Influence of the porosity and pore size on the compressive and splitting strengths of cellular concrete with millimeter-size pores

Construction and Building Materials 235 (2020) 117508

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Construction and Building Materials journal homepage: www.elsevier.com/locate/conbuildmat

Influence of the porosity and pore size on the compressive and splitting strengths of cellular concrete with millimeter-size pores Sherong Zhang a,b, Kelei Cao a,b, Chao Wang a,b,⇑, Xiaohua Wang a,b, Genhua Deng b, Peiyong Wei a,b a b

State Key Laboratory of Hydraulic Engineering Simulation and Safety, Tianjin University, Tianjin 300072, China School of Civil Engineering, Tianjin University, Tianjin 300072, China

h i g h l i g h t s
Segregation degree is remarkably dependence on the volume fraction and size of Sat-SAP.
Porosity and pore size have obvious influence on the mechanical properties.
The modified stress-strain model accurately describes the compression behavior.
Correlations between porosity, pore size and mechanical behaviors are investigated.
The relationship between compressive and splitting-tensile strengths is studied.

a r t i c l e

i n f o

Article history: Received 31 May 2019 Received in revised form 1 November 2019 Accepted 5 November 2019

Keywords: Cellular concrete Porosity Pore size Segregation characteristic Compressive strength Splitting tensile strength

a b s t r a c t Cellular concrete with millimeter-sized pores has recently been suggested as an attractive material in engineering fields because of its specific material properties. However, few studies have assessed the effects of porosity, pore size and their interaction on the splitting tensile strength of cellular concrete. A comprehensive understanding of the mechanical properties of cellular concrete is needed prior to its application in engineering. The effects of porosity (0, 5, 10, 15, 20, 25 and 30%) and pore size (0, 1, 4.5, 7.5, 9.5 and 12.0 mm) on the segregation, compressive strength and splitting tensile strength of cellular concrete fabricated using a saturated superabsorbent polymer (Sat-SAP) and their relationships are investigated in this study. The incorporation of SAP significantly increases the segregation characteristic of cellular concrete. The segregation is particularly high for porosities higher than 15%, and the lowest segregation is observed for pore sizes of 4.5–7.5 mm. The compressive and splitting tensile strengths of cellular concrete decrease gradually with increasing porosity and pore size, but the effect of pore size on the mechanical properties of cellular concrete is very weak at pore sizes greater than 9.5 mm and porosities less than 10%. Moreover, a modified method for describing the stress-strain curve and calculating the elastic modulus of cellular concrete is suggested. Based on these results, some empirical formulas are proposed or modified to better analyze the segregation characteristics, accurately describe the stress-strain curve, and effectively predict the mechanical properties of cellular concrete with different porosities and pore sizes and their relationship. Ó 2019 Elsevier Ltd. All rights reserved.

1. Introduction Cellular concrete, a type of lightweight concrete material with a large number of closed pores, consists of a mixture of cement, aggregates, water and voids. The entrapment of air in voids makes the material lightweight but may also compromise its mechanical properties [1,2], such as its compressive strength and durability,

⇑ Corresponding author at: State Key Lab of Hydraulic Engineering Simulation and Safety, School of Civil Engineering, Tianjin University, Tianjin 300072, China. E-mail address: [email protected] (C. Wang). https://doi.org/10.1016/j.conbuildmat.2019.117508 0950-0618/Ó 2019 Elsevier Ltd. All rights reserved.

hindering its application and popularization in engineering. However, due to its unique material properties of a low density, sound insulation, thermal insulation and energy absorption, cellular concrete has been widely applied for structural shock absorption, insulated wall panels, mine backfilling, flame resistance and antiknock protection in structures [2,3]. Typically, the porous structure of cellular concrete is obtained by introducing air voids produced by foaming agents in mechanical or chemical methods; however, the uniformity of the shape, size, quantity and distribution of the voids is usually difficult to control, mainly due to the stability of foam [4–6]. Some scholars have tried

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to use superabsorbent polymers (SAPs) with uniform shapes, sizes and densities to prepare cellular concrete in an attempt to compensate for the deficiency associated with the foaming method of the foaming agent. During the fabrication process, the saturated SAP is mixed directly with the concrete matrix and play a role more like the coarse light-weight aggregate (LWA) for lightweight concrete due to its relatively low specific gravity (close to 1). Furthermore, the use of a saturated SAP as an internal curing agent to form cellular concrete not only reduces costs and simplifies the preparation process but also improve the autogenous shrinkage, permeability and microscopic pore characteristics of the concrete. Therefore, in recent years, the potential application of SAPs in cellular concrete has been attention by some scholars, and have achieved certain results in preparation method, static mechanical property and dynamic mechanical property [7–9]. Existing studies on the concrete containing SAPs have mostly focused on the effects of the water absorption/desorption kinetics of the SAPs, the SAP content and mixture design on selfdesiccation, autogenous shrinkage, hydration degree, and other properties. The internal curing effect of SAPs not only provides additional water to promote the hydration of the cement-based matrix and thereby alleviate the effect of self-desiccation in dry environments [10,11], but also improves the microscopic properties of the concrete matrix to improve the durability and tensile strength of the concrete [12]. In addition, other studies have identified a small range in which the improvement of mechanical properties provided by the internal curing does not compensate for the extra porosity of the SAP [13–16]. However, the effect of pores introduced by the SAP on the mechanical properties of concrete has not been studied extensively, and further research is warranted. Some researchers have expounded on the importance of pore properties in determining the mechanical properties of concrete, and some conclusions have been generated from laboratory tests and numerical simulations [17,18]. However, these achievements have been sporadic, and the results have been less than satisfactory. To date, most researchers have utilized the methods of water displacement, mercury intrusion and vacuum saturation to measure the total porosity [19,20], and the distribution characteristics of pores at different scales in a concrete matrix have been studied by combining fractal theory with scanning electron microscopy (SEM) and image analysis techniques [21–23]. Furthermore, different types of mathematical prediction models or regression models have been generated to establish the relationship between the strength and porosity based on the results of different tests [24,25], and prediction results have been compared with test results to verify the accuracy of prediction models. However, previous studies have been limited to examining the effect of porosity in hydrated cement paste on the mechanical properties. Few research studies have extended beyond simple analyses of a specific material containing different fractions of macroscopic pores to explore the separate associations between the porosity, pore size, and pore shape and mechanical properties (compressive strength and tensile strength) of the studied material, and studies that involve cellular concrete utilizing a saturated SAP are particularly scarce. Compressive strength and splitting strength are important design parameters in civil engineering, and their values are substantially modulated by the pore characteristics. To date, research on the mechanical properties, particularly the splitting tensile strength, of the novel cellular concrete prepared using saturated SAP, has been very limited; moreover, correlations between the mechanical properties and pore properties of the novel cellular concrete have not been reported. Therefore, a study of the material properties of the novel cellular concrete is warranted. This study analyzed the effects of porosity (0, 5, 10, 15, 20, 25 and 30%) and

pore size (0, 1, 4.5, 7.5, 9.5 and 12.0 mm) on the segregation characteristics and compressive and splitting tensile behaviors of cellular concrete. The applicability of the identified strength-porosity relationships for cellular concrete predictions was examined/ extended and compared with experimental results. The investigation provides a fundamental understanding of the segregation characteristics and compressive and splitting tensile behaviors of cellular concrete with millimeter-sized pores. This study not only provides a reference for selecting the appropriate volume fraction and diameter of the SAP in the mixture design but also provides useful information for the application of cellular concrete in civil engineering construction projects. 2. Experimental investigation 2.1. Materials Portland cement (PC) with a 42.5 grade and a Blaine fineness of 330 m2/kg was used in this experiment. The river sand (RS) was fine sand with a moisture content of 2.8% and a fineness modulus of 2.2. The coarse aggregates with continuous particle sizes less than or equal to 10 mm were used and had a specific gravity of 2.76. Silica fume (SF) and waste marble powder (WMP) were added as partial replacements for the cement in concrete, and the specific gravities of the SF and WMP were 1.9 g/cm3 and 2.21 g/cm3, respectively. Tap water (W) was used for mixing, and the workability of the fresh concrete was ensured by the polycarboxylate superplasticizer (SP). The chemical compositions of the cement, SF and WMP are shown in Table 1. Saturated superabsorbent polymer (Sat-SAP) with five diameters of 1.0, 4.5, 7.5, 9.5 and 12.0 mm were obtained with different concentrations of a sodium chloride (NaCl) preleaching solution, i.e., 1.95, 1.00, 0.70, 0.50 and 0.15 mol/L, respectively. The water absorption capacities of the SAP beads in the different concentrations of NaCl solution are shown in Fig. 1(a), with the maximum water absorptivity values of 7.6, 14.16, 16.7, 18.0 and 18.8. An image of the spherical saturated SAP with a diameter of 4.5 mm is shown in Fig. 1(b). 2.2. Mixture proportions According to previous studies, SAP beads tend to float when mixed with concrete, mainly because the specific gravity of the saturated SAP is approximately 1. This floating was observed during the concrete compactness vibration process in the present study. The consistency of the concrete paste was increased to reduce the floating of SAP beads and obtain spherical millimeter-sized pores that were uniformly distributed in the cellular concrete [8]. The incorporation of both SF and WMP in this study not only improved the consistency of the concrete paste, thereby reducing the floatation of the SAP beads, but also improved the mechanical, environmental and economic performance of the cellular concrete [26]. The selected mass ratio was SF: WMP: SP: (SF + WMP + PC) = 0.1: 0.05: 0.007: 1. The proportions of the concrete mixtures are presented in Table 2; all of the concretes had the same mortar matrix with a water-to-binder (SF + WMP + PC) ratio (W/B) of 0.33. The mix designated SAP0 in Table 3 was considered the control group, and did not contain the saturated SAP. The different concrete mixtures are denoted by the volume fraction and diameter of the saturated SAP used in this study, for example, SAP5-1.0. Furthermore, the volume fraction and diameter of the spherical Sat-SAPs in concrete are defined as the porosity P (excluding the capillary pores) and pore size D. The experiment investigated the effects of different SAP porosities (i.e., 0%, 5%, 10%, 20%, 25% and 30%) and pore sizes

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S. Zhang et al. / Construction and Building Materials 235 (2020) 117508 Table 1 Chemical compositions of the cement, silica fume and waste marble powder. Test Material

SiO2 (%)

Fe2O3 (%)

Al2O3 (%)

CaO (%)

MgO (%)

K2O (%)

Na2O (%)

Alkali Content (%)

SO3 (%)

Loss on Ignition (%)

Cement Silica fume Waste marble powder(WMP)

19.92 96.74 8.38

5.32 0.08 0.65

3.55 0.32 0.67

57.48 0.11 61.83

4.07 0.10 14.36

0.57 – 0.07

0.14 0.09 0.60

0.52 – –

2.20 0.00 0.33

1.34 – 13.02

Fig. 1. The millimeter-sized spherical SAP: (a) Water absorptivity of the SAP beads incubated with different concentrations of the NaCl solution for different times (b) Image of the spherical saturated SAP with a diameter of 4.5 mm.

Table 2 Proportions of the cellular concrete mixtures (kg/m3). Component Cement SF WMP Water Sand Aggregate

SAP0

7–10 mm 4–7 mm <4mm

Superplasticizer Superabsorbent polymer (SAP)

484.3 57.0 28.5 188.0 772.8 406.8 280.2 119.4 3.9 0.0

SAP51.0/4.5/7.5/

SAP101.0/4.5/7.5/

SAP151.0/4.5/7.5/

SAP201.0/4.5/7.5/

SAP251.0/4.5/7.5/

SAP301.0/4.5/7.5/

9.5/12.0

9.5/12.0

9.5/12.0

9.5/12.0

9.5/12.0

9.5/12.0

460.1 54.1 27.1 178.6 734.2 386.5 266.2 113.4 3.7 49.4

435.9 51.3 25.6 169.2 695.5 366.1 252.2 107.5 3.5 98.8

411.7 48.4 24.2 159.8 656.9 345.8 238.2 101.5 3.3 148.2

387.4 45.6 22.8 150.4 618.3 325.4 224.2 95.5 3.1 197.6

363.2 42.7 21.4 141.0 579.6 305.1 210.2 89.5 2.9 247.0

339.0 39.9 19.9 131.6 541.0 284.8 196.2 83.6 2.7 296.5

(i.e., 0, 1.0, 4.5, 7.5, 9.5 and 12.0 mm) on the workability, segregation, compressive strength and splitting tensile strength of cellular concrete in this paper. 2.3. Specimen preparation All cellular concrete specimens were mixed in a 50 dm3 laboratory gravity type mixer. Homogeneous mixtures were obtained with the feeding sequence described below. First, the powder components and aggregates were placed in the mixer and premixed together without water for 1 min. Then, the saturated SAP beads were introduced and mixed for approximately 1 min at a slow speed. Finally, the water and PS were added, and all the materials were mixed for 5 min to ensure that the concrete became homogeneous. After mixing, the fresh concrete mixture was poured into 150  150  150 mm cubic plastic molds and compacted by a vibrating tube to remove air voids, as shown in Fig. 2. The total time required to produce a specimen was approximately 1– 3 min. When cement mortar appeared on the surface of the specimen, the compactness of the concrete was considered to have met

the requirements, and the compaction was complete. Previously, the research team had observed that the floating capacity of SAP beads in cellular concrete was substantially reduced when using the vibrating tube (relative to using the shaking table) for vibration compaction, which improved the uniformity of the concrete mixture. Then, all the specimens with each mix were cured in a standard curing room (20 ± 2 °C, 95% relative humidity (RH)) for the segregation and mechanical property tests.

2.4. Experimental methods 2.4.1. Slump Slump is an important index for evaluating the workability of fresh concrete, particularly for cellular concrete, because it is affected by the flotation of SAP beads in concrete. All workability tests were conducted immediately after the mixing was complete, i.e., within 10 min, to avoid a loss of workability. The method for testing slump was performed in strict accordance with the Chinese standard (GB/T 50081-2002) [27].

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Table 3 Specific density values of the slices. Slice density (kg/m3)e

Mix ID

SAP0 SAP5-1.0 SAP10-1.0 SAP15-1.0 SAP20-1.0 SAP25-1.0 SAP30-1.0 SAP5-4.5 SAP10-4.5 SAP15-4.5 SAP20-4.5 SAP25-4.5 SAP30-4.5 SAP5-7.5 SAP10-7.5 SAP15-7.5 SAP20-7.5 SAP25-7.5 SAP30-7.5 SAP5-9.5 SAP10-9.5 SAP15-9.5 SAP20-9.5 SAP25-9.5 SAP30-9.5 SAP5-12.0 SAP10-12.0 SAP15-12.0 SAP20-12.0 SAP25-12.0 SAP30-12.0 e

Specimen density q0(kg/m3)

Specific density of slice

q1

q2

q3

q4

qS1

qS2

qS3

qS4

2324 2061 1966 1838 1654 1512 1387 2067 2002 1961 1703 1628 1504 2115 2050 2010 1801 1634 1546 2076 2019 1935 1752 1643 1510 2064 2015 1942 1704 1609 1508

2347 2161 2134 1945 1795 1646 1550 2222 2107 2060 1892 1708 1640 2160 2173 2113 1942 1747 1638 2234 2162 2053 1860 1724 1630 2251 2162 2059 1820 1735 1628

2368 2389 2224 2059 2042 1865 1754 2244 2177 2114 1958 1864 1800 2274 2181 2127 1993 1843 1750 2245 2171 2059 1955 1881 1803 2253 2181 2120 1931 1888 1803

2383 2414 2323 2230 2134 2128 2077 2339 2209 2158 1965 2088 2003 2384 2248 2188 2066 2019 2008 2313 2216 2121 2130 2108 2041 2331 2228 2294 2153 2204 2133

0.991 0.906 0.913 0.903 0.859 0.835 0.817 0.933 0.941 0.945 0.902 0.891 0.865 0.943 0.953 0.956 0.921 0.901 0.897 0.936 0.943 0.95 0.905 0.891 0.871 0.925 0.937 0.927 0.891 0.866 0.855

1.001 0.950 0.991 0.956 0.932 0.909 0.913 1.003 0.990 0.993 1.002 0.935 0.943 0.963 1.010 1.005 0.993 0.963 0.950 1.007 1.010 1.008 0.961 0.935 0.940 1.009 1.005 0.983 0.952 0.934 0.923

1.010 1.050 1.033 1.012 1.06 1.03 1.033 1.013 1.023 1.019 1.037 1.02 1.035 1.014 1.014 1.012 1.019 1.016 1.015 1.012 1.014 1.011 1.010 1.02 1.040 1.010 1.014 1.012 1.010 1.016 1.022

1.016 1.061 1.079 1.096 1.108 1.175 1.223 1.056 1.038 1.04 1.041 1.143 1.152 1.063 1.045 1.041 1.056 1.113 1.165 1.043 1.035 1.041 1.100 1.143 1.177 1.045 1.036 1.095 1.126 1.186 1.209

The density value is reported as the average density of three specimens.

Fig. 2. The specimen preparation process.

2345 2275 2153 2035 1926 1811 1698 2215 2128 2075 1888 1827 1739 2243 2151 2102 1956 1814 1724 2218 2141 2037 1936 1844 1734 2231 2151 2095 1912 1858 1764

S. Zhang et al. / Construction and Building Materials 235 (2020) 117508

2.4.2. Density test Due to the light weight of the SAP beads, they tend to float to the top surface of the mixture during specimen preparation and setting, resulting in the uneven distribution of the mixture and an increased risk of segregation. Li [28,29] analyzed the specific density to evaluate the segregation degree and to understand the distribution of the homogeneity of SAP-containing cellular concrete during setting. After 28 days of standard curing, the specimens were placed in a room and subjected to natural air drying for 24 h. Then, the weight and density (q0) of each air-dried specimen were measured, and q0 was defined as a reference to calculate the specific density. The volume of each air-dried specimen was defined as 150  150  150 mm3. Each specimen was uniformly cut into four slices along the direction of casting by a cutting machine and the volume of each slice was measured using Archimedes’ principle. The densities of the four slices of each specimen, from top to bottom, were denoted q1, q2, q3 and q4, as shown in Fig. 3. The specific density of the slices was determined by the

Fig. 3. Schematic depicting the slice model of a specimen.

5

equation qsn = qn/q0, where n = 1, 2, 3, or 4. The specific densities were used to evaluate the degree of segregation of the concrete with different volume fractions and diameters of the SAPs. The wet density of each mixture was obtained by averaging the values of three specimens after smoothing the top with a spatula. 2.4.3. Compressive strength test The prepared specimens were loaded to characterize the mechanical behaviors under a quasi-static loading rate of 1 mm/s using a microcomputer-controlled electrohydraulic servo loading testing machine. Grease was applied to the interfaces between the specimen and apparatus before the quasi-static compressive tests to minimize the end friction. The loads and strains of the specimens with various proportions were measured using an internal force transducer and an internal linear variable differential transducer (LVDT) during the compressive test under quasi-static loading. The standard procedure for conducting the compressive strength test was performed in strict accordance with the Chinese standard (GB/T 50081-2002) [27]. 2.4.4. Splitting tensile strength test The splitting tensile strength tests were performed using the microcomputer-controlled electrohydraulic servo loading testing machine. A schematic of the splitting tensile test is shown in Fig. 4. A high-strength steel fixture was placed between the apparatus platen and specimen, to allow uniformly distributed pressure to be applied along the midline of the two relative surfaces. During the splitting tensile strength tests, the loading was controlled at a constant loading rate of 1 mm/s, and the crack opening displacement was obtained by displacement transducers placed on both sides of the tested specimen. An effect of the width of the loading strip on the splitting-tensile strength has been reported [30,31], and the width of the loading strip should not exceed 4% of the diameter of the specimen [32]. Therefore, the width of the loading strip was set to 5 mm, and allowing uniformly distributed tensile stress to be produced in the vertical plane under the action of external forces. The Chinese standard (GB/T 50081-2002) was also used to assess the splitting tensile strength [27].

Fig. 4. Schematic of the splitting tensile strength test.

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3. Experimental results 3.1. Properties of fresh concrete Fig. 5 shows the effects of SAP porosity and pore size on the wet density and slump of all mixes. As shown in Fig. 5(a), the wet density of a specific mixture was not substantially affected by the pore size, but it was substantially affected by porosity. The wet density of cellular concrete decreased gradually with increasing porosity because the density of the Sat-SAP is less than the mixture. When the porosity was 30%, the wet density of cellular concrete decreased by approximately 20–24% compared to specimens without the Sat-SAP. The wet density of the specimens without Sat-SAP was 2371 kg/m3. However, the slump of fresh concrete, as shown in Fig. 5(b), was substantially affected by the porosity and pore size of the SAP. The slump of cellular concrete increased linearly with increasing porosity but showed a negative correlation with the SAP pore size. The slump of fresh concrete with a SAP diameter of 1 mm was greater than fresh concrete with SAPs of other diameters in the same mixture, and the maximum slump value was 235 mm at a volume fraction of 30%. The differences in the slump within each mixture are mainly attributed to the ball-bearing effect of the SAP beads in the fresh concrete. Generally, the incorporation of millimeter-sized spherical Sat-SAP is beneficial for improving concrete workability. 3.2. Segregation characteristics of cellular concrete 3.2.1. Specific density distribution of the specimen Table 3 presents the specific density values of all specimens, which reflect the effects of SAP porosity and pore size on the segregation. As shown in Table 3, the specific density of cellular concrete ranged from 2345 to 1698 kg/m3. The specific densities were significantly different among specimens with different SAP porosities and different SAP pore sizes. The specific density of the specimens containing different mixes ranged from 0.817 to 1.223. A curve diagram of the specific density distribution with the barycenter coordinate of the slice as the Z-coordinate was drawn [28] according to the results shown in Table 3, and the curves of slices of specimens of different mixes are shown in Fig. 6. As shown in Fig. 6, regardless of the volume fractions and diameter of the SAPs introduced in the concrete, the specific densities of the specimens increased gradually from the top to the bottom along the direction of casting. This phenomenon was potentially attributed to the segregation of the SAP beads. In general, differences in slice density appear along the direction of casting as the

SAP beads float to the surface; thus, the density of the bottom slice was higher than the top slice because of the sinking of the coarse aggregates under the action of dead weight. No segregation was observed at a specific density of 1 (reference value). As shown in Fig. 6, the introduction of SAPs significantly increased the segregation of concrete. The segregation was particularly high for porosities greater than 15%, and pore sizes between 4.5 and 7.5 yielded the highest segregation. Based on these findings. Based on these findings, the effects of the volume fraction and diameter of SAPs on the segregation characteristics of cellular concrete should be considered when producing cellular concrete. 3.2.2. Segregation index In this study, the segregation index (SI) was used to evaluate the degree of segregation in the mixtures [28]. The value of SI was obtained from the average slope of the linear regression curve for the specific density distribution and the barycenter coordinate of the slice. The SI was calculated using the formula SI = dq/dz. In theory, the average slope of the linear regression curve for the specific density distribution and the barycenter coordinate of the slice was equal to zero for uniform cellular concrete. When the mixture segregated, the value of SI was greater than zero. Therefore, the SI was adopted to intuitively and effectively evaluate the degree of segregation in a mixture The SI values of the specimens with different SAP porosities and pore sizes are shown in Fig. 7. SI increased with the increasing volume fractions of SAP, and high SI values were obtained for the specimens with SAP diameters ranging from 1 to 7.5 mm. The smallest SI value of 0.22 was obtained for the specimen without SAP, whereas the maximum SI value of 3.61 was obtained for the specimen SAP30-1. As an SI value of 0 would have been almost impossible because of experimental error, the SI range of 0 to 0.22 was defined as the category of no segregation in the present study. The SI values of all mixes were significantly greater than 0.22, indicating that the introduction of SAP was not conducive to producing a mixture with a uniform distribution. In practical engineering applications, measures such as increasing the consistency of the concrete paste, controlling the volume fraction and diameter of SAP and performing compacting vibration with a vibrating tube, can be used to improve the segregation of cellular concrete. Consistent results were obtained using the special density method, which confirmed the effectiveness of the SI method in evaluating the segregation of cellular concrete. A new exponential equation for the SI was developed to obtain a better understanding of the segregation indexes of specimens with different SAP porosities and pore sizes, as shown in Fig. 7. The SIs of all mixes increased with increasing porosity in exponential form.

Fig. 5. Effects of SAP porosity and pore size on the properties of fresh concrete, including the (a) wet density and (b) slump.

S. Zhang et al. / Construction and Building Materials 235 (2020) 117508

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Fig. 6. Specific density vs. Z-coordinate curves of the specimens containing different porosities and pore sizes of SAPs.

The following equation was used to calculate the empirical relationship curve obtained using a regression analysis: 0

SI ¼ a0  expðb  PÞ

ð3Þ

ð1Þ

where P represents the porosity (the volume fraction of the SAPs) 0 and the fitting parameters a0 and b are related to the particle size of the SAP. Using a nonlinear regression analysis, the relationships between the fitting parameters and pore size D of SAP were established as described in Eqs. (2) and (3). The correlation coefficients for Eqs. (2) and (3) are 0.9735 and 0.9635, respectively. Based on the results of this analysis, the pore size of the SAP is closely related to the SI of the cellular concrete.

a0 ¼ 0:0125  D2  0:1822  D þ 1:1146

0

b ¼ 0:0238  D2 þ 0:3810  D þ 4:0254

ð2Þ

3.3. Compressive behaviors of cellular concrete under quasi-static loading 3.3.1. Compressive behaviors of cellular concrete A stress-strain curve directly reflects the deformation and strength characteristics of a material. Experimental stress-strain curves of the specimens with different SAP porosities and pore sizes are presented in Fig. 8. The stress-strain curves of cellular concrete exhibited similar shapes and trends regardless of the porosity and pore size of the SAP. Compared with the specimen lacking the SAP, namely, normal concrete, the stress-strain curves of cellular concrete containing the SAP exhibited a slowly increas-

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S. Zhang et al. / Construction and Building Materials 235 (2020) 117508

Fig. 7. Segregation indexes of different mixtures.

ing trend during the initial stage. The shallow slopes of the stressstrain curves in the initial stage were explained by the collapse and compaction of surface pores or defects. The collapse of surface pores gradually increased with increasing stress. The slow growth stage of the stress-strain curve was defined as the ‘‘initial compaction stage”. As shown in Fig. 8(a) and (b), the peak stress and peak strain of the specimens decreased continuously with increasing porosity. These phenomena were mainly observed because the ease with which cracks form between the pores increases as the number of pores increases, eventually resulting in failure. After peak stress, the stress began to decrease and then leveled off until it reached a plateau. During this process, the residual strength of

cellular concrete tended to be constant while the strain increased. Moreover, an increase in pore size reduces the peak stress and peak strain of cellular concrete. These phenomena were attributed to t an earlier occurrence of the softening phenomenon as the pore size of a particular mixture increases. Moreover, for the descending branch of the stress-strain curve, the strain continues to increase, resulting in a sharp decrease in the stress. However, the influences of pore size on peak stress and peak stain were almost insignificant at the pore sizes of 9.5 and 12.0 mm and at porosities less than 10%. These finding were mainly attributed to the interaction of the positive and negative factors of pore size and pore number at the same porosity. In general, the peak stress and peak strain of cellular concrete decreased with an increase in the porosity or pore size. The compressive strengths of the specimens with different porosities and pore sizes are shown in Fig. 8(d). The compressive strength of cellular concrete decreased with increasing porosity or pore size mainly because a larger pore size (or higher the porosity) increases the ease at which penetrating cracks form in the concrete matrix, resulting in damage. At pore sizes greater than 9.5 mm, the effect of pore size on the compressive strength of specimens with a porosity less than 10% becomes increasingly weaker, which was mainly attributed to the offset of the effect of pore size and pore number. Therefore, the effects of porosity and pore size on the compressive behaviors of cellular concrete should be considered when designing the mixture. 3.3.2. Statistical analysis of the characteristic indexes of compressive behaviors The peak strain (strain corresponding to the peak stress), the peak stress, the ultimate strain and the elastic modulus are generally considered the characteristic indices that affect the stress-strain curve of a material. The strain corresponding to a compressive stress value of approximately 85% of the peak stress

Fig. 8. Compressive behaviors of specimens with different porosities and pore sizes of SAP: (a) specimens with different porosities and a pore size of 4.5 mm, (b) specimens with different porosities and a pore size of 7.5 mm, (c) specimens with different pore sizes and a porosity of 10%, (d) variability in compressive strength.

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was defined as the ultimate strain [33]. In this investigation, the mean values of the peak strain, the peak stress and the ultimate peak strain under uniaxial compression tests are summarized in Table 4. The peak stresses of the specimens were substantially affected by the porosity or the pore size of the SAP, as shown in Table 4. The greatest decrease in compressive strength compared to the specimen without SAP was observed at a porosity of 30% and a pore size of 12 mm. The ultimate strain tended to decrease gradually as the porosity or pore size increased. The ranges of peak strain and ultimate strain of the specimens with different porosities or pore sizes were 0.00085–0.00247 and 0.00101–0.00267, respectively. Moreover, as shown in Table 4, the coefficient of variation (COV) for the compressive strength of cellular concrete was generally less than 10%, indicating insignificant variability in compressive strength among the three specimens of the same mixture. The low COV suggests that the mean values are appropriate for inclusion in subsequent statistical analyses. Based on the statistical analyses, the relationships between the characteristic indices and the porosity/pore size of the cellular concrete were calculated using Eqs. (4) and (5), and the results are shown in Fig. 9.

ec ¼ c0  expðd0  PÞ

ð4Þ

ecf ¼ c00  expðd00  PÞ

ð5Þ 0

00

where P represents the porosity and the constants c0 , c00 , d and d represent the fitting parameters. 0 00 The fitting parameters c0 , c00 , d and d were markedly altered by the pore size D, and these parameters were calculated from D in Eqs. (6)–(9). The correlation coefficients for Eqs. (6)–(9) are 0.9874, 0.9097, 0.9945 and 0.9351, illustrating that the character-

istic index of compressive behavior is strongly associated with the pore size of cellular concrete.

c0 ¼ 0:0026  D2 þ 0:0269  D þ 2:3833

ð6Þ

0

d ¼ 1:6146  expð0:0543  DÞ

ð7Þ

c0 ¼ 0:0005  D2  0:0011  D þ 2:6585

ð8Þ

00

d ¼ 0:0055  D2 þ 0:0704  D þ 1:6265

ð9Þ

The linear regression model of peak stress and peak strain/ultimate strain is expressed in Eq. (10) and Eq. (11), and all the model parameters were obtained by performing a regression analysis.

ec ¼ e0  f c þ g0

ð10Þ

ecf ¼ e00  f c þ g00

ð11Þ 0

00

0

where f c represents the compressive strength, and e , e , g and g00 represents the fitting parameters of Eq. (10) and Eq. (11). 3.4. Splitting tensile behaviors of cellular concrete 3.4.1. Splitting tensile behavior of cellular concrete The load-displacement curves for the splitting tensile strength tests of the specimens with different porosities and pore sizes are shown in Fig. 10. Fig. 10(a)–(c) show similar shapes and trends for the load-displacement curves of cellular concrete, regardless of the porosity and pore size of the SAP. The load-displacement curve for the control specimen without SAP displayed a linear behavior until the peak/cracking load was achieved. Unlike normal concrete without SAP, the cellular concrete containing millimeter-

Table 4 The characteristic indices of cellular concrete. Mix ID SAP0 SAP5-1.0 SAP10-1.0 SAP15-1.0 SAP20-1.0 SAP25-1.0 SAP30-1.0 SAP5-4.5 SAP10-4.5 SAP15-4.5 SAP20-4.5 SAP25-4.5 SAP30-4.5 SAP5-7.5 SAP10-7.5 SAP15-7.5 SAP20-7.5 SAP25-7.5 SAP30-7.5 SAP5-9.5 SAP10-9.5 SAP15-9.5 SAP20-9.5 SAP25-9.5 SAP30-9.5 SAP5-12.0 SAP10-12.0 SAP15-12.0 SAP20-12.0 SAP25-12.0 SAP30-12.0

f c (MPa)

ec ð103 Þ

ecf ð103 Þ

Std. Dev (kN)

Cov (%)

49.04 46.43 37.29 34.46 28.98 26.60 24.44 44.00 33.48 29.16 23.39 20.02 16.85 42.27 30.82 26.24 20.71 16.61 13.16 39.59 28.96 20.98 14.90 10.80 7.60 38.70 27.63 20.40 14.33 10.42 7.30

2.47 2.23 1.96 1.83 1.73 1.63 1.52 2.07 1.82 1.74 1.66 1.50 1.45 2.18 1.76 1.65 1.49 1.29 1.10 1.85 1.61 1.48 1.32 1.20 1.01 1.82 1.58 1.33 1.21 1.06 0.85

2.67 2.43 2.22 2.08 1.86 1.71 1.61 2.36 2.00 1.86 1.79 1.65 1.53 2.29 1.90 1.75 1.58 1.37 1.16 2.06 1.77 1.60 1.45 1.33 1.08 2.04 1.67 1.51 1.30 1.20 1.01

0.10 3.11 1.80 0.91 0.62 1.16 0.78 1.23 0.79 0.81 2.76 1.10 1.61 1.06 3.17 0.57 0.35 0.78 0.93 0.80 1.12 0.55 0.59 1.20 2.90 0.44 2.28 0.95 1.14 0.85 0.90

0.20 6.70 4.83 2.64 2.14 4.36 3.19 0.20 2.8 2.36 2.78 11.80 5.49 2.51 10.29 2.17 1.69 4.70 7.07 1.97 3.87 2.33 3.28 8.84 28.54 1.143 9.67 5.18 9.62 10.66 17.94

*Note: f c represents the mean peak stress (compressive strength), ec represents the mean peak strain, ecf represents the mean ultimate strain, Std. Dev represents the standard deviation in the compressive strength of different specimens of each mixture, Cov represents the coefficient of variation for the compressive strength of different specimens of each mixture.

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S. Zhang et al. / Construction and Building Materials 235 (2020) 117508

Fig. 9. Characteristic indices of cellular concrete with different porosities and pore sizes. (a) Plot of peak strain versus pore characteristics, (b) plot of ultimate strain versus pore characteristics, (c) plot of peak strain versus peak stress, and (d) plot of ultimate strain versus peak stress.

Fig. 10. Splitting tensile behavior of specimens with different porosities and pore sizes: (a) specimens with different porosities and a pore size of 4.5 mm, (b) specimens with different porosities and a pore size of 7.5 mm, (c) specimens with different pore sizes and a porosity of 10%, and (d) variability in the splitting tensile strength.

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S. Zhang et al. / Construction and Building Materials 235 (2020) 117508

sized Sat-SAP exhibited a compaction stage with a low increase in the rate at the initial stage of the load-displacement curves. After the initial stage, displacement increased linearly with the load, similar to the specimen without SAP. The initial compaction stage of the stress-strain curve and load-displacement curve of the millimeter-sized cellular concrete have a certain degree of dependence on the pore size and porosity. Thereafter, the loaddisplacement curves of the specimens exhibited softening behavior as the specimens collapsed, indicating that the specimens displayed little resistance to the applied displacement. After achieving the peak load, the load exhibited an initial linear decrease, followed by a stepwise descent. The load of individual curve tended to be stable after it decreased to a certain value, which was mainly caused by the anti-down block bars on both sides of the tensile test device. As shown in Fig. 10(d), the splitting tensile strength of cellular concrete decreased as the porosity or pore size increased. This decrease mainly occurred because an increase in pore size/porosity increased the interfacial defect of the sample, resulting in the formation of an increasing number of macroscopic cracks and likely interface failure of the specimen. At a specific porosity, the splitting tensile strength of cellular concrete greatly decreased substantially as the pore size increased from 4.5 to 7.5 mm and from 1 to 4.5 mm. As the porosity increased from 0% to 30% for all mixtures, the splitting tensile strength of the specimens ranged from 5.02 to 0.43. As shown in Fig. 10(c), the effects of pore size on the peak load and peak displacement of the specimen were negligible when the pore size was greater than 9.5 mm. Thus, the splitting tensile behavior of cellular concrete was affected by the porosity and pore size.

3.4.2. Statistical analysis of the characteristic indexes of splitting tensile behavior Table 5 shows the results of the splitting tensile strength tests of cellular concrete with different porosities and pore sizes. The

peak load, splitting tensile strength and peak displacement observed in the splitting tensile strength tests of cellular concrete decreased with increasing porosity and pore size. The greatest decreases in the peak load, splitting tensile strength and peak displacement from the values for the specimen without SAP were observed for the specimens with a porosity of 30% or a pore size of 12 mm. Moreover, at a porosity of less than 10%, the peak load and peak displacement of the specimen with a pore size of 9.5 were approximately equivalent to the values of the specimen with a pore size of 12.0 mm, as listed in Table 5. This phenomenon was also caused by the interaction of the positive and negative factors of pore size and pore number. The regression models calculated using Eqs. (12) and (13) that describe the relationships between peak displacement and peak stress and between peak load and porosity were fitted based on the test results presented in Table 5, and the data are shown in Fig. 11. 0

0

sst ¼ h  Fkst 00

ð12Þ 00

sst ¼ h  expðk  PÞ

ð13Þ

where F st represents the mean peak stress, P represents the poros0 00 0 00 ity, and the constants h , h , k and k represent the model parameters. As shown in Fig. 11, the parameters in the model of the relationships between peak load and porosity were clearly affected by the particle size D. Therefore, the following equations were developed 00 00 to calculate the parameters h and k . The correlation coefficients for Eqs. (14) and (15) were 0.9867 and 0.9439, respectively. 00

ð14Þ

00

ð15Þ

h ¼ 0:0002  D þ 0:1111 k ¼ 0:9212  expð0:1204  DÞ

Table 5 Summary of the results of splitting-tensile strength tests of cellular concrete. Mix ID

F st (kN)

Sst (mm)

f st (MPa)

Std. Dev (MPa)

SAP0 SAP5-1.0 SAP10-1.0 SAP15-1.0 SAP20-1.0 SAP25-1.0 SAP30-1.0 SAP5-4.5 SAP10-4.5 SAP15-4.5 SAP20-4.5 SAP25-4.5 SAP30-4.5 SAP5-7.5 SAP10-7.5 SAP15-7.5 SAP20-7.5 SAP25-7.5 SAP30-7.5 SAP5-9.5 SAP10-9.5 SAP15-9.5 SAP20-9.5 SAP25-9.5 SAP30-9.5 SAP5-12.0 SAP10-12.0 SAP15-12.0 SAP20-12.0 SAP25-12.0 SAP30-12.0

177.32 151.71 134.61 121.26 110.87 92.85 80.44 125.87 103.49 85.56 75.234 54.05 49.59 114.62 94.22 70.74 58.68 40.43 39.26 100.45 81.00 53.54 42.15 26.86 26.04 98.90 76.95 36.07 28.93 15.02 13.65

0.1090 0.1034 0.1005 0.0963 0.0897 0.0805 0.0763 0.1028 0.0980 0.0931 0.0824 0.0765 0.0734 0.1000 0.0881 0.0826 0.0730 0.0686 0.0567 0.0865 0.0796 0.0650 0.0543 0.0473 0.0405 0.0856 0.0787 0.0602 0.0503 0.0400 0.0310

5.02 4.30 3.81 3.43 3.14 2.63 2.28 3.57 2.93 2.42 2.13 1.53 1.40 3.25 2.67 2.00 1.66 1.14 1.11 2.84 2.29 1.52 1.19 0.76 0.74 2.80 2.18 1.02 0.82 0.43 0.39

0.05 0.27 0.28 0.28 0.33 0.16 0.11 0.33 0.14 0.11 0.07 0.10 0.20 0.55 0.30 0.16 0.24 0.08 0.15 0.43 0.10 0.04 0.38 0.20 0.14 0.11 0.24 0.11 0.05 0.10 0.03

*Note: F st represents the mean peak load, Sst represents the mean displacement, f st represents the mean peak stress (splitting tensile strength), Std. Dev represents the standard deviation of the splitting tensile strength of different specimens of each mixture.

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S. Zhang et al. / Construction and Building Materials 235 (2020) 117508

Fig. 11. Relationships with the splitting-tensile behaviors of cellular concrete: (a) peak displacement and peak load, and (b) peak load and porosity.

3.5. Analysis of the fracture characteristics of cellular concrete

4. Discussion

Research on the fracture characteristics of concrete has resulted in a better understanding of the law of propagation of cracks during the process of concrete material failure and methods to control and prevent the fracture and failure of concrete structures. Some scholars have generalized the fracture characteristics of concrete into three types [34,35], as illustrated in Fig. 12: 1) binder fracture between aggregates, 2) interface fracture at the interface between the aggregate and binder, and 3) aggregate fracture. The fracture characteristics of cellular concrete with millimeter-sized pores were studied by photographing and observing the fractured cross-sections. Images of the partial fracture zones of the fractured cross-sections of cellular concrete are presented in Fig. 13. The binder fracture is not addressed in this paper because it occurs in any fracture form. The fracture of the specimen without SAP, namely, normal concrete, was mainly caused by the fracturing of the binder and the fracturing of the interface between the binder and the aggregate (Fig. 13(a)). As shown in Fig. 13(b) and Fig. 13(c), fracturing in the weak areas where the pores (SAP) are located increased with the introduction of SAPs, and this fracture characteristic was attributed to an aggregate fracture path through the SAPs. The numbers of aggregate fractures (containing SAP and aggregate) of specimens SAP20-7.5 and SAP20-9.5 were much higher than specimen SAP0. This result was mainly attributed to the lack of capacity of the SAP itself to bear a load and the lack of cohesive force between the binder and SAP, which facilitates fracture.

4.1. Development of the stress-strain model Fig. 14(a) shows the stress-strain curves of normal concrete and cellular concrete. The normalized stress-strain curves of cellular concrete with different porosities and pore sizes are shown in Fig. 14(b)–(d). Notably, the stress-strain curve of cellular concrete is significantly different from normal concrete. A calculation the elastic modulus of cellular concrete using the method for calculation the elastic modulus of ordinary concrete is meaningless [27], mainly because the ‘‘initial compaction stage” of cellular concrete is not the real elastic stage, and the maximum stress in the ‘‘initial compaction stage” is usually greater than the initial stress value of 0.5 MPa used to calculate the elastic modulus of ordinary concrete. Therefore, a new method using the stress and strain values at the end point of the initial compaction stage of the cellular concrete as the initial value (namely, the value of point o00 in Fig. 14(a)) for calculating the elastic modulus of cellular concrete is proposed in this paper. Using this method, the stress-strain model of Xiao [33] is modified by the authors of this paper according to characteristics of the stress-strain curve of cellular concrete and the principle of image translation in mathematics (m is the translation distance in Fig. 14(a)). The modified model not only better describe the stress-strain characteristics of cellular concrete, but also obtain the parameter v required to calculate the elastic modulus of cellular concrete. The modified model for describing the stress-strain curve of cellular concrete is listed in Eq. (16) as follows:

Fig. 12. Illustrations of concrete fracture characteristics: (a) binder fracture, (b) interface fracture, (c) pore/aggregate fracture [34].

S. Zhang et al. / Construction and Building Materials 235 (2020) 117508

13

Fig. 13. Fracture characteristics of cellular concrete: (a) SAP0, (b) SAP20-7.5, (C) SAP20-9.5. Note: ‘‘h”represents an interface fracture and ‘‘s”represents pore (SAP) fracture.

Fig. 14. Comparisons of the modified model and the experimental results of cellular concrete in the analyzed groups.

f ¼ fc

(

g  ðg  vÞ þ ð3  2  gÞ  ðg  vÞ þ ðg  2Þ  ðg  vÞ ; 2

g

3

0g<1

; jðg1Þ2 þg ð16Þ

where f represents the compressive stress and f c is the mean peak stress, namely axial compressive strength. g ¼ ec =ecf , where ec is the strain in the compression and ecf is the peak strain. Additionally, gand j represent the model parameters, and v is a constant that can be calculated as v ¼ m=ec . The stress-strain model of Eq. (16) was applied to the test results of the analyzed groups to verify the applicability of the modified stress-strain model for porous concrete with different porosities and pore sizes under uniaxial compression. The theoretical stress-strain curves of the specimens in the analyzed groups

were compared with the experimental results as illustrated in Fig. 14. Furthermore, the calculated results for the analyzed groups are listed in Table 6. As shown in Table 6, the correlation coefficients of the ascending branches (0  g < 1) are all greater than 0.9946, illustrating that Eq. (16) is suitable for calculating the compressive behavior of cellular concrete with different porosities and pore sizes under uniaxial compression. For the descending branches, the correlation coefficients are greater than 0.8167 for all of the specimens, except SAP10-4.5 and SAP20-4.5, indicating that Eq. (16) can be used to describe the compressive behavior of cellular concrete. In addition, the elastic modulus calculated using the traditional method is approximately the same as the value obtained with the calculation method developed in this paper at low porosity. However, the difference between the two elastic

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S. Zhang et al. / Construction and Building Materials 235 (2020) 117508

Table 6 The calculated results for the analyzed groups. MIX ID

Ascending branch

SAP0 SAP5-4.5 SAP10-4.5 SAP15-4.5 SAP20-4.5 SAP25-4.5 SAP30-4.5 SAP5-7.5 SAP10-7.5 SAP15-7.5 SAP20-7.5 SAP25-7.5 SAP30-7.5 SAP10-1 SAP10-9.5 SAP10-12.0

Descending branch 2

g

v

R

0.57400 0.08124 0.33860 0.31680 0.12640 0.30130 0.10360 0.00309 0.20840 0.21350 0.15550 0.25950 0.16040 0.17940 0.23550 0.16690

0.00000 0.00060 0.05685 0.05178 0.00711 0.05984 0.03005 0.00671 0.05005 0.06656 0.03256 0.04848 0.07541 0.00055 0.08551 0.06029

0.9998 0.9996 0.9982 0.9980 0.9994 0.9982 0.9992 0.9996 0.9969 0.9965 0.9976 0.9963 0.9946 0.9997 0.9957 0.9994

m

Elastic modulus (GPa)

2

j

R

52.04 50.47 39.40 60.14 72.29 19.27 22.14 104.6 65.54 65.72 74.57 76.26 238.4 22.84 24.56 54.24

0.9453 0.9560 0.7679 0.8898 0.7235 0.9249 0.8621 0.9308 0.8223 0.8885 0.8066 0.9574 0.8669 0.9410 0.9742 0.8167

Traditional method

Method developed in this paper

21.35 18.86 16.90 15.09 13.00 12.23 10.24 15.63 14.92 13.83 12.05 11.18 9.69 18.83 15.53 15.16

21.35 18.86 17.20 15.41 13.01 12.93 10.46 15.80 15.29 14.25 12.16 11.28 9.88 18.84 16.35 16.41

0.000000 0.000001 0.000103 0.000090 0.000010 0.000090 0.000040 0.000010 0.000090 0.000110 0.000049 0.000063 0.000083 0.000001 0.000139 0.000100

Note: R2 is the correlation coefficient.

modulus calculation methods becomes greater as the porosity and pore size increase, and the maximum difference is 1.25 GPa. Therefore, the method used in this paper is recommended for calculating the elastic modulus of cellular concrete. 4.2. Relationships between porosity/pore size and the mechanical properties of cellular concrete The compressive strength and splitting tensile strength are important design parameters in civil engineering, and the values of these strength parameters generally depend on the internal structural characteristics of the concrete materials. As reported, the pore characteristics (porosity, pore size and pore distribution) affect the mechanical properties of concrete, such as compressive strength, splitting tensile strength and elastic modulus [18,28,36]. Several empirical models have been proposed to describe the relationship between the strength and porosity, to obtain a better understanding of the effects of pore characteristics on the compressive strength and splitting tensile strength and to improve predictions [37,38,39]. However, few studies have been conducted on the applicability of these statistical models to materials with different pore sizes, particularly millimeter-sized pores. In this paper, a few statistical models are used to verify the predicted effects of pore characteristics on the compressive strength and splitting tensile strength of cellular concrete with pore sizes of 1, 4.5, 7.5, 9.5 and 12.0 mm, as listed in Table 7. The two main limitations of the empirical model proposed by Roy [18] are that:

1) the maximum porosity must be obtained by testing, and 2) the strength does not depend on the sphere size when it approaches zero. Based on the results of the present study, the pore size exerts a pronounced effect on the compressive strength of cellular concrete. Therefore, the empirical model proposed by Roy is extended by the authors of the present study to describe the relationship between the mechanical properties and the porosities of cellular concrete. This extension avoids the limitations of the original model, while improving the simplicity and generality of the model. Fig. 15 shows the relationships between the porosity/pore size and compressive strength of cellular concrete, which have all have correlation coefficients of determination greater than 0.9591, illustrating that the models presented in Table 7 are useful for predicting the compressive strength of cellular concrete with different porosities. However, the predictions obtained using the models presented by Hasselman [42] and Chen [38] for the porosities of 0% and 30% display a poorer fit of the data than the other models, as shown in Fig. 15(d) and Fig. 15(e). Therefore, the integral absolute error (IAE) is used to further evaluate the gap between the test results and predictions. The IAE values for the different prediction models are shown in Table 7. The IAE values of the predictions for materials with different porosities and pore sizes are all less than 10%, except for values obtained using the prediction models established by Hasselman and Chen to predict the compressive strength of cellular concrete with pore sizes of 9.5 and 12 mm. Thus, the prediction models proposed by Balshin [40], Ryshkevitch [41]

Table 7 Relationship between the porosity and compressive strength of concrete materials. Derivations

Proposed equation

Fitting SAP-1.0

SAP-4.5

SAP-7.5

SAP-9.5

SAP-12.0

Balshin [40]

r ¼ r0  ð1  PÞa

fc f cf fc f cf fc f cf fc f cf

2.99 1.74 2.70 1.67 3.94 2.54 3.04 1.18

3.09 6.70 2.65 5.37 6.03 10.49 3.78 7.97

2.72 8.15 2.61 6.32 7.50 15.64 4.73 13.24

1.94 13.85 2.45 12.51 11.56 18.63 5.37 16.01

4.91 11.77 6.30 9.97 12.85 29.30 10.49 27.01

fc f cf

3.23 1.51

2.79 4.49

2.60 4.69

2.15 8.06

3.81 4.64

Ryshkevitch [41] Hasselman [42] Chen [38]

Our model

arepresents the model parameter r ¼ r0  expðbPÞ for ceramics and rocks (a represents the model parameter) r ¼ r0  cP for different refractory materials (c represents the model parameter) h i1=2 r ¼ r0 ðPcPP Þ  ð1  P2=3 Þ c for cement mortar (Pc represents the model parameter)  

r ¼ r0 ðr  ð1  uPÞÞ=ðr þ uPÞ (r, urepresent the model parameters related to pore size)

Note: IAE ¼

IAE

P 0:5 P ðððAi  Bi Þ2 Þ = Ai  100Þ, Ai represents the experimental result, Bi represents the model predicted value, and P represents the porosity.

S. Zhang et al. / Construction and Building Materials 235 (2020) 117508

15

Fig. 15. Relationships between the porosity/pore size and compressive strength of cellular concrete: (a) SAP-1, (b) SAP-4.5, (c) SAP-7.5, (d) SAP-9.5, and (e) SAP-12.0.

and the extended models developed in the present study are more suitable than the models established by Hasselman and Chen for predicting the compressive strength of cellular concrete with different porosities and pore sizes. The relationships between the porosity/pore size and splitting tensile strength of cellular concrete are shown in Fig. 16. The correlation coefficients of determination are all greater than 0.8396, and the deviation between the predictions (obtained from prediction models established by Hasselman and Chen) and the test results increases with increasing pore size. Furthermore, the correlation coefficient of the models proposed by Hasselman and Chen also decreases as the pore size increases from 1.0 mm to 12.0 mm. The IAE values of the splitting tensile strength are used to evaluate the predictive effects of the proposed models. Ten IAE values in Table 7 are greater than 10%, and the largest value is 29.30%. Therefore, the prediction models recommended to obtain the splitting tensile strength of cellular concrete are the models proposed and developed by Ryshkevitch and the authors of the present study. In addition, the maximum value of the IAE of the

prediction model developed in this paper is only 8.06 for all mixtures, indicating a high fitting accuracy to the experimental results of cellular concrete with all mixtures. Based on the correlation coefficient and IAE results described above, the prediction model developed in this paper is useful for predicting not only the compressive strength of concrete, but also the splitting strength of concrete, as it has a high prediction accuracy for cellular concrete with different porosities and pore sizes. Therefore, the model developed in this paper is recommended for predicting the compressive strength and splitting tensile strength of cellular concrete with different porosities and pore sizes. Furthermore, the model parameters of compressive strength (r,u) and splitting tensile strength (r0 , u0 ) can be expressed with D as follows in Eqs. (17)–(20). The correlation coefficient for Eqs. (17)–(20) are 0.9474, 0.9979, 0.8761 and 0.9667, illustrating that the mechanical properties of cellular concrete are closely related to pore size.

r ¼ 1:261  D0:329

ð17Þ

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S. Zhang et al. / Construction and Building Materials 235 (2020) 117508

Fig. 16. Relationships between the porosity/pore size and splitting tensile strength of cellular concrete: (a) SAP-1, (b) SAP-4.5, (c) SAP-7.5, (d) SAP-9.5, and (e) SAP-12.0.

u ¼ 0:0285  D þ 0:7681

ð18Þ

r0 ¼ 1:3689  D0:836

ð19Þ

u0 ¼ 0:0317  D þ 0:7959

ð20Þ

4.3. Relationship between the compressive and splitting tensile strength of cellular concrete Compared with compressive strength tests, flexural and splitting tensile tests have the advantages of being simple and quickly attainable at low cost. To date, most reports have been limited to the compressive strength of cellular concrete [2,43,44], although some have focused on the splitting tensile strength. The relationship between the compressive strength and splitting tensile strength of cellular concrete with different porosities and pore sizes are discussed in this study. Most of the empirical relationships have been proposed for conventional and porous concrete

[45,46], but it is unknown whether the formulas can describe the relationship between the compressive strength and splitting tensile strength of cellular concrete with different porosities and pore sizes, especially that with different pore sizes. The few formulas that express the relationship between compressive strength and splitting tensile strength are listed in Table 8, and the predictive effect is assessed by the correlation coefficient and IAE values. The empirical relationships between the compressive strength and splitting tensile strength of cellular concrete with different porosities are shown in Fig. 17. The models of ACI 318, ACI 363, Crouch et al. and Gaedicke et al. do not accurately predict the splitting tensile strength of cellular concrete with different porosities and pore sizes. The predictive ability of these models is reflected by the IAE values, as shown in Table 8. The IAE values of the models of ACI 318, ACI 363, Crouch et al and Gaedicke et al are generally greater than 20% and the maximum IAE values are 62.72%, illustrating that the prediction results of these models do not meet the criteria of prediction accuracy. Using the results of the present paper, the relationships between the compressive strength and

17

S. Zhang et al. / Construction and Building Materials 235 (2020) 117508 Table 8 Empirical formulas expressing the relationship between compressive strength and splitting tensile strength. Derivations

Proposed equation SAP-1.0

SAP-4.5

SAP-7.5

SAP-9.5

0:5

12.15

24.26

34.10

50.69

58.48

0:5

10.13

28.09

38.09

55.02

62.72

0:55

43.86

33.10

30.95

33.26

41.28

0:875

16.09

33.34

39.13

51.92

54.52

5.38 4.23

7.43 7.19

7.97 9.21

8.84 12.13

8.75 15.86

ACI 318 [45]

f st ¼ 0:56  f c

ACI 363 [47]

f st ¼ 0:59  f c

Crouch et al. [46] Gaedicke et al. [48] Model in this paper

IAE

f st ¼ 0:28  f c

f st ¼ 0:181  f c f st ¼ n  expðq  f c Þ k

f st ¼ h  f c

SAP-12.0

Fig. 17. Relationships between the compressive strength and splitting tensile strength of cellular concrete with different porosities and pore sizes: (a) SAP-1, (b) SAP-4.5, (c) SAP-7.5, (d) SAP-9.5, and (e) SAP-12.0.

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S. Zhang et al. / Construction and Building Materials 235 (2020) 117508

splitting strength of cellular concrete with different porosities and pore sizes are fitted with various regression models. The fitting results indicate that good relationships between the compressive strength and splitting tensile strength of cellular concrete with different porosities and pore sizes are obtained from the exponential and power models. The exponential and power models, which are fitted for cellular concrete with different porosities and pore sizes by the authors of the present paper, are expressed in Eqs. (21) and (22), respectively. The correlation coefficients are all greater than 0.9449, indicating that the two models possess a good predictive ability. Moreover, the predicted effects of cellular concrete with different porosities and pore sizes are evaluated based on the IAE values. As shown in Table 8, the IAE values obtained from the exponential model for cellular concrete with different pore characteristics are generally smaller than the power model. This observation indicates the superior accuracy of the prediction results obtained from the exponential model to the power model. Therefore, the exponential model is recommended for predicting the splitting tensile strength of cellular concrete with different porosities and pore sizes. Notably, the two models fitted by the authors of this paper display high prediction accuracy for cellular concrete with pore sizes less than 7.5 mm.

f st ¼ n  expðq  f c Þ

ð21Þ

k

ð22Þ

f st ¼ h  f c

where f st represents the splitting tensile strength, f c represents the compressive strength, and the constants n, q, h, and k represent the model parameters related to particle size. The model parameters (n,q) of the exponential model are expressed with D using Eqs. (23) and (24). The correlation coefficients for Eqs. (23) and (24) are 0.9859 and 0.977, respectively, indicating that the relationship between mechanical properties is closely associated with the pore size.

n ¼ 1:4718  expð0:112  DÞ

ð23Þ

q ¼ 0:0257  expð0:0608  DÞ

ð24Þ

5. Conclusions This paper investigates the effects of porosity and pore size on the segregation characteristics and compressive and splitting strengths of cellular concrete utilizing millimeter-sized spherical Sat-SAPs. The effects of the volume fraction and diameter of the Sat-SAP on the segregation characteristics of cellular concrete are analyzed, and the effects of different porosities and pore sizes on the mechanical properties of cellular concrete are discussed. The main conclusions are listed below. 1) The volume fraction and diameter of Sat-SAP exert significant effects on the segregation characteristic of millimetersized cellular concrete. The segregation is particularly high at porosities greater than 15%, and the lowest segregation is observed for pore sizes of 4.5–7.5 mm. Moreover, the segregation characteristic of millimeter-sized cellular concrete is reduced by controlling the volume fraction and diameter of Sat-SAP in engineering applications. 2) Porosity and pore size exert significant effects on the compressive and splitting tensile behaviors of millimeter-sized cellular concrete. The initial compaction stage of the stress-strain curve and load-displacement curve of cellular concrete have a certain degree of dependence on the pore size and porosity, which is explained by the collapse and compaction of surface pores or defects. Furthermore, the

peak stress, compressive strength, peak strain, ultimate strain, peak load, splitting tensile strength and peak displacement of the cellular concrete all decreased continuously with increasing porosity and pore size, but the mechanical behaviors of cellular concrete are not substantially affected by pore size when the pore size is greater than 9.5 mm and the porosity is less than 10%. Notably, the weak effect was mainly attributed to the interaction of the positive and negative factors of pore size and pore number at the same porosity. 3) A modified stress-strain model is suggested by the authors of this paper based on the characteristics of the stressstrain curve of cellular concrete and the principle of image translation in mathematics. The predictions obtained using the modified stress-strain model display good agreement with the test results of cellular concrete with different porosities and pore sizes. In addition, a new method is proposed in which the stress and strain values at the end point of the initial compaction stage of the cellular concrete are used as the initial values to calculate the elastic modulus of millimeter-sized cellular concrete because the initial compaction stage is not the real elastic stage. The difference between the traditional method and new method becomes greater as the porosity and pore size increase, and the maximum difference is 1.25 GPa. 4) Porosity and pore size exert important effects on the compressive strength and splitting-tensile strength of cellular concrete. The theoretical results obtained from the existing models do not meet the accuracy requirements for the prediction because they ignore the effect of the pore size. The modified model proposed by the authors of the present paper provides good predictions of the compressive and splitting tensile strengths of cellular concrete with different porosities and pore sizes. 5) Based on the results of the experiments described in the present study, a new model for predicting the splitting tensile strength of cellular concrete with different porosities and pore sizes is proposed, and its IAE value is obviously lower than the previous prediction model, indicating its higher prediction accuracy and superiority.

Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This research was supported by the Natural Science Foundation of China (Nos. 51779168 & 51979188), the Foundation for Innovative Research Groups of the National Natural Science Foundation of China (No. 51621092). The corresponding author acknowledges the support from Open Foundation (No. 2017SGG02) from State Key Laboratory of Water Resources and Hydropower Engineering Science, Wuhan University. References [1] P.J. Tikalsky, J. Pospisil, W. MacDonald, A method for assessment of the freezethaw resistance of preformed foam cellular concrete, Cem. Concr. Res. 34 (2004) 889–893, https://doi.org/10.1016/j.cemconres.2003.11.005. [2] D.K. Panesar, Cellular concrete properties and the effect of synthetic and protein foaming agents, Constr. Build. Mater. 44 (2013) 575–584, https://doi. org/10.1016/j.conbuildmat.2013.03.024.

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