Integrative modeling on self-desiccation and moisture diffusion in concrete based on variation of water content

Cement and Concrete Composites 97 (2019) 322–340

Contents lists available at ScienceDirect

Cement and Concrete Composites journal homepage: www.elsevier.com/locate/cemconcomp

Integrative modeling on self-desiccation and moisture diffusion in concrete based on variation of water content

T

Xiaoping Ding, Jun Zhang∗, Jiahe Wang Key Laboratory of Safety and Durability of Structual Engineering of China Education Ministry, Department of Civil Engineering, Tsinghua University, Beijing, 100084, China

ARTICLE INFO

ABSTRACT

Keywords: Concrete Self-desiccation Moisture diffusion Model

In the present paper, an integrative moisture distribution model that can catch both self-desiccation and moisture diffusion in concrete simultaneously is developed. In the modeling, self-desiccation due to cement hydration and moisture diffusion are simulated with the same relation of water content in cement paste and internal relative humidity of concrete. The above relationship of water content and internal relative humidity of concrete is developed based on Kelvin's law and capillary pore distribution in the cement paste that can be determined by test. The water content in cement paste is used as the critical parameter to predict the reduction of internal relative humidity in concrete, which may be resulted either by cement hydration or by environmental dying. Using the model, the moisture distribution of three concrete, C30, C50 and C80 that represent low, middle and high strength concrete in practice, is simulated respectively. The effect of curing condition on the development of internal relative humidity inside of concrete is analyzed. The model can be used for moisture field calculation of concrete elements under any given curing strategy, such as sealing and drying strategy, as well as drying conditions.

1. Introduction It is well understood that shrinkage of concrete is closely linked to the loss of water in concrete [1–5]. The content of water in concrete differs from surface to center and moisture gradient exists before the hygrometric equilibrium with the surroundings is reached. Therefore, shrinkage strain develops near the drying surface much faster than that in the center of a concrete element [5,6]. The evaluation of the shrinkage induced stress in the structure requires the knowledge of the distribution of shrinkage deformation, which, in turn needs the information of water, or called moisture, distribution first. In addition, moisture content of pores directly affects strength, thermal and creep properties, as well as the rate of cement hydration. Therefore, the moisture content and its distribution inside of concrete, especially in early-ages, are critically needed to calculate the shrinkage induced stresses, creep and deflection in concrete structure, and further to predict the formation of cracks, durability and the service-life of the structures. Regarding the moisture distribution in concrete, a number of references are found in literature. Andrade et al. measured internal relative humidity and temperature of matured concrete cylinders exposed to the outdoor climate [7]. Parrott [8] and Nilsson [9] conducted experimental studies on the internal relative humidity of matured ∗

concrete specimens exposed to natural weathering or in contact with seawater. Recent years, Huang, Qi and Zhang [10] experimentally investigated the development of relative humidity inside of concrete at early-age. Based on the experimental findings, a mathematical modeling on moisture distribution and its variation with time was carried by Zhang et al. [11]. In this simulation, constant moisture capacity (slope of moisture content and relative humidity curve) of concrete was assumed and this assumption may induce some error on relatively low strength concrete and/or high water to cement ratio concrete. This is because obviously nonlinear performance on sorption isotherm in the high humidity zone was displayed on these concrete [9,12]. Meanwhile, a fit based model to calculate the moisture reduction due to cement hydration was used in the simulation. The self-desiccation model used in the simulation may still lack universality, because the model can apply only to a fixed strategy of drying. More recently, an improved model for moisture distribution of concrete is developed, in which selfdesiccation due to cement hydration and non-linear moisture capacity of cementitious materials are considered [13]. However, a fit based selfdesiccation model was still used in the simulation that makes the model can only be used in some specific situations. Therefore, more universal or mechanism based model for prediction of moisture distribution of concrete, especially for early-age concrete, is still needed.

Corresponding author. E-mail address: [email protected] (J. Zhang).

https://doi.org/10.1016/j.cemconcomp.2019.01.008 Received 19 August 2018; Received in revised form 2 January 2019; Accepted 14 January 2019 Available online 18 January 2019 0958-9465/ © 2019 Elsevier Ltd. All rights reserved.

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

The aim of this paper is to establish an integrative moisture distribution model that can catch both self-desiccation and moisture diffusion in concrete simultaneously. The model should be universal, and can be used in any given strategy of environmental drying. In the modeling, the two critical issues, self-desiccation due to cement hydration and moisture diffusion are simulated with the same relationship between water content and internal relative humidity of concrete, which is normally called moisture capacity or adsorption-desorption law. The constitutive law of water content and internal relative humidity of cement paste is developed based on Kelvin's law and capillary pore distribution that can be determined by test. The water content in cement paste is used as the critical parameter to predict the reduction of internal relative humidity in concrete, which may be resulted either by cement hydration or by environmental dying. By using water content as the linking parameter of moisture diffusion and self-desiccation, the developed model can be used for moisture field calculation of concrete elements under any given curing strategy, such as sealing and drying strategy, as well as drying conditions. Using the model, the moisture distribution of three concrete, C30, C50 and C80 that represent low, middle and high strength concrete in practice, is simulated respectively. The effect of curing condition on the development of internal relative humidity inside of concrete is analyzed.

Data Collector

Temperature & Humidity Transmitter Sealant 1

Plastic Pipe

2

Sealing Ring

3

PMMA Mould Unit: mm

50

dry

50

100

20

Concrete

50

50

350 (a) Data Collector Temperature & Humidity Transmitter

Sealant 1

Plastic Pipe

3

Sealing Ring

PMMA Mould Unit: mm

50

seal

2. Experimental findings on self-desiccation and moisture diffusion in concrete

50

100

20

Concrete

100

350

Fig. 1. Schematic diagram of concrete humidity measurement set-up, (a) drying (b) Sealing.

Recently, Zhang et al. [13] carried out series tests on moisture movement under drying and sealed conditions of concrete by measuring internal relative humidity at selected locations in the sample. In their tests, a mould with the inner dimension of 100×100×400 mm and a removable plastic filler of 100×100×50 mm in size was used to cast the specimen. Therefore, the actual dimension of the specimen was 100×100×350 mm. For each concrete, two parallel specimens were cast at the same time. One used for drying test and the other used as a reference specimen under complete plastic film sealing. In drying specimen, three humidity sensors were used to measure the relative humidity and temperature, which were located along the center axial of long direction of the specimen and at the places from the drying surface of 20 mm, 70 mm and 120 mm respectively. For sealed specimen, only two sensors were used, which were located at places from the long-end of the specimen 20 mm and 120 mm respectively. The locations of the sensor in drying and sealed specimens, as well as the dimension of the specimen and mould, are shown in Fig. 1a and b respectively. In the tests, resistance based digital sensor was used for relative humidity and temperature measurements. The accuracy of the relative humidity and temperature measurements is 1–1.5% and 0.1 °C respectively. In their tests, three concrete mixtures with water to binder ratio of 0.62, 0.43 and 0.30, representing low (C30), middle (C50) and high strength (C80) concrete in practice respectively, were used. Portland cement was used as cementing binder. Fly ash and silica fume were used respectively in C30, C50 and C80 concrete. The chemical compositions of cement, fly ash and silica fume, as well as the mineral compounds of the cement are listed in Table 1. Natural sand and crushed limestone with a maximum particle size of 5 mm and 20 mm, respectively, were used as normal fine and coarse aggregates. A superplasticizing admixture was used in the mixtures to ensure all the concrete has a comparable slump in 80–100 mm. Concrete mix proportions and compressive strength at 28 days are given in Table 2. Fig. 2 displays the test results of the development of internal relative humidity with ages since concrete cast under plastic film sealed and drying conditions of the three concrete. In these figures, the

temperature measured by the sensor is present as well. From the test results, first we can observe that the progress of the internal relative humidity of concrete since cast can be described by a water vapor saturated stage (relative humidity H = 1, stage I) followed by a stage in which the relative humidity is decreasing with age gradually (relative humidity H < 1, stage II) [5,10,11]. The length of the stage I is significantly influenced by the strength and/or water to binder ratio of concrete. The lower the water to binder ratio, the shorter the length of stage I should be present. The length of moisture saturated stage measured from sealed tests is 336.5 h, 177.4 h and 62.4 h for C30, C50 and C80 concrete respectively. Second, a similar variation trend is observed on the two sensors under sealing status, which suggests that a uniform humidity reduction is resulted by cement hydration throughout the specimen. The lower the water to binder ratio, the higher the humidity reduction due to cement hydration is produced. In the present study, the internal relative humidity at 60 days since casting is 0.93, 0.85 and 0.76 respectively for C30, C50 and C80 respectively under sealed condition. As one-face drying starts, obvious humidity gradient along the drying direction is observed in the three kinds of concrete. Under one face drying, the internal relative humidity of the position of 20 mm from the drying surface at 60 days since casting is 0.76, 0.72 and 0.64 respectively for C30, C50 and C80 respectively. Previous studies have provided explanations on the mechanism of the above findings on the progress of internal relative humidity of concrete under sealed or drying [13]. It may be summarized as the following points: (1) After mixing of all consists of concrete, the solid cementitious particles are covered with a layer of water and formed flowable cement paste that endues fresh concrete with fluidity. Within an initial short period after concrete cast, the voids of the solid skeleton are filled with liquid water and cement particles are connected to each other through the surface water-layer. In this stage, a continuous liquid network is existed in the paste, and such continuity results in the relative humidity inside of concrete is equal or close to 1 theoretically

323

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

Table 1 Oxide of cement, silica fume, fly ash and compound compositions of cement. Oxide

Cement

CaO SiO2 Al2O3 Fe2O3 MgO K2O Na2O SO3 Loss Density (g·cm−3)

Silica Fume

Fly Ash

Percentage

Compound

Percentage

Percentage

Percentage

63.91 21.08 4.55 3.95 1.16 0.73 0.58 2.06 1.98 3.13

C3S C2S C3A C4AF CS – – – –

58.01 16.80 5.35 12.09 4.42 – – – –

0.36 94.56 0.33 0.76 0.85 0.28 – – 1.31 2.22

3.89 57.93 21.31 7.68 2.16 1.53 1.58 – 1.87 2.20

(Stage I), even the salt content in concrete may lead the internal relative humidity dropping little from 1, which normally depends on the salt content in pore water [14]. In the present study, the effect of salt content in the water on saturated humidity is neglected. (2) With progress of cement hydration, the cement particles in the skeleton are gradually enlarged and the neighboring cement particles gradually contact each other through breaking part of the surface water-layer. As the connection between solid particles is sufficiently large to support concrete self-weight, setting of concrete occurs. It should be noted that the partial connection of hydrated cement particles at concrete set does not mean the continuity of liquid water in the network is already broken down. Therefore, internal relative humidity is still close to 1 at the moment of the concrete set. It may well explain why the time of concrete set is ahead of the point of relative humidity starting to drop from the saturated value [15]. In stage I, cement hydration resulted chemical shrinkage takes place continuingly and it should partially transfer into macro shrinkage of concrete due to the self-restraint of hardened concrete. (3) With persistent hydration of cement, the continuity of the pore water is finally broken and the pore water is isolated into individual pocket lakes without direct connection between each others. Meanwhile, cement hydration resulted chemical shrinkage and moisture diffusion resulted water loss takes place continuously. To compensate volume reduction in the cement paste, some empty capillary pores with curved liquid/vapor interface are gradually formed within the pore water. The formation of curved liquid/vapor interface should lead the decrease of internal relative humidity of concrete. Thus, the moment when the formation of empty capillary pore is the time when the internal relative humidity of concrete starts to drop from 1. After this moment, the advance of internal relative humidity goes into the humidity reduction stage (Stage II). With increase of pore water loss resulting by either cement hydration or moisture diffusion, the meniscus radius of the empty capillary pores is decreased, and the relative humidity of concrete decreases as well until the water loss is stopped.

continuity of the pore water is finally broken down. The chemical shrinkage of cement hydration and diffusion of water will lead to the formation of empty capillary pores, and simultaneously lead to the decrease of internal relative humidity of concrete (stage II). The diffusion of water vapor through continuous vapor space becomes the dominant mechanism for moisture transfer at this stage in concrete. The transformation point from stage I to stage II is controlled by the connectivity of water in the capillary pores in the matrix, which can be quantitatively determined by water content at the end of Stage I, be called critical water content, Wc. Parameter Wc should be a function of water to binder ratio of concrete, and can be calculated from Powers volumetric model [12,16] as long as the cement hydration degree is known. 3.1. Water consumption due to cement hydration The progress of cement hydration in concrete can be monitored by adiabatic test [11,17]. Through measuring the adiabatic temperature rise of concrete, the cement hydration degree can be estimated by:

=

Tad (t ) Tad ( )

(1)

Where Tad (t) is the adiabatic temperature rising at time t. Tad(∞) is the adiabatic temperature rise as the cement is completely hydrated that can be estimate according to cement composition and mixture proportion of concrete. Details for determination of Tad(∞) of concrete used in the present study can be found in author's previous publication [17]. Thus, cement hydration degree in concrete under adiabatic temperature condition can be determined. However, the temperature history for a given position in concrete structures is varied and that should be different to the adiabatic condition. To calculate the hydration degree under different temperature history, a concept called equivalent age is used [11,18]. The equivalent age concept assumes that samples of a concrete mixture of the same equivalent age will have the same mechanical properties or cement hydration degree, regardless of the combination of time and temperature yielding the equivalent age. Based on the above definition, the equivalent age te can be expressed as

3. Modeling on self-desiccation and moisture diffusion based on variation of water content As described above, the loss of water in early-age concrete is normally caused by either cement hydration or water diffusion. At the initial period after the concrete cast, the water in concrete pores are connected with each others and the relative humidity in concrete is almost equal to 1 (Stage I). With persistent hydration of cement, the

t

te =

exp 0

1 Uar Rg 293

UaT 273 + T

dt

(2)

Where te is the equivalent age at the reference temperature (here the

Table 2 Mix proportions (kg/m3) and compressive strength of the three concrete. No.

Water/binder ratio

Cement

Water

Sand

Stone

Fly ash

Silica Fume

Compressive strength at 28d (MPa)

C30-OC C50-OC C80-OC

0.62 0.43 0.30

240 345 450

186 185 150

750 685 580

1150 1090 1140

60 85 –

– – 50

34.1 54.7 84.3

324

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al. 100

50

45

80

40

40

35

30

20

Relative Humidity

60

20

40

40

35

40 30 20

20 0

RH-C30-1 RH-C30-3 Temperature-C30-1 Temperature-C30-3 Environmental humidity

60

25

0

45

Seal-C30

25

0

60

Temperature (OC)

RH-C30-1 RH-C30-2 RH-C30-3 Temperature-C30-1 Temperature-C30-2 Temperature-C30-3 Environmental humidity

50

80

Dry-C30

Temperature (OC)

Relative Humidity

100

20 0

20

Time (days)

40

60

Time (days)

(a) 100

50

45

80

40

40

35

30

20

Relative Humidity

60

40

20

40

40

35

30

25

0

20 0

RH-C50-1 RH-C50-3 Temperature-C50-1 Temperature-C50-3 Environmental humidity

60

20

25

0

45

Seal-C50

20 0

60

Temperature (OC)

RH-C50-1 RH-C50-2 RH-C50-3 Temperature-C50-1 Temperature-C50-2 Temperature-C50-3 Environmental humidity

50

80

Dry-C50

Temperature (OC)

Relative Humidity

100

20

40

60

Time (days)

Time (days)

(b) 100

50

45

80

40

40

35

30

20

Relative Humidity

60

Seal-C80

40

20

40

35

30

20 0

40

RH-C80-1 RH-C80-3 Temperature-C80-1 Temperature-C80-3 Environmental humidity

60

20

25

0

45

25

0

60

Temperature (OC)

RH-C80-1 RH-C80-2 RH-C80-3 Temperature-C80-1 Temperature-C80-2 Temperature-C80-3 Environmental humidity

50

80

Dry-C80

Temperature (OC)

Relative Humidity

100

20 0

20

Time (days)

40

60

Time (days)

(c) Fig. 2. Development of internal relative humidity of concrete under drying and sealing conditions, (a) C30, (b) C50, (c) C80.

reference temperature is equal to 20 °C is assumed). Uar and UaT are the apparent activation energy (J·mol−1) at reference and actual temperature respectively. Rg is the universal gas constant, 8.314J·mol−1·K−1. T is temperature in Celsius (oC). Regarding apparent activation energy, a number of researchers have concluded that it could not be considered as a constant independent of time except during the beginning of cement hydration [19,20]. Based on these findings, the apparent activation energy of concrete is expressed as a function of temperature and curing time as [21]:

Ua = (42830

43T )exp( 0.00017Tt )

Where A, B and t0 are three empirical constants, which can be determined by fitting adiabatic test results, equation (1), and the model result, equation (4). αu is the ultimate degree of cement hydration for given concrete mixture, which is normally a function of water to cement ratio (w/c) for pure Portland cement concrete as [18]: u

u

exp

A te + t 0

1.031w/ c 0.194 + w / c

(5)

For concrete with fly ash or silica fume addition, the value of αu needs to be revised due to their filling action and pozzolanic activity. Schindler et al. [22] investigated the effect of fly ash on the ultimate degree of cement hydration by experiments. Their results show that the effect of fly ash on αu may behave as the reduction on the thickness of the water layer formed at the surface of cement particles on one hand, which should reduce cement hydration degree, and increasing long term cement hydration degree due to its pozzolanic activity on the other hand. Therefore, concrete with fly ash addition, the cement

(3)

Where T is curing temperature (oC) and t is curing time in days. Based on the equivalent age, the hydration degree of cement defined in equation (1) can be simulated by Refs. [11,17]:

=

=

B

(4)

325

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

hydration degree αu is revised as [22]: u

1.031w /(c + f ) + 0.50f /(c + f ) = 0.194 + w/(c + f )

respectively in the paste that can be expressed as:

1.0

Vw =

(6)

Where f is mass content of fly ash. For concrete with silica fume addition, such as C80 series in the present study, the effect of silica fume on the ultimate degree of cement hydration is preliminary displayed as the action of reducing the thickness of the water layer at surface of cement particles due to its extremely high fineness. For example, experimental result of Luzio et al. [23] shows that the addition of silica fume reduces the ultimate cement hydration degree of cement. Therefore, for silica fume added concrete (mass of silica fume less than 10% of the total mass of cementitious materials), the ultimate cement hydration degree αu is revised as: u

1.031w/(c + s ) = 0.194 + w /(c + s )

Vc = 1

Vc

c w

0.401

w/c +

Let p =

w / f )(f / c )

(10)

w /c c + (

w/

1 1 + ( c / f )(f /c )

w / f )(f / c )

w/c , w / c + w / c + ( w / c )(f / c )

1

k= 1+

c / f (f / c )

(11)

. Using (10) and

(11) in (9), we can obtain the capillary water as:

Vcw = p

0.401k (1

p)

c

(12)

w

(7)

Similarly, the chemical shrinkage of cement hydration, Vcs can be calculated by:

Vcs = 0.056 c k (1

(13)

p)

It is understood that in moisture saturated stage (Stage I), no empty capillary pores are formed, and the formation of empty capillary pores is occurred in moisture decreasing stage (Stage II). The chemical shrinkage occurred in the moisture saturated stage is directly transferred into macro shrinkage of the cement paste. While, the chemical shrinkage occurred in the moisture decreasing stage is transferred into empty capillary pores. Assuming the cement hydration degree as internal relative humidity stats to dropping from moisture saturated value is αc that can be determined from humidity test on sealed specimen, the volume of the empty capillary pores resulting from chemical shrinkage, Vcse can be expressed as:

(8)

Where wt is the weight of chemical and gel water used for per gram of completely hydrated cement. x is the weight ratio of each chemical phase of cement. According the composition of the cement list in Table 1, the chemical and gel water used for per gram of hydrated cement is 0.401 g. Meanwhile, according to the composition of the cement used in the present study, and test results of Powers et al. [12] and Brouwers [24], the chemical shrinkage of the cement used in the present study is between 0.055 and 0.057 cm3 for per gram cement reacted. In addition, Jensen's study showed silica fume in concrete will joint the hydration as well even in early age [16]. The experimental results showed that the contribution of silica fume to C-S-H gel water is 0.50 g for per gram of reacted silica fume, while without contribution to chemically bound water of hydrated products. Contribution of silica fume to chemical shrinkage of cement paste is 0.22 cm3 for per gram of reacted silica fume. In the present study, the composition of cement paste of C30 and C50 concrete is water, cement and fly ash. For C80 concrete, the composition of cement paste is water, cement and silica fume. In the following, volumetric model will be used to derive the content of capillary water and capillary porosity of the cement paste used in the study. For C30, C50 concrete, assuming the fly ash does not joint the reaction in early age, such as less than 60 days that is used in the present test. Certainly, the addition of fly ash may influence the hydration of cement that is reflected in the cement hydration model described in the last section of the paper. Assuming the total volume of water, cement and fly ash in the paste is 1 cm3. The weight of water, cement and fly ash in the paste (1 cm3) is w, c and f (in gram). Density of water, cement and fly ash is ρw, ρc and ρf (in g·cm−3). Referencing Jensen's derivation [16], for given cement hydration degree, α, the volume of capillary water in the cement paste, Vcw can be calculated by subtracting the water used for cement hydration from the initial water content as:

Vcw = Vw

w/c +(

w/ c

And

The water consumption due to cement hydration can be estimated through Powers' volumetric model [12]. Power and Brownyard's initial measurements [12], and Brouwer's later measurements [24], showed that chemical and C-S-H gel water for per gram of cement reacted during hydration are almost constant and can be related to cement composition as:

wt = 0.334x C3 S + 0.374x C2 S + 1.410x C3 A + 0.471x C4 AF + 0.261x CS

w/c +

Vcse =

0 for 0.056 c k (1

c

p)(

c)

for

>

(14)

c

The total capillary pore volume, Vp can then be obtained by summing (12) and (14). For concrete with silica fume addition, such as C80 concrete used in the present work, similar derivation can be done just replacing fly ash with silica fume (weight fraction s, density ρs), meanwhile taking the reaction of silica fume into consideration. If assuming the silica fume reacts proportionally to the cement [16], the volume of capillary pore water per 1 cm3 cement paste, Vcw can be expressed as:

Vcw = p

0.401 + 0.50

s k (1 c

p)

c

(15)

w

Where p = w / c + / + ( / )(s / c ) , k = 1 + ( / )(s / c ) . The volume of w c w c c s the empty capillary pores resulting from chemical shrinkage of reaction of cement and silica fume, Vcse can be calculated by: w/c

Vcse =

0 for

1

c

(0.056 + 0.22 ) s c

c k (1

p)(

c)

for

>

c

(16)

Thus, for a given cement hydration degree, the amount of capillary water can then be calculated from (12) and (15) respectively for concrete with fly ash (C30, C50) and with silica fume (C80) addition. Meanwhile, the total volume of capillary pores can be obtained by summing (12) and (14), (15) and (16) respectively for fly ash concrete and silica fume concrete.

(9)

Where Vw and Vc are the initial volume fraction of water and cement

326

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

3.2. Water loss due to diffusion

through concrete are so slow that various phases of water in each pore (vapor, capillary water and absorbed water) remain almost in thermodynamic equilibrium at any time [29]. A number of experimental studies [9,11,13,29,30] have shown that the magnitude of moisture diffusivity strongly depends on the pore humidity. In the present work, humidity dependent diffusivity is modeled by multiple-linear functions, as showed in Fig. 3. The continuous relationship between diffusion coefficient and pore humidity is divided into N sections according to humidity values. In each section, the coefficient is modeled by a linear function as

3.2.1. Water loss on drying surface In the present experimental configuration, the moisture transport through the sample is one-dimensional. On the drying surface, the gradient of vapor pressure between the top of the drying material and the ambient will drive the water to evaporate into the environment. The mass flux, J (kg/m2s) of water vapor at the drying surface can be expressed as:

J = am (H1

(17)

He )

Dj = kj H + dj

Where H1 and He are pore relative humidity of the drying surface of the material and the ambient respectively. am is called surface factor (kg·m−2·s−1) that is normally governed by the environmental conditions and surface status of the drying materials [25,26]. Recently, Zhang et al. [27] made a comprehensive study on am by weighting the weight of concrete samples under drying. It is concluded that am is a function of air flow rate, air temperature and concrete element size along air flow direction. am can be expressed as:

am = 0.380e17.4 ln(Ta) am = e17.4 ln(Ta)

107.6

Va L

0.338L 1)

for Rel

Rec

2H wd = D 2d + k t x

r=

t

=

x

D

(H

Hs ) x

(20)

j = 1 to N

Hd x

2

(21)

2 w Mw cos w RT

1 ln H

(22) −1

Where Mw is the molar weight of water (0.01802 kg mol ), ρw is the density of water (kg·m−3) and R is ideal gas constant (8.314 J mol−1 K−1). θ is the contact angle between water and hydration products of cement and normally θ ≈ 0. On the other hand, the water content in the pores for per unit volume of cement paste (1 cm3) with radius from zero to the specific value of rc, that can be related to internal relative humidity H by equation (22), can be expressed as:

3.2.2. Water loss inside of concrete In stage II, the moisture is transported through the water vapor and the liquid in pores acts mainly as a source for water vapor generation. Thus, the driving force of water diffusion in concrete should be capillary potential that can directly relate to local relative humidity (H) of the concrete [29,30]. If one-dimensional water diffusion is considered (along the coordinate direction x), according to the second Fick's law, the moisture content balance requires:

ws )

H < Hj + 1

For given water loss Δws, the corresponding humidity reduction, ΔHs depends on the relationship between them, which is called desorption isotherms [11. 29]. In the present work, above relationship is developed with capillary theory and pore structure of cement paste. In moisture progressing stage II, from Kelvin's law, the relationship between radius of meniscus r (m) and relative humidity H, surface tension of water γw (N/m) and absolute temperature T (K) can be expressed as:

(18)

Where Va is the air flow rate (m·s−1), Ta is air temperature (K), L is the length (m) of the drying element along air flow direction. Rec is the Reynolds number at which the transition from the laminar to the turbulent flow taking place. Generally, the value of Rec varies with experimental conditions and in general condition, Rec = 1.00 × 105 [28]. Rel is Reynolds number of the drying element and can be calculated by VaLρ/μ. Here ρ and μ are density and dynamic viscosity respectively of the flow humid air, and ρ = 1.212 kg m−3 and −5 −1 −1 μ=1.81 × 10 kg m s .

(w

Hj

Where kj and dj are the constants that can be determined from experimental results. As long as N is sufficiently large, the variations of the diffusion coefficient with humidity can well be simulated. Using (20) in (19), and let Hd=H−ΔHs, wd=w−Δws, equation (19) can be rewritten as:

for Rel < Rec

107.6 (0.582V 0.8L 0.2 a

for

rc

w=

w Vp

v (r ) dr

(23)

0

Where Vp is the total volume of capillary pores of cement paste that can be obtained by summing equations (12) and (14) or (15) and (16) respectively for concrete with fly ash and silica fume. v(r) is pore volume density function of cement paste. Accumulated pore volume of cement paste can be obtained from mercury penetration experiments on cement paste. The pore volume density of cement paste can well be simulated by Raleigh distribution function as [11]:

(19)

Where, w is water content and Δws is water consumed by cement hydration. D is the moisture diffusion coefficient depending on the pore humidity and on the composition of concrete. ΔHs is the humidity reduction due to cement hydration. Both cement hydration and diffusion

rc

v (r ) dr = 1 0

exp(

rc )

(24)

Where β is a parameter reflecting the influence of cement hydration that can be reflected by cement hydration degree, α. It can be simulated as β = μeλα, μ and λ are two experimental determined constants. Using equations (22) and (24) in (23), the relationship between water content and internal relative humidity in stage II can be obtained as:

Fig. 3. The simulation of humidity dependent diffusivity by multiple-linear functions.

Fig. 4. Discrete node elements of concrete suffering to one-face drying. 327

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

Fig. 5. Types of nodes used in humidity field analysis.

w w Vp

=1

exp

µ exp(

2 w Mw

)

w RT

1 ln H

(25)

Thus, as long as the cement hydration degree is know, water consumption due to cement hydration resulted humidity reduction can be calculated by equation (25). After that, water loss due to diffusion can be solved. If the water content after cement hydration and moisture diffusion is known, the final local relative humidity can then be calculated as well with (25). As a calculation example, one end of the concrete element suffers to dry, as showed in the test program, is assumed. Along the drying direction, the specimen is divided into n discrete nodes with a constant node distance h, see Fig. 4. In the present work, finite differential method was used to solve equation (21) numerically. In the mesh, there are three kinds of nodes, i.e. the surface node, internal node and the bottom node, which are displayed in Fig. 5. To the exposed surface node, the water is transferred directly into the environment through the interfacial layer. If the environmental humidity, He and the surface factor, am that can be calculated by (18) are known, the variation of water content in the surface node element within the time interval, Δt since time t due to diffusion can be obtained by:

W1d, t +

t

=

am (H1,s t +

D (H1,s t + t )(H1,s t +

He)

t

t

H2,s t + t )/ h

h/2

t

(26)

Where is the humidity value of node i after suffering hydration period of Δt from the initial time t. For internal node element, from (21), the variation of water content within the time interval, Δt since t due to diffusion can be expressed as:

His, t + t

Wid, t +

= D (His, t + t )

t

+ ki

His

2His, t +

His

t

1, t + t h2

His+ 1, t +

1, t + t

t

His+ 1, t +

t

Fig. 6. Flow chart of calculation of internal relative humidity of concrete.

t

Using the developed model, the complete humidity field in concrete since casting, coupling with both cement hydration and environmental drying, can be obtained according to the following algorithm by stepby-step integration in time. For given time increment (Δt) from initial time t, consumed water due to cement hydration at each node element, Δwis,t+Δt is calculated first from (12) or (15). The current water content, wi,t-Δwis,t+Δt is compared to the critical water content, Wc. If wi,tΔwis,t+Δt > Wc, the local relative humidity is set to 1. Otherwise, the corresponding humidity reduction due to cement hydration, ΔHis,t+Δt is calculated using (25) and the current humidity value, His,t-ΔHis,t+Δt, is updated. Then using His,t-ΔHis,t+Δt as the initial humidity condition, the water loss, ΔWid,t+Δt due to surface drying and diffusion after the time interval Δt since t of each node element is calculated using equations (26)-(28) respectively. The current water content, wis,t+Δt-Δwid,t+Δt of each node element is updated and compared to the critical water content again. If wis,t+Δt-Δwid,t+Δt > Wc, the local relative humidity is set to 1. Otherwise, the local relative humidity is calculated with (25).

2

(27)

2h

Further, assuming there is a node, n+1 outside of another end node, n and Hn+1 = Hn, the variation of water content for node element n can be calculated by:

Wnd, t +

t

= D (Hns, t + t ) + kn

Hns

Hns, t +

t

Hns

1, t + t

h2 1, t + t

2h

Hns, t +

t

t

2

(28)

Thus, the water content in each node element can be updated with (26) to (28) respectively in each time step and the corresponding internal relative humidity can be calculated from equation (25) after diffusion.

328

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

Table 3 Input parameters for humidity field calculation. No.

Cement hydration parameter A

B

t0(hours)

μ

λ

Wc(g·cm-3)

am ( × 10−5) (kg·m−2·s−1)

He

0.8853 0.8106 0.6014

30.58 33.29 36.77

1.17 1.03 0.99

10.00 10.80 11.60

0.00027 0.0019 0.0090

8.30 7.10 5.87

0.3601 0.2566 0.1895

3.75 3.75 3.75

0.3 0.3 0.3

1.0

100

0.8

80

0.6

0.4 C30 C50 C80 Model Result

0.2

60 C30 1d 3d 7d 28d 60d 90d Model

40

20

0 1

0.0 0

Environmental parameters

αu

Density of pore volume (%)

Degree of hydration

C30-OC C50-OC C80-OC

Pore structure parameter

100

200

300

400

10

500

Equivalent Age (hours)

100 1000 10000 100000 1000000 Pore radius (nm) (a)

100

(a)

Density of pore volume (%)

1.0

Degree of hydration

0.8

0.6

0.4 C30 C50 C80 Model Result

0.2

80

60 C50 1d 3d 7d 28d 60d 90d Model

40

20

0 1

10

0.0 0

500

1000

1500

2000

2500

100 1000 10000 100000 1000000 Pore radius (nm) (b)

3000

Equivalent Age (hours)

100

(b)

Density of pore volume (%)

Fig. 7. Degree of cement hydration versus equivalent age diagrams of three concrete, (a) results of the initial 500 h, (b) overall results.

Afterwards, the calculation goes back to the next time step until the time is approaching the expected age. The calculation flow chart of the above process is illustrated in Fig. 6. 4. Model results and analyze 4.1. Parameters for model inputs The related parameters used in the moisture field calculation of C30, C50 and C80 concrete are listed in Table 3. The cement hydration parameters of each concrete were obtained by adiabatic tests [17]. The pore distribution parameters were obtained by mercury penetration tests on the cement paste. Fig. 7 displays the model and test results of

80

60 C80 1d 3d 7d 28d 60d 90d Model

40

20

0 1

10

100 1000 10000 100000 1000000 Pore radius (nm) (c)

Fig. 8. Model and test results of pore volume density of C30 (a), C50 (b) and C80 (c) concrete at selected age. 329

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

1.0

Normalized water content

C30: 90, 60, 28, 7, 3, 1 days 0.8

0.6

0.4

0.2

0.0 0

20

40

60

80

100

80

100

80

100

Relative humidity (%) (a)

Fig. 9. Capillary pore volume and age diagrams of C30, C50 and C80 concrete.

1.0

cement hydration degree of the three concrete in terms of hydration degree and equivalent age diagrams, in which (a) displays the results of the initial 500 h and (b) displays the overall results. From Fig. 7, it can be seen that the cement hydration degree model can well catch the progress of cement hydration in concrete. The critical water content, Wc (g·cm−3) of each concrete was calculated with equation (12) or (15) respectively for C30, C50 and C80 concrete according to the time of internal relative humidity starting to drop from the initial saturated value, and the corresponding cement hydration degree. It can be found that the Wc of concrete is almost proportionally decreased with the decrease of water to binder ratio. The value of Wc may indirectly reflect the connectivity of capillary pore in the paste and it will be used as a criterion to justify if the concrete is in moisture saturated stage (H = 1) or not (H < 1). Fig. 8 displays the test and model results of pore volume density of each concrete at some selected ages. From Fig. 8, it can be observed that overestimation of model prediction to test result can be produced as the capillary pore radius is larger than 100 nm. However, from (17), it can be found as the pore radius is larger than 100 nm, the corresponding pore humidity is higher than 99% that is already close to the moisture saturated state. Therefore, above error may not influence the calculation of the content of capillary pore water under a given pore humidity in the humidity decreasing stage. Using cement hydration degree, α in (12) and (13) or (14) and (15), the content of capillary pore, Vp in the cement paste of each concrete can be obtained. Fig. 9 displays the relationship between capillary pore volume (in cm3 per cm3 cement paste) and ages of C30, C50 and C80 concrete respectively. In the figure, three lines are displayed, in which the value of αc was set to be different value in (13) or (15). From Fig. 9, it can be seen if αc is set to be zero (assume all chemical shrinkage is transferred into empty capillary pores) or to the hydration degree at set of concrete (assume the chemical shrinkage after set of concrete starts to transfer into empty capillary pores), a higher capillary pore volume is obtained than that assuming only the chemical shrinkage in the humidity decreasing stage can be transferred into the empty pores (αc is equal to the hydration degree at end of moisture saturated stage). Above selections of αc will influence the capillary pore content, which in turn will influence the model results of self-desiccation. Using the related parameters in equation (25), the relationship between content of capillary pore water, w and the corresponding relative humidity, H can be calculated, as showed in Fig. 10 for C30, C50 and C80 concrete

Normalized water content

C50: 90, 60, 28, 7, 3, 1 days 0.8

0.6

0.4

0.2

0.0 0

20

40

60

Relative humidity (%) (b)

Normalized water content

1.0

0.8 C80: 90, 60, 28, 7, 3, 1 days 0.6

0.4

0.2

0.0 0

20

40

60

Relative humidity (%) (c)

Fig. 10. Relationship between capillary pore water content and the corresponding relative humidity of concrete at selected ages, (a) C30, (b) C50 and (c) C80.

330

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

Fig. 11. Effect of node distance (a) and time interval (b) on numerical results of the model.

respectively. In the figure, the relations of w and H at age of 1, 3, 7, 28, 60 and 90 days after casting are displayed. From Fig. 10, it can be observed, in an early age, more water loss is needed for the same internal relative humidity decreasing than that in a later age. In other words, in a later age, the same amount of water loss will lead more reduction on internal relative humidity. This is reasonable because the capillary pore becomes finer and finer with the increase of age. The same water loss will result in more reduction on radius of meniscus of capillary water with increase of age, which in turn, leads more reduction on internal relative humidity. The relationship between capillary pore water content and internal relative humidity should be regarded as a material constitutive relation of concrete that is principally governed by the pore structure of the cement paste in the concrete. It tends to be a steady state relation with increase of age of concrete. Apart from the material parameters described above, the mesh parameters for solving the differential equation, including node distance, h and time interval, Δt should influence the calculation results as well. Reasonable h and Δt for model calculation should be determined first. As a calculation example, C80 concrete with one face drying, environmental humidity He = 0.3, surface factor

Fig. 12. Model and test results of the development of internal relative humidity of concrete under sealed state since casting, (a) C30, (b) C50, (c) C80.

331

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

100

C80 C50 C30

5

90

Relative Humidity (%)

Moisture diffusivity (x10-10 kg/ms)

6

4

3

2

80

70 Dry-C30 RH-1 RH-2 RH-3 Model

60

1

0 0.3

50

0.4

0.5

0.6

0.7

0.8

0.9

0

1

20

40

60

40

60

40

60

Age (days)

Relative humidity

(a)

Fig. 13. Moisture dependent diffusivity of concrete.

100

am = 3.75 × 10−5 kg m−2 s−1 [27], and moisture dependent diffu9(H 1) ] kg·m−1·s−1 [13] are sivity, D = 4.28 × 10 12 + 5.77 × 10 10 [2 10 used to study the effect of node distance and time interval on the numerical results. The same drying starting age as used in tests is used. Fig. 11a displays the model results in terms of internal relative humidity at the position of 2 cm from drying face and age diagram as using different node distance from h = 0.01 m to 0.001 m and a constant time interval, Δt = 2 min. It can be seen that for a given age, with decrease of node distance, the internal relative humidity gradually increases and the difference between each pair of adjacent data points is gradually decreased. This means the effect of node distance on model results is decreased with decrease of node distance. Fig. 11b displays the model results as using different time interval from Δt = 200 min to 0.5 min and a constant node distance, h = 0.0025 m. From the results, we can see that for a given age, with decrease of time interval, the internal relative humidity gradually increases. Again, the difference in internal relative humidity between each pair of adjacent data points is gradually decreased as well. The effect of time interval on model results is decreased with decrease of Δt. Based on above trial calculations, considering the time consumption of the calculation as well, h = 0.0025 m and Δt = 2 min are used in the following calculations.

Relative Humidity (%)

90

80

70 Dry-C50 RH-1 RH-2 RH-3 Model

60

50 0

20

Age (days) (b)

100

Relative Humidity (%)

90

4.2. Model prediction on self-desiccation It is well known the self-desiccation of cementitious material is an important phenomenon that may greatly influence the service life of concrete structures. The first application of the developed model is the prediction of self-desiccation representing by the progress of internal relative humidity of concrete. Fig. 12 displays the model and test results of development of internal relative humidity of the three concrete under sealed status since casting. In the figure, three calculation lines are displayed, in which the value of αc used in (14) or (16) was set to be zero, hydration degree at the set of concrete and hydration degree at the end of moisture saturated stage respectively. Clearly, model prediction is overestimated comparing to the test results as αc was set to be zero and the hydration degree at set of concrete, even though the second setting is little bit better that the first one. By contrast, model prediction is agreed well with test results as αc was set to be the

80

70 Dry-C80 RH-1 RH-2 RH-3 Model

60

50 0

20

Age (days) (c)

Fig. 14. Comparison between model and test results on internal relative humidity of concrete, (a) C30, (b) C50 and (c) C80.

332

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

Table 4 Parameters of moisture diffusivity of different concrete.

0.62 0.43 0.30

D0 (kg·m

·s

−12

19.3 × 10 8.0 × 10−12 4.3 × 10−12

)

γ 10.22 14.62 17.63

hydration degree at the point that the internal relative humidity starts to drop from the initial moisture saturated value. This interesting result indicates that only the chemical shrinkage occurred in the humidity decreasing stage (Stage II) is transformed into empty capillary pore. While chemical shrinkage occurred in moisture saturated stage is directly transferred into macro shrinkage of concrete. The present model can well predict the self-desiccation of concrete ranging from water to binder ratio of 0.62 to 0.30, which may represent normal and high strength concrete in practice.

C80 C50 C30 Model

5 2

4

Moisture diffusivity (x10-10 kg/ms)

C30-OC C50-OC C80-OC

Water/binder ratio

−1 −1

Moisture diffusivity (x10-10 kg/ms)

Concrete

6

3

2

C80 C50 C30 Model

1

1 0 0

0.2

0.4

0.6

0.8

1

Relative humidity

0 0

0.2

0.4

0.6

0.8

1

Relative humidity

4.3. Moisture diffusivity based on inverse analyzing

Fig. 15. Fitting model of moisture diffusivity of concrete.

The second application of the developed model is to solve moisture diffusivity of concrete. In the present work, inverse approach is used. The test results of internal relative humidity at three positions in concrete under one face drying (Fig. 2a) are used as test input in the analyses. Inverse analyzing consists of trial and error selections of various expressions for D, such as different selections on ki and di in equation (20) was used to solve moisture diffusivity. For each selection, the nonlinear diffusion equation was solved numerically to identify which selection best fits the test results. The optimum value of ki and di in each calculation step can be obtained by an optimization algorithm that minimizes the sum of squared deviation from test data. The related parameters used in humidity field calculation are listed in Table 3. Using the described methodology, the determined moisture diffusivity of C30, C50 and C80 as a function of relative humidity is displayed in Fig. 13. The humidity field in terms of development of internal relative humidity and age at three selected positions in the specimen is presented in Fig. 14, in which the corresponding test results are displayed for comparison with model predictions. Apparently, the selection of moisture diffusivity can well capture the progress of internal relative humidity of concrete. From Fig. 13, first we can see the moisture diffusivity is strongly dependent on the moisture content in concrete, which was noted in previous researches as well [9,11,13,17,29,30]. The diffusivity is first fast decreasing with the decrease of internal relative humidity from 100% to 80%. After that, it goes into a relatively stable stage with a small decreasing rate with decrease of internal relative humidity. Second, the higher the concrete strength (or the lower the water to binder ratio), the lower the moisture diffusivity. For example, the diffusivity at H = 1 is 5.148 × 10−10 kg m−1 s−1, 3.528 × 10−10 kg m−1 s−1 and 3.134 × 10−10 kg m−1 s−1 respectively for C30, C50 and C80 concrete. As H = 0.3, the diffusivity becomes 0.195 × 10−10 kg m−1 s−1, 0.081 × 10−10 kg m−1 s−1 and 0.043 × 10−10 kg m−1 s−1 for C30, C50, and C80 respectively. The ratio of diffusivity between moisture saturated state (H = 1) and dried state (H = 0.3) is 26.5, 43.6 and 73.2 respectively for C30, C50 and C80 concrete. The effect of capillary water on moisture diffusivity in high strength concrete (or low water to binder ratio concrete) is more notable than that in normal strength concrete. It may be caused by the difference on the overall path length of liquid water between normal and high strength concrete (or low water binder ratio concrete) under the same internal relative humidity. In addition, it should also be noted

that the above diffusivity is a dynamically coupled parameter with age and water content, which may reflect the variation of the pore content and structure as well as water saturation degree in the pores. This is the actual state of concrete in the early age when the concrete experiences significant variation in pore content and structure, and water content in it. Generally, the diffusivity of concrete should be a function of pore content and distribution, as well as water content in the pores. Assume the moisture diffusivity may be composed by two parts as: (29)

D = D0 [1 + f (D0 ) g (H )]

Where D0 is the diffusivity as water content is equal to zero theoretically. The second item in the bracket is the contribution of pore water to the moisture diffusivity. f(D0) and g(H) are two functions that may reflect the effect of pore content and structure, and water in the pores on the diffusivity respectively. By fitting the data showed in Fig. 13, f(D0) and g(H) are obtained and equation (29) can be rewrote as:

D = D0 1 + 13.035 +

234.500 × 10 D0

12

e

(H 1)

(30)

Parameter γ may reflect the influence of pore shape on liquid water path in the pore, which in turn influences the diffusivity of concrete under a given pore humidity. The values of D0 (kg·m−1·s−1) and γ for C30, C50 and C80 concrete are listed in Table 4. They can be related with water to binder ratio (w/b) respectively by:

D0 = e 4.636(w / b)+ 0.087 × 10

12

(31)

And

=

23.16(w/ b) + 24.58

(32)

Thus, equation (30) provides an expression of moisture diffusivity of concrete that takes the influences of pore content and structure, as well as water content in the pores into consideration. The comparison between model and test data of moisture diffusivity of C30, C50 and C80 concrete is shown in Fig. 15. It should point out that above fitting constants may be only suitable for the concrete with water to binder ratio of 0.30–0.62. Additional verification tests may be needed for the

333

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

concrete out of above rang. Further applications of the developed model on analyses of the effects of curing and environmental conditions on the humidity field of concrete are described in Appendix.

humidity in concrete, which may be resulted either by cement hydration or by environmental drying. By using water content as the linking parameter of moisture diffusion and self-desiccation, the model can be used for moisture field calculation of concrete elements under any given curing strategy and drying conditions. The model includes two kinds of input: material compositions, including cement composition and concrete mix proportion; and experimentally determined parameters, including cement hydration parameters and pore distribution parameters. As applications of the model, moisture dependent diffusivity of three concrete, C30, C50 and C80 that represent low, middle and high strength concrete in practice, is solved. Moisture distribution in the above three concrete under varied curing condition is simulated and analyzed.

5. Summery and conclusions In the present paper, an integrative model for moisture distribution prediction of concrete is developed. The model takes self-desiccation and moisture diffusion of concrete into account simultaneously. In the modeling, the two critical issues, self-desiccation and moisture diffusion are simulated with the same relation between water content and internal relative humidity of concrete, which is normally called moisture capacity or adsorption-desorption law. The constitutive law of water content and internal relative humidity of cement paste is developed based on Kelvin's law and capillary pore distribution that can be experimentally determined. The water content in cement paste is used as the critical parameter to predict the reduction of internal relative

Acknowledgement This work has been supported by grants from the National Science Foundation of China (51678342, 51878380) to Tsinghua University.

Appendix A. Supplementary data Supplementary data to this article can be found online at https://doi.org/10.1016/j.cemconcomp.2019.01.008. Appendix Curing of concrete after casting is needed to ensure there is sufficient water to joint cement hydration. The main function of curing is to avoid concrete suffering to dry. Therefore, surface sealing with membrane or chemical agent is the ordinary method of curing. However, curing time used in practice is generally varied. In this section, the model will be used to simulate moisture field of concrete as concrete suffers to dry at different age after casting. It is assumed that concrete suffers to one-face dry at 1, 3, 7, 14 and 21 days after casting respectively. The same length of concrete along the drying direction as used in the tests, i.e. 300 mm is used. The surface factor, am = 3.75 × 10−5 kg m−2 s−1 and environmental humidity, He = 0.3 are used. Figs.A1 present the model results of the progress of internal relative humidity at positions, 0 m (drying surface) and 0.02 m from drying surface in figure (a) and (b), and humidity distribution along the drying direction at selected ages after casting in figure (c) of the three concrete respectively. From figures (a) and (b), first it can be observed that at a given age, the earlier the drying starting, the higher the internal humidity reduction. Second, as expected, the decreasing rate on internal relative humidity of the drying surface is obviously higher than that inside of concrete. The progress of internal relative humidity at drying surface displays as three-stage manner: first slowly decreasing along the selfdesiccation path of the cement paste, then stepping into a fast decreasing stage within a relatively short period, after that it goes into a gradually decreasing stage towards to a steady state value that is close to environmental humidity. By contrast, the decreasing pattern of relative humidity inside of concrete, such as position of 0.02 m from the drying surface, becomes more gradually to decrease. Mode lower reducing rate on humidity can be observed than that of drying surface. This disparity is principally controlled by the difference of moisture diffusivity of external (surface to atmosphere) and internal (inside of concrete). Third, as expected, the higher the concrete strength (or lower the water to binder ratio), the higher the humidity reduction inside of concrete. However, in the area close to the drying surface, the humidity reduction is more significant for low strength concrete. This aspect may be observed in Figs.A1(c) to A3(c). From figure (c), we can note that the humidity distribution along the specimen length from the drying surface is nonuniform and obvious humidity gradient is existed in each concrete. In this case, humidity distribution of concrete shows a plateau region apart from the drying surface, but, high humidity gradient occurs at the out layer near the exposed surface. The developed model can predict the progress of internal relative humidity in concrete at arbitrary curing and drying strategy. As an essential parameter of the environment, environmental humidity will greatly influence the moisture loss of concrete. To investigate the effect of environmental humidity on the progress of internal relative humidity of concrete under drying, environmental humidity, He = 0.1, 0.3, 0.5, 0.7, 0.9, 1.0 were used in the model. Three days sealed curing after concrete casting was assumed. The humidity field of C30, C50 and C80 concrete under one-face drying at 28 days after casting is shown in Fig.A5 (a) to (c) respectively. From Fig.A4, first it can be seen that the increase of environmental humidity will reduce the humidity reduction in the zone where is close to the drying surface. While, for the place where is far from the drying surface, little effect of environmental humidity on the moisture state of concrete is observed. The size of influencing zone of environmental drying may be governed by the moisture diffusivity of concrete. The model can well reflect the effect of environmental humidity on the moisture distribution of concrete. Another parameter of environment that may influence the moisture distribution of concrete under drying should be the surface factor. As showed in equation (18), surface factor that is principally governed by air temperature, air flow rate and element size. To investigate the effect of surface factor on the development of internal relative humidity of concrete under drying, am = 3.75 × 10−5 kg m−2 s−1 (indoor, Va = 0.1 m/s), am = 12.08 × 10−5 kg m−2 s−1 (outside, Va = 1 m/s) and am = 30.76 × 10−5 kg m−2 s−1 (outside, Va = 5 m/s) are used in the model calculation. Three days sealing before drying is assumed. The environmental relative humidity is assumed to be equal to 0.3. The humidity field of C30, C50 and C80 concrete at 28 days under one-face drying is displayed in Fig.A5 (a) to (c) respectively, in which the results with different surface factor are present. First, as expected, the impact of surface factor on moisture distribution of concrete under drying is also limited within the zone where is close to the drying surface. For given age and poison in drying influencing zone, the higher the surface factor, the lower the relative humidity. Second, the effect of surface factor on moisture distribution is more pronounced for low strength concrete than that of high strength concrete. This is principally due to the difference on moisture diffusivity between low and high strength concrete.

334

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

100 Dry-C30 Position: 0 m

Relative Humidity (%)

80

60

40

Sealed age: 1, 3, 7, 14, 21 days 20

0 0

20

40

60

Age (days) (a)

100 Dry-C30 Position: 0.02 m

Relative Humidity (%)

90

80

70 Sealed age: 1, 3, 7, 14, 21 days 60

50 0

20

40

60

Age (days) (b)

100

Relative Humidity (%)

80 Concrete age: 60, 28 14 days Sealed age: 1, 3, 7, 14, 21 days 60

40

Dry-C30 14 days 28 days 60 days

20

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Position (m) (c)

Fig. A1. Effect of curing age on moisture distribution of C30 concrete, (a) relative humidity at the drying surface, (b) relative humidity at 0.02 m from drying surface, (c) moisture distribution at selected age along the drying direction.

335

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

100 Dry-C50 Position: 0 m

Relative Humidity (%)

80

60

40

20

Sealed age: 1, 3, 7, 14, 21 days

0 0

20

40

60

Age (days) (a)

100

Relative Humidity (%)

Dry-C50 Position: 0.02 m 90

80

Sealed age: 1, 3, 7, 14, 21 days 70

60 0

20

40

60

Age (days) (b)

100

Relative Humidity (%)

80 Concrete age: 60, 28 14 days Sealed age: 1, 3, 7, 14, 21 days 60

40

Dry-C50 14 days 28 days 60 days

20

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Position (m) (c)

Fig. A2. Effect of curing age on moisture distribution of C50 concrete, (a) relative humidity at the drying surface, (b) relative humidity at 0.02 m from drying surface, (c) moisture distribution at selected age along the drying direction.

336

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

100 Dry-C80 Position: 0 m

Relative Humidity (%)

80

60

40

Sealed age: 1, 3, 7, 14, 21 days

20

0 0

20

40

60

Age (days) (a)

100 Dry-C80 Position: 0.02 m

Relative Humidity (%)

90

80

Sealed age: 1, 3, 7, 14, 21 days

70

60

50 0

20

40

60

Age (days) (b)

100

Relative Humidity (%)

80 Concrete age: 60, 28 14 days Sealed age: 1, 3, 7, 14, 21 days 60

40 Dry-C80 14 days 28 days 60 days

20

0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Position (m) (c)

Fig. A3. Effect of curing age on moisture distribution of C80 concrete, (a) relative humidity at the drying surface, (b) relative humidity at 0.02 m from drying surface, (c) moisture distribution at selected age along the drying direction.

337

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

100

Relative Humidity (%)

80

He=0.1, 0.3, 0.5, 0.7 0.9, 1.0 60

40

20 Dry-C30 Age: 28 days 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Position (m) (a)

100

Relative Humidity (%)

80

He=0.1, 0.3, 0.5, 0.7 0.9, 1.0 60

40

20 Dry-C50 Age: 28 days 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Position (m) (b)

100

Relative Humidity (%)

80

He=0.1, 0.3, 0.5, 0.7 0.9, 1.0 60

40

20 Dry-C80 Age: 28 days 0 0

0.05

0.1

0.15

0.2

0.25

0.3

Position (m) (c)

Fig. A4. Effect of environmental humidity on moisture distribution of concrete, (a) C30, (b) C50, (c) C80.

338

0.35

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al.

100

Relative Humidity (%)

80 am=3.75x10-5; 12.08x10-5; 30.76x10-5 kg/m2s

60

40

20 Dry-C30 Age: 28 days 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Position (m) (a)

100

Relative Humidity (%)

80 am=3.75x10-5; 12.08x10-5; 30.76x10-5 kg/m2s 60

40

20 Dry-C50 Age: 28 days 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Position (m) (b)

100

Relative Humidity (%)

80 am=3.75x10-5; 12.08x10-5; 30.76x10-5 kg/m2s 60

40

20 Dry-C80 Age: 28 days 0 0

0.05

0.1

0.15

0.2

0.25

0.3

0.35

Position (m) (c)

Fig. A5. Effect of surface factor on moisture distribution of concrete, (a) C30, (b) C50, (c) C80.

[2] V. Baroghel-Bouny, M. Mainguy, T. Lassabatere, O. Coussy, Characterization and identification of equilibrium and transfer moisture properties for ordinary and highperformance cementitious materials, Cement Concr. Res. 29 (8) (1999) 1225–1238. [3] F.H. Wittmann, G. Martinola, H. Sadouki, Material properties influencing early age cracking of concrete, in: H. Mihashi, F.H. Wittmann (Eds.), Proceedings of International Workshop on Control of Cracking in Early Age Concrete, Swets &

References [1] B. Bissonnette, P. Pierre, M. Pigeon, Influence of key parameters on drying shrinkage of cementitious materials, Cement Concr. Res. 29 (10) (1999) 1655–1662.

339

Cement and Concrete Composites 97 (2019) 322–340

X. Ding et al. Zeitlinger, Lisse, 2002P.3-18. [4] Z.B. Zhang, J. Zhang, Experimental study on the relationship between shrinkage strain and environmental humidity of concrete, J. Build. Mater. 6 (6) (2006) 720–723 In Chinese. [5] J. Zhang, D.W. Hou, W. Sun, Experimental study on the relationship between shrinkage and interior humidity of concrete at early age, Mag. Concr. Res. 62 (3) (2010) 191–199. [6] T. Ayano, F.H. Wittmann, Drying, moisture distribution, and shrinkage of cementbased materials, Mater. Struct. 35 (247) (2002) 134–140. [7] C. Andrade, J. Sarria, C. Alonso, Relative humidity in the interior of concrete exposed to natural and artificial weathering, Cement Concr. Res. 29 (8) (1999) 1249–1259. [8] L.J. Parrott, Influence of cement type and curing on the drying and air permeability of cover concrete, Mag. Concr. Res. 47 (171) (1995) 103–111. [9] L.O. Nilsson, Long-term moisture transport in high performance concrete, Mater. Struct. 35 (2002) 641–649. [10] Y. Huang, K. Qi, J. Zhang, Development of internal humidity in concrete at early age, J. Tsinghua Univ. 47 (3) (2007) 309–312 In Chinese. [11] J. Zhang, K. Qi, Y. Huang, Calculation of moisture distribution in early-age concrete, ASCE J. Eng. Mech. 135 (8) (2009) 871–880. [12] T.C. Powers, T.L. Brownyard, Studies of physical properties of hardened Portland cement paste, Am. Concr. Inst. J. 1947 (1946) 101–132 249-336, 469-504, 549-602, 669-712, 845-880, 933-992. [13] J. Zhang, J. Wang, Y. Gao, Moisture movement in early ages concrete under cement hydration and environmental drying, Mag. Concr. Res. 68 (8) (2016) 391–408. [14] O.M. Jensen, P.F. Hansen, Water-entrained cement-based materials: I. Principles and theoretical background, Cement Concr. Res. 31 (4) (2001) 647–654. [15] J. Zhang, D. Hou, Y. Han, Micromechanical modeling on autogenous and drying shrinkages of concrete, Constr. Build. Mater. 29 (3) (2012) 230–240. [16] O.M. Jensen, Autogenous Deformation and RH-Change−self-Desiccation and Self-

[17] [18] [19] [20] [21] [22] [23] [24] [25] [26] [27] [28] [29] [30]

340

Desiccation shrinkage, Building Materials Laboratory, Technical University of Denmark, Lyngby, Denmark, 1993 R 285/93. J. Wang, J. Zhang, J.J. Zhang, Study on cement hydration rate of normal and internally cured concretes, J. Adv. Concr. Technol. 16 (2018) 306–316. E. Rastrup, Heat of hydration, Mag. Concr. Res. 6 (17) (1954) 127–140. K. Kjellsen, R.J. Detwiler, Later-age strength prediction by a modified maturity model, ACI Mater. J. 90 (3) (1993) 220–227. G. Chanvillard, L. Daloia, Concrete strength estimation at early age: modification of the method of equivalent age, ACI Mater. J. 94 (6) (1997) 220–227. J.K. Kim, Estimation of compressive strength by a new apparent activation energy function, Cement Concr. Res. 31 (2) (2001) 217–225. A.K. Schindler, K.J. Folliard, Heat of hydration models for cementitious materials, ACI Mater. J. 102 (1) (2005) 24–33. G.D. Luzio, G. Cusatis, Hygro-thermo-chemical modeling of high performance concrete. I: Theory, Cement Concr. Compos. 31 (2009) 301–308. H.J.H. Brouwers, The work of powers and brownyard revisited: Part 1, Cement Concr. Res. 34 (9) (2004) 1697–1716. K.A. Sakata, Study on moisture diffusion in drying and drying shrinkage of concrete, Cement Concr. Res. 13 (3) (1983) 164–170. Y. Gao, Studies on Shrinkage and Shrinkage Induced Stresses of Concrete Structures under Dry-Wet Cycles. Ph.D Thesis, Tsinghua University, 2013. J. Zhang, J. Wang, X. Ding, Test and simulation on moisture flow in early-age concrete under drying, Dry. Technol. 36 (2) (2018) 221–233. M. Bakhshi, B. Mobasher, M. Zenouzi, Model for early-age rate of evaporation of cement-based materials, J. Eng. Mech. 138 (11) (2012) 1372–1380. Z.P. Bazant, L.J. Najjar, Nonlinear water diffusion in nonsaturated concrete, Mater. Struct. 5 (25) (1972) 3–20. Y. Xi, Z.P. Bažant, H.M. Jennings, Moisture diffusion in cementitious materialsAdsorption isotherms, Adv. Cem. Base Mater. 1 (6) (1994) 248–257.