Modelling of two-dimensional chloride diffusion concentrations considering the heterogeneity of concrete materials

Construction and Building Materials 243 (2020) 118213

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Modelling of two-dimensional chloride diffusion concentrations considering the heterogeneity of concrete materials Linjian Wu a,⇑, Yuanzhan Wang b, Yuchi Wang c, Xueli Ju a, Qingmei Li b a National Engineering Research Center for Inland Waterway Regulation, School of River and Ocean Engineering, Chongqing Jiaotong University, 66 Xuefu Road, Nan’an District, Chongqing 400074, People’s Republic of China b State Key Laboratory of Hydraulic Engineering Simulation and Safety and Collaborative Innovation Center for Advanced Ship and Deep-Sea Exploration, Tianjin University, 135 Yaguan Road, Jinnan District, Tianjin 300072, People’s Republic of China c Tianjin Research Institute for Water Transport Engineering, Ministry of Transport, China 2618 Xingang 2nd Road, Binhai New District, Tianjin 300000, People’s Republic of China

h i g h l i g h t s
An indoor test for 2D chloride diffusion under real-time tidal cycle is studied.
Quantified index of the heterogeneity of concrete materials (HCMs) is provided.
Effect of heterogeneity of concrete materials on 2D chloride diffusion is explored.
Time & HCMs-dependent 2D chloride diffusion coefficient model is established.
Model of 2D chloride diffusion concentration considering the HCMs is proposed.

a r t i c l e

i n f o

Article history: Received 6 September 2019 Received in revised form 15 January 2020 Accepted 17 January 2020

Keywords: Concrete Heterogeneity of material Chloride Two-dimensional diffusion Prediction model

a b s t r a c t Chloride ingression into reinforced concrete (RC) is currently considered an important reason behind the deterioration of RC structures exposed to aggressive environments. Previous studies have been mostly devoted to investigations on the one-dimensional transport characteristics of chloride in concrete; however, for the RC structures exposed to marine environment, the corners and edges of various components are often subjected to two-dimensional chloride penetration. Moreover, concrete is generally considered a typical heterogeneous material because of its coarse aggregate being randomly distributed in a cement paste. Both the two-dimensional chloride diffusion behaviour and the heterogeneity of concrete materials have large influences on the chloride concentration distribution in concrete. For the investigations in this paper, using the coarse aggregate volume fraction (CAVF) to quantify the heterogeneity of concrete materials, an indoor experiment for exploring the two-dimensional chloride diffusion behavior of concrete under real-time tidal cycles in a marine environment was carried out. The two-dimensional chloride diffusion concentrations within concrete specimens cast using different CAVFs of Vca = 0, 0.2, 0.3, 0.4, and 0.5 were tested at various exposure periods of t = 30, 70, 100, 140, and 180 days. The decreased percentages for the tested two-dimensional chloride diffusion concentrations increased with increasing CAVF, and the percentage values decreased from 4.95%, 5.22%, 6.29%, and 7.46% to 65.52%, 73.68%, 91.56%, and 97.22% for Vca = 0.2, 0.3, 0.4, and 0.5 in relation to Vca = 0 in the diagonal sample holes of the concrete specimens, respectively. The quantitative influence of Vca = 0.5 on the two-dimensional chloride diffusivity showed an average reduction of approximately 50.79% for each exposure period in relation to the specimen values for Vca = 0. On the basis of the two impact factors related to the exposure period (If (t)) and the CAVF (If (Vca)), a time-dependent model for predicting the two-dimensional chloride diffusion concentration in concrete by accounting for the heterogeneity of concrete materials was developed and compared with those determined by the meso-scopic numerical simulation method and physical experiment. The comparisons exhibited that the two-dimensional chloride concentrations assessed by the model and numerical simulation were almost within a ±20% error margin, validating the accuracy, correctness and reasonability of the prediction model developed in this paper. Ó 2020 Elsevier Ltd. All rights reserved.

⇑ Corresponding author. E-mail addresses: [email protected] (L. Wu), [email protected] (Y. Wang), [email protected] (Y. Wang), [email protected] (X. Ju), [email protected] (Q. Li). https://doi.org/10.1016/j.conbuildmat.2020.118213 0950-0618/Ó 2020 Elsevier Ltd. All rights reserved.

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1. Introduction Chloride ingression into RC covers is currently treated as an important reason for the deterioration of RC structures exposed to aggressive environments [1,2]. Consequently, it is vital to deeply investigate the chloride diffusion behaviour in RC structures and further develop an applicable chloride transport model that can provide significant technical support for reasonably predicting the durability of reinforced concrete components in marine structures [3]. In the early 1960s, chloride transport into concrete materials became a topic of interest for the durability of concrete structures. Recently, a large number of conclusions related to the chloride transport behavior in concrete have been acquired and successfully used in predicting the service life of RC structures [4–12], and methods for calculations and kinds of concrete materials, such as closed-form solutions, silica-fume concrete, and fly ash concrete, were referred [13,14]. Nevertheless, the previous achievements mostly corresponded to one-dimensional chloride diffusion in concrete. For RC structures exposed to marine environments, special boundary locations such as the corners and edges of cross beams and square piles of high-piled wharfs (as shown in Fig. 1) are often subjected to two-dimensional chloride penetration. Therefore, the application of the one-dimensional characteristics of chloride to estimate the concentration distribution at the corner or edge area of concrete structures is not reasonable. Moreover, the corner location of RC components is generally a stress concentration area. This stress would substantially increase the vulnerability of the cover concrete with corner- or edge-located steel bars with side-located reinforcement [15]. Thus, the steel reinforcements embedded in the corner or edge locations of the concrete were corroded more easily than those of the side locations [16]. As a result, during the process of predicting the service life for marine RC structures, the influence of two-dimensional chloride diffusion on the chloride concentration distribution in the concrete, especially at the corner and edge areas of the concrete, cannot be neglected. Some efforts involving experimental studies and numerical simulations were devoted to exploring the two-dimensional chloride diffusion characteristics in the concrete cover. For the experimental studies, Zhang et al. [16] developed an experimental

method to investigate the accelerating effect of flexural stress on one-dimensional and two-dimensional chloride diffusion in concrete with various water-to-binder ratios (w/bs) and fly ash contents (FAs). Based on the literature [16], Zhang et al. [17] further studied the interaction effect due to two-dimensional and threedimensional chloride diffusion in concrete with different FAs, w/bs, and curing times by comparing the measured results of one-dimensional chloride ingression. Wu et al. [18] experimentally explored the coupling effects of the service environment and crack width on the one- and two-dimensional chloride diffusion characteristics of RC columns under freeze-thaw cycles and seawater immersions. Subsequently, an indoor experiment for the influences of various low fatigue load levels on the one- and two-dimensional chloride diffusion behaviors of RC beams under wetting-drying cycles of sea water was carried out by Wu et al. [19]. For the numerical simulation, Jin et al. [15] used the finite element method to explore the influence of corner located reinforcement corrosion on the cracking behavior of the concrete cover when subjected to two-dimensional chloride ion attack. Hu et al. [20] established a finite element model (FEM) for concrete to investigate the effects of exposure time, temperature, and relative humidity on oneand two-dimensional chloride diffusion diffusivity. Moreover, other numerical simulation methods, including the finite difference method [21–23], the boundary element method [24], and the meshless method [25,26], were employed to estimate the two-dimensional chloride diffusion concentration distributions in an assumed homogeneous concrete material, and the initiation corrosion time of the steel bars and the durability service life of RC structures, were predicted using these aforementioned methods. To sum up, the conclusions showed that the chloride concentrations in the corner and edge areas of concrete accumulated due to the two-dimensional chloride diffusion behaviour, which resulted in significantly higher chloride concentration values in these areas than those determined under one-dimensional chloride diffusion. This discrepancy would shorten the corrosion initiation time of corner-located and edge-located rebar embedded in concrete covers and reduce the service life for RC structures. Also, concrete has generally been deemed as a typical heterogeneous material in which the main constituents are cement, water, and aggregates (including the coarse aggregate and fine aggregate).

Fig. 1. RC structure exposed to marine environment: (a) reinforcement drawing for the cross beam of high-piled wharf; (b) reinforcement drawing for square pile of highpiled wharf; (c) Two-dimensional chloride ingress into reinforced concrete specimens exposed to marine environment.

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The heterogeneity of concrete materials is mainly attributed to the volume fraction, the grading, and the shape, as well as the random distribution of aggregate particles (including the coarse and fine aggregates) embedded in cement-based materials [27]. These aforementioned two influencing factors, i.e., the shape and particle random distribution of the aggregates, are mainly dependent on the essential properties of the natural aggregates (stone and sand) and preparation technology used during the production of the concrete matrix; thus, the shape and random particle distribution are hardly quantified for controlling the heterogeneity of concrete materials. In comparison, the volume fraction of the aggregates is a controllable parameter and can be used as a quantitative index to conveniently assess the heterogeneity of materials for concrete with a consistent grading of aggregates [28]. Moreover, coarse aggregates and fine aggregates embedded in concrete can change the capillary porosity and the internal pore structure of concrete and thus alter its chloride diffusivity [29]. In fact, on account of the sizes of capillary and gel pores, as well as the diameter range of fine aggregate particles being considerably smaller than that of coarse aggregates [30], the coarse aggregate and its properties (involving type, volume fraction, shape, grading, random distribution, etc.) can be treated as the critical factor dictating the heterogeneity of concrete materials; thus, the influence of aggregates on chloride ion transport behaviour for concrete mixtures can hence be attributed to the coarse aggregate [31]. Coarse aggregates generally occupy more than 40% of the concrete volume; accordingly, it is extremely important to investigate the influence of coarse aggregate on the chloride transport characteristics in concrete. In summary, for this paper’s investigations, the volume fraction of coarse aggregate is ultimately used to quantify the heterogeneity of concrete materials. A number of previous efforts devoted to researching the aforementioned problems used mathematical models [32–38], physical experiments [28,39–44], and numerical simulation methods [27,30,31,45–49], which were elaborated upon in the literature [29]. In summary, the above achievements revealed that the chloride diffusivity importantly relies on the aggregate volume fraction and that the chloride diffusivity declines as the volume fraction of aggregate increases [30,31,35,38,39,42,43]. In summary, the two-dimensional chloride transport behaviors and the heterogeneity of concrete materials can both affect chloride concentration distributions in concrete. Despite all the laudable findings of the previous research, the existing investigations related to the aforementioned two factors were either devoted to only studying the two-dimensional chloride diffusion concentration distributions in an assumed homogeneous concrete media [15,20–26,50], or focused on exploring the effect of the heterogeneous properties of concrete materials attributed to coarse aggregates on the one-dimensional chloride ion permeation behavior in concrete mixtures [27,28,30–49]. The effects of concrete material heterogeneity on the two-dimensional chloride diffusion characteristics in concrete need to be thoroughly discussed. During this paper’s exploration, using the CAVF to quantify the heterogeneity of concrete materials, a physical experiment for exploring the two-dimensional chloride diffusion behaviour in concrete specimens was carried out under real-time marine tidal cycles. Experimental concrete specimens with different CAVFs of Vca = 0, 0.2, 0.3, 0.4, and 0.5 were cast to investigate the influence of the heterogeneous properties of concrete materials on the twodimensional chloride diffusion characteristics in concrete. The two-dimensional chloride diffusion concentrations for the whole concrete specimens were tested for various exposure periods of t = 30, 70, 100, 140, and 180 days. In terms of the experimental measurements, two impact factors related to the exposure period and the CAVF were determined and utilized to complete the expression for two-dimensional chloride diffusivity, i.e., the

diffusion coefficient, in a closed-form solution to Fick’s second law. Ultimately, a mathematical model taken into account the heterogeneity of concrete materials was developed for predicting the two-dimensional chloride diffusion concentration in concrete mixtures. The accuracy of the developed prediction model was validated using a comparison between the two-dimensional chloride diffusion concentrations estimated by the proposed model and those determined by meso-scopic numerical simulation method and physical experiment. 2. Experiments 2.1. Raw materials To concentrate on the effect of heterogeneity of concrete materials on the two-dimensional chloride concentration distribution in concrete, experimental concrete specimens with a series of CAVFs were produced during this paper’s investigation. Ordinary Portland cement (OPC, P. O. 42.5) produced by Tianjin Cement Plant, with a density of 3.1  103 kg/m3, was deemed as the binding material for all the concrete specimens. The properties of ordinary Portland cement (OPC) are enumerated in Table 1. Crushed limestone with an apparent density of qca = 2.69  103 kg/m3 served as the coarse aggregate for concrete specimens, and the nominal size range for the coarse aggregate was 5–20 mm. Natural river sand, with an apparent density of qfa = 2.61  103 kg/m3 and a fineness modulus of fm = 2.7 (containing no initial chloride concentration), was treated as the fine aggregate for the experimental concrete specimens. The gradation curves for the fine and coarse aggregates employed during this paper’s experimental study are illustrated in Fig. 2, and we can observe that the fine and coarse aggregates meet the requirements for continuous grading. Moreover, distilled water was applied during the casting, curing, and testing processes for all the concrete specimens to exclude the chloride in the raw materials of concrete from influencing the final tested chloride concentration results. 2.2. Mixing details For all the mixture proportions of this paper’s experimental concrete specimens with a size of 100  100  400 mm2, a water-cement ratio (w/c) of 0.54 was used on the basis of literatures [3,51–54], and the CAVFs were prepared as Vca = 0, 0.2, 0.3, 0.4, and 0.5. The mixing details for all experimental concrete specimens are presented in Table 2. For the investigations of this paper, the meso-scopic structure of concrete is considered an appropriate

Table 1 Properties of ordinary Portland cement, OPC (P. O. 42.5). Properties Chemical analyzes (%) Calcium oxide (CaO) Silicon dioxide (SiO2) Ferric oxide (Fe2O3) Aluminum oxide (Al2O3) Magnesium oxide (MgO) Sulfur trioxide (SO3) Potassium oxide (K2O) Sodium oxide (Na2O) Titanium dioxide (TiO2) Manganese oxide (MnO) Initial chloride ions (Cl) (weight % cement) Bogue Composition (%) Tricalcium silicate C3S Dicalcium silicate C2S Tricalcium aluminate C3A Tetracalcium aluminoferrite C4AF

Cement 63.7 20.9 4.05 4.82 1.94 2.13 0.56 0.21 0.32 0.08 0.021 57.8 19.1 7.8 10.6

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designs for concrete with various CAVFs were reported in the literature [29,42,55]. 2.3. Concrete specimen preparation All experimental specimens, sized 100  100  400 mm3, were cast in plastic moulds and subsequently compacted using a vibrating table. After the fresh concrete mixtures were left for 24 h at 20 ± 5 °C and a relative humidity of approximately 50–55%, the concrete mixtures were subsequently demoulded and transferred to water saturated with calcium hydroxide for 28 days of curing at an environmental temperature of 20 ± 3 °C. Next, the specimens were sealed on four surfaces using a polyurethane-based epoxy coating to expose two adjacent surfaces, perpendicular to each other, (sized 100  400 mm2) to ensure the two-dimensional chloride diffusion into concrete, as illustrated in Fig. 3. For the investigation in this paper, the compressive strengths after 28 days (fc28) for the concrete specimens with Vca = 0, 0.2, 0.3, 0.4, and 0.5 were measured according to three parallel samples in accordance with EN 12390–3-2001 [56], and their results are illustrated in Fig. 4. From this figure, we can observe that the almost linear variation trend for the tested fc28 increases as the CAVF increases. The quantitative influence of Vca = 0 on fc28 is an average reduction of approximately 65% in relation to the value for Vca = 0.5, indicating that the increased CAVF can result in improvements to the concrete compressive strength. 2.4. Simulation of marine environment

Fig. 2. Gradation curves of aggregates: (a) fine aggregate; (b) coarse aggregate.

three-phase composite material, including mortar, coarse aggregates, and an interfacial transition zone (ITZ) [46]. During the design and preparation of the experimental concrete specimens, the mixture proportions of the cement, water, and fine aggregates for all of the experimental groups in Table 2 remain constant, i.e., mc: mw: mf = 1: 0.54: 1.82, to ensure the consistent mixture ratio of the mortar phase within the concrete composite material. Thus, the diffusion medium of the chloride ions in the meso-scopic structure of the concrete, i.e., the mortar phase, is consistent. Based on the aforementioned considerations, the mixture details for the experimental concrete specimens with different CAVFs were hence determined under the preconditions of the consistent mortar phase, as exhibited in Table 2. Additionally, similar mixture

To accurately simulate the real-time marine tidal cycles during the experiment procedure, an automatic device for marine environment simulation was used. This automatic device included components such as storage and test tanks with water level indicators; control systems related to the temperature and the relative humidity; spray system for the sprinklers and the splashes; flow rate control system with solenoid values, water pumps, and flow meters; a human operating system; and others. The connections among these components were relied on the system with a programmable logic controller (PLC). The schematic and a real product photo of the automatic device for the marine environment simulation are illustrated in Fig. 5(a) and (b), and the automatic device in operation and after the arrangement of the experimental specimens are exhibited in Fig. 5(c) and (d). Compared to the traditional mode of wetting-drying cycles, where the wet-dry ratio is 1:1 for concrete specimens [29,31,38], the automatic device for marine environment simulation can realistically, automatically, and effectively simulate real-time tidal cycling during the experimental process. A series of important control parameters, such as high-tidal and low-tidal levels, the amount of tidal cycles, the relative humidity and temperature in the test tank, etc., can be conveniently inputted through the human operating system of the automatic device. Subsequently, the automatic device can fulfil the command through the PLC system. In summary, the automatic device for marine environment simulation can not only simulate the marine

Table 2 Mixing details for concrete specimens. No.

w/c

Water (kg/m3)

Cement (kg/m3)

Fine aggregate (kg/m3)

Coarse aggregate (kg/m3)

Vca

1 2 3 4 5

0.54 0.54 0.54 0.54 0.54

345 277 242 208 173

639 512 448 384 320

1173 932 816 699 583

0 538 807 1076 1345

0 0.2 0.3 0.4 0.5

Vca denotes the coarse aggregate volume fraction (CAVF).

L. Wu et al. / Construction and Building Materials 243 (2020) 118213

Fig. 3. Concrete specimens were sealed on four surfaces using a polyurethanebased epoxy coating to expose two adjacent surfaces 1 and 2, perpendicular to each other, (sized 100  400 mm2): (a) prism of concrete specimen; (2) expanded view for concrete specimen.

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centrations of chloride ions appear in the regions along the 0.25– 0.5 elevation from the bottom to the top of this RC wharf [57]. That is, the real-time marine tidal cycles have an observable influence on the chloride concentration distributions within RC structures along the elevation direction. Consequently, for this paper’s experimental study, when each required exposure period of chloride diffusion was completed, the corresponding concrete specimens were taken out of the test tank, and subsequently, they were separated into average four cubes, which were labelled as I, II, III, and IV from the top to the bottom. Particularly, Part III of the concrete cube, located at the area of 0.25–0.5 elevation for the experimental specimen, was utilized as a typical sample to drill and obtain the powders, as illustrated in Fig. 6(a). To ensure the accuracy of the experimental measurements, the cubes of Part III for the concrete specimens were further divided into two parts, numbered III-1 and III-2 (shown in Fig. 6(a)), and the concrete powders from Part III-1 and Part III-2 were drilled and obtained from the same sample holes using a concrete drilling machine (CDM), as exhibited in Fig. 6(b). The diameter of the drilled core for the CDM is approximately 3 mm, and the depth of all the sample holes for obtaining the required concrete powders is almost 30 mm, as shown in Fig. 6(c). The distribution of the drilling sample holes for Part III1 and Part III-2 is illustrated in Fig. 6(d), and the final drilled section of the concrete sample, for example Vca = 0.4, is photographed in Fig. 6(e). The chloride ion results were determined by testing the two concrete powders acquired from the sample holes, which were located at consistent positions for the specimens of Part III-1 and Part III-2, and the average values of the aforementioned two chloride concentrations were regarded as the ultimate tested chloride concentrations. The concrete powders were required to be dried for two hours in a drying box maintained at a temperature of 105 ± 5 °C and then passed through a 0.63 mm sieve; ultimately, the free chloride concentrations, i.e., water-soluble chloride concentrations, contained in the sieved concrete powders from different sample holes were tested and obtained according to the ASTM C1218 [58].

3. Results and analysis Fig. 4. Compressive strength after 28 days of concrete specimens with different volume fractions of coarse aggregate.

tidal level to gradually go up and down but also regulate the environmental relative humidity and temperature, which can realize automation for the experiment and greatly improve the physical experimental efficiency. A NaCl solution configured as 3.5% was utilized as man-made seawater. The concrete specimens were all placed on corrosionresistant tables vertically located in the test tanks for twodimensional chloride diffusion; the height of the tables was approximately 200 mm. According to the geometry of the experimental concrete specimens (sizes: 100  100  400 mm4), the high-tidal level was set as 600 ± 20 mm, and the low-tidal level was considered as 200 ± 20 mm. Moreover, a single marine tidal cycle was equal to 24 h, involving 12 h for rising tide and the remainder 12 h for ebbing tide, and the exposure periods of all concrete specimens were 30, 70, 100, 140, and 180 consecutive days. 2.5. Chloride concentration test By means of a field test for chloride detection within a certain RC wharf, the chloride measurements increased initially and then decreased as the elevation of the wharf structure increased. In addition, the tested results show that the most unfavourable con-

3.1. Tested two-dimensional chloride diffusion concentrations The two-dimensional chloride diffusion concentration distributions of concrete specimens with various CAVFs (Vca = 0, 0.2, 0.3, 0.4, and 0.5) for different exposure periods (t = 30, 70, 100, 140, and 180 days) were tested during the experimental procedure. The representatives corresponding to the exposure period t = 180 days are applied to show the chloride concentration isolines in the oxy plane due to the two-dimensional chloride diffusion in the concrete specimen, as drawn in Fig. 7. The free (watersoluble) chloride concentrations exhibited in Fig. 7 and all of this paper’s following figures are denoted as a percentage of the concrete mass (%). From Fig. 7, we can observe that the chloride transport is more severe near the corner region on account of its proximity to both external surfaces of the concrete cover, which is different from the one-dimensional chloride diffusion profile in concrete, and the two-dimensional chloride diffusion concentration distributions in the oxy plane of concrete present the ‘‘L” form. Nevertheless, the measured chloride concentrations in this paper are unobvious at the position of the rebar, i.e., cover thickness c = 50 mm [4,19], due to the relatively short tested exposure time, i.e., the maximum chloride diffusion time is t = 180 days during the experiment. For the RC structures exposed to marine environment, because the cover concrete with corner-located steel bars is attacked by two-dimensional chloride penetration compared to that with the side-located reinforcement that is attacked by only

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Fig. 5. Automatic device for marine environment simulation: (a) components of this device; (b) real product photo; (c) automatic device in operation; (d) automatic device after arrangement of experimental specimens.

one-dimensional chloride diffusion, the steel reinforcements embedded in the corner locations of the concrete were corroded more easily than those of the side locations during a long-term service time [16]. The two-dimensional chloride diffusion concentration must be more severe than that of the one-dimension chloride diffusion concentration at the position of the rebar inside the concrete when the exposure time is sufficient. Moreover, as expected, the tested two-dimensional chloride diffusion concentrations in concrete specimens at t = 180 days declined with the diffusion depth and the increase in CAVF Vca. The two-dimensional chloride diffusion behaviour and the CAVF (corresponding to the heterogeneity of concrete materials) both have great influences on concentration distribution of chloride ions within concrete specimens. To directly and conveniently analyze the experimental results of the two-dimensional chloride diffusion into concrete, a geometrical parameter, Lh, which is defined as the distance between the centre point of each sample hole and the origin o in the oxy plane, is hence expressed as:

period, and CAVF, and it illustrates that the two-dimensional chloride diffusion concentrations increase with the exposure period, whereas there is no significant variation in the trend between the two-dimensional chloride diffusion concentration and Lh. Moreover, to quantitatively analyze the influence of the CAVF on the tested two-dimensional chloride diffusion concentrations, the decreased percentages of the chloride concentration between Vca = 0.2, 0.3, 0.4, 0.5 and Vca = 0 in the diagonal sample holes of the concrete section (oxy plane), including Lh(x1, y1), Lh(x2, y2), Lh(x3, y3), Lh(x4, y4), Lh(x5, y5), and Lh(x6, y6), are determined and exhibited in Fig. 9(f) at an exposure time of t = 180 days. On the basis of Fig. 9(f), the decreased percentages for the chloride concentration increase with increasing CAVF, and the percentage values decrease from 4.95%, 5.22%, 6.29%, and 7.46% to 65.52%, 73.68%, 91.56%, and 97.22% for Vca = 0.2, 0.3, 0.4, and 0.5 in relation to Vca = 0, respectively. This further indicates that the CAVFs have a large influence on the two-dimensional chloride diffusion concentrations in concrete specimens.


 qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Lh xi ; yj ¼ x2i þ y2j

3.2. Fundamental two-dimensional chloride diffusion model

ð1Þ

where xi and yj denote the abscissa and ordinate for an arbitrary sample hole, and the geometrical relationship between Lh and (xi, yj), for instance, Lh(x5, y8), Lh(x6, y6), and Lh(x7, y3), is as shown in Fig. 8. According to the aforementioned definition, the test results of the two-dimensional chloride diffusion concentrations in concrete with different CAVFs and for various exposure periods versus Lh are plotted in Fig. 9. Fig. 9 intuitively reflects the relationship among the two-dimensional chloride diffusion concentration, Lh, exposure

The tested two-dimensional chloride diffusion concentrations in Fig. 9 required quantitative analysis, and Fick’s second law of diffusion can be used to model the performance of the twodimensional chloride diffusion distributions in concrete. The partial differential equation (PDE) for Fick’s second law is expressed as:

    @C @ @C @ @C ¼ D þ D @t @x @x @y @y

ð2Þ

L. Wu et al. / Construction and Building Materials 243 (2020) 118213

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Fig. 6. Concrete specimen processing: (a) separation of concrete specimen; (b) concrete drilling machine and drilling section; (c) drilled core and concrete powder; (d) drilling sample holes distributed in Part III; (e) drilled section of concrete sample (Vca = 0.4).

where C denotes the two-dimensional chloride diffusion concentration (% weight of concrete); D is the two-dimensional chloride diffusivity (m2/s); t means the exposure period (s); and x and y denote the chloride diffusion depth in the x direction (abscissa in oxy plane) and y direction (ordinate in oxy plane), respectively (mm). For an assumed constant D, the closed-form solution to Eq. (2) based on the actual initial condition C = C0 = 0 (when x > 0, and t = 0) and the boundary condition C = Cs (when x = 0, and t  0) is determined as [59]:

     x y C ðx; y; t Þ ¼ C s 1  erf pffiffiffiffiffiffi  erf pffiffiffiffiffiffi 2 Dt 2 Dt

ð3Þ

where C(x, y, t) denotes the two-dimensional chloride diffusion concentration at the position coordinates (x, y) in the oxy plane of a concrete specimen when the exposure period is equal to t (% weight of concrete), Cs is the surface chloride concentration (% weight of concrete), and the erf() refers to the Gaussian error function.

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Fig. 7. The experimental two-dimensional chloride diffusion isoconcentration contours for concrete specimens: (a) Vca = 0 (t = 180 days), (b) Vca = 0.2 (t = 180 days), (c) Vca = 0.3 (t = 180 days), (d) Vca = 0.4 (t = 180 days), and (e) Vca = 0.5 (t = 180 days).

The two-dimensional chloride diffusivity (D) and the surface chloride concentration (Cs) in Eq. (3) have been treated as the two vital parameters that are affected by the heterogeneity of concrete materials attributed to the CAVFs. The influences of concrete material heterogeneity on these two parameters, D and Cs, of twodimensional chloride diffusion are deeply explored in this paper’s following sections. During this paper’s analysis, the values of D and Cs for different CAVFs and exposure periods are confirmed

through fitting Eq. (3) to the measured two-dimensional chloride diffusion concentrations using regression analysis [3,29,31]. 3.3. Two-dimensional chloride diffusivity Based on the nonlinear regression analysis by means of the method of least squares fit mentioned above, the regression results for the two-dimensional chloride diffusivity (D) as factors of the

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2

0

1

x 6 B C C ðx; y; t Þ ¼ C s 41  erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA 1 m 2 Dr  ð1  mÞ  ðtr =t Þ  t 0 13 y B C7 erf @ qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiA5 1 m 2 Dr  ð1  mÞ  ðt r =t Þ  t

ð6Þ

In terms of the equivalent relationship between Eq. (3) and Eq. (6), the expression for two-dimensional chloride diffusivity is determined as:

D ¼ Dr  ð1  mÞ1  ðt r =t Þm

ð7Þ

where the Dr with Vca = 0 (mortar specimen) (illustrated in Fig. 11 (a)) is treated as the reference two-dimensional chloride diffusivity within Eq. (7); that is to say, Dr = Dr(Vca = 0) = 7.3608  1012 m2/s. On the basis of Eq. (7), the impact factor If (t) is thus expressed as:

If ðt Þ ¼ ð1  mÞ1  ðt r =t Þm Fig. 8. Geometrical relationship between Lh and (xi, yj).

exposure period and the CAVF of concrete are determined, as exhibited in Fig. 10. On the basis of this figure, it can be observed that the D values decline as the CAVF and exposure period increase. Concrete specimens, which were composed with lower CAVFs, presented higher D results, particularly, at the initial exposure period t = 30 days. The quantitative influence of Vca = 0.5 on the twodimensional chloride diffusivity (D) is an average reduction of approximately 50.79% for each exposure period in relation to the specimen values for Vca = 0 (mortar), further validating the significant influence of the heterogeneity of concrete materials on the two-dimensional chloride diffusivity for concrete. Moreover, the relationship between the two-dimensional chloride diffusivity and the factors of exposure period and the CAVF of concrete required further quantitative modeling. To quantify the effect of double factors, i.e., the exposure period and heterogeneity of concrete materials, on the two-dimensional chloride diffusivity (D), the mathematical model for the timedependent two-dimensional chloride diffusivity for concrete, obtained by consideration of the heterogeneity of concrete materials, is required to be established. The general expression for D is modelled using the multifactor method [29,50,60] as follows:

Dðt; V ca Þ ¼ Dr  If ðtÞ  If ðV ca Þ

ð4Þ

where D (t, Vca) denotes the two-dimensional chloride diffusivity for concrete accounting for the influences of the exposure period (t) and the heterogeneity of concrete materials attributed to the CAVF (Vca) (m2/s), Dr defines the reference two-dimensional chloride diffusivity at a reference time (m2/s), and If (t) and If (Vca) are the two impact factors of the exposure period and the CAVF. (1) Reference two-dimensional chloride diffusivity (Dr) and the impact factor (If (t)) In general, the reference time was developed by means of the 28 days proposed in the literature [52]. When only considering the time-dependent two-dimensional chloride diffusivity, D(t) was modelled as:

Dðt Þ ¼ Dr  ðtr =t Þm

ð5Þ

where tr denotes the reference time, tr = 28 days, and m is the aging factor. Substituting the time-dependent D(t) (Eq. (5)) into Eq. (2), the analytical solution for Eq. (2) according to the aforementioned consistent initial and boundary conditions is expressed as:

ð8Þ

Moreover, the regression values of the aging factor (m) versus Vca are shown in Fig. 11(b). From this figure, we can observe that the scatters of m appear irregular as Vca increases. Additionally, the relative errors between the average value of m and the scatter points are all within ±10%, therefore, it is fair to determine that no prominent relationship exists between Vca and m. Finally, the average value of the aging factor m = 0.3067 is submitted to Eq. (8), and the expression for If (t) can be adjusted as If (t) = 1.4424(tr / t)0.3067. (2) Impact factor If (Vca) The impact factor If (Vca), which considers the influence of CAVF on two-dimensional chloride diffusivity, is determined as:

If ðV ca Þ ¼ Dr ðV ca Þ=Dr

ð9Þ

where Dr(Vca) is the reference two-dimensional chloride diffusivity obtained for a series of CAVFs, as shown in Fig. 11(a), and Dr = Dr(Vca = 0) = 7.3608  1012 m2/s is portrayed in Fig. 11(a) as well. Substituting the values of Dr(Vca) into Eq. (9), the impact factors (If (Vca)) are calculated and plotted in Fig. 12, as are the fitted curve and formula for If (Vca) determined using the linear regression analysis. From the aforementioned description, the time-dependent model for the two-dimensional chloride diffusivity considering the heterogeneity of concrete materials is established comprehensively as follows:

8 Dðt; V ca Þ ¼ Dr  If ðtÞ  If ðV ca Þ > > > > < D ¼ 7:3608  1012 m2 =s r > If ðtÞ ¼ 1:4424  ðt r =tÞ0:3067 > > > : If ðV ca Þ ¼ 1  0:9427  V ca

ð10Þ

It is worth noting that the chloride diffusivities for concrete with different CAVFs are determined by fitting Eq. (3) to the measured two-dimensional chloride diffusion concentrations using regression analysis. During the regression process, the concrete material with each CAVF is considered as a homogeneous, macroscopic and overall matrix. Nevertheless, the chloride diffusivities determined based on various CAVFs exhibit difference and decrease as CAVF increases. It indicates that the difference of CAVFs can obviously affect the chloride diffusivities for concrete. Therefore, for this paper’s study, the CAVF can be used as the quantitative standard for the heterogeneity of concrete materials, and the impact factor of coarse aggregate volume fraction If(Vca) = 1– 0.9427Vca is adopted to quantify the influence of heterogeneity of concrete materials, i.e., CAVF, on two-dimensional chloride diffusivity of concrete.

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Fig. 9. The experimental two-dimensional chloride diffusion concentration in concrete specimens versus Lh: (a) Vca = 0, (b) Vca = 0.2, (c) Vca = 0.3, (d) Vca = 0.4, (e) Vca = 0.5, and (f) decreased percentage of chloride concentrations due to the influence of CAVF at t = 180 days.

3.4. Surface chloride concentration In Eq. (3) and Eq. (6), apart from the chloride diffusivity (D), the surface chloride concentration (Cs) is also an important parameter for evaluating the two-dimensional chloride diffusion concentrations in concrete. By fitting Eq. (3) to the measured twodimensional chloride diffusion concentrations of the concrete specimens (plotted in Fig. 9) and using the regression analysis mentioned in Section 3.2, the values for the surface chloride concentration Cs for various CAVFs after different exposure periods are confirmed, as revealed in Fig. 13. In terms of Fig. 13, we can observe that the all Cs values increase with the exposure period, thus, Cs can be considered as a time-dependent variable, namely,

Cs = Cs (t). Moreover, although the scatter in Cs for different exposure periods demonstrates a random fluctuation with the CAVF variation, the relative errors between the average value of Cs and the scatter points are almost within a ±5% range. Consequently, we can confirm that the values of Cs have no specific relation to the CAVF. The relative conclusion is consistent with the literature [29]. In other words, the average Cs values for each exposure period are considered as the representatives, and the fitted curve and formula of Cs(t) using logarithmic form [3,29,31] are shown in Fig. 13 and expressed as [3]:

C s ðtÞ ¼ 0:6537  lnðt Þ  1:8324; t P 30 d

ð11Þ

L. Wu et al. / Construction and Building Materials 243 (2020) 118213

Fig. 10. Time-dependent two-dimensional chloride diffusion coefficients for various volume fractions of coarse aggregate.

11

Fig. 12. Impact factors of the coarse aggregate volume fractions and their fitted curve (If (Vca)).

Fig. 13. Surface chloride concentrations and their fitted curve (Cs(t)).

3.5. Modelling of the two-dimensional chloride diffusion concentration in concrete To conveniently and rapidly estimate the two-dimensional chloride diffusion concentrations in concrete with various CAVFs at every exposure period (t) and plane position (x, y), the established model of the two-dimensional chloride diffusivity for concrete, D (t, Vca) (Eq. (10)), and the time-dependent surface chloride concentration using the logarithmic form, Cs(t) (Eq. (11)), are both inserted into Eq. (3). Subsequently, a mathematical model for predicting the two-dimensional chloride diffusion concentration in concrete based on Fick’s second law is established as follows:

     8 > y x > pffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  erf pffiffiffiffiffiffiffiffiffiffiffiffiffiffi C ð x; y; t; V Þ ¼ C ð t Þ  1  erf > ca s > 2 Dðt;V ca Þt 2 Dðt;V ca Þt > > > > > ð t Þ ¼ 0:6537  ln ð t Þ  1:8324; t P 30 d C > s > < Dðt; V ca Þ ¼ Dr  If ðtÞ  If ðV ca Þ > > > Dr ¼ 7:3608  1012 m2 =s > > > > > I ðtÞ ¼ 1:4424  ðt r =tÞ0:3067 > f > > : If ðV ca Þ ¼ 1  0:9427  V ca

Fig. 11. Regression results for (a) reference two-dimensional chloride diffusion coefficients and (b) ageing factors.

ð12Þ

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3.6. Validation of the model In this paper, the numerical simulation method based on the meso-scopic finite element model (FEM) of concrete is used to validate the accuracy of the proposed prediction model, Eq. (12). Ultimately, the chloride concentrations assessed according to the developed model Eq. (12) are compared with those produced by the FEM and experiment. At the meso-scopic level, concrete is regarded as a three-phase composite material composed of mortar, coarse aggregates, and an ITZ [31,46]. During the numerical simulation, the mortar and ITZ phases are usually considered as a homogeneous media for chloride transportation, and the time-dependency of the chloride diffusion coefficients for the mortar and ITZ are expressed as Dm(t) = DrIf(t) and DITZ(t) = NITZDm(t), respectively, where Dr is the reference two-dimensional chloride diffusion coefficient when Vca = 0 and Dr = 7.3608  1012 m2/s. If(t) is the impact factor for the exposure time, and If(t) = 1.4424(tr/t)0.3067, and tr = 28 days. NITZ denotes the normalized reference chloride diffusion coefficient of the ITZ, and the empirical formula for NITZ, considering the influences of the ITZ thickness dITZ and CAVF Vca, is determined based on the consistent method reported in the literature [29] and expressed as:

8

BðV ca Þ d > > ; V ca ¼ AðV ca Þ  tdITZ < NITZ dITZ ca ca AðV ca Þ ¼ 0:0976  0:044  V ca > > : BðV ca Þ ¼ 0:9581  0:096  V ca

ð13Þ

where dca is the average diameter of the coarse aggregates (mm), and dca = 11.92 mm for this experiment; dITZ/dca denotes the normalized ITZ thickness; and A(Vca) and B(Vca) are the regression parameters corresponding to Vca. Additionally, for the meso-scopic model of the concrete, the coarse aggregate phase is modeled by means of threedimensional spheres [30,61] that are in line with the essential characteristics of the random distribution in the mortar phase with suitable locations ensuring the nonoverlap between the adjacent spherical coarse aggregates, and continuous grading with a size range of 5–20 mm. The aforementioned properties for the spherical coarse aggregates can be simulated by means of the Monte Carlo method [62]. Moreover, the coarse aggregate phase is generally considered a very dense material in which the chloride ions cannot diffuse [29]. Therefore, the chloride diffusion coefficient of the coarse aggregates can be reasonably assumed to be Dca = 0 m2/s. In summary, meso-scopic numerical models of concrete with Vca = 0, 0.2, and 0.4 in the form of a cube of 100  100  100 mm3 are built, as exhibited in Fig. 14. During the finite element analysis, the COMSOL Multiphysics software is adopted to simulate the chloride diffusion in the concrete models [29,31,63]. To ensure the two-dimensional diffusion of the chloride ions during the meso-scopic numerical simulation, the gamma free flux is set on four surfaces of all the concrete models to expose the other adjacent two surfaces, as illustrated in Fig. 14 (sealed surface). The time-dependent surface chloride concentration Cs(t) = 0.6 537ln(t)  1.8324, expressed in Eq. (11), is used as the boundary condition of the two-dimensional chloride diffusion for all mesoscopic models of the concrete. The chloride diffusion coefficients for the mortar and coarse aggregate phases have already been determined according to the aforementioned discussions. For the ITZ phase, the normalized reference chloride diffusion coefficients of the ITZ are equal to NITZ(Vca = 0.2) = 15.152 and NITZ(Vca = 0.4) = 12.288 under the premise of dITZ = 50 lm [46], respectively. That is, the chloride diffusion coefficients for the ITZ phase based on the

Fig. 14. Meso-scopic numerical models for concrete: (a) Vca = 0; (b) Vca = 0.2; (c) Vca = 0.4.

exposure time, ITZ thickness, and CAVF can be written as DITZ(t, dITZ, Vca) = NITZ(dITZ, Vca)Dm(t) = NITZ(dITZ, Vca)DrIf(t). The three-phase components, i.e., mortar phase, coarse aggregate phase, and ITZ phase, included in the meso-scopic numerical

L. Wu et al. / Construction and Building Materials 243 (2020) 118213

13

model of the concrete are separately meshed, and the initial, step, and terminal computation times during the simulation analysis are set as ti = 0 day, ts = 1 day, and tt = 200 days, respectively. When the simulation process is completed, the two-dimensional chloride diffusion concentrations at various sample positions in the oxy plane of meso-scopic numerical models of the concrete are determined, and the distributions for the sample positions are all consistent with those of the experiment, as shown in Fig. 6(d). Finally, the exposure period t, the CAVF Vca, and the position coordinates (x, y), which are consistent with part of the conditions of the experiment, i.e., t = 100, 140, and 180 continuous days; Vca = 0, 0.2, and 0.4; and the values of (x, y) exhibited in Fig. 6(d), are used in Eq. (12). Subsequently, the two-dimensional chloride diffusion concentrations at different positions in the oxy plane of the concrete can be assessed by means of Eq. (12), and these estimated chloride results are used to compare the chloride concentrations tested by the experiment (shown in Fig. 9) to that of those based on the numerical simulation using the FEM, as illustrated in Fig. 15. These graphs describe the variations in the twodimensional chloride diffusion concentrations in concrete mixtures for various exposure periods against Lh. According to Fig. 15, we can observe the followings: (1) The magnitudes of the two-dimensional chloride diffusion concentrations evaluated by the numerical simulation method using the FEM agree well with the results determined by the physical experiment, validating the accuracy and reliability of the FEM method. (2) Additionally, the estimated two-dimensional chloride diffusion concentrations, in terms of the prediction model using Eq. (2) proposed by this paper, fit well with the results confirmed by the physical experiment and numerical simulation program using the FEM, further validating the accuracy of the developed prediction model using Eq. (2). Furthermore, the evaluated two-dimensional chloride diffusion concentrations in terms of the prediction model using Eq. (12) and the numerical simulation method versus those from the experiment are provided in Fig. 15. All of the data plotted in Fig. 16 are derived from Fig. 15. The tested two-dimensional chloride diffusion concentrations for concrete (the black scatters in Fig. 15) are used as the abscissa in Fig. 16, and the evaluated chloride concentrations obtained using Eq. (12) (the red full lines in Fig. 15) and the numerical simulation (the green imaginary lines in Fig. 15) are considered the ordinate in Fig. 16. Using this figure type, the relative errors among the two-dimensional chloride diffusion concentrations acquired via Eq. (12), the numerical simulation method and the experimental measurements can be intuitively analyzed and described, as portrayed in the literature [3,11,29,31,60]. Fig. 16 shows that the two-dimensional chloride concentrations assessed by Eq. (12) and the numerical simulation are almost within a ±20% error margin, further validating the accuracy, correctness and reasonability for the prediction model using Eq. (12). In summary, the model using Eq. (12) can correctly predict and assess the two-dimensional chloride diffusion concentration in the concrete with an arbitrary time by taking into account the heterogeneity of the concrete materials attributed to the CAVFs. Nevertheless, the model (Eq. (12)) proposed in this paper is appropriate for estimating and predicating the two-dimensional chloride diffusion concentrations in concrete with different CAVFs when exposed to a marine tidal environment, with w/c = 0.54, and with the corresponding concrete mixtures in Table 2. It cannot be used for other conditions, such as, other values for w/c or w/b with different supplementary cementitious materials of fly ash (FA), slag, silica fume (SF), etc., as well as for other service environments

Fig. 15. Comparison of the two-dimensional chloride diffusion concentrations determined by the prediction model (Eq. (12)), meso-scopic numerical simulation and experiment: (a) Vca = 0, (b) Vca = 0.2, (c) Vca = 0.4.

of immersion, splash, spray, atmosphere, freeze, etc. Therefore, more studies are required in the future. 4. Conclusions During this paper’s exploration, the two-dimensional chloride diffusion behaviour in concrete, taking into account the hetero-

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developed and compared with those determined by the meso-scopic numerical simulation method and physical experiment. The comparisons indicated that the twodimensional chloride concentrations estimated by the proposed model and numerical simulation are almost within a ± 20% error margin, validating the accuracy, correctness and reasonability of the prediction model developed in this paper. Moreover, future works can be further investigated on the basis of literature [51], as follows:

Fig. 16. Comparison of the two-dimensional chloride diffusion concentrations determined by prediction model Eq. (12) and meso-scopic numerical simulation method versus those of experiment.

geneity of concrete materials via an indoor experiment of real-time tidal cycles, was studied. A time-dependent model for predicting two-dimensional chloride diffusion concentrations in concrete by considering the influence of the CAVF was developed. The important points are listed as follows: (1) Based on the experiment, the tested two-dimensional chloride diffusion concentrations are more severe near the corner region of the concrete due to its proximity to both of the external surfaces of the concrete cover. The tested chloride results increase with the increase in the exposure period, whereas the tested chloride results decline as the depth and the coarse aggregate volume fraction increases. The decreased percentages for the tested two-dimensional chloride diffusion concentrations increase with increasing CAVF, and the percentage values decrease from 4.95%, 5.22%, 6.29%, and 7.46% to 65.52%, 73.68%, 91.56%, and 97.22% for Vca = 0.2, 0.3, 0.4, and 0.5 in relation to Vca = 0 in the diagonal sample holes of the concrete specimens, respectively. The two-dimensional chloride diffusion behavior and the heterogeneity of the concrete materials (corresponding to CAVF) both have significant influences on the two-dimensional chloride concentration distributions in the concrete. (2) The variations in the two-dimensional chloride diffusivity and surface chloride concentration are both determined through regression analysis. The two-dimensional chloride diffusivity decreases as the exposure period and CAVF increases, and the quantitative influence of Vca = 0.5 on the two-dimensional chloride diffusivity shows an average reduction of approximately 50.79% for each exposure period in relation to the specimen values for Vca = 0; however, because the relative errors between the average value of the surface chloride concentration Cs and the scatter points for Cs are almost within a ± 5% range, the surface chloride concentration Cs has no specific relationship to the CAVF. (3) Two impact factors related to the exposure period (If (t)) and the CAVF (If (Vca)) are presented to quantify the influences of the exposure time and CAVF on the two-dimensional chloride diffusivity. On the basis of the aforementioned If (t) and If (Vca), a time-dependent model for predicting the two-dimensional chloride diffusion concentration in concrete considering the heterogeneity of concrete materials is

(1) Indoor experiments can be further carried out to consider the influence of rebar with various diameters on the twodimensional chloride diffusion concentrations in concrete, and the major influence area and secondary influence area could be divided to describe the chloride ion concentration distribution characteristics. (2) In terms of the experimental chloride measurements, the coupling influences of the heterogeneity of the concrete materials, i.e., the volume fraction of the coarse aggregate, and blocking effects of the steel reinforcement on the twodimensional chloride diffusion behaviors in the concrete could be quantitatively analyzed. Subsequently, a prediction model for the two-dimensional chloride diffusion concentrations in concrete that considers the coupling influences of the coarse aggregates and steel reinforcement would be proposed. CRediT authorship contribution statement Linjian Wu: Conceptualization, Data curation, Writing - original draft. Yuanzhan Wang: Methodology, Resources, Supervision. Yuchi Wang: Software, Validation. Xueli Ju: Writing - review & editing. Qingmei Li: Visualization, Investigation. 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. Acknowledgments The authors would like to appreciate the insightful and constructive comments of three anonymous reviewers. This research was supported by the National Natural Science Foundation of China (51979191), the National Key Research and Development Program of China (2016YFC0802204), the Project Funded by China Postdoctoral Science Foundation (2019M653824XB), and the Project Funded by Chongqing Postdoctoral Science Foundation (cstc2019jcyj-bshX0063). References [1] C.E.T. Balestra, T.A. Reichert, W.A. Pansera, G. Savaris, Chloride profile modeling contemplating the convection zone based on concrete structures present for more than 40 years in different marine aggressive zones, Constr. Build. Mater. 198 (2019) 345–358. [2] C.E.T. Balestra, T.A. Reichert, G. Savaris, Contribution for durability studies based on chloride profiles analysis of real marine structures in different marine aggressive zones, Constr. Build. Mater. 206 (2019) 140–150. [3] Y.Z. Wang, X.L. Gong, L.J. Wu, Prediction model of chloride diffusion in concrete considering the coupling effects of coarse aggregate and steel reinforcement exposed to marine tidal environment, Constr. Build. Mater. 216 (2019) 40–57. [4] S.-W. Pack, M.-S. Jung, H.-W. Song, S.-H. Kim, K.Y. Ann, Prediction of time dependent chloride transport in concrete structures exposed to a marine environment, Cem. Concr. Res. 40 (2) (2010) 302–312.

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