numerical approach towards aging management of concrete

Numerical Approach towards Aging Management of ConcreteStructures: Material Strength Evaluation in a MassiveConcrete Structure under One-Sided HeatingIppei Maruyama, Go Igarashi

Journal of Advanced Concrete Technology, volume ( ), pp.13 2015 500-527

Origin of Drying Shrinkage of Hardened Cement Paste: Hydration PressureIppei MaruyamaJournal of Advanced Concrete Technology, volume ( ), pp.8 2010 187-200

Cement reaction and resultant physical properties of cement pasteIppei Maruyama , Go IgarashiJournal of Advanced Concrete Technology, volume ( ), pp.12 2014 200-213

Numerical Study on Drying Shrinkage of Concrete Affected by Aggregate SizeIppei Maruyama , Ai SugieJournal of Advanced Concrete Technology, volume ( ), pp.12 2014 279-288

Mechanism of Change in Splitting Tensile Strength of Concrete Under Heating or Drying Up to 90 °CMao Lin , Mitsuki Itoh Ippei Maruyama,Journal of Advanced Concrete Technology, volume ( ), pp.13 2015 94-102

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Journal of Advanced Concrete Technology Vol. 13, 500-527, November 2015 / Copyright © 2015 Japan Concrete Institute 500

Scientific paper

Numerical Approach towards Aging Management of Concrete Structures: Material Strength Evaluation in a Massive Concrete Structure under One-Sided Heating Ippei Maruyama1* and Go Igarashi2

Received 1 April 2015, accepted 20 October 2015 doi:10.3151/jact.13.500

Abstract For the purpose of performance evaluation of an existing reinforced concrete member, a chemo-thermo-hygro model to predict the spatial and temporal changes of physical properties of concrete in the member, named the “Computational Cement-Based Material Model (CCBM)”, was proposed. This proposed simulation model includes models of rate of hydration of cement minerals, phase composition, and resultant properties of cement paste (i.e., compressive strength, Young’s modulus, Poisson’s ratio, thermal expansion coefficient, autogenous shrinkage, drying shrinkage, heat capacity, heat transfer coefficient, water vapor sorption isotherms, and water transfer coefficient). Furthermore, the model for compressive strength of concrete considered the variation in cement paste strength due to its colloidal features as well as micro-defects produced around aggregate due to differences in volume between aggregates and mortar upon heating and drying. The concrete properties of spatial distribution and temporal changes were evaluated by coupling these models with heat and water transport. Validation of these models was achieved by using existing experimental data. Using this CCBM, a thick concrete wall made with moderate Portland cement with a water-to-cement ratio of 0.55 under one-sided heating was simulated and potential problems that can arise during an integrity evaluation were discussed. If the required compressive strength, which was assessed within 91 days of placement, remains unchanged, an additional hy-dration process can build an adequate strength margin to overcome the risk of strength reduction due to heat and drying. However, in the case that the required strength is increased due to a re-evaluated risk, such as the magnitude of an earthquake, performance evaluation is not trivial as the core sample taken from the side where the execution of sampling is possible could exhibit a greater strength than the average strength of the target concrete member. Therefore, numerical evaluation might aid in this kind of situation.

1. Introduction

Effective utilization of existing reinforced concrete structures is an important issue in Japan from both an environmental and economical perspective. For aging management of concrete structures, their current and future performances until the end of the service or maintenance period should satisfy required performances. Therefore, the development of a numerical simulation model to predict changes in properties and experiments contributing to the development of a computational model have been examined for several decades in the research and engineering fields. Japan, which is affected by a large number of earthquakes, requires evaluation of structural and seismic performance, and the compressive strength and Young’s modulus of concrete in a structural member are important material factors for this per-

formance evaluation (Building Research Institute 2001; Architectural Institute of Japan 2008).

There is a need to evaluate the performance of a con-crete member that is difficult to access. In an engineering plant, there are concrete members that adjoin a reactor or pipes, which create a heating and drying environment, but such concrete members that are on the side with severe conditions cannot be accessed and evaluated di-rectly via core sample tests or non-destructive tests. Despite such difficulties, the performance evaluation of such a concrete member is necessary for assessing the structural performance of the structure. In the present study, in order to overcome this difficulty, a computa-tional simulation model, which deals with “virtual” compressive strength in a concrete member, named the “Computational Cement-Based Material model (CCBM)”, is proposed. This model is designed for in-terpolation of concrete data between standard curing sample testing results or non-destructive results of a structure, and a concrete member under severe condi-tions.

Many rate of hydration models (Kondo and Ueda 1968; Pommersheim and Clifton 1979; Bezjak and Jelenić 1980; van Breugel 1995b; Navi and Pignat 1996; Bentz 1997; Tomosawa 1997; Maekawa et al. 2003b; Thomas 2007; Bishnoi and Scrivener 2009), phase composition and microstructural models (Jennings and

1Associate Professor, Graduate School of EnvironmentalStudies, Nagoya University, Nagoya, Japan. *Corresponding author, E-mail: [email protected] 2Doctoral student, Graduate School of Environmental Studies, Nagoya University, Nagoya, Japan. Current position: Assistant Professor, Graduate School of Engineering, Tohoku University, Sendai, Japan.

I. Maruyama and G. Igarashi / Journal of Advanced Concrete Technology Vol. 13, 500-527, 2015 501

Johnson 1986; van Breugel 1995; Navi and Pignat 1996; Bentz 1997; Maekawa et al. 2003b; Lothenbach and Winnefeld 2006; Bullard et al. 2011; Thomas et al. 2011), and physical property models (van Breugel 1995a; Tomosawa 1997; Bentz et al. 1998; Maekawa et al. 2003a; Lothenbach et al. 2008) exist. Based on these existing research studies, causal relations between ce-ment hydration and physical properties of hardened ce-ment paste (hcp) or concrete are re-arranged and nu-merically modeled. A view outlining the proposed model is shown in Fig. 1.

One of the characteristics of the proposed model is that the model reflects the colloidal feature of cement hy-drates, which alter microstructure, water vapor sorption isotherms, and strength of hardened cement paste due to drying or heating. In addition, the proposed model is based on recent experimental works in which cement paste and concrete properties under equilibrium at dif-ferent relative humidity and temperature conditions were collected. In the present paper, each component of the model is compared with experimental data and some are validated by comparison with existing experimental works by different researchers.

Finally, a thick concrete wall under a one-sided heat-ing condition is simulated and issues that might arise in the case of soundness evaluation are discussed.

2. Modeling of cement hydration

2.1 Rate of hydration A rate of hydration model is firstly introduced. In general, the hydration process of Portland cement is not scien-tifically clear. There are discussions on areas including the dormant period mechanism (Kondo and Daimon 1969; Garrault and Nonat 2001; Garrault et al. 2005), the impact of minor chemical components in cement miner-als on their rate of hydration (Mascolo and Ramachandran 1975), difference of rate of hydration of polymorphism (Link et al. 2015), mutual interactions of the hydration process of cement minerals through ion concentration in a solution (Gartner and Jennings 1987, Maruyama and Igarashi 2011a), and impact of precipita-tion sites and regions (Zhou et al. 2006). Therefore, in this section, the rate of hydration model is based on ex-perimental data and many parts of the model are em-pirical rather than theoretical.

It has been proposed that the hydration process is di-vided into three different rate-controlling steps (Parrott and Killoh 1984) on which more detailed divisions and discussions exist (Kondo and Ueda 1968). The three steps are categorized as 1) the nucleation and growth period, 2) the diffusion period, and 3) the period for formation of a hydration shell, and this concept is gen-erally accepted as being useful to apply numerical mod-eling (Lothenbach et al. 2008). Thus, this concept is applied for the proposed model; meanwhile, in the dif-fusion period, the representing equation is not directly

Rate of cement hydration

Phase composition

Sorption isotherm

Moisture transferCoefficient

Heat releaseWater consumption

Water content

TEC Shrinkage

Strength

Shrinkage

Damage in concrete

Concrete physical properties

Cement paste

C-S-H aging

BET surface area

Heat transferWater transfer

Young’s modulusPoisson’s ratio

Heat transfer Coefficient

Heat capacity

Temperature

Aggregates

Heat transfer Coefficient

Heat capacity

2.1

: Section in the paper

2.2

3.13.2

3.3

3.4

Note: TEC is thermal expansion coefficient Fig. 1 An outline of the proposed model and related sections in the paper.


employed from Parrott and Killoh (1984). Equations (1) ~ (3) correspond to the rate-controlling

steps 1) - 3). Nucleation and growth period:

( ) ( )( )( ) ( ),21,1,1

,1

( )1 ( ) 1 ln 1 ( ) , 1 ( )iNii

i i i i

i

Ktt t t

t Nα

α α α α−∂= − − − ≤ <

∂ (1)

Diffusion period (only for C3S and C3A):

( ) ( ),2

, ,2 ,1 ,2

( )0.1 1 ( ) , ( )iNi

i w srf a i i i i i

tK t t

tα

γ γ γ α α α α∂

= ⋅ ⋅ ⋅ ⋅ − ≤ <∂

(2)

Formation of hydration shell period:

( ) ( ),2

, ,2 ,2

( )1 ( ) , ( )iNi

i w srf a i i i i

tK t t

tα

γ γ γ α α α∂

= ⋅ ⋅ ⋅ − <∂

(3)

where Ki,1, Ki,2, Ni,1 and Ni,2: coefficients for rate of hy-dration of each cement mineral at 293 K (-); αi (t): the degree of hydration of each cement mineral (-); i: type of cement mineral (i.e., alite (notation: C3S), belite (nota-tion: C2S), aluminate phase (notation: C3A), and ferrite phases (notation: C4AF) (here, the notations obey the following rules of cement chemistry: C: CaO, S: SiO2, A: Al2O3, F: Fe2O3); αi ,1, αi ,2: degree of hydration of each cement mineral from period 1) to 2) and from period 2) to 3); γi,w: the coefficient of stagnation of hydration as a function of water content in cement paste, which attrib-utes to the precipitation of hydrates (cm3/cm3); γsrf: the coefficient of rate of hydration due to the difference in surface area of cement particles (2); γa: the coefficient of belite hydration, which is affected by the reaction of alite. Note that this term is only valid for the hydration of belite.

Each coefficient is calculated by the following equa-tions, which are also presented in Table 1:

,1 ,0 3 20.25 0.23 ( , )i i ia v r i C S C S= − ⋅ ⋅ = (4)

2

3 3

3 3

,0,0,2 ,

,0 ,0

1.0 0.4 C SwC S C S t

C S C S

vvK v

v v

⎛ ⎞= ⋅ +⎜ ⎟⎜ ⎟

⎝ ⎠ (5)

2 2

2

,0,2 ,

,0

0.03 wC S C S t

C S

vK v

v= (6)

3 3,2 ,00.3C A C AK v= (7)

4 4,2 ,00.1C AF C AFK v= (8)

, , , ,0/i w w i t wv vγ = (9)

3

3, , 11% , ,/ C S Hw C S t RH t w CSH t

CSH

h Vv w v

Vρ

⋅= − ⋅ (10)

2

2, , 11% , ,/ C S Hw C S t RH t w CSH t

CSH

h Vv w v

Vρ

⋅= − ⋅ (11)

3 4

3 4, , , , 11% , ,/ C AorC AF H

w C A t w C AF t RH t w CSH t

CSH

h Vv v w v

Vρ

⋅= = − ⋅ (12)

0/srf A Aγ = (13)

3

3 3

3

0.2 ( ( ) 0.75)0.2 ( ( ) 0.75) (0.75 ( ) 0.83

1.0 ( ( ) 0.83)

C S

a C S C S

C S

tt t

t

αγ α α

α

<= + − < ≤

>

⎧⎪⎨⎪⎩

(14)

where vw,i,t: volume of water available for the hydration process of cement mineral i at age t (cm3/cm3); vi,0 : vol-ume of cement mineral i just after mixing in the unit volume of cement paste (cm3/cm3); vw,0: volume of water just after mixing in the unit volume of cement paste (cm3/cm3); w11％RH,t: evaporable water over 11% RH at 293 K in cement paste (g/cm3); vCSH,t: volume of calcium silicate hydrate (C-S-H) at age t in the unit volume of hardened cement paste (cm3/cm3), while the density and chemically bound water of C-S-H are assumed to be the values at equilibrium to 11% RH at 293 K; Vj : molar volume of substance j (cm3/mol), while j is either C-S-H (notation: C-S-H) or H (H2O); ρw: density of water (g/cm3); ρi: density of cement mineral i (g/cm3); A: Blain value of cement (cm3/g); A0: reference Blain value (3300 cm3/g); and hi: coefficient representing adsorbed water which does not contribute to the additional hydration of cement mineral i (mol/mol).

During the formulation of Equations (4) ~ (9), the dormant period of alite and belite was a function of cal-cium ion in the solution, but this was considered indi-rectly by using a function of the degree of hydration of each cement mineral. Both the aluminate and ferrite phases also have a dormant period coexisting with SO4

2- ion in the solution, but such data is not sufficient, and it is assumed that their dormant periods continue until the degree of hydration of each mineral attains 0.10. Re-

Table 1 Factors and parameters for the rate of hydration model. Alite (C3S) Belite (C2S) Aluminate phase (C3A) Ferrite phase (C4AF) Ki,1 0.167 0.021 0.042 0.015 Ni,1 0.7 1.0 0.85 0.7 Ki,2 Equation (4) Equation (5) Equation (6) Equation (7) Ni,2 1.5 3.0 1.5 3.5 αi ,1 Equation (8) Equation (8) 0.10 0.10 αi ,2 αi ,1＋0.03 αi ,1 αi ,1＋0.03 αi ,1 Ei (kJ/mol) 30 56 45 32 hi 3.0 2.5 2.5 2.5


garding the alite hydration process, parts of the surface of belite are used as the precipitation sites for C-S-H (Kishi et al. 2008); consequently, the volumetric ratio of alite to belite is considered as a factor affecting alite hydration.

During the formulation of Equations (9) ~ (14), the following phenomena were taken into account:

The impact of the water-to-cement ratio on the rate of hydration appeared as a difference in the total amount of precipitation from evaporable water. Less evaporable water in cement paste causes the stagnation of hydration through a reduction of the rate of precipitation. This especially holds in the period of formation of the hydra-tion shell, since the precipitation process controls the rate of hydration during that period. This has been suggested by existing experimental data and hydration models (Taplin 1959; Kishi and Maekawa 1995; van Breugel 1995a, 1995b). Therefore, the rate of hydration is mod-eled as a function of water content using the factor γw. In addition, some parts of evaporable water cannot be used in the cement reaction since adsorbed water is retained from the surface, and the solution and precipitation process are repressed (Dove and Rimstidt 1994). This effect of adsorption water on the rate of hydration varies for each cement mineral due to the differences in their reaction enthalpies. This difference is empirically con-sidered by using coefficient hi defined for fitting the experimental data. It should be emphasized that these parameters affects distribution of degree of hydration in a concrete member by coupling with moisture transfer phenomenon of which details of numerical modelling are introduced in 3.2.

The rate of hydration in cement is affected by the total surface area or particle size distribution (Uchida 1987). This tendency is empirically taken into account by using Equation (13) and the factor γsrf.

There is a clear strong correlation between the degrees of hydration of alite and belite, which is not affected by curing temperature (Maruyama et al. 2007; Maruyama and Igarashi 2011a). It is suggested that during the alite hydration process, belite hydration is retarded. The key is the difference of the solubility curves of alite and belite. The solubility curve of alite, which is a function of CaO and SiO2 concentration, is on the larger side of CaO and SiO2 concentration compared to the solubility curve of belite, and during the hydration process of alite, the CaO and SiO2 concentration in the pore solution is on the curve of the solubility curve of alite. During such cases, the reaction of belite is quite slow as the condition of the pore solution is far from the solubility curve of belite (Gartner and Jennings 1987). The proposed model takes this phenomenon into account by using the factor γa.

The temperature dependency of cement hydration is considered by using the Arrhenius law, and the reference temperature is set as 293 K. This dependency is ex-pressed by Equation (15):

0293

( ) ( ) 1 1expi i i

T T T K

t t Et t R T T

α α

= =

⎛ ⎞⎛ ⎞∂ ∂= − −⎜ ⎟⎜ ⎟⎜ ⎟∂ ∂ ⎝ ⎠⎝ ⎠

(15)

where Ei: activation energy of cement mineral i (J/mol･K); R: gas constant (8.314 J/mol･K); and T0: reference temperature (293 K). This equation also gives the dis-tribution of degree of hydration in a concrete member by coupling with heat transfer phenomenon of which details of modeling is introduced in 3.1.

These factors and parameters are determined by ex-periments in which the degree of hydration of each ce-ment mineral in cement paste with different wa-ter-to-cement ratios and different types of Portland ce-ment is examined (Maruyama and Igarashi 2011a, 2014). The applied temperature histories of the experiments are shown in Fig. 2. The results of the factors and parameters for the rate of hydration model are summarized in Table 1. The premised values of cement hydrates, such as mo-lar volume, density, and chemical composition under different drying conditions, are listed in Table 2 (Maruyama and Igarashi 2014). These are based on the following references: (Schwiete and Iwai 1964; Kuzel 1969, Schwiete and Ludwig 1969; Allmann 1977; Feldman and Ramachandran 1982; Fischer and Kuzel 1982; Taylor 1986; Pöllmann et al. 1990; Motzet and Pöllmann 1999; Jennings 2000; Tennis and Jennings 2000; Zhou and Glasser 2001; Brouwers 2005).

A comparison of the calculated results using the pro-posed parameters and the results of experimental data are shown in Fig. 3. It is confirmed that the overall trend of each degree of hydration of cement minerals under dif-ferent water-to-cement ratios and different temperature histories are reproduced.

2.2 Phase composition The degree of hydration of each cement mineral, the water-to-cement ratio, and the mineral composition of initial cement can provide reliable phase composition information in cement paste by using thermodynamic equilibrium calculations with a proper set of thermody-namic data (Lothenbach and Winnefeld 2006). On the contrary, the thermodynamic equilibrium calculations require large computational times and have some limita-

Fig. 2 Temperature histories applied to cement paste; 20 ºC constant is denoted as 20c and the elevated tem-perature history with a maximum temperature of 60 ºC in the early age is denoted as 60.


tions, such as the drying state calculation and adopting the colloidal feature of C-S-H. Consequently, we tenta-tively take into account the phase composition using a classical approach that is the premise of the chemical reaction equations. Based on experimental data (Maruyama and Igarashi 2011a, 2014) and vast existing discussions regarding cement reaction (Osbaeck 1992; Bentz 1997; Taylor 1997), the following equations are applied to the proposed phase composition model:

Alite:

3 2 2.5

3

C S 3.5H C SH CH544 (J/g C S)H

+ → +

Δ = − − (16)

Belite:

2 1.5 2.5

3

C S 3.0H C SH CH544 (J/g C S)H

+ → +

Δ = − − (17)

Table 2 Chemical composition of cement hydrates assumed in the proposed model. Molar mass Density Hydrates Drying state H/C Composition for-

mula (g/mol) (g/cm3) Note

1000 °C 0 CxSH0 155 ―

105 °C 0.88 CxSH1.5 182 2.60 2) 11% RH, 20 °C 1.24 CxSH2.5 200 2.41

3) C-S-H 1)

Saturated 1.24 CxSH2.5 200 2.41 4) 1000 °C 0 C3A･3(C S )･H0 679 ―

105 °C 2.00 C3A･3(C S )･H12 895 2.38 5)

11% RH, 20 °C 5.33 C3A･3(C S )･H32 1255 1.78 6) Ett

Saturated 5.33 C3A･3(C S )･H32 1255 1.78

1000 °C 0 C3A･C S ･H0 406 ―

105 °C 2.00 C3A･C S ･H8 550 2.40 7)

11% RH, 20 °C 3.00 C3A･C S ･H12 623 2.02 8) Ms

Saturated 3.00 C3A･C S ･H12 623 2.02

1000 °C 0 C3AH0 270 ― 105 °C 2.00 C3AH6 378 2.52 9)

11%RH, 20 °C 2.00 C3AH6 378 2.52 HG

Saturated 2.00 C3AH6 378 2.52 9) 1000 °C 0 C4AH0 326 ― 105 °C 1.75 C4AH7 452 2.52 10)

11%RH, 20 °C 3.25 C4AH13 561 2.05 C-A-H

Saturated 3.25 C4AH13 561 2.05 11), 12) 1000 °C 0 C4FH0 384 ― 105 °C 1.75 C4FH7 510 2.84 13)

11%RH, 20 °C 3.25 C4FH13 618 2.16 C-F-H

Saturated 3.25 C4FH13 618 2.16 14)

Notation C: CaO, S: SiO2, H: H2O, A: Al2O3, F: Fe2O3, S: SO3 1) Molecular weight of C-S-H when the Ca/Si atomic ratio is assumed to be 1.7 is listed 2) Jennings (2000), Taylor (1986) 3) Feldman and Ramachandran (1982) 4) In the present study, the adsorbed water at the condition above 11% RH is assumed to behave like bulk water 5) Zhou and Glasser (2001), Tennis and Jennings (2000) 6) Tennis and Jennings (2000), Pöllmann et al. (1993) 7) Motzet and Pöllmann (1999), Tennis and Jennings (2000) 8) Allmann (1977) 9) Kuzel (1969) 10) Fischer and Kuzel (1982), Schwiete and Ludwig (1969) 11) Fischer and Kuzel (1982) 12) This has been reported to become C4AH19 at a RH of 88% or higher, but C4AH13 is used for the present study due to

its easy decomposition. 13) Due to missing data, the density was calculated by assuming the same crystal structure as C4AH7. While it is generally

accepted that Fe forms a liquid solution partly substituting with Al in C-A-H, they are separated for convenience in this study.

14) Schwiete and Iwai (1964)


Aluminate with gypsum in the system:

3 2 3 32

3

C A 3CSH 26H C A 3CS H1745 (J/g C A)H

+ + → ⋅ ⋅Δ = − − (18)

Aluminate without gypsum in the system:

3 3 32

3 12 4 13

3

4C A C A 3CS H 2CH 28H3C A CS H 2C AH

510 (J/g C A)H

+ ⋅ ⋅ + +→ ⋅ ⋅ +

Δ = − − (19)

Ferrite with gypsum in the system:

4 2

3 32

4

C AF 6CSH 2CH 50H2C (A,F) 3CS H

1340 (J/g C AF)H

+ + +→ ⋅ ⋅

Δ = − − (20)

Ferrite without gypsum in the system:

4 3 32

3 12 4 13

4

2C AF C (A,F) 3CSH 6CH 24H3C (A,F) CS H 2C (A,F)H

607 (J/g C AF)H

+ ⋅ + +→ ⋅ ⋅ +

Δ = − − (21)

The heat of hydration of each equation is, in general, affected by the temperature and pressure, but in the pro-posed model, constant values are used for the first ap-proximation. The reaction passes of both the aluminate and ferrite phases have a temperature dependency (Matschei et al. 2007; Lothenbach et al. 2008) and those passes are a function of coexisting gypsum, hemihydrate, and ion. In the present model, Equation (19) is consid-

ered to be applicable to the case above 323 K. An example of the calculated phase composition of

N5520c (notaion is shown in the caption of Fig. 3) compared with experimental results are shown in Fig. 4, and in this calculation, C4AH13 and C4(A,F)H13 are con-sidered as the amorphous phase. Comparisons of chemically bound water and amount of portlandite be-tween the calculated results and the experimental data are shown in Fig. 5 and Fig. 6, respectively.

3. Modeling of cement paste and concrete properties

3.1 Heat-related properties The temperature history in a concrete member has a large impact on cement hydration and the hydration process produces heat, consequently resulting in a feedback system in the concrete. In order to numerically evaluate this feedback system in concrete, the following govern-ing equation (Equation 22) for heat transfer should be solved. This requires the mathematical modeling of the heat capacity of concrete, density of concrete, and the heat transfer coefficient, which are functions of the de-gree of hydration and water content.

rcon con con

QT Tc r 1t x x t

∂∂ ∂ ∂⎛ ⎞= +⎜ ⎟∂ ∂ ∂ ∂⎝ ⎠ (22)

where ccon: heat capacity of concrete (J/g･K); ρcon: den-sity of concrete (g/cm3); λcon: thermal conductivity of concrete (W/m･K); and Qr: heat production or con-

Fig. 3 Comparison of the results of the proposed model with the experimental degree of hydration data. (N: ordinary Portland cement; L: low heat Portland cement; XXYY: XX is a percentage of the water-to-cement ratio; YY denotes that the temperature history 20c is a constant temperature at 20 ºC and 60 is temperature history simulating massive con-crete.)


sumption due to the reaction of the concrete component (J).

The heat capacity of concrete is calculated via Equa-tion (23).

con cem cem hyd hyd wat wat agg aggc W c W c W c W c= ⋅ + ⋅ + ⋅ + ⋅ (23)

where ccon: heat capacity of concrete (J/g･K); Wi: mass of concrete component i in unit mass of concrete, where i = cem (unhydrated cement), hyd (cement hydrates), wat (evaporable water above 105 ºC), and agg (aggregate); and ci: heat capacity of concrete component i, where i = cem, hyd, wat, and agg (J/g･K). While the heat capacity of substances related to the cement hydration system has been reported by Lothenbach et al. (2008), based on a former experimental study (Maruyama et al. 2006), the heat capacity values of 0.78, 1.205, 4.185, and 0.70 ~ 1.00 are used for unreacted cement, cement hydrates (under the dried condition at 105 ºC), evaporable water, and aggregate, respectively.

Figure 7 shows plots of both the experimental and calculated results of the relationship between the degree of hydration and the heat capacity of dried hardened cement paste. The calculated results show the general trend of experimental data, while a certain range of scattering is observed in the experimental results.

Thermal conductivity is evaluated by a parallel model as the arithmetic mean value of the concrete component (Maruyama and Igarashi 2014) :

1 . . .( )hcp p s s p l l p g g1 g w V 1 V 1 V 1= ⋅ + + (24)

, ,con c hcp hcp c agg aggV Vλ λ λ= + (25)

where Vs: volume of solids in unit volume of hardened cement paste (vol. / vol.); Vl: volume of evaporable water in unit volume of hardened cement paste (vol. / vol.); Vg: volume of gas in unit volume of hardened cement paste (vol./vol.); Vhcp: volume of hardened cement paste in unit volume of concrete (vol. / vol.); Vagg: volume of aggre-gate in unit volume of concrete (vol. / vol.); λs: thermal conductivity of solid in hardened cement paste (=1.54 W/m･K); λl: thermal conductivity of evaporable water in hardened cement paste (=0.60 W/m･K); λg: thermal conductivity of gas in hardened cement paste (=0.026 W/m･K); λhcp: thermal conductivity of hardened cement paste (W/m･K); λcon: thermal conductivity of concrete (W/m･K); and λagg: thermal conductivity of aggregate (W/m･K). The factor γλ (Rw) represents the reduction ratio of thermal conductivity of hardened cement paste due to a decrease in the connectivity of solid and liquid phases in hardened cement paste caused by agglomera-tion of colloidal C-S-H and the resultant increase of macro pores. This factor, as presented in Equation (26), is represented by a function of relative water content Rw, where it is the ratio of evaporable water w (g/g-dried hcp) to water content at the saturated condition w0 (g/g-dried hcp).

( ) 0.70 0.30h wg w R= + (26)

By using these equations, the calculated results of thermal conductivity of hardened cement paste as a function of time and thermal conductivity of hardened

Fig. 4 Comparison between the calculated phase com-position of N5520 (right) and experimental results (left).Experimental data were obtained from Maruyama and Igarashi (2014).

Fig. 5 Comparison between experimental results and calculated results regarding chemically bound water (chemically bound water is water released from 105 ºC to 1000 ºC). Experimental data were obtained from Maruyama and Igarashi (2014).

Fig. 6 Comparison between experimental results and calculated results of the amount of Portlandite in cement paste. Experimental data were obtained from Maruyama and Igarashi (2014).


cement paste as a function of water content are compared with experimental data in Fig. 8 and Fig. 9, respectively.

The heat capacity model and the heat of hydration model are used to simulate the adiabatic temperature rise of concrete, which is affected by the temperature at the fresh state. These results are shown in Fig. 10. During modeling, the impact of superplasticizer on the rate of hydration is unknown; therefore, the rate of hydration is set as 0.8 times that of the original rate of hydration until the end of the dormant period. In addition, the heat ca-pacity of aggregates is set as 0.93 J/g･K in the calcula-tion. The general trend of the adiabatic curve is conse-quently reproduced.

3.2 Moisture transport and water vapor sorption Water is the key substance in the performance of concrete structures and it connects almost all the phenomena of concrete properties. In this section, the moisture transport

phenomenon is investigated. Water consumption due to cement hydration as well as environmental humidity condition causes the moisture transport in concrete member, therefore, there is a feedback system of cement hydration and moisture transport in concrete.

A general expression of moisture transport in concrete is presented in Equation (27):

,

1 w w r

eff hcp w

p p ww Kwp t x x tρ

∂ ∂ ∂∂ ∂ ⎛ ⎞= − +⎜ ⎟∂ ∂ ∂ ∂ ∂⎝ ⎠ (27)

where ww p∂ ∂ : water capacity; w: water content (g/cm3); pw: water vapor pressure (MPa); Kw: water transfer coefficient (m2/s･g/cm3･1/MPa); and wr: water production or consumption due to a hydration or dehy-dration process (g/cm3). In Equation (27), the potential of moisture transport is water vapor in the gaseous phase, and it is premised that the total pressure of gas phase is constant (i.e., 1 atm).

Fig. 7 Comparison between the experimental results and the calculated heat capacity of D-dried hardened cement paste as a function of the degree of hydration. In this calculation, the drying degree of a D-dried sample is assumed to be equivalent to that of a 105 ºC dried sample. N5010c denotes that the cement paste with a water-to-binder ratio of 0.50 cured under a 10 ºC constant temperature condition. The letter L denotes low heat Portland cement. Experimental data were obtained from Maruyama et al. (2006).

Fig. 8 Comparison of the calculated heat conductivity of sealed hardened cement paste as a function of age. Experimental data were obtained from Maruyama and Igarashi (2014).

Fig. 9 Comparison between experimental data and cal-culated heat conductivity of hardened cement paste af-fected by water content. Experimental data were ob-tained from Kishi and Maruyama (2009).


The moisture transport in a porous media is derived by pressure in general, vapor is moved by gas pressure and capillary condensed water is moved by negative pressure built by the meniscus. However, we use the diffusion equation for the moisture transport. The driving force of other water transport phenomena, such as film diffusion, is also represented by chemical potential, which has units of pressure. In the proposed model, the pressure of water vapor is considered as a representative potential for moisture transport. In the case of capillary water, the direction of moisture transport is identical to the trans-port direction of water vapor when the equilibrium be-tween capillary water and water vapor is assumed to be always satisfied. With this assumption, there is no con-tradiction with regard to the moisture transport direction by using water vapor potential. This is a very important aspect for the governing equation when there is a varia-tion in the microstructure and temperature in an ana-lytical target body.

Firstly, water vapor sorption isotherm is discussed for modeling water capacity in Equation (27). Water vapor Brunauer–Emmett–Teller (BET) surface area (SH2O) is an important parameter for composing a model of water vapor sorption isotherms. There are many discussions with regard to SH2O. Based on the hysteresis under a low RH range, Feldman and Sereda (1968) stated that water vapor adsorbed molecules which are related to the BET surface area calculation correspond to the re-hydration of cement hydrates. In addition, SH2O is affected by pre-treatment of the sample before the measurement process and by the drying history of the sample (Tomes et al. 1957). This impact of the drying process of the sample on SH2O is caused by the colloidal feature of C-S-H (Maruyama et al. 2014c).

The proposed model is based on water vapor isotherms data, which were obtained within a few days since C-S-H in the sample was expected to show a relatively small colloidal alternation during the water vapor isotherm processes.

SH2O of hardened cement paste is represented as a function of the amount of the amorphous phase (Maruyama and Igarashi 2014):

( )( )20 170 ,500 0.35H C S H C S HS Max R R− − − −= − (28)

where SH2O: water vapor BET surface area (m2/g-dried hcp) with pretreatment under a 105 ºC vacuum condition for 30 minutes; and RC-S-H: amount of amorphous phase in hardened cement paste under a 105 ºC vacuum condi-tion for 30 minutes (g/g-dried hcp), and currently, the amorphous phase amount is defined by the summation of C-S-H, C4AH13, and C4(A,F)H13.

A comparison between experimental data and results calculated using Equation (28) is shown in Fig. 11.

Water vapor isotherm of hardened cement paste is modeled based on SH2O, which confirm the following experimental characteristics: 1) long-term desorption data under a 40% RH range obtained by a gravity method is reversible (Helmuth and Turk 1967; Maruyama 2010), and it is almost identical to that of the desorption process obtained by a volumetric method; 2) the water vapor BET surface area in the adsorption process has a linear relation with that under the desorption process by a volumetric method (Maruyama and Igarashi 2011b); 3) long-term drying affects SH2O (Maruyama et al. 2014c); 4) temperature affects SH2O; and 5) a sorption amount at 98% RH in the desorption branch is affected by the drying or heating condition of a sample (Igarashi and Maruyama 2012). Background data and resultant fitted data with regard to 3) and 4) are shown in Fig. 12. The first desorption model of hardened cement paste is as follows: First desorption process:

98, 0 98,

RH100% ~ 98% :( )( 0.98) / 0.02de dew w w w h= + − − (29)

98, 40, 40,

RH98% ~ 40% :( ) ( 0.40) / 0.58de de dew w w h w= − ⋅ − +

(30)

Fig. 10 Comparison of the calculated results of the rise in adiabatic temperature of concrete. Three different tem-perature conditions at a fresh state were assumed. The experimental data were obtained from Kishi and Maekawa (1995).

Fig. 11 Calculated water vapor BET surface area of hardened cement pastes compared with experimental results. Experimental data were obtained from Maruyama and Igarashi (2014).


( )( ) 31 2

RH40% ~ 0% :ln ln( ) 10dew S L L h= ⋅ − ⋅ (31)

98, 0 ( 0.0044( 293) 1)dew w T= − − + (32)

20 ,2min min

1.44 exp( 0.017( 293))( 0.479 1.107 0.373)

de H eff hcpS S r Th h

= ⋅ ⋅ ⋅ − −⋅ − + +

(33)

where h: relative humidity (-); w: water content in hardened cement paste (g/cm3-dried hcp); w 0: water content at 100 % RH (g/cm3-dried hcp); w98,de: water content at 98% RH in the first desorption process (g/ cm3

-dried hcp); w40,de: water content at 40% RH in the first desorption process (g/ cm3 -dried hcp); Sde: water vapor BET surface area per unit volume of hardened cement for the desorption process (m2/cm3); L1, L 2: parameters for statistical thickness of adsorption (nm), and are L 1 = 0.39 and L 2 = 0.12, respectively (Maruyama and Igarashi 2011b); T: absolute temperature of hardened cement paste (K); hmin: minimum equilibrium relative humidity of hardened cement paste in its history (-); and ρeff,hcp: nominal density of hardened cement paste following vacuuming under a 105 ºC condition for 30 minutes (g/cm3).

The re-adsorption branch which corresponds to the minimum value of sorption curves is then modeled as follows: Re-adsorption branch:

,

RH40% ~ 98% :

(1 )(1 ( 1) )m BET

ad

v Ckhw S

kh C kh=

− + − (34)

98, 0 98,

RH98% ~ 100% :( )( 0.98) / 0.02ad adw w w w h= + − − (35)

20 , exp( 0.017( 293))ad H eff hcpS S r T= ⋅ ⋅ − − (36)

where vm,BET, C BET, k BET: coefficients for modified BET

theory applied by Xi et al. (1994), and vm,BET = 2.72･10-4, C BET = 18.0, and k BET = 0.83; Sad: water vapor BET surface area for the re-adsorption branch per unit volume of hardened cement paste (m2/cm3); and w98,ad : water content at 98% RH in re-adsorption branch in hardened cement paste (g/cm3-dried hcp) obtained by Equation (34), where h = 0.98.

The transition process between the first desorption branch and the re-adsorption branch is assumed to have a linear relation of water content with h, and is modeled as follows: Transition process:

( )ad or de ad or dew w w h h= + Δ − (37)

00.33w wΔ = (38)

where Δw: water content in the transition process be-tween the first desorption branch and the re-adsorption branch (g/g - dried hcp); and wad or de, h ad or de: water content and relative humidity at the point of intersection between the first desorption branch or re-adsorption branch and the transition process line.

The sorption process calculated with a randomly ap-plied relative humidity history at 293 K is shown in Fig. 13 with integration of existing data obtained by the au-thors. Meanwhile, the desorption processes at different temperatures are shown with the calculated results in Fig. 14. A satisfactory reproduction of experimental data was confirmed by these results.

The nonlinear diffusion coefficient of water transfer in concrete was first addressed by Bažant and Najjar (1972). It was confirmed by numerical analysis that a constant diffusion coefficient for moisture transport in a ce-ment-based material is unsuitable, and the diffusion coefficient as a function of water content or equilibrium relative humidity reproduces the time dependent mois-ture content change in the target specimen. Subsequently, Sakata (1983) experimentally proved that the moisture diffusion coefficient of concrete depends on water con-

(a) Relative humidity dependency (b) Temperature dependency

Fig. 12 Water vapor BET surface area of hardened cement paste, which reflects the colloidal feature of C-S-H affected by relative humidity and temperature. (a) SBET as a function of equilibrium relative humidity, and (b) SBET as a function of temperature. The data of (a) are from Maruyama et al. (2014c) and the data of (b) are from Igarashi and Maruyama (2012). The equations shown in the figures are used in Equation (33).


tent by using the Boltzmann-Matano method (Matano 1933).

In the experiment regarding the moisture transport in concrete evaluated by the Boltzmann-Matano method, water content or relative water content in concrete as a diffusion potential is mathematically identical. However, these considerations are valid only if the temperature is constant and there is no microstructural distribution. In reality, however, there is microstructural distribution and a temperature gradient; consequently, the water vapor pressure or chemical potential is better for the modeling of moisture transport. In the proposed model, as water movement from a higher temperature to a lower tem-perature is experimentally confirmed in concrete (England and Ross 1972), water vapor pressure is chosen as a potential of moisture movement. It should be noted that in a recent study, the water in hardened cement paste showed very little capillary condensation (Muller et al. 2012); therefore, the moisture movement in aged con-crete might be governed by the vapor transport.

The moisture transport coefficient in the proposed model is based on experimental data at constant tem-perature conditions (Maruyama et al. 2011a; Lin et al. 2013).

3,100 0 20/ / 10w w Ht w Sρ= ⋅ (39)

,100 2

8

1exp(7.17 )5.0 9.1 4.15

(1.0 0.1( 293.15)) 10

w ww w

K tR R

T −

= ⋅− +

⋅ + − ⋅ (40)

where tw,100: statistical thickness of adsorption in satu-rated hardened cement paste (Badmann et al. 1981) (nm); wρ : density of water (g/cm3); Rw: relative water content (w / w0); and Kw: moisture transport coefficient (m2/s･g/cm3･1/MPa). The calculated results of water content change in the specimen under different tem-perature conditions are compared with experimental data in Fig. 15.

Moisture transport under changes in microstructure

due to hydration is simulated in Fig. 16. In the experi-ment with regard to Fig. 16, ordinary Portland cement concrete whose water-to-cement ratio is 0.6 is dried just after placing. In this specimen, stainless wires were each placed at a depth of 20 mm and the specimen was split according to the measuring timings. The amount of evaporable water in the split specimens was then col-lected. The environmental conditions were 293 K and 40% RH. In the analytical simulation, after 4 hours of concrete aging, the target specimen was open to envi-ronmental conditions, and the moisture transport coeffi-cients of hardened cement paste and concrete were as-sumed to be identical (Lin et al. 2013). Both calculations reproduced the trends of experimental data, and the proposed sorption isotherm model and moisture transport coefficient was validated with limited conditions.

(a) Calculated results (b) Experimental data Fig. 13 Calculation results of the sorption model using Equations 27 - 36 and experimental data integrated from Maruyama (2010) and Maruyama and Igarashi (2010).

Fig. 14 Comparison between experimental data and calculated results of the first desorption of hardened cement paste under different temperature conditions. Experimental data were obtained from Igarashi and Maruyama (2012). The origin point of sorption was the condition at 65 ºC under N2 gas flow, and those data are re-calculated so that the origin point is under a 105 ºC vacuum condition.


3.3 Modulus of elasticity, shrinkage, and ther-mal expansion coefficient The volume change of a concrete component has a sig-nificant role on the strength of concrete (Lin et al. 2015; Maruyama et al. 2014b). The Young’s modulus and Poisson’s ratio of hardened cement paste under sealed conditions are first addressed. Many research studies exist on the relationship between microstructure and Young’s modulus (Sun et al. 2004; Haecker et al. 2005; Šmilauer and Bittnar 2006; Chamrova 2010), and some models simulate the Young’s modulus of hardened ce-ment paste by considering the phase composition and Bulk modulus of cement hydrates with the general rule of mixtures or a discrete spatial distribution such as the finite element method. In the proposed model, more simplified and empirical approaches are used for predic-tions. It has been reported that the Young’s modulus and Poisson’s ratio of hardened cement paste are a function of the gel-to-space ratio in hardened cement paste under

a 11% RH condition (Maruyama and Igarashi 2014). Based on this study, Equations (41) and (42) given below are used to predict the Young’s modulus and Poisson’s ratio:

MAX(21.0 11.2,0)p pE xρ= − (41)

( )5.40.50 0.24 exp 0.20(0.46 / )pv x= − (42)

where x: gel-to-space ratio, which was originally pro-posed by Powers (1958) (-); and Ep, vp: Young’s modulus (GPa) and Poisson’s ratio (-) of hardened cement paste, respectively.

A comparison between experimental data and calcu-lated results of the Young’s modulus and Poisson’s ratio are shown in Fig. 17 and Fig. 18, respectively. The ac-curacy of low heat Portland cement paste is worse than that of ordinary Portland cement paste; however, in general, the calculated results reproduced the trends of

Fig. 15 Comparison between calculated results and ex-perimental data of the change in water content in hard-ened cement paste specimens made with ordinary Port-land cement with w/c=0.55 under a 60% RH condition at 20 °C and 40 °C. Data were obtained from Maruyama et al. (2011b)

Fig. 16 Comparison between experimental data and calculated results of water content distribution in the concrete specimen, which was open to the environment at 293 K and 60% RH just after placing. Experimental data was obtained from Hashida et al. (1990).

Fig. 17 Comparison between experimental data and calculated Young’s modulus results of hardened cement paste.


the experimental data. The Young’s modulus of mortar or concrete might be

predicted via a model by considering the impact of an interfacial transition zone (ITZ) and damage in concrete due to an uneven volume change of cement paste, mortar, and aggregates by the homogenization theory (Nilsen and Monteiro 1993; Lutz and Monteiro 1994; Kawakami 1997; Tsukahara and Uomoto 2000). However, this topic is not investigated in this paper.

The volume change of hardened cement paste due to a change in water content is introduced in this section. The shrinkage mechanism of hardened cement paste is still under discussion, and several theories such as the surface energy theory, capillary tension theory, and disjoining (or hydration pressure) theory exist, with a combination of them having been proposed thus far (Powers 1965; Feldman and Sereda 1968; Wittmann 1968; Hansen 1987; Beaudoin et al. 2010; Maruyama 2010). In addi-tion to these theories, hardened cement paste under the first desorption process shows irreversible shrinkage due to colloidal alternation of C-S-H (Helmuth and Turk 1967; Maruyama et al. 2015). This irreversible shrinkage strain is affected by the water-to-cement ratio and the cement type, and in a range of water-to-cement ratios of ordinary strength concrete, the irreversible shrinkage exceeds more than half the total drying shrinkage (Helmuth and Turk 1967; Maruyama et al. 2014c). Therefore, the new approach is introduced for modeling the drying shrinkage of hardened cement paste as follows. It has been reported that the drying shrinkage of matured hardened cement paste under the first desorption and subsequent re-adsorption process exhibits a linear rela-tion with statistical thickness of adsorption (Maruyama 2010). This experimental phenomenon suggests that shrinkage of hardened cement paste is governed by a change in the number of water molecules adsorbed in the calcium silicate layers of C-S-H whose thickness is dy-namically varied. Based on this hypothesis, shrinkage of hardened cement paste due to drying is modeled as a

function of statistical thickness of adsorption:

( )( )( )

,

1.0 ( 0.8)1.0 exp 0.017( 293) (2.5 2.0) 1.0 (0.8 0.4)

exp 0.017( 293) ( 0.4)S T

hC T h h

T h

≥= − − − − + > >

− − ≤

⎧⎪⎨⎪⎩

(43)

320 ,/ / / 10w w H S Tt w S Cρ= ⋅ (44)

( )( )

,100,

,40 , ,40

0.00388 ( 0.4)0.00912 ( 0.4)

w wp Dsh

w w p sh

t t ht t h

εε

⎧ − ≥⎪= ⎨ − + <⎪⎩ (45)

where CS,T: factors of SH2O affected by the temperature; ε p ,Dsh: drying shrinkage strain of hardened cement paste whose original point is the saturated condition (-); ε

p,sh,40 : drying shrinkage strain of hardened cement paste at 40% RH; tw: statistical thickness of adsorption when the water content is w (nm); and tw,40: statistical thickness of adsorption at 40% RH.

Drying shrinkage strain under the first desorption process and the subsequent re-adsorption process of hardened cement paste with different water-to-cement ratios is calculated in Fig. 19, and the drying shrinkage under the first desorption process under different tem-perature conditions is calculated in Fig. 20. Both figures exhibit satisfactory results.

Autogenous shrinkage, which is derived by self-desiccation due to cement hydration, is generally believed to possess the same mechanism as that of drying shrinkage. However, the location of water consumption (i.e., the spacing between calcium layers or capillary condensed water which can exist only in early ages) may have a different impact on the shrinkage strain. In other words, the hypothesis that the shrinkage strain of hard-ened cement paste is a function of the change in inter-layer spacing of C-S-H requires the following new and additional hypothesis: the water consumption from cap-illary water due to cement hydration has a smaller impact on the shrinkage strain of hardened cement paste. The controversial nature of this hypothesis is obvious, and

Fig. 18 Comparison between experimental data and calculated results of Poisson’s ratio of hardened cement paste.


there have been no discussions thus far. However, it should be noted that drying shrinkage, which is observed normally when the RH condition is less than 70%, is rather a severe condition. The RH during self-desiccation is caused by cement hydration and is normally above 70% RH. Therefore, this new concept cannot be wholly denied due to the difference in the dominant RH range. In the proposed model, autogenous shrinkage is modeled using statistical thickness of adsorption as follows:

Min(0.008exp( / 0.9),0.00388)Ash wC t= − (46)

3, 20 ,/ / / 10p Ash Ash hyd w H S TC w S Cε ρΔ = ⋅ Δ ⋅ (47)

where CAsh: coefficient representing the impact of change in statistical thickness of adsorption on autogenous shrinkage strain; ΔεAsh,p: incremental shrinkage strain per unit time; and Δwhyd : incremental water consumption due

to cement hydration per unit time. Using Equations (46) and (47), the autogenous

shrinkage of hardened cement paste with wa-ter-to-cement ratios of 0.55 and 0.40 is calculated and compared with experimental data in Fig. 21.

Meanwhile, with regard to the volume change of mortar in concrete, fine aggregates should be taken into account. Sakaida et al. (2014) measured the shrinkage of mortar with different aggregate sizes, particle size dis-tributions, and volumetric ratios in mortar. Their results showed that the aggregate volume in mortar has the largest impact on mortar shrinkage and the impact of aggregate size and particle size distribution on mortar shrinkage are rather small. Based on these experimental results, in order to predict the volume change of mortar, the coefficient as a function of aggregate volume ratio is multiplied by shrinkage of hardened cement paste:

Fig. 19 Comparison between experimental data and calculated drying shrinkage values under the first desorption and re-adsorption branch. Data was obtained from Maruyama (2010).

Fig. 20 Comparison between experimental data and calculated results of shrinkage strain of hardened cement paste under the first desorption process with different temperatures. Experimental data was obtained from Maruyama and Kishi (2011).

Fig. 21 Comparison between experimental data and calculated autogenous shrinkage values of cement paste with different water-to-cement ratios. Data were obtained from Maruyama and Teramoto (2011). The first expan-sion within 10 hours after casting expectedly produced by ettringite formation was ignored.


( ), , , ,m sh m sh m agg p shC Vε ε= ⋅ (48)

where C m ,sh (Vm,agg): ratio of mortar shrinkage to hard-ened cement paste shrinkage affected by volumetric ratio of fine aggregates in mortar; for ordinary strength con-crete, the value is about 0.30 ~ 0.35.

Thermal expansion coefficient (TEC) of cement paste at a saturated condition exhibits a magnitude between 15 ~ 19 μ/K (Loser et al. 2010; United States Department of Transportation- Federal Highway Administration 2011; Maruyama et al. 2014), and it has been reported that the TEC of hardened cement paste depends on water content or equilibrium relative humidity. Further, in the case of relative humidity, the TEC of hardened cement paste shows a maximum value at 60 ~ 80% RH with an addi-tional 8 ~ 15 μ/K to the value at the saturated condition (Mayer 1950; Sellevold and Bjøntegaard 2004; Maruyama et al. 2009). While several explanations exist for this mechanism (Sellevold and Bjøntegaard 2004; Grasley and Lange 2007; Wyrzykowski and Lura 2013), they are not clear. Therefore, in the proposed model, the hypothesis that the mechanism of TEC is identical to that of shrinkage is used for modeling:

( ){ 1.0 exp( 0.0017) / 0.6 ( 1.0) 1.0 ( 0.4)exp( 0.0017) ( 0.4)TEC

h hC h− − ⋅ − + ≥=

− < (49)

( )2, 20 min min

1/ / / 0.479 1.10 0.373

1w TEC w HTEC

t w S h hC

ρΔ = ⋅ − + +−

⎛ ⎞⎜ ⎟⎝ ⎠

(50)

( ), , , 40, , ,00.00582TEC p eff hcp w TEC de TEC pt w wα ρ α= Δ − + (51)

where αTEC, p: TEC of cement paste (/K); CTEC: coeffi-cient of impact of statistical thickness of adsorption on TEC; Δt w,TEC : incremental statistical thickness of ad-sorption for calculation of TEC (nm); and αTEC, p,0: TEC

of hardened cement paste at the saturated condition (0.000017/K).

Calculation results of the difference in thermal ex-pansion coefficient of hardened cement paste from that of the saturated condition are plotted as a function of equilibrium relative humidity and compared with ex-perimental data in Fig. 22.

The TEC of mortar or concrete should take the TEC of aggregates and aggregate volume into account. Follow-ing the general rule of mixtures equation, equation (52), which has been validated via experiments of high-strength mortar and concrete (Teramoto et al. 2007), is applied for the present study.

,, ,

, ,

11TEC c or m

m agg m agg

TEC p TEC agg

V Vα

α α

=−

+ (52)

where αTEC, c or m: TEC of mortar or concrete (/K); Vm,agg: volumetric ratio of aggregates in concrete or mortar; and αTEC, agg: TEC of aggregate (/K). 3.4 Compressive strength of cement paste and concrete The famous relationship between the strength of a ce-ment-based material and the microstructure of a binding cement paste matrix is called the “gel-to-space ratio”, and was proposed by Powers (1958). This relationship was also confirmed by the authors’ experiment of hard-ened cement paste, and their relationships were identical among cement paste at a sealed condition with different water-to-cement ratios and degrees of hydration (Maruyama and Igarashi 2014); however, this relation-ship is not applicable to hardened cement paste under a dried condition (Maruyama et al. 2014c). It has been suggested that in a high relative humidity range (e.g., more than 40% RH at room temperature), the strength of hardened cement paste is affected by bridging of pre-cipitated cement hydrates due to drying, and increases in the volume of macropores due to compaction of colloidal C-S-H globules under drying. Further, in cases with more severe drying conditions, strengthened C-S-H due to thinning of the basal spacing of calcium silicate layers due to drying causes an increase in the strength of hardened cement paste. Based on these experimentally confirmed trends, the strength of hardened cement paste at the saturated condition was modeled as a function of the gel-to-space ratio (Equation 53) and the impact of drying was modeled by Equations (54) and (55).

( ) 3.5, ,250p Fp h Fp tF C C x= ⋅ + ⋅ (53)

2,

31.1052 1.4536 3.65082.302 ( 0.40)

Fp hC h hh h

= − +− >

(54)

( ), ,40 ,40 ,0.50 ( )Fp t w w w w lC t t t t t= ⋅ − > > (55)

where x: gel-to-space ratio of hardened cement paste at

Fig 22 Comparison between experimental data and cal-culated incremental thermal expansion coefficients of hardened cement paste as a function of relative humidity.The original base line denotes thermal expansion coeffi-cients of a saturated sample. Data were obtained from Mayer (1950).


11% RH and 293 K condition (-); CFp,h , CFp,t: factor for impact of drying on compressive strength of hardened cement paste; and tw,l: lower limit statistical thickness adsorption that corresponds to an increase in the strength of hardened cement paste due to drying (0.25 nm).

By using Equations (53)-(55), the compressive strength of hardened cement paste with different wa-ter-to-cement ratios under different temperature histories is calculated and the results are compared with experi-mental results in Fig. 23. This figure shows that trends are reproduced satisfactorily.

The compressive strength of a concrete cylinder specimen is affected by the strength of hardened cement paste, maximum aggregate size, particle size distribution of aggregates, strength and stiffness of aggregates, ITZ, and other properties related to mixture proportion and their components. The proposed model is able to repro-duce the compressive strength of hardened cement paste, but unable to predict the absolute compressive strength of a concrete cylinder specimen according to the infor-mation of mixture proportion and properties of compo-nents due to insufficient scientific knowledge and back-ground data. Therefore, the strategy utilized in this work is to use the compressive strength of the cylinder specimen at 28 days under a standard curing condition as a benchmark for predicting the concrete strength.

The mechanism concerning heating or drying on concrete compressive strength was recently studied and several data are available. The compressive strength of concrete is affected by the strength of hardened cement paste that is altered by drying and heating conditions and the damage to mortar produced around aggregates caused by a difference in volume change between the aggregates and mortar. Based on these mechanisms, the impact of heating or drying on concrete compressive strength is numerically modeled as follows:

( ), ,1.0c Fc h Fc T Fc p pF C C C F−= + + ⋅ ⋅ (56)

( ) ( ), , , , ,200 / e e eFc h c c agg c m Agg sh m sh ITZC V V−= ⋅ ⋅ − − (57)

( ) ( ), , , , ,200 /Fc T c c agg c m Agg T m TC V V ε ε−= ⋅ ⋅ − (58)

where Fc: compressive strength of cylinerical concrete specimen (MPa); CFc-p : benchmark factor Fp,28/Fc,28 from compressive strength of hardened cement paste Fp,28 to compressive strength of concrete at 28 days under a saturated condition Fc,28,; CFc,h: factor representing the impact of damage around aggregates on compressive strength of concrete caused by a difference in volume change between aggregates and mortar under drying; CFc,T: factor representing the impact of temperature on concrete strength; Vc,c-agg: unit volume of a coarse ag-gregate (cm3/cm3); Vc,m: unit volume of mortar (cm3/cm3); εAgg,sh: shrinkage of coarse aggregates as a function of relative humidity; ITZε : representative strain effect of the ITZ corresponding to the drying shrinkage strain of the mortar around aggregates (e.g., if the di-ameter of a coarse aggregate is 20 mm and the thickness of the ITZ is 5 μm, the representative strain effect of the ITZ is calculated as 250 μ.); ε Agg,T,: thermal strain of a coarse aggregate; and ε m,T : thermal strain of mortar. In the present study, shrinkage of coarse aggregates is as-sumed to have a linear relation with relative humidity.

Experimental results of the change in compressive strength of concrete during drying or heating are com-pared with calculated results using Equations (56) – (58) in Fig. 24. It should be noted that all the data are under equilibrium for each drying or heating condition; in other words, the mass change of specimens attained the ulti-mate value in each experiment. The concrete for the

Fig. 23 Comparison between experimental data and calculated results of compressive strength of hardened cement paste. Experimental data was obtained from Maruyama and Igarashi (2014).

Fig. 24 Comparison between experimental data and calculated Fc/Fc,o of mortar, concrete with limestone, and concrete with shrinking sandstone. All the specimens are under equilibrium drying conditions. The calculated parameters are listed in Table 3. The experimental data was obtained from (Maruyama et al. 2014b). The mixture proportions of concrete or mortar is the same as those shown in the reference.


experimental results used high early strength Portland cement with a water-to-cement ratio of 0.55 and with different types of coarse aggregates. The assumed values for the calculation are summarized in Table 3. The trend of the ratio of Fc (compressive strength of concrete after drying or heating) to Fco (compressive strength of con-crete under sealed conditions) is affected by the type of aggregate and these trends are reproduced by the pro-posed model. Therefore, the proposed model can evalu-ate the change in concrete strength due to drying or heating.

3.5 Validation of proposed models. Finally, for the validation of the proposed models, the concrete strength in a massive specimen, which was investigated by Shiire et al. (1987), is evaluated by using the proposed model. Prediction of concrete strength development of both standard saturated curing specimen and massive concrete specimen which exposed to the ordinary environmental condition is very challenging, because, in case of massive concrete, heat production, heat transfer, resultant thermal deformation of inside of concrete, moisture movement due to temperature gradi-ent as well as self-desiccation due to hydration and re-sultant shrinkage of mortar cause the deterioration of concrete strength. Therefore, this evaluation process is comparable to one of the validation of holistic approach and this process is significant in this paper.

In this experiment, the water-to-cement ratio of con-crete was 0.50 and moderate heat Portland cement was used. The size of the massive specimen was 1200 mm × 1500 mm × 1500 mm, and it was placed during winter. The core-drilled samples, whose dimensions were ø 100 × 200 mm, were obtained from the specimen at different concrete ages and the compressive strength was meas-ured. At the same time, from the same batch as the mas-

sive specimen, the standard curing specimens with the same dimensions as the core-drilled specimens were prepared and the development of compressive strength was recorded. The details of the massive specimen are shown in Fig. 25. The experimental data of the devel-opment of concrete strength under standard curing (293 K, submerged in lime-saturated water) and in the massive specimen are shown by the calculated results in Fig. 26. In the calculation, the mixture proportion, which was shown in the reference, is used. Regarding the mineral composition of moderate heat Portland cement, an av-erage value for commercial cement, as shown in Table 4, was used. Regarding the coarse aggregate, it was shown that sandstone is used in the reference, therefore, the value in Table 3 was used for the calculation. The tem-perature condition around the specimen was re-calculated from a maturity value listed in the literature, and the relative humidity was set as 0.60 regardless of the temperature. In addition, the value of CFc-p was 0.87. According to Fig. 25, the strength development of both the massive specimen and the standard curing specimen was reproduced by the proposed model. It is concluded that the proposed model is beneficial for discussing the long-term trend of strength change in a real structure.

4. Case studies and discussion

Structural performance evaluation and aging manage-ment based on current performance and predicted future performance are crucial problems for long-term service life of important concrete structures. We possess a seis-mic evaluation method for existing reinforced concrete buildings (Building Research Institute 2001) where the distribution of strength in a massive reinforced concrete member is not considered and there is no comprehensive rule to evaluate the strength results of core-drilled sam-ples. To discuss this issue, the strength development of massive reinforced concrete structures under one-sided heating, which is uncommon for normal residential or

Table 4 Mineral composition of moderate heat Portland cement.

Alite Belite Aluminate phase Ferrite phase Gypsum48.1% 32.8% 3.4％ 12.4％ 3.3％

Fig. 25 Schematic of the specimen used in the experiment by Shiire et al. (1987).

Table 3 Parameters of limestone concrete and sandstone concrete.

Limestone Sandstone Maximum shrinkage strain at 0% RH

0.000600 0.0

Thermal expansion coeffi-cient (/ºC)

0.000009 0.000003

εITZ 0.0 0.000400


office buildings and is a characteristic environmental condition for industrial plants, was simulated by using the model proposed in the preceding chapters.

In case of nuclear power plant, especially the case of pressurized water reactor (PWR), there is a biological shielding wall (BSW) which supports a reactor pressure vessel (RPV). This important member is one of the major one-sided heated RC member due to gamma-heating and radiation heat from a reactor core. As the heating and resultant drying causes the change in concrete strength, and one-sided heating condition promote the moisture movement in massive concrete, the strength distribution of BSW is significant for evaluating the structural per-formance of this member. In Japan, a new nuclear reactor regulation law came into force on July 8, 2013 after the lessons from the Fukushima-1 nuclear power plant ac-cident, and an approval system of extension of operation period of nuclear power plants is introduced (Nuclear Regulation Authority. 2013). Power companies are al-lowed to operate nuclear power plants for up to 40 years and after that they can apply permission to extend op-eration period just one time. In this context, the strength evaluation of one-sided heated member is crucial. In addition, this member is almost impossible to access and to take a core from the side of reactor, therefore, the evaluation process needs discussion.

In the present discussion, we also address a relaxation of the temperature regulation (American Concrete Institute 2012). In ACI-349 committee, relaxation of the concrete temperature for long-term operation has been proposed. This discussion is based on 1) a margin be-tween design strength for mixture proportion and design strength for the structure, 2) a margin probably produced by additional hydration in massive concrete, and 3) de-terioration ratio of strength due to elevated temperature derived from the concrete data base compiled by Naus (2010). But in case of Japan, required design strength has been upgraded because expected energy of coming earthquake was increased by the researchers, and resul-tantly, some margins discussed in the ACI 349 can not be

taken into account. Therefore, this discussion is neces-sary here.

A massive member, 2500 mm in width, under one-sided heating conditions was simulated. It was as-sumed that the concrete of the member contained mod-erate heat Portland cement and a water-to-cement ratio of 0.55, and the design strength was 24 MPa. The mineral composition of the cement is shown in Table 4, and the Blaine value was set as 3200 cm2/g. The mixture pro-portion of the concrete is shown in Table 5. Limestone was used as the coarse aggregate and parameters for strength impact were set from Table 3. With regard to time flow, placing was done up to 2 hours after mixing, and the member was demolded at 14 days. Then, 1 year after mixing, the member was exposed to air with 20 ºC (293 K) and 60% RH. The member then has a one-sided heating condition until 61 years after mixing. The heat-ing side environment was 45 ºC (318 K), 65 ºC (338 K), and 90 ºC (363 K) and it is exposed to ventilated air and its vapor pressure was the same as that of 60% RH at 20 ºC. The temperature of 45 ºC reflects the condition of a biological shielding wall in an anonymous Japanese nuclear power plant; meanwhile, the temperature of 65 ºC reflects the temperature regulations (ACI-ASME Joint Committee 2001); finally, the temperature of 90 ºC re-flects a recent ACI discussion with regard to the relaxa-tion of the temperature regulation (American Concrete Institute 2012).

On the other side of the member, it is assumed that the surface is exposed to the human work environment which is 20 ºC and 60% RH. Note that in the case where the member was heated at 45 ºC, there was an additional calculation where the member was open to 20 ºC and had

Fig. 26 Comparison between experimental data and the calculated temperature history of the specimen and compressive strength development of standard curing samples and core-drilled samples from the mass block. Data was obtained from Shiire et al. (1987)

Table 5 Mixture proportion of concrete.

Unit weight (kg/m3-conc.) W/C G max s/a

Water Cement Sand GravelAir

0.55 20 mm 0.42 154 280 806 1071 4.5％


an impermeable condition. The calculated results are shown in Figs. 27 - 30. Firstly, the strength development in the member is

viewed broadly. At the age of 28 days, the strength of concrete at the center part of the member is larger than

that at the periphery. This occurs as the hydration process is enhanced by the elevated temperature due to the heat of hydration. Meanwhile, the cause of a very low con-crete strength at the surface is the stagnation of cement hydration due to drying after demolding.

Fig. 27 Calculation results of transition and distribution of compressive strength, water content, degree of hydration, and temperature in a 2500 mm-thick concrete wall heated from a side with a 45 ºC condition. Its opposite side was open to 20 ºC and 60% RH.



At 182 days, it was confirmed that there is an opposite curvature of concrete strength distribution to that at 28 days, such that the strength at the center part exhibits the minimum value. The reason for this trend is due to damage around aggregates resulting from the difference in volume change between mortar and aggregates. It

should be noted that in some experiments, a similar concrete strength distribution was obtained after 13 weeks (Ozaki et al. 2001). Another explanation for this trend is stagnation of hydration at elevated temperatures related to aluminate and ferrite phases (Lee 1997); therefore, further investigation is required for a better

Fig. 29 Calculation results of transition and distribution of compressive strength, water content, degree of hydration, and temperature in a 2500 mm-thick concrete wall heated from a side with a 65 ºC condition. Its opposite side was open to 20 ºC and coated by a non-permeable resin.



understanding of this trend. The difference of concrete strength between that at the

center and that at the periphery decreases as time in-creases from 28 days to 1 year. This is caused by a counterbalance of the additional hydration process at the periphery and the stagnation process by a decrease in available water for cement hydration. In the present calculation, the compressive strength of a cylinder con-crete specimen under standard curing was 26.2 MPa and 34.3 MPa at 28 and 91 days, respectively. On the contrary, the strength of concrete at the center part of the member at 91 days was 32.1 MPa due to the impact of elevated temperature. Even if an impact of elevated temperature exists, the design strength of concrete is wholly satisfied at 91 days.

One year after mixing, one-sided heating commenced. The evaporable water at the heated side in the member was moved to both the heated surface and inside the member and the concrete at the heated periphery was dried rapidly, and this drying process made the rate of hydration slower. On the contrary, the inner part of the member whose temperature increased showed an in-crease in hydration and resultantly, an increase in strength. Some of the water from the heated side moved to areas with lower temperatures. The strength distribu-tion in the member after a 60 year service period is de-termined by a balance among the stagnation of hydration due to heating, additional increase in hydration under elevated temperatures on the condition that there is water available for cement hydration, , and the strength dete-rioration due to heating and drying. Strength develop-ment continues in areas with low temperature that re-ceive a water supply from the heated area. The strength of concrete at the side opposite to the heated side is also very low due to the stagnation of hydration due to drying. Therefore, the local maximum strength is observed 2000 mm from the heated surface. This trend is common to cases with one-sided heating and that are open to the air without an impermeable coating. If the heating tem-perature is higher, the region in which the strength of concrete is increased at the lower temperature condition becomes larger.

In cases with an impermeable coating on the surface of the low temperature side, vapor is supplied from the high temperature side to low temperature areas and the water condenses in the concrete in the low temperature area. This mechanism continues until 50 years when equilib-rium is almost attained, while complete equilibrium is no longer attained because water consumption continues as a result of cement hydration. It must be noted that re-hydration exists and a resultant strength increase is exhibited in the side that has a low temperature. There is insufficient confirmation of experimental results stating that additional hydration upon drying by supplying water increases the strength of concrete; thus, this trend re-quires further validation. The reason for the gap between 0.33 g/cm3-hcp and 0.42 g/cm3-hcp in Fig. 29 (c), which illustrates the water content, is explained by the model

that reflects the occurrence of densification and seg-mentation of C-S-H globules due to elevated tempera-tures and a resultant increase in the volume of macro-pores.

The evaluation method for integrity of the member is discussed in this section. After 28 days, the strength at the center part of the member is 28.1 MPa and exceeds the strength of concrete under standard curing (28.1 MPa) and the design strength of 24 MPa. Further, at a depth of only a few centimeters from the surface, the strength of concrete is smaller than that of standard cur-ing. Additional hydration and the resultant strength in-crease is shown in Figs. 27 - 30, and after 91 days, the strength in the member exceeds the design strength and has a 20% margin to the design strength. Therefore, in the case that the inspection of concrete strength is carried out at 28 days, a 20% margin is maintained. In ACI 349-12 (American Concrete Institute 2012), a 15% compressive strength margin to the design strength is required for relaxation of regulation of temperature from 65 ºC to 82 ºC. Therefore, the condition that the design strength is satisfied at 28 days by using moderate heat or low heat Portland cement concrete probably procures additional hydration and a 20% margin of compressive strength, and resultantly meets the ACI requirement.

Countries that frequently face earthquakes or are ex-posed to terrorist attacks may review the risk of failure of structures and require higher concrete strengths than that at the design stage. In this case, a more complicated evaluation process is required. To confirm the perform-ance of a concrete member, the strength of the concrete is directly evaluated by core-drilled samples. However, there is a possibility that a reactor, pipe, or another fa-cility is located behind the one-sided heating surface of the concrete member, and for this reason, core-drilling is impossible. In this case, a core-drilled sample will be taken from the low temperature side. It should be noted that the strength of concrete in the low temperature re-gion of the member with one-sided heating shows a continuous increase. Therefore, the core-drilled sample might show a larger compressive strength than the av-erage of the concrete member. In addition, the later this evaluation process is, the larger the difference between concrete strength in the heated region. Also, because the hydration process in the concrete in the heated region is stagnated and damage around aggregates deteriorates the concrete strength, differences in concrete strength at both surfaces become greater due to the continuous hydration process in the lower temperature surface side region. Thus, it should be noted that the core-drilled sample from the low temperature side in the member with one-sided heating might result in an overestimation. The impact of heating or drying can be obtained by additional experi-ments from core-drilled samples, but the impact due to the difference in hydration degree cannot be estimated experimentally. For this reason, the proposed numerical simulation is essential for understanding this trend.

The authors intend to investigate real structures and


members in depth to validate and enhance the proposed model in the near future. A research project that uses Hamaoka Nuclear Power Station Unit 1 is expected to contribute to this issue (Chubu Electric Power Co. 2014).

5. Summary

For the purpose of performance evaluation of an existing reinforced concrete member, a computational simulation model named the “Computational Cement-Based Mate-rial Model (CCBM)” is proposed. This numerical simu-lation framework aims to aid the evaluation of a concrete member, which has an inaccessible part and/or a large section by interpolating the distribution and development of concrete properties. This model is based on the rate of a cement hydration model, and resultant phase composi-tion and properties of hardened cement paste are pre-dicted. The virtual compressive strength of a cylinder concrete specimen under both standard curing and heat-ing and drying conditions can be simulated by consid-ering the change of the nature of hardened cement paste and the damage accumulated around coarse aggregates caused by a difference in volume change between the aggregates and mortar. Since it is difficult to predict the absolute strength value due to a lack of scientific knowledge and background data, the compressive strength under standard curing is used as a benchmark in the proposed model. Therefore, the proposed chemo-thermo-hygro modeling can evaluate the impact of temperature and drying conditions on a massive con-crete member, and based on this calculation, the integrity evaluation method of a massive concrete member under a one-sided heating condition is discussed. This discussion clearly shows the issues of integrity evaluation of a massive concrete structure with upgraded requirements caused by social needs, and also helps to identify direc-tions of future research. In conclusion, the necessity of a numerical approach is thus validated. Acknowledgements This research was supported by the Nuclear Regulation Authority (NRA, Japan), and we thank the NRA for their sponsorship. The information presented in this paper is the sole opinion of the authors and does not necessarily reflect the views of the sponsoring agencies.

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Notation: Hydration model: αi (t) Degree of hydration of each cement mineral (-),

where i: type of cement mineral (i.e., alite (nota-tion: C3S), belite (notation: C2S), aluminate phase (notation: C3A), and ferrite phase (notation: C4AF))

Ki,1, Ki,2,Ni,1,Ni,2 Coefficients for rate of hydration of each cement mineral at 293 K (-)

αi ,1, αi ,2: Degree of hydration of each cement mineral from period 1) to 2) and from period 2) to 3), re-spectively

γi,w Coefficient of stagnation of hydration as a func-tion of water content in cement paste (cm3/cm3)

γsrf Coefficient of rate of hydration due to differences in the surface area of cement particles

γa Coefficient of belite hydration, which is affected by the reaction of alite

vw,i,t Volume of water available for the hydration process of cement mineral i at age t (cm3/cm3)

vw,0 Volume of water just after mixing in the unit volume of cement paste (cm3/cm3)

vi,0 Volume of cement mineral i just after mixing in the unit volume of cement paste (cm3/cm3)

vCSH,t Volume of calcium silicate hydrate (C-S-H) at age t in the unit volume of hardened cement paste (cm3/cm3); the density and chemically bound water of C-S-H is assumed to be the value at equilibrium to 11% relative humidity (RH) at 293 K

w11％RH,t Evaporable water above 11% RH at 293 K in cement paste (g/cm3)

Vj Molar volume of substance j (cm3/mol), while j is C-S-H (notation: C-S-H) or H (H2O)

ρw Density of water (g/cm3) ρi Density of cement mineral i (g/cm3) A, A0 Blain value of the cement (cm3/g) (the reference

blain value is 3300 cm3/g) hi Coefficient representing adsorbed water which

does not contribute to the additional hydration of cement mineral i (mol/mol)

Ei Activation energy of cement mineral i (J/mol･K) R Gas constant (8.314 J/K•mol) T, T0 Absolute Temperature (K) (the reference tem-

perature is 293 K) Heat transfer: ρcon Density of concrete (g/cm3) Qr Heat production or consumption due to the reac-

tion of the concrete component (J) Wi Mass of concrete component i in unit mass of

concrete, where i = cem (unhydrated cement), hyd (cement hydrates), wat (evaporable water above 105 ºC), and agg (aggregate)

ci Heat capacity of concrete component i, where i = cem, hyd, wat, and agg (J/g･K)

λi Thermal conductivity of hardened cement paste component or concrete component i, where i = s (solid in hcp (=1.54 W/m･K)), l (evaporable wa-ter in hcp (=0.60 W/m･K)), g (gas in hcp (=0.026 W/m･K)), hcp (hardened cement paste in con-crete), agg (aggregate in concrete), and con (concrete)

Vp,i Volume of hardened cement paste component i in unit volume of hcp, i = s (solids), l (evaporable water), and g (gas) (cm3 / cm3)

Vc,i Volume of concrete component i in unit concrete volume, i = agg (aggregate) and hcp (hardened cement paste) (cm3 / cm3)

γλ (Rw) Reduction ratio of thermal conductivity of hard-ened cement paste due to drying by the aggrega-tion of colloidal C-S-H that produces large pores in hardened cement paste and resultantly de-creases the connectivity of the solid and liquid phases in hardened cement paste

w Evaporable water w (g/g-dried hcp); the equilib-rium dried state is 105 ºC below the nitrogen flow

w0 Water content at the saturated condition (g/g-dried


hcp) Rw Ratio of evaporable water w (g/g-dried hcp) to

water content at the saturated condition w0 (g/g-dried hcp)

Moisture transfer:

/ ww p∂ ∂ Water capacity wv Volumetric water content in unit hardened cement

paste volume (cm3/cm3) pw Water vapor pressure (MPa) Kw Water transfer coefficient (m2/s･g/cm3･1/MPa) wr Water production or consumption due to the hy-

dration or dehydration process (g/cm3) SH2O Water vapor BET (Brunauer–Emmett–Teller)

surface area (m2/g-dried hcp) with pretreatment under 105 ºC vacuum conditions for 30 minutes

RC-S-H Amount of amorphous phase in hardened cement paste under 105 ºC vacuum conditions for 30 minutes (g/g-dried hcp), and in the present defi-nition of amorphous phase, is the summation of C-S-H, C4AH13, and C4(A,F)H13

h Relative humidity (-) hmin Minimum equilibrium relative humidity of

hardened cement paste in its history (-) w98,de Water content at 98% RH in the first desorption

process (g/ cm3 -dried hcp) w40,de Water content at 40% RH in the first desorption

process (g/ cm3 -dried hcp) Sde Water vapor BET surface area per unit volume of

hardened cement for the desorption process (m2/cm3)

L1,L2 Parameters for statistical thickness of adsorption (nm), which equate to L1 = 0.39 and L2 = 0.12

ρeff,hcp Nominal density of hardened cement paste after vacuuming at a temperature of 105 ºC for 30 minutes (g/cm3)

vm,BET , C BET, k BET Coefficients for modified BET theory and vm,BET = 2.72･10-4, C = 18.0, and k = 0.83

Sad Water vapor BET surface area for re-adsorption branch per unit volume of hcp (m2/cm3)

w98,ad Water content at 98% RH in the re-adsorption branch in hardened cement paste (g/cm3-dried hcp) obtained by Equation (34), where h = 0.98

Δw Water content in the transition process between the first desorption branch and the re-adsorption branch (g/g - dried hcp)

wad or de, h ad or de Water content and relative humidity at the point of intersection between the first desorp-tion branch or the re-adsorption branch and the transition process line

tw,100 Statistical thickness of adsorption in saturated hardened cement paste (nm)

wρ Density of water (g/cm3) Volume change of concrete components: x Gel-to-space ratio (-) Ep Young’s modulus of hardened cement paste (GPa)

vp Poisson’s ratio of hardened cement paste (-) CS,T Factors of SH2O affected by the temperature ε p,Dsh, Drying shrinkage strain of hardened cement paste

whose original point is the saturated condition (-) ε p,Ash Autogenous shrinkage strain of hardened cement

paste whose original point has unreacted ε p,sh,40 Drying shrinkage strain of hardened cement paste

at 40% RH ε p,sh Whole shrinkage strain of hardened cement paste ε m,sh Whole shrinkage strain of mortar tw Statistical thickness of adsorption when the water

content is w (nm) tw,40 Statistical thickness of adsorption at 40% RH

(nm) CAsh Coefficient representing the impact of change in

statistical thickness of adsorption on the autoge-nous shrinkage strain

Δεp,Ash Incremental shrinkage strain per unit time Δwhyd Incremental water consumption due to cement

hydration per unit time C m ,sh (Vm,agg) Ratio of mortar shrinkage to hardened

cement paste shrinkage affected by volumetric ratio of fine aggregates in mortar, which for or-dinary strength concrete takes a value of about 0.30 ~ 0.35

Vm,agg Volumetric ratio of aggregates in concrete or mortar

αTEC, p Thermal expansion coefficient of cement paste (/K)

αTEC, p,0 Thermal expansion coefficient of hardened cement paste at the saturated condition (0.000017/K)

αTEC, c or m Thermal expansion coefficient of mortar or concrete (/K)

αTEC, agg Thermal expansion coefficient of aggregates (/K)

CTEC Coefficient of impact of statistical thickness of adsorption on the thermal expansion coefficient

Δtw,TEC Incremental statistical thickness of adsorption for the calculation of the thermal expansion coef-ficient (nm)

Strength of hcp and concrete: Fp Strength of hcp at the saturated condition (MPa) CFp,h Factor for impact of drying on the compressive

strength of hardened cement paste CFp,t Factor for impact of heating on the compressive

strength of hardened cement paste tw,l Lower limit statistical thickness adsorption that

corresponds to an increase in the strength of the hardened cement paste due to drying (0.25 nm).

Fc Compressive cylinder strength of concrete (MPa) CFc-p Benchmark factor Fp,28/Fc,28 from the compressive

strength of hardened cement paste Fp,28 to the compressive strength of concrete at 28 days under the saturated condition Fc,28

CFc,h Factor representing the damage around aggre-gates on the compressive strength of concrete


caused by a difference in the volume change be-tween aggregates and mortar from drying

CFc,T Factor representing the impact of temperature on the strength of concrete

Vc,c-agg Unit volume of coarse aggregates (cm3/cm3) Vc,m Unit volume of mortar (cm3/cm3) εAgg,sh Shrinkage of coarse aggregates as a function of

relative humidity ε ITZ Representative strain of effect of the interfacial

transition zone (ITZ) compensating the damage around the aggregates due to drying (for e.g., if the diameter of a coarse aggregate is 20 mm and the ITZ thickness is 5 μm, the representative strain of effect of the ITZ is calculated to be 250 μ.)

ε Agg,T, Thermal strain of coarse aggregates ε m,T Thermal strain of mortar

numerical approach towards aging management of concrete

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