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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
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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
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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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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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