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

theoretical and applied f~dure

mechanlcs

Fractal analysis of fracture in concrete

Jun Peng a,*, Zhimin Wu b, Guofan Zhao 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. Introduction

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 of 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-

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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. Fractal characteristics of concrete fracture

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 o f 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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136 J. Peng et al. / Theoretical and Applied Fracture Mechanics 27 (1997) 135-140

.gg ,eg. ,e i'7 ok,.to Aggregate~ ~ ,, cement

Cement Cement

l l l l t t t t t t

Aggregat Crack into ', ¢ aggregate

Cement

It tt Fig. 1. Crack growth paths in concrete: (a) crack at interlace, (b)

crack into cement, and (c) c rack into aggregate .

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

Table 1

Composi t ions of concrete specimens ( k g / m 3)

Group Cement Coarse aggrega te (mm) Fine Wate r

8 ~ 20 20 ~ 40 40 ~ 80 80 ~ 150 aggrega te

A 427 1214 - - - - - - 654 205

B 386 685 685 - - - - 560 185

C 344 504 504 505 - - 478 165

D 312 402 403 402 403 428 150

D

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 )

I n ( 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

~ Camera

Pol]rlz

~Ouarter-wave

\ ° ' T " ' D--/

\~j/ / S Reflecting surface

III/r'll//]/lIII /I/~//// Specimen

Fig. 2. Schemat ic d iagram of photoelast ic coat ing set-up.

J. Peng et al. / Theoretical and Applied Fracture Mechanics 27 (1997) 135-140 137

(a) (b) (c)

(d) (e) (f) Fig. 3. Crack growth paths shown by photoelastic coating method for P/Pmax equals to (a) 0.555, (b) 0.615, (c) 0.704, (d) 0.778, (e) 0.859 and (f) 0.926.

technology, the fractal dimension of concrete frac- ture surfaces in our tests is from 1.15 to 1.24.

3. Dynamic model of microcrack

Under external loadings, the coalescence of neighboring microcracks depends not only on the physical and mechanical properties of concrete, but also on the relative distribution of the cracks, and sizes of specimens. A simple mechanics analysis will be given to account for microcrack coalescence.

Fig. 4 shows that two cracks with length c are collinear, they are spaced at a distance d apart and subjected to stress o- 0. Dimensional analysis leads to expression for the crack tip stress [16]

o" 0

where r is the distance from the crack tip. The average stress in the ligament d is

°v 1 loaf( r d ) ( d ) °0 ~ , dr = F (6)

ttttti°tttti C C

1111111111 %

Fig. 4. Coalescence of two collinear cracks.

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

o r

c = - = c ( 7 ) c ~o-0]

For two microcracks with length c t and c 2, the resulting stress will be shared by the neighboring media. Taking into account the stress intensity factor a , the averaged stress in the ligament d is given by

(c , /2 ) 4Oo + ( c2/2) ,~Oo + dOo o-~= d

(oc ) =° ' ° a + l , (8)

in which

C I + C 2

Cv 2 (9)

it follows that

d a o o L . . . . (10)

C v O" v -- Or 0

using a stress criterion, the microcracks will assume to coalesce when o-v _> o-~ with o-~ being the critical stress. The microcrack coalescence threshold can be given as

t~ tY 0

L < L c ° ' c - ° - o (11)

For different O-v/O- ~ and stress intensity factor o~, the coalescence threshold L c will be different. Test results give: % = 0.55 to 0.81 G, a = 1.8 to 6.5, so that L~ = 2.2 to 27.7.

4. Statistical fractal model

Studies on the formation mechanism of fractal surfaces have led to a number of macroscopic geo- metrical fractal models, such as intergranular fracture model for metals and transgranular fracture model for geomaterials. However, the dynamic process of damage and fracture were not considered.

Consider the fractal characteristics of concrete, it can be assumed that

d L

z cCvC-- - L c (12)

Fig. 5. The statistical fractal model.

The microcrack coalescence threshold can be ex- pressed as follows:

77" 7r 0 _ < z < l - ~ - ~ 0 , 0 1 , 0 2 < ~- (13)

As shown in Fig. 5. the 'practical measurable' length of coalesced crack is

lp = ClCOS 0 1 + C2COS 0 2 + d (14)

The corresponding effective length is

C e : CICOS 01 -[- C2COS 0 2 "[- dcos 0 (15)

The number of microcracks is very large and differ- ent forms of coalescence may exist. From the statisti- cal point of view, the practical and effective lengths of cracks can be taken as follows:

1 ~'7r/2 , '¢r /2 rTr /2 f l ip-- 7 L ,2L o, Lo,do lp dz d0 d0 , d 0 2

= o ~ - + ( 1 6 ) 7"/"

i f ,2 f , "Ce = - " ~ - ¢ r / 2 " - ~ / 2 " - ~ / 2 " 0 G dz dO~ d02 dO

( 4 + L ~ ) = c, [ - - - ~ . (17)

From Eq. (2), the dimension of fracture surfaces is given by

D = ln(~p/Ov) l n ( 4 / = + L J 2 ) = ( 1 8 ) ln(~¢/ov) ln(4/~+LJ~)

Tests showed that the coalescence threshold can be assumed L¢ = 2.2 to 27.7, this gives fractal dimen- sion of 1.17 to 1.27, which is in agreement with the experimental results.

5. Discussion and conclusion

The experimental results of fractal dimension and fracture energy Gf of concrete are shown in Fig. 6.

J. Peng et al. / Theoretical and Applied Fracture Mechanics 27 (1997) 135-140 139

z

=..E

.=

5.3

0.10 O. 5 0.20 0.25

Fractal dimension D

Fig. 6. Fracture energy versus fractal dimension for concrete (present work).

are closely related to the dynamics of fracture. This has also been shown in [6].

The present statistical fractal model can explain the above experimental results. As shown in Fig. 8, fractal dimension vs. coalescence threshold 1 / L c is a unimodal curve. The curve in Fig. 8 is in agree- ment with those in Figs. 6 and 7.

To conclude concrete fracture surface can be re- garded as fractal statistically. The formation of frac- tal surfaces can be attributed to the dynamic process of damage evolution.

0.6

0.4

0.2

/ .N / x\

/ / 4340 st \ \

lumines e % 300 grad meraglng ",~.= steel ~.~.

Titanium\~ / e,oye ¢~"

" " 10 12 14 16

Specific energy In ~ IN/m)

Fig. 7. Fractal dimension versus surface energy [17].

Note that the curve attains a unimodal relation. Similar trend is found for the experimental results in [17] that summarizes the fractal dimension D as a function of the surface energy r for different metal alloys as illustrated in Fig. 7.

Poor explanation for these phenomena were at- tributed to the lack of experimental accuracy of the fractal dimensions. The present study shows that the formation of fractal surfaces and dimensional changes

0.28

g 0.24

~3 ~ 0.20

0.18, 014 0'.8 1.2 118 ;.0 Coalescence threshold l l L c

Fig. 8. Variations of fractal dimension with inverse of coalescence threshold parameter.

Acknowledgements

This work is financially supported by the National Science Foundation of China. The authors would like to express their thanks to the engineers Xiwen Wang and Wanmin Yu for their assistance in experimental work.

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