An analytical approach to predict fracture parameters of coral aggregate concrete immersed in seawater

Ocean Engineering 191 (2019) 106508

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An analytical approach to predict fracture parameters of coral aggregate concrete immersed in seawater Shutong Yang a, b, *, Xianshun Zhang a, Miao Yu a, Jie Yao a a

Department of Civil Engineering in College of Engineering, Ocean University of China, Qingdao, 266100, PR China Cooperative Innovation Center of Engineering Construction and Safety in Shandong Blue Economic Zone, Qingdao University of Technology, Qingdao, 266033, PR China

b

A R T I C L E I N F O

A B S T R A C T

Keywords: Coral aggregate concrete Fracture parameters Average aggregate size Maximum fracture load Immersion in seawater

An analytical approach is proposed to predict the fracture parameters of coral aggregate concrete (CAC). Both the size-independent tensile strength and fracture toughness are related to the maximum fracture load linearly based on the boundary effect model by incorporating the average aggregate size. Moreover, an explicit expression is derived to correlate the maximum fracture load with the local fracture energy at the crack-tip region. The local fracture energy distribution and size-independent fracture energy are then determined by virtue of the maximum fracture load. Four groups of three-point-bending notched CAC beams are tested by considering two ages and two environmental conditions (immersion in seawater or not) and the initial crack length-to-beam depth ratios are set from 0.1 to 0.7 in each group. Results show that the failure modes of all the specimens are coral coarse aggregate fracture without interfacial debonding between the aggregate and surrounding mortar. The average values of tensile strength, fracture toughness and fracture energy can be obtained in each group by using the experi­ mentally measured maximum fracture loads and the experimental scatters were analyzed based on normal distribution analysis. All the fracture parameters increase with the age and become larger if the specimens were immersed in seawater.

1. Introduction Island constructions have been carried out recently with the rapid development of marine exploitation. If all the raw materials for mixing concrete, such as cement, river sands, crushed stones, freshwater and so on, have to be obtained from the mainland, however, many un­ certainties may be encountered due to the long distance ship trans­ portation and the construction cost and period would be increased consequently. Therefore, it is better to seek locally available raw ma­ terials. There are abundant coral reefs in the islands of South China Sea, which are formed by the dead anthozoans after thousands of years. The coral reefs can be crushed into particles and replace the conventional fine and coarse aggregates in the light of particle sizes. The seawater can be adopted locally instead of freshwater in concrete mixing. If so, more than 70% of the raw materials could be given locally without long dis­ tance ship transportation and the construction cost and period would be reduced significantly. Thus, it is necessary to analyze the mechanical properties of CAC systematically. The main mineral components of coral aggregates are different from

those in natural aggregates, and more than 96% are calcium carbonate (Chen et al., 2008). Coral aggregates are generally porous and should be presoaked before concrete mixing due to their high water absorption (Arumugam and Ramamurthy, 1996). Early strength of coral aggregate concrete (CAC) develops faster due to the chloride effects from the corals compared to ordinary concrete (OC) (Wang and Fan, 2015; Zhao et al., 2011). The ratios of 7-day strength to 28-day strength even exceed 80% (Yang et al., 2018). Besides, the values of uniaxial compressive strength are closer to those of cubic compressive strength in CAC compared to OC (Yang et al., 2018). The splitting tensile strength and elastic modulus are larger than those of OC with the same strength grade because of the better bond performance between the porous coral aggregate and sur­ rounding mortar (Wang and Fan, 2015). Tan et al. (2018) evaluated the compressive strength of CAC by using non-destructive techniques and presented the correlation between the compressive strength and ultra­ sonic pulse velocity. CAC shows a more significant rate-dependence in compressive strength than OC when it is subjected to dynamic compression (Ma et al., 2019). Moreover, it is found that CAC has remarkable brittle properties especially when it is used as columns under

* Corresponding author. Department of Civil Engineering in College of Engineering, Ocean University of China, Qingdao, 266100, PR China. E-mail address: [email protected] (S. Yang). https://doi.org/10.1016/j.oceaneng.2019.106508 Received 25 July 2019; Received in revised form 14 September 2019; Accepted 28 September 2019 Available online 14 October 2019 0029-8018/© 2019 Elsevier Ltd. All rights reserved.

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Fig. 1. Stress distribution indicating nominal strength in the cracked section.

compressive action (Da et al., 2016a, 2018). Therefore, the CAC columns should be enveloped by FRP (fiber-reinforced polymer) tubes to prevent spalling failure (Wang et al., 2017; Zhang et al., 2019). Since CAC is especially used in ocean engineering, the durability is so important in structural application. Cheng et al. (2017, 2018) proposed a proper mix proportion of CAC by adding blast furnace slag and met­ akaolin instead of part of Portland cement. The chloride diffusion co­ efficient, capillary water absorption and carbon depth are then reduced, and the microstructure is improved (Cheng et al., 2018). Da et al. (2016b) found that the use of magnesium sulfate cement instead of or­ dinary Portland cement in CAC mixing can improve the resistance to chloride diffusion. Alkali-activated geopolymer cement can be also adopted to improve the stability of cement hydration products (Wang et al., 2018). Lower water-to-binder ratio is suggested to increase the compressive strength and reduce the surface free Cl concentration and apparent Cl diffusion coefficient (Yu et al., 2017). Moreover, it is well acceptable that FRP bars have good resistance to chloride corrosion and can be adopted as reinforcements in CAC structures. However, FRP materials may be damaged by hydroxide ions from concrete pore solu­ tion transmitted by externally permeating water (Robert and Benmok­ rane, 2013). Cracks in the concrete would provide channels for the harmful ions and water penetration and then much affect the durability of concrete structures. Therefore, it is necessary to study the fracture properties of CAC under ocean environment. To our best knowledge, there is no literature aimed at the fracture behavior of CAC so far. The intention of this paper is to determine the fracture parameters of CAC immersed in seawater. Tensile strength, fracture toughness and fracture energy are three fundamental parameters in describing the fracture behavior of concrete. However, the parameters are found to be size-dependent and analyzed based on size effect law by Ba�zant et al. (1984, 1990). Hu et al. (1992, 2000; 2002; 2004; 2007; 2008) developed boundary effect model (BEM) to reveal the mechanism of size effect which is induced by the interaction between the fracture process zone (FPZ) ahead of crack-tip and boundaries of specimens. A nominal strength σn by considering the crack length is presented and related analytically to an equivalent crack length ae indicating the distance from the crack-tip to the nearest boundary (Hu et al., 2000, 2002; 2007, 2008). The BEM was recently improved to obtain the size-independent uniaxial tensile strength ft and fracture toughness KIC based on the linear regression method by virtue of the experimentally measured maximum fracture load Fmax (Hu et al., 2017; Wang et al., 2016, 2017). A simplified linear equation between Fmax and ft is derived by consid­ ering the effect of average grain size G (Guan et al., 2018, 2019a; 2019b; Han et al., 2019; Zhang et al., 2018). The values of ft and KIC for each

specimen can be given analytically by using the tested Fmax and the experimental scatters were evaluated based on normal distribution analysis. Moreover, Hu and Wittmann (1992) pointed out that size effect in fracture energy of cementitious materials is mainly due to non-uniform local fracture energy gf distribution along the ligament of specimen. The gf distribution can be simulated by a bi-linear model indicating the size-independent or maximum fracture energy GF and back boundary effect (Abdalla and Karihaloo, 2003; Duan et al., 2003). Yang et al. (2011, 2014) proposed a maximum fracture load model (MFLM) to correlate the Fmax with the gf at the crack-tip region, i.e., gf-tip. The variation of gf-tip with the initial crack length a0 is obtained by comparing the analytically predicted Fmax with the experimentally measured Fmax and then a tri-linear relationship is deduced for the gf distribution indicating both the front and back boundary effects. The MFLM is further improved by combing with the BEM to obtain the size-independent ft, KIC and GF for alkali-activated slag CAC (Xu et al., 2019). Although the MFLM attempts to establish the correlation between the Fmax and gf-tip, no explicit analytical expression is derived, which results in difficulties in the application of MFLM. Thus, the first intention of this study is to determine an explicit analytical relationship between the Fmax and gf-tip to improve the MFLM further. Three-point-bending tests are then performed on CAC immersed in seawater for two different ages. CAC beams cured under normal drying environment are also prepared as control specimens. The ft, KIC and GF determined by the experimental Fmax of CAC in different conditions are analyzed and discussed. 2. Analytical modeling 2.1. Determination of ft and KIC When the maximum fracture load Fmax is reached, the stress distri­ bution indicating nominal strength σ n in the cracked section of a stan­ dard three-point-bending notched beam can be assumed as shown in Fig. 1 (Guan et al., 2018, 2019a; 2019b; Han et al., 2019; Zhang et al., 2018). Herein, b, h and L represent the width, height and span of beam, respectively, and L ¼ 4h. Moreover, when the applied load F is increased from the initial cracking load Fini to the Fmax, the crack propagation process should not be continuous but discrete related to the maximum aggregate size dmax. In other words, the critical crack propagation length Δac should be integer or semi-integer multiples of dmax. Δac ¼ 1.0dmax can gives the best estimation according to Hu et al. (2017) and dmax�1.5G. 2

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Fig. 2. Stress distribution in the cracked section when the maximum fracture load is reached.

Based on the assumption of linear strain distribution along the uncracked part of critical section, the equilibrium condition of forces in Fig. 1 gives 1 2bðh Fmax þ W ¼ 2

a0 Þðh a0 þ 3GÞ σn 3L

fracture toughness KIC can be determined by using Eqs. (8) and (9) conveniently. 2.2. Determination of gf-tip by improving the MFLM (Yang et al., 2011, 2014)

(1)

In Eq. (1), W is the self-weight of beam. According to the BEM (Hu et al., 2000, 2002; 2007, 2008), the nominal strength σ n can be expressed as follows (Guan et al., 2018, 2019a; 2019b; Han et al., 2019; Zhang et al., 2018). f

When the applied load F attains the Fmax, the stress distribution along the beam depth is shown in Fig. 2 according to the MFLM (Yang et al., 2011, 2014). The critical crack propagation length Δac is still assumed to be 1.5G. The stress in the fictitious crack here is not constant as the σ n but distributes linearly with the ft at the fictitious crack-tip. The rela­ tionship between the cohesive stress σ w and crack width w is simulated by a bi-linear model as indicated by Abdalla and Karihaloo (2004), which reads !� � 28ft 5gf tip σ w ¼ ft 1 w 0�w� (10a) 25gf tip 7ft

(2)

σ n ¼ qffiffiffiffitffiffiffiffiffiffiffiffi 1 þ aa*e

∞

ae ¼

α¼

!2

αÞ2 YðαÞ

ð1

1:12

⋅a0

a0 h

YðαÞ ¼

(3) (4)

σw ¼ 1:99

a*∞ ¼ 0:25

αð1 αÞð2:15 3:93α þ 2:7α2 Þ pffiffiffi πð1 þ 2αÞð1 αÞ3=2

� �2 KIC � 3G ft

(5)

Substituting Eqs. (2) and (6) into Eq. (1), we have (7a)

A ¼ bh

(7b)

� � 2hð1 αÞ 1 α þ 3G h pffiffiffiffiffiffiffiffiffiaffiffieffi gðh; a0 ; GÞ ¼ 3L 1 þ 3G

hc �

Fmax þ 12 W A⋅gðh; α; GÞ

pffiffiffiffiffiffi KIC ¼ 2ft 3G

w tip

!� 5gf tip 45gf �w� 7ft 7ft

� tip

(10b)

h

a0 2

1:5G

(11)

According to the equilibrium conditions of forces in Fig. 2 as indi­ cated in the MFLM (Yang et al., 2011, 2014), the maximum bending moment Mmax can be given as follows. # " ðh a0 þ 3GÞðh a0 1:5GÞ2 1 21ft Gwtc ðh a0 GÞ Mmax ¼ ft b � gf tip 50 6ðh a0 3GÞ

(7c)

Then ft ¼

7ft 45gf

Moreover, the area under the adopted bi-linear curve is defined as the local fracture energy gf and it is assumed constant as the one at the crack-tip region, i.e., gf-tip, since the critical crack propagation length is only 1.5G or 1.0dmax (Yang et al., 2011). The crack surface is assumed to be plane, i.e., the crack opening width w distributes linearly along the beam depth. Moreover, the assumption of linear distribution of strains in the un-cracked part of critical section is still introduced. The height hcþ1.5G of tensile region is about half of the depth h-a0, i.e.,

(6)

1 Fmax þ W ¼ ft Agðh; α; GÞ 2

9 ft 1 40

(8)

(12)

(9)

Herein, wtc is the critical crack-tip opening displacement. It should also be the average grain size G multiplied by a discrete number similar to the Δac. The wtc can be then expressed with a discrete number βw as follows.

Therefore, once the maximum fracture load Fmax is obtained for each specimen from the test, the size-independent tensile strength ft and 3

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Table 1 Chemical components of artificial seawater (g/L). NaCl

MgCl2

Na2SO4

CaCl2

KCl

NaHCO3

22.16

5.265

3.861

1.082

0.745

0.207

wtc ¼

βw � 1:5G 1000

(13)

Inserting Eq. (13) into Eq. (12) gives " ðh a0 þ 3GÞðh a0 1:5GÞ2 1 β � 63ft G2 ðh a0 Mmax ¼ ft b � w gf tip 100000 6ðh a0 3GÞ

# GÞ

3. Experimental programme

(14) Moreover, the maximum fracture load Fmax can be given by Fmax ¼

4Mmax L

W 2

3.1. Materials (15)

Cement used in the test is P.O.42.5 ordinary Portland cement (Chi­ nese Standard GB 175, 2007). All the water used for mixing concrete and immersing specimens is artificial seawater according to the water in South China Sea with detailed chemical components as shown in Table 1. Fig. 3(a) shows the used coral sand with apparent density and bulk density of 2517 kg/m3 and 1415 kg/m3, respectively. Coarse ag­ gregates used in the test are columnar coral particles with the maximum diameter of 10 mm and porous properties as shown in Fig. 3(b). The apparent and bulk densities are 1899 kg/m3 and 918 kg/m3, respec­ tively. The water absorption of coral stones used in this test is 15.1%. Moreover, polycarboxylate-based high-range water reducer (HRWR) was used in mixing concrete.

By combing Eq. (14) with Eq. (15), the gf-tip is obtained analytically as follows. gf

tip

¼g

1

g2

ft Fmax þW=2 ft Ag2

(16a)

α þ 3G=hÞð1 α 1:5G=hÞ2 3Lð1 α 3G=hÞ

(16b)

where g1 ¼

g2 ¼

2hð1

βw � 63G2 ð1 α 25000L

G=hÞ

It can be seen from Eq. (16a) that the gf-tip can be given directly once the maximum fracture load is measured from the test and the ft is determined based on the improved BEM (Guan et al., 2018, 2019a; 2019b; Han et al., 2019; Zhang et al., 2018) for all of the specimens with different initial crack lengths. The improved model in this paper avoids the estimation of gf-tip based on the comparison between the analytical and experimental Fmax, and is more simple and useful in application. According to the above analysis, the ft, KIC and gf-tip can be derived by virtue of the proposed analytical approach. The values need to be determined subsequently in combination with the test of three-pointbending notched beams where the Fmax has to be given.

(16c)

Fig. 3. Coral aggregates used in the test. (a) Coral sand (b) Coral stones. 4

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Fig. 3. (continued). Table 2 Mix proportion of CAC (kg/m3). Seawater

Cement

Coral sand

Coral stones

HRWR

190

500

556

712

2.5

3.2. Mix proportion All the coarse aggregates were immersed in seawater for 23 h and exposed to drying environment for 1 h before concrete mixing. The detailed mix proportion of CAC is seen in Table 2. The mass of either coral sands or coral stones is the net mass of dry corals. Moreover, the slump values of fresh concrete from different batches are kept between 110 mm and 125 mm by adding the HRWR. 3.3. Three-point-bending test Four groups of beams with sizes of 100 � 100 � 515 mm3 were pre­ pared in the test. In each group, the initial crack length-to-beam depth ratios (a0/h) are set from 0.1 to 0.7. Three samples are prepared for each ratio. Therefore, there are totally 84 beams to be tested. All the speci­ mens were first cured in standard environment with temperature of 20 � C and relative humidity of 95% for 28 days. Then the beams in two groups denoted by Z28 and Z90 were exposed to normal drying envi­ ronment for 28 and 90 days, respectively. The specimens in two groups

Fig. 4. Test set-up.

denoted by J28 and J90 were immersed in 20 � C seawater for 28 and 90 days, respectively, until the test begun. Moreover, six cubic specimens and six prisms are made to determine the cubic compressive strength fcu and elastic modulus Ec of CAC in the same condition with the beams in each group. 5

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Table 3 Cubic compressive strength and elastic modulus of CAC.

Table 4 Summary of ft and KIC.

Nos. of groups

fcu (MPa)

Ec (GPa)

Nos. of groups

α

Fmax (kN)

ft (MPa)

KIC (MPa∙m1/2)

Z28 J28 Z90 J90

32.7 (1.6) 34.1 (2.1) 35.9 (3.3) 36.8 (2.2)

21.4 (0.4) 22.3 (0.3) 23.9 (0.2) 24.5 (0.4)

Z28

0.10 0.11 0.09 0.24 0.21 0.20 0.30 0.35 0.32 0.42 0.40 0.43 0.51 0.58 0.54 0.65 0.58 0.57 0.73 0.70 0.71

3.820 3.889 3.499 3.325 3.477 2.993 2.801 2.692 2.747 2.031 1.832 1.970 1.573 1.375 1.292 0.878 1.310 1.242 0.463 0.666 0.376

2.60 2.71 2.32 3.13 3.05 2.57 2.91 3.33 3.14 3.05 2.68 3.14 3.14 3.80 2.87 3.05 3.36 3.07 2.48 2.97 1.76

0.752 0.784 0.673 0.907 0.885 0.746 0.844 0.965 0.911 0.885 0.777 0.910 0.909 1.101 0.832 0.883 0.974 0.891 0.720 0.861 0.509

J28

0.15 0.11 0.10 0.22 0.22 0.23 0.30 0.31 0.31 0.40 0.42 0.40 0.50 0.50 0.52 0.59 0.59 0.58 0.76 0.69 0.65

2.924 3.383 3.757 3.593 3.493 3.736 2.671 2.678 2.838 2.872 1.999 2.139 1.842 2.066 2.032 1.182 1.375 1.331 0.732 0.714 0.722

2.15 2.36 2.55 3.09 3.14 3.43 2.84 2.99 3.17 4.06 3.01 3.12 3.54 3.96 4.16 3.16 3.66 3.41 2.68 3.02 2.53

0.624 0.683 0.740 0.897 0.909 0.994 0.824 0.867 0.918 1.176 0.871 0.905 1.026 1.148 1.207 0.916 1.061 0.989 0.778 0.875 0.733

Z90

0.10 0.10 0.10 0.21 0.19 0.21 0.30 0.29 0.30 0.40 0.39 0.40 0.50 0.51 0.52 0.60 0.61 0.61 0.71 0.69 0.71

4.680 4.269 4.167 3.879 4.095 3.503 3.199 3.445 2.971 2.395 2.061 2.188 1.622 1.939 1.832 1.808 1.570 1.417 0.845 0.745 0.770

3.17 2.84 2.83 3.40 3.36 3.07 3.40 3.41 3.39 3.39 2.93 3.02 3.13 3.85 3.76 4.97 4.51 4.08 3.92 3.14 3.59

0.920 0.824 0.820 0.986 0.973 0.891 0.984 0.988 0.982 0.984 0.848 0.876 0.906 1.115 1.090 1.441 1.306 1.182 1.136 0.911 1.041

J90

0.14 0.12 0.13 0.22 0.19 0.19 0.31 0.30 0.35 0.40

3.920 4.320 4.964 3.982 3.788 3.889 3.209 3.370 3.179 3.263

2.87 3.02 3.47 3.57 3.17 3.26 3.58 3.66 3.92 4.60

0.832 0.874 1.007 1.035 0.920 0.944 1.036 1.061 1.137 1.334

Fig. 5. Typical F-CMOD curves.

Fig. 6. Typical fracture surface.

The fracture test set-up is shown in Fig. 4. The test machine has a maximum range of 2000 kN. In each beam, two strain gauges are set horizontally at the same height with the crack-tip to monitor the initial cracking state. A load cell with a maximum range of 70 kN is adopted to detect the applied load F, and a clip gauge having a maximum range of 4 mm is used to measure the crack mouth opening displacement (CMOD). All the readings from the above cells were collected by a data acquisition system. The application of load is controlled by the vertical displacement of loading head and the rate is kept as 0.2 mm/min.

(continued on next page)

6

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4. Results and discussions

Table 4 (continued ) Nos. of groups

α

Fmax (kN)

ft (MPa)

KIC (MPa∙m1/2)

0.37 0.39 0.49 0.48 0.48 0.60 0.56 0.58 0.71 0.75 0.69

2.913 3.122 2.144 2.468 2.593 1.549 1.622 1.505 0.759 0.830 0.942

3.70 4.29 3.86 4.43 4.65 4.27 3.71 3.85 3.54 4.82 3.93

1.072 1.242 1.118 1.283 1.347 1.239 1.075 1.115 1.027 1.396 1.139

4.1. Cubic compressive strength and elastic modulus The average values of fcu and Ec of CAC in all the four groups are summarized in Table 3. The derivations are attached in the brackets. Apparently, both the fcu and Ec increase with the period in different environments. The strengths of CAC after seawater immersion are slightly higher than those exposed in drying condition. It has been mentioned that all the coral coarse aggregates are presoaked in seawater before mixing concrete. As the age increases, the absorbed water in the coral particles is gradually released. It then results in further hydration reaction of surrounding paste. The interface between the coral particles and cement mortar becomes denser. Therefore, the strengths of CAC increase as the time is longer. Moreover, the subsequent hydration

Fig. 7. Normal distributions of ft (21 test results): (a) Z28 (b) J28 (c) Z90 (d) J90. 7

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Fig. 7. (continued).

Fig. 8. Normal distributions of KIC (21 test results). (a) Z28 (b) J28 (c) Z90 (d) J90.

reaction would be more sufficient under seawater immersion conditions. Thus, the strengths of CACs after immersion in seawater would be higher.

4.2. Analysis of crack propagation process The crack propagation process of CAC is very similar to that of OC even after immersion in seawater. It still includes the initial cracking 8

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Fig. 9. Variations of σ n with ae. (a) Z28 (b) J28 (c) Z90 (d) J90.

state, stable crack propagation stage and unstable crack propagation process. The typical variations of the applied load F with CMOD in all the four groups are shown in Fig. 5. Fig. 5 gives the typical F-CMOD curves in the four groups with a0/ h ¼ 0.3. The maximum fracture load Fmax increases significantly with the age. The Fmax in the beams after seawater immersion are higher than those in the specimens exposed to drying environment. It is also due to the subsequent hydration reaction of cement paste in CAC and more sufficient reaction under seawater immersion condition. Moreover, the failure modes of all the beams are fracturing of columnar coral aggregates in the cracked section as shown in Fig. 6. It is mainly because cement mortar in CAC has good bond performance with coral particles due to the porous properties of the latter. The failure mode tends to be aggregate fracture rather than interfacial debonding. As the age increases and the condition is changed from drying envi­ ronment to seawater immersion, the interfacial transition zones become denser.

(8) and (9). It is approximately dmax/1.5 according to Han et al. (2019). Thus, G ¼ 7 mm is adopted since the dmax is 10 mm for the CAC in the present study. The effect of G will be discussed later. The results of all the beams are then listed in Table 4 as follows. In each group, the values of either ft or KIC from the measured Fmax have certain experimental scatters. They should follow normal distri­ bution. Thus, the scatters can be analyzed by virtue of the normal dis­ tribution functions as follows. The mean μf and standard deviation σf of the ft and the corresponding ones μK and σ K of the KIC are obtained based on the normal distribution analysis. Moreover, Figs. 7 and 8 show the normal distributions of ft and KIC. The means and the upper and lower bounds with 95% reliability are predicted directly. Moreover, the rela­ tive errors are 29.6%, 34.3%, 31.6% and 28.3% for Z28, J28, Z90 and J90, respectively. 2

½ftðiÞ μf � h i 1 2σ2 f f ftðiÞ ¼ pffiffiffiffiffi e ði ¼ 1 2πσ f

4.3. Determination of ft and KIC with Fmax from the test

(17a)

21Þ

2

½KICðiÞ μK � � � 1 2σ 2 K ði ¼ 1 f KICðiÞ ¼ pffiffiffiffiffi e 2π σ K

Both the ft and KIC can be determined from Eqs. (8) and (9) for each specimen once the Fmax is measured from the fracture test. However, the average grain size G has to be estimated before the application of Eqs.

21Þ

(17b)

It can be seen from Figs. 7 and 8 that both the ft and KIC are 9

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Fig. 10. Variations of gf-tip with a0. (a) Z28 (b) J28 (c) Z90 (d) J90.

improved by 20% as the period is increased from 28-day to 90-day. When the ages are the same, the increases of ft and KIC are 10% if the CAC is immersed in seawater compared to that in drying environment. The reasons are explained as follows. As the period is longer, the hy­ dration reaction is more sufficient accompanied by subsequent hydra­ tion reaction due to the water release from the porous coral aggregates. Moreover, the reaction would develop more sufficiently when the CAC is immersed in seawater. The ft is much dependent on the bridging action

of coral aggregates and tensile resistance of cement paste according to the failure modes. Therefore, the tensile resistance TR related to ft can be given as follows. TR ¼ TR

M

þ TR

A

(18)

Herein, TR-M and TR-A are the tensile resistance attributed to the cement mortar and columnar coral aggregates in the CAC, respectively. Since the types and amounts of coral aggregates are the same for the four 10

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Fig. 10. (continued).

groups of CAC, the increase of ft is mainly induced by the higher tensile resistance TR-M of mortar. According to Eq. (9), the increase of ft also results in larger values of KIC. By virtue of the means, upper and lower bounds of ft, the variations of σn with ae are shown in Fig. 9 for all of the four CACs. The transition regions from strength-controlled failure to fracture toughness-controlled failure can be clearly identified in Fig. 9. It can be seen that all the test results just fall in the transition regions. Besides, the values of ae are significantly smaller than those of a*∞ . Therefore, it is inclined to

strength-controlled failure although the CAC shows quasi-brittle frac­ ture irrespective of the period and environment. 4.4. Determination of GF with Fmax from the test For each group, the gf-tip corresponding to different initial crack length a0 can be directly obtained from Eq. (16a) once the Fmax is measured from the test and the ft is determined by Eq. (8). Moreover, the discrete number βw related to the wtc is assumed to be 2.0 here. The rationality of the adopted value will be discussed later. Thus, the 11

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also indicates the distance towards the back boundary. When the a0 is shorter than 30 mm, the gf-tip is much affected by the front boundary effect. The value is increased as the crack-tip is far away from the front boundary. When the a0 is longer than 60 mm, the crack-tip is gradually approaching the back boundary. The gf-tip shows a decrease due to the back boundary effect. Therefore, the local fracture energy gf follows a tri-linear distribution as indicated by Yang et al. (2011, 2014) based on the new model. Fig. 11 shows a tri-linear gf distribution for CACs in different conditions. The transition lengths due to the front and back boundary effects are 30–40 mm and about 5.5 times of the average aggregate size G (�5.5G). Moreover, the maximum value of gf or gf-tip is actually the sizeindependent fracture energy GF (Hu and Wittmann, 1992). Similar to the ft and KIC, the values of GF from all the beams in each group also follow the normal distribution. The mean μG and standard deviation σG of GF can be derived from Eq. (19) as follows. The normal distributions of GF in the four groups are shown in Fig. 12. The upper and lower

Fig. 11. Local fracture energy distribution in CACs under different conditions.

variations of gf-tip with a0 are shown in Fig. 10 for all of the four groups. From Fig. 10, it can be seen that the varying tendencies of the gf-tip with the a0 are similar in the four groups. When the a0 does not exceed 30 mm, the gf-tip shows an apparent increase. After that, the values of gftip almost keep constant until the a0 is longer than 60 mm. Then the gf-tip begins to decrease continuously. In fact, the value of a0 directly reflects the distance between the crack-tip and front boundary of specimen and

Table 5 Summary of fracture parameters. Nos. of groups

ft (MPa)

KIC (MPa∙m1/2)

GF (N/m)

Z28 J28 Z90 J90

2.91 3.15 3.48 3.82

0.844 0.912 1.01 1.106

74.2 86.8 87.5 100.9

Fig. 12. Normal distributions of GF (21 test results): (a) Z28 (b) J28 (c) Z90 (d) J90. 12

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Ocean Engineering 191 (2019) 106508

Table 6 Analysis of effects of G. Nos. of groups

G ¼ 6 mm

Z28 J28 Z90 J90

G ¼ 7 mm

ft (MPa)

KIC (MPa ∙m1/2)

GF (N/m)

ft (MPa)

KIC (MPa ∙m1/2)

GF (N/m)

ft (MPa)

KIC (MPa ∙m1/2)

GF (N/m)

3.08 3.39 3.69 4.04

0.827 0.893 0.99 1.085

67.5 78.6 80.1 88.6

2.91 3.15 3.48 3.82

0.844 0.912 1.01 1.106

74.2 86.8 87.5 100.9

2.76 2.99 3.31 3.62

0.856 0.926 1.025 1.123

75.4 88 88.2 105.8

concrete. Thus, βw ¼ 2.0 is more rational in the present study.

Table 7 Analysis of effects of βw.

5. Conclusions

Nos. of groups

βw ¼ 1.0 ft (MPa)

KIC (MPa∙m1/2)

GF (N/ m)

ft (MPa)

KIC (MPa∙m1/2)

GF (N/ m)

Z28 J28 Z90 J90

2.91 3.15 3.48 3.82

0.844 0.912 1.01 1.106

37.1 43.4 43.8 50.4

2.91 3.15 3.48 3.82

0.844 0.912 1.01 1.106

74.2 86.8 87.5 100.9

βw ¼ 2.0

An analytical approach is proposed to determine the sizeindependent tensile strength ft, fracture toughness KIC and fracture en­ ergy GF of CAC. The ft and KIC are directly determined using the maximum fracture load Fmax based on the improved BEM (Guan et al., 2018, 2019a; 2019b; Han et al., 2019; Zhang et al., 2018) by incorpo­ rating the average aggregate size G. Besides, the Fmax is first correlated with the crack-tip local fracture energy gf-tip in the form of an explicit analytical solution by improving the MFLM (Yang et al., 2011, 2014). The local fracture energy gf distribution and GF can be then obtained directly. Moreover, three-point-bending tests were carried out on CAC beams with different a0/h. Four different conditions are designed, i.e., drying environment for 28 days, drying environment for 90 days, im­ mersion in seawater for 28 days and immersion in seawater for 90 days. The values of fracture parameters from all the specimens in each con­ dition were obtained and evaluated based on normal distribution anal­ ysis. The main conclusions are drawn as follows.

bounds are given with reliability of 95%. The relative errors are 22.4%, 28.8%, 24.5% and 22.8% for Z28, J28, Z90 and J90, respectively. � � 1 f GFðiÞ ¼ pffiffiffiffiffi e 2πσ G

G ¼ 8 mm

½GFðiÞ

μG

2σ2 G

�

2

ði ¼ 1

21Þ

(19)

As the period is longer, the GF is increased by 16–18%. When the condition is changed from dying environment to seawater immersion, the GF also shows an increase which is 15–17%. The varying tendencies are similar to those of ft and KIC.

1. Both the compressive strength fcu and elastic modulus Ec increase with the period. The strengths of CAC immersed in seawater are higher compared to those exposed to drying environment. 2. The failure modes of all the beams are fracturing of columnar coral aggregates which have good bond performance with the surrounding mortar due to their porous properties. 3. Based on the proposed approach, the means and upper and lower bounds with reliability of 95% can be obtained analytically for all of the fracture parameters. 4. The gf distributions of CACs under different conditions still follow trilinear variations indicating both the front and back boundary effects. The transition lengths due to the boundary effects are about 5.5G. 5. Both the tensile strength and fracture toughness are improved as the period is longer due to the subsequent hydration reaction in the paste. When the CAC is immersed in seawater, the hydration reaction may be more sufficient and all the parameters show an increase compared to the CAC in drying environment.

4.5. Further discussions Table 5 summarizes the average values of ft, KIC and GF from all of the four groups. It can be seen that all the parameters increase with the period. If the CACs are immersed in seawater, both the strength and fracture toughness are improved further compared to those in drying environment. It demonstrates that the environment of seawater im­ mersion would not deteriorate the mechanical performance of CAC ac­ cording to the results from the present study. It has been mentioned that the G is adopted as 7 mm in the above calculations. In fact, it is too difficult to be estimated accurately. The size distributions of aggregates may be varied in different batches. To analyze the sensitivity of the results to the variations of G, G ¼ 6 mm and G ¼ 8 mm are also introduced here. The fracture parameters with different values of G are then given in Table 6 as follows (see Table 7). As the G increases, the concrete is more coarse-structured since the sizes of specimens are invariable. The ft is then reduced but the fracture toughness is increased. However, the relative errors (relative to the re­ sults with G ¼ 7 mm) are smaller than 7% for the ft and 2% for the KIC. The maximum relative error occurs in the GF of J90 CAC between G ¼ 6 mm and G ¼ 7 mm, and attains 12%. Others are all below 10%. Therefore, G ¼ 7 mm is accurate enough. Moreover, the βw is adopted as 2.0 in the above calculation. It is a discrete number and may be between 1.0 and 2.0. Herein, the fracture parameters based on βw ¼ 1.0 are given and compared to the results based on βw ¼ 2.0 as follows. According to Eqs. (1), (8), (9), (16a) and (16c), the value of βw has no effect on the ft and KIC but much affects the GF. When the βw decreases from 2.0 to 1.0, the value of GF is reduced by 50%. Besides, it is found that the maximum value of GF is around 50 N/m with βw ¼ 1.0 and 100 N/m with βw ¼ 2.0. Previous work by the Xu et al. (2019) concluded that the GF is 100 N/m for alkali-activated slag seawater coral aggregate

In summary, analytical solutions of fracture parameters can be explicitly given only using the maximum fracture load which is very easy to be determined in test. Thus, the approach proposed in this paper is simple and applicable. Moreover, seawater immersion is merely one of the simulated marine environments. The present study is a pilot work only aimed at one common marine environment and could provide some reference for the application of CAC in ocean engineering. At least, no deterioration is found in the mechanical properties of CAC immersed in seawater according to the results in this paper. Further studies, such as long-term properties in other marine environments, will be performed in the future. Acknowledgements The authors gratefully acknowledge funding from the National Natural Science Foundation of China (Grant 51778591) and Educational 13

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Ocean Engineering 191 (2019) 106508

Innovation and Research Foundation of Graduate Student in Shandong Province of China (Grant HDJG17006).

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