A numerical method for predicting Young’s modulus of heated cement paste

Construction and Building Materials 54 (2014) 197–201

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

A numerical method for predicting Young’s modulus of heated cement paste Jie Zhao a,⇑, Jian-Jun Zheng b, Gai-Fei Peng a a b

School of Civil Engineering, Beijing Jiaotong University, Beijing 100044, PR China School of Civil Engineering and Architecture, Zhejiang University of Technology, Hangzhou 310014, PR China

h i g h l i g h t s  A numerical method is developed for predicting the Young’s modulus of heated cement paste.  Thermal decomposition analysis of cement paste is conducted.  The effects of thermal decomposition and microcracking on the Young’s modulus of heated cement paste are considered.

a r t i c l e

i n f o

Article history: Received 2 August 2013 Received in revised form 10 December 2013 Accepted 17 December 2013 Available online 14 January 2014 Keywords: Cement paste Young’s modulus High temperature Thermal decomposition Microcracking

a b s t r a c t The evaluation of the mechanical properties of heated cement paste is essential to the safety assessment of concrete structures exposed to elevated temperatures. A numerical method is developed in this paper for predicting the Young’s modulus of heated cement paste with or without silica fume up to 600 °C. The initial volume fractions of various constituents and the thermal decomposition of hydration products in cement paste are approximately estimated. A two-phase composite sphere model is then built and a twostep approach is applied to evaluating the Young’s modulus of heated cement paste with thermal decomposition and microcracking effects. Finally, the validity of the proposed numerical method is verified with three sets of experimental data collected from the literature. Ó 2013 Elsevier Ltd. All rights reserved.

1. Introduction Young’s modulus of heated cement paste is essential to the stiffness evaluation of concrete structures under high temperature conditions and therefore has been studied experimentally and theoretically. Dias et al. [1] showed that the mechanical properties of heated cement paste are characterized by the highest temperature ever experienced by the material. At the temperatures of 300 and 600 °C, two remarkable drops in Young’s modulus were observed. Using the non-destructive ultrasonic technique, Masse et al. [2] found a decreasing trend of the Young’s modulus of cement pastes at different curing ages with the heating temperature. Odelson et al. [3] and Kerr [4] observed that the stiffness loss of heated cement paste occurs predominantly below 120 °C and concluded that the loss is mainly attributed to microcraking. Padeveˇt and Zobal [5] found a drastic decrease in Young’s modulus of cement paste at a

⇑ Corresponding author. Tel.: +86 10 51684866. E-mail address: [email protected] (J. Zhao). 0950-0618/$ - see front matter Ó 2013 Elsevier Ltd. All rights reserved. http://dx.doi.org/10.1016/j.conbuildmat.2013.12.070

heating temperature of 450 °C. Compared with experimental investigations, theoretical analyses are relatively few. Ulm et al. [6] analyzed the Channel Tunnel fire using a linear relationship between the dehydration degree and Young’s modulus of concrete. In Lee et al.’s model [7], the decomposition degree of various constituents in cement paste was taken as a linear function of temperature and the Young’s modulus of heated cement paste was estimated by applying the theory of composite damage mechanics. However, microcracking of cement paste is not considered explicitly. All of the aforementioned research clearly shows that it is still highly desirable that a numerical method can be available for evaluating the Young’s modulus of heated cement paste with thermal decomposition and microcracking effects. The purpose of this paper is to predict the Young’s modulus of heated cement paste with or without silica fume up to 600 °C. After the initial volume fractions of various constituents in cement paste are formulated in an approximate manner, the thermal decomposition of cement paste is analyzed. By applying a two-phase composite sphere model and a two-step procedure, the effects of thermal decomposition and microcracking on the Young’s modulus of heated cement paste are modeled. Finally, the validity of the proposed method is verified with three sets of experimental data.

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2. Initial volume fractions of various constituents in cement paste To predict the thermal decomposition, the initial volume fractions of various constituents in cement paste have to be estimated. Usually, they can be expressed as a function of the water to binder ratio, the degree of hydration, and the chemical composition of cement. When silica fume is added, the pozzolanic reaction needs to be taken into account. It has been shown that, at complete hydration, the volume of cement gel generated from 1 g of cement is 0.682 cm3 [8]. By taking the specific gravity of Portland cement as 3.15, the volume fractions of unhydrated cement and cement gel in hardened cement paste are given by

funhyC

0:32ð1  ac Þ ¼ w=c þ 0:32

ð1Þ

fgel ¼

0:68ac w=c þ 0:32

ð2Þ

where ac is the degree of hydration of cement and w/c is the water to cement ratio. If a mass fraction S of cement is replaced with silica fume and its degree of hydration is taken as zero, funhyC and fgel are modified as

funhyC ¼

fgel ¼

0:32ð1  ac Þð1  SÞ w=c þ 0:32ð1  SÞ þ S=qSF

0:68ac ð1  SÞ w=c þ 0:32ð1  SÞ þ S=qSF

ð3Þ

ð4Þ

where qSF is the density of silica fume and w/c is converted into the water to binder ratio. According to Bentz and Garboczi [9], each volume unit of C3S produces 1.7 volume units of C–S–H and 0.61 volume units of CH; each volume unit of C2S produces 2.39 volume units of C–S– H and 0.191 volume units of CH; and each volume unit of C3A produces 1.69 volume units of hydrated aluminates. It is assumed in this paper that each unit volume of C4AF produces the same amount of hydration products as C3A. With these parameters, the CH AL volume fractions of CH (fgel ), hydrated aluminates (fgel ), and C–S– CSH H (fgel ) in hydration products can be evaluated. For silica fume, which consists primarily of amorphous silicon dioxide, a k-value of 1.0 is usually applied. The pozzolanic reaction of silica fume can be expressed as [10]

1:1CH þ S þ 2:8H ! C1:1 SH3:9 1:35

1

1:87

ð5Þ

3:77

where the number below each reactant represents the volume stoichiometry. Thus, the initial volume fractions of pozzolanic C–S–H 0 0 0 (fpCSH ), CH (fCH ), conventional C–S–H (fCSH ), and hydrated aluminates 0 (fAL ) can be respectively estimated for a given degree of hydration of silica fume as as follows 0 fpCSH

3:77Sas =qSF ¼ w=c þ 0:32ð1  SÞ þ S=qSF

0 CH fCH ¼ fgel  fgel 

1:35Sas =qSF w=c þ 0:32ð1  SÞ þ S=qSF

ð6Þ

ð7Þ

0 CSH fCSH ¼ fgel  fgel

ð8Þ

0 AL fAL ¼ fgel  fgel

ð9Þ

It should be mentioned that Eqs. (1)–(9) are only valid for traditional cement paste or silica fume blended cement paste. If fly ash or other types of supplementary cementitious materials are added, a new set of expressions needs to be established.

3. Thermal decomposition analysis of cement paste With the decomposition prediction method proposed by Zhao et al. [11], the conversion degree of each constituent in cement paste can be determined. Thus, the volume fractions of decomposed constituents fid are equal to

fid ¼ fi0 ai

ð10Þ

where ai and fi0 are the conversion degree and initial volume fraction of reactant i (i = CH, AL, and C–S–H), respectively. The water decomposed from the reactants is regarded as additional pores and the volume fraction fiw is given by

fiw ¼ fid nw i

qi =Mi qw =Mw

ð11Þ

where nw i is the amount of water in mole decomposed per mole of reactant i, qw and Mw are the density and molar mass of water, and qi and Mi are the density and molar mass of reactant i, respectively. It was showed from a detailed analysis [11] that, nw CH ¼ 1:0, qCH ¼ 2:24 g=cm3 , and MCH = 74 g/mol for CH; nwAl ¼ 20, qAL = 1.8 g/cm3, and MAL = 1255 g/mol for hydrated aluminates; and 3 w nw CSH ¼ npCSH ¼ 3:0, qCSH = qpCSH = 1.75 g/cm , and MCSH = MpCSH = 365 g/mol for conventional and pozzolanic C–S–H. The initial volume fractions of the solid phase in conventional and pozzolanic C–S–H are respectively equal to gelp 0 s fCSH ¼ ð1  fCSH ÞfCSH

ð12Þ

s 0 fpCSH ¼ ð1  0:19ÞfpCSH

ð13Þ

gelp where fCSH is the volume fraction of gel pores in conventional C–S–H and given by

gelp CSH fCSH ¼ 0:28=fgel

ð14Þ

Thus, the total initial volume fraction of the solid phase in C–S– H is 0s s s fCSH ¼ fCSH þ fpCSH

ð15Þ

Substitution of Eq. (15) into Eq. (11) yields the volume fraction of water released from the decomposition of C–S–H. Division of the volume fraction of water released from each reactant by the volume fraction of the converted reactant yields the porosities of decompopp pp sition products of hydrated aluminates (fAL ), CH (fCH ), conventional pp pp C–S–H (fCSH ), and pozzolanic C–S–H (fpCSH ) pp fAL ¼ nw AL

qAL =MAL ¼ 0:52 qw =Mw

ð16aÞ

pp fCH ¼ nw CH

qCH =MCH ¼ 0:54 qw =Mw

ð16bÞ

pp fCSH ¼ nw CSH

qCSH =MCSH 0:28 þ CSH ¼ 0:58 qw =Mw fgel

pp fpCSH ¼ nw CSH

qCSH =MCSH þ 0:19 ¼ 0:40 qw =Mw

ð16cÞ

ð16dÞ

It should be mentioned that, since the volume fraction of C–S–H CSH in gel is around 0.65 for most commonly used cement, fgel is taken as 0.65 in Eq. (16c). 4. Young’s modulus of heated cement paste In predicting the Young’s modulus of heated cement paste, two main factors are considered: one is the thermal decomposition of hydration products and the other is cracks formed in the cement

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paste due to high temperatures. Their effects on the Young’s modulus of cement paste will be evaluated using a two-step procedure. Since the porosities of most decomposition products are larger than 50% as seen from Eq. (16) and the solid phase in decomposition products becomes granular as shown in Fig. 1 [12–14], the volume fraction of the solid granular, which is called the packing density in granular mechanics, is smaller than 50%. According to granular mechanics, when the packing density is smaller than the random loose-packed limit of 56%, the stiffness of the granular assembly is equal to zero [15]. Thus, the cement paste can be modeled as a two-phase composite material with spherical decomposition products of zero bulk modulus as shown in Fig. 2. If the Young’s modulus, shear modulus, bulk modulus, and Poisson’s ratio of cement paste at room temperature are denoted by E0, G0, K0, and l0, respectively, the bulk modulus of cement paste with the thermal decomposition effect, K0,t is given by [16,17]

K 0;t ¼ K 0 

K 0 fin 1  ð1  fin ÞK 0 =ðK 0 þ 4G0 =3Þ

ð17Þ

where fin is the volume fraction of decomposition products and is equal to

fin ¼

X

fid

Decomposition products

Fig. 2. Two-phase composite sphere model.

in gel pores, resulting in drying shrinkage. The thermal–mechanical loading includes thermal stresses and the build-up vapor pressure. To evaluate the Young’s modulus of heterogeneous solids with microcracks, Feng and Yu [20] developed an effective medium approach. One of the advantages of the approach is that only one parameter is needed to quantify the effect of microcracks on Young’s modulus. For heated cement paste, it is reasonable to assume that microcracks are uniformly distributed and oriented. Thus, the Young’s modulus of cement paste with thermal decomposition and microcracking effects, E0,tc, is given by [20]

ð18Þ

According to the theory of elasticity [18], G0 and K0 are related to E0 and l0 by

E0 G0 ¼ ; 2ð1 þ l0 Þ

E0 K0 ¼ 3ð1  2l0 Þ

ð19Þ

K 0;t ¼ K 0 

K 0 fin 1  ð1  fin Þð1 þ l0 Þ=½3ð1  l0 Þ

ð20Þ

Considering that the Poisson’s ratio of cement paste varies slightly during the thermal decomposition of hydration products and has a small effect on the Young’s modulus, it is assumed in this paper that the Poisson’s ratio of cement paste with thermal decomposition effect is still equal to l0. It follows from Eqs. (19) and (20) that the Young’s modulus of cement paste with the thermal decomposition effect, E0,t, is related to E0 by

E0 fin 1  ð1  fin Þð1 þ l0 Þ=½3ð1  l0 Þ

ð21Þ

It should be pointed out that the proposed model is only valid for the heating temperature lower than 600 °C. When the heating temperature is higher than 600 °C, most of C–S–H, which makes up around 70% of the volume of hydration products, decomposes [12,19]. In this case, the whole solid structure of cement paste becomes different and a more advanced model needs to be built for predicting the Young’s modulus. It has been experimentally confirmed that, besides thermal decomposition, heated cement paste undergoes microcracking due to drying shrinkage and thermal–mechanical loading [1– 5,14]. The high temperature and the decomposition of hydration products accelerate the release of the physically adsorbed water

Void Decomposed particles Weak connections

E0;tc ¼

lD ¼

ð22Þ

16ð1  x2 Þð10  3xÞ 45ð2  xÞ

ð23Þ

½10  3l0 þ gðl0 Þxl0 ð10  3l0 Þ½1 þ gðl0 Þx

ð24Þ

Thus, the Young’s modulus of heated cement paste can be estimated from Eqs. (21) and (22). 5. Experimental verification To verify the proposed numerical method, three sets of experimental data collected from the literature are used for comparison. The first one is taken from Masse et al. [2]. In their experiment, cylindrical specimens with a diameter of 15 mm and a height of 15–19 mm were made with Portland cement CEM I 52.5. After 28-day curing at room temperature, three types of specimens with water to cement ratios of 0.25, 0.35, and 0.5 were exposed to elevated temperatures. The target temperatures were 80, 120, 150, 200, 250, 300, 400, 500, 600, 700, and 800 °C and the heating rate was fixed as 0.2 °C/min. The exposure duration at each target temperature was 10 h. After the specimens were naturally cooled to room temperature, the non-destructive ultrasonic technique was used to measure the Young’s modulus. The results are shown in Fig. 3. It should be noted that, since the ultrasonic technique was used and the specimens were tested after cooling, the measured modulus is the residual dynamic modulus, which is not generally transferable to the static Young’s modulus. Due to the scarcity of the experimental data on heated cement paste, this set of experimental data is still selected to verify the numerical method. To evaluate the Young’s modulus of heated cement paste, the degree of hydration needs to be known, but was not provided in the experiment. Thus, the Parrot and Killoh [21] approach is used to estimate the degree of hydration in this paper. In the approach, the rate of hydration of a particular clinker phase Ri,t is expressed by a set of three equations as follows

Ri;t ¼ Fig. 1. Geometric schematic of decomposition products.

E0;t 1 þ gðlD Þx½1 þ gðl0 Þx

where x is the conventional scalar microcrack density parameter and g(x) and lD are defined respectively as

gðxÞ ¼

Substitution of Eq. (19) into Eq. (17) yields

E0;t ¼ E0 

Cement paste

K1 1N ð1  ai;t Þ½ lnð1  ai;t Þ 1 for nucleation and growth N1

ð25aÞ

J. Zhao et al. / Construction and Building Materials 54 (2014) 197–201

Young’s modulus (GPa)

200

Ri;t ¼

Exp.

40

Numer.

1  ð1  ai;t Þ1=3

for diffusion

ð25bÞ

Ri;t ¼ K 3 ð1  ai;t ÞN3 for formation of hydration shell 0.25

30 0.35 0.5

0

100

200

300

400

500

ai;tþDt ¼ ai;t þ DtRi;t

600

Temperature (°C) Fig. 3. Comparison of numerical results and experimental results of Masse et al. [2].

Exp.(0.3)

50

Exp.(0.4) 40

Exp.(0.5) Numer.

30 20 10 0

0

100

200

300

400

500

600

Temperature (°C) Fig. 4. Comparision of numerical results and experimental results of Padeveˇt and Zobal [5].

40

Young’s modulus (GPa)

Young’s modulus (GPa)

Numer.

30 20 10

0

100

300

200

Numer.

20 10

0

100

200

300

Temperature (°C)

Temperature (°C)

(a) Specimen S1

(b) Specimen S2

40 Exp. Numer.

30 20 10

0

Exp. 30

0

400

Young’s modulus (GPa)

Young’s modulus (GPa)

Exp.

0

ð26Þ

where Dt is the time interval for integration. The overall degree of hydration a is calculated as a weighted average of the degrees of hydration of the clinker phases. The effects of temperature and water to cement ratio on the rate constants can be accounted for by introducing two factors [21]. With this approach, the degrees of hydration ac are estimated to be 53%, 63%, and 75% for water to cement ratios of 0.25, 0.35, and 0.50, respectively. The initial volume fractions of various constituents in cement paste can then be estimated using Eqs. (6)–(9). According to the heating process, a thermal decomposition analysis is conducted and the extent of thermal decomposition of cement paste is described by the conversion degrees of the hydration products. Since E0 and l0 are not provided in their experiment [2], E0 is evaluated from the initial volume fractions of various constituents [17] and l0 is taken as 0.25 [22]. Based on the thermal decomposition analysis, E0,t can be obtained from Eq. (21). The determination of x is generally difficult, and experimental calibration seems to be the only feasible, practical method. In this paper, the measured Young’s modulus of heated cement paste with a water to cement ratio of 0.35 is used to calibrate x. With ac, E0, E0,t, l0, and x known, the Young’s moduli of heated cement paste with water to cement ratios of 0.25 and 0.50

40

0

100

200

ð25cÞ

where ai,t is the degree of hydration of clinker phase i (C3S, C2S, C3A, and C4AF) at time t (in days), K1, N1, K2, K3, and N3 are empirical constants [21], and the lowest value of Ri,t at any time is taken as the rate controlling step and used to calculate the instantaneous degree of hydration. The degree of hydration at time t + Dt is then expressed as

20

10

Young’s modulus (GPa)

K 2 ð1  ai;t Þ2=3

300

400

400

40 Exp.

30

Numer.

20 10 0

0

100

200

300

Temperature (°C)

Temperature (°C)

(c) Specimen S3

(d) Specimen S4

Fig. 5. Comparision of numerical results and experimental results of Kerr [4].

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can be evaluated using the proposed method as shown in Fig. 3, indicating that the numerical results are in good agreement with the experimental results with correlation coefficients of 0.942 and 0.986 for w/c = 0.25 and 0.50, respectively. The second set of experimental data is taken from Padeveˇt and Zobal [5]. In their experiment, cylindrical cement paste specimens with water to cement ratios of 0.3, 0.4, and 0.5 were made with Portland cement CEM I 42.5R. All the cylindrical specimens were 25 mm long and 10 mm in diameter. After curing at room temperature for 25 days and air drying for one day, the specimens were heated to the target temperatures of 200, 300, 450, and 600 °C within 1 h. The exposure duration at each target temperature was 2.5 h. After naturally cooling to room temperature, the Young’s modulus was measured by compression tests. The results are shown in Fig. 4. It is noted that the measured Young’s modulus belongs to the residual mechanical properties of cement paste. As in the last example, the degrees of hydration of the three types of specimens are estimated to be 57%, 66%, and 74% for water to cement ratios of 0.3, 0.4, and 0.5, respectively [21]. This experiment also provided the value of E0, and l0 is still taken as 0.25. The measured Young’s modulus of heated cement paste with a water to cement ratio of 0.4 is used to calibrate the value of x. The predicted Young’s moduli of heated cement paste with water to cement ratios of 0.3 and 0.5 are compared with the experimental results as shown in Fig. 4. As can be seen from Fig. 4, the numerical results are again in good agreement with the experimental results. The correlation coefficients between them are 0.947 and 0.962 for w/c = 0.3 and 0.5, respectively. The third set of experimental data is taken from Kerr [4]. In his experiment, different cement pastes with water to binder ratios ranging from 0.22 to 0.40 were made with ASTM type III cement, fly ash, and silica fume. At the curing age of 90–100 days, 1.5  10  55 mm plate specimens were tested under displacement-controlled oscillatory bending in a dynamic mechanical analyzer. During the test, the specimens were heated from room temperature to 400 °C at a constant heating rate of 2 °C/min. At 400 °C, temperature was held constant for 30 min. Again, it should be noted that the measured value is the dynamic Young’s modulus. Since the proposed method is only valid for traditional cement paste or silica fume blended cement paste, the test results of four types of cement paste specimens, S1, S2, S3, and S4, are selected for verification. In specimens S1 and S2, no mineral admixtures were added and the water to cement ratios were 0.4 and 0.5, whereas in S3 and S4, 10% cement by mass was replaced with silica fume and the water to binder ratios were 0.22 and 0.30, respectively. The results are shown in Fig. 5. In a similar manner, ac is estimated as 70%, 78%, 52%, and 59% for specimens S1, S2, S3, and S4, as is estimated as 0.50 for specimens S3 and S4, respectively [21,23], and l0 is still taken as 0.25. x is calibrated using the experimental results of specimens S1 and S3. With these inputs, the Young’s moduli of specimens S2 and S4 are evaluated using the proposed method. The results are shown in Fig. 5. It can be seen from Fig. 5 that the numerical results are well correlated with the experimental results with correlation coefficients of 0.963 and 0.948 for specimens S2 and S4, respectively. Therefore, the validity of the proposed method is verified. 6. Conclusions A numerical method has been developed in this paper for predicting the Young’s modulus of heated ordinary Portland cement

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paste with or without silica fume up to 600 °C. In the method, the effects of thermal decomposition and microcracking on the Young’s modulus of heated cement paste have been taken into account. Finally, the validity of the numerical method has been verified with three sets of experimental data. Acknowledgments The financial support from the Fundamental Research Funds for the Central Universities with Grant No. 2013JBM068, the Natural Science Foundation of Zhejiang Province with Grant No. LY12E08022, and the National Natural Science Foundation with Grant No. 51278048, of the People’s Republic of China, is greatly acknowledged. References [1] Dias WPS, Khoury GA, Sullivan PJE. Mechanical properties of hardened cement paste exposed to temperatures up to 700 °C (1292 °F). ACI Mater J 1990;87(2):160–6. [2] Masse S, Vetter G, Boch P, Haehnel C. Elastic modulus changes in cementitious materials submitted to thermal treatments up to 1000 °C. Adv Cem Res 2002;14(4):169–77. [3] Odelson JB, Kerr EA, Vichit-Vadakan W. Young’s modulus of cement paste at elevated temperatures. Cem Concr Res 2007;37(2):258–63. [4] Kerr EA. Damage mechanisms and repairability of high strength concrete exposed to elevated temperatures, PhD thesis. USA: School of Civil Engineering and Geological Sciences, the University of Notre Dame; 2007. [5] Padeveˇt P, Zobal O. Comparison of material properties of cement paste at high temperatures. In: Proc. 3rd WSEAS Int. Conf. On Engineering Mechanics, Structures, Engineering Geology. Corfu Island, Greece; 2010. p. 402–7. [6] Ulm FJ, Coussy O, Bazˇant ZP. The ‘Chunnel’ fire. I: chemoplastic softening in rapidly heated concrete. J Eng Mech 1999;125(3):272–82. [7] Lee J, Xi Y, Willam K, Jung Y. A multiscale model for modulus of elasticity of concrete at high temperatures. Cem Concr Res 2009;39(9):754–62. [8] Hansen TC. Physical structure of hardened cement paste: a classical approach. Mater Struct 1986;19(6):423–36. [9] Bentz DP, Garboczi EJ. Percolation of phases in a three-dimensional cement paste microstructural model. Cem Concr Res 1991;21(2–3):325–44. [10] Bentz DP, Jensen OM, Coats AM, Glasser FP. Influence of silica fume on diffusivity in cement-based materials I. Experimental and computer modeling studies on cement pastes. Cem Concr Res 2000;30(6):953–62. [11] Zhao J, Zheng JJ, Peng GF, van Breugel K. Prediction of thermal decomposition of hardened cement paste. J Mater Civil Eng 2012;24(5):592–8. [12] Lin WM, Lin TD, Powers-Couche LJ. Microstructures of fire-damaged concrete. ACI Mater J 1996;93(3):199–205. [13] Pourchez J, Valdivieso F, Grosseau P, Guyonnet R, Guilhot B. Kinetic modelling of the thermal decomposition of ettringite into metaettringite. Cem Concr Res 2006;36(11):2054–60. [14] DeJong MJ, Ulm FJ. The nanogranular behavior of C–S–H at elevated temperatures (up to 700 °C). Cem Concr Res 2007;37(1):1–12. [15] Jaeger HM, Nagel SR. Physics of the granular state. Science 1992;255(5051):1523–31. [16] Christensen RM. Mechanics of composite materials. New York: John Wiley & Sons; 1979. [17] Zheng JJ, Zhou XZ, Shao L, Jin XY. Simple three-step analytical scheme for prediction of elastic moduli of hardened cement paste. J Mater Civil Eng 2010;22(11):1191–4. [18] Timoshenko SP, Goodier JN. Theory of elasticity. 3rd ed. Singapore: McGrawHill; 1984. [19] Peng GF, Huang ZS. Change in microstructure of hardened cement paste subjected to elevated temperatures. Constr Build Mater 2008;22(4):593–9. [20] Feng XQ, Yu SW. Estimate of effective elastic moduli with microcrack interaction effects. Theor Appl Fract Mech 2000;34(3):225–33. [21] Parrot LJ, Killoh DC. Prediction of cement hydration. British Ceramic Proceedings 1984;35:41–53. [22] Li GQ, Zhao Y, Pang SS. Four-phase sphere modelling of effective bulk modulus of concrete. Cem Concr Res 1999;29(6):839–45. [23] Lu P, Sun G, Young JF. Phase composition of hydrated DSP cement pastes. J Am Ceram Soc 1993;76(4):1003–7.