Study on the Effect of Cracking Parameters on the Migration Characteristics of Chloride Ions in Cracked Concrete

Abstract

In engineering, concrete often develops cracks due to various reasons, which accelerate the erosion rate of chloride ions in concrete and consequently expedite the degradation of the mechanical properties of concrete structures. This study simplifies the four-phase model into a two-phase model using homogenization methods. Based on this, numerical simulations are employed to investigate the influences of dimensionless structural parameters and material parameters of cracks on the equivalent diffusion coefficient of cracked concrete, and a theoretical model for the equivalent diffusion coefficient of cracked concrete is established according to Fick’s diffusion law. The research findings indicate that when cracks are positioned in the middle of the boundary through which chloride ions enter and exit the concrete, and the direction of the cracks is parallel to the diffusion direction of chloride ions; this scenario is the most detrimental to the durability of concrete. For n cracks (n ≥ 2), when they are parallel to the x-axis and symmetrical about the x-axis, and the spacing between cracks equals 1/n times the width of the concrete, this scenario is the most detrimental to the durability of concrete containing multiple cracks. Whether for a single crack or multiple cracks, when they are in the most unfavorable condition, the “parallel-then-series” theoretical model can accurately predict the equivalent diffusion coefficient of cracked concrete.

1. Introduction

Concrete, as the primary material for marine engineering structures, directly influences the service life and performance of these crucial infrastructures due to its material properties and time-dependent behaviors. Chloride ion ingress is the primary cause of reinforcing steel corrosion for concrete structures in coastal areas [1,2,3]. Chloride ions penetrate concrete through pores, leading to depassivation and corrosion of steel reinforcement from the outside to the inside [4,5,6]. Corrosion-induced expansion causes delamination of the concrete cover, ultimately resulting in the durability deterioration of reinforced concrete components [7,8]. Cracking is an inevitable phenomenon in reinforced concrete structures, which may occur due to various reasons such as plastic shrinkage, restraint shrinkage, thermal loading, alkali–aggregate reaction, and improper design. When concrete cracks, the migration of substances such as chloride ions, moisture, and oxygen is accelerated, leading to the increased corrosion rates and affecting the normal serviceability or premature failure of concrete structures [9]. Therefore, studying the diffusion behavior of chloride ions in cracked concrete is of great significance.

Currently, research on chloride ion diffusion in cracked concrete mainly focuses on experimental studies and numerical simulations. In terms of experiments, some studies have induced artificial cracks using specimens of different sizes [10,11,12]. Other experiments induce cracks through splitting tensile tests [13,14,15,16], three/four-point bending tests [17], central expansion tests [18], wedge splitting tests [19], etc., and then conduct traditional immersion tests or accelerated diffusion tests to measure chloride ion penetration depth, mass fraction, or diffusion coefficients. Djerbi et al. [13] obtained cracks with average widths ranging from 30 to 250 μm through splitting tensile tests. The study revealed that the chloride ion diffusion coefficient increased with crack width within the range of 30–80 μm, and remained almost constant when the crack width exceeded 80 μm. Jang et al. [14] found that the threshold crack width for diffusion was approximately 55–80 μm, beyond which the diffusion coefficient increased with crack width. Sahmaran et al. [20] generated cracks with widths ranging from 29 to 390 μm using a bending load method and studied the relationship between crack width and chloride ion diffusion coefficient. The results showed that for cracks narrower than 135 μm, the crack width had little effect on the effective diffusion coefficient of mortar, while for cracks wider than 135 μm, the effective diffusion coefficient increased rapidly. Cracks narrower than 50 μm exhibited self-healing. These experiments may have certain biases because crack width is measured in the unloaded state, while the crack width under loading conditions is larger than the measured value. Due to differences in concrete composition, experimental methods, and diffusion coefficient measurement methods, the above studies have not reached a consensus.

However, experimental studies have drawbacks such as long-time requirements, high costs, and difficulties in determining the independent influence of certain parameters. Therefore, some researchers have turned to numerical simulation methods for related research. Bentz et al. [21] established a cracked concrete model using finite element methods to predict the service life of cracked concrete. Liu et al. [22,23] employed a stochastic aggregate model consisting of mortar matrix, coarse aggregate, cracks, and a interfacial transition zone (ITZ) to study multi-component ion transport through coupling mass conservation and Poisson’s equation. Wu et al. [24,25] adopted discrete cohesive fracture approaches to couple fracture with diffusion multiphase fields for studying the diffusion process of chloride ions. Šavija et al. [26,27] developed a three-dimensional lattice model to simulate chloride ion diffusion in saturated cracked concrete. In their study, concrete was considered a three-phase composite material comprising aggregate particles, cement paste matrix, and the interface transition zone. Du et al. [28] considered the heterogeneity of concrete and regarded cracked concrete as a composite material composed of four phases: cement mortar matrix, aggregate particles, cracks, and an interface transition zone (ITZ). They investigated the effects of artificial cracks and flexural cracks on chloride ion diffusion coefficients. The results showed that the width of artificial cracks was the main factor affecting chloride ion diffusion coefficients in concrete, while both the width and depth of flexural cracks influenced chloride ion diffusion coefficients. Cheng et al. [29] simulated concrete as a four-phase composite material consisting of aggregate, mortar, cracks, and interface transition zone (ITZ), and studied the effects of crack width, crack number, and erosion time on chloride ion migration behavior and characteristics. Yu et al. [30] developed a three-dimensional diffusion mechanics model for reinforced concrete with transverse cracks, considering the influences of crack deflection angle and width on the diffusion process.

The literature review above indicates that research on chloride ion transport behavior in cracked concrete is relatively scattered. In contrast to most existing works, this study employs a homogenization approach to transform the four-phase model into a two-phase model containing only cracks and mortar. Using COMSOL Multiphysics 6.0 the study investigates the influence of factors such as crack width, crack length, crack position, crack angle, crack shape, and crack frequency on the chloride ion transport performance in cracked concrete, providing a more comprehensive evaluation of the impact of cracks on concrete durability.

2. Chloride Diffusion Theory

2.1. Homeostasis Studies

The theoretical methods in this paper consist of two approaches. One involves the use of Fick’s first law to evaluate the equivalent diffusion coefficient of cracked concrete under steady-state conditions, while the other utilizes Fick’s second law to study chloride ion diffusion with time-dependent parameters.

Assuming that the diffusion flux is linearly related to the concentration gradient, i.e., in a diffusion system, the mass entering and leaving per unit volume at any moment remains equal, and it falls under steady-state diffusion. The study in COMSOL focuses on steady-state diffusion, which can be described using Fick’s first law:

$$J=-D∆C$$

(1)

where $J$ is the flux of chloride ions, $D$ is the diffusion coefficient, and $C$ is the concentration of chloride ions.

When the entire system reaches steady-state, the flux of chloride ions at the outflow surface remains constant. According to the definition of the diffusion coefficient, when the flux of chloride ions at the outflow surface is known, the diffusion coefficient of concrete can be calculated using the following equation:

$$D=J\frac{L}{∆C}$$

(2)

where $L$ is the length of the specimen and $∆C$ is thechloride concentration difference between the inlet and outlet surfaces.

2.2. Transient Studies

The diffusion properties of chloride ions in cement-based composite materials are typically described using Fick’s second law, expressed as the following:

$$\frac{\partial C\left(x,t\right)}{\partial t}=\frac{\partial }{\partial x}\left(D\frac{\partial C\left(x,t\right)}{\partial x}\right)$$

(3)

Performing a Laplace transform on Equation (3) and obtaining its analytical solution yields the following:

$$C\left(x,t\right)=C_0+\left(C_s-C_0\right)\cdot \left[1-erf\left(\frac{x}{2\sqrt{Dt}}\right)\right]$$

(4)

where $C\left(x,t\right)$ represents the concentration of chloride ions in the concrete at depth $x$ and time $t$, $C_0$ is the initial chloride ion concentration in the concrete, $C_s$ is the surface chloride ion concentration, erf denotes the error function defined as $erf\left(z\right)=\frac{2}{\sqrt{\pi }}{\int }_0^z{e}^{-t^2}dt$, and $D$ is the diffusion coefficient.

3. Numerical Model of Cracked Concrete

3.1. Four-Phase Model versus Two-Phase Model

Two methods can be used to establish numerical models of cracked concrete: the four-phase model and the two-phase model. One is a microscale model composed of aggregates, cement mortar matrix, the interfacial transition zone (ITZ), and cracks, while the other is a macroscale model consisting of cement mortar matrix and cracks. The advantages and disadvantages of the two-phase model and the four-phase model are shown in

Table 1. Four-phase model versus two-phase model.

	Four-Phase Model	Two-Phase Model
Constitute	Mortar matrix, coarse aggregates, ITZ, cracks.	Mortar matrix, cracks.
Advantages	The influences of the four phases on the diffusion properties of concrete can be considered. The study on ion transport can delve into a microscopic level, enabling analysis of the distribution and transport processes of ions in localized regions of concrete.	Only the influence of the two phases on the diffusion properties of concrete can be considered. The study of ion transport is limited to a macroscopic level, allowing analysis of the distribution and transport patterns of ions in the entirety of concrete.
Disadvantages	Modeling complexity, large computational requirements, and required parameters are numerous and difficult to obtain.	Simple modeling, low computational effort, fewer parameters required.

below.

The two-phase model can be simplified from the four-phase model by considering the mortar matrix, coarse aggregates, and interfacial transition zone (ITZ) as a whole, and employing a homogenization method to determine the equivalent diffusion coefficient of these three phases. This equivalent diffusion coefficient is then used as the effective diffusion coefficient of the matrix phase in the two-phase model.

The calculation results of the four-phase model and the two-phase model are compared by the following arithmetic example.

The schematic diagram of cracked concrete is shown in Figure 1. The concrete dimensions are 100 mm × 100 mm, with a crack width b = 0.2 mm and a crack length a = 40 mm. The crack is located at the midpoint of the left boundary of the concrete, running parallel to the upper and lower boundaries and penetrating into the interior of the concrete.

In the four-phase model, aggregates are assumed to be circular, and according to previous studies, the shape of aggregates has a minimal impact on chloride ion diffusion in concrete [31,32]. The Walraven formula [33] is used to generate aggregates with diameters ranging from 5 mm to 15 mm, with a volume percentage of 50%. The initial concentration of concrete is 0 mol/m³, the concentration at the left boundary is fixed at 520 mol/m³, the concentration at the right boundary is fixed at 0 mol/m³, and there are no fluxes at the upper and lower boundaries. The main parameters of the four-phase model compared to the two-phase model are shown in

Table 2. Model parameters.

	Four-Phase Model	Two-Phase Model
Aggregate size range ($d$)	5 mm–15 mm
Aggregate volume fraction ($V_a$)	50%
Interface transition zone thickness ($d_{\mathrm{I}\mathrm{T}\mathrm{Z}}$)	100 μm
Diffusion coefficient in the interfacial transition zone $(D_{\mathrm{I}\mathrm{T}\mathrm{Z}}$)	$4.74605\times {10}^{-12} {\mathrm{m}}^2/\mathrm{s}$
Mortar diffusion coefficient $(D_{cr}$)	$3.32224\times {10}^{-11} \mathrm{m}/\mathrm{s}$	$2.0822\times {10}^{-12} {\mathrm{m}}^2/\mathrm{s}$
Crack diffusion coefficient $(D_{cr}$)	$1.23665\times {10}^{-9} {\mathrm{m}}^2/\mathrm{s}$	$1.23665\times {10}^{-9} {\mathrm{m}}^2/\mathrm{s}$

.

The actual thickness of the interfacial transition zone (ITZ) in ordinary concrete ranges from 20 to 50 μm [34,35,36]. In this study, the ITZ is set to 100 μm. To compensate for the increased thickness of the ITZ, a reduction in the diffusion coefficient is applied when determining the value of the diffusion coefficient in the ITZ.

The diffusion coefficient of chloride ions in mortar is related to the water–cement ratio of concrete. For the selection of the diffusion coefficient in cement mortar, this study adopts the calculation formula provided by Wang [37]:

$$D_{cp}=\left[15.93\left(\frac{w}{c}\right)-2.98\right]\times {10}^{-12}$$

(5)

where $D_{cp}$ is the diffusion coefficient of chloride ions in cement mortar, and w/c is the water–cement ratio of concrete.

In the equation, $D_{cp}$ represents the diffusion coefficient of cement mortar (in m²/s); w/c is the water–cement ratio of concrete. In this study, the water–cement ratio is taken as 0.485. The diffusion coefficient of mortar in the four-phase model can be obtained using Equation (5), while in the two-phase model, the diffusion coefficient of mortar is determined using the homogenization method.

The Interfacial Transition Zone (ITZ) exists between the bulk cement paste and aggregates and is typically regarded as a weak layer surrounding each aggregate [38]. Numerous experiments have indicated that the diffusion coefficient of the ITZ is approximately 1.3 to 16 times higher than that of the cement mortar matrix [39,40]. A compromise value of eight times the diffusion coefficient of the cement mortar matrix is chosen, and considering the increased thickness of the ITZ, the diffusion coefficient of the ITZ in this study is taken as seven times that of the cement mortar diffusion coefficient [22].

The diffusion coefficient of cracks is determined by averaging the data provided by Şahmaran [20] and Djerbi et al. [13].

Şahmaran [20] provided a fitting relationship between crack width and crack diffusion coefficient:

$$D_{cr1}=\left(34.58+0.002w_{cr}^2\right)\times {10}^{-11}$$

(6)

The experimental data from Djerbi et al. [13] were fitted using MATLAB R2022b to obtain the following:

$$D_{cr2}=-3.511\ast {10}^{-14}{w}_{cr}^2+1.314\ast {10}^{-11}{w}_{cr}+1.039\ast {10}^{-10}$$

(7)

In Equations (6) and (7), $D_{cr1}$ and $D_{cr2}$ are in m²/s, and w_cr is in μm, with a fitting R² value of 0.979.

$$D_{cr}={(D}_{cr1}+D_{cr2})/2$$

(8)

In this study, to reduce the variability in experimental results, the final value of the crack diffusion coefficient $D_{cr}$ is obtained by averaging the sum of Equations (6) and (7), as shown in Equation (8).

Figure 2 depicts the geometric model of the four-phase model, while Figure 3 illustrates the mesh of the four-phase model. The green shaded area represents the Interfacial Transition Zone (ITZ), the red area represents cracks, the gray portion represents the cement mortar matrix, and the blank areas represent aggregates. Since the aggregate regions are considered impermeable, they are not meshed, and chloride ion diffusion occurs only in the ITZ, cracks, and cement mortar matrix.

Figure 4 illustrates the geometric model of the two-phase model, while Figure 5 shows the mesh of the two-phase model. The red area represents cracks and the gray area represents the cement mortar matrix.

The calculated results of the equivalent diffusion coefficients for the four-phase model and the two-phase model are shown in

Table 3. Equivalent diffusion coefficient values.

	Four-Phase Model	Two-Phase Model
$\mathrm{Equivalent} \mathrm{diffusion} \mathrm{coefficient} \mathrm{value} (D_{cr}$)	$2.2968\times {10}^{-12} {\mathrm{m}}^2/\mathrm{s}$	$2.2921\times {10}^{-12} {\mathrm{m}}^2/\mathrm{s}$

. The concentration profiles with depth are illustrated in Figure 6.

From Table 3, it can be observed that the difference in the calculated equivalent diffusion coefficient values between the four-phase model and the two-phase model is minimal, with an error of only 0.2%.

From Figure 6, it can be observed that the overall trends of chloride ion concentration decrease are similar in both the two-phase and four-phase models, but there is a difference in the transportation speed between them. Specifically, in the two-phase model, the transportation speed of chloride ions is slower, while it is relatively faster in the four-phase model. Despite the existence of significant errors in local concentrations, it is feasible to simplify the four-phase model into a two-phase model using homogenization methods to calculate the overall equivalent diffusion coefficient of cracked concrete at a macroscopic level.

3.2. Validation of the Simulation Model

To validate the accuracy of the current model, it was compared with Şahmaran’s experimental data. The water-to-cement ratio (w/c) in the experiments was 0.485, and the diffusion coefficient of mortar can be calculated using Equation (5). The simulated results after exposing uncracked mortar to 3% NaCl solution for 30 days and 90 days were compared with Şahmaran’s experimental values. As shown in Figure 7, there is good consistency between the experimental and simulated data. Next, numerical studies on chloride ion transport behavior in cracked concrete will be conducted based on this model.

4. Simulation Results and Discussion

In concrete, a representative volume element (RVE) is selected, and homogenization methods are employed to determine the equivalent diffusion coefficients for the mortar matrix, coarse aggregates, and interfacial transition zone. Theoretically, these equivalent diffusion coefficients should be equal to the equivalent diffusion coefficient of the concrete matrix in the two-phase model. The two-phase model is then utilized to investigate the influence of cracks on the ion transport properties of concrete, and the overall diffusion performance of concrete containing cracks is characterized using the equivalent diffusion coefficients. Finite element simulations are conducted using the commercial software COMSOL, with steady-state calculation methods employed for calculations involving equivalent diffusion coefficients.

Assuming that both the concrete matrix and the cracks are rectangular, a Cartesian coordinate system is established with the center of the concrete as the origin and the concrete symmetry axis as the coordinate axis, as shown in Figure 8. The main geometric and material parameters involved in the study are listed in

Table 4. Geometric and material parameters.

Parameters	Connotation
$A$	Length of the concrete matrix
$B$	Width of the concrete matrix
$a$	Length of the crack
$b$	Width of the crack
$\theta$	Angle rotated counterclockwise from the positive x-axis to the major axis of the crack
$\left(x,y\right)$	Coordinates of the center of the crack
$\left|x_{max}\right|,\left|y_{max}\right|$	Maximum distances from the crack center to the y-axis and the x-axis
$D_m$	Diffusion coefficient of the concrete matrix
$D_c$	Diffusion coefficient of the crack
$D_x$	Equivalent diffusion coefficient in the x-direction for concrete containing cracks

. To explore the general rules governing the influence of cracks on the ion transport properties of concrete, further dimensionless parameters were defined and the dimensionless parameters are shown in

Table 5. Dimensionless parameters.

Dimensionless Parameters	Formulas
Crack location parameters	$\left(x_p=\frac{x}{\left|x_{max}\right|},y_p=\frac{y}{\left|y_{max}\right|}\right)$
Crack angle	$\theta$
Crack width	$W_i=\frac{b}{B}$
Crack length coefficient	$L_i=\frac{a}{A}$
Aspect ratio of the crack	$S_c=\frac{a}{b}$
Crack area ratio	$A_r=\frac{ab}{AB}=L_i·W_i$
Dimensionless crack diffusion coefficient	$R_d=\frac{D_c}{D_m}$
Dimensionless equivalent diffusion coefficient	$R_e=\frac{D_x}{D_m}$

.

4.1. Crack Location Parameters

Cracks may occur at different locations within concrete. For instance, cracks often initiate from the surface when concrete is subjected to bending, while temperature-induced cracks formed during concrete casting tend to be located internally. Assessing the impact of crack locations on the durability of concrete can be achieved by calculating the concrete’s equivalent diffusion coefficient, assuming the dimensions of both the concrete and cracks remain constant, and the angle of the crack θ = 0 is fixed while the crack can be positioned arbitrarily within the concrete. Hence, a numerical model of concrete with cracks is established using COMSOL to simulate the transport of ions within it. The equivalent diffusion coefficient $D_x$ of the concrete is calculated. The variation in $D_x$ with the parameters of crack position $\left(x_p,y_p\right)$ is studied, and the results are shown in Figure 9. In the figure, ${\left(D_x\right)}_{min}$ represents the minimum value of the equivalent diffusion coefficient, and $D_x/{\left(D_x\right)}_{min}$ represents the ratio of the equivalent diffusion coefficient to the minimum value at any position.

Figure 9a illustrates the spatial surface of $D_x/{\left(D_x\right)}_{min}$ concerning variations in $x_p$ and $y_p$, which resembles a saddle shape with a relatively flat central region and steep slopes around it. Figure 9b represents the projection of Figure 9a onto the x-y plane, depicting contour lines symmetrically about the x-axis, y-axis, and origin. It can be observed from both figures that when the crack is positioned at the midpoint of the left and right boundaries of the concrete, the equivalent diffusion coefficient reaches its maximum. Conversely, when the crack is located at the midpoint of the upper and lower boundaries, the equivalent diffusion coefficient reaches its minimum. The ratio of the maximum value to the minimum value is approximately 1.02, indicating that although there are variations in the equivalent diffusion coefficient with changes in crack position, the overall change is minor. Figure 9c depicts the curve of $D_x/{\left(D_x\right)}_{min}$ with variations in $x_p$ when $y_p=0$. From the figure, it can be observed that the equivalent diffusion coefficient changes gradually when the crack is located in the central region, whereas it increases rapidly when the crack approaches the left or right boundaries. Figure 9d illustrates the curve of $D_x/{\left(D_x\right)}_{min}$ with variations in $y_p$ when $x_p=-1$. It can be seen from the figure that the equivalent diffusion coefficient changes gradually when the crack is situated in the central region, while it decreases rapidly when the crack approaches the upper or lower boundaries.

In general, cracks pose the greatest risk to concrete when they are close to the inlet and outlet surfaces, while their impact on concrete is minimized when they are near the upper and lower boundaries. The influence of cracks on concrete is relatively minor when they move within the central region.

4.2. Crack Angle Parameters

In concrete, cracks often exist with different orientations. By setting various angles for the cracks and employing numerical simulation methods to analyze the changes in ion concentration distribution within the concrete, as well as variations in the equivalent diffusion coefficient, one can assess the impact of crack orientation on the ion transport properties of concrete.

Figure 10 presents contours of ion concentration in concrete under three different crack angles. This figure reflects the variation in ion concentration distribution with crack angle. It can be observed from the figure that along the direction of the crack, the contour lines bend noticeably towards the lower concentration direction. When the crack orientation aligns with the direction of ion diffusion, the contour lines exhibit the most significant curvature. Conversely, when the crack orientation is perpendicular to the direction of ion diffusion, the contour lines are largely unaffected. This phenomenon occurs because ions diffuse faster within the crack than within the matrix, resulting in higher ion concentration within the crack compared to the matrix on the same cross-section.

Figure 11 illustrates the variations in the equivalent diffusion coefficient in concrete with respect to crack angle. In the figure, ${\left(D_x\right)}_{90}$ represents the equivalent diffusion coefficient along the x-direction when $\theta =90°$, and $D_x/{\left(D_x\right)}_{90}$ represents the ratio of the equivalent diffusion coefficient at any angle to that at $90°$. When $x_p=-1$ and $y_p=0$, the crack is precisely located on the symmetry axis of the concrete. When $x_p=-0.625$ and $y_p=0.5$, the crack is positioned in the upper left part of the concrete.

From Figure 11, it can be observed that regardless of the crack’s position, the curve representing the variation in the equivalent diffusion coefficient with crack angle resembles a cosine curve. The only difference lies in the amplitude of the curve. When $\theta =90°$ (i.e., when the crack is perpendicular to the direction of ion diffusion), the equivalent diffusion coefficient is minimal. Conversely, when $\theta =0°$ or $\theta =180°$ (i.e., when the crack is parallel to the direction of ion diffusion), the equivalent diffusion coefficient is maximal.

Figure 10 and Figure 11 indicate that the orientation of cracks not only affects the distribution of ion concentration in localized areas within concrete but also significantly influences the overall diffusion performance of concrete.

Considering the combined effects of crack position and crack angle, when the crack is positioned in the middle of the boundaries where ions enter and exit the concrete, and the crack orientation is parallel to the direction of ion transport, the equivalent diffusion coefficient of the concrete is maximal. This implies that this situation is the most detrimental to the durability of concrete. Therefore, in the following studies, the crack is assumed to be in the most unfavorable condition. Further analysis will be conducted to examine the effects of various parameters on the ion transport properties of concrete based on this assumption.

4.3. Crack Width Coefficient and Crack Length Coefficient

The study on the effects of crack position and crack angle reveals that the most detrimental position for cracks in concrete is at the midpoint of the inlet edge in the $\theta =0°$ direction. Therefore, the subsequent research will focus on this position.

Assuming the dimensions of the concrete matrix are 100 mm × 100 mm, and the crack is in the most unfavorable state with a ratio of crack diffusion coefficient to matrix diffusion coefficient $R_d=200$, the influence of the crack width coefficient $W_i$ and crack length coefficient $L_i$ on the ion transport properties of concrete are further investigated. The dimensionless equivalent diffusion coefficient $R_e$, obtained through numerical simulations, is depicted in the contour in Figure 12. From the figure, it can be observed that with increases in $W_i$ and $L_i$, the contour lines become denser, and they gradually trend towards the vertical direction. This indicates that as the crack dimensions increase, the variations in the equivalent diffusion coefficient intensify, and the equivalent diffusion coefficient becomes more sensitive to changes in the length coefficient.

Figure 13 depicts the curves of the dimensionless equivalent diffusion coefficient with respect to the length coefficient and width coefficient. From Figure 13a, it can be observed that with a constant $L_i$, $R_e$ increases as $W_i$ increases, and except for $L_i=1$, the increasing trend gradually slows down. The larger the value of $L_i$, the greater the magnitude of $R_e$ variation with $W_i$. When $L_i=0.1$, the variation in $R_e$ with $W_i$ is minimal, while when $L_i=1$, $R_e$ increases linearly with $W_i$. From Figure 13b, it can be seen that with a constant $W_i$, $R_e$ increases as $L_i$ increases, and when $L_i$ approaches 1, $R_e$ increases sharply. The larger the value of $W_i$, the greater the magnitude of $R_e$ variation with $L_i$.

4.4. The Aspect Ratio of the Crack and Crack Area Ratio

The shape of cracks and crack area are also important geometric parameters affecting the ion transport properties of cracked concrete. The dimensionless equivalent diffusion coefficient $R_e$ with respect to the crack aspect ratio $S_c$ and crack area ratio $A_r$ is depicted in the contour in Figure 14. From the figure, it can be observed that the shapes and orientations of the contour lines are generally similar. $R_e$ increases with the increases in $S_c$ and $A_r$, and as $S_c$ and $A_r$ increase, there is a tendency for the contour lines to become closer together, indicating that the magnitude of $R_e$ variation increases.

Figure 15 illustrates the curves of $R_e$ with respect to $S_c$ and $A_r$. From Figure 15a, it can be observed that when $A_r$ is constant, $R_e$ increases rapidly with the increase in $S_c$ at first, then the growth rate slows down. When $S_c$ is approaching its maximum value (when the crack penetrates the concrete, $L_i=1$), $R_e$ increases rapidly again. Different $A_r$ values correspond to different ranges of $S_c$. Larger $A_r$ values lead to smaller ranges of $S_c$, faster variations in $R_e$ with $S_c$, and higher maximum values of $R_e$. From Figure 15b, it can be seen that when $S_c$ is constant, $R_e$ initially increases linearly with the increase in $A_r$, and then increases rapidly. When $A_r$ is approaching its maximum value (when the crack penetrates the concrete, $L_i=1$), the growth rate of $R_e$ further increases. Different $S_c$ values correspond to different ranges of $A_r$. Larger $S_c$ values lead to smaller ranges of $A_r$, and the initial slope of the curve is larger. However, the maximum value of $R_e$ is smaller.

4.5. Dimensionless Crack Diffusion Coefficient

The dimensionless crack diffusion coefficient is an important material parameter affecting the ion transport performance of cracked concrete. When the geometric parameters of cracks are fixed, the dimensionless equivalent diffusion coefficient $R_e$ of cracked concrete does not vary linearly with the dimensionless crack diffusion coefficient $R_d$. Different geometric parameters have varying degrees of influence on the curve of $R_e$ with respect to $R_d$, as shown in Figure 16. From the figure, it can be observed that as $R_d$ increases, $R_e$ increases nonlinearly, and the rate of increase gradually diminishes. Figure 16a,b reflect the influence of the crack width coefficient $W_i$ on the curve changes. When $R_d$ is small, the wider the crack, the greater the rate of change in $R_e$. However, as $R_d$ increases, the rate of change in $R_e$ decreases gradually. Moreover, the wider the crack, the smaller the rate of change in $R_e$. This indicates that the larger the $R_d$ value, the wider the crack, and the less sensitive $R_e$ is to changes in $R_d$. Figure 16c,d reflect the influence of the crack length coefficient $L_i$ on the curve changes. The longer the crack, the more sensitive $R_e$ is to changes in $R_d$. However, as $R_d$ increases, the rate of change in $R_e$ for longer cracks decreases more rapidly, leading to a gradual reduction in the difference in the rate of change in $R_e$ with respect to $R_d$ caused by different length coefficients. Figure 16e,f reflect the influence of the crack aspect ratio $S_c$ on the curve changes. The thinner and longer the crack, the more sensitive $R_e$ is to changes in $R_d$. Moreover, as $R_d$ increases, the rate of change in $R_e$ for thinner and longer cracks decreases more slowly, leading to a gradual increase in the difference in the rate of change in $R_e$ with respect to $R_d$ caused by different aspect ratios. Figure 16g,h reflect the influence of the crack area ratio $A_r$ on the curve changes. The larger the crack area, the more sensitive $R_e$ is to changes in $R_d$. However, as $R_d$ increases, the rate of change in $R_e$ for larger crack areas decreases more rapidly, leading to a gradual reduction in the difference in the rate of change in $R_e$ with respect to $R_d$ caused by different area ratios.

4.6. Crack Frequency

In the previous study, the position where a single crack poses the greatest threat to concrete was identified. In this section, the impact of multiple cracks on the durability of concrete is investigated. Figure 17 shows the relationship between crack spacing and the diffusion coefficient, where the horizontal axis d represents the crack spacing. When there are two cracks present, the crack spacing will affect the equivalent diffusion coefficient. From the graph, it can be observed that the equivalent diffusion coefficient reaches its maximum value when the spacing between two cracks is 50 mm. Conversely, the equivalent diffusion coefficient reaches its minimum value when the crack spacing is either 0 or at its maximum value. From the figure showing the relationship between three cracks and their spacing, it is observed that when there are three cracks, the maximum value of the equivalent diffusion coefficient occurs at a spacing of 33.33 mm. Similarly, when there are four cracks, the maximum value occurs at 25 mm. The pattern summarized in Figure 18 indicates that when the side length of the concrete inlet is B and the number of cracks is n, the maximum value of the concrete’s equivalent diffusion coefficient occurs when the crack spacing is at the B/n position and the distance between the cracks at the top and bottom ends and the concrete edge is B/2n. At this point, the damage to the durability of the concrete is maximized.

4.7. The Theoretical Solution for the Equivalent Diffusion Coefficient

4.7.1. Parallel Model and Series Model

When the ions diffuse along the x-direction, assuming the ion concentrations at the left and right boundaries are constant, denoted as ${\mathrm{c}}_{\mathrm{L}}$ and ${\mathrm{c}}_{\mathrm{R}}$, respectively, and there are no fluxes at the top and bottom boundaries, with the concrete having a length of A and a width of B where the red area represents the crack with length a and width b, under these conditions, the equivalent diffusion coefficient ${\mathrm{D}}_{\mathrm{x}}$ in the x-direction is derived; the computational model is depicted in Figure 19a. The total amount of ions at the right boundary is given by the following:

$$JA=J_{1}b+J_2\left(B-b\right)$$

(9)

where $J$ is the average flux at the right boundary, and $J_1$ and $J_2$ represent the fluxes at the right boundary corresponding to the crack area and the mortar area, respectively.

$$\left\{\begin{array}{c}J=D_x\frac{c_L-c_R}{A}\\ J_1=D_m\frac{c_L-c_R}{A}\\ J_2=D_c\frac{c_L-c_R}{A}\end{array}\right.$$

(10)

Substituting Equation (10) into Equation (9) and simplifying yields gives the following:

$$D_x=\frac{\left(B-b\right)}{B}{D}_m+\frac{b}{B}{D}_c$$

(11)

Using simplified parameterization yields:

$$D_x=\left(1-W_i\right)D_m+W_i{D}_c$$

(12)

Here, Equation (12) represents the formula for calculating the equivalent diffusion coefficient in parallel connection.

The equivalent diffusion coefficient ${\mathrm{D}}_{\mathrm{y}}$ in the y-direction is derived under the following conditions: The ions diffuse along the y-direction, assuming the ion concentrations at the upper and lower boundaries are constant, denoted as ${\mathrm{c}}_{\mathrm{u}}$ and ${\mathrm{c}}_{\mathrm{d}}$, respectively, and there are no fluxes at the left and right boundaries. Additionally, the ion concentrations on the upper and lower sides of the crack layer are denoted as ${\mathrm{c}}_1$ and ${\mathrm{c}}_2$, respectively; the series connection model is depicted in Figure 19b. Since the fluxes at the individual cross-sections perpendicular to the y-axis are equal, the total amount of ions passing through the lower boundary is given by the following:

$$J=D_m\frac{c_u-c_1}{\frac{B-b}{2}}=D_c\frac{c_1-c_2}{b}=D_m\frac{c_2-c_d}{\frac{B-b}{2}}$$

(13)

The following equation can be obtained from Equation (13):

$$\left\{\begin{array}{c}{c}_u-c_1=\frac{\frac{J\left(B-b\right)}{2}}{D_m}\\ c_1-c_2=\frac{Jb}{D_c}\\ c_2-c_d=\frac{\frac{J\left(B-b\right)}{2}}{D_m}\end{array}\right.$$

(14)

For the overall analysis, the flux at the lower boundary can also be expressed as the following:

$$J=D_y\frac{c_u-c_d}{B}=D_y\frac{\left(c_u-c_1\right)+\left(c_1-c_2\right)+\left(c_2-c_d\right)}{B}$$

(15)

Substituting Equation (14) into Equation (15) and simplifying yields gives the following:

$${\left(D_y\right)}^{-1}=\frac{\left(B-b\right)}{B}{\left(D_m\right)}^{-1}+\frac{b}{B}{\left(D_c\right)}^{-1}$$

(16)

Using simplified parameterization yields:

$${\left(D_y\right)}^{-1}=\left(1-Wi\right){\left(D_m\right)}^{-1}+Wi{\left(D_c\right)}^{-1}$$

(17)

Equation (17) represents the formula for calculating the equivalent diffusion coefficient in series connection.

Sometimes, the crack does not fully penetrate. In such cases, it is necessary to combine both parallel model and series model. The specific method, using the parallel model first followed by the series model, is illustrated in Figure 20: first, segregate the model with the crack using the parallel model in Figure 20b to calculate $D^{\prime }$. Then, consider the entire region in Figure 20b as the crack area and use the series model in Figure 20c to calculate the overall equivalent diffusion coefficient $D_x$.

Substituting into Equation (11) yields the following:

$$D^{\prime }=\frac{B-b}{B}{D}_m+\frac{b}{B}{D}_c$$

(18)

Substituting into the series model yields the following:

$${\left(D_x\right)}^{-1}=\frac{A-a}{A}{\left(D_m\right)}^{-1}+\frac{a}{A}{\left(D^{\prime }\right)}^{-1}$$

(19)

The approach employing the series-then-parallel model as depicted in Figure 21 is as follows: First, the model containing cracks is distinguished and utilized to compute $D^{\prime }$ using the series model depicted in Figure 21b. Then, the entire region in Figure 21b is considered as the cracked area, and the overall equivalent diffusion coefficient ${\mathrm{D}}_{\mathrm{x}}$ is obtained using the parallel model illustrated in Figure 21c.

Substituting into Equation (16) yields the following:

$${\left(D^{\prime }\right)}^{-1}=\frac{\left(A-a\right)}{A}{\left(D_m\right)}^{-1}+\frac{a}{A}{\left(D_c\right)}^{-1}$$

(20)

Substituting into the parallel model yields the following:

$$D_x=\frac{\left(B-b\right)}{B}{D}_m+\frac{b}{B}{D}^{\prime }$$

(21)

4.7.2. Verification of Theoretical Solutions

To validate the correctness of the theoretical formulas, both the series and parallel models are verified. A COMSOL simulation is conducted for a through-crack with widths ranging from 0.1 to 0.2 mm and a length of 100 mm. As shown in

Table 6. Validations of theoretical solutions for the parallel model and series model.

Parallel Model			Series Model
Theoretical Solution (m2/s)	Real Value (m2/s)	Errors (%)	Theoretical Solution (m2/s)	Real Value (m2/s)	Errors (%)
5.8953 × 10−12	5.8953 × 10−12	0	4.70469 × 10−12	4.7047 × 10−12	2.92057 × 10−6
6.13436 × 10−12	6.1344 × 10−12	6.52061 × 10−6	4.70562 × 10−12	4.7056 × 10−12	5.23485 × 10−6
6.37342 × 10−12	6.3734 × 10−12	3.13804 × 10−6	4.70656 × 10−12	4.7066 × 10−12	7.78033 × 10−6
6.61248 × 10−12	6.6125 × 10−12	3.02457 × 10−6	4.7075 × 10−12	4.7075 × 10−12	5.31844 × 10−7
6.85154 × 10−12	6.8515 × 10−12	5.83814 × 10−6	4.70844 × 10−12	4.7084 × 10−12	8.92046 × 10−6
7.0906 × 10−12	7.0906 × 10−12	0	4.70938 × 10−12	4.7094 × 10−12	3.84899 × 10−6

, when the crack is fully penetrating, the relative errors of the theoretical solution for the equivalent diffusion coefficient obtained using both parallel and series models are almost close to 0. Hence, using this method to calculate the overall equivalent diffusion coefficient of cracked concrete with penetrating cracks is feasible.

In general, cracks do not fully penetrate concrete structures. In such cases, it is necessary to use either the parallel-then-series model or the series-then-parallel model to calculate the overall equivalent diffusion coefficient of cracked concrete. The results from conducting COMSOL simulations for cracks with widths ranging from 0.1 to 0.2 mm and a length of 50 mm are depicted in Figure 22, where the theoretical solutions obtained from the two models are compared with the simulation values obtained from COMSOL.

Table 7. Validations of the theoretical solutions of the parallel-then-series model and the series-then-parallel model.

Parallel-Then-Series Model			Series-Then-Parallel Model
Theoretical Solution (m2/s)	Real Value (m2/s)	Errors (%)	Theoretical Solution (m2/s)	Real Value (m2/s)	Errors (%)
5.29125 × 10−12	5.1988 × 10−12	1.778206	4.75076 × 10−12	5.1988 × 10−12	8.618147
5.38559 × 10−12	5.2649 × 10−12	2.292374	4.7517 × 10−12	5.2649 × 10−12	9.747541
5.47579 × 10−12	5.3255 × 10−12	2.822126	4.75264 × 10−12	5.3255 × 10−12	10.756855
5.56212 × 10−12	5.3814 × 10−12	3.35816	4.75359 × 10−12	5.3814 × 10−12	11.666376
5.64481 × 10−12	5.4331 × 10−12	3.896616	4.75453 × 10−12	5.4331 × 10−12	12.489599
5.72409 × 10−12	5.4811 × 10−12	4.433233	4.75547 × 10−12	5.4811 × 10−12	13.238774

illustrates the maximum and minimum errors for both approaches. The maximum error for the series-then-parallel model is 13.2%, with a minimum error of 8.6%. On the other hand, the maximum error for the parallel-then-series model is 4.4%, with a minimum error of 1.77%. It is evident that the errors of the series-then-parallel model are significantly smaller than those of the parallel-then-series model. Therefore, when dealing with general cracks, it is preferable to use the series–parallel model for calculation. The average relative error of the series-then-parallel model is 3.09%, indicating a certain level of feasibility in obtaining theoretical solutions for cracked concrete using this model.

5. Conclusions

This paper employs a homogenization method to simplify the four-phase model into a two-phase model. Based on COMSOL software, the study investigates the influences of factors such as crack location, crack angle, crack length and width, crack shape, and crack frequency on the chloride ion transport properties in cracked concrete. According to the simulation results of the model, the following conclusions can be drawn.

(1)

When the crack position varies along the x-direction, the change in the equivalent diffusion coefficient in the middle portion of the concrete is gradual, while it rapidly increases near the left and right boundaries. When the crack position varies along the y-direction, the trend is opposite to that of the x-direction. In the middle portion, the equivalent diffusion coefficient changes gradually, while it rapidly decreases near the top and bottom boundaries. Generally, cracks pose the greatest threat to concrete when they are close to the inlet and outlet surfaces, and the least threat when they are close to the top and bottom boundaries. The impact on concrete is relatively small when cracks move in the middle positions.

(2)

Under the influence of the crack angle, when θ = 90° (i.e., the crack is perpendicular to the direction of chloride ion diffusion), the damage to concrete is minimized. However, when θ = 0° or θ = 180° (i.e., the crack is parallel to the direction of chloride ion diffusion), the damage to concrete is maximized.

(3)

When the crack length is about to penetrate the concrete, the equivalent diffusion coefficient of cracked concrete increases sharply. Under the condition of equal crack area, the thinner and longer the crack, and the greater the equivalent diffusion coefficient of cracked concrete. The increase in the equivalent diffusion coefficient of cracked concrete is non-linear with the increase in the crack diffusion coefficient, and the trend of increase gradually diminishes. Different crack structural parameters lead to different trends in the variation in the equivalent diffusion coefficient of cracked concrete with the crack diffusion coefficient.

(4)

Through studying multiple cracks, the following pattern is discovered. When the side length of concrete is B and the number of cracks is n (n ≥ 2), if the spacing between cracks is located at B/n, and the distance between the cracks at the top and bottom ends and the concrete edge is B/2n, the equivalent diffusion coefficient reaches its maximum value. At this point, the damage to the concrete is maximized.

(5)

Based on Fick’s diffusion law, a theoretical model for the equivalent diffusion coefficient of cracked concrete was established. A comparison between theoretical and numerical solutions shows that the maximum error in the equivalent diffusion coefficient calculated using the “parallel-then-series” model is 4.4%, while for the “series-then-parallel” model, it is 13.2%. Both models can effectively predict the equivalent diffusion coefficient of cracked concrete in real projects, with the “parallel-then-series” model exhibiting higher accuracy.