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E L S E V I E R

Theoretical and Applied Fracture Mechanics 27 (I 997) 135-140

t h e o r e t ic l n d

a p p l ie d f ~ d u r e

mech nlcs

rac ta l ana lys i s o f f rac ture in concre te

J u n P e n g a , , Z h i m i n W u b, G u o f a n Z h a o b

a Department of Civil and Constructing Engineering, Dalian University, Dalian 116622, PR China

b Department of Cioil Engineering, Dalian University of Technology, Dalian 116023, PR China

Abstract

Experimental results indicate that propagation paths of cracks in concrete are often irregular, producing rough fracture

surfaces which are fractal. Based on dynamic analysis of microcrack coalescence, this paper presents a statistical fractal

model to describe the damage evolution of concrete. The model demonstrates that the mechanism of fracture surfaces formed

in concrete is closely related to the dynamic processes of the cascade coalescence of microcracks. A unimodal relation

between the fractal dimension and the coalescence threshold can qualitatively explain the relation between fractal dimension

and fracture energy.

I I n t r o d u c t i o n

Fracture mechanics was first applied to study the

failure behavior of concrete in [1]. It was based on

the assumption that cracks are smooth and straight.

This, however, is contrast to the fact that cracks in

concrete follow a zigzag pattern. To better under-

stand the fracture behavior of concrete, irregularity

in the crack path should be considered.

In recent years, fractal geometry has been widely,

used to describe some irregular phenomena in the

fracture behavior of materials [2-8]. However, the

correlation between fractal dimension and fracture

energy was contrary to experimental observation

[3,5,9]. One of the reasons is that most o f the works

were limited to geometrical description of fractal

surfaces. Dynamic effects related to the formation of

fractal surfaces were neglected as a rule.

In what follows, the fracture behavior of concrete

is analyzed by fractal geometry. Based on experi-

Corresponding author. Fax: + 86-411-3633080.

mental and analytical results that include the dynam-

ics of microcrack coalescence, the crack formation

process is first studied. The fractal dimension as

affected by the fracture energy is then explained

qualitatively.

2 F r a c t a l c h a r a c t e r i s t ic s o f c o n c r e t e f r a c t u r e

It was pointed out in [10,11] that the microstruc-

ture of concrete contains a huge number of cracks

prior to any loading. These microcracks are usually

formed by the hydration and segregation process.

Their subsequent nucleation, growth, and interaction

are responsible for the macroscopic failure of the

solid [12-14].

2.1. Description of concrete specimen

Concrete is a multiphase material composed of

coarse and fine aggregates, cement and water. Cracks

in concrete usually propagate along three paths as

shown in Fig. l(a-c) which correspond respectively,

to kinking along the interfaces between aggregate

0167-8442/97/$17.00 Copyright 1997 Elsevier Science B.V. All rights reserved.

PII

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J. Peng et al. / Theoretical and Applied Fracture Mechanics 27 (1997) 135- 140

. g g e g . e i 7 o k , . t o

A g g r e g a t e ~

~ ,,

cement

Cement Cement

l l l l t t t t t t

Aggregat

Cracknto

',

aggregate

Cement

I t t t

F i g . 1 . C r a c k g r o w t h p a t h s i n c o n c r e t e : a ) c r a c k a t i n t e r l a c e , b )

c r a c k i n to c e m e n t , a n d c ) c r a c k i nt o a g g r e g a t e .

and cement paste, kinking into the cement paste and

into the aggregate. Aggregates usually consist of

crushed stones, the surfaces of which are fractal [15].

For a certain kind of concrete, the physical and

mechanical properties of the cement paste are fixed.

Subjected to external loadings, the crack propagating

paths may be assumed as self-similar. Attempts will

be made to verify such a behavior by experiments.

Because the maximum coarse aggregate sizes are

different the specimens are divided into four groups

(A to D). The cement type is No. 425 common

portland cement. Coarse aggregates are mechanical

crushed limestone with maximum sizes of 20, 40, 80

and 150 mm. Fine aggregates are washed river sand

with finess modulus of 3.20. The compositions of the

concrete specimens in Table 1 are determined such

that the uniaxial compressive strengths (6 12 in.

cubic strength) and the uniaxial tensile strength of

specimens in four groups are the same (30.4 MPa

and 2.87 mPa).

The sizes of specimens in four groups are (L X h

t) 450 x 450 x 450 mm (18 X 18 18 in.), with

a / h

= 0.4, where a is the length of precast crack.

2.2. Fractals

Shown in Fig. 2 is a set-up for photoclastic

coating which is applied to observe the crack paths

in the splitting-tensile tests. The results are given in

Fig. 3(a) to (f), inclusive for different ratios of

P/Pmax

where P is the current load while Pmax is

the maximum load. Crack propagation paths are

shown in these photos. When the color of strips

changes from green to yellow, the microcracks which

are invisible by naked eyes coalesce and become

T a b l e 1

C o m p o s i t i o n s o f c o n c r e t e s p e c i m e n s k g / m 3 )

G r o u p C e m e n t C o a r s e a g g r e g a t e m m ) F i n e W a t e r

8 ~ 2 0 2 0 ~ 4 0 4 0 ~ 8 0 8 0 ~ 1 5 0 a g g r e g a t e

A 4 2 7 1 2 1 4 - - - - - - 6 5 4 2 0 5

B 3 8 6 6 8 5 6 8 5 - - - - 5 6 0 1 8 5

C 3 4 4 5 0 4 5 0 4 5 0 5 - - 4 7 8 1 6 5

D 3 1 2 4 0 2 4 0 3 4 0 2 4 0 3 4 2 8 1 5 0

o r

macroscopic in size. Direct observation demonstrates

that portion of the crack paths are self-similar.

For curves with fractal characteristic, they can be

described as

U r ) - r - D

1)

o r

L r )

=N( r) -r r -D 2)

In this way D can be obtained as follows:

In N(r)

In(I / r ) 3)

In

L r )

D = 1 + ln(1/r-----~ (4 )

where r is a measurement scale with dimension of

length,

N r )

is number of measurements,

L r )

is the

length of the corresponding curve and D is dimen-

sion of fractal. By means of the graphics analysis

~ C a m e r a

P o l ] r l z

~ O u a r t e r - w a v e

\ ' T '

D - - /

\~j/ / S R e f le c t in g u r f a c e

I I I / r l l / / ] / l I I I / I / ~ / / / / S p e c i m e n

F i g . 2 . S c h e m a t i c d i a g r a m o f p h o t o e l a s t i c c o a t i n g s e t- u p .

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J. Peng et al. / Theoretical and Applied Fracture Mechanics 27 (1997) 135-140

137

a ) b ) c )

d ) e ) f )

F i g . 3 . C r a c k g r o w t h p a t h s s h o w n b y p h o t o e l a st i c c o a ti n g m e t h o d f o r

P/Pmax

e q u a l s t o ( a ) 0 .5 5 5 , ( b ) 0 . 6 1 5 , ( c ) 0 . 7 0 4 , ( d ) 0 . 7 7 8 , ( e ) 0 . 8 5 9

a n d ( f ) 0 . 9 2 6 .

t e c h n o l o g y , t h e f r a c t a l d i me n s i o n o f c o n c r e t e f r a c -

ture surfaces in our tes ts i s from 1.15 to 1 .24.

3 D y n a m i c m o d e l o f m i c r o c ra c k

U n d e r e x t e r n a l l o a d i n g s , t h e c o a l e s c e n c e o f

n e i g h b o r i n g mi c r o c r a c k s d e p e n d s n o t o n l y o n t h e

p h y s i c a l a n d me c h a n i c a l p r o p e r t i e s o f c o n c r e t e , b u t

a l so on the re l a t ive d i s t r ibu t ion o f the c racks , and

s i z e s o f s p e c i me n s . A s i mp l e me c h a n i c s a n a l y s is w i l l

b e g i v e n t o a c c o u n t f o r m i c r o c r a c k c o a l e s c e n c e .

Fig . 4 shows tha t two cracks wi th l eng th c a re

co l l inear , t hey a re spaced a t a d i s t ance d apar t and

sub jec ted to s t res s o -0 . Dimens iona l ana lys i s l eads to

express ion fo r the c rack t ip s t res s [16]

o 0

w h e r e r i s t h e d i s t a n c e f r o m t h e c r a c k t i p . T h e

average s t res s in the l igament d i s

v 1 l o a f ( r d ) ( d )

0 ~ , d r = F ( 6 )

t t t t t t t t t

F i g . 4 . C o a l e s c e n c e o f t w o c o l l i n e a r c r a c k s .

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138 J. Peng et al. / Theoretical and Applied Fracture Mechanics 27 (1997) 135-14 0

o r

c = - = c 7 )

c ~ o - 0 ]

F o r t w o m i c r o c r a c k s w i t h l e n g t h c t a n d c 2 , th e

r e s u l t i n g s t r e s s w i l l b e s h a r e d b y t h e n e i g h b o r i n g

m e d i a . T a k i n g i n t o a c c o u n t t h e s t r e s s i n t e n s i t y f a c t o r

a , t h e a v e r a g e d s t re s s in t h e l i g a m e n t d i s g i v e n b y

c , / 2 ) 4 O o + c 2 / 2 ) ,~ O o + dO o

o-~= d

o c )

= a + l , ( 8 )

i n w h i c h

C I + C 2

C v 2 ( 9 )

i t f o l l o w s t h a t

d a o o

L . . . . ( 1 0 )

C v O -- Or0

u s i n g a s t r e s s c r i t e r i o n , t h e m i c r o c r a c k s w i l l a s s u m e

to co ale sc e w he n o-v _> o-~ w ith o-~ bein g th e cr i t ic al

s tr e ss . T h e m i c r o c r a c k c o a l e s c e n c e t h r e s h o l d c a n b e

g i v e n a s

t~ tY0

L

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J. Peng et al. / Theoretical and Applied Fracture Mechanics 27 (1997) 135-140

139

z

= E

=

5 . 3

0 . 1 0

O . 5 0 . 2 0 0 . 2 5

Fractal d imension D

Fig. 6. Fracture energy versus fractal dimension for concrete

present work).

a r e c l o s e l y r e l a t e d t o t h e d y n a mi c s o f f r a c t u r e . T h i s

h a s a l s o b e e n s h o w n i n [6 ].

T h e p r e s e n t s t a t i s t i c a l f r a c t a l mo d e l c a n e x p l a i n

t h e a b o v e e x p e r i me n t a l r e s u l t s . A s s h o w n i n F i g . 8

f r a c t a l d i me n s i o n v s . c o a l e s c e n c e t h r e s h o l d 1/L c is

a u n i mo d a l c u r v e . T h e c u r v e i n F i g . 8 i s i n a g r e e -

ment wi th those in F igs . 6 and 7 .

T o c o n c l u d e c o n c r e t e f r a c t u r e s u r f a c e c a n b e r e -

garded as f rac ta l s t a t i s t i ca l ly . The fo rmat ion o f f rac-

t a l s u r fa c e s c a n b e a t t ri b u t e d to t h e d y n a m i c p r o c e s s

o f d a ma g e e v o l u t i o n .

0 . 6

0 . 4

0 . 2

/ .N

/ x \

/ / 4340 st \ \

l um i n e s e %

300 grad

me rag l n g ,~ .=

steel

~ ~

T i t an i u m \ ~

/ e o y e ~

1 0 1 2 1 4 1 6

Specif ic energy In ~ IN /m )

Fig. 7. Fractal dim ension versus surface energy [17].

Note tha t the cu rve a t t a ins a un imodal re l a t ion .

S i mi l a r t r e n d i s f o u n d f o r t h e e x p e r i me n t a l re s u l ts i n

[17] tha t summar izes the f rac ta l d imens ion D as a

f u n c t i o n o f th e s u r f a c e e n e r g y r f o r d i f f e r e n t me t a l

al loys as i l lus t rated in Fig. 7 .

Po o r e x p l a n a t i o n f o r t h e s e p h e n o me n a w e r e a t -

t r i b u t e d t o t h e l a c k o f e x p e r i me n t a l a c c u r a c y o f t h e

f r a c ta l d i me n s i o n s . T h e p r e s e n t s t u d y s h o w s t h a t t he

f o r ma t i o n o f f ra c t a l s u r f a c e s a n d d i me n s i o n a l c h a n g e s

0 . 2 8

g 0.24

~ 3

~ 0 2 0

0 . 1 8 , 0 1 4 0 . 8 1 . 2 1 1 8 ; . 0

Coalescence threshold l l L c

Fig. 8. Va riat ions of fractal dim ension with inverse of coalescence

threshold parameter.

cknowledgements

T h i s w o r k i s f i n a n c i a l l y s u p p o r t e d b y t h e N a t i o n a l

Sc i e n c e Fo u n d a t i o n o f Ch i n a . T h e a u t h o r s w o u l d l i k e

t o e x p r e s s t h e i r t h a n k s t o t h e e n g i n e e r s X i w e n W a n g

a n d W a n mi n Y u f o r t h e i r a s s i s t a n c e i n e x p e r i me n t a l

w o r k .

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