Numerical Simulation of Deterioration Process in Reinforced Concrete Based on COMSOL Multiphysics

Reinforced concrete is deteriorated by chloride attack and dry-wet cycles to suffer serious durability problems. Reinforced concrete degradation typically occurs in two distinct stages: initially through chloride erosion, followed by the electrochemical corrosion of rebars. In this research, COMSOL Multiphysics was employed to model how various crack widths impact the degradation of reinforced concrete. Findings indicated that the deterioration process of reinforced concrete could be better simulated by COMSOL Multiphysics, and the degradation of reinforced concrete was significantly influenced by crack widths. As crack widths widen, chloride diffusion accelerated, leading to a rise in the corrosion current density of rebars. Notably, a crack width increased from 0.1 mm to 0.4 mm resulted in a 227.53% higher corrosion current density in rebars after 150 days of exposure. This study provides a theoretical basis for predicting reinforced concrete deterioration in the future.


Introduction
Reinforced concrete structures, favoured in civil engineering for their material superiority [1][2], suffer from a critical vulnerability to corrosion and degradation [3], especially under harsh environmental conditions like dry-wet cycles.This susceptibility significantly compromises their durability [4], drawing widespread concern in the field.
The primary factor contributing to the durability decline of concrete in harsh environments is chloride-induced corrosion of rebars [5].High chloride levels migrate to the rebar surface through concrete's internal pores, which will cause corrosion damage to the rebars by pitting and cratering in a highly alkaline environment [6].This corrosion generates expansive by-products that weaken the bond between reinforcement and concrete, thereby compromising the structure's durability [7,8].Hence, researching the internal chloride migration and electrochemical corrosion of rebars in reinforced concrete is crucial.
The traditional method of measuring the chloride diffusion and the rebar's corrosion has a long test period, which is troublesome to operate and requires multiple groups of parallel tests.In recent years, the evolution of computer technology has fostered the widespread adoption of numerical simulation in researching chloride diffusion and the corrosion patterns of rebars [9].Compared with the traditional test method, numerical simulation has many advantages, such as fast, convenient, accurate calculation results, etc [10].However, few studies on numerically simulating chloride diffusion and rebar corrosion in concrete exposed to chloride attacks and dry-wet cycles are limited.Consequently, modelling these processes in such environments through numerical simulation is very crucial.
In this study, COMSOL Multiphysics was used to simulate the effect of different crack widths on the chloride concentration distribution and the corrosion current density of rebars, and further explore the diffusion of chloride and the corrosion behavior of rebars under chloride attack and dry-wet cycles.The strength grade of concrete specimens as C30 was designed, and 6% NaCl solution for the erosion solution was set.Drying-wetting cycle times were exposure for 30 d and 150 d, respectively.
In this study, COMSOL Multiphysics facilitated the simulation of how crack widths impact chloride concentration and rebar corrosion current density, delving into chloride diffusion and rebar corrosion under chloride attack and dry-wet cycles.The strength grade of concrete specimens as C30 was designed, and 6% NaCl solution for the erosion solution was set.Dry-wet cycles were conducted over periods of 30 and 150 days.

Chloride transporting model
The transport mechanism of chloride mainly includes diffusion, convection and physical adsorption, which can be described by Eq. (1).Besides, cracks have an important impact on the chloride diffusion, and their width is critical to the chloride diffusion coefficient.Prior research has determined a correlation between the chloride diffusion coefficient and crack width, as defined in Eq. ( 2) [12].
Where Cc (%) represents the chloride concentration; Dc (m 2 /s) denotes the chloride diffusion coefficient; Ccb (%) refers to the concentration of bound chloride.
Where Dcr (m 2 /s) indicates the chloride diffusion coefficient within cracks; ѡcr (μm) refers to the crack width.

Rebar electrochemical corrosion model 2.2.1 Potential distribution in concrete.
The rebar's corrosion rate in concrete is represented by current density, calculable from the potential distribution (E, V) at the reinforced concrete interface.
Where ρ (Ω•m) signifies concrete's effective resistivity.The Laplace equation can be used to ascertain potential distribution: The macrocell current density (imac) flowing through the interface at the reinforced concrete interface can be expressed by Eq. ( 5) to Eq. ( 6).
Activation zone: The electric flux of the remaining boundary is 0, which can be expressed by Eq. (7).

Corrosion polarization equation.
Chloride attack and dry-wet cycles primarily induce electrochemical corrosion reactions on rebar surfaces, driven by chloride-induced anodic corrosion.This process is governed by the anodic Tafel polarization curve's slope, along with anodic potential and anodic current.The Butler-Volmer equation and polarization theory articulate these reactions at the anode and cathode on rebar surfaces through Eq. ( 8) ~Eq.(9) [13].
Polarization reaction of anode corrosion: Polarization reaction of cathodic corrosion: Where Fe 0 and O 2 0 (A/m 2 ) represent the exchange current density of anode and cathode, respectively; Fe 0 and O 2 0 (V) denote their respective equilibrium potentials; Fe and O 2 refer to the anodic and cathodic Tafel polarization curve slopes; L (A/m 2 ), the limiting current density, is derivable from Eq. (10).

Effect of crack widths on chloride concentration on concrete
Figure 2 shows the three-dimensional concentration distribution of chloride in concrete after 30 d and 150 d under different crack widths.The chloride's distribution would be influenced by crack widths, and it showed a sharp diffusion phenomenon.Smaller crack widths exhibited limited evidence of this sharp diffusion phenomenon, attributed to the fewer available paths for Cl -migration into the concrete [13].Conversely, as crack widths increased, this phenomenon was more obvious due to the expanded diffusion pathways for chloride, expediting its penetration into the concrete.Additionally, as exposure duration extended, the pronounced concentration distribution of chloride at the cracks indicated an accelerated transport of chloride over time.The combined effects of chloride attack and dry-wet cycles amplified the impact of crack width on chloride transport.This occurred because the widening cracks and extended exposure period provided more channels for chloride diffusion into the concrete, ultimately hastening the process.

Effect of Cracks widths on corrosion current density on rebar
Figure 3 illustrates the variation in corrosion current density of rebars within concrete as influenced by differing crack widths.This trend revealed a gradual escalation in corrosion current density correlating with increased crack width.Specifically, when comparing a crack width of 0.4 mm to one of 0.1 mm, there was a notable 227.53% rise in corrosion current density in rebars after 150 days of exposure.The primary reason for this phenomenon was the enhanced permeability of Cl -into the concrete's interior with wider cracks, resulting in higher corrosion current density.Furthermore, as the exposure duration extends, there's a noticeable increment in the rebars' corrosion current density.This increase was attributed to the synergistic impact of chloride penetration and the alternating dry-wet cycles, which collectively exacerbate the corrosion damage in rebars.

Results
In this study, COMSOL Multiphysics was utilized to simulate the impact of varying crack widths on the degradation process of reinforced concrete.The main conclusions were abstained as follows: (1) COMSOL Multiphysics effectively replicated the degradation process in reinforced concrete, and the results obtained by COMSOL Multiphysics were similar to those obtained by many researchers in experimental studies.

Where O 2
denotes the valency of cathodic reaction; F (C/mol) represents the faraday constant; O 2 (m 2 /s) refers to the diffusion coefficient of O2 oxygen; O 2 b (mol/m 3 ) is the concentration of O2 on the surface of concrete; L (m) indicates the distance between the concrete and rebar surfaces.Rebar electrochemical corrosion encompasses two mechanisms: microcell and macrocell corrosion.The total corrosion current density (itotal) within the rebar's activation zone combines macrocell current density (imac) in this zone and macrocell current density (imic) in the passivation zone, as calculated by Eq. (11)~Eq.(14).

2. 2 . 3
Model foundation.COMSOL Multiphysics was used to simulate the concentration of chloride in concrete, and the corrosion current and local corrosion current density of rebars under different crack widths.Initially, the two modules within COMSOL Multiphysics: the secondary current distribution module and the dilute matter transfer in porous media module for research.Secondly, assuming that concrete is homogeneous, a three-dimensional model with a basic plane size of 100 mm × 100 mm × 100 mm is established, and the model is meshed by free tetrahedral mesh, as shown in figure1.

Figure 3 .
Figure 3. Corrosion current density of rebars after different days.

Figure 4
displays the local corrosion current density of rebars within concrete subjected to various crack widths over different exposure periods.It could be seen that the local corrosion current density of rebars reached a peak at the crack.The peak value was increasing with the increase of crack widths.Besides, this local corrosion current density showed an upward trend with prolonged exposure.For crack widths of 0.1 mm, 0.2 mm, 0.3 mm and 0.4 mm, the peak local corrosion current densities of rebars after 150 d were 328.85%, 376.89%, 390.95% and 420.94% higher than that after 30 d, respectively.The localized corrosion area gradually expanded over the duration of exposure.