 ] i n a p o l y e t h y l e n e fi b r e.W e h a v e a s s u m e d h e r e t h a t t h e b o n d s t r e n g t h r e m a i n s

c o n s t a n t a t ~ o. F r o m a n i n d e p e n d e n t t e st se r ie s e m p l o y i n gc o n t i n u o u s a l i g n e d f ib r e s o f th e s a m e k i n d , S t a n g [ 2 7 ]i n f e r r e d Vo t o b e i n t h e r a n g e o f 0 .4 t o 1 .8 M P a , b a s e do n t h e b e n d , o v e r p o i n t o f th e u n i a x i a t s t r e s s - s tr a i n c u r v ea n d o n a n a n a l y t i c a l m o d e l o r i g i n a t e d b y A v e s t o n e t a l .[ 2 8 ] . B e c a u s e t h e a l i g n e d f i b r e s p e c i m e n s e m p l o y a ne x t r e m e l y w e l l - p a ck e d n e a t c e m e n t m a t r i x p r o c e s s e d b ya p u l t r u s i o n t e c h n i q u e , w e e x p e c t t h e a c t u a l b o n ds t r e n g th f o r t h e p r e se n t F R C s ( w i t h s h o r t r a n d o m f i b re s )p r o c e s s e d b y c o n v e n t i o n a l v i b r a t i o n c a s t i n g t o b e i n t h el o w e n d o f t h i s r a n g e . A v a l u e o f 0 .8 M P a h a s b e e na d o p t e d f o r u s e i n th e p r e s e n t m o d e l .

T h e s n u b b i n g c o e f f i c i e n t f f o r s t e e l f i b r e s i n a c o n c r e t em a t r i x h a s b e e n s t u d i e d b y L i e t a l . [ 1 1 ] . T h e y i n f e r r e df r o m t h e m a x i m u m p o s t - p e a k s t r e s s o f a t e st s er i e s b yV i s a lv a n i c h a n d N a a m a n [ 2 9 ] t h a t f l ie s s o m e w h e r eb e t w e e n 0 . 5 a n d 1 . 0 . A n a v e r a g e v a l u e o f 0 . 7 5 h a s b e e na d o p t e d f o r t h e p r es e n t m o d e l. N o c o r r e s p o n d i n g d a t ae x i s t f o r p o l y p r o p y l e n e f i b re , a n d f = 0 h a s b e e n a s s u m e dfor th i s ty pe o f f ibre .

F i n al ly , t h e C o o k - G o r d o n p a r a m e t e r ~ p re s e n ts t h eg r e a t e s t u n c e r t a i n t y . I t w o u l d s e e m n o t u n r e a s o n a b l e t og a u g e i t s v a l u e f r o m t h e m a g n i t u d e o f t h e d i s t a n c eb e t w e e n a d j o i n i n g c r a c k s o b s e r v e d t o z i g - z a g a t f i b r es i t e s [ 1 7 ] , w h i c h w o u l d s u g g e s t t h a t e s h o u l d f a l l i n t h er a n g e o f o n e t o s e v e r a l t e n s o f t h e f i b r e d i a m e t e r . B e n t u r[40 ] ha s sugge s ted ~ = 2df - 10dr for s t ee l f ibre in nea tc e m e n t p a s t e. F o r s te e l f i b re i n a m o r e h e t e r o g e n e o u sc o n c r e t e m a t r i x , w e m a y e x p e c t l a r g e r v a l u e s o f ~. W eh a v e a d o p t e d ~ = 1 5d r f o r l a c k o f m o r e p r e c i s e e x p e r i -m e n t a l e v i d en c e ,

4.2 M odel resultsF i g . 4 a - e s h o w s t h e p r e d i c t e d G c - w c u r v e s f o r t h e s t e e lF R C w i t h Vf = 0 , 0 . 5, 1 . 0 , 1 .5 a n d 2 . 0 ~ , r e s p e c t i v e l y ,t o g e t h e r w i t h t h e e x p e r i m e n t a l d a t a o b t a i n e d f r o m t h eu n i a x i a l t e s t p r o c e d u r e d e s c r i b e d i n s e c t i o n 3 . I n a l l f i v ep l o ts , th e f i b re a n d c o n c r e t e m a t r i x d a t a f o r m o d e l i n p u ta r e f i x e d a s d e s c r i b e d a b o v e i n t h e s e c t i o n 4 . 1 . O n l y t h ef i b re v o l u m e is c h a n g e d . T h e 0 ~ d a t a a n d c u r v e a r ei n c l u d e d t o s h o w t h a t t h e m o d e l r e m a i n s v a l i d e v e n w h e nt h e f i b r e v o l u m e f r a c t i o n a p p r o a c h e s z e r o . I n b o t he x p e r i m e n t a l d a t a a n d t h e o r e t i c a l p r e d i c ti o n s , th e s e a c - wc u r v e s a r e c h a r a c te r i z e d b y a n i n it ia l l o a d d r o p f o l l o w e db y a s t r e s s r i s e a n d a s u b s e q u e n t c l o s e t o l i n e a r d e c a y .T h e m a x i m u m s t re s s a t w = 0 i n c r e a s e s b y a s m a l la m o u n t , w i t h a w e a k d e p e n d e n c e o n V f. T h e m a g n i t u d e

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M a t e r i a l s a n d S t r u c t u r e s 4 91

M o d e l v e r s u s e x p e r i m e n t o l r e s u l ts8.00OG .b 6.00

H i g h s t r e n g t h c o n c r e t eV f = O . O 0

(a )

4.00 L

2.00

0.00 . . . . . . . . . ~ . . . . . . . . . J . . . . . . . . . i . . . . . . . . . i . . . . . . . . . , . . . . . . . . .O .O O E+ O 5 . 0 0 E - 5 1 . 0 0 E - 4 1 . 5 0 E - 4 2 . 0 0 E - 4 2 . 5 0 E - 4 3 . 0 0 E - 4(m )

M o d e l v e r s u s e x p e r i m e n t o l r e s u l t s8.00013..z;vb 6.00

Stee l f ibers in h igh s t reng th concre teV,=O.O05

4 . 0 0 ~ 2 ~ oo O O O o o o

o~OOoo~ 2 o ~ Oo . . .2.00 o o o~ 9 " , ~ . / - o ~ ~ 1 7 6 1 7 6 oo o o0 " 0 0 ~ j o 0 % ~ O g n 0 0 0

0 0 O~ 0

0 . 0 0 . . . . . . . . . ~ . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . .O .O O E +O 5 . 0 0 s 1 . 0 0 E - 4 1 . 5 0 E - 4 2 . 0 0 E - 4 2 . 5 0 s , 3 .0 0 E - 4w ( m )b)

M o d e l v e r s u s e x p e r i m e n t e l r e s u l ts8.00O0..2~b 6.00

Stee l f i be r s i n h i gh s t r eng th c onc r e teV f = O . O 1)

~ o ~ %00 ~8 oOO:oo %0

0 0 0 04.00 ~0~0~00 (~ O~ 0 O0 0 O0 0 0 0~ " ~ ' ~ - ~ o ~ ~ 1 7 6 o o ~ % ~ o% .. .. o o ~ o ~ 8 ~ ~- o o u ~ ~ o o O o2.oo ~ o o ~ o- Z, 1 7 - . . - ~ - . . ~ . . ~

o o o ~ o ~ , o 6 ) o

0.00O.OOE+O' ' "5 .00k ~5 ' ' ' 1 ' .00k ~4 ' ' ' 1 ' .50E~4 ' ' ' 2 .00 F" - ' 4 " ' 2 . 50 E- 4 ' ' ' , ,3 .00E-4w ( m )c )

M o d e l v e r s u s e x p e r i m e n t o l r e s u l t s8.00oO_b 6.00

00~ ~ o ~ o4 . 0 0 o o ~ ~ o ~ o o o o o oo ~ o o o o o o ~ o o o o~176 ~ ~

o o -Goo ~ _ o o o o ~o ~ ~oa0eOoo Oo ~176 ~ Ooo o o o2 .0 0 o o o o o ~ 1 7 6 1 7 6 1 7 6

Stee l f i be r s i n h i gh s t r eng th c onc r e teV~= 0 .01 50 .0 0 . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . . . . . . . . . . .O .O O E +O 5 . 0 0 E - 5 1 . 0 0 ' E - 4 . 1 . 5 0 E - 4 2 . 0 0 E - 4 2 . 5 0 E - 4 3 . 0 0 E - 4

( d) w (m )

8.000O _

b 6.00

M o d e l v e r s u s e x p e r i m e n t o l r e s u l t s

~ o ~ o ~o Oooo

% ~ - ~= O o o o o o oO _aoD 00o o 0 u 0 o 04.00 ~- ~-a~o~o ~

o o O o ~o o o ~o o o2.00

S t e e l f i b e r s in h i g h s t r e n g t h c o n c r e t eV~=O.020 . 0 0O .O O E +O 5 . 0 0 E - 5 1 . 0 0 E - 4 1 . 5 0 E - 4 2 . 0 0 E - 4 2 . 5 0 E - 4 3 . 0 0 E - 4(e) w (m )

Fig . 4 Compar i son of ao-w cur ves ob ta ined f r om uniax ia l t ens ion tes ts and f r om mod e l p r ed ic t ions of s tee l FRC f or(a) Vr = 0.0, (b) V = 0.5, (c) V = 1.0, (d) Vf = 1.5 an d (e) V = 2.0% .

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

F i g . 5 a - d s h o w s a s i m i l a r s e t o f d a t a , b u t f o rp o l y p r o p y l e n e F R C . T h e f i b re v o l u m e p e r c e n t a g e sa r e 0 .0 , 1 . 0 , 2 . 0 a n d 3 .0 , r e s p e c t i v e l y . T h e g e n e r a lt r e n d b o t h e x p e r i m e n t a l a n d t h e o r e t i c a l , o f t h e ac-w

c u r v e s i n d i ca t e s a s m o o t h e r d e c a y i n c o m p a r i s o n t ot h o s e o f s t e e l f i b r e - r e i n f o r c e d c o n c r e t e . A g a i n , t h ep a r a m e t r i c v a l u e s u s e d i n t h e m o d e l a r e f i x e d a n dd e s c r i b e d i n t h e p r e v i o u s s e c t i o n , a n d o n l y t h e f i b r ev o l u m e f r a c t i o n i s c h a n g e d f o r m o d e l i n p u t f o r t h e s ef igur es .

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4 9 2 L i, S t a n g a n d K r e n c h e l

(a )

M o d e l v e r s u s e x p e r i m e n t o l r e s u l ts~ ' - 5 . 0 0O0_z;v

4 . 0 0b3 . 0 0

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M o d e l v e r s u s e x p e r i m e n t o l r e s u l t s" 6 "o _ 5 . 0 0 ]P p . f i b e r s it" n o r m a l s t r e n g t h c o n c r e t e

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~ o ~ 1 7 6 o o o o .o o dP ~ c~ d~ c9 c~ ~ @ ~O O O o o o o o o o o o o o o o o o oc~ 17 61 76 o o o o ~ ~ ^ ~ o

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5 . 0 0013.-

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M o d e l v e r s u s e x p e r i m e n t o l r e s u l ts

o o o o O O O o o o O O O o o o o o o O o o o o o co % O o : o : ooooo 0o 72 . 0 0 ~ 1 ~ : : ~ - ~ - - ~ . . .. . . . . : : - . . . . . . . . . . . . . . o

0 o 0 0 8 @ ~ o 0 0 0 8 0 ~ 8 ~ 8 ~ ~ c x ~O c ~ oc ~~ ~ -- OOoooooooooooooooooooo (

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P p . f i b e r s in n o r m o l s t r e n g t h c o n c r e t e% = 0 0 20 . 0 0 . . . . : . . . . J . . . . . . . . . ] . . . . . . . . . , . . . . . . . . . I . . . . . . . . . , . . . . . . . . .O .O O E +O 5 . 0 0 E - 5 1 . 0 0 E - 4 1 . 5 0 E - 4 2 0 0 E - 4 2 . 5 0 E - 4 3 . 0 0 E - 4

w ( m )c )

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o ~ 1 7 6 1 7 6 1 7 6o o ~ : - - flllil~m ~oooo ~.,,.,-~o o o o ~ o o g o ~ o % a % ~ 8 L

P p . f i b e r s i n n o r m a l s t r e n g t h c o n c r e t e% = 0 . 0 30 . 0 0 . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . .0 . 0 0 s 5 . 0 0 E - 5 1 . 0 0 E -4 1 . 5 0 E - 4 2 . 0 0 s 2 . 5 0 E - 4 3 . 0 0 E - 4

( d) w (m )F ig. 5 C o m p a r i so n o f a ~ - w curves obta ined from uniax ia l tension tests and from mode l predic t ions o f po lypropylene FRC for(a) V~ = 0.0, (b ) l~} = 1.0, (c) Vr = 2.0 an d (d) Vf = 3.0%

5 . D I S C U S S I O N O F M E C H A N I S M SG O V E R N I N G T H E c rc-w C U R V E S

T o g a i n p h y s i c a l i n s i g h t i n t o t h e g o v e r n i n g m e c h a n i s m so f t h e a c - w c u r v e s , w e h a v e s e p a r a t e l y p l o t t e d i n F i g . 6t h e i n d i v i d u a l c o n t r i b u t i o n s o f th e a g g r e g a t e b r i d g i n ga c t i o n , t h e f i b r e b r i d g i n g a c t i o n a n d t h e f i b r e p r e s t r e s s( u s i n g V = 2 % st e e l f i br e a s i l l u s t r a t i o n ) , r e p r e s e n t e d b yE q u a t i o n s 1 , 4 , 6 a n d 1 4 , r e s p e c t i v el y . T h e s e t h r e eb r i d g i n g a c t i o n s c o n t r o l t h e b r o a d p i c tu r e o f t h e s h a p eo f t h e a c - w c u r v e s . I n d e e d , t h e m a j o r d i f f e r e n c e b e t w e e nt h e a c - w r e l a ti o n s o f s t ee l a n d p o l y p r o p y l e n e F R C s l ie si n t h e f i b r e b r i d g i n g a c t i o n , a s e x p e c t e d , a n d F i g . 6i n d i c a te s t h a t t h e e la s t ic m o d u l u s o f t h e fi b re d o m i n a t e st h e l o c a t i o n o f t h e p e a k ( a t 6 * ) i n t h e f ib r e b r i d g i n ga c t i o n . T h u s , t h e s t if fe r s t e e l f i br e s c r ea t e a s t r o n g e r b u m pi n t h e a c - w c u r v e , a t s m a l l e r c r a c k o p e n i n g d u e t o t h es m a l l e r 6 * . T h e m o r e g e n t l e f ib r e b r i d g i n g a c t i o n o f t h el o w e r - m o d u l u s p o l y p r o p y l e n e f i b r e s i n t u r n e x p l a i n st h e s m o o t h e r a ~ - w c u r v e s o b s e r v e d w h e n t e s t i n g t h ec o r r e s p o n d i n g F R C . T h e p r e s t r e s s i n g m e c h a n i s m i s

8 . 0 0D e c o m p o s i t i o n o f t h e m o d e l

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b 6.ooS t e e l f i b e r s i n h i g h s t r e n g t h c o n c r e t e% = 0 . 0 2

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, , , ~ [ '/ ~ ~ . . A g g r e g a t e b r i d g i n g, _ . r " > - , , - -. . . . : : " : ' - . . . . . . . . . . . . . . . . , 7 , - - , , :: : . . . . . . . . . . . . . . .. 0 0O .O O E +O 5 . 0 0 E - 5 I . OO E -4 1 . 5 0 E - 4 2 . 0 0 E - 4 2 . 5 0 E - 4 3 . 0 0 sW ( m )

F i g . 6 I n d i v i d u a l c o n t r i b u t io n s o f a g g r e g at e b r i d g i n g a c t i o na n d f i b r e b r i d g i n g a c t i o n t o a ~ - w c u r v e s . I l l u s t r a t e d a l s o i st h e f ib r e p r e - s tr e s s in g e f fe c t . S t e e l f i b r e s in h i g h - s t r e n g t hconcrete, V r = 2 . 0 ~

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S t e e l f i b e r s i n h i g h s t r e n g t h c o n c r e t eV f=O.020 . 0 0 . . . . . . . . . . . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . . , . . . . . . . . .O .O O E +O 5 . 0 0 E - 5 1 . 0 0 E - 4 1 . 5 0 E - 4 2 . 0 0 E - 4 2 . 5 0 E - 4 3 . 0 0 E - 4

, , , ( r - n )Fig, 7 Influence of Cook-Gordon effect on the ac-w curves .Frictio n bo nd deca y is also illustrated. Steel fibres inhigh-strength concrete; Vf = 2.0% .

r e s p o n s i b l e f o r t h e d e f i n it e , a lb e i t s m a l l, t e n s i l e s t r e n g t hi n c r e a s e w i t h s t e e l f i b r e v o l u m e f r a c t i o n a t w = 0 .

F i g . 7 sh o w s t h e c o m b i n e d i n f l u e n c e o f a g g r e g a t eb r i d g i n g a n d f i b r e b r i d g i n g , w i t h a n d w i t h o u t t h eC o o k - G o r d o n e ff ec t. I t is c l ea r th a t w h e n t h e C o o k -G o r d o n e f f e c t i s a b s e n t , t h e o v e r e s t i m a t e d i n i t i al f i b r ebr idg ing s t if fnes s causes a r i s e in the c urv e a t w _~ 0 wh ichis n o t o b s e r v e d i n a n y o f t h e e x p e r i m e n t a l d a t a s e ts . T h u st h e C o o k - G o r d o n e ff ec t o f a n i n it ia l d e b o n d e d s e g m e n ti s i n d i r e c t l y r e v e a l e d i n t h e G c - w m e a s u r e m e n t s . T h ei n f lu e n c e o f f r ic t io n a l b o n d d e c a y w i t h s l i p p ag e ( E q u a t i o n7) is a lso i l lus t rated.

6 . C O N C L U S I O NW e h a v e p r e s e n t e d a m i c r o m e c h a n i c a l l y b a s e d a n a l y t i c a lm o d e l w h i c h d e s c r i b e s t h e p o s t - c r a c k b r i d g i n g s t r e s s -c r a c k w i d t h r e l a t i o n s h i p i n F R C m a t e r i al s . T h i s r e l a t io n -s h i p is b e s t s u m m e d u p i n t h e f o l l o w i n g fo r m :

a c = F ( w , c o n c r e t e p a r a m e t e r s , f i b re p a r a m e t e r s ,i n t e r f a c e p a r a m e t e r s ) ( 1 9 )

w h e r e t h e c o n c r e t e p a r a m e t e r s c o n s i s t o f E m , a ~ , p a n dwo , t h e f i b r e p a r a m e t e r s c o n s i s t o f V f, E l , L f a n d dr, a n dt h e i n t e r f a c e p a r a m e t e r s c o n s i s t o f z , f a n d ~ . A r m e dw i t h t h e se c o n s t i t u e n t m a t e r ia l p a r a m e t e r s , t h e c o m p o s i t e~ c - w r e l a t i o n s h i p , h i g h l y n o n - l i n e a r a s i t m a y b e , c a n b ee a s i l y c a l c u l a t e d s o l o n g a s t h e a s s u m p t i o n s b u i l t i n t ot h e m o d e l a r e s a t i s f i e d . W i t h t w o h i g h l y d i f f e r e n t f i b r et y p e s , s t ee l a n d p o l y m e r , a s r e i n f o r c e m e n t s f o r a c o n c r e t ec o m p o s i t e , a n d w i t h f o u r a n d t h r e e d i f fe r e n t f i b re v o l u m ef r a c ti o n s e a c h , F ig s 4 a n d 5 d e m o n s t r a t e t h a t t h ep r o p o s e d m o d e l c a n b e a p p l i e d w i t h s o m e c o n f i d e n c e .E q u a t i o n 1 9 i m p l ie s t h a t t h is i m p o r t a n t c o m p o s i t ep r o p e r t y c a n b e e n g i n e e r e d b y c o n t r o l l i n g t h e c o n c r e t e ,f i br e a n d i n t e r f a c e p a r a m e t e r s , s o t h a t a c o m p o s i t e w i tht h e d e s i r a b l e a r r e l a t i o n s h i p c a n b e d e s i g n e d .

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

i n t h e f o r m o f E q u a t i o n 1 9, is in t h e d e s i g n o r p r e d i c t i o no f s t r u c t u r a l r e s p o n s e , w h e n t h e m a t e r i a l i s a f i b r e -r e i n fo r c e d c o n c r e t e w i t h o u t p s e u d o - s t r a i n - h a r d e n i n g . A tp r e s e n t , a c o n s t r a i n i n g f a c t o r i n t h e a p p l i c a t i o n o ff i b r e - r e i n f o r c e d c o n c r e t e t o p r a c t i c a l s t r u c t u r e s i s t h e l a c ko f d e s i g n t o o l s t o a l l o w t h e s t r u c t u r a l e n g i n e e r t o p r e d i c tt h e s t r u c t u r a l r e s p o n s e . F o r t h i s p u r p o s e , E q u a t i o n 1 9s e r v e s a s a c o n s t i t u t i v e r e l a t i o n , s i m i l a r t o t h o s e c u r r e n t l yu s e d i n g o v e r n i n g t h e e l a s t i c o r n o n - l i n e a r e l a s t i cb e h a v i o u r o f co n c r e t e , b u t s p e c i a li z e d f o r h a n d l i n g t h ep o s t - c r a c k r e g i m e i n a f i b r e - r e i n f o r c e d c o n c r e t e s t r u c t u r e .F u r t h e r , a s a l l u d e d t o i n s e c t i o n 1 , f o r th e p u r p o s e o fl e a k a g e c o n t r o l i n f l u i d - c o n t a i n i n g c o n c r e t e v e s s el s, o r i nd u r a b i l i t y d e s i g n f o r g e n e r a l c o n c r e t e s t r u c t u r e s , p a r t i c u -l a r l y t h o s e e x p o s e d t o a g g r e s s i v e e n v i r o n m e n t s s u c h a si n co a s t a l r e g io n s o r p a r k i n g d e c k s, c r a c k w i d t h c o n t r o lw i l l b e p a r a m o u n t i n m a i n t a i n i n g s t r u c t u r a l f u n c t i o n a l i t y .A g a i n , E q u a t i o n 1 9, i n c o m b i n a t i o n w i t h f in i t e -e l e m e n ta n a l y s i s , w o u l d s e r v e a s a n a n a l y t i c a l t o o l f o r c r a c k w i d t hc o n t r o l , t o b e c a r r i e d o u t b e f o r e t h e s t r u c t u r e i s b u i l t a n dp u t i n t o s e r v i c e .

M i c r o m e c h a n i c a l m o d e l s s u c h a s E q u a t i o n 1 9 a l lo wt h e s t r u c t u r a l d e s i g n e r a n d / o r m a t e r i a l s e n g i n e e r t oc h o o s e a n a p p r o p r i a t e m a t e r i a l s r e c e i p e w h i c h w il l b e s tm e e t t h e r e q u i r e d s t r u c t u r a l p e r f o r m a n c e .

A C K N O W L E D G E M E N T ST h i s w o r k w a s p e r f o r m e d w h i l e V i c to r C . L i w a s aV i s it in g P r o f e s s o r a t t h e D e p a r t m e n t o f S t r u c t u r a lE n g i n e e r i n g a t T h e T e c h n i c a l U n i v e r s i ty o f D e n m a r k( D T H ) i n J u n e - A u g u s t , 1 99 2. F u n d i n g b y D T H f o r t h eV i s i t i n g P r o f e s s o r s h i p i s g r a t e f u l l y a c k n o w l e d g e d . V . C .L i w o u l d l ik e t o t h a n k h is c o l le a g u e s a t D T H f o r t h e irh o s p i t a l i t y a n d c o o p e r a t i o n d u r i n g h i s s t a y i n D e n m a r k .D r H e n r i k S t a n g a n d P r o f e s s o r H e r b e r t K r e n c h e la c k n o w l e d g e s u p p o r t f r o m t h e R e se a r ch P r o g r a m m e o nF i b r e R e i n f o r ce d C e m e n t i ti o u s C o m p o s i t e s s p o n s o r e d b yT h e D a n i s h C o u n c i l f o r S c i en t if i c a n d I n d u s t r i a l R e s e a r c ha n d T h e D a n i s h M i n i s t r y f o r I n d u s t r y . D i s c u s s io n s w i t hA r n o n B e n t u r h a v e b e e n p a r t i c u l a r l y h e lp f u l w i t h r e g a r dt o t he C o o k - G o r d o n e ffe ct.REFERENCES

1. Stang, H . , 'Ev aluat ion of propert ies of cemen ti t iousmaterials ' , in "High Performance Fiber ReinforcedCement Composi tes ' , Proceedings of Internat ionalRILE M/A CI W orkshop, ed it ed by H. W. Re inhard t andA. E. Naam an (Spon, 1992) pp. 388-406.2. Li, V. C. and Liang, E., 'Fr ac tur e processes in con crete andfibre-reinforced cementitious composites ' , A S C E J . E n g .M e c h . 12(6 ) (1986) 566-586.3. Stang, H. and A arre, T. , 'Evalu at ion of crack width in FR Cwith convent ional reinforcement ' J. Cem. Concr . Compos.14(2) (1992) 143-154.4. Li , V. C. and Wu, H . C. , 'Condi t ion s fo r pseudostrain-hardening in f iber reinforced brit t le matrix com -posites', J . A pp l . M ech . R ev . 45(8) (1992) 390-398.5. Li, V. C. and Leung, C. K. Y., "Th eory of steady state and

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494 Li , Stang and Krenchelm u l t i p le c r a c k i n g o f r a n d o m d i s c o n t i n u o u s f i be r re -i n f o r c e d b r i t t l e m a t r i x c o m p o s i t e s ' , ASCE J . EngngMech. 118(11) (1992) 2246-2264 .6 . C o o k , J . a n d G o r d o n , J . E . , ' A m e c h a n i s m f o r t h e c o n t r o lo f c r a c k p r o p a g a t i o n i n a ll b r i tt l e s y s t e m s ' , Proc. Roy.Soc. 2 8 2 A ( 1 9 6 4 ) 5 0 8 - 5 2 0 .

7 . S t a n g , H . , L i , V , C . a n d K r e n c h e l , H . , ' D e s i g n a n d s t r u c t u r a la p p l i c a t io n s o f s t r e s s - c r a c k w i d t h r e l a t i o n s i n f i b e rr e i n f o r c e d c o n c r e t e ' , i n p r e p a r a t i o n .

8 . O r t i z , M . , ' M i c r o c r a c k c o a l e s c e n c e a n d m a c r o s c o p i cc r a c k g r o w t h i n i t i a t i o n i n b r i t t l e s o l i d s ' , Int. J. SolidsStruct. 24 (1988) 231-250 .

9 . H o r i i , H ., H a s e g a w a , A . a n d N i s h i n o , F . , ' F r a c t u r e p r o c e s sa n d b r i d g i n g z o n e m o d e l a n d i n f l u e n c i n g f a c t o r s i nf r a c t u re o f c o n c r e t e ' , in ' F r a c t u r e o f C o n c r e t e a n d R o c k ' ,e d i t e d b y S . P . S h a h a n d S . E . S w a r t z ( S p r i n g e r , N e wY o r k , 1 9 8 9) , p p . 2 0 5 - 2 1 9 .

1 0. K a r i h a l o o , B . L . a n d H u a n g , X . , ' T e n s i l e r e s p o n s e o fq u a s i - b r i t t l e m a t e r i a l s ' , Pure Appl . Geophys . 137(4)(1991) 461- 427 .1 1. L i , V . C . W a n g , Y . a n d B a c k e r , S . , ' E f f e c t o f i n c l i n i n ga n g l e , b u n d l i n g , a n d s u r f a c e t r e a t m e n t o n s y n t h e t i c f i b e rp u l l - o u t f r o m a c e m e n t m a t r i x ' , J. Compos. 21(2) (1990)1 3 2 - 1 4 0 .

1 2 . L i , V . C . , ' P o s t - c r a c k s c a l i n g r e l a t i o n s f o r f i b e r r e i n f o r c e dc e m e n t i t i o u s c o m p o s i t e s ' , AS CE J. Mater . C ivi l Engng4 ( I ) , ( 1 9 9 2 ) 4 1 - 5 7 .

1 3. W a n g , Y . , L i, V . C . a n d B a c k e r , S . , ' M o d e l i n g o ff i b er p u ll - o u t f r o m a c e m e n t m a t r i x ' , Int . J . Cem.Compos. Lightwt Concr. 10(3) (1988) 143-149 .

1 4 . A a r r e , T . , ' T e n s i l e C h a r a c t e r i s t i c s o f F R C w i t h S p e c i a lE m p h a s i s o n i ts A p p l i c a b il i ty i n a C o n t i n u o u s P a v e -m e n t ' , P h D t h es is , D e p a r t m e n t o f S t r u c t u r a l E n g i n e e r -i ng , T e c h n i c a l U n i v e r s i t y o f D e n m a r k ( 19 9 2) .

1 5. G l a v i n d , M . , ' E v a l u a t i o n o f t h e C o m p r e s s i v e B e h a v i o r o fF i b e r R e i n f o r c e d H i g h S t r e n g t h C o n c r e t e ' , P h D t h e s i s ,D e p a r t m e n t o f S t r u c t u r a l E n g i n e e r i n g , T e c h n i c a l U n i -v e r s it y o f D e n m a r k ( 19 9 2) .

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1 7. B e n t u r , A . , D i a m o n d , S . a n d M i n d e s s , S ., ' C r a c k i n gp r o c e s s i n s t e e l f i b e r r e i n f o r c e d c e m e n t p a s t e ' , Cem.Concr. Res. 1 5 ( 1 9 8 5 ) 3 3 1 - 3 4 2 .18 . ldem, ' T h e m i c r o s t r u c t u r e o f s t ee l f i b e r - c e m e n t i n t e r f a c e ',J. Mater. Sci. 2 0 ( 1 9 8 5 ) 3 6 1 0 - 3 6 2 0 .

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3 0. B e n t u r , A ., p e r s o n a l c o m m u n i c a t i o n .

R E S U M EM icrom6 caniqu e de la repr is e de f i s s urat ions dans l e b6tonrenfore6 de fbresOn a dk terminO de faqon expkr imen ta le la r e la t ioncont ra in te - larg eur de f i s sure pou r des bk tons r enforcks pardeu x t ypes de f ibres - a c ier e t polypro py lkne - avecdi f fk rentes propor t ions en volume de f ibres . On p ropose unmodk le thdor ique bask sur la micromkcanique , qui prenden compte l e s carac tkr i s t iques es sent i e l l e s des r e la t ionscont ra in te - largeu r de fi s sure po ur des largeurs l imi t kes( < 0 , 3 m m ) . L e s m i c r om bc an i sm e s p r i s e n c ons i d& a t i on

comp rennent l ' ef f et de pon tage , l ' e ff e t Coo k-G ord on bl ' in t er face e t la prkcont ra in te des f ibres . L ' e f f e t d e pontag edes f ibres im pl ique un f ro t t em en t en r e la t ion avec l eg l i s sement e t un f ro t t em ent qui r e lkve l e s f ibres s e cro i santd angles obl iques . La comparai son ent re l e s prkdic t ionsth~oriques basbes sur des pararn ktres ind kpe nda nts et lesmesures expkr imenta les de la r e la t ion cont ra in te - largeurde f issure es t sat is faisante. Le s rOsul tats de cet te recherc hedonn ent conf iance dans l 'u t i l i sa t ion du m odk le proposb pourdes matkr iaux de gbnie c i v i l des t inds des per formancess tructurel les prescr i tes .