Simulation of Macroscopic Chloride Ion Diffusion in Concrete Members

Abstract

To quantitatively analyze the macroscopic diffusion process of chloride ions in existing concrete members, the Peridynamic Differential Operator (PDDO) was introduced to formulate a discrete format for Fick’s second law, and a simulation model was established and validated. Subsequently, the influence of specific or randomly distributed defects in the concrete is reflected by adjusting the coefficients in the model’s global matrix. Moreover, the complex geometry of concrete members is captured by employing a point set-based spatial discretization approach. The model also accommodates for the complex corrosion conditions encountered in practice by imposing different boundary conditions. These features allowed for the simulation and validation of chloride ion diffusion experiments on concrete under natural environmental conditions. The study further analyzed how factors such as defects, diffusion coefficients, boundary conditions, and the geometric shape of members influence the macroscopic diffusion process. The findings indicate that the numerical model based on the PDDO can effectively quantify the macroscopic diffusion of chloride ions in existing concrete members. It provides fundamental data for the durability maintenance of concrete infrastructures and potentially reduces their carbon footprint by preventing unnecessary rehabilitation or reconstruction.

1. Introduction

Chloride ion ingress to the steel reinforcement surface of existing concrete members, initiating corrosion, is a significant factor in the degradation of their structural durability [1,2,3,4]. In China, conditions such as the coastal salt spray environment, the extensive saline soil conditions in the western regions, and the widespread use of deicing salts in practical engineering scenarios present considerable opportunities for chloride ingress. This issue poses a risk to the durability of numerous significant infrastructures, including coastal and high-altitude buildings, high-speed rail lines, docks, sea-crossing bridges, breakwaters, and undersea tunnels [5,6]. Maintaining the durability of these infrastructures can postpone the rehabilitation or reconstruction of them, which will result in huge greenhouse gas emissions. Therefore, to support the durability maintenance of infrastructures and then to significantly reduce their carbon footprint, it is essential to quantitatively analyze the chloride ion transport process in existing concrete members.

Apparent diffusion is the macroscopic mechanism governing the transport of chloride ions within concrete, which can be described using Fick’s second law and its modified models [7,8,9,10,11]. This framework is characterized by its simplicity and clear meaning, allowing for the coupling effects of other transport mechanisms to be reflected through the model parameters, thereby effectively addressing engineering problems [12,13,14]. However, during quantitative analysis, the inherent pores and microcracks in concrete materials (collectively referred to as defects) lead to discontinuities in the solution domain. The varying geometries of different concrete members result in irregular solution domains, while differing environmental influences contribute to complex boundary conditions. These macro features render the analytical solutions of Fick’s second law inapplicable and necessitate the use of numerical methods for resolution. The finite volume method (FVM) can effectively simulate chloride diffusion under complex boundary conditions, enabling accurate predictions of the chloride distribution and service life of concrete structures exposed to chloride environments [12]. The finite element model (FEM) can effectively be used to solve for multiphase chloride diffusion in concrete, incorporating the influence of the interfacial transition zone (ITZ) [15]. The FEM can be further used to solve three-phase and multi-component ionic transport models [16].

Although the aforementioned methods are applicable and can give meaningful results, they assume the solution domain is a continuous medium or a medium with minimal defects at specified locations. Such an assumption diverges significantly from the characteristics of existing concrete materials and members. Furthermore, these methods discretize the solution domain into a grid, making them less adaptable to complex geometries and resulting in pronounced grid dependency in the solution outcomes. Additionally, difference methods often struggle to satisfy the conservation principles expressed in integral form [17].

Under conditions of discontinuity in the medium (solution domain), employing nonlocal numerical methods to solve differential equations offers distinct advantages [18,19]. In scenarios in which the geometries of the solution domain and the boundary conditions are complex, nonlocal numerical methods based on point cloud discretization provide better adaptability compared to local methods that rely on grid discretization [20]. Researchers have utilized the nonlocal bond-based peridynamics (BBPD) method to conduct in-depth analyses of the diffusion process of chloride ions in saturated concrete at the mesoscale [21,22,23]. The approach based on the Peridynamic Differential Operator (PDDO) [24] retains the characteristics and advantages of the aforementioned peridynamic methods: it does not require the assumption of medium continuity; it introduces nonlocal (non-contact) interactions between material points, using the weighted averages of differences within neighborhoods to represent differentiation, thereby transforming differential equations into integral equations; and it effectively computes derivative values within the domain, whether at smooth, continuous, or discontinuous points. Furthermore, this method is more flexible than other implementations of peridynamics, allowing for irregular family shapes, or influence domains, of points and not requiring specialized modifications for boundaries. The computational process preserves the original macroscopic parameters and their physical significance, avoiding the introduction of difficult-to-calibrate mesoscopic parameters [25,26].

This study attempts to apply the PDDO to quantify the macroscopic diffusion process of chloride ions based on a modified form of Fick’s second law. Initially, we provide a brief introduction to the theory of the PDDO. Subsequently, we simulate the three-dimensional diffusion process in concrete under ideal conditions and validate the reliability of the PDDO-based method by comparing it with analytical solutions. Building on this foundation, we consider engineering factors such as the discontinuity of concrete materials, the complex geometries of diffusion regions, and the diversity of diffusion boundary conditions to quantify the macroscopic diffusion phenomena of chloride ions in concrete members under practical engineering scenarios. Finally, based on the simulation results from the PDDO-based method, we analyze the characteristics of chloride ion diffusion within concrete members. The results indicate that the proposed PDDO-based method can quantify macroscopic chloride ion diffusion in concrete. It can aid in the durability maintenance of infrastructures and potentially help to reduce their carbon footprint. To the best of the authors’ knowledge, this is the first attempt to apply the PDDO to a simulation of macroscale chloride diffusion under engineering scenarios.

2. Discrete Solution Format of Fick’s Second Law Using PDDO

2.1. Peridynamic Differential Operator

As illustrated in Figure 1, the solution domain can be discretized into a set of PD points. Each PD point $\mathit{x} ({\mathrm{x}}_1, {\mathrm{x}}_2, {\mathrm{x}}_3)$ has multiple member points within its influence domain $H_{\mathbf{x}}$ (shaded gray area). Interactions occur through bonds between the PD point $\mathit{x}$ and any member point ${\mathbf{x}}^{\prime }({\mathrm{x}}_1^{\prime },{ \mathrm{x}}_2^{\prime },{ \mathrm{x}}_3^{\prime })$, regardless of whether they are in contact. The initial length of the bond is defined as the magnitude of the initial relative position vector between the two points. The characteristic length (radius) of the influence domain is referred to as the horizon.

The Peridynamic Differential Operator (PDDO) for the field function $f\left(\mathrm{x}\right)$ at point x is defined as follows [25]:

$${\displaystyle \frac{{\partial }^{p_1+p_2+p_3}f\left(x\right)}{\partial x_1^{p_1}\partial x_2^{p_2}\partial x_3^{p_3}}} ={\int }_{H_x}f\left(x_1^{\prime }, x_2^{\prime }, x_3^{\prime }\right)g_n^{p_1{p}_2{p}_3}\left(\mathsf{\xi }\right)dV$$

(1)

where $n$ is the sum of the highest order of the differential in the $x_1, x_2, x_3$ dimension; $p_1, p_2,{ p}_3$ are the order of the differential in the $x_1,{ x}_2,{ x}_3$ dimension, respectively; $0\le p_1 +{ p}_2 + p_3 \le n$; $dV = d{\mathsf{\xi }}_{1}d{\mathsf{\xi }}_{2}d{\mathsf{\xi }}_3$ denotes the micro-volume, represented by the member point $(x_1^{\prime }, x_2^{\prime },{ x}_3^{\prime })$; and $g_n^{p_1{p}_2{p}_3}\left(\mathsf{\xi }\right)$ is the peridynamic (PD) function as follows:

$$g_n^{p_1{p}_2{p}_3}\left(\mathsf{\xi }\right) =\sum _{q_1=0}^n\sum _{q_2=0}^{n-q_1}\sum _{q_3=0}^{n-q_1-q_2}{a}_{q_1{q}_2{q}_3}^{p_1{p}_2{p}_3}{\omega }_{q_1{q}_2{q}_3}\left(\left|\mathsf{\xi }\right|\right){\mathsf{\xi }}_1^{q_1}{\mathsf{\xi }}_2^{q_2}{\mathsf{\xi }}_3^{q_3}$$

(2)

where $a_{q_1{q}_2{q}_3}^{p_1{p}_2{p}_3}$ are the undetermined coefficients, $\left|\mathsf{\xi }\right|$ is the magnitude of the relative position vector, and ${\omega }_{q_1{q}_2{q}_3}\left(\right|\mathsf{\xi }\left|\right)$ is the weight function. The weight function represents the physical characteristic parameters of the material and reflects the intensity of interactions between material points, providing a theoretical foundation for handling discontinuities in the solution domain. Based on its statistical mechanical meaning and the required conditions of non-negativity and normalization, the form selected in this study is as follows [24]:

$$\mathsf{\omega }(|\mathsf{\xi }\left|\right) ={ \mathsf{\pi }}^{-{\displaystyle \frac{l}{2}}}({\displaystyle \frac{s}{\mathsf{\delta }}}{)}^l\exp \left(-{\displaystyle \frac{s^2}{{\delta }^2}}{\left|\mathsf{\xi }\right|}^2\right)$$

(3)

where l denotes the number of dimensions of the problem, with l = 1, 2, or 3, and $s > 0$ is the shape parameter, which is set to 4 in this study.

To determine the undetermined coefficients $a_{q_1{q}_2{q}_3}^{p_1{p}_2{p}_3}$, Equation (2) can be substituted into the orthogonality condition as follows:

$${\displaystyle \frac{1}{n_1!n_2!n_3!}}{\int }_{H_{\mathbf{x}}}{\mathsf{\xi }}_1^{n_1}{\mathsf{\xi }}_2^{n_2}{\mathsf{\xi }}_3^{n_3}{g}_n^{p_1{p}_2{p}_3}\left(\mathsf{\xi }\right)d{\mathsf{\xi }}_{1}d{\mathsf{\xi }}_{2}d{\mathsf{\xi }}_3 ={ \delta }_{n_1{p}_1}{\delta }_{n_2{p}_2}{\delta }_{n_3{p}_3}$$

(4)

where $n_1, n_2,{ n}_3$, respectively, represent the highest order of expansion in the $x_1, x_2,{ x}_3$ dimension after neglecting the remainder, and δnp is the Kronecker delta.

Then, by rearranging this, a system of linear equations is obtained as follows:

$$\sum _{q_1=0}^n\sum _{q_2=0}^{n-q_1}\sum _{q_3=0}^{n-q_1-q_2}{A}_{\left(n_1,n_2,n_3\right)\left(q_1,q_2,q_3\right)}{a}_{q_1{q}_2{q}_3}^{p_1{p}_2{p}_3} = b_{n_1{n}_2{n}_3}^{p_1{p}_2{p}_3}$$

(5)

where the shape coefficients $A_{(n_1,n_2,n_3)(q_1,q_2,q_3)}=$${\int }_{H_x}{\omega }_{q_1{q}_2{q}_3}\left(\right|\mathsf{\xi }\left|\right){\mathsf{\xi }}_1^{n_1+q_1}{\mathsf{\xi }}_2^{n_2+q_2}{\mathsf{\xi }}_3^{n_3+q_3}d{\mathsf{\xi }}_{1}d{\mathsf{\xi }}_{2}d{\mathsf{\xi }}_3$, and on the right-hand side, $b_{n_1{n}_2{n}_3}^{p_1{p}_2{p}_3} = n_1!n_2!n_3!{\delta }_{n_1{p}_1}{\delta }_{n_2{p}_2}{\delta }_{n_3{p}_3}$.

After solving Equation (5) to obtain all the coefficients $A_{(n_1,n_2,n_3\left)\right(q_1,q_2,q_3)}$, we can substitute them into Equation (2) to determine the PD function $g_n^{p_1{p}_2{p}_3}\left(\mathsf{\xi }\right)$. Consequently, the PDDO form of the differential operator can be established from Equation (1).

It is important to note that as the order of differentiation n increases, the shape coefficient matrix defined by Equation (5) may become ill-conditioned, leading to significant rounding errors. To achieve convergence and sufficient accuracy within a reasonable computation time, it is essential to ensure that the size of the horizon is appropriately chosen.

2.2. Spatial Discretization of Fick’s Law Using PDDO

Fick‘s second law, which describes the transient diffusion process of chloride ions in saturated concrete, is given as follows:

$${\displaystyle \frac{\partial c(x_1,{ x}_2,{ x}_3, t)}{\partial t}} = D\left[{\displaystyle \frac{{\partial }^{2}c\left(x_1, x_2,{ x}_3, t\right)}{\partial x_1^2}}+{\displaystyle \frac{{\partial }^{2}c\left(x_1,{ x}_2,{ x}_3, t\right)}{\partial x_2^2}}+{\displaystyle \frac{{\partial }^{2}c\left(x_1,{ x}_2, x_3, t\right)}{\partial x_3^2}}\right]$$

(6)

where c is the time-dependent distribution function of the chloride ion concentration, D is the apparent diffusion coefficient of chloride ions in concrete, and t denotes time.

The conditions for determining solutions include the initial and boundary conditions. The initial condition is as follows:

$$c(x_1, x_2, x_3, 0) ={ c}_0$$

(7)

where c₀ is the initial concentration.

The boundary conditions are specified as follows:

$$c\left({\overline{x}}_1,{\overline{ x}}_2,{\overline{ x}}_3, t\right) = \overline{c}\left(t\right)$$

(8)

where ${\overline{x}}_i(i = 1~3)$ represents the spatial coordinates of the boundaries of the solution domain, and $\overline{c}\left(t\right)$ is the chloride ion concentration measured at the boundaries.

By applying the third-order PDDO as given in Equation (1), the spatial differential Equation (6) can be expressed in the following differential/integral equation:

$${\displaystyle \frac{\partial c\left(x_1, x_2, x_3, t\right)}{\partial t}} = D{\int }_{H_x}c(x_1^{\prime },{ x}_2^{\prime },{ x}_3^{\prime }, t\left)\right[g_3^{200}({\mathsf{\xi }}_1,{ \mathsf{\xi }}_2,{ \mathsf{\xi }}_3)+g_3^{020}({\mathsf{\xi }}_1,{ \mathsf{\xi }}_2, {\mathsf{\xi }}_3)+g_3^{002}({\mathsf{\xi }}_1,{ \mathsf{\xi }}_2,{ \mathsf{\xi }}_3\left)\right]dV$$

(9)

Discretizing the spatial domain into a set of PD points results in the discrete form of Equation (9), given as follows:

$${\displaystyle \frac{\partial c\left(x_1,{ x}_2, x_3, t\right)}{\partial t}} = D\sum _{i=1}^{N}c(x_{i1}, x_{i2},{ x}_{i3}, t\left)\right[g_3^{200}({\mathsf{\xi }}_{i1},{ \mathsf{\xi }}_{i2},{ \mathsf{\xi }}_{i3})+g_3^{020}({\mathsf{\xi }}_{i1}, {\mathsf{\xi }}_{i2},{ \mathsf{\xi }}_{i3})+g_3^{002}({\mathsf{\xi }}_{i1}, {\mathsf{\xi }}_{i2}, {\mathsf{\xi }}_{i3}\left)\right]V_i$$

(10)

where N is the total number of member points within the near field of PD point x, $x_{i1},{ x}_{i2},{ x}_{i3}$ represent the coordinates of the i-th member point, ${\mathsf{\xi }}_{i1},{ \mathsf{\xi }}_{i2},{ \mathsf{\xi }}_{i3}$ denote the components of the initial relative position vector between the PD point and the i-th member point, and V_i is the representative volume of the i-th member point.

For each spatially discretized PD point, Equation (10) can be established. By assembling the equations corresponding to all discretized points, we obtain the following:

$${\displaystyle \frac{\partial }{\partial t}}\mathit{c} = \mathit{Tc}$$

(11)

where c is the vector composed of chloride ion concentration values at all discretized points, and T is the coefficient matrix formed by the results of the right-hand side of Equation (10).

2.3. Time Domain Difference

Equation (11) shows the trend of the diffusion: it increases if the production of the apparent diffusion coefficients, which form the matrix T, and the current concentration is high. To quantify the trend, the time domain is discretized as follows.

By applying finite differences to Equation (11) in the time domain, we obtain the following:

$${\displaystyle \frac{{\mathit{c}}^{t+\mathsf{\Delta }t}-{ \mathit{c}}^t}{\mathsf{\Delta }t}} = \mathit{T}{\mathit{c}}^{t+\mathsf{\Delta }t}$$

or

$$\left(\mathit{I}-\mathsf{\Delta }t\mathit{T}\right){\mathit{c}}^{t+\mathsf{\Delta }t}={ \mathit{c}}^t$$

(12)

where ${\mathit{c}}^{t+\mathsf{\Delta }t}$ and ${\mathit{c}}^t$ are the column vectors of chloride ion concentration values at times $t+\mathsf{\Delta }t$ and $t$, respectively, $\mathsf{\Delta }t$ is the time step, and $\mathit{I}$ is the identity.

The Lagrange multiplier is employed to incorporate the boundary conditions (8) [27]:

$$\left[\begin{array}{cc}\mathit{I}-\mathsf{\Delta }\mathit{t}\mathit{T}& {\mathit{I}}^{\mathit{T}}\\ \mathit{I}& \mathbf{0}\end{array}\right]\left\{\begin{array}{c}{\mathit{c}}^{\mathit{t}+\mathsf{\Delta }t}\\ \mathit{\lambda }\end{array}\right\}=\left\{\begin{array}{c}{\mathit{c}}^{\mathit{t}}\\ {\overline{\mathit{c}}}^{\mathit{t}}\end{array}\right\}$$

(13)

where $\mathit{\lambda }$ is the Lagrange multiplier.

Then, based on the initial condition (7), we can solve Equation (13) step by step. In the calculations at each time step, this study utilizes the Intel Math Kernel Library (MKL) to implement heterogeneous parallel computing on shared or distributed memory systems, effectively enhancing the efficiency of solving Equations (5) and (12). Figure 2 illustrates the workflow for solving the macroscopic diffusion problem of chloride ions.

3. Validation of the PDDO Discrete Solution Format

3.1. Determination of the Horizon

The horizon comprehensively reflects the microscopic characteristics of the material and the specific features of the problem being solved. It determines the integration range that represents the cross-scale influence of microscopic features on macroscopic performance [28,29].

When spatially discretizing the solution, a horizon that is too small will lead to too few member points within the neighborhood, resulting in an ill-conditioned shape coefficient matrix in Equation (5), and may also produce unrealistic zero-energy deformation modes. Conversely, a horizon that is too large will significantly increase the computational workload without necessarily improving the accuracy. A horizon value equal to 3.015 times the distance between discretized points has been proposed [30]. For diffusion differential equations, after preliminary calculations and corrections, a value of four times the distance between discretized points is adopted for the horizon.

3.2. Diffusion Analysis Under Ideal Conditions

Figure 3a shows a cubic concrete specimen (150 mm × 150 mm × 150 mm) with three surfaces (the left side, bottom, and back) exposed to a chloride salt environment, where the chloride ion concentration at the exposed surfaces is 5% and the diffusion coefficient D = 3 × 10⁻⁶ mm²/s. The specimen is discretized into PD points with a spacing of 6 mm, and the representative volume for each point is 216 mm³, as illustrated in Figure 3b.

This example features a regular shape and simple boundary conditions. Assuming that the solution domain is continuous, the analytical solution for the diffusion Equation (6) can be readily obtained as follows [12]:

$$c(x_1, x_2,{ x}_3, t) = c_0(1-erf({\displaystyle \frac{x_1}{2\sqrt{Dt}}})erf({\displaystyle \frac{x_2}{2\sqrt{Dt}}})erf({\displaystyle \frac{x_3}{2\sqrt{Dt}}}\left)\right)$$

(14)

where $erf(.) = {\displaystyle \frac{2}{\sqrt{\mathsf{\pi }}}}{\int }_0{e}^{-t^2}dt$ is the error function.

The spatial diffusion process of chloride ions is calculated using Equation (12). By placing the coordinate origin at the lower-left corner of the model, the computed results along the z-axis at x = 75 mm and y = 75 mm are shown in Figure 3c. It demonstrates an ideal accuracy when comparing the analytical results and the test results from the literature [31], in which the test was conducted according to the standard for test methods of long-term performance durability of concrete [32].

It can be proven that as the horizon decreases (while keeping the number of member points constant or increasing), the peridynamic solution converges to the analytical solution of the general local theory [24,33]. Therefore, simulating the diffusion process of chloride ions in concrete using the PDDO-based method has a reliable theoretical foundation.

4. Simulation of Macroscopic Chloride Ion Diffusion in Concrete

In practical engineering scenarios, the apparent diffusion coefficient of chloride ions in concrete varies over time, the shapes of members are diverse, and the erosion conditions are complex, with defects such as pores and microcracks present. This results in a significant deviation from the boundary conditions and analytical solutions discussed in the previous section, making it impossible to obtain analytical solutions. In such cases, the PDDO model can still effectively simulate practical engineering situations.

Taking a concrete slab exposed to a chloride salt environment as an example, Figure 4 illustrates the slab’s side edges being exposed to the chloride salt environment in different configurations, neglecting the effects in the thickness direction. The initial apparent diffusion coefficient of chloride ions is 3 × 10⁻⁶ mm²/s. The slab is discretized into PD points with a spacing of 2 mm, and the representative area for each point is 4 mm².

4.1. Influence of Time-Dependent Diffusion Coefficient

The apparent diffusion coefficient is a macroscopic expression of various transport mechanisms and exhibits significant time-dependent characteristics, which can be represented by Equation (15) [34]:

$$D = D_R({\displaystyle \frac{t_R}{t}}{)}^m$$

(15)

where D_R is the initial (reference) apparent diffusion coefficient, t_R and t represent the reference age and the age at which the solution is sought, respectively, with t_R set to 1 year, and m is the deterioration coefficient, taken as 0.19. Figure 5 presents the computational results considering the time-dependent behavior of the apparent diffusion coefficient as per the above equation. It is clear that the diffusion slightly relives considering its time-dependent nature. Therefore, improving the apparent diffusion coefficient of concrete should be considered in maintenance strategies.

4.2. Influence of Complex Boundary Conditions

Due to varying environmental conditions, the different ways in which members are exposed to corrosive environments, and the differing corrosion protection measures, there exist various erosion surfaces in practical engineering scenarios. For cases in which the surface chloride ion concentration varies according to a known pattern, this can be abstracted as Dirichlet boundary conditions for Fick’s second law. For cases in which the flux of chloride ions across the surface changes according to a known pattern, this can be abstracted as Neumann boundary conditions for Fick’s second law. In the PDDO-based computational model represented by Equation (12), these two types of boundary conditions can be uniformly introduced using the Lagrange multiplier [24].

Figure 6 presents the computational results considering multiple boundary conditions. It shows that it is unfavorable if the chloride exposure surfaces are near or adjacent to each other. An appropriate coating strategy that separates the chloride exposure surfaces should be considered.

4.3. Influence of Shape of Members

The PDDO discretizes the solution domain into a point cloud rather than a grid, providing excellent adaptability to the solution domain and being largely unaffected by the complexity of its geometric shape. This allows for effective simulations of concave and convex polyhedral as well as simply and multiply connected domains. Additionally, the PDDO relaxes the requirement in conventional PD models that the influence domains of PD points must be symmetric around themselves, and it does not necessitate the special treatment of boundary points as some meshless methods do, further enhancing the flexibility of modeling.

Figure 7 presents the computational results for irregular regions. It indicates that the corner of members results in the superimposition of diffusion zones and the accumulation of chloride ion concentration. Therefore, the corners should be carefully protected in maintenance strategies.

4.4. Influence of Defects

Defects such as pores and cracks are randomly distributed in concrete. A porous hardened paste with large and less tortuous capillary pores presents an easier path for chloride ion diffusion. Meanwhile, the cracks act as preferential, high-speed pathways for the ingress of chlorides, water, and oxygen, allowing these aggressive agents to bypass the protective concrete cover. When simulating the diffusion of chloride ions, the defects can be abstracted as discontinuities within the solution domain. Discontinuities are present at multiple locations throughout the solution domain, and this characteristic can be reflected by adjusting the weight function in the PDDO or the coefficients of the matrix T in Equation (12).

When discussing the influence of a small number of defects with known locations, defect planes can be set at the corresponding positions in the model, as illustrated in Figure 8. When the bond between a PD point and its member point traverses a defect, the weight function value corresponding to that bond is modified based on the blocking, delaying, or accelerating effects of the solution or air within the defect on the diffusion process.

When discussing the impact of the random distribution of defects on the macroscopic diffusion process, the matrix T in Equation (12) can be directly utilized. According to the physical meaning of the macroscopic diffusion equation, each coefficient t_ij in T represents the intensity of the diffusion influence between material point i and its j-th member point [12]. Therefore, a probabilistic model can be specified based on the random distribution pattern of the defects, allowing for the sampling of coefficients to adjust t_ij. For concrete with normal quality, the volume of defects is relatively small compared to the member, so the adjustment will not lead to an ill-conditioned matrix T.

Neither of these methods increase the complexity of the model, require re-discretization during the calculation process, or increase the number of computations; therefore, they do not add to the overall accumulated error.

Figure 8a and Figure 8b present the computational results for defects with specified locations and randomly distributed defects, respectively. A normal distribution with a mean of 1 and a standard deviation of 5 × 10⁻⁵ is used to sample coefficients for adjusting t_ij. It indicates that random distributed defects will push the peak of the chloride ion concentration forward into the concrete. That means the chloride ions will penetrate the concrete cover and reach the rebars faster. This quantitively demonstrates the importance of using good-quality concrete with fewer defects.

5. Simulation and Prediction of Macroscopic Chloride Ion Diffusion in Concrete Members Under Natural Environments

5.1. Simulation of Experiments

Using the method outlined in Section 3, the experiments from reference [30] that were conducted under natural environments were simulated. Considering both computational accuracy and scale, the spacing between discretized points was set to 10 mm. The intrusion surfaces were subjected to Dirichlet boundary conditions based on on-site measurements. The apparent diffusion coefficient was determined according to reference [30], as shown by Equation (15) with D_R = 2.96 × 10⁻⁶ mm²/s, t_R = 1.3 years, and m = 0.19. The value is calibrated by field measurements and includes the influence of defects. Therefore, the distribution of defects is not explicitly considered again in this validation case. The time-dependent boundary chloride ion concentration was also calculated using the experimental results in reference [30], as shown by Equation (16):

$$c({\overline{x}}_1,{ \overline{x}}_2, {\overline{x}}_3, t) ={ c}_0{t}^n$$

(16)

where c₀ and n were determined through a regression after excluding outliers: c₀ = 3.72% and n = 0.31, yielding a coefficient of determination of R² = 0.88 with the experimental results.

Figure 9a illustrates the computational model, while Figure 9b presents a comparison of the simulation results with the experimental results.

The results from Figure 5, Figure 6, Figure 7, Figure 8 and Figure 9 validate the flexibility and fidelity of simulating the macroscopic diffusion of chloride ions in concrete using the PDDO, demonstrating that this method can offer quantitative data to aid in explaining the mechanisms of the diffusion process and can also reliably predict the diffusion process.

5.2. The Impact of Boundary Conditions

Building upon Section 5.1, we consider the differences in chloride ion concentration between the windward surface and the side surfaces. We reduce the chloride ion concentration on the side surfaces, denoting it as $c = 0.0372t^{0.31}\mathsf{|}cos \mathsf{\theta }\mathsf{|}$, where θ is the angle between the windward surface and the side surface, which is 120° in this case.

The calculation results are shown in Figure 10: after reduction, the rate of change of the chloride ion concentration in the direction of the vertical intrusion surface is significantly slowed down. It shows that the windward surfaces that are exposed to the chloride ion attack directly are more vulnerable than other surfaces. They determine the locations of the peak chloride ion concentration in existing concrete members and they are the key factor in service life predictions for existing concrete members. Appropriate coating or other measures should be applied in the case of these windward surfaces. Such maintenance measures can significantly reduce the chloride ion attack, delay the diffusion, and extend the members’ service life.

5.3. The Impact of Concrete Quality

It has been proposed that the diffusion of chloride ions will influence the structural or deformation parameters of concrete [35,36,37]. Building upon Section 4.1, we consider the random distribution of defects in concrete. We generate coefficients by sampling from a normal distribution with a mean of 1 and a standard deviation of 0.004 to adjust the coefficients t_ij in Equation (12).

The calculation results are illustrated in Figure 11: the influence of random defects causes significant fluctuations in the chloride ion concentration values at the same depth. The peaks of the chloride ion concentration shift forward in the depth direction, and this trend becomes more pronounced with higher concentration levels. The situation is in agreement with that revealed in Section 4.4: it potentially hastens the erosion of rebars in existing concrete members and shortens their service life. Therefore, maintenance would only be effective if based on good commerce, i.e., the members to be maintained use good-quality concrete with fewer defects.

6. Conclusions

This paper introduces the PDDO to establish a peridynamic model, taking into account the complex geometry of concrete members, different corrosion conditions, and the effects of random defects in concrete to quantitatively simulate the diffusion process of chloride ions. The research findings are as follows:

• Comparisons with analytical or experimental results show the peridynamic model based on the PDDO can effectively simulate the macroscopic diffusion process of chloride ions in concrete. Moreover, the model can be conveniently applied to concrete members with complex geometries and diverse corrosion conditions, while these conditions would result in challenges for other grid-dependent methods;
• The peridynamic model based on the PDDO can effectively quantify the impacts of known and random defects in concrete without the need to re-discretize the solution domain, and it does not increase computational iterations or cumulative errors. This could be difficult for other methods based on the assumption of continuous fields;
• The simulation based on the PDDO suggests specific measures in maintenance strategy for concrete members, including an improvement of the apparent diffusion coefficient of concrete, an appropriate coating that separates adjacent chloride exposure surfaces, and the careful protection of corners in concrete members;
• It is quantitively demonstrated that the defects in concrete result in the peaks of chloride ion concentration shifting forward in the depth direction, and this trend becomes more pronounced with higher concentrations. Under such circumstances, the chloride ions will penetrate the concrete cover and reach the rebars faster. Therefore, to slow the erosion of rebars in existing concrete members and extend their service life, it is important to use good-quality concrete with fewer defects.

The peridynamic model based on the PDDO provides an effective tool for simulating the macroscopic diffusion process of chloride ions in concrete members, thereby offering foundational data for the durability maintenance of a wide range of existing concrete structures. Proper maintenance will significantly reduce their carbon footprint by preventing unnecessary rehabilitation or reconstruction.