Advances in Materials Science and Engineering

Volume 2015 (2015), Article ID 510845, 10 pages

http://dx.doi.org/10.1155/2015/510845

## On the Durability of Sealable Circular Concrete Structures under Chloride Environment

^{1}Jiangsu Key Laboratory of Engineering Mechanics, Department of Engineering Mechanics, Southeast University, Nanjing, Jiangsu 210096, China^{2}Department of Mechanical Engineering, University of Houston, Houston, TX 77204, USA

Received 3 March 2015; Accepted 26 May 2015

Academic Editor: Ana S. Guimarães

Copyright © 2015 Changwen Mi et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Reinforcement corrosion resulting from chloride attack is one major mechanism that compromises concrete durability. Numerical methods were commonly used for tackling Fick’s diffusion equations. In this paper, we developed a Crank-Nicolson based finite difference scheme suitable for circular concrete structures. Both a time-dependent surface chloride model and diffusivity were considered. The impact of an ideal sealer on chloride redistribution was further investigated. Results suggested that the chloride threshold depth in a concrete structure is greatly affected by the radius of curvature, environment severity, and diffusivity. For sealable concrete structures, both the sealer application timing and location are of great importance.

#### 1. Introduction

The reinforcement steel in concrete structures is protected from adverse environments by a thin layer of passive oxide film. Exposure to aggressive species, for example, chloride anions, may result in the breakdown of the naturally formed oxide [1, 2]. The volume expansion due to corrosion products causes an excessive stress state. In severe cases, cracking and even spalling of the concrete cover occur. The durability of a concrete structure is thus compromised due to the possible strength degradation of rusted reinforcements [3].

A threshold level of chlorides (CTL) is one of the necessary conditions for reinforcement corrosion [4]. The specific values defined for the CTL, however, vary from one another in the literature. For example, a threshold value of 0.594 kg/m^{3} was used by Kassir and Ghosn [5] and Phurkhao and Kassir [6]. Moriwake [7] defined a larger value of 2.0 kg/m^{3} for crack initiation due to steel corrosion. Song et al. [3] used both values as well as an intermediate CTL of 1.2 kg/m^{3} for predicting the service life of repaired concrete structures. As Alonso et al. [8] summarized, the CTL of a concrete structure is affected by many factors such as the oxide condition of reinforcements, concrete quality, and environmental severity.

Chloride ingress in concretes is primarily associated with two mechanisms: diffusion [9] and hydraulic solution diffusivity [10]. Concentration gradient and capillary suction would be the driving force of these two mechanisms in absence of pressure in unsaturated concrete. Under pressure phenomena would be even more complex. For each mechanism, a certain extent of chloride binding may occur, depending on the microstructural porosity and the moisture state of a concrete structure [11, 12]. Nonetheless, quantitative models are still being developed [13, 14] and the detailed mechanism of chloride binding on the durability of concrete structures remains unclarified [4].

Without considering the effect of chloride binding, chloride diffusion in concrete is relatively a well-understood mechanism, benefitted from Fick’s fundamental diffusion laws [15]. A large amount of literature studies was devoted to this line of research. Analytical solutions have been developed for simple geometric domains, boundary conditions, initial chloride distribution, and diffusivity [5, 6, 15]. They are often used as benchmark examples for calibrating the reliability of more complicated solutions.

A large portion of the relevant literature studies is concerned with one-dimensional diffusion problems [3, 14, 16, 17]. More recently, considerable efforts have also been made to numerically resolve two-dimensional problems of different domains. The numerical methods that were employed include finite difference method [18], finite element method [13, 19, 20], finite point (meshless) method [21], cellular automata [22], and boundary element method [23]. Analytical solutions were also being under development for cases with an apparent diffusion constant [24, 25].

Less effort has been directed towards the study of circular columns, even though they form a very common type of structural elements in civil infrastructures, as opposed to rectangular and elliptic ones. The chloride diffusion inside a two-dimensional circular domain remains elusive. This is the goal of the present study.

When compared to analytical solutions, one distinct advantage of a numerical method is its capability to accommodate complex diffusivity model, surface chloride model, and initial chloride distribution. Experimental data from field studies has revealed that both the diffusivity [26] and surface chloride concentration [27] vary with concrete age. In the present work, we borrowed the time-dependent diffusivity of Song et al. [3], which was slightly adapted from Bentz and Thomas [28].

For surface chloride evolution, the situation is more complicated. Environmental factors that affect the evolutionary history of the surface chloride include geographic location of a concrete structure, accessibility to rinsing precipitations, relative location inside an infrastructure, season, and wetting and drying cycles. A few regression models, including exponential [5], ramp-type [6], logarithmic [3, 29], and power functions [30], have been proposed in the literature by fitting to experimental measurements in the least squares fashion. While each of these surface chloride models can be incorporated into the present study, we specifically focused on the ramp-type [6] and log-type [31]. They represent a deicing salt induced chloride and marine environment, respectively.

Among all available solution strategies, the finite difference method is the simplest one in terms of the easiness in mathematical implementation [3, 23, 32]. In this work, we developed in detail a Crank-Nicolson based finite difference scheme to numerically solve the axially symmetric diffusion equation defined in a circular domain. The CTL value of 1.2 kg/m^{3} [3] was used. In addition to the chloride diffusion in an original unpolluted concrete, numerical experiments were also performed for the chloride redistribution following a sealer application. The sealer was applied at either the outer surface or at a certain depth.

Simulation results indicated that the chloride distribution strongly depends on concrete size, surface chloride evolution, diffusivity, initial chloride distribution, and sealer application timing and location. The most effective means for slowing down chloride diffusion is to elevate concrete quality. To prevent chlorides from further penetrating, a sealer application is the best option. In this case, the sealer should be applied early and as close to the outermost reinforcements as possible.

The remainder of this paper is structured as follows. In Section 2 the numerical scheme that we proposed for the analysis of chloride diffusion in circular concrete structures is presented in detail. In Section 3 a variety of numerical experiments detailing the chloride diffusion in both unsealed and sealed concretes are reported and discussed and finally, in Section 4, conclusions are provided and future works described.

#### 2. Method of Solution

##### 2.1. Crank-Nicolson Scheme for Chloride Diffusion in a Circular Domain

According to Fick’s first law [15], the concentration gradient of chlorides serves as the driving force to their spatial diffusion in concrete structures:where is the diffusion flux of chloride anions, the chloride concentration, and the diffusivity. Generally, both **J** and are functions of space and time whereas is typically treated as a time-dependent variable only.

In the absence of chloride binding the conservation of mass inside a differential volume dictates that the rate of change of chlorides must be balanced by the divergence of the diffusion flux. The resultant partial differential equation is referred to as Fick’s second law of diffusion:

In this work, we focused on circular and homogeneous concretes. From the practical point of view, it seems reasonable to assume that the chloride diffusion process is also symmetric about the centroidal axis. Under these assumptions, (2) can be rewritten as

Unlike the one-dimensional diffusion problem, the analytical solution to (3) is unavailable in the literature. Numerical alternatives are employed instead. For the reasons stated in Section 1, we decided to employ the finite difference method and aimed to develop a numerical model on the basis of Crank-Nicolson scheme [32, 33].

We divided the radius of a circular domain into segments, each with the same length. As a result, there are a total number of () spatial nodes. The central idea of the Crank-Nicolson scheme is to consider the diffusion equation (3) at an arbitrary nonboundary node and halfway between two adjacent temporal steps:where the subscripts () and () denote the spatial and temporal indices, respectively. They are related to the real coordinates through and , where and are the spatial and temporal step increments. The temporal derivative is approximated by the centered difference scheme:The first- and second-order spatial derivatives in (4) are approximated by temporally averaging a one-sided and centered difference scheme over two contiguous temporal steps, respectively,

Substituting (5) and (6) back into (4) and solving for the concentrations at the th temporal step, we havewhere , , and are functions of the diffusivity (), the Courant number (), and the spatial node index (). For brevity, they are tabulated in Appendix A.

This numerical scheme is implicit since at an arbitrary temporal step the concentrations at three neighboring nodes are always coupled. Thus, care should be practiced for the boundary nodes. The chloride concentration at the outer boundary is governed by the temporal evolution of surface chlorides:where represents a time-dependent surface concentration model. As a result, the equation of (7) must be reformulated to reflect this condition:

Given the symmetry property of the present problem, the concentration gradient vanishes at . In mathematical context, this represents a Neumann boundary condition [32]:

At (), we may start from a centered approximation for both spatial derivativesThe equality to zero in the first-order derivative is obviously a result of the Neumann boundary condition (10). To proceed, those concentrations that correspond to the undefined node () need to be eliminated. Multiplying the first equation by and adding to the second one, we managed to derive a new finite difference approximation for the second-order spatial derivative at :

Now consider the original diffusion equation (3) for . The term involving the first-order spatial derivative is addressed by limit analysis. Upon application of L’Hôpital’s rule for , this term’s limit turns out to be another second-order spatial derivative. Replacing in (3) the temporal derivative with (5) and the second-order spatial derivatives with (12), an evolutionary scheme for can be developed:where and are functions of the diffusivity and the Courant number. Both are tabulated in Appendix A.

Up to the present, we constructed linear equations regarding the chloride concentration at an arbitrary temporal step, that is, () equations from (7), equations (9) and (13). The total number of unknowns is the same; that is, . Recall that the chloride concentration at the largest node number () is dictated by the surface chloride evolution. To facilitate the solution procedure, this system of linear equations was reformulated into conventional matrix form. The reader is kindly invited to refer to Appendix A for further details.

##### 2.2. Sealer Application

In engineering practice, sealers, coating, and membranes are often employed as means of maintenance. The idea is to prevent or at least to slow down chloride ingress toward reinforcement steels inside a concrete structure. In physical context, the prevention of chloride transport denotes that the diffusion flux is zero across a sealing interface. In view of (1), a sealer application annihilates the concentration gradient of chloride ions in position. Such a condition is mathematically identical to Neumann boundary condition that was applied at the center of a circular concrete.

Therefore, the approximation schemes used for the concrete center , that is, (11) through (13), can be adapted for a sealing interface. If a sealer is applied at the outer surface of the concrete, only a single spatial interval needs to be considered. Otherwise, two separate regions must be taken into account. As a result, the numerical scheme designed for an original concrete structure must be revised to reflect Neumann boundary condition associated with the sealing interface. For conciseness, the revised scheme is outlined in Appendix B.

##### 2.3. Time-Dependent Surface Chloride Evolution

To implement the numerical scheme developed in the previous section, we first need to quantify an evolution model for the surface chloride concentration, an initial chloride distribution, and a time-dependent diffusivity model. As briefly discussed in Section 1, several regression models have been proposed in the literature to simulate the temporal evolution of surface chloride ions [3, 5, 6, 29, 30]. Although any of them can be implemented, we specifically focused on two surface chloride models. The first is a ramp-type surface concentration model [6]:This model predicts that surface concentration increases linearly as exposure time for the first () years of a fresh concrete structure, beyond which the surface concentration remains saturated ( kg/m^{3}). The two parameters were extracted by fitting the proposed function to the measured surface chloride data from 15 bridge decks in the snow belt region of the United States [27]. The source of chloride ions is due primarily to deicing salts applied on the bridges during snow seasons. Given the initially increasing and subsequently fluctuating nature of experimental surface chloride measurements [27], (14) seems to represent an improved regression over its exponential counterpart [5].

For comparison purpose, the logarithmic model designed for concrete structures under marine environment was also employed [3, 31]:where and are fitting parameters that are functions of the distance from seawater. For the case of zero distance, kg/m^{3} and year^{−1}. The surface concentration predicted by (15) monotonically increases as exposure time but at a decreasing rate, for example, . Such a model is appropriate for concrete structures subjected to wetting and drying cycles.

##### 2.4. Initial Chloride Distribution and Time-Dependent Diffusivity

For concretes free of sealer applications, an initially chloride-free condition was always assumed. The chloride distribution immediately prior to a sealer application action was undoubtedly treated as the initial condition of the subsequent chloride evolution.

As evidenced by Fick’s laws (1) and (2), the evolution of chloride concentration interior of a concrete structure is governed by the time-dependent diffusivity. Mathematically, diffusivity is the linear proportionality between diffusion flux and concentration gradient. In physical terms, it means how difficult or how easy chlorides can spatially redistribute. It is proposed by Bentz and Thomas [28] that chloride diffusivity decays as a power function for the first 25 years and stays constant afterwards. Such a dependent pattern is believed to be a result of the void growth and coalescence in a concrete structure due to cement hydration [31].

In this work, a slightly modified version [3] of the time-dependent diffusivity [28] is used:where is the reference diffusivity when years and a function of the water to cementitious material ratio [3]:Typical range of is between and . In the absence of fly ash and slag, the decaying rate of diffusivity was assumed as 0.2 [3].

#### 3. Results and Discussion

Based on the finite difference scheme developed in the previous section, we performed a variety of numerical experiments for investigating the influential factors on chloride diffusion in circular concrete structures. These factors include curvature of radius, water to cement ratio, surface concentration model, and sealer application timing and location.

To proceed, numerical examples were first implemented to examine the reliability of the proposed numerical scheme with respect to both spatial and temporal step sizes. Figure 1 reports chloride concentration as a function of spatial step for two distinct temporal steps. Concentrations were all sampled at the perimeter 0.1 m distant from the outer surface of a column of radius 0.4 m, after 20 years of exposure to chloride environment. The ramp-type surface concentration model (14) was employed. Spatial steps considered range from 10 *μ*m to 0.1 m. For both temporal steps, that is, day and 10 days, the chloride concentration remains convergent for mm. For larger spatial steps, the chloride concentration diverges and eventually becomes unbounded.