Simulation of chloride ion transport in concrete under the coupled effects of a bending load and drying–wetting cycles

Construction and Building Materials 241 (2020) 118045

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Simulation of chloride ion transport in concrete under the coupled effects of a bending load and drying–wetting cycles Cao Tongning a, Zhang Lijuan b, Sun Guowen a,c,⇑, Wang Caihui a, Zhang Ying a, Wang Pengshuo a, Xu Aoxue d a

School of Materials Science and Engineering, Shijiazhuang Tiedao University, Shijiazhuang 050043, China Foundational subjects department, Shijiazhuang TieDao University SiFang College, Shijiazhuang 051132, China Key Laboratory of Smart Material and Structures Mechanics, Hebei Province, Shijiazhuang Tiedao University, Shijiazhuang 050043, China d School of Foreign Studies, Hebei Normal University, Shijiazhuang 050024, China b c

h i g h l i g h t s
A variable coefficient constitutive model for chloride ion transport under coupled bending and drying–wetting conditions is proposed.
The model considers the key factors such as the porosity, tortuosity and saturation changes during the transport of chloride ions in concrete.
The two-dimensional variable coefficient constitutive equations of water and chloride ion transport are solved by the alternating implicit finite

difference method.

a r t i c l e

i n f o

Article history: Received 23 September 2019 Received in revised form 23 December 2019 Accepted 2 January 2020

Keywords: Concrete Bending load Drying–wetting cycle Variable coefficient of chloride ion transport Tortuosity

a b s t r a c t The transport of chloride ions in concrete under the coupled effects of a bending load and drying–wetting cycles accelerates the degradation of marine concrete. Based on the microstructure characteristics of concrete, the models of the tortuosity of concrete under a bending load were built. Considering the interaction between the transport of water and the tortuosity in the concrete microstructure, a nonlinear diffusion-convection model of the water and chloride ions transport in unsaturated conditions were also proposed in this paper. Finally, a two-dimensional variable coefficient model of chloride ion transport in concrete under the coupled effects of a bending load and drying–wetting cycles was also established based on Fick0 s second law, which was numerically solved using the alternating direction implicit (ADI) method. The results show that the values predicted agree with those of experiments and those reported in the literatures. Ó 2020 Elsevier Ltd. All rights reserved.

1. Introduction The study of transport properties of chloride ions in concrete under the coupled effect of bending load and drying–wetting cycles is an urgent issue in marine concrete structures, which has been studied for many years. Two methods, which are traditional laboratories experimental method and numerical simulation method, are widely used to in this field. The experimental method are more reliable, however, it usually succumbs to the test conditions and environmental conditions, which makes the test results ⇑ Corresponding author. E-mail addresses: [email protected] (C. Tongning), [email protected] (Z. Lijuan), [email protected] (S. Guowen), [email protected] (W. Caihui), [email protected] (Z. Ying), [email protected] (W. Pengshuo), [email protected] (X. Aoxue). https://doi.org/10.1016/j.conbuildmat.2020.118045 0950-0618/Ó 2020 Elsevier Ltd. All rights reserved.

relatively discrete. Therefore, it is difficult to reflect the transport characteristics of chloride ions in concrete objectively, and further to accurately predict the service life of structural concrete by means of experimental method. On the other hand, the numerical simulation method is also frequently used in predicting the chloride profile of concrete. Scholars have established a prediction model to understand the transport characteristics and laws of chloride ions in real time, for the transport of chloride ions in concrete under a load, under alternate drying and wetting or under a combination of the two effects. Zhang et al. [1] conducted long-term exposure tests on reinforced concrete beams subjected to different bending loads in a tidal environment. They established a chloride ion transport model under different combinations of stress and chloride ion concentration considering the effect of the chloride ion diffusion coefficient over time. Guan et al. [2] explored the influence of the coefficient of the

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C. Tongning et al. / Construction and Building Materials 241 (2020) 118045

dynamic bending load on chloride ion diffusion coefficient and hydraulic diffusion coefficient according to the relationship between the dynamic bending load of the concrete and the chloride ion diffusion coefficient, and on the basis of the chloride ion transport model in an environment with drying and wetting cycles. Then they proposed a model of chloride ion transport under the action of a dynamic bending load and drying–wetting cycles. Xu et al. [3] studied the effect of continuous compressive loading on the chloride ion penetration of concrete. Under different axial stress levels and different cycle lengths of drying and wetting, they considered the time dependence and stress dependence of concrete. They put forward a predictive model of the chloride ion permeability characteristics of concrete under axial compression. Bao et al. [4] built a quantitative relationship between water, chloride ion diffusivity and compressive stress level by fitting experimental data. Based on the convection–diffusion equation, a prediction model for chloride ion transport under continuous compressive loading of unsaturated concrete was presented. Petcherdchoo et al. [5] proposed a new, closed solution of the chloride ion function of a linear cyclic surface, which was used to predict the transport of chloride ions in unsaturated concrete, and then further applied the solution to concrete undergoing chloride ion erosion and compressive loading. They set up a bilinear model of each parameter in the solution. However, their model did not directly establish the relationship between the transport characteristics of the chloride ions and the concrete microstructure. Fu et al. [6] studied the permeability of chloride ions in concrete under uniaxial tensile fatigue loading, and established a transport model of chloride ions under fatigue loading based on the influence of residual strain on the apparent chloride ion diffusivity of fatiguedamaged concrete. Based on the multi-scale transition method, Sun et al. [7] established a modern concrete nano-micromicroscopic-macro multi-scale transmission model by means of numerical calculation to obtain the constitutive relationship between concrete microstructure and chloride ion transport. Wang et al. [8] proposed a theoretical framework of one-dimensional movement of chloride ions for unsaturated concrete under tensile stress which is closely related to capillary water absorption, based on a quantitative relationship between the change of water or chloride ion diffusivity, the permeability profile and tensile stress level of unsaturated concrete. Wang et al. [9] used experimental data to fit the surface chloride ion concentration and chloride ion diffusion coefficient of the analytical chloride ion inflow model as random variables in order to evaluate the effect of load on the modeling of chloride ion influx. The random variables identified were then used to determine the distribution of the corrosion onset for the current experimental configuration and to assess the long-term probability of corrosion initiation under other environmental exposures and coverage conditions. Petcherdchoo et al. [10] researched the service life and environmental impact due to repairs by metakaolin concrete after chloride attack. Xu et al. [11] established a analytical model for load dependence of chloride penetration into concrete. However, these above models do not consider the microstructure change of structural concrete caused by the fatigue loading. Meanwhile, the key indicators involved in the model, such as the hydraulic diffusion coefficient and chloride diffusion coefficient are fixed at constants. Moreover, the predicted distribution of chloride ion concentration in the model is for the one-dimensional transport of concrete, and does not consider the transport of chloride ions in two or three dimensions. Whether it is the tensile load or the bending load that causes damage to the structural concrete, the damage provides a convenient routes for chloride ion. The tortuosity of the pore is usually measured to character the change of the chloride transport paths [10]. The nature of the damage to structural concrete under

drying–wetting cycle conditions is the change of pores and the saturation concentration of pores or microcracks, resulting in changes in the transport rate of chloride ions. Therefore, this paper establishes a two-dimensional variable-coefficient model of a chloride ions transport in concrete under a bending load and drying–wetting cycles, in which the transport coefficient of the chloride ion is directly related to the parameters of the damage characteristics of the concrete, i.e., tortuosity, porosity, and saturation. 2. Theoretical modeling process The transport process of chloride ion in concrete during a drying and wetting cycle is actually the coupled processes of the transport of water and chloride ion in concrete under unsaturated conditions. When the concrete in the alternating dry-wet environment is drying, the water is transported outward and evaporates. When the concrete is being wet, the water is transported to the interior of the concrete. Because of the alternate wetting and drying, the water transport rate in the surface layer of the concrete is high, and the proportion of water transport in the whole chloride ion transport process is also relatively high. Under the alternate drying–wetting process, water in concrete is in both gaseous and liquid states. Therefore, both a liquid water diffusion coefficient model and a gaseous water diffusion coefficient model are needed. In addition, as the pore is always in an unsaturated state under drying–wetting cycle conditions and the diffusion of concentration can only occur in pore fluid, the chloride ion diffusion coefficient in unsaturated state needs to be converted according to the pore saturation in saturated state [13]. The internal damage of the tension zone of structural marine concrete is accelerated under a bending load, which changes the chloride ion transport path. The question is thus how to transform the mechanical factors of the structure into physical factors. This paper converts the transport path of structural concrete into a quantitative relationship between the tortuosity and the macroscopic stress In addition, the transport model of a chloride ion in concrete under the combined action of a bending load and drying–wetting cycles is mostly one-dimensional. In fact, marine concrete components are eroded by chloride ions, including walls and plates, beams, columns and piles are eroded on multiple faces. As shown in Fig. 1, this erosion mainly involves two-dimensional situations. For example, the interactions in the two directions (x, y) are not considered in depth. Therefore, this paper focuses on the two-dimensional transport variable coefficient model of the chloride ion in concrete under the combined action of a bending load and drying–wetting cycles. Because the concrete components are mostly rectangular sections, the simulation process is mainly based on the transport in rectangular components.

2.1. Constitutive equation of water transport under unsaturated conditions The water vapor flows and the condensed liquid water within the concrete are caused by the vapor pressure differences and capillary pressures differences, respectively. At high relative humidity (RH), the liquid flow significantly accelerates the process of moisture transport in concrete exposed to the atmospheric air. Darcy0 s law can be extended to water transport in unsaturated concrete. Since water is transported in concrete in both gaseous and liquid states, the water diffusion coefficient should also include the contribution of these two states. The gas–liquid transport equation under isothermal can be expressed as [13]:



 @w @ @p @p @ @p @p ¼ KL  c þ Kv  v þ KL  c þ Kv  v @t @x @y @x @x @y @y

ð1Þ

C. Tongning et al. / Construction and Building Materials 241 (2020) 118045

3

Fig. 1. Schematic diagram of two-dimensional model of chloride ion eroded concrete.

Where w is the mass of water; KL is the permeability of liquid water, dependent on relative humidity h; and Kv is the permeability of water vapor in unsaturated concrete. According to the Kelvin equation under thermodynamic equilibrium conditions, the relationship between the value of capillary pressure and relative humidity is as follows [13]:

pc ¼ 

qL RTlnh Mw

pv ¼ hpv s

arMw qL RTlnh

where kv is the water vapor permeability of single-hole [14]; lm is the free path of the gaseous water molecules; re is the effective radius, the size of which is equal to the actual radius minus the thickness of the adsorbed water layer; Dva is the diffusion coefficient of water vapor in air [12]; D0 is the water vapor diffusion coefficient at pressure P0 and temperature T0 (D0 = 21.6  106 m2/s, P0 = 11325 Pa, T0 = 273.16 K); rc is the capillary condensate radius; a denotes a factor considering different liquid–gas interfaces before adsorption (a is equal to 1) and desorption (a is equal to 2) processes, respectively. According to Eq. (7), the water vapor permeability (KV) of concrete under unsaturated conditions can be derived as follows:

ð4Þ

As expressed in Eqs. (5) and (6), the distribution of pores follows the Rayleigh-Ritz (R-R) model [15], which uses two parameters— porosity and the pore radius corresponding to the logarithmic scale of the peak to reflect the distribution of pores of different radius:

KV ¼

ð6Þ



1:88 /D0 P0 T 1 M 2aBrM exp 1 þ lm =2re RT sP T 0 qRTlnh

ZAc ZAc KL ¼

g 0




kL ¼

kL

Z

rc 0

kv

s

r 2 f d ðr Þdr

ð7aÞ

ð8Þ

dpab

ð9aÞ

0

q req 2 g 8



dpab ¼ dAa dAb 2.2.1. Determination of gaseous water permeability With the increase of effective pore radius (rc), water vapor permeability increases and becomes closer to air permeability. Considering the tortuosity of pore s, the water vapor permeability (KV) of unsaturated concrete is obtained based on the integral expressed as follows:

ð7dÞ

2.2.2. Determination of liquid water permeability To facilitate the derivation of liquid water permeability, it is assumed that the pores are cylindrical and well connected to each other, and that the liquid water does not slip with the tube wall during transport in the capillary [16]. The liquid water permeability (KL) can be derived as follows:

ð5Þ

where ut represents the total porosity in concrete. The pore size distribution is represented by a probability density function fd, which is equal to the derivative of the porosity corresponding to a given pore radius.

KV ¼

ð7cÞ

ð3Þ

2.2. Model of water diffusion coefficient under unsaturated conditions

f d ðr Þ ¼ /t BexpðBr Þ


1:88 P0 T P T0

rc ¼ 

@w @ ¼ @t @x

/ ¼ /t ð1  expðBrÞÞ

Dv a ¼ D0

ð7bÞ

ð2Þ

where Mw is the molar mass of water; qL is the density of water; R is a gas constant; pc represents the capillary pressure of liquid water; pv represents vapor pressure; pvs represents saturated vapor pressure; T is the temperature. KL and KV are functions of relative humidity h. The goal in this paper is to establish a chloride ion transport model with water as the carrier. Using Eqs. (1)–(3), the equation of water transport in unsaturated concrete can be derived as follows:


  q RT @h K L  L þ K v pv s Mh @x 
  @ qL RT @h þ KL  þ K v pv s @y Mh @y


 Dv a M 1 þ lm =2r e RT

kv ¼

ð9bÞ ð9cÞ

where Ac is the standard area of the critical hole; kL is the permeability of the liquid water passing through each cylindrical hole calculated by Hagen-Poiseuille0 s law [16]; g is the viscosity of liquid water; req (r2eq ¼ ra  r b ) is obtained based on the equivalent of the pore volume and the proportional relationship between the pore length and the radius; dpab is the probability of liquid penetration of different scale pores (a and b); dAa and dAb are the areas of hole A and hole B, respectively.

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C. Tongning et al. / Construction and Building Materials 241 (2020) 118045

According to Eq. (9), considering the porosity u and tortuosity s of the pores, the liquid water permeability (KL) can be derived as follows:

KL ¼



2 /2 q 1 1 2arM 2aBrM exp   B qRTlnh 8s2 g B qRTlnh

ð10Þ

quality of the water molecule monolayer adsorbed by unit cement slurry mass; t0 is the curing time; w and c are the quality of the water and cement, respectively; Vct is the influence coefficient of the cement type; T represents temperature; and Nct is the cement type influence coefficient. 2.3. Tortuosity and porosity model of mortar and concrete

2.2.3. Determination of the relationship between relative humidity and saturation The functional relationship between relative humidity and saturation in the drying process is different from that of relative humidity and saturation in the wetting process. A large number of experiments have shown that the saturation corresponding to the same relative humidity in the drying process is higher than that in the wetting process which is calledb the hysteresis effect [17]. The symbols hs and hdes represent the relative humidity in the wetting and drying processes, respectively. The hysteresis effect is caused mainly by the difference in the meniscus of the capillary pore in the wetting and drying process. If the influence of the aggregate and interfacial transition zone on the internal saturation of the concrete is neglected, the relationship between relative humidity and saturation during the wetting cycle can be expressed as follows [17,18]:

( hs ¼ ln



a  a2  4ð1  C E Þ (

ð11aÞ 1=2 )



a  a2  4ð1  C E Þ

ð11bÞ

2kt ð1  C E Þ




 Vm a¼2þ  1 CE W 20 S W¼

ð11cÞ

C E kt V m  h ð1  kt hÞ½1 þ ðC E  1Þkt h

 V m ¼ 0:068 

ð11dÞ

  0:22 w V ct  0:85 þ 0:45 c ðt 0 =86400Þ

ð11eÞ


 C E0 C E ¼ exp T kt ¼

ð11fÞ

sm ¼ sc ¼

sm þ km lm þ km sc þ k c lc þ kc

scp

sm

ð12aÞ

ð12bÞ

where sm is the mean tortuosity of mortar; sc is the mean tortuosity of concrete. 2.3.1. Tortuosity model of mortar Mortar can be regarded as composed of pore, hardened cement matrix and sand aggregate. The sand aggregate is regarded as spherical, and rm represents the radius of sand. The volume fraction (fsa) of spherical sand aggregate in mortar can be calculated by mixing ratio. The volume (Vm) and surface area (Sm) of spherical aggregate can be expressed as:

4 pr m 3 3

ð13Þ

Sm ðrm Þ ¼ 4prm 2

ð14Þ

V m ðr m Þ ¼

In a cement matrix, the number of spherical aggregates (Nm) can be expressed as:

ð1  1=ns ÞC E  1 CE  1


ns ¼ 2:5 þ

by the sum of the average chord length (l) and the average spacing of the aggregate particle size(k). The schematic diagram of aggregate particles is shown in Fig. 2.

1=2 )

2kt ð1  C E Þ

hdes ¼ 2ln

The tortuosity is usually defined as the ratio of the average diffusion path of a particle to the shortest distance of its straight line [12]. Because of the different particle sizes of the aggregate particles and their random distribution in the concrete, the tortuosity is expressed by the mean tortuosity (s) at any section, as shown in Eq. (12). In this paper, sand is regarded as spherical aggregate and gravel as ellipsoidal aggregate. The mean tortuosity is the sum of the average arc length (s) and the average spacing of the aggregate particle size of any two adjacent aggregates (k), divided

ð11 gÞ

 15 ð0:33 þ 2:2w=cÞNct t 0 =86400

ð11 hÞ

where W is the water quality fraction relative to the cement slurry; CE is the energy parameter, with CE0 = 855; kt is a constant; Vm is the

(a)Tortuosity of spherical aggregate

Nm ¼ P

f sa L3 aðr m ÞV m ðrm Þ

ð15Þ

where L is the side length of the sample; a(rm) represents the sieve percentage of each grade of sand. The average value of surface spacing between adjacent spherical sand aggregate particles can be expressed by mean free path

(b)Tortuosity of ellipsoidal aggregate

Fig. 2. Definition of the mean tortuosity.

5

C. Tongning et al. / Construction and Building Materials 241 (2020) 118045

(km) [12,19], which is directly affected by sand volume fraction (fsa) and the solid surface area (SV), as shown in Eqs. (16) and (17):

P

SV ¼ km ¼

N m aðrm ÞSm ðr m Þ

ð16Þ

L2 4ð1 - f sa Þ SV

ð17Þ

The average radius of spherical sand aggregate (lm ) and the average arc length (sm ) can be expressed as:

lm ¼

X

2aðr m Þr m

ð18Þ


sm ¼ plm

ð19Þ

By substituting Eqs. (17), (18) and (19) into Eq. (12), the average tortuosity (sm) of the mortar can be expressed as:

sm ¼

lm þ 4ð1 SV f sa Þ lm þ 4ð1 SV f sa Þ

p

h ha ¼ 1  0:5 ð1 þ 1:67t Þ

 0:6

þ ð1 þ 0:29tÞ

ð21aÞ  0:48

i

1=2 ðp  2Þ 1  /uhcp ¼1þ  1=2 2  31=2 p



1

1  1  /hcp

1=2

h



1=2 i2 1 1=2 1  1  /hcp þ 4

ð21bÞ

ð21cÞ

ð21dÞ

where ha is the degree of hydration of the cement; suh-cp and sh-cp are the tortuosity of the stacked cement particles and the tortuosity of the fully hydrated cement particles, respectively; and uuh-cp and uh-cp are the tortuosity of the stacked cement particles and the tortuosity of the fully hydrated cement particles, respectively, with uuh-cp = 0.58. gc and gr are the shape correction coefficient of unhydrated cement particles and the shape correction coefficient of fully hydrated cement particles, respectively. These two parameters are obtained based on the principle that the irregular cement particles are equal in volume to ideal spherical and cubic cement particles. wwc is a correction factor that considers the effect of the water-cement ratio on the tortuosity of the hardened cement slurry. The tortuosity model of the concrete transport path is similar to that of mortar. It can be assumed that concrete is composed of pores, hardened mortar matrix, and coarse aggregate stone particles. The stone particles are regarded as rotating ellipsoidal particles with different diameters, embedded in the mortar matrix. Assuming that the cross-sectional shape of any stone aggregate is approximately elliptical, k is the ratio of length to diameter of a revolving ellipsoid, representing the shape parameters of the ellipsoid. When k = a/b > 1, the ellipsoid is obtained by rotating around the X axis with b as its radius, while when k = a/b < 1, the ellipsoid can be obtained by rotating around the Y axis with a as its radius. The volume (V) and the surface area (S) of the rotating ellipsoid is as follows:

V ðbÞ ¼

8 < :

3 4p kb 3

k>1

4p 2 3 k b 3

k<1

ð23Þ

2 2

4pk b k < 1

For ellipsoidal aggregates such as pebbles, the equivalent diameter (Deq) of an aspheric particle is defined as the diameter of a sphere which equals to the volume of the aspheric particle. In other words, by the equivalent diameter, the particle size distribution of the non-spherical aggregate particles can be correlated with the particle size distribution of the equivalent spherical particles. For a rotating ellipsoid,

(

Deq ¼

1

2bk3 k > 1 2bk

23

ð24Þ

k<1

The rotating ellipsoid volume and surface area of the equivalent diameter are expressed, respectively:



(



pD 3

k>1

pD 3

k<1

eq

6

ð20Þ

scp

scp ¼ gc ð1  ha Þsuhcp þ ha wwc gr shcp

shcp ¼

2

4pkb k > 1

SðbÞ ¼

V c Deq ¼

The tortuosity (scp) and porosity of cement paste are detailed as follows [12]:

suhcp

(

eq

6

 Sc Deq ¼

(

ð25Þ

pk3 Deq 2 k > 1 2 pk3 Deq 2 k < 1

ð26Þ

This paper mainly discusses two special grades of aggregate: Fuller grade aggregates and equal volume grade aggregates. The particle size grading of ellipsoidal aggregates in concrete is also typically characterized by two distribution functions. The two size distribution functions represent the upper and lower limits of aggregate size grading in concrete. The elliptical aggregate size distribution function fd(x) is shown in Eq. (27) [19]:

q f d ðxÞ ¼  qþ1 q Dmaxeq  Dq mineq Deq



q ¼ 2:5 ! fuller q ¼ 3 ! EVF

ð27Þ

The volume fraction of coarse aggregate in concrete can be calculated by concrete mix ratio. When the edge length of concrete blocks is known to be L, the number of ellipsoidal aggregates (N) can be expressed as follows:

f st  L3

N¼Z

Dmaxeq

Dmineq

ð28Þ

f d ðxÞ  V c ðxÞdx

The average value of surface spacing between adjacent ellipsoidal aggregates can be expressed by the mean free path (kc ), which is directly affected by sand volume fraction (fst) and solid specific surface area (SV), as shown in Eqs. (29) and (30):

Z

Dmineq

SV ¼ kc ¼

Dmaxeq

Nf d ðxÞSc ðxÞdx ð29Þ

L3 4ð1  f st Þ SV

ð30Þ


The average short half-axis length of ellipsoidal aggregate (b) and the average chord length (
k) can be calculated as follows:

 b Deq ¼

8 1 > < Deq k3 > :

2 2 Deq k3

2


¼ b

Z

Deqmax

k>1

ð31Þ

k<1

f d ðxÞbðxÞdx

ð32Þ

Deqmin

ð22Þ


l
c ¼ 2kb

ð33Þ

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C. Tongning et al. / Construction and Building Materials 241 (2020) 118045

The average arc length of ellipsoidal aggregate (
s) is:

sc ¼ 2pb þ 4bðk  1Þ

ð34Þ

By substituting Eqs. (32), (33) and (34) into Eq. (12b), the expression of the average tortuosity of concrete (sc) is as follows:

sc ¼


þ 4b
ðk  1Þ þ 4ð1f st Þ 2p b SV sm
þ 4ð1f st Þ 2kb

ð35Þ

SV

Porosity (u) is an important parameter reflecting concrete micro-structure and affecting ion diffusion performance in concrete. The formula of porosity is shown as follows [20]:


 w=c  0:39ha / ¼ Max f v c ;0 w=c þ 0:32

ð36Þ (a) Relationship between water diffusion coefficient and saturation

2.4. Water diffusion coefficient For the wetting process, according to Eqs. (8), (11a), (35), and (36), the liquid water diffusion coefficient (Dlw) can be obtained:

qRT @hs

Dlw ð/; s; SÞ ¼ K L

ð37Þ

Mhs @S

According to Eqs. (10), (11a), (35), and (36), the gaseous state water diffusion coefficient (Dvw) of the wetting process can be obtained:

Dv w ð/; s; SÞ ¼ K V pv s

@hs @S

ð38Þ

The water diffusion coefficient (Dmw) of the wetting process can be obtained from Eqs. (37) and (38):

Dmw ð/; s; SÞ ¼ K L

qRT @hs Mhs @S

þ K V pv s

@hs @S

(b) Relationship between water diffusion coefficient and tortuosity

ð39Þ

For the drying process, according to Eqs. (8), (11b), (35), and (36), the liquid water diffusion coefficient (Dld) can be obtained:

Dld ð/; s; SÞ ¼ K L

qRT @hdes Mhdes @S

ð40Þ

According to Eqs. (10), (11b), (35), and (36), the gaseous state water diffusion coefficient (Dvd) of the drying process can be obtained:

Dv d ð/; s; SÞ ¼ K V pv s

@hdes @S

ð41Þ

The water diffusion coefficient (Dmd) of the drying process can be obtained from Eqs. (40) and (41):

Dmd ð/; s; SÞ ¼ K L

qRT @hdes @hdes þ K V pv s Mhdes @S @S

ð42Þ

The derived water diffusion coefficient equations, Eqs. (39) and (42), show that water is mainly controlled by key factors such as porosity, tortuosity, and saturation of the concrete during the transport process. Therefore, the established water diffusion coefficient model is a variable coefficient model with independent variables of pore saturation, tortuosity and porosity, rather than the usual constant coefficient transport model. The effects of saturation, tortuosity and porosity on the water diffusion coefficients of the wetting and drying processes are simulated according to Eqs. (39) and (42). The relationship between water diffusion coefficient and saturation, tortuosity and porosity are shown in Fig. 3. The water diffusion coefficient in the wetting process is larger than that in the drying process because of the hysteresis effect in the water transport process [22]. Specifically, for a given tortuosity and porosity, the water diffusion coefficient varies with pore water saturation, as shown in Fig. 3a. The figure shows that when the saturation of the pore water is less than 0.75, the diffusion coefficients

(c) Relationship between water diffusion coefficient and porosity Fig. 3. Relationship between water diffusion coefficient and saturation, tortuosity, and porosity.

of the drying and wetting processes are basically the same. However, when the saturation of the pore water is greater than 0.75, the water diffusion process of the wetting and drying processes is significantly affected by the saturation. As the saturation increases, the diffusion coefficient also increases. Fig. 3(b) shows the effect of the change of tortuosity on the water diffusion coefficient. For a given water saturation and porosity, the water diffusion coefficient decreases with the tortuosity increases. It shows the effect of the change in porosity on the water diffusion coefficient in Fig. 3(c). The figure shows that for a given water saturation and tortuosity, the water diffusion coefficient increases as the porosity increases.

C. Tongning et al. / Construction and Building Materials 241 (2020) 118045

2.5. Hydraulic diffusion coefficient under a bending load The stress generated by a bending load creates microcracks inside concrete, which changes the microstructural characteristics of the concrete. Especially in the tension zone, the microcracks will increase. With the increase of bending load, the expansion of the microcracks leads to a decrease in the tortuosity of the concrete. When the load in the tension zone increases from 0 to the tensile strength of the concrete, the tortuosity of the concrete decreases from sc to 1. According to the phenomenological method, the porosity and tortuosity of concrete under stress can be expressed as follows [21]: The tortuosity of concrete under stress is

sc r ¼ sc  sc 1  f t ½

r signðrÞþ1 2

þ1

ð43Þ

The porosity of concrete under stress is

ur ¼ u1  f t ½

r signðrÞþ1 2



ð44Þ

where sc r is the tortuosity of concrete under stress conditions; sc is the initial tortuosity of concrete in the unstressed state; ft is the tensile strength of the concrete; and r is the stress applied to structural concrete. The sign is a symbol function which can be expressed as follows:

8 > < 1; sign ¼ 0; > : 1;

r>0 Tensile stress r¼0 Stress free r < 0 Compressive stress

ð45Þ

2.6. Constitutive equation of chloride ion transport under the coupled effect of the bending load and drying–wetting cycles The pores in the area of concrete exposed to alternate drying and wetting are always in an unsaturated state. Since concentration diffusion can occur only in the pore liquid, it is necessary to reduce the diffusion coefficient in the saturated state according to the pore saturation. The chloride ion diffusion coefficient is generally considered to be proportional to the pore saturation, and the chloride ion diffusion flux (JCl) is [15]

J Cl ¼ Ds  S  gradðC Þ C¼

U qcon uS

ð46aÞ ð46bÞ

where qcon represents the density of concrete, and U represents the content of chloride ions in concrete. Since the action of the drying and wetting cycles, the pore saturation distribution of the concrete is always in a non-uniform state from the surface to the interior, thus forming a pore saturation distribution field. The pore fluid seeps because of the field, so the dissolved chloride ions form a convection phenomenon with the pore liquid inside the concrete, with which the diffusionconvection equation can be expressed as [15]

J Cl ¼ Ds  S  gradðC Þ þ C  J l

ð47Þ

Combining with the conservation of mass of the chloride ions, the constitutive equation of chloride ion transport in the unsaturated state can be expressed as [22]

@C ¼ div ðDs  S  gradðC Þ þ C  Dl ðu; s; SÞ  gradðSÞÞ @t

ð48Þ

According to Eqs. (43)~(45), Eqs. (37)~(42) and Eq. (48), the constitutive equation of chloride ion transport under the coupled effect of bending load and drying–wetting cycle can be obtained:

 @C @ @C @S ¼ Ds  S  þ C  Dl ðr; SÞ  @t @x @x @x
 @ @C @S Ds  S  þ C  Dl ðr; SÞ  þ @y @y @y

7




ð49Þ

2.7. Boundary conditions In the tension zone of concrete, the applied stress should not exceed the concrete of tensile strength. The applied load of this model is 0.4 times of the tensile strength. During the drying and wetting cycles, the boundary conditions of the wetting process are relatively simple. After contacting the chloride ion solution, the pores in the surface of the concrete are considered as saturated, and the chloride ion concentration in the pore water is the same as that in the boundary environment. During the drying process, the water in the pores of the concrete surface evaporates at a fixed velocity (v), while the chloride ions in the water remain in the concrete surface. This state can be expressed by the second type of boundary condition of the partial differential equation.

Dmd ðr; SÞ  gradðSÞ ¼ v

ð50Þ

gradðC Þ ¼ 0

ð51Þ

For the drying process, the change of pore water saturation and chloride ion concentration in the surface concrete is related to the water evaporation rate. If the position of the concrete environmental boundary is set to x = 0, the continuity condition of the water transport rate (I) at the boundary is expressed as:

Ix¼0 ¼ Ix¼0þ 

ð52Þ +

where 0 and 0 represent the boundary of the ambient air and concrete, respectively. If the concrete is indoors, the wind speed in the room is very low. Therefore, water vapor convection from the concrete surface to the environment can be ignored. The diffusion flow of water vapor in the atmosphere at the boundary can be approximated as follows [12]:

Ix¼0 ¼ 

Dva M w pv s hs  he  RT d

ð53Þ

where Dva represents the water vapor diffusion coefficient, Mw represents the molar mass of water, hs and he represent the relative humidity of the concrete surface and the environment, respectively. d is the thickness of the boundary layer of the air velocity on the concrete surface, averaging about 25 mm. The water transport rate on the concrete side of the boundary can be expressed as:

Ix¼0þ ¼ Dmd ðr; SÞ

@S @x

ð54Þ

Combining Eq. (50) through Eq. (54), the equation for the boundary conditions of the water transport in the x direction is:

Dmd ðr; SÞ

@S Dv a Mw pv s ðhs  he Þ ¼ @x RTd

ð55Þ

The equation for the boundary conditions of the chloride transport equation in the x direction is:

@C ¼0 @x

ð56Þ

The calculation of boundary conditions for the y direction is the same as for the x direction. If the position of the concrete environmental boundary is set to y = 0, the continuity condition of the water transport rate (J) at the boundary is expressed as follows:

J x¼0 ¼ J x¼0þ

ð57Þ

8

C. Tongning et al. / Construction and Building Materials 241 (2020) 118045

The equation for the boundary conditions of the water transport in the y direction is:

Dmd ðr; SÞ

@S Dv a M w pv s ðhs  he Þ ¼ @y RTd

ð58Þ

The equation for the boundary conditions of the chloride ion transfer in the y direction is:

@C ¼0 @y

ð59Þ

So far, a multi-scale model for predicting chloride diffusion coefficient of concrete is established, and its prediction process is shown in Fig. 4. 2.8. Numerical calculation process of variable coefficient of chloride ion transport model by MATLAB programming The conventional finite difference method is simple in calculation, but the convergence is weak and the calculation results are not accurate enough. Although the results of the parallel finite element method and the radial integral method are accurate enough, the calculation cost is high and these methods are impractical. In this paper, the alternating implicit finite difference (ADI) method, which is a kind of finite difference method, is used to solve the chloride ion transport control equation and shows small calculation error and unconditional convergence. After defining the integral step size, one equation is divided into two equations, which are implicit in the x direction and the y direction. 2.8.1. Discretization of the water transport equation Based on Eqs. (4), (39) and (42), two-dimensional partial differential equation of concrete pore water saturation can be expressed as:



 @S @ @S @ @S þ ¼ Dmwx ðx; y; t Þ Dmwy ðx; y; t Þ @t @x @x @y @y

ð60Þ

2.8.2. Solution of the water transport equation First, mesh the solution area, so that the space increment is Dh = l/M, the time increment is Dt = T/K, and the parallel line cluster is given as follows [23]:

xi ¼ ih; yj ¼ jh; ði; j ¼ 0; 1; :::M Þ

ð61Þ

tk ¼ kDt ðk ¼ 0; 1; :::K Þ

Introducing the transition layer k + 1/2 layer, the transition happens from the k layer to the k + 1 layer in two steps: The first step is to use the implicit format in the x direction and the explicit in the y direction from the k layer to the k + 1/2 layer. That is, @ 2 S=@x2 is replaced by the difference quotient on the k + 1/2 layer, and@ 2 S=@y2 is replaced by the difference quotient on the k layer. After replacement,

@Sði; j; k þ 1=2Þ Sði; j; k þ 1=2Þ  Sði; j; kÞ ¼ @t Dt=2  Þ =@x ¼ h12 ½Dmwx ði þ 1=2; j;k þ 1=2Þ @ Dmwx @Sði;j;kþ1=2 @x

ð62Þ

ð63Þ

ðSði þ 1; j; k þ 1=2Þ  Sði; j; k þ 1=2ÞÞ @Dmwx ði  1=2; j; k þ 1=2ÞðSði; j;k þ 1=2Þ  Sði  1;j; k þ 1=2ÞÞ  i;j;kÞ =@y ¼ h12 @ Dmwy @Sð@y  Dmwy ði; j þ 1=2; kÞðSði; j þ 1; kÞ  Sði; j; kÞÞ  Dmwy ði; j  1=2; kÞ  ðSði; j; kÞ-Sði; j  1; kÞÞ

ð64Þ Substituting Eqs. (62), (63) and (64) into Eq. (60) is the first step in the ADI solution method for the two-dimensional variable-factor diffusion equation. In the second step, the explicit format is used in the x direction from the k + 1/2 layer to the k + 1 layer, and implicit is used in the y direction. That is,@ 2 S=@x2 is replaced by the difference quotient on the k + 1/2 layer, and@ 2 S=@y2 is replaced by the difference quotient on the k + 1 layer.

@Sði; j; k þ 1Þ Sði; j; k þ 1Þ  Sði; j; k þ 1=2Þ ¼ @t Dt=2  . Þ @x @ Dmwx @Sði;j;kþ1=2 @x

ð65Þ

¼ h12 ½Dmwx ði þ 1=2; j; k þ 1=2ÞðSði þ 1; j; k þ 1=2Þ  Sði; j; k þ 1=2ÞÞ  Dmwx ði  1=2; j; k þ 1=2ÞðSði; j; k þ 1=2Þ  Sði; j; k þ 1=2Þ ð66Þ  . Þ @ Dmwy @Sði;j;kþ1 @y @y  ¼ h12 Dmwy ði; j þ 1=2; k þ 1ÞðSði; j þ 1; k þ 1Þ  Sði; j; k þ 1ÞÞ

ð67Þ

 Dmwy ði; j  1=2; k þ 1ÞðSði; j; k þ 1Þ  Sði; j  1; k þ 1ÞÞ

Substituting Eqs. (65), (66) and (67) into Eq. (60) is the second step in the ADI solution format for the two-dimensional variablefactor diffusion equation. Among them,

Dmwx ði þ 1=2; j; k þ 1=2Þ ¼ ðDmwx ði; j; k þ 1=2Þ þ Dmwx ði þ 1; j; k þ 1=2ÞÞ  1=2

ð68aÞ

Dmwx ði  1=2; j; k þ 1=2Þ ¼ ðDmwx ði; j; k þ 1=2ÞDmwx ði  1; j; k þ 1=2ÞÞ  1=2

ð68bÞ

 Dmwy ði; j þ 1=2; kÞ ¼ Dmwy ði; j; kÞ þ Dmwy ði; j þ 1; kÞ  1=2

ð68cÞ



Fig. 4. Flow chart of chloride diffusion coefficient of concrete.

 Dmwy ði; j  1=2; kÞ ¼ Dmwy ði; j; kÞ þ Dmwy ði; j-1; kÞ  1=2

ð68dÞ

9

C. Tongning et al. / Construction and Building Materials 241 (2020) 118045

2.8.3. Discretization of the chloride ion transport equation The partial differential equation for the two-dimensional transport of chloride ions of Eq. (49) can be expressed as follows:

 @S þ C  Dlw ðx; y; t Þ  @x @C @ Ds ðx; y; t Þ  S  @C @x ¼ @t @x  @S @ Ds ðx; y; t Þ  S  @C þ C  Dlw ðx; y; tÞ  @y @y þ @y

ð69Þ

2.8.4. Solution of the chloride ion transport equation As with the method of solving the saturation S, the ADI method is used in calculating the chloride ion transport [23]. First, mesh the solution area, so that the space increment is Dh = l/M, the time increment is Dt = T/K, and the parallel line cluster is given as follows:

xi ¼ ih; yj ¼ jh; ði; j ¼ 0; 1; :::M Þ t k ¼ kDtðk ¼ 0; 1; :::K Þ

ð70Þ


 @C ði; j;kÞ @Sði;j; kÞ @ Ds ði; j;kÞ  Sði; j;kÞ  þ C ði;j; kÞ  Dlw ði;j; kÞ  @y @ @ 
 1 U ði; j þ 1;kÞ U ði;j; kÞ  ¼ 2 Ds ði;j þ 1=2;kÞSði; j þ 1=2; kÞ Sði; j þ 1; kÞ Sði; j;kÞ h
 U ði; j þ 1=2;kÞ þ Dlwy ði; j þ 1=2; kÞðSði; j þ 1;kÞ  Sði; j; kÞÞ Sði; j þ 1=2; kÞ
 U ði; j; kÞ U ði  1; j  1;kÞ   Ds ði;j  1=2;kÞSði; j  1=2; kÞ Sði; j;kÞ Sði  1; j  1; kÞ
 U ði;j  1=2;kÞ þ Dlwy ði; j  1=2; kÞðSði; j;kÞ  Sð1;j  1; kÞÞ Sði; j  1=2; kÞ ð73Þ In the second step, the explicit format is used in the x direction from the k+1/2 layer to the k+1 layer, and implicit is used in the y direction.

@C ði; j; k þ 1Þ ¼ @t

U ði;j;kþ1Þ Sði;j;kþ1Þ

i;j;kþ1=2Þ  USðði;j;kþ1=2 Þ

Dt=2


 @C ði; j; k þ 1=2Þ @Sði; j; k þ 1=2Þ þ C ði; j; k þ 1=2Þ  Dlw ði; j; k þ 1=2Þ  @x @ Ds ði; j; k1=2Þ  Sði; j; k þ 1=2Þ  @x @x 
 1 U ði þ 1; j; k þ 1=2Þ U ði; j; k þ 1=2Þ  ¼ 2 Ds ði þ 1=2; j; k þ 1=2ÞSði þ 1=2; j; k þ 1=2Þ  Sði þ 1; j; k þ 1=2Þ Sði; j; k þ 1=2Þ h
 U ði þ 1=2; j; k þ 1=2Þ þ Dlwx ði þ 1=2; j; k1=2ÞðSði þ 1; j; k þ 1=2Þ  Sði; j; k þ 1=2ÞÞ Sði þ 1=2; j; k þ 1=2Þ
 U ði; j; k þ 1=2Þ U ði  1; j; k þ 1=2Þ   Ds ði  1=2; j; k þ 1=2ÞSði  1=2; j; k þ 1=2Þ  Sði; j; k þ 1=2Þ Sði  1; j; k þ 1=2Þ
 U ði  1=2; j; k þ 1=2Þ þ Dlwx ði  1=2; j; k þ 1=2ÞðSði; j; k þ 1=2Þ  Sði  1; j; k þ 1=2ÞÞ Sði  1=2; j; k þ 1=2Þ

ð74Þ

ð75Þ

Introducing the transition layer k + 1/2 layer, the transition happens from the k layer to the k + 1 layer in two steps: The first step is to use the implicit format in the x direction and the explicit in the y direction from the k layer to the k + 1/2 layer. That is, @ 2 S=@x2 is replaced by the difference quotient on the k + 1/2 layer, and@ 2 S=@y2 is replaced by the difference quotient on the k layer. After replacement,

 @C ði; j; k þ 1=2Þ U ði; j; k þ 1=2Þ U ði; j; kÞ ¼  Dt=2 @t Sði; j; k þ 1=2Þ Sði; j; kÞ

ð71Þ



 @C ði; j; k þ 1=2Þ @Sði; j; k þ 1=2Þ @ Ds ði; j; k þ 1=2Þ  Sði; j; k þ 1=2Þ  þ C i; j; k þ 1=2  Dlw ði; j; k þ 1=2Þ  @x @x @x 
 1 U ði þ 1; j; k þ 1=2Þ U ði; j; k þ 1=2Þ  ¼ 2 Ds ði þ 1=2; j; k þ 1=2ÞSði þ 1=2; j; k þ 1=2Þ  Sði þ 1; j; k þ 1=2Þ Sði; j; k þ 1=2Þ h
 U ði þ 1=2; j; k þ 1=2Þ þ Dlwx ði þ 1=2; j; k þ 1=2ÞðSði þ 1; j; k þ 1=2Þ  Sði; j; k þ 1=2ÞÞ Sði þ 1=2; j; k þ 1=2Þ
 U ði; j; k þ 1=2Þ U ði  1; j; k þ 1=2Þ   Ds ði  1=2; j; k þ 1=2ÞSði  1=2; j; k þ 1=2Þ  Sði; j; k þ 1=2Þ Sði  1; j; k þ 1=2Þ
 U ði  1=2; j; k þ 1=2Þ þ Dlwx ði  1=2; j; k þ 1=2ÞðSði; j; k þ 1=2Þ  Sði  1; j; k þ 1=2ÞÞ Sði  1=2; j; k þ 1=2Þ

ð72Þ

10

C. Tongning et al. / Construction and Building Materials 241 (2020) 118045


 @C ði; j; k þ 1Þ @Sði; j; k þ 1Þ þ C ði; j; k þ 1Þ  Dlw ði; j; k þ 1Þ  @y @ Ds ði; j; k þ 1Þ  Sði; j; k þ 1Þ  @y @y 
 1 U ði; j þ 1; k þ 1Þ U ði; j; k þ 1Þ  ¼ 2 Ds ði; j þ 1=2; k þ 1ÞSði; j þ 1=2; k þ 1Þ  Sði; j þ 1; k þ 1Þ Sði; j; k þ 1Þ h
 U ði; j þ 1=2; k þ 1Þ þ Dlwy ði; j þ 1=2; k þ 1ÞðSði; j þ 1; k þ 1Þ  Sði; j; k þ 1ÞÞ Sði; j þ 1=2; k þ 1Þ
 U ði; j; k þ 1Þ U ði  1; j  1; k þ 1Þ  Ds ði; j  1=2; k þ 1ÞSði; j  1=2; k þ 1Þ   Sði; j; k þ 1Þ Sði  1; j  1; k þ 1Þ
 U ði; j  1=2; k þ 1Þ þ Dlwy ði; j  1=2; k þ 1ÞðSði; j; k þ 1Þ  Sði  1; j  1; k þ 1ÞÞ Sði; j  1=2; k þ 1Þ

ð76Þ

In the formula,

Dlwx ði þ 1=2;j;k þ 1=2Þ ¼ ðDlwx ði;j; k þ 1=2Þ þ Dlwx ði þ 1;j;k þ 1=2ÞÞ  1=2 ð77aÞ Dlwx ði  1=2;j;k þ 1=2Þ ¼ ðDlwx ði;j; k þ 1=2Þ þ Dlwx ði  1;j;k þ 1=2ÞÞ  1=2 ð77bÞ

 Dlwy ði; j þ 1=2; kÞ ¼ Dlwy ði; j; kÞ þ Dlwy ði; j þ 1; kÞ  1=2

ð77cÞ

 Dlwy ði; j  1=2; kÞ ¼ Dlwy ði; j; kÞ þ Dlwy ði; j  1; kÞ  1=2

ð77dÞ

Similarly, the expression of the equation of the chloride ion convection diffusion in the boundary layer during the drying process can be obtained:

2.8.5. Discretization of boundary condition models During the drying process, the water in the pores of the concrete surface evaporates at a fixed velocity (v), while the chloride ions in the water remain in the concrete surface. This can be expressed by the second type of boundary condition of the partial differential equation, which is,

Dmd ðr; SÞ  gradðSÞ ¼ v

v¼

Dv a M w pv s ðhs  he Þ RTd

gradðC Þ ¼ 0

ð78Þ ð79Þ ð80Þ

In the model calculation, I = 1 is used as the boundary layer of concrete. When dealing with such boundary conditions, it is assumed that there is an i = 0 layer, and Eqs. (78) and (80) can be expressed as

Sð2; j; kÞ  Sð0; j; kÞ v ¼ 2h Dmd ð1; j; kÞ

ð81Þ

C ð2; j; kÞ  C ð0; j; kÞ ¼ 0

ð82Þ

The water diffusion equation is developed in the i = 1 layer in the explicit format, and obtain the pore saturation calculation format in the boundary layer during the drying process:

Dt ½Dmd ðSð3=2; j;kÞÞ þ Dmd ðSð1=2;j;kÞÞ h

Dmd ðSð1=2;j;kÞÞ  2hv  ðSð2; j;kÞ  Sð1; j;kÞÞ  Dmd ðSð1;j; kÞÞ ð83aÞ

Sð1; j;k þ 1=2Þ ¼ Sð1;j;kÞ þ

Dt ½Dmd ðSð3=2;j; k þ 1=2ÞÞ h þ Dmd ðSð1=2; j;k þ 1=2ÞÞ  ðSð2; j;k þ 1=2Þ

Dmd ðSð1=2; j;k þ 1=2ÞÞ  Sð1;j; k þ 1=2ÞÞ   2hv Dmd ðSð1;j; k þ 1=2ÞÞ

Fig. 5. Algorithm flowchart of MATLAB.

Table 1 Mix design of concrete (kg/m3).

Sð1; j;k þ 1Þ ¼ Sð1; j;k þ 1=2Þ þ

ð83bÞ

Number

Cement

Sand

Stone

Water

1 2 3

614 478 391

671 733 772

788 860 906

215 215 215

C. Tongning et al. / Construction and Building Materials 241 (2020) 118045


 U ð1; j; k þ 1=2Þ U ð1; j; kÞ Dt U ð2; j; kÞ U ð1; j; kÞ ¼ þ Sð3=2; j; kÞDs  Sð1; j; k þ 1Þ Sð1; j; kÞ h Sð2; j; kÞ Sð1; j; kÞ
 U ð1; j; kÞ U ð2; j; kÞ   Sð1=2; j; kÞDs Sð1; j; kÞ Sð2; j; kÞ  2hv =ðDld ðSð1; j; kÞÞÞ
 1 U ð1; j; kÞ U ð2; j; kÞ þ þ Dld ðSð3=2; j; kÞÞðSð2; j; kÞ  Sð1; j; kÞÞ 2 Sð1; j; kÞ Sð2; j; kÞ
 1 U ð1; j; kÞ U ð2; j; kÞ þ  Dld ðSð3=2; j; kÞÞ 2 Sð1; j; kÞ Sð2; j; kÞ  2hv =ðDld ðSð1; j; kÞÞÞ   ðSð1; j; kÞ  Sð2; j; kÞ þ 2hv =ðDld ðSð1; j; kÞÞÞ ð84aÞ U ð1; j; k þ 1Þ U ð1; j; k þ 1=2Þ ¼ Sð1; j; k þ 1Þ Sð1; j; k þ 1=2Þ 
 Dt U ð2; j; k þ 1=2Þ U ð1; j; k þ 1=2Þ Sð3=2; j; k þ 1=2ÞDs  þ h Sð2; j; k þ 1=2Þ Sð1; j; k þ 1=2Þ
U ð1; j; k þ 1=2Þ  Sð1=2; j; k þ 1=2ÞDs Sð1; j; k þ 1=2Þ  U ð2; j; k þ 1=2Þ  Sð2; j; k þ 1=2Þ  2hv =ðDld ðSð1; j; k þ 1=2ÞÞÞ
 1 U ð1; j; k þ 1=2Þ U ð2; j; k þ 1=2Þ þ þ 2 Sð1; j; k þ 1=2Þ Sð2; j; k þ 1=2Þ

 Sð2; j; k þ 1=2Þ þ 2hv =ðDld ðSð1; j; k þ 1=2ÞÞÞ

Among them,

Dld ðSð1=2; j; kÞÞ ¼ ðDld ðSð1; j; kÞÞ þ Dld ðSð0; j; kÞÞÞ  1=2

ð85aÞ

Dld ðSð3=2; j; kÞÞ ¼ ðDld ðSð1; j; kÞÞ þ Dld ðSð2; j; kÞÞÞ  1=2

ð85bÞ

In the j = 1 layer, the expansion mode is the same as in the i = 1 layer. In the model calculation, j = 1 is used as the boundary layer of concrete. When dealing with such boundary conditions, it is assumed that there is an i = 0 layer, and Eqs. (78) and (80) can be expressed as

Sði; 2; kÞ  Sði; 0; kÞ v ¼ 2h Dmd ði; 1; kÞ

ð86Þ

C ði; 2; kÞ  C ði; 0; kÞ ¼ 0

ð87Þ

The water diffusion equation is developed in the j = 1 layer in the explicit format, and obtain the pore saturation calculation format in the boundary layer during the drying process:

Dt ½Dmd ðSði; 3=2;kÞÞ þ Dmd ðSði;1=2; kÞÞ h

Dmd ðSði;1=2; kÞÞ ð88Þ  2hv  ðSði; 2; kÞ  Sði; 1;kÞÞ  Dmd ðSði; 1;kÞÞ

Sði;1;k þ 1Þ ¼ Sði; 1; kÞ þ

 Dld ðSð3=2; j; k þ 1=2ÞÞðSð2; j; k þ 1=2Þ  Sð1; j; k þ 1=2ÞÞ
 1 U ð1; j; k þ 1=2Þ U ð2; j; k þ 1=2Þ  þ 2 Sð1; j; k þ 1=2Þ Sð2; j; k þ 1=2Þ  2hv =ðDld ðSð1; j; k þ 1=2ÞÞÞ  Dld ðSð3=2; j; k þ 1=2ÞÞ  ðSð1; j; k þ 1=2Þ

11

 ð84bÞ

Similarly, the expression of the equation of the chloride ion convection–diffusion in the boundary layer during the drying process can be obtained:


 U ði; 1; k þ 1Þ U ði; 1; kÞ Dt U ði; 2; kÞ U ði; 1; kÞ ¼ þ Sði; 3=2; kÞDs  Sði; 1; k þ 1Þ Sði; 1; kÞ h Sði; 2; kÞ Sði; 1; kÞ
 U ði; 1; kÞ U ði; 2; kÞ   Sði; 1=2; kÞDs Sði; 1; kÞ Sði; 2; kÞ  2hv =ðDld ðSði; 1; kÞÞÞ
 1 U ði; 1; kÞ U ði; 2; kÞ  Dld ðSði; 3=2; kÞÞðSði; 2; kÞ  Sði; 1; kÞÞ þ þ 2 Sði; 1; kÞ Sði; 2; kÞ
 1 U ði; 1; kÞ U ði; 2; kÞ þ  2 Sði; 1; kÞ Sði; 2; kÞ  2hv =ðDld ðSði; 1; kÞÞÞ   Dld ðSði; 3=2; kÞÞðSði; 1; kÞ  Sði; 2; kÞ þ 2hv =ðDld ðSði; 1; kÞÞÞ ð89Þ Among them,

Fig. 6. Apparatus for determining the ion transport of concrete under dynamic bending loading and drying–wetting cycles.

Dmd ðSði; 3=2; kÞÞ ¼ ðDmd ðSði; 1; kÞÞ þ Dmd ðSði; 2; kÞÞÞ  1=2

ð90aÞ

Dmd ðSði; 1=2; kÞÞ ¼ ðDmd ðSði; 1; kÞÞ þ Dmd ðSði; 0; kÞÞÞ:1=2

ð90bÞ

During programming, the MATLAB main program first calculates the water saturation in the concrete through the water transfer control equation, and substitutes the obtained saturation value into the chloride ion transfer control equation to find the free chloride ion concentration. Then the next cycle will be sought again. The obtained free chloride ion concentration value is back to the

Fig. 7. Example of a sampling position map.

12

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water transfer control equation. According to this model, the free chloride ion concentration in each layer of concrete can be obtained. The specific calculation process is shown in Fig. 5.

3. Experimental design and verification To verify the rationality of the model of the transport of chloride ions in concrete under the coupled effects of a bending load and

(a) Distribution of chloride ion content by simulation for No.1 samples

(b) Comparison of simulated and experimental results at the diagonal for No.1 samples

(c) Distribution of chloride ion content by simulation for No.2 samples

(d) Comparison of simulated and experimental results at the diagonal for No.2 samples

(e) Distribution of chloride ion content by simulation for No.3 samples

(f) Comparison of simulated and experimental results at the diagonal for No.3 samples

Fig. 8. Results of distribution of chloride ion content obtained by experiment and simulation.

C. Tongning et al. / Construction and Building Materials 241 (2020) 118045

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drying–wetting cycles, a test process was used. The test process was shown as follows. 3.1. Materials for testing and their mix ratio The coarse aggregate have a particle size of 5 mm to 20 mm and an apparent density of 2700 kg/m3. The fine aggregate is river sand with an apparent density of 2620 kg/m3. The cement is 42.5 ordinary Portland cement produced by China United Cement Group. The water-to-binder ratios of the concrete were 0.35, 0.45 and 0.55, as shown in Table 1. 3.2. Test process For the convenience of loading, the test pieces of each group numbered in Table 1 were molded into nine test pieces. The molded part is 70 mm  70 mm  280 mm. After standard curing for 28 days, three test pieces of each group were used to test flexural strength. The maximum load stress of the bending load did not exceed 0.4 times the flexural strength of the specimen. The designed device to apply the bending load was integrated in Fig. 6. The NaCl solution was placed at a concentration of wt. 5%. Two 70 mm  70 mm faces of each test piece were coated with epoxy resin, while the remaining four faces were reserved. Then each test piece was mounted and placed in a drying–wetting cycle tester. In this experiment, one cycle was 14 days, with a 7 days drying and a 7 days wetting. The experiment was carried out for two cycles. At the end of each drying–wetting cycle, three samples were taken from each set of test pieces. The samples were taken from within the top quarter of the 140 mm depth, down from the 70 mm  70 mm surface. As shown in Fig. 7, seven holes were drilled uniformly along the straight AE direction from point A in the test piece and the depth of each hole was set to 36 mm. The diameter of the hole is 12 mm. Sampling was done seven times at every 3 mm in each hole, and then the powder samples from the same set of three test pieces and from the same depth were mixed and placed in an oven. After baking at 50 °C for 24 h, 4.0 g of concrete sample from each depth was weighed and 40 mL of deionized water was used to form a solution. After fully shaking and standing for 24 h, the chloride ion concentration in the solution was measured by a chloride ion analyzer. In this paper, the effect of stress on chloride ion transport in concrete was transformed into the influence of stress on the tortuosity and porosity of the concrete. These parameters were substituted into the equation of water diffusion coefficient and that of the boundary conditions. Numerical simulations were performed using the model designed based on MATLAB software in this paper. MATLAB integrates numerical analysis, matrix calculations, scientific data visualization, and modeling and simulation of nonlinear dynamic systems into one easy-to-use window environment for scientific research, engineering design, and many sciences that must perform effective numerical calculations. The field provides a comprehensive solution that represents the state of the art in today’s international scientific computing software. Since the chloride ions were symmetrically diffused along the square cross section, one quarter of the square area (L  L) needed to be solved. The parameters entered in the model were shown as follows: 1. The concentration of chloride ions in the boundary solution is 5%; 2. The bending load is 0.4 times than the flexural strength; 3. The length of a drying and wetting cycle is 14 days, with the drying and wetting processes of 7 days, respectively; 4. The diffusion coefficient of a chloride ion in the saturated state is 5  1010 m2/s;

Fig. 9. Comparison of simulated results with experimental results [2].

Coordination is as shown in Table 1. The above conditions were substituted into the model. Fig. 8 shows that the simulated results agree well with the experimental data, and the R-square is 0.9631, which indicates that the proposed model is reasonable. To further verify the reliability of the variable coefficient transport model established in this paper, relevant experimental data were found in the literature. Guan et al. [2] conducted a series of tests to characterize the concentration distribution of chloride ions transported in concrete under the combined action of drying–wetting cycles and bending loads. The water-to-binder ratio of concrete in the experiment was 0.4; the dimensions of the test piece were 100 mm  100 mm  400 mm; the maximum stress level ratios applied were 0.6 and 0.4 times the tensile strength; one cycle was 48 h, with the drying and wetting time 24 h each; and the concentration of sodium chloride solution at the boundary was 8%. The distribution of the chloride ion concentration in the concrete under a dynamic bending load and drying–wetting cycle was measured after 30 days. The boundary conditions of this experiment were brought into the model proposed in this paper to simulate the concentration distribution of chloride ions. Fig. 9 shows that simulated results are basically the same as the experimental results. Through error analysis, the R-square is 0.9753, indicating that the model is reasonable. Chen et al. [24] also conducted a series of experiments to test the distribution of chloride ion concentration under a bending load and drying–wetting cycle. The strength of the concrete used in this test was C50, and the size of the test piece was 70 mm  70 mm  280 mm. After 28 days of curing, the sample was placed in a 5% NaCl solution with a wetting time of 24 h and a drying time of 48 h. Long-term loading was performed at many different stress levels using a four-point bending loading device until the end of the experiment. The numerical simulation was carried out using the model constructed in this paper under the same conditions. Fig. 10 shows that the simulation results agree with the experimental results. The R-square is 0.9565, again indicating that the model is reasonable. 4. Model parameter sensitivity analysis 4.1. Influence of the proportion of time of the wetting and drying processes in the cycle During cycles of drying and wetting, the transport of chloride ions in concrete is mainly controlled by diffusion and convection.

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To facilitate the investigation of the influence of drying and wetting cycles on the transport of chloride ions in concrete, the time ratio (D/W) of the drying and wetting processes is used. It is assumed that D/W = 1, 2 and 3. The ratio of the bending load to the statistical average bending strength that the specimen can

withstand is 0.4. The surface saturation is 0.5. These parameters are substituted into Eq. (49) and the simulation results are shown in Fig. 11. As D/W increased from 1 to 3, the chloride content increased by 5.1% and 6.3%, respectively. The two-dimensional map of chloride ion distribution in Fig. 11 (a)–(c) shows that a diffusion mechanism dominates the transport of chloride ions in concrete at low D/W. This diffusion mechanism gradually transforms into a convective mechanism with the increase of D/W. The main reason is that when D/W increases, the saturation of the pore solution in the surface concrete is reduced and the capillary pressure is increased at the same time, which accelerates the migration speed of the chloride ions from the pore liquid to the surface layer concrete. Therefore, the amount of chloride ions transported to the concrete surface layer significantly increases, as shown in the yellow area in Fig. 11(a)–(c). Correspondingly, the content of chloride ions is reduced inside the concrete, as indicated by the blue-green area in the figures. In addition, as D/W increases, the depth of chloride ion transport becomes shallower at the intersection of the two planes in the X and Y directions, with the content in the concrete decreases. Fig. 11(d) shows the specific results. 4.2. Influence of bending load ratio

Fig. 10. Comparison of simulated results with experimental results [24].

The bending load affects the internal microstructure of the concrete, which essentially changes the transport path of chloride ions in the concrete. Here the bending load ratio is defined by the ratio

(a)Distribution of chloride ion at D/W of 1:1

(b) Distribution of chloride ion at D/W of 2:1

(c) Distribution of chloride ion at D/W of 3:1

(d)Results of chloride ions at the diagonal

Fig. 11. Distribution of chloride ions content at different drying–wetting cycles.

C. Tongning et al. / Construction and Building Materials 241 (2020) 118045

(a) Distribution of chloride ion at σ=0.5

(b)Distribution of chloride ion at σ=0.4

(c)Distribution of chloride ion at σ=0.3

(d)Results of chloride ions at the diagonal

Fig. 12. Distribution of chloride ions content at different stress ratios.

(a) Distribution of chloride ion at S0= 0.7

(b) Distribution of chloride ion at S0= 0.5

(c) Distribution of chloride ion at S0= 0.3

(d)Results of chloride ions at the diagonal

Fig. 13. Distribution of chloride ions content at different Surface saturation.

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of the bending load applied to the statistical average bending strength that the specimen can withstand, expressed as r, where r = 0.3, 0.4, 0.5, and the surface saturation is given as 0.5 and 1, respectively. These parameters are substituted into Eqs. (4), (43), (44), and (49). Fig. 12 shows the simulation result. As r increased from 0.3 to 0.5, the chloride content increased by 3.2% and 3.5%, respectively. The two-dimensional map of the distribution of chloride ions shown in Fig. 12(a)–(c), indicate that with the increase of r, the tortuosity of the chloride ion transport path is decreased while the number of microcracks generated inside the concrete increases and the content of chloride ions transported into the concrete are increased. Furthermore, the chloride ion content increases at the junction of the X and Y direction planes as r increases. The specific results are shown in Fig. 12(d). 4.3. Surface concrete saturation during drying During a drying and wetting cycle, the saturation of the wetting process is 1, while the saturation of the surface concrete during the drying process is controlled by the relative humidity of the outside environment. In the process of corrosion of concrete by chloride salt, the saturation of the surface concrete is an important factor affecting the distribution of chloride ions in the concrete. In this test, the surface saturation is assumed to be S = 0.3, 0.5, 0.7, with a D/W of 1 and r of 0.4. The parameters are substituted into Eqs. (4), (49), (50) and (51). The simulation results are presented in Fig. 13. As S0 increased from 0.7 to 0.3, the chloride content decreased by 9.2% and 11.5%, respectively. As shown in the yellow area of Fig. 13(a)–(c) with the decrease of saturation of the surface concrete, the pressure of the capillary pores of the surface concrete is decreased, which drives the migration of chloride ions from the pore liquid to the surface concrete. Thus, the content of chloride ion transported to the concrete surface layer is increased. Correspondingly, the chloride ion content is reduced inside the concrete, as shown in the blue-green area of Fig. 13(a)–(c). Moreover, at the junction of the X and Y planes, as the surface concrete saturation increases, the transport mechanism of chloride ion in the concrete gradually changes from convection to diffusion, which lead to the increase of the content of chloride ions in the concrete, as shown in Fig. 13(d). 5. Conclusions (1) Based on Darcy0 s law and Fick0 s second law, the twodimensional variable coefficients of the constitutive equation of water and of chloride ion diffusion-convection in concrete in the unsaturated state are firstly established. Then, a model of the relationship between the bending load and tortuosity and porosity is built. Finally, a variable coefficient constitutive model for chloride ion transport under coupled bending and drying–wetting conditions is proposed, considering the key factors such as the porosity, tortuosity and saturation changes during the transport of chloride ions in concrete in this model. (2) The reliability of the model of chloride ion transport under a bending load and drying–wetting cycles is verified by experiment and by datas from the literatures. The results predicted by the model in this paper are consistent with the experimental results. Theoretical studies show that the two-dimensional variable coefficient constitutive equations of water and chloride ion transport can be solved by the alternating implicit finite difference (ADI) method, which satisfies the unconditional convergence condition. The ADI discretized equation is programmed and calculated with MATLAB software. The operating efficiency is significantly

improved and the correlation between simulation results and experimental results is good (3) The effects of the ratio of drying time to wetting time (D/W), the stress ratio, and the surface concrete saturation on the concentration distribution of chloride ions in concrete are analyzed theoretically. The results show that the chloride ion content in concrete decreases with the increase of D/W, the total chloride ion content in concrete increases with an increase in the bending load and surface concrete saturation. CRediT authorship contribution statement Cao Tongning: Conceptualization, Methodology, Software, Investigation, Writing - original draft. Zhang Lijuan: Validation, Formal analysis, Visualization, Software. Sun Guowen: Validation, Formal analysis, Visualization. Wang Caihui: Resources, Writing review & editing, Supervision, Data curation. Zhang Ying: Resources, Writing - review & editing, Supervision, Data curation. Wang Pengshuo: Writing - review & editing. Xu Aoxue: Writing - review & editing. Declaration of Competing Interest The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. Acknowledgements This work was supported by the National Natural Science Foundations of China (No. 51478278), (No. 51778378) and Higher Education Science and Technology Research Project of Hebei Province (No. ZD2016065). References [1] J. Zhang, Y. Zheng, J. Wang, Y. Zhang, Y. Gao, Chloride transport in concrete under flexural loads in a tidal environment, J. Mater. Civ. Eng. 30 (2018) 04018285, https://doi.org/10.1061/(asce)mt. 1943-5533.0002493. [2] B. Guan, J. Wu, T. Yang, A. Xu, Y. Sheng, H. Chen, Developing a model for chloride ions transport in cement concrete under dynamic flexural loading and dry-wet cycles, Math. Probl. Eng. 2017 (2017) 1–13, https://doi.org/10.1155/ 2017/5760512. [3] J. Xu, F. Li, J. Zhao, L. Huang, Model of time-dependent and stress-dependent chloride penetration of concrete under sustained axial pressure in the marine environment, Constr. Build. Mater. 170 (2018) 207–216, https://doi.org/ 10.1016/j.conbuildmat.2018.03.077. [4] J. Bao, L. Wang, Combined effect of water and sustained compressive loading on chloride penetration into concrete, Constr. Build. Mater. 156 (2017) 708– 718, https://doi.org/10.1016/j.conbuildmat.2017.09.018. [5] A. Petcherdchoo, Closed-form solutions for modeling chloride transport in unsaturated concrete under wet-dry cycles of chloride attack, Constr. Build. Mater. 176 (2018) 638–651, https://doi.org/10.1016/j.conbuildmat. 2018.05.083. [6] C. Fu, H. Ye, X. Jin, D. Yan, N. Jin, Z. Peng, Chloride penetration into concrete damaged by uniaxial tensile fatigue loading, Constr. Build. Mater. 125 (2016) 714–723, https://doi.org/10.1016/j.con- buildmat.2016.08.096. [7] G. Sun, Y. Zhang, W. Sun, Z. Liu, C. Wang, Multi-scale prediction of the effective chloride diffusion coefficient of concrete, Constr. Build. Mater. 25 (2011) 3820– 3831, https://doi.org/10.1016/j.con-buildmat.2011.03.041. [8] L. Wang, J. Bao, Investigation on chloride penetration into unsaturated concrete under short-term sustained tensile loading, Mater. Struct. Constr. 50 (2017) 1–15, https://doi.org/10.1617/s11527-017-1095-6. [9] X.H. Wang, E. Bastidas-Arteaga, Y. Gao, Probabilistic analysis of chloride penetration in reinforced concrete subjected to pre-exposure static and fatigue loading and wetting-drying cycles, Eng. Fail. Anal. 84 (2018) 205–219, https:// doi.org/10.1016/j.engfailanal.2017.11.008. [10] A. Petcherdchoo, Service life and environmental impact due to repairs by metakaolin concrete after chloride attack, RILEM Bookseries 10 (2015) 35–41. [11] X. Jun, F.M. Li, Analytical model for load dependence of chloride penetration into concrete, ASCE J. Mater. Civil Eng. 29 (5) (2017).

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