Experimental study on the effect of aggregate content on fracture behavior of concrete

Engineering Fracture Mechanics 67 (2000) 65±84

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Experimental study on the e€ect of aggregate content on fracture behavior of concrete Felix E. Amparano a, Yunping Xi b,*, Young-Sook Roh b a New Jersey Division, Federal Highway Administration, West Trenton, NJ 08628-1019, USA Department of Civil, Environmental, and Architectural Engineering, University of Colorado at Boulder, Campus Box 428, Boulder, CO 80309-0428, USA

b

Received 13 August 1998; received in revised form 21 March 2000; accepted 25 March 2000

Abstract E€ect of aggregate content on fracture behavior of concrete is studied by testing on 48 geometrically similar three-point bend concrete beams. The results are analyzed by using a size e€ect method, in which the fracture behavior of concrete is characterized by two parameters: fracture energy Gf and e€ective fracture process zone cf . Test results showed that with increasing volume fraction of aggregate in the range 45±75%: (1) the compressive strength of concrete decreases slightly (15%), and can be practically considered to be a constant; (2) fracture energy Gf varies within 25%, and there is not critical volume fraction which gives the maximum Gf ; and (3) the size of the fracture process zone decreases, which may be explained by the change in coarseness of grain structures de®ned in terms of mosaic patterns. 7 2000 Elsevier Science Ltd. All rights reserved. Keywords: Fracture; Concrete; Aggregates; Mosaic patterns; Size e€ect; Beam tests

1. Introduction The e€ect of inclusions on fracture behavior of composites has been a major research topic for many years, and has been studied from di€erent points of view. First of all, it has been found that the properties of matrix and inclusions in a composite material have dominant e€ects on the e€ective properties of composite. Secondly, bond behavior in the interface between the matrix and the inclusion has strong e€ect on the properties of composites since the e€ectiveness of the reinforcement provided by the addition of particles depends on the interfacial bond. Thirdly, the size distribution, volume fraction, and shape of the particles have a considerable e€ect on some properties of composite materials, for * Corresponding author. Tel.: +1-303-492-8891; fax: +1-303-492-7317. E-mail address: [email protected] (Y. Xi). 0013-7944/00/$ - see front matter 7 2000 Elsevier Science Ltd. All rights reserved. PII: S 0 0 1 3 - 7 9 4 4 ( 0 0 ) 0 0 0 3 6 - 9

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instance, viscosity and fracture toughness. This is due to the fact that the size, shape, and content of the particles predominantly control the morphological features of the internal structure of composite. Systematic experiments have been performed to evaluate the e€ect of morphology of the internal structure on fracture property of various composites with brittle particulate and ductile inclusions in brittle matrix [16,17,35], rigid particles in epoxy resins [36,37], and a model food system with sephadex particles in a protein matrix [24]. Concrete is a speci®c type of composite material, which consists of three phases: cement paste as matrix, interface transition zone (ITZ), and aggregates. Many studies about the ITZ characterization have been conducted. Speci®cally, Garboczi and Bentz [13,14] developed analytical models based on spherical shape of aggregates for the e€ect of ITZ on properties of concrete. Buyukozturk and Lee [6] studied the behavior of mortar±aggregate interfaces. Lee and An [26] studied the factors in¯uencing fracture toughness of mortar±aggregate interface. Kawakami [23] and Kan and Swartz [22] conducted systematic testing for the e€ect of di€erent types of aggregates on fracture properties of concrete. Issa et al. [21] and Zollinger et al. [46] studied the e€ect of the aggregate size on the fracture toughness and fracture process zone of concrete. Feng et al. [11] investigated the e€ects of various parameters of the concrete mix design including water to cement ratio, size of coarse aggregate, weight ratio of coarse to ®ne aggregate, and the volume content of coarse aggregate. El-Sayed et al. [10] conducted a series of tests (compression, tension, and three-point bend) to study the in¯uences of the various aggregate shapes on the fracture behavior of concrete. It has been found in various composite materials that a certain amount and proper size of the aggregate are bene®cial to the strength and fracture energy of the composite [7,19,32]. Both the critical stress intensity factor and the fracture energy increase with the addition of inclusions, at least for low volume fractions of inclusions. Feng et al. [11] conducted a preliminary study and tested concrete with various volume fractions of coarse aggregates. They found that both the critical stress intensity factor and the fracture energy increase monotonically with increasing coarse aggregate contents from 0% (pure cement paste [20]) up to 75% of the concrete volume. This, of course, needs to be con®rmed by a more systematic experimental study. Moreover, in concrete industry, coarse aggregates are used almost always along with ®ne aggregates. Therefore, the total aggregate content rather than the coarse aggregate content should be the testing parameter. To this purpose, the e€ect of total aggregate content on the fracture energy of concrete will be the ®rst focus of the present study. In particular, we are going to examine whether there is a critical volume fraction of the total aggregate content for concrete. The size of the fracture process zone (FPZ) in a composite material depends on the coarseness of the internal structure of the material. Among cement paste, mortar, and concrete, the size of the FPZ of cement paste is smaller than that of mortar, which in turn is smaller than that of concrete. This is because the size of the coarse aggregate in concrete is in the range of centimeter, the size of the ®ne sand in mortar is in the range of millimeter, and the particle size of the Portland cement in cement paste is in the range of micrometer. In concrete mixing design, gradation requirement results in a mixture of many di€erent sizes of aggregates, which can be represented by an average aggregate size. Test results showed that, at a ®xed value of total aggregate content, larger average aggregate size corresponds to a larger size of the FPZ [46]. Then the next question will be, at a ®xed average aggregate size, what is the e€ect of the total aggregate content on the size of the FPZ? This will be the second focus of the present study. One of the diculties involved in experimental determination of the fracture toughness of concrete is the fracture size e€ect: the measured fracture toughness varies with the size of the specimen used. The fracture size e€ect is quite di€erent from the so-called strength size e€ect, which has usually been explained by the Weibull statistical theory [39]. In order to deal with the fracture size e€ect, some theories have been developed. One of the theories, called size e€ect method developed by Bazant [2] will

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67

be used in the present study. The size e€ect method extrapolates the test results obtained from a ®nite specimen size to an in®nite specimen size. The remaining part of this paper is organized in the following way. The previous work will be brie¯y reviewed ®rst, which includes the various procedures to handle the e€ect of volume fraction of inclusion on the fracture properties of various composites and the size e€ect method used in the present study. And then, the specimen preparation, the experimental procedure, and the testing parameters of concrete beams for determining fracture properties of concrete will be described. Subsequently, the test results will be analyzed and discussed. And ®nally, the relationship between the internal structure and the fracture properties of concrete will be discussed, based on a morphological model.

2. Review of the previous work 2.1. E€ect of the volume fraction of inclusions The enhancement of fracture properties of composites due to added particles can be explained by several mechanisms. One of them is the microcrack shield, which causes a reduction of stress in the FPZ. Another one is the crack bridging, which provides closing pressure in the FPZ. Also, the interlocking of the particles between the crack surfaces consumes energy and thus enhances the fracture resistance. The most generally accepted mechanism is that when a crack meets an array of impenetrable obstacles it becomes pinned. In order to pass the obstacles the crack would have to bow out and thus make the crack surface rougher. This would lead to an increase in fracture energy. This is true, however, only for static loading. Under dynamic loading with di€erent loading rates, the roughness of fracture surface of composites vary. The change in fracture toughness due to added particles has been correlated to a quantity called interparticle separation, Ds, for polymers, which is a function of both the particle diameter dp and the volume fraction of particles Vp (reviewed by Spanoudakis and Young [36,37]). ÿ
2dp 1 ÿ Vp …1a† Ds ˆ 3Vp Later on, Spanoudakis and Young [36,37] and Langley et al. [24] suggested that the fracture toughness of polymers is actually related to the ratio dp =Ds , which is a simple function of the volume fraction, Vp given by, dp 3Vp
ˆ ÿ Ds 2 1 ÿ Vp

…1b†

With increasing Vp, the ratio dp =Ds increases, and the fracture toughness increases, at least in the range of low volume fraction. However, the physical meaning of dp =Ds has not been clearly understood as to why it a€ects the fracture toughness. On the other hand, Eqs. (1a) and (1b) cannot be directly applied to concrete because they were developed for composites with polymer matrix, which are quite di€erent from fracture behavior of concrete. The potential of the fractal geometry as means of describing the scaling of surface roughness has been explored by many researchers [5,18,25,27]. Mandelbrot et al. [27] were the ®rst to suggest a linkage between fractal dimension of fracture surface and fracture toughness. Since the roughness of the cracked surface may play the role of the intrinsic parameter at the microstructure level, the roughness of the fractured surface and the fractal theory have been used to quantitatively characterize the coarseness of

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internal structure of concrete [12,28,30,45] ÿ
D2 ÿ2 ÿ
S Z 2 ˆ S0 Z 2 2

…2†

in which Z = mesh size for measurement of a fracture surface; S0 is a constant; S…Z 2 † = area measured with the mesh size Z; and D2 = surface fractal dimension. D2 ˆ 2 means that the measured surface area is a constant S0 , which is independent of the mesh size (or scale level). This occurs when the fracture surface is very smooth. When D2 > 2, the measured surface area is not a constant S0 , but a function of mesh size Z: With a constant Z, a higher fractal dimension D2 corresponds to a rougher fracture surface. One needs to realize that, for any composite material, the degree of roughness of a fracture surface and the degree of heterogeneity of the internal structure are two di€erent concepts. Heterogeneity of the internal structure represents the morphological information of the mixture structure and the constituent phases of the composite. While the fractal dimension, although de®ned as a geometrical parameter of fracture surface, is measured after a destructive loading test, and thus includes not only the geometrical feature of the internal structure of the composite material but also the loading e€ect. For instance, completely di€erent roughnesses of fracture surfaces occur under di€erent loading rates for the same internal structure of concrete (from the same mixing design). Therefore, it is much easier to establish a relationship between mix parameters of concrete and morphological parameters of the concrete, because the e€ects of loading con®guration and specimen geometry are not involved. In the later sections, parameters for concrete mix design, such as average aggregate size and volume fraction, will be related to a morphological characteristic (called coarseness) of concrete. It will be used to interpret the present results of fracture test. 2.2. The size e€ect method Geometrically similar specimens of quasibrittle materials, such as concrete, rock, and ceramics exhibit a pronounced size e€ect on their failure loads. With the same concrete composition, the fracture toughness obtained from concrete specimens of di€erent sizes are di€erent. In fact, with increasing size of specimens, the failure load increases, but the obtained nominal stresses (the load divided by cross section area) decreases. The reduction in the nominal stress depends on the size of the specimen. As shown in Fig. 1, the reduction of the nominal stress of high performance concrete (HPC) is less than the reduction predicted by linear elastic fracture mechanics (LEFM), which corresponds to slope of ÿ 12 : On the other hand, the reduction is higher than the statistical size e€ect predicted by Weibull's weakest link theory, which corresponds to the slope of ÿ m2 , where m is the Weibull modulus [4]. The value of m is about 12 for concrete in bending, and therefore, the slope for the Weibull's theory is ÿ 16 : Such a size e€ect phenomenon, the transition between LEFM, and the statistical size e€ect are important manifestations of the in¯uence of the heterogeneous internal structure of concrete. It has been described by the size e€ect method proposed by Bazant [2] Bfu sN ˆ p 1‡b

bˆ

d d0

…3†

in which sN = nominal stress at the maximum load Pu …sN ˆ cn Pu =bd †); b = thickness of the specimen; d = size or characteristic dimension of the specimen (e.g., beam depth in bending test); cn = coecient introduced for convenience; B and d0 are two parameters determined experimentally; and fu is a measure of material strength, which can be taken as the tensile strength of the concrete. In Eq. (3), the Weibull's statistical e€ect, as shown in Fig. 1, is not included. One can see later that when the specimen size is large enough, size e€ect of LEFM is dominant, and Eq. (3) gives accurate prediction for the reduction

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69

in the nominal strength. For geometrically similar specimens of di€erent sizes, Eq. (3) can be algebraically transformed to a linear relationship Y ˆ AX ‡ C

…4†

in which X ˆ d and Y ˆ …fu =sN † 2 : The slope A and intercept p C in the plot of Y vs. X can be used to determine the two parameters B and d0 in Eq. (3), B ˆ 1= C; and d0 ˆ C=A: And then, the two fracture parameters Gf and cf can be determined [3] based on B and d0 Gf ˆ

…Bfu † 2 d0 g… a0 † cn2 E

…5†

cf ˆ

d0 g… a0 † g 0 … a0 †

…6†

in which Gf and cf are de®ned as the energy required for crack growth and the e€ective length of the fracture process zone (FPZ) in an in®nitely large specimen, respectively. E = Young's modulus of concrete; a0 = initial relative crack length …a0 ˆ a0 =d); a0 = initial notch length; g…a† = geometrydependent function of relative crack length a ˆ a=d, and a = actual crack length. The ®nite size of FPZ, cf , is a fracture parameter which characterizes the di€erence between the fracture behavior of concrete and those which can be described by LEFM. Eqs. (4)±(6) can be used to determine Gf and cf , when a series of fracture tests of concrete beams is conducted. An important consequence of the ®nite size of FPZ is the toughening of the fracture resistance curve or R-curve. It was found that the R-curve for concrete strongly depends on the specimen geometry. A general expression of the R-curve equation based on the size e€ect parameters [3] is

Fig. 1. Size e€ect curve for four high performance concrete beams [6].

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g 0 …g † c g 0 … a0 † c f

…7a†

  g 0 …a0 † g…g † c ÿ g ‡ a0 ˆ cf g… a0 † g 0 … g †

…7b†

R…c† ˆ Gf

in which g is an internal variable of the R-curve. For a given value of g, Eq. (7b) can be used for determining the parameter c, which is the crack extension, and then Eq. (7a) can be used to determine the R-curve. The size e€ect method, reviewed above, will be used in the later sections to determine the fracture properties, Gf and cf . The e€ect of aggregate contents on Gf and cf will be examined by testing concrete beams with various aggregate contents. 3. Experiments Three-point bend beam specimens were used in the present study. The geometry of the beam specimens is shown in Fig. 2. The ratio of the span to the depth of the beam (L/d ) was 2.5 for all the specimens. The ratio of the notch length to the beam depth …a0 =d† was 0.25 for all the specimens. The ratio of the total length to the beam span was 1.2. In this way, all the beams were geometrically similar in the two dimensions (see Fig. 2). The thickness of all beams was 63.5 mm (2.5 in.), which was determined by considering the maximum aggregate size (19 mm) to make sure that the thickness of the beams is three times larger than the maximum aggregate size. In the literature, constant thickness of specimens has been adopted in all other series tests [2±5,7,15,46] using the size e€ect method proposed by Bazant [2]. Four di€erent beam sizes were tested. The four beam depths were: 76.2, 114.3, 152.4, and 228.6 mm (3, 4.5, 6, and 9 in.). To examine the e€ect of aggregate contents, four di€erent volume fractions were used: 45, 55, 65, and 75% (as the percentage of the total concrete volume). Four sizes of beams were tested for each volume fraction of aggregate, and three specimens for each size, and thus, totally 48 beams were tested. The coarse and ®ne aggregates used in the test were gravel and river sand, respectively. As discussed

Fig. 2. Specimen geometry for the three-point bend beams.

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71

earlier, the aggregate size, the gradation curve, and the aggregate content all have e€ects on the fracture properties of concrete. The present study focused only on the e€ect of the aggregate content. Therefore, the maximum aggregate size used was 19 mm (3/4 in.) for all sample beams. The volume ratio of coarse to ®ne aggregate was 2 (Test results [11] showed that the fracture energy of concrete reaches its maximum value at the volume ratio around 2). Using a ®xed volume ratio for coarse and ®ne aggregate in this study, was, in order to keep the same gradation curve and the same average aggregate size for all sample beams, so that the e€ect of the aggregate content can be singled out. It should be pointed out that varying aggregate contents result in some changes in other mix parameters and properties of fresh concrete. In fact, the total mass of aggregates increased with increasing aggregate contents. The water± cement ratio was 0.5 for all the beams. Along with the beam specimens, three 76.2  152.4 mm (3  6 in.) cylinders were cast for the test of compressive strength under the same conditions as the other beam specimens. All the specimens were tested after 14 days of curing in a fog room. Table 1 lists the concrete compositions for the four di€erent mixes used in the present study. The three-point bend beam tests were performed on an Instron close-loop control testing system with crack mouth opening displacement (CMOD) control. The loading rate was controlled such that the beams reached their maximum loads in about 10 min. It was carefully examined that there was no cracking at the notch tip prior to the start of testing. 4. Test results and analyses Typical Load±CMOD curves, obtained from the test, are shown in Fig. 3(a)±(d). All the Load± CMOD curves of the concretes with various volume fractions of the aggregate show similar pattern. One can see that the slopes of the initial part of the curves are almost the same, which means that the sti€nesses of the beams are almost the same. After a short linear portion of the Load±CMOD curves, the deviation from the linear curves was observed, which is close to the onset of crack initiation at the tip of the notches. The post-peak (after maximum loads) section of the curves exhibit decline of the load with a steady and smooth pattern. No drastic change in the load is observed after the peak load (maximum load). This is the expected result, when the CMOD control is used in the test. The slopes of the post-peak part of the curves are almost constant. The values of the peak load of the concrete beams are listed in Table 2. From the recorded peak loads in Table 2 and following the procedure of Eq. (4), the linear regression analysis can be carried out. The two obtained two parameters, B and d0, for the concrete beams with various volume fractions of aggregates are listed in Table 3. One of the linear regression results for Va ˆ 55% is plotted in Fig. 4.

Table 1 Concrete compositionsa Va (%)

Weight of cement (kg)

Weight of water (kg)

Weight of gravel (kg)

Weight of sand (kg)

45 55 65 75

39.4 32.1 24.8 17.5

19.7 16.1 12.4 8.8

57.2 69.9 82.6 95.3

23.6 28.6 34.1 39.0

a Bulk speci®c gravity (BSG) of gravel is 2.99; BSG of sand is 2.45; and BSG of cement is 3.15. The absorption of gravel is 0.8%; and the absorption of sand is 2.4%.

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F.E. Amparano et al. / Engineering Fracture Mechanics 67 (2000) 65±84

4.1. Compressive strength Fig. 5 is the result of f'c vs. Va (volume fraction of the aggregate in %), in which, the strength values are the average of the three test samples. One can see that f'c decreases slightly with increasing Va. From the concrete placement point of view, as the aggregate content increases under ®xed water±cement ratio, the workability of the concrete mixes becomes lower. The reduced workability may be one of the

Fig. 3. Load±CMOD curves for concrete beams with: (a) Va ˆ 0:45%; (b) Va ˆ 0:55%; (c) Va ˆ 0:65%; (d) Va ˆ 0:75%:

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73

Fig. 3 (continued)

reasons responsible for the decreasing compressive strength with increasing aggregate content. On the other hand, in the case of the very high strength concrete, cement paste may have higher strength than the aggregate used, and the addition of more aggregate may weaken the concrete. In the present study, the compressive strength of concrete is not very high, in the range of 27.17 to 34.55 MPa (about 4000 to 5000 psi) (see Table 4). Therefore, the aggregate used can be considered to be much stronger than the cement paste, and the added aggregate, is supposed to strengthen the concrete, not to weaken it. Obviously, this result needs more explanation since it is completely di€erent from the prediction of

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composite theories, according to which, the strength of composites should be enhanced by the added strong particles [8]. One of the possible explanations for the present result is that the added aggregates introduce interface transition zone (ITZ) into the concrete. ITZ between the aggregate and the surrounding cement paste is a weak zone and has a very important e€ect on all kinds of properties of concrete. The porosity of ITZ is much higher than that of bulk cement paste, and thus the strength of the cement paste of ITZ is lower. As the volume and surface area of the aggregate increases, the volume fraction of ITZ also increases. High volume fraction of the aggregates in concrete results in small spacing between adjacent aggregates. At a certain volume fraction, the cement paste matrix in the concrete, with high volume of aggregate, may be percolated. The particular volume fraction has been experimentally studied by mercury intrusion tests on concretes with various aggregate contents [40]. The results showed that at 50%, or sometimes at even lower values of the aggregate content, the percolation clusters are forming, which may be regarded as the complete connection between the ITZ of the aggregates. In normal concrete, the aggregate volume fraction is almost always higher than 50%, which means that large portion of cement paste in normal concrete are ITZs. The above argument is also supported by other experimental evidences. In general, the thickness of ITZ is considered to be 30±50 mm [13,14,33,40]. Based on the statistical analysis, the average clear spacing between aggregate particles in concrete has been found to be about 100 mm [9,31]. From another point of view, this means that a large portion of the paste is composed of ITZ. The composite e€ect induced by the added strong aggregate, strengthens the concrete. The cement paste matrix, on the other hand, is weakened due to the large amount of interface area created by the added aggregate. These two opposite e€ects o€set each other, and the combination of them leads to the slightly declined strength as shown in Fig. 5. There are other factors that may cause the reduction of strength, such as the change in workability due to the change in mix design. Nevertheless, it is important to note that this is a completely di€erent phenomena compared to other composite systems, such as solid particles-®lled polymer matrix in which, the composite theory can be used to predict the reinforcement resulting from the inclusions, quite well. As a result, the theories developed for regular composites, such as Eqs. (1a) and (1b), may not be used directly for concrete, because in those theories the properties of the matrix are considered to be independent of the amount of inclusions added, which is not true for concrete.

4.2. Modulus of elasticity Accurate determination of the modulus of elasticity of concrete, E, is very important in the present study, since the modulus of elasticity plays an important role in the calculation of other fracture parameters using the size e€ect method. Di€erent methods for determining modulus of elasticity have been proposed in the literature. In the present study, the modulus of elasticity, E, was obtained from the initial linear portion of the Load±CMOD curves [1,29,34]. CMOD is a function of applied load, crack length, structural geometry, and modulus of elasticity of the material. In the case of three-point bend beams, CMOD is [34], CMOD ˆ

  4sa a g2 E d

…8a†

where, g2 …a=d† depends on the ratio of the span to the depth of the beam and is given by the following formula, when L ˆ 2:5d,

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Table 2 Maximum loads of the concrete beams Depth d (mm)

Span l (mm)

76.2 76.2 76.2 114.3 114.3 114.3 152.4 152.4 152.4 228.6 228.6 228.6

190.5 190.5 190.5 285.7 285.7 285.7 381.0 381.0 381.0 571.5 571.5 571.5

g2 …a=d† ˆ

Maximum load Pu (N) Va ˆ 45%

Va ˆ 55%

Va ˆ 65%

Va ˆ 75%

2505 2332 2687 4539 4442 3278 5428 5106 4255 6574 6039 5429

3703 3458 2886 4060 3828 3523 5196 4396 5029 6606 6077 6258

3035 2936 2920 4525 4102 3163 4665 5130 4648 5848 6514 6107

4410 3739 3433 4527 5434 4443 5672 4738 5107 7238 7536 7666

1:73 ÿ 8:56…a=d† ‡ 31:2…a=d† 2 ÿ46:3…a=d†3 ‡25:1…a=d†4 …1 ÿ a=d†3=2

…8b†

By using the slopes of Load±CMOD curves shown in Fig. 3(a)±(d), the modulus of elasticity, E, can be obtained from Eq. (8a). The results of the calculated E are listed in Table 4 and plotted in Fig. 6. As can be seen in Fig. 6, the modulus of elasticity increases as the volume fraction of the aggregate increases. The variation is less than 25%. The increase in modulus of elasticity can be attributed to the added strong particles in the concrete. On the other hand, it seems that the e€ect of ITZ on the sti€ness of the concrete, is not as strong as for the strength of the concrete. 4.3. Fracture energy The fracture energy, Gf , can be calculated by using Eq. (5) together with the values of B, d0, and E of each group of specimens with a speci®c Va. The function g…a0 † in Eq. (5) can be found in some handbooks on fracture mechanics [34]. The values of Gf are listed in Table 4 and plotted in Fig. 7 in which one important result may be noticed, that is, the absence of the critical volume fraction of the aggregate which gives the maximum value of the fracture energy. As one can see, there is no peak (maximum) value of Gf . In fact, Gf ®rst drops slightly from Va = 45 to 65% and then starts to increase. The variation of Gf in the range of aggregate content tested is about 25%. This result indicates that the enhancement of Gf by the added aggregates is not as e€ective as those observed in other Table 3 Result of the regression analysis Va (%)

B

d0

R 2 coecient

45 55 65 75

0.7642 1.1494 1.0561 1.4633

128.744 54.453 72.306 53.678

0.8008 0.9337 0.9874 0.8392

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Fig. 4. Test data and the regression analysis for concrete beams with Va ˆ 55%:

composite systems, such as glass particle-®lled epoxy resin, where 100±130% increase in Gf was reported [36,37]. On the basis of Fig. 7, we may say that the critical volume fraction corresponding to the maximum value of Gf does not exist for concrete in the range of Va = 45±75%. It is important to note that the present experimental study considers only the e€ect of the aggregate content on Gf at ®xed aggregate gradation and water±cement ratio. More experimental studies are absolutely necessary in order to examine the combined e€ect of the aggregate content and other factors on Gf . 4.4. Fracture process zone (FPZ) The size of the FPZ (i.e., cf ) is determined by Eq. (6). The result is plotted in Fig. 8, which shows that the size of FPZ decreases initially with increasing volume fraction of aggregate and then tends to level o€. To explain this test result, we need to identify the dominant parameters that control the size of FPZ. First, let us compare the e€ect of aggregate size on cf . As discussed in the introduction, cementitious materials with ®ne grain structures tend to have small FPZ. For concrete, it has been experimentally proven, by the test results of Zollinger et al. [46], that the size of FPZ of concrete increases with increasing average aggregate size. From this point of view, the coarseness of grain

Table 4 Results of tests and the size e€ect analysis Va (%)

fc' a (Mpa)

E (GPa)

Gf (N/m)

cf (mm)

l

45 55 65 75

34.55 31.93 30.62 29.17

10.50 10.44 11.92 13.07

21.59 18.00 18.83 25.39

26.09 11.04 14.65 10.88

0.0477 0.0583 0.0750 0.1050

a

f'c is the average value of three specimens.

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Fig. 5. The e€ect of the aggregate content on compressive strength.

structure of concrete should be considered as one of the dominant parameters a€ecting the size of the FPZ. Now, in this study, the size of the aggregate is kept constant and only the volume fraction of aggregate is considered to be a variable. To this end, two related questions must be answered. The ®rst one is, how to de®ne the coarseness of heterogeneous internal structures like concrete? and the second is, how to determine the change of the coarseness due to a change of the volume fraction of the aggregate? The answers to these two questions will be presented in the next section, and then we will be able to explain why cf decreases with increasing Va. 4.5. R-curves and the size e€ect Based on the values of Gf and cf , the R-curves evaluated by Eqs. (7a) and (7b) are plotted in Fig. 9. With increasing crack extension, the asymptotic values shown in the curves are Gf for the concretes with various aggregate contents. The size e€ect plot is shown in Fig. 10 for the concrete beams with all four aggregate contents, 45, 55, 65, and 75%. From Fig. 10, one can see that the size e€ect, i.e. the decline of the nominal stress with increasing depth of concrete beam, is less than ÿ 12 , which corresponds to the size e€ect of LEFM. Obviously, the present test results fall into the range of the transition zone between the fracture mechanics size e€ect and the horizontal line representing the yielding criterion [2]. Comparing the present test data in Fig. 10 with those in Fig. 1 clearly shows a shift from the left to the right side, as the volume fraction of aggregate varies from 45 to 75%. The shift represents the transition of the controlling failure mechanisms from yielding criterion to LEFM. From the morphological point of view, with increasing volume fraction of aggregate, the grain structure of concrete becomes ®ner (see explanation in Section 5), and thus, the FPZ is smaller. When FPZ is small, the failure pattern is more brittle and towards the region of LEFM, which is, manifested as the shift to the right side on the size e€ect plot (Fig. 10). Similar shift in the size e€ect plot was obtained in the study by Chang et al. [7], in which, the size of the aggregate, instead of the volume fraction, was the experimental parameter. In the study of Chang et

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Fig. 6. The e€ect of the aggregate content on modulus of elasticity.

al., the test data for the concrete with large aggregate (and thus coarse internal structure) fall in the transition zone between LEFM and the strength criterion; while the test data of cement pastes with small aggregates (and thus ®ne internal structure) shift to the right. From LEFM point of view, this means that cement paste is more brittle than concrete, and can be characterized by LEFM.

Fig. 7. The e€ect of the aggregate content on fracture energy.

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79

Fig. 8. The e€ect of volume fraction of the aggregate on FPZ.

5. Relationship of cf and coarseness of internal structure The increase of the aggregate content from 45 to 75% results in a decrease in cf (Fig. 8). This may be due to a change in the coarseness of the internal structure of concrete compared to the e€ect of aggregate size on the coarseness (large aggregate leads to coarse structure). However, the e€ect of aggregate content on coarseness of heterogeneous internal structure may not be seen explicitly, and has

Fig. 9. R-curves of concrete beams with various aggregate contents.

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Fig. 10. Size-e€ect plot.

not been quantitatively evaluated. A theoretical model is apparently needed to characterize the coarseness of the random internal structure in terms of the aggregate content. To this end, a morphological model, called mosaic pattern, is used in the present study. Mosaic pattern is one of the morphological models developed for characterizing the heterogeneous features of random grain structures. Di€ering from the previously reviewed methods (Eqs. (1) and (2), in the framework of mosaic pattern, a morphological parameter, called coarseness l, is de®ned to characterize the coarseness of the internal structure of concrete. l has a very clearly de®ned physical meaning and has been discussed in detail in the theory of mathematical morphology [38,41,42]. More importantly, l has been formulated in terms of the combined e€ect of volume fraction and average size of the aggregate. For the reader's convenience, the model of mosaic pattern will be brie¯y introduced. Fig. 11 shows a computer generated two-phase mosaic pattern to simulate a two-phase heterogeneous internal structure. Although the internal structure of concrete consists of three phases as previously mentioned, the interface transition zone may be combined with the bulk cement paste matrix, and the concrete can then be considered as a simpli®ed two-phase structure that can be characterized by the mosaic pattern. There are two independent mosaic parameters that can be used to characterize the morphological features of the two-phase structure. One is the volume fraction of a constituent phase, and the other one is the coarseness l of the grain structure. The white matrix and black particles in Fig. 11 are composed of basic cells of the mosaic pattern. The basic cells are the random polygons of Fig. 11. Any black particle or white matrix may be composed of more than one basic cell. Coarseness l represents the density of basic cells of the mosaic pattern. Large l corresponds to high density of basic cells (more basic cells per unit area) and thus ®negrained structures, while small l represents low density of basic cells and therefore, coarse-grained structures. In this sense, it would be more appropriate to call l the ®neness parameter. However, l has been used as coarseness indicator in many other research ®elds for years, and its naming convention was simply adopted in the present study. More detailed discussion on mosaic patterns and computer generated mosaic patterns for simulating concrete internal structures with various l and Va are shown elsewhere [41]. l can be either evaluated by an image analysis method [41], or evaluated in terms of average aggregate size Dave and volume fraction Va for a two phase composite.

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Fig. 11. A two-phase mosaic pattern.

lˆ

1 Dave …1 ÿ Va †

…9†

This formula is very useful especially when the image analysis method is not available. The important feature of Eq. (9) is that it incorporates the e€ects of both Dave and Va on the coarseness of random grain structures. With a ®xed volume fraction, for a smaller size of aggregates, l becomes higher, which corresponds to a ®ner grain structure, and accordingly, the concrete exhibits a smaller size of FPZ. This was experimentally con®rmed by Zollinger et al. [46] On the other hand, at a ®xed average size of the aggregate, with increasing volume fraction of the aggregate, l increases (see Eq. (9)). This means that higher volume fraction of the aggregate leads to a ®ner grain structure, and thus a smaller size of FPZ.

Fig. 12. The e€ect of coarseness of grain structures on FPZ.

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This is experimentally con®rmed by the present results. The size of FPZs evaluated, based on the present test results, are plotted with coarseness l in Fig. 12. From the above discussion, one can see that parameter l may be considered to be a comprehensive measure of coarseness of the grain structures of concrete. It captures qualitatively the general relationship of FPZ and morphological features of the grain structure of concrete. Obviously, a quantitative model to correlate cf and l based on the morphological theory and fracture mechanics is needed. A preliminary study was conducted by the second author [43,44], in which an empirical model for cf , as a function of l, was developed. One important application of the quantitative model for cf is that, with cf being evaluated by the model, fracture toughness Gf becomes the only unknown parameter to be determined by fracture tests (see Eqs. (5) and (6)). Therefore, a specimen of just one size instead of di€erent sizes will be sucient to determine Gf . A preliminary study along this direction has been published elsewhere [43,44].

6. Conclusions The fracture properties of concrete with various aggregate contents were tested by geometrically similar three-point bend concrete beams. The results were analyzed by using the size e€ect method, in which the fracture behavior of concrete are characterized by two parameters, fracture energy Gf and e€ective fracture process zone cf . Fracture properties of concrete depend strongly on both total aggregate content and the coarse to ®ne aggregate ratio. The present study focus on the e€ect of total aggregate content, and the volume ratio of coarse to ®ne aggregate is set to be 2. The following conclusions are obtained for total aggregate content from 45 to 75% of the concrete volume. 1. The test results showed that with increasing volume fraction of aggregate, the compressive strength of the concrete decreases, which is di€erent from the prediction of conventional composite theories. The possible explanation of this result is based on the interface transition zone (ITZ) around the aggregate, which is considered to be the weak zone in concrete. With more aggregate added into concrete mixtures, more interfaces are formed in the hardened concrete. The added aggregate strengthens the composite, and the associated interface weakens it. These two opposite e€ects o€set each other, and the combination of them leads to the declined strength. On the other hand, the increase of aggregate content reduces the workability of a concrete mix and thus reduces the strength. 2. The fracture energy Gf exhibits variations in the range of 25%, which indicate that the e€ect of aggregate on Gf is not as signi®cant as the enhancement of fracture energy for other types of composites. Gf decreases with an increasing aggregate content and reaches a minimum value by Va ˆ 65%, then starts to increase thereafter. This means that the critical volume fraction of the aggregate, corresponding to the peak value of Gf , may not exist for the concrete used in the present study (with water-to-cement ratio 0.5 and a ®xed aggregate gradation). 3. On the size e€ect plot, the nominal strength of concrete beams shift from the left to the right with increasing volume fraction of the aggregate, which means that the failure mechanisms of the concrete used in the present study shift toward the region of linear elastic fracture mechanics. 4. The coarsenesof random grain structure of concrete may be de®ned, based on a morphological model: mosaic pattern. Coarseness l of the mosaic pattern can be formulated in terms of both the average size and the volume fraction of the aggregate. l interprets the morphological features of the internal structure quantitatively. With increasing volume fraction of the aggregate, l increases, which corresponds to a ®ne grain structure, and thus small cf . With smaller cf , the failure pattern tends to be dominated by linear elastic fracture mechanics, which is also con®rmed by the shift to the right in the size e€ect plot.

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