Fracture of concrete and mortar under uniaxial impact tensile loading

FRACTURE OF CONCRETE AND MORTAR

UNDER

FRACTURE OF CONCRETE AND MORTAR

UNDER

UNIAXIAL IMPACT TENSILE LOADING

o •o o o UJ o . itltl I, I II!! I Hi , 111"! I'll iii i f i ill I III III ¡¡1 it i l l ! II I i i i! i i i i •o M -4 - J 0 0 B I B L I O T H E E K T U Delft P 1718 4 2 7 7 C 8 0 6 9 4 4

FRACTURE OF CONCRETE A N D MORTAR

UNDER

UNIAXIAL IMPACT TENSILE LOADING

PROEFSCHRIFT ter verkrijging van

technische wetenschappen

aan de Technische Hogeschool Delft,

op gezag van de rector magnificus,

voor een commissie aangewezen

door het college van dekanen

te verdedigen op

dinsdag 23 november 1982

te 14.00 uur door

ANDRZEJ JERZYZIELINSKI

civiel ingenieur

geboren te Warszawa

Delft University Press /1982

de graad van doctor in de

de promotor: PROF.DR.-ING. H.W. REINHARDT

ACKNOWLEDGEMENT

The a u t h o r w i s h e s , t o thank a l l the members o f the C o n c r e t e S t r u c t u r e s Group f o r t h e i r c o l l a b o r a t i o n and encouragement. E s p e c i a l l y the e n c o u -ragement and s u p p o r t r e c i e v e d f r o m P r o f . D r . I r . A.S.G. B r u g g e l i n g are g r a t e f u l l y a p p r e c i a t e d as wel as h i s i n i t i a t i v e and a r r a n g e m e n t s w h i c h made i t p o s s i b l e f o r the a u t h o r to p a r t i c i p a t e i n the r e s e a r c h p r o j e c t

"Impact l o a d i n g " c a r r i e d out i n the S t e v i n L a b o r a t o r y a t the D e l f t U n i v e r s i t y o f T e c h n o l o g y .

P a r t o f the e x p e r i m e n t a l i n v e s t i g a t i o n was p e r f o r m e d w i t h f i n a n c i a l s u p p o r t and under s u p e r v i s i o n o f CUR-VB ( N e t h e r l a n d s Committee f o r R e s e a r c h , Codes and S p e c i f i c a t i o n s f o r C o n c r e t e ) . The t e s t s on c o n c -r e t e we-re c a -r -r i e d out j o i n t l y w i t h I -r . H.A. K ö -r m e l i n g whose v a l u a b l e c o l l a b o r a t i o n and d i s c u s s i o n s a r e g r a t e f u l l y acknowledged.

The a u t h o r i s g r e a t l y i n d e b t e d to the S t i c h t i n g P r o f e s s o r B a k k e r f o n d s f o r the f i n a n c i a l s u p p o r t he r e c i e v e d t h r o u g h o u t the whole r e s e a r c h program and i n p a r t i c u l a r f o r p u b l i c a t i o n o f t h i s d o c t o r a l t h e s i s .

CONTENTS

1. INTRODUCTION 1

2. LITERATURE SURVEY 5 2.1. U n i a x i a l t e n s i l e t e s t s on c o n c r e t e 5

2.2. A g g r e g a t e s and t h e i r bond w i t h cement m a t r i x 12

2.3. F r a c t u r e m e c h a n i c s 16 2.3.1. B a s i c c o n c e p t s 16 2.3.2. A p p l i c a t i o n t o cement-based m a t e r i a l s 19

3. MODEL FOR TENSILE FRACTURE OF CONCRETE 29

3. 1. B a s i c i d e a s 29 3.2. G e o m e t r i c a l i d e a l i z a t i o n o f a c o m p o s i t e m a t e r i a l 33 3.3. S t r e s s s i t u a t i o n i n an e l a s t i c m a t r i x w i t h a c i r c u l a r i n c l u s i o n 39 3.3.1. Bonded i n c l u s i o n o f v a r y i n g s t i f f n e s s 39 3.3.2. P a r t i a l l y debonded i n c l u s i o n 46 3.3.3. Remarks w i t h r e s p e c t t o a c t u a l cement-based

com-p o s i t e s 48 3.4. F r a c t u r e due t o s l o w l y i n c r e a s i n g l o a d i n g 52 3.4.1. F r a c t u r e o f a homogeneous cement m a t r i x 52 3.4.2. F r a c t u r e o f a h e t e r o g e n e o u s p a r t i c u l a t e c o m p o s i t e 53 3.4.3. C o m p a r i s o n o f f r a c t u r e e n e r g i e s 55 3.5. F r a c t u r e due t o impact l o a d i n g 56 3.5.1. F r a c t u r e o f a homogeneous cement m a t r i x 56 3.5.2. F r a c t u r e o f a h e t e r o g e n e o u s p a r t i c u l a t e c o m p o s i t e 59 3.5.3. C o m p a r i s o n o f f r a c t u r e e n e r g i e s 67 3.6. F i n a l remarks 71 4. EXPERIMENTAL PROGRAM 73 4.1. T e s t method and equipment 73

4.2. Scope 77 4.3. M a n u f a c t u r e o f s p e c i m e n s , t e s t i n g p r o c e d u r e s and p r o p e r

5.1. R e l a t i o n s h i p between t e n s i l e s t r e n g t h and s t r e s s r a t e 81 5.1.1. C o n c r e t e 81 5.1.2. M i c r o c o n c r e t e 92 5.1.3. M o r t a r 93 5.2. Impact and s t a t i c s t r e s s - s t r a i n b e h a v i o u r 94 5.2.1. M o r t a r 94 5.2.2. M i c r o c o n c r e t e 96 5.2.3. C o n c r e t e 97 5.3. D i s c u s s i o n and c o m p a r i s o n w i t h model 98 5.4. P r a c t i c a l s u g g e s t i o n s prompted by t h e e x p e r i m e n t a l r e s u l t s and t h e model 105

6. EXPERIMENTAL RESULTS OF REPEATED IMPACT TENSILE LOADING TESTS H I 6.1. R e l a t i o n s h i p between impact f a t i g u e t e n s i l e s t r e n g t h and t h e number o f l o a d i n g c y c l e s 111 6.1.1. C o n c r e t e 111 6.1.2. M i c r o c o n c r e t e 116 6.1.3. M o r t a r 116 6.2. Impact f a t i g u e s t r e s s - s t r a i n b e h a v i o u r 117 6.2.1. M o r t a r 117 6.2.2. M i c r o c o n c r e t e 118 6.3. D i s c u s s i o n 120

7. RETROSPECTIVE VIEW AND CONCLUSIONS 124

SUMMARY 128

SAMENVATTING j 3Q

NOTATION j 32

REFERENCES j 35

1

-1. INTRODUCTION

C o n c r e t e i s a m u l t i p h a s e complex m a t e r i a l c o n s i s t i n g o f a g g r e g a t e p a r t i c l e s o f v a r i o u s s i z e s and i r r e g u l a r shape, w h i c h a r e d i s p e r s e d and embedded i n h a r d e n e d cement p a s t e . A i r v o i d s , i n t e r f a c i a l bond m i c r o c r a c k s and m i c r o -c r a -c k s w i t h i n t h e -cement m a t r i x a r e i n h e r e n t i n t h i s m a t e r i a l due t o manu-f a c t u r i n g p r o c e d u r e s and p h y s i c o - c h e m i c a l p r o c e s s e s t a k i n g p l a c e d u r i n g h a r d e n i n g o f cement-based c o m p o s i t e s .

Some f a c t o r s a f f e c t i n g t h e m e c h a n i c a l b e h a v i o u r o f c o n c r e t e a r e r e v i e w e d , a f t e r Newman and Newman [ 7 4 ] , i n T a b l e 1.1.

The above i n d i c a t e s t h a t any a t t e m p t t o p r e d i c t and d e s c r i b e t h e p r o p e r -t i e s and b e h a v i o u r o f s u c h a h i g h l y h e -t e r o g e n e o u s m a -t e r i a l as c o n c r e -t e , under v a r i o u s l o a d i n g t y p e s must meet many d i f f i c u l t i e s .

S e v e r a l t h e o r i e s o f t e n s i l e f a i l u r e have b e e n p r o p o s e d b o t h a t t h e s t r u c -t u r a l and p h e n o m e n o l o g i c a l l e v e l s . Hansen [34] h a s d i s c u s s e d some models i n w h i c h c o n c r e t e h a s b e e n c o n s i d e r e d as a two-phase m a t e r i a l w i t h an a g g r e g a t e phase and a l e s s s t i f f and s t r o n g cement m a t r i x phase. These p h a s e s have been a n a l y s e d a s s e r i e s and p a r a l l e l m a c r o s y s t e m s . Such models c a n n o t be t r e a t e d as s a t i s f a c t o r y , s i n c e t h e s t r e n g t h o f m a t e r i a l i s a s t r u c t u r e - s e n s i t i v e p r o p e r t y . F o r s e r i e s systems t h e t e n s i l e f r a c t u r e o f c o n c r e t e i s g o v e r n e d by f a i l u r e o f t h e weaker cement m a t r i x phase and f o r p a r a l l e l systems t h e f r a c t u r e r e s u l t s f r o m t h e f a i l u r e o f t h e s t r o n g e r a g g r e g a t e phase. M o d e l s i n w h i c h c o n c r e t e i s c o n s i d e r e d a s a two-phase m a t e r i a l s u c h t h a t t h e a g g r e g a t e p a r t i c l e s phase i s d i s p e r s e d i n t h e cement m a t r i x phase and t h e s t r e s s s t a t e i s d e t e r m i n e d a r o u n d a g g r e g a t e p a r t i c l e s (see f o r i n s t a n c e [63]) a r e a l s o o f l i t t l e p r a c t i c a l i m p o r t a n c e . The r e a s o n f o r t h i s f a c t i s t h a t t h e s t r e s s s t a t e i n an e l a s t i c medium c a n be d e s c r i b e d f o r one o r a few i n c l u s i o n s b u t f o r a more complex s y s t e m o f i n c l u s i o n s t h i s t a s k , even w i t h p o w e r f u l computers, i s e x t r e m e l y l a b o r i o u s . A n o t h e r p o i n t i s t h a t t h e i n i t i a t i o n o f f r a c t u r e i n a r e a s where t h e s t r e s s r e a c h e s t h e l o c a l s t r e n g t h o f t h e m a t e r i a l does n o t n e c e s s a i r l y l e a d t o f r a c t u r e o f t h e e n t i r e c r o s s - s e c t i o n . Here t h e c r a c k a r r e s t i n g p r o c e s s e s and r e l a x a t i o n o f m a t e r i a l i n t h e z o n e s n e x t t o t h e c r a c k a r e o f e s s e n t i a l i m p o r t a n c e . Most i m p o r t a n t , however, i s t h e e n e r g y r e q u i r e m e n t i n t h e f r a c t u r e p r o c e s s . Even a h i g h c o n c e n t r a t i o n o f s t r e s s i n c o n j u n c t i o n w i t h l o w e n e r g y i n p u t and s h o r t time d u r i n g w h i c h t h e e x t e r n a l l o a d i s a p p l i e d may n o t c a u s e f r a c t u r e o f t h e s e c t i o n o f m a t e r i a l . These f a c t s h a v e been r e c o g n i z e d by G l u c k l i c h [ 2 9 ] , who c o n c l u d e d t h a t t h e o n l y p r o p e r a p p r o a c h t o t h e b e h a v i o u r o f c o n c r e t e must be b a s e d on e n e r g y c o n s i d e r a t i o n s and s h o u l d t a k e a c c o u n t o f t h e s t r u c

-c o m b i n i n g a s t a t i s t i -c a l a p p r o a -c h t o t h e m i -c r o s t r u -c t u r e o f m a t e r i a l s w i t h i d e a s o f f r a c t u r e m e c h a n i c s . The a p p l i c a t i o n o f s u c h models t o c o n c r e t e m a t e r i a l s has r e c e n t l y been shown by M i h a s h i [64] who s t u d i e d a l s o t h e i n f l u e n c e o f t h e r a t e o f l o a d i n g upon t h e s t r e n g t h o f c o n c r e t e [ 6 5 ] . A l t h o u g h m o d e l s m e n t i o n e d o f f e r a g r e a t l o n g r a n g e p r o m i s e f o r t h e a n a l y s i s o f t h e b e h a v i o u r o f v a r i o u s m a t e r i a l s under d i f f e r e n t l o a d i n g c o n d i t i o n s , T a b l e 1.1 F a c t o r s a f f e c t i n g the m e c h a n i c a l b e h a v i o u r o f c o n c r e t e C h a r a c t e r i s t i c s o f C o n c r e t e S t r u c t u r e Volume f r a c t i o n s o f a g g r e g a t e and p a s t e . A g g r e g a t e p a r t i c l e s i z e and d i s t r i b u t i o n . Degree o f m a t u r i t y o f cement p a s t e . P r o p e r t i e s o f C o n s t i t u e n t s M e c h a n i c a l , p h y s i c a l and c h e m i c a l p r o p e r t i e s o f a g g r e g a t e and cement p a s t e . P r o p e r t i e s o f a g g r e g a t e - p a s t e i n t e r f a c e . Specimen and E n v i r o n m e n t a l C h a r a c t e r i s t i c s M o i s t u r e and T e m p e r a t u r e C o n d i t i o n s A v e r a g e , and d i s t r i b u t i o n s o f , m o i s t u r e c o n t e n t and t e m p e r a t u r e w i t h i n t e s t s p e c i m e n . R e l a t i v e m o i s t u r e c o n t e n t s and t e m p e r a t u r e s o f s p e c i m e n and e n v i r o n m e n t .

Specimen S i z e and Shape

I n f l u e n c e upon s t r e n g t h , s h r i n k a g e and c r e e p . S t a t e o f S t r e s s o r S t r a i n M a g n i t u d e o f A p p l i e d S t r e s s o r S t r a i n V o l u m e t r i c and d e v i a t o r i c components o f s t r e s s o r s t r a i n . A c t u a l D i s t r i b u t i o n o f S t r e s s o r S t r a i n U n i f o r m o r n o n - u n i f o r m d i s t r i b u t i o n o f a p p l i e d s t r e s s o r s t r a i n . E f f e c t o f s p e c i m e n , p l a t e n s and machine upon t h e a c t u a l d i s t r i b u

-t i o n o f s -t r e s s o r s -t r a i n . Method o f L o a d i n g Type o f L o a d i n g Rate o f i n c r e a s e o f s t r e s s o r s t r a i n . L o n g - t e r m s t a t i c l o a d i n g S h o r t - t e r m s t a t i c l o a d i n g Dynamic l o a d i n g P a t t e r n o f L o a d i n g L o a d i n g i n c r e a s e d m o n o t o n i c a l l y t o f a i l u r e ( s t a t i c o r d y n a m i c ) . S t e a d y , s t a t i c o r c r e e p l o a d i n g . R e p e t i t i v e ( f a t i g u e ) , p u l s e o r a l t e r n a t i n g l o a d i n g .

- 3 —

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

T h i s r e s e a r c h work i s f o c u s e d on one p a r t i c u l a r p r o b l e m , namely t h e behavi o u r o f c o n c r e t e under behavimpact t e n s behavi l e l o a d behavi n g . T h behavi s p r o b l e m behavi s o f c o n s behavi d -e r a b l -e i m p o r t a n c -e i n -e n g i n -e -e r i n g p r a c t i c -e and c a l l s f o r s p -e c i a l a t t -e n t i o n . N o r m a l l y t h e t e n s i l e b e h a v i o u r o f c o n c r e t e i s n e g l e c t e d i n s t r u c t u r a l a n a -l y s i s , b u t i n d i r e c t -l y e v e r y e -l e m e n t r e -l i e s on i t , s i n c e t e n s i -l e s t r e n g t h g r e a t l y i n f l u e n c e s c r a c k i n g o f c o n c r e t e , bond p r o p e r t i e s o f r e i n f o r c i n g s t e e l and b e h a v i o u r under s h e a r f o r c e s .

The t e n s i l e s t r e n g t h i s a l s o one o f the f u n d a m e n t a l p a r a m e t e r s o f two- and t h r e e d i m e n s i o n a l f a i l u r e e n v e l o p e s f o r c o n c r e t e w h i c h a r e o f g r e a t i m p o r -t a n c e f o r a d v a n c e d a n a l y s i s o f 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 s . The -t e n s i l e s t r e n g t h o b t a i n e d f r o m s t a t i c t e s t s i n w h i c h f r a c t u r e o f c o n c r e t e s p e c i m e n s o c c u r s w i t h i n a few m i n u t e s , however, c a n n o t be u s e d f o r t h e p u r p o s e o f a n a l y s i n g t h e b e h a v i o u r o f c o n c r e t e e l e m e n t s s u b j e c t e d t o i m p a c t l o a d i n g c o n d i t i o n s .

Some modern s p e c i a l s t r u c t u r e s may be s u b j e c t e d t o l o a d i n g a t v e r y h i g h s t r e s s r a t e s . O f f s h o r e s t r u c t u r e s , n u c l e a r r e a c t o r c o n t a i n m e n t s , highway s t r u c t u r e s and o t h e r s must w i t h s t a n d i m p a c t l o a d i n g s c a u s e d by e a r t h q u a k e s , e x p l o s i o n s and s h i p , a i r c r a f t o r v e h i c l e c o l l i s i o n s . I n v a r i o u s s t r u c t u r a l e l e m e n t s t e n s i l e s t r e s s can o c c u r as a r e s u l t o f t h e a p p l i e d t e n s i o n , and due t o b e n d i n g , s h e a r o r p u n c h i n g . I n some e l e m e n t s , f o r i n s t a n c e i n d r i v e n c o n c r e t e p i l e s , a c o m p r e s s i v e s t r e s s wave can be r e f l e c t e d as a t e n s i l e s t r e s s wave and can c a u s e c r a c k i n g and f a i l u r e o f c o n c r e t e . The a n a l y s i s o f t h e b e h a v i o u r o f c o n c r e t e e l e m e n t s u n d e r i m p a c t l o a d i n g i n v o l v e s , on t h e one hand, the i n f l u e n c e o f the way o f l o a d i n g upon t h e s r e s s s t a t e i n the e l e m e n t and, on the o t h e r hand, t h e i n f l u e n c e o f h i g h s t r e s s r a t e s upon t h e b e h a v i o u r o f t h e m a t e r i a l as s u c h . A l t h o u g h t h e f i r s t p r o b l e m i s o f g r e a t i m p o r t a n c e f o r t h e a n a l y s i s o f s t r u c t u r a l members, i t i s n o t d e a l t w i t h i n t h i s r e s e a r c h program.

The aim o f t h e r e s e a r c h work p r e s e n t e d h e r e i s t o p r o v i d e more i n f o r m a t i o n and t o g i v e b e t t e r i n s i g h t i n t o the b e h a v i o u r o f c o n c r e t e m a t e r i a l s s u b -j e c t e d t o t e n s i l e l o a d i n g a t h i g h s t r e s s r a t e s . T h i s i s a t t e m p t e d t h r o u g h c o n s i d e r a t i o n s b a s e d on f r a c t u r e m e c h a n i c s a p p l i e d t o an i d e a l i z e d p a r t i c -u l a t e c o m p o s i t e m a t e r i a l c o n s i s t i n g o f a g g r e g a t e p a r t i c l e s d i s p e r s e d and embedded w i t h i n the m a t r i x p h a s e . The e n e r g y a b s o r b e d d u r i n g f r a c t u r i n g o f t h e m a t e r i a l i s m a i n l y a s s o c i a t e d w i t h c r a c k e x t e n s i o n i n t h e m a t r i x ,

a g g r e g a t e and i n t e r f a c i a l bond z o n e s . The d e g r e e o f m u l t i p l e c r a c k i n g o f m a t e r i a l i s assumed t o be s t r e s s r a t e d e p e n d e n t . A l s o , f o r a g i v e n c o n f i g -u r a t i o n o f a g g r e g a t e p a r t i c l e s w i t h i n t h e m a t r i x , a p a t h o f a s i n g l e c r a c k i s assumed t o be i n f l u e n c e d by the r a t e o f l o a d i n g . T h e r e f o r e the f r a c t u r e e n e r g y e x p r e s s e d as a sum o f p r o d u c t s o f the s p e c i f i c s u r f a c e e n e r g i e s o f the m a t r i x , a g g r e g a t e and i n t e r f a c i a l bond zones w i t h a p p r o p r i a t e a r e a s of t h e s e zones w h i c h c o n t r i b u t e t o the t o t a l f r a c t u r e d a r e a , i s s t r e s s r a t e d e p e n d e n t .

An e x t e n s i v e e x p e r i m e n t a l program was c a r r i e d o u t i n o r d e r t o o b t a i n more d a t a on t h i s s u b j e c t and t o v e r i f y the p r o p o s e d m o d e l . The s c o p e o f t h i s i n v e s t i g a t i o n i s s c h e m a t i c a l l y shown i n F i g . 1.1. CONCRETE FRACTURE MECHANICS IDEALIZATION OF COMPLEX MATERIAL APPLICATION TO HETEROGENEOUS MATERIAL

INFLUENCE OF LOADING RATE UPON

FRACTURE MECHANISMS

MODEL FOR TENSILE FRACTURE OF CONCRETE

I

5 -2. LITERATURE SURVEY 2.1 U n i a x i a l t e n s i l e t e s t s on c o n c r e t e T h i s p a r t o f l i t e r a t u r e s t u d y was ment to c o l l e c t f u n d a m e n t a l i n f o r m a t i o n on the b e h a v i o u r o f c o n c r e t e a t h i g h r a t e s o f t e n s i l e l o a d i n g . The b a s i c q u e s t i o n s were:

how does t h e i n c r e a s i n g r a t e o f t e n s i l e l o a d i n g i n f l u e n c e the s t r e s s -s t r a i n b e h a v i o u r o f c o n c r e t e ?

- what i s t h e e f f e c t o f changes i n t h e mix and o t h e r p a r a m e t e r s on t h i s b e h a v i o u r ?

I t a p p e a r e d t h a t the number o f p u b l i c a t i o n c o n c e r n e d w i t h u n i a x i a l i m p a c t t e n s i l e l o a d i n g o f c o n c r e t e i s v e r y l i m i t e d . A c t u a l l y , t h e r e s u l t s o f t e s t s p e r f o r m e d i n s e v e n r e s e a r c h p r o j e c t s can be b r i e f l y d i s c u s s e d . The a t t e n -t i o n o f r e s e a r c h e r s was f o c u s e d m a i n l y on -the r e l a -t i o n s h i p be-tween -t e n s i l e s t r e n g t h and s t r e s s r a t e .

I n F i g . 2.1. t h e r e s u l t s o b t a i n e d a r e p l o t t e d i n t h e same d i a g r a m w i t h on the h o r i z o n t a l a x i s the s t r e s s o r s t r a i n r a t e and on the v e r t i c a l a x i s t h e impact t e n s i l e s t r e n g t h s t a n d a r d i z e d t o the s t a t i c s t r e n g t h . , f/f„ - o K o m l o s o H e i l m a n n * T a k e d a A K v i r i k a d z e - 0 S n e i k i n - A H a t a n o B i r k i m e r t r a f f i c e a r t h q u a k e g a s e x p t . a i r p l a n e i pile d r i v i n g 10* Ö ( N / m m2 m s ) 1 0 ' ( 1 / s ) F i g . 2.1. S u r v e y o f r e s u l t s o f e x p e r i m e n t s on t e n s i l e s t r e n g t h r e l a t e d t o t h e l o a d i n g r a t e ( a f t e r [ 5 6 ] ) .

A s s u m i n g a c o n s t a n t modulus o f e l a s t i c i t y , the l o a d i n g r a t e s a p p l i e d a r e e q u a l t o s t r a i n r a t e s v a r y i n g f r o m 10 ^ to 1 p e r s e c o n d . Impacts c a u s e d by t r a f f i c , e x p l o s i o n s , a i r p l a n e c o l l i s i o n s and p i l e d r i v i n g a r e i n c l u d e d i n t h i s l o a d i n g r a t e r a n g e . B e c a u s e t h e s t a t i c t e n s i l e s t r e n g t h was n o t a l -ways w e l l d e f i n e d o r e v e n not r e p o r t e d , t h e l o w e s t l o a d i n g r a t e v a l u e g i v e n i n e a c h i n v e s t i g a t i o n i s t a k e n as s t a t i c . The i n c r e a s e i n t e n s i l e s t r e n g t h o f c o n c r e t e w i t h i n c r e a s i n g s t r e s s r a t e i s c l e a r l y m a n i f e s t . The r e s e a r c h by Komlos [55] c o n c e r n s a l o a d i n g r a t e r a n g e o f 2.5 x 10 to 8.5 x 10 N/mm ms ( s t r a i n r a t e i s 0.8 x 10 to 30 x 10 Is). H e i l m a n n , H i l s d o r f and F i n s t e r w a l d e r [39] u s e d s t r a i n r a t e s between 10 and 10 ' / s . I n a p u b l i c a t i o n o f K v i r i k a d z e [58] r e f e r e n c e to r e s u l t s o f _ Q

S n e i k i n and N i k o l a e b i s made, w i t h l o a d i n g r a t e s between 2.7 x 10 and

—6 2 —5 4.3 x 10 N/mm ms and o f Hatano w i t h l o a d i n g r a t e s between 2.7 x 10

-1 2

and 1.4 x 10 N/mm ms. I n h i s own r e s e a r c h a s t r e s s r a t e between — 7 —6 2

1.8 x 10 and 2.7 x 10 N/mm ms i s u s e d . R e a l l y h i g h s t r a i n r a t e s a r e r e p o r t e d by B i r k i m e r and Lindemann [ 8 ] , who r e a c h e d s t r a i n r a t e s between 3 and 25/s.

The most e x t e n s i v e p r o g r a m i s r e p o r t e d by H e i l m a n n , H i l s d o r f and F i n s t e r -w a l d e r [ 3 9 ] . They -were c o n f r o n t e d -w i t h t h e f a c t t h a t the t e n s i l e s t r e n g t h o f c o n c r e t e seemed to be g r e a t l y dependent on the t e s t method. They i n -v e s t i g a t e d t h e m u t u a l i n f l u e n c e s o f age, s t r a i n r a t e , c o n c r e t e q u a l i t y and h u m i d i t y . Much a t t e n t i o n was d e v o t e d to o b t a i n i n g and c o n t r o l l i n g a u n i f o r m l y d i s t r i b u t e d g a i n i n s t r e s s d u r i n g t h e t e s t . I t t u r n e d out t h a t the u n i a x i a l t e n s i l e t e s t s showed an i r r e g u l a r s t r a i n d i s t r i b u t i o n w h i c h might be c a u s e d by n o n - u n i f o r m growth o f c r a c k s . I n t h e t e s t t h e s t r a i n , measured on two s i d e s o f t h e s p e c i m e n , was m a i n t a i n e d a t t h e same growth r a t e . The i n f l u e n c e o f t h e s t r a i n r a t e was d e t e r m i n e d on s p e c i m e n s o f

2 t h r e e d i f f e r e n t q u a l i t i e s ( c o m p r e s s i v e s t r e n g t h s 14, 22.5 and 35 N/mm ) and f o u r d i f f e r e n t ages ( 3 , 7, 28 and 90 d a y s ) . The s t r a i n r a t e s i n v o l v e d were 1.3 x 1 0 ~9, 2.1 x 1 0 ~8, 1.7 x 1 0 ~7 and 1.3 x 1 0 ~6/ s .

2

As an example, the s t r e s s - s t r a i n c u r v e f o r c o n c r e t e o f 22.5 N/mm 28-day s t r e n g t h a t d i f f e r e n t s t r a i n r a t e s i s g i v e n i n F i g . 2.2.

H i g h e r s t r a i n r a t e s seemed t o r e s u l t i n h i g h e r t e n s i l e s t r e n g t h . They f o u n d t h a t the r e l a t i v e i n c r e a s e was i n d e p e n d e n t o f t h e c o n c r e t e q u a l i t y and t h e age o f t e s t i n g .

The modulus o f e l a s t i c i t y d e t e r m i n e d f o r a s t r a i n o f 0 . 0 5 ° / i n c r e a s e d w i t h oo

i n c r e a s i n g s t r a i n r a t e . I t i s t o be n o t e d t h a t t h e s t r e s s - s t r a i n c u r v e i s a l m o s t l i n e a r t i l l j u s t b e f o r e f a i l u r e .

7 -0 ( N / m m2)

i

V

2

About t h e same t i m e , Komlos [55] p u b l i s h e d a p a p e r i n w h i c h he d e s c r i b e d the s t r e s s - s t r a i n c u r v e s f o r c o n c r e t e i n t e n s i o n a t a l o a d i n g r a t e o f

- 5 - 5 2

2.5 x 10 and 8.3 x 10 N/mm ms. He chose t h e a g g r e g a t e - c e m e n t r a t i o as a p a r a m e t e r .

r e s u l t s show an agreement w i t h t h o s e o f H i l s d o r f . He c o n c l u d e d t h a t an i n c r e a s e i n t h e l o a d i n g r a t e r e s u l t e d i n a d e c r e a s i n g s t r a i n f o r maximum s t r e s s e s and i n i n c r e a s e i n t h e maximum s t r e s s . As r e g a r d s t h e cement c o n t e n t i t was c o n c l u d e d t h a t more cement made c o n c r e t e l e s s s e n s i t i v e t o l o a d i n g r a t e v a r i a t i o n s . However, h a v i n g r e g a r d t o F i g . 2.3., d i m i n i s h i n g f r a c t u r e s t r a i n v a l u e s f o r i n c r e a s i n g l o a d i n g r a t e s a r e h a r d l y c r e d i b l e . F i g . 2.3. I n f l u e n c e o f t h e l o a d i n g r a t e on t h e s t r e s s - s t r a i n b e h a v i o u r o f c o n c r e t e w i t h d i f f e r e n t a g g r e g a t e - c e m e n t r a t i o s ( a f t e r Komlos [ 5 5 ] ) . W i t h t h e a i d o f r e s u l t s f r o m h i s own r e s e a r c h and r e s u l t s o f S n e i k i n , N i k o l a e b and H a t a n o , K v i r i k a d z e [58] d e t e r m i n e d e x p e r i m e n t a l f o r m u l a s i n w h i c h t h e s t r e n g t h o r modulus o f e l a s t i c i t y i s r e l a t e d t o t h e s t r e s s r a t e : § = 1 + 0.1 l o g

%

9

-Takeda and T a c h i k a w a [94] t e s t e d t h e i n f l u e n c e o f t h e s t r a i n r a t e upon the c o m p r e s s i v e s t r e n g t h i n an e x t e n s i v e program. O n l y one s e r i e s was p e r f o r m

-• —6

¿ = 2 . 3 l o " ^

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E =

2.3 , 0 "

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i t c a n be s e e n t h a t f o r t e n s i l e t e s t s P o i s s o n ' s r a t i o i n c r e a s e d f o r h i g h e r s t r a i n r a t e s , whereas the r e v e r s e was t r u e f o r c o m p r e s s i v e t e s t s . The s t r e s s l e v e l examined was 40% o f t h e s t r e n g t h . V

0.3

0.2

0.1

— •

and w i t h Watson [46] n o t e d more e x t e n s i v e c r a c k i n g o f c o n c r e t e a t h i g h

r a t e s o f c o m p r e s s i v e l o a d i n g , w h i c h p r e s u m a b l y s h o u l d c o r r e s p o n d to h i g h e r v a l u e s o f P o i s s o n ' s r a t i o a t t h e i n s t a n t o f f a i l u r e .

The h i g h e s t s t r a i n r a t e s were a p p l i e d i n t h e t e s t s o f B i r k i m e r and L i n d e -mann [ 8 ] . F o r p l a i n c o n c r e t e a maximum s t r a i n r a t e o f 25/s was a p p l i e d . T h i s was p o s s i b l e by u s i n g a p r i s m a t i c b a r i n w h i c h a c o m p r e s s i v e wave h a d been i n t r o d u c e d by s t r i k i n g one end o f t h e b a r . The f r e e o p p o s i t e end o f

the b a r r e f l e c t e d t h i s c o m p r e s s i v e wave as a t e n s i l e wave. The a m p l i t u d e o f t h e t e n s i l e wave was l a r g e enough t o e x c e e d t h e u l t i m a t e s t r e n g t h o f t h e c o n c r e t e s p e c i m e n . They p r o v e d t h a t t h e amount o f e n e r g y n e c e s s a r y t o p r o d u c e f a i l u r e o f t h e c o n c r e t e c a n be r e g a r d e d as a c o n s t a n t v a l u e . T h i s v a l u e i s dependent on v a r i o u s p a r a m e t e r s g i v e n i n t h e f o r m u l a : A c U = — i ~ f e ( f ) RT (2.6) U"c i s t h e c r i t i c a l o r f a i l u r e e n e r g y , A i s t h e c r o s s s e c t i o n o f t h e s p e c i -men, cL i s t h e l o n g i t u d i n a l wave v e l o c i t y i n the s p e c i m e n , f and e ( f ) a r e t h e s t r e n g t h and t h e c o r r e s p o n d i n g v a l u e o f s t r a i n a t t h e i n s t a n t o f f a i l -u r e , and RT i s t h e r i s e time o f t h e s t r e s s p -u l s e f r o m z e r o t o t h e f a i l -u r e s t r e s s . A t t h e i n s t a n t o f f a i l u r e t h e s t r e s s - s t r a i n d a t a f o r d i f f e r e n t s t r a i n r a t e s a r e shown i n F i g . 2.5. I t i s r e m a r k a b l e t h a t t h e modulus o f e l a s t i c i t y a t t h e f a i l u r e moment i s l a r g e l y i n d e p e n d e n t o f t h e s t r a i n r a t e .

20

15

10

5

) t

11 -I n a d d i t i o n t o e x p e r i m e n t a l i n v e s t i g a t i o n s t h e o r e t i c a l s t u d i e s o n t h e s t r e s s r a t e e f f e c t c a n be f o u n d . A c o n c e p t b a s e d on f r a c t u r e m e c h a n i c s combined w i t h a s t a t i s t i c a l a p p r o a c h c o n c e r n i n g t h e d e v e l o p m e n t o f c r i t i c a l c r a c k s l e d M i h a s h i and I z u m i [64] t o t h e f o l l o w i n g r e l a t i o n : 1 1 + S | = (2.8) fo °o i n w h i c h f i s t h e s t a t i c t e n s i l e s t r e n g t h a t t h e s t a t i c s t r e s s r a t e aQ, f i s t h e t e n s i l e s t r e n g t h a t t h e s t r e s s r a t e a and B i s a m a t e r i a l p a r a -m e t e r . T h i s r e l a t i o n s h i p i s a l s o v a l i d f o r c o -m p r e s s i o n . T a k i n g t o g e t h e r a l l t h e r e s u l t s f o u n d i n t h e l i t e r a t u r e i t c a n be c o n -c l u d e d t h a t an i n -c r e a s e i n s t r e s s r a t e r e s u l t s i n a h i g h e r t e n s i l e 2

s t r e n g t h . Below s t r e s s r a t e s o f 10 N/mm ms a r e l a t i o n between s t r e n g t h and the l o g a r i t h m o f s t r e s s o r s t r a i n r a t e , as g i v e n i n f o r m u l a ( 2 . 2 ) , a p p e a r s s u i t a b l e .

F o r h i g h e r s t r e s s r a t e s a r e l a t i o n as d e r i v e d b y M i h a s h i a p p e a r s t o g i v e the b e s t f i t . T h i s i s t r u e a l s o f o r lower s t r e s s r a t e s .

The r e s u l t s o f t h e f r a c t u r e s t r a i n a r e c o n f u s i n g . Komlos and H e i l m a n n i n t h e i r o b s e r v a t i o n zone i n f e r r e d a d e c r e a s e o f f r a c t u r e s t r a i n a t i n c r e a -s i n g -s t r e -s -s r a t e -s . B i r k i m e r e t a l . on t h e o t h e r hand gave a r e l a t i o n between f r a c t u r e s t r a i n and s t r a i n r a t e f r o m w h i c h an i n c r e a s i n g f r a c t u r e s t r a i n v a l u e c o u l d be e x p e c t e d f o r i n c r e a s i n g s t r a i n r a t e s . The modulus o f e l a s t i c i t y a p p e a r s t o be r a t e - d e p e n d e n t a c c o r d i n g t o f o r m u l a (2.5) o f Takeda and T a c h i k a w a . W i t h t h i s s t a t e m e n t a g r e e a l l o t h e r r e s e a r c h -e r s , -e x c -e p t B i r k i m -e r , who f o u n d a t d i f f -e r -e n t h i g h l o a d i n g r a t -e s a c o n s t a n t E v a l u e a t t h e maximum s t r e s s . I t i s n o t c l e a r i f t h e e x p l a n a t i o n o f h i s r e s u l t s must be sought i n t h e e x t r e m e l y h i g h s t r a i n r a t e s t h a t were u s e d . B e t t e r agreement c a n be f o u n d when t h e shape o f t h e s t r e s s - s t r a i n c u r v e i s o b s e r v e d . N e a r l y a l l t h e r e s e a r c h e r s f o u n d a l i n e a r c u r v e up t o j u s t b e -f o r e -f a i l u r e -f o r a l l l o a d i n g r a t e s .

I t c a n be c o n c l u d e d t h a t i n f o r m a t i o n g i v e n i n t h e l i t e r a t u r e i s n o t s u f -f i c i e n t -f o r a p r o p e r u n d e r s t a n d i n g o -f t h e b e h a v i o u r o -f c o n c r e t e m a t e r i a l s under h i g h r a t e s o f t e n s i l e l o a d i n g .

The phenomena a s s o c i a t e d w i t h i m p a c t t e n s i l e f r a c t u r e o f c o n c r e t e a r e n o t y e t f u l l y r e c o g n i z e d and t h e r e i s a l a c k o f a r e a l i s t i c model f o r s t r e s s -r a t e dependent b e h a v i o u -r o f cement-based m a t e -r i a l s . T h e r e f o r e more e x p e r i m e n t a l d a t a a r e r e q u i r e d f o r b e t t e r i n s i g h t i n t o the b e h a v i o u r o f c o n c r e t e m a t e r i a l s under t e n s i l e l o a d i n g a t v a r i o u s s t r e s s r a t e s .

A l s o , more t h e o r e t i c a l work on a c o n s i s t e n t model i n c o r p o r a t i n g the complex s t r u c t u r e o f c o n c r e t e and d i f f e r e n t mechanisms o f i t s f r a c t u r e u n d e r t e n -s i l e l o a d i n g i -s needed f o r p r a c t i c a l a p p l i c a t i o n -s .

2.2. A g g r e g a t e s and t h e i r bond w i t h cement m a t r i x

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

I n n o r m a l w e i g h t c o n c r e t e s w i d e l y u s e d i n e n g i n e e r i n g p r a c t i c e g r a v e l and sand come from v a r i o u s m i n e r a l o g i c a l s o u r c e s . Rock commonly u s e d as c o n c r e t e a g g r e g a t e s i s s t i f f e r and s t r o n g e r t h a n h a r d e n e d cement p a s t e . The i n t e r -f a c i a l bond zones u s u a l l y -f o r m the w e a k e s t l i n k i n cement-based c o m p o s i t e s . T h e r e f o r e t h e p r o p e r t i e s o f a g g r e g a t e s and t h e i r bond w i t h the cement m a t r i x w i l l be d i s c u s s e d i n t h i s s e c t i o n .

The i m p o r t a n c e o f i n t e r f a c i a l bond between a g g r e g a t e and cement m a t r i x was r e c o g n i z e d i n t h e 1960s, and s i n c e t h e n many i n v e s t i g a t i o n s have b e e n c a r r i e d out i n o r d e r to d e t e r m i n e t h e bond p r o p e r t i e s and t h e i r i n f l u e n c e on t h e s t r e n g t h o f c o n c r e t e . V a r i o u s t e s t methods have b e e n a p p l i e d and a r e r e v i e w e d by A l e x a n d e r e t a l . [ 1 ] .

Swamy [92] p o i n t e d out t h a t the bond f o r c e s between a g g r e g a t e and h a r d e n e d cement p a s t e may be s o l e l y p h y s i c a l , due t o a d h e s i o n and m e c h a n i c a l i n t e r -l o c k , o r p u r e -l y c h e m i c a -l , due to f o r m a t i o n o f new p r o d u c t s o f s u r f a c e r e a c t i o n s . I n most c a s e s , however, p h y s i c a l and c h e m i c a l f o r c e s a r e comb i n e d . The t o p o g r a p h y and d e g r e e o f s u r f a c e r o u g h n e s s o f the a g g r e g a t e i n -f l u e n c e t h e a d h e s i v e -f o r c e s , w h i l e t h e m i n e r a l o g i c a l o r i g i n o -f a g g r e g a t e d e t e r m i n a t e s the c h e m i c a l bond.

The i n t e r f a c i a l bond between a g g r e g a t e p a r t i c l e s can be b r o k e n i n two ways, as s c h e m a t i c a l l y shown i n F i g . 2.6.

I n t h e c a s e o f t e n s i l e bond f a i l u r e , s e p a r a t i o n a l o n g the i n t e r f a c e , b u t a l s o t e n s i l e f r a c t u r e o f t h e cement m a t r i x o r t h e a g g r e g a t e may o c c u r . The a c t u a l f a i l u r e mode w i l l depend on t h e p r o p e r t i e s o f t h o s e t h r e e p h a s e s .

13

I n the c a s e o f s h e a r bond f a i l u r e t h r e e t y p e s o f m a t e r i a l p h a s e s a r e l i k e -w i s e i n v o l v e d i n s l i d i n g a t the i n t e r f a c e .

F i g . 2.6. Mechanisms o f i n t e r f a c i a l bond f a i l u r e .

The t e n s i l e bond s t r e n g t h between f o u r t y p e s o f a g g r e g a t e and m o r t a r o r cement p a s t e has been i n v e s t i g a t e d by Hsu and S l a t e [41] on specimens as shown i n F i g . 2.7.

F i g . 2.7. Specimen u s e d by Hsu and S l a t e [ 4 1 ] .

The most i m p o r t a n t c o n c l u s i o n s f r o m t h i s r e s e a r c h p r o g r a m a r e t h a t the p a s t e - a g g r e g a t e t e n s i l e bond s t r e n g t h v a r i e s f r o m 41 t o 91 p e r c e n t o f t h e t e n s i l e s t r e n g t h o f p a s t e and the m o r t a r - a g g r e g a t e t e n s i l e bond s t r e n g t h v a r i e s f r o m 33 t o 67 p e r c e n t o f t h e t e n s i l e s t r e n g t h o f m o r t a r . The bond s t r e n g t h depends on r o c k t y p e , s u r f a c e r o u g h n e s s , w a t e r - c e m e n t r a t i o and cement-sand r a t i o f o r m o r t a r . Those i n f l u e n c e s a r e shown i n F i g .

0

c e m e n t m a t r i x

a g g r e g a t e

2.8.

Rehm e t a l . [81] showed t h a t the bond s t r e n g t h can be i n c r e a s e d 100 t o 300 p e r c e n t by the a p p l i c a t i o n o f some a d m i x t u r e s , w h i c h r e s u l t s i n a s t r u c

-t u r a l s -t r e n g -t h e n i n g o f -t h e c o n -t a c -t a r e a o r r e p l a c e m e n -t o f -t h e c o n -t a c -t zone by a n o t h e r s y s t e m . f t N / m m2)

>

N

\

k

h

J <PASTL^ I ^ MORTAR 1:2 > | ^ MORTAR 1:3 > | q / g

T e n s i l e bond s t r e n g t h o f p a s t e and m o r t a r t o p l a n e s u r f a c e o f c o a r s e a g g r e g a t e , and t e n s i l e s t r e n g t h o f p a s t e and m o r t a r ,

( a f t e r H s u and S l a t e [ 4 1 ] ) .

T a y l o r and Broms [ 9 5 ] , i n t h e i r e x t e n s i v e r e s e a r c h , i n v e s t i g a t e d s h e a r bond s t r e n g t h between a g g r e g a t e and cement p a s t e o r m o r t a r on s p e c i m e n s a s shown i n F i g . 2.9. a g g r e g a t e s l a b m o r t a r

i

>1W

J

15

-T h e i r r e s u l t s a r e shown i n F i g . 2.10 and F i g . 2.11 and i n d i c a t e t h a t t h e s h e a r bond s t r e n g t h i s a l m o s t p r o p o r t i o n a l t o t h e n o r m a l s t r e s s a t t h e a g g r e g a t e - p a s t e o r m o r t a r i n t e r f a c e . O n l y s m a l l v a r i a t i o n s a r e due t o changes i n water-cement o r cement-sand r a t i o s . The somewhat s u r p r i s i n g l y s m a l l i m p o r t a n c e o f water-cement r a t i o w i t h r e g a r d t o s h e a r bond s t r e n g t h , as compared w i t h t e n s i l e bond s t r e n g t h , was p r o b a b l y c a u s e d by f r e e w a t e r movement a t t h e i n t e r f a c e w h i c h i n f l u e n c e d t h e p r o p e r t i e s o f the bond zone.

0 10 20 30 40 50 0 ( N / m m2) F i g . 2.10. Shear bond r e l a t i o n -s h i p -s f o r -s a n d -s t o n e ( a f t e r T a y l o r and Broms [ 9 5 ] ) . T l N / m m2) ~1 LIMES" "ONE _1:0c/J < 1:0 0.3S w/c 1:2c/s 0.55 w; 1:2c/s 0.36w/ c ~ 0 3C/S °" 75w/c 3c/s 55w/c 0 10 20 30 40 50 0 | N / m m2) F i g . 2.11. Shear bond r e l a t i o n -s h i p -s f o r l i m e -s t o n e ( a f t e r T a y l o r and Broms [ 9 5 ] ) .

The f r i c t i o n a n g l e i s a p p r o x i m a t e l y 30°-35 and 35 - 4 0 ° f o r s a n d s t o n e and l i m e s t o n e , r e s p e c t i v e l y .

The c o h e s i o n i n t e r c e p t s were n o t c o n s i s t e n t l y i n f l u e n c e d by any p a r t i c u l a r t y p e o f a g g r e g a t e . C o m p a r i s o n w i t h r e s u l t s o f Hsu and S l a t e on t e n s i l e bond s t r e n g t h l e a d s t o t h e c o n c l u s i o n t h a t s h e a r f o r c e s l e a d t o i n t e n s i f i -c a t i o n o f p r i n -c i p a l s t r e s s e s and may t h e r e f o r e a -c -c e l e r a t e t e n s i l e - b o n d f a i l u r e .

No r e p o r t s on t e n s i l e s h e a r t e s t s p e r f o r m e d on bond zones between a g g r e gate and h a r d e n e d cement p a s t e have been f o u n d i n t h e l i t e r a t u r e . N e v e r -t h e l e s s , i -t c a n be deduced -t h a -t a -t a low -t e n s i l e s -t r e s s l e v e l a s h e a r f a i l u r e w i l l o c c u r u n d e r a s h e a r s t r e s s a p p r o x i m a t e l y e q u a l t o t h e t e n s i l e s t r e s s i n t h e case o f a p u r e t e n s i o n .

2.3. F r a c t u r e m e c h a n i c s

2.3.1 B a s i c c o n c e p t s

The b a s i c i d e a s o f f r a c t u r e m e c h a n i c s w i l l h e r e be b r i e f l y r e v i e w e d , s i n c e the b e h a v i o u r o f cementbased m a t e r i a l s s u b j e c t e d t o s t a t i c and impact u n i -a x i -a l t e n s i l e l o -a d i n g w i l l be d i s c u s s e d i n f u r t h e r c h -a p t e r s w i t h t h e -a i d o f f r a c t u r e m e c h a n i c s c o n c e p t s . A more c o m p r e h e n s i v e s t u d i e s o n f r a c t u r e m e c h a n i c s a r e t o be found i n v a r i o u s s p e c i a l i s t i c p u b l i c a t i o n s among w h i c h [60] c a n be recommended. The i n f l u e n c e o f a s i n g l e e l l i p t i c a l t h r o u g h t h i c k n e s s c r a c k w i t h r a d i u s o f c u v a t u r e o f t h e c r a c k t i p p , on t h e s t r e s s s t a t e i n an i n f i n i t e i d e a l Hookean p l a t e s u b j e c t e d t o t e n s i o n ( s e e F i g . 2.12) was f i r s t a n a l y s e d by I n g l i s [ 4 7 ] . F i g . 2.12. S t r e s s c o n c e n t r a t i o n o a t t h e t i p o f t h e l o n g e l l i p t i c a l c r a c k ( a c c o r d i n g t o I n g l i s [ 4 7 ] ) . T h e r e a f t e r , v a r i o u s s t a t e s o f s t r e s s i n p l a t e s o f f i n i t e d i m e n s i o n s w i t h e i t h e r a s i n g l e c r a c k o r s e v e r a l c r a c k s o f d i f f e r e n t l e n g t h and c o n f i g u r a t i o n were s t u d i e d by r e s e a r c h e r s who c o n s i d e r e d s e v e r a l l o a d i n g c o n d i -t i o n s . Some r e s u l -t s o f -t h e s -t r e s s a n a l y s i s m e n -t i o n e d a r e summarized i n [85] and d i s c u s s e d i n terms o f t h e s t r e s s i n t e n s i t y f a c t o r K f o r t h r e e modes o f c r a c k s u r f a c e d i s p l a c e m e n t s w h i c h a r e shown i n F i g . 2.13. The s t r e s s i n t e n s i t y f a c t o r i s a measure t h a t c h a r a c t e r i s e s t h e e l a s t i c s t r e s s and s t r a i n f i e l d a t a l l p o i n t s i n t h e v i c i n i t y o f t h e c r a c k t i p , and i s r e l a t e d o n l y t o t h e l o a d i n g and geometry o f t h e s y s t e m .

0

t t l t t t t

0

17

I - o p e n i n g mode I I - s l i d i n g mode I I I - t e a r i n g mode

F i g . 2.13 B a s i c modes o f c r a c k s u r f a c e d i s p l a c e m e n t . G r i f f i t h , i n h i s t h e o r y o f f r a c t u r e [31], u s e d r e s u l t s o b t a i n e d by I n g l i s . He s t u d i e d t h e f r a c t u r e o f an i n f i n i t e t w o - d i m e n s i o n a l i d e a l homogeneous, i s o t r o p i c and b r i t t l e m a t e r i a l i n c o n s e q u e n c e o f p r o p a g a t i o n o f a s i n g l e s h a r p c r a c k under e x t e r n a l l y a p p l i e d t e n s i o n . He e x p r e s s e d t h e t o t a l e n e r g y of t h e s y s t e m as f o l l o w s : U = (-W + U£) + us (2.9) where: u - t o t a l e n e r g y o f t h e system W - work o f a p p l i e d l o a d o r d i s p l a c e m e n t o f m a t e r i a l b o u n d a r i e s U - s t r a i n p o t e n t i a l e n e r g y B Us - e n e r g y r e q u i r e d f o r c r a c k e x t e n s i o n Under i n c r e a s i n g e x t e r n a l l o a d t h e s t r e s s i n t e n s i t y a t t h e c r a c k t i p w i l l i n c r e a s e . The c r a c k w i l l become u n s t a b l e and r e s u l t i n s p o n t a n o u s f r a c t u r e when t h e d e c r e a s e i n m e c h a n i c a l e n e r g y (-W + U ) e q u a l s o r e x c e e d s t h e s u r f a c e e n e r g y o f t h e newly formed c r a c k s u r f a c e . F o r a t h i n p l a t e ( p l a n e s t r e s s ) u n d e r c o n s t a n t l o a d t h e t o t a l e n e r g y becomes: 2 2 TTC 0 + k c y (2.10) where: a - a p p l i e d t e n s i o n 2c - c r a c k l e n g t h y — s u r f a c e e n e r g y ( p e r u n i t a r e a ) E - Young's modulus

The e q u i l i b r i u m c o n d i t i o n dU/dc = 0 i m p l i e s t h a t t h e c r a c k w i l l become un-s t a b l e f o r c r i t i c a l un-s t r e un-s un-s 0 :

a » / 2 E I ' (2.11) " ' 7T c I f the l o a d i s r e l e a s e d b e f o r e s p o n t a n e o u s c r a c k e x t e n s i o n o c c u r s , the c r a c k w i l l t e n d to c l o s e up r e v e r s i b l y . The e n e r g i e s a s s o c i a t e d w i t h a c r a c k a r e g r a p h i c a l l y shown i n F i g . 2.14. c r a c k e n e r g y 0 Us • y s S \ c r a c k N J e n g t h N N J e n g t h N \ \ U \ \ \ \ \ \ \ s F i g . 2.14 E n e r g y b a l a n c e a c c o r d i n g t o G r i f f i t h .

G r i f f i t h ' s c o n c e p t was e x t e n d e d by I r w i n [49] and Orowan [76] f o r non-b r i t t l e m a t e r i a l s i n w h i c h p l a s t i c zones a r e formed a t t h e c r a c k t i p . The s u r f a c e e n e r g y f o r such m a t e r i a l s V can be e x p r e s s e d as f o l l o w s :

r - y + yp (2.12)

where y i s a term r e f l e c t i n g d i s s i p a t i o n o f e n e r g y i n p l a s t i c p r o c e s s e s . B e c a u s e o f the e n e r g y d i s s i p a t i o n the s t a b l e c r a c k , when u n l o a d e d , w i l l n o t c l o s e up r e v e r s i b l y . S t a g e o f u n s t a b l e c r a c k growth i n w h i c h t h e c r a c k v e l o c i t y r a p i d l y i n -c r e a s e s , i s a s s o -c i a t e d w i t h the -c r i t i -c a l s t r e s s i n t e n s i t y d e s -c r i b e d by K^. The c r i t i c a l s t r e s s i n t e n s i t y f a c t o r K i s o f t e n r e g a r d e d as f r a c t u r e t o u g h n e s s o f m a t e r i a l as a m a t e r i a l c o n s t a n t r e q u i r e s geometry i n d e p e n -dence - and i s r e l a t e d t o t h e c r i t i c a l s t r e s s a : c r T = Y a /c I c c r (2.13)

19

-where Y i s a shape f a c t o r t a k i n g a c c o u n t o f s p e c i m e n and c r a c k g e o m e t r y . F o r the c o n s i d e r e d s h a r p t h r o u g h t h i c k n e s s c r a c k Y = /TT .

The g r e a t e r t h e v a l u e o f t h e f r a c t u r e t o u g h n e s s o r s p e c i f i c s u r f a c e e n e r g y , s i n c e f o r p l a n e s t r e s s :

i = K2 /2E (2.14)

I c

the more r e s i s t a n t i s t h e m a t e r i a l t o the o c c u r r e n c e o f s p o n t a n e o u s c r a c k p r o p a g a t i o n .

When a c r a c k r e a c h e s t h e p o i n t o f i n s t a b i l i t y , o r when t h e l o a d i n g i s ap-p l i e d r a ap-p i d l y , t h e c r a c k w i l l grow v e r y f a s t and t h e volume e l e m e n t s n e x t t o the c r a c k s u r f a c e w i l l be a c c e l e r a t e d . T h e r e f o r e f o r dynamic l o a d i n g t h e k i n e t i c e n e r g y component U s h o u l d be added t o the t o t a l e n e r g y o f the s y s t e m :

U = (-W + U£) + Ug + UR (2.15)

Under i m p u l s i v e l o a d i n g t h e d u r a t i o n o f the s t r e s s p u l s e i s v e r y i m p o r t a n t , s i n c e t h e a p p l i e d s t r e s s must a c t f o r some minimum time i n o r d e r t o i n i t i -a t e c r -a c k growth o r t o d r i v e -a c r -a c k t o f u r t h e r e x t e n s i o n [ 1 6 ] . Under r a p i d ( i m p a c t ) l o a d i n g i n w h i c h much e n e r g y i s i n t r o d u c e d i n t o m a t e r i a l i n a v e r y s h o r t time c r a c k b r a n c h i n g may o c c u r , t h i s b e i n g due t o t h e dynami-c a l i n t e r a dynami-c t i o n between t h e dynami-c r a dynami-c k f r o n t and some m i dynami-c r o s t r u dynami-c t u r a l inhomogen i t i e s c a u s i inhomogen g c r a c k t i p f i e l d d i s t o r t i o inhomogen o r due t o i inhomogen i t i a t i o inhomogen o f s e c o inhomogen d a -r y f -r a c t u -r e s .

2.3.2 A p p l i c a t i o n t o cement-based m a t e r i a l s

G l u c k l i c h [28] was the f i r s t who p o i n t e d o u t d i f f i c u l t i e s i n t h e d i r e c t a p p l i c a t i o n o f the a b o v e m e n t i o n e d c o n c e p t s t o the a n a l y s i s o f t h e b e

-h a v i o u r o f c o n c r e t e . Cement-based c o m p o s i t e s a r e a n i s o t r o p i c , -h e t e r o g e n e o u s , m u l t i p h a s e systems w i t h i n h e r e n t m i c r o c r a c k s i n the cement m a t r i x p h a s e , i n t e r f a c i a l bond m i c r o c r a c k s and a i r v o i d s .

A n o t h e r p o i n t i s t h a t even h a r d e n e d cement p a s t e i s a c t u a l l y a h e t e r o g e -neous m a t e r i a l i f i t s m i c r o s t r u c t u r e i s t a k e n i n t o c o n s i d e r a t i o n .

The complex m i c r o s t r u c t u r e o f cement p a s t e changes w i t h t i m e and may be i n f l u e n c e d by a number o f t e c h n o l o g i c a l and e n v i r o n m e n t a l p a r a m e t e r s .

T h e r e f o r e t h e f r a c t u r e d s u r f a c e o f cement p a s t e may c o n s i s t o f v a r i o u s e l e m e n t s as shown i n F i g . 2.15 a f t e r Grudemo [ 3 2 ] . C l e a v a g e s u r f a c e s i n l a m e l l a r CH p h a s e , r u p t u r e d zones o f c o n t a c t between g e l a g g r e g a t e s , r a d i a l c l e a v a g e f r a c t u r e s t h r o u g h o u t e r and i n n e r g e l s u r f a c e s o f r e m a i n -i n g anhydrous n u c l e -i and -i n t e r n a l s u r f a c e s o f -i n n e r g e l a f t e r r e m o v a l o f t h e anhydrous n u c l e u s c a n be d i s t i n g u i s h e d . F i g . 2.15. P o s s i b l e e l e m e n t s o f t h e f r a c t u r e d s u r f a c e c a u s e d by t e n s i l e f r a c t u r e i n cement p a s t e ( a f t e r Grudemo [ 3 2 ] ) .

Grudemo p o i n t e d o u t t h a t a l s o i n cement p a s t e c r a c k s may be a r r e s t e d , as shown i n F i g . 2.16.

4 4 ^

I t +

F i g . 2.16. Mechanisms f o r a r r e s t i n g t h e growth o f m i c r o c r a c k s i n cement p a s t e ( a f t e r Grudemo [ 3 2 ] ) .

The growth o f m i c r o c r a c k s i n t h i s m a t e r i a l c a n be a r r e s t e d i f t h e c r a c k t i p e n c o u n t e r s m a t e r i a l zones o f h i g h e r c a p a c i t y t o t a k e up t h e c o n c e n t r a -t i o n o f -t e n s i l e s -t r e s s e s i n -t h e c r a c k -t i p , o r -t h e s -t r e s s c o n c e n -t r a -t i o n d e c r e a s e s due t o a r o u n d i n g - o f f o f t h e c r a c k f r o n t .

21

-The m i c r o c r a c k s i n c o n c r e t e a r e p o t e n t i a l s o u r c e s o f c r a c k i n i t i a t i o n i n f r a c t u r e p r o c e s s . The t o u g h a g g r e g a t e p a r t i c l e s embedded i n t h e cement m a t r i x and a i r v o i d s may p e r f o r m a c r a c k - a r r e s t i n g a c t i o n . Some mechanisms of c r a c k a r r e s t i n c o n c r e t e have been r e c o g n i z e d by Shah and McGarry [88] and a r e shown i n F i g . 2.17. The c r a c k a p p r o a c h i n g an a g g r e g a t e p a r t i c l e can be s t o p p e d s i n c e t h e a g g r e g a t e i s t o u g h e r t h a n t h e cement m a t r i x and i t s f r a c t u r i n g r e q u i r e s more e n e r g y ( a ) . I f t h e c r a c k tends t o e x t e n d around the p a r t i c l e (b) more e n e r g y w i l l be n e c e s s a r y , s i n c e t h e s u r f a c e o f t h e c r a c k i n c r e a s e s i n c o m p a r i s o n w i t h t h a t o f a s t r i g h t c r a c k i n t h e a b s e n c e o f a g g r e g a t e p h a s e . I n t h e c a s e ( c ) t h e c r a c k j o i n s t h e e x i s t i n g i n t e r -f a c i a l bond m i c r o c r a c k and i t s l e n g t h i n c r e a s e s r a p i d l y ; t h e m a t e r i a l i n t h a t zone w i l l become more r e s t r a i n e d . T h e r e f o r e t h e a p p l i e d l o a d h a s t o be i n c r e a s e d i n o r d e r t o f o r c e t h e c r a c k t o f u r t h e r e x t e n s i o n . When t h e c r a c k e n t e r s t h e zone between two p a r t i a l l y debonded a g g r e g a t e p a r t i c l e s

(d) i t w i l l v e r y p r o b a b l y be s t o p p e d due t o t h e low s t r e s s i n t e n s i t y . The c r a c k e n t e r i n g an a i r v o i d (e) i s a l s o supposed t o s t o p , s i n c e t h e r a d i u s o f a c r a c k t i p w i l l i n c r e a s e and t h e r e f o r e t h e s t r e s s c o n c e n t r a t i o n w i l l d e c r e a s e . F i g . 2.17. The c r a c k a r r e s t i n g f u n c t i o n o f a g g r e g a t e p a r t i c l e s . I n t h e f r a c t u r e a n a l y s i s o f c o n c r e t e m a t e r i a l s a v e r y i m p o r t a n t f a c t i s t h a t t h e c r a c k i n g p r o c e s s i s n o t l i m i t e d o n l y t o t h e major c r a c k b u t a l s o i n c l u d e s e x t e n s i v e m i c r o c r a c k i n g i n t h e h i g h l y s t r e s s e d c r a c k t i p z o n e s . A l l t h e a b o v e - m e n t i o n e d phenomena l e a d t o t h e c o n c l u s i o n t h a t t h e e n e r g y r e q u i r e d p e r u n i t a r e a o f c r a c k e x t e n s i o n i n c o n c r e t e i n c r e a s e s w i t h i n -c r e a s i n g -c r a -c k l e n g t h as i n d i -c a t e d i n F i g . 2.18.

F i g . 2.18. E n e r g y b a l a n c e f o r c o n c r e t e .

Many a t t e m p t s have been made t o a n a l y s e the b e h a v i o u r o f c o n c r e t e w i t h the a i d o f a p p r o a c h e s b a s e d on f r a c t u r e m e c h a n i c s .

R e c e n t l y , the i d e a s o f Dugdale [21] and B a r e n b l a t t [ 4 ] , who i n t r o d u c e d models b a s e d on e x t e n s i o n o f f i c t i t i o u s c r a c k a b l e t o t r a n s f e r t e n s i l e s t r e s s e q u a l t o t h e y i e l d s t r e s s , have b e e n f u r t h e r d e v e l o p e d by Modeer, P e t e r s s o n and H i l l e r b o r g [37, 38, 71, 79] f o r c o n c r e t e .

They r e p r e s e n t e d the f r a c t u r e zone o f c o n c r e t e by a f i c t i t i o u s c r a c k t h a t i s a b l e t o t r a n s f e r s t r e s s i n s u c h a way t h a t i t s s t r e s s - t r a n s f e r r i n g c a p a b i l i t y d e c r e a s e s w i t h i n c r e a s i n g c r a c k w i d t h ( s e e F i g . 2.19). F i g . 2.19 a) The f r a c t u r e zone i n a r e a l m a t e r i a l . b) I n FCM t h e f r a c t u r e zone i s r e p r e s e n t e d by a c r a c k t h a t can t r a n s f e r s t r e s s . c) F E M - r e p r e s e n t a t i o n o f FCM.

23

-They combined t h e f i c t i t i o u s c r a c k model (FCM) w i t h t h e f i n i t e e l e m e n t method (FEM) f o r the p r e d i c t i o n o f m a t e r i a l b e h a v i o u r - m a i n l y i n t e n s i o n and b e n d i n g . They assumed t h a t t h e f r a c t u r e zone s t a r t s d e v e l o p i n g a t one p o i n t when the f i r s t p r i n c i p a l s t r e s s r e a c h e s the t e n s i l e s t r e n g t h . The t e n s i l e s t r e n g t h and the f r a c t u r e e n e r g y s h o u l d be o b t a i n e d f r o m the c o m p l e t e s t r e s s - s t r a i n d i a g r a m s . The a s c e n d i n g p a r t o f t h e o - e - c u r v e t a k e s a c c o u n t o f t h e b e h a v i o u r of m a t e r i a l o u t s i d e t h e f r a c t u r e zone and the d e s c e n d i n g p a r t o f t h e a - e - c u r v e d e t e r m i n e s a r e l a t i o n between s t r e s s and a b s o l u t e d e f o r m a t i o n o f t h e f r a c t u r e d zone. T h i s model i s c o n v e n i e n t f o r a n a l y s i s o f b e h a v i o u r o f c o n c r e t e e l e m e n t s b u t c a n n o t be u s e d f o r a n a l y s i n g f r a c t u r e mechanisms a s s o c i a t e d w i t h t h e c r a c k e x t e n s i o n w i t h i n a complex s t r u c t u r e o f cement-based m a t e r i a l s . F o r t h e l a t t e r p u r p o s e not o n l y t h e l o c a l s t r e s s has t o be c o n s i d e r e d , b u t c r a c k a r r e s t i n g p r o c e s s e s and t h e e f f e c t s o f t h e way o f l o a d i n g upon t h e c r a c k p r o p a g a t i o n s h o u l d a l s o be t a k e n i n t o c o n s i d e r a t i o n . I t i s b e l i e v e d t h a t an a p p r o a c h b a s e d on c o n s i d e r a t i o n s o f f r a c t u r e e n e r g y i s more a d e q u a t e f o r c o n c r e t e a n a l y s i s t h a n any a p p r o a c h b a s e d on l o c a l s t r e s s as a f r a c t u r e c r i t e r i o n . I n t h e f r a c t u r e o f c o n c r e t e the s u r f a c e e n e r g y o f t h e main c r a c k i n v o l v e s e n e r g i e s o f t h e cement m a t r i x p h a s e , i n t e r f a c i a l bond zones between ag-g r e ag-g a t e p a r t i c l e s and cement m a t r i x , and a ag-g ag-g r e ag-g a t e . Any chanag-ges i n t h e c o m p o s i t i o n o f the c o n c r e t e and 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 s u s e d w i l l i n f l u e n c e t h e e n e r g y a s s o c i a t e d w i t h c r a c k growth. T h i s i s t h e r e a s o n f o r the wide s c a t t e r o f t h e r e s u l t s o f i n v e s t i g a t i o n s on f r a c t u r e p a r a m e t e r s o f m o r t a r s and c o n c r e t e s d e t e r m i n e d by v a r i o u s r e -s e a r c h e r -s [52, 63, 53, 101, 88, 100, 9, 80, 67, 73, 68, 23, 35, 27, 36, 32, 78, 103, 104, 10] ( i n c h r o n o l o g i c a l o r d e r ) who a p p l i e d d i f f e r e n t t e s t methods.

The i n v e s t i g a t i o n s on the f r a c t u r e o f cement-based c o m p o s i t e s have been r e v i e w e d and d i s c u s s e d by Naus e t a l . [ 7 3 ] , Z i e g e l d o r f [103] and r e c e n t l y by Swamy [ 9 3 ] . I t was c o n c l u d e d t h a t the a p p l i c a t i o n o f f r a c t u r e m e c h a n i c s t o c o n c r e t e m a t e r i a l s i s v e r y much i n i t s i n f a n c y and i s s t i l l q u e s t i o n a b l e , s i n c e the f r a c t u r e p r o c e s s i n s u c h m a t e r i a l s i s o f a g r e a t c o m p l e x i t y and h e t e r o g e n i t y . However, i t i s a g r e e d t h a t t h e a p p l i c a t i o n o f the l i n e a r e l a s t i c f r a c t u r e m e c h a n i c s to cement p a s t e a p p e a r s j u s t i f i e d , s i n c e t h i s m a t e r i a l i s b r i t t l e and r a t h e r homogeneous.