In this section, the corrosion-induced cracking model is developed. In detail, this model is mainly involved in four aspects, including the determination of threshold pressure, the corrosion rate, weight loss and corrosion current density for steel bars in concrete. The development of the corrosion-induced cracking model is described as follows:
2.1.1. The Threshold Pressure of the Initial Crack and Cover Crack
At the interface between the reinforced bar and concrete, the emanation of the initial defect is mainly attributed to the concrete settlement [
22]. In this work, for these initial defects, a semi-elliptical crack existed in a thick-walled cylinder is adopted for modeling such an initial defect [
32,
33]. Additionally, in order to ensure the accuracy of relative analysis, the effects caused by these initial defects on the growth process of the corrosion-induced crack are also considered. In detail,
Figure 1 shows an idealized section of a concrete cylinder with an initial defect which is assumed to be a three-dimensional semi-ellipsoid and can be simplified for plane problems. Specifically, in
Figure 1,
c denotes the half length of the semi-elliptical crack;
a denotes the radial depth of the initial defect through the concrete cover;
R denotes the inner steel radius;
C denotes the concrete cover depth; the angle
β is utilized to describe the position around the semi-elliptical crack varies between 0 ≤
β ≤ π, and
A and
B denote the deepest points of the crack in vertical and horizontal directions, respectively.
The theory of fracture mechanics is first introduced succinctly here in order to investigate the corrosion-induced cracking process in RC structures incorporating the effects of initial defects. In order to develop the model for predicting the time of corrosion-induced cracking on concrete cover, a model proposed by Liu and Weyers [
6] is utilized, and relative parameters such as stress intensity factors are obtained from the literature [
32,
33]:
For the deepest of B:
where
p denotes the uniform internal pressure on the inner surface, and
FA and
FB are the boundary correction factor of the initial defect which can be expressed as follows:
where
The parameters
MiA were obtained by
The geometry correction factors,
Y0 and
Y1 are expressed as
where
Ai and
Bi are gained by
Q is the elliptical crack shape factor:
where the value of the stress intensity factor reaches the fracture toughness, the crack starts to propagate. Double
K fracture criterion for mode I crack of concrete can be represented as follows [
39,
40]:
where
KI denotes the stress intensity factor appeared in Equation (1);
and
denote the initial fracture and the unstable fracture toughness, respectively (named double-K fracture parameters).
Generally, the double-K fracture parameters are utilized to analyze concrete problems involved with the crack initiation and growth [
41], and a variety of studies associated to experimental observations and analytical methods attempting to determine these parameters can be traced [
41,
42,
43]. However, due to absence of the consideration on the coupling of size and boundary effects, utilization of the double-K fracture parameters is limited in the corrosion-induced cracking process. Therefore, Zhang et al. [
22] adopted a correction method by treating the reinforced concrete thick wall cylinder as a three-point bending notched beam. The relationship between the modified stress intensity factor and the experimental data is expressed as [
39]:
where
h,
V, and
KIc are the height, volume and fracture toughness of a three-point bending beam, respectively;
is the Weibull modulus related to the variation of experimental results, which can be obtained by
Based on the theory of fracture mechanics, the stress intensity of the initial defect will increase with the growth of the rust expansion force generated by the expansion of steel corrosion products. When the increase in the stress intensity factor is equal to the initial fracture toughness of the concrete material, the location of the initial defects first started to crack, namely the first phase of the concrete crack, crack length increases as the corrosive force continues to increase, eventually forming a well versed in the crack of the concrete surface; this phase is the concrete cover cracking criterion completely. For the semi-elliptical flaw, the equilibrium
a/2
c always increases to a limiting value of 0.36 [
44]. This results from a variation in the stress intensity factor
KI along the surface of the ellipse. When
β = 90°,
KI is maximized, but is smallest when
β = 0°. Hence, the crack will grow fastest when
β = 90°.
It can be known from the above discussion, when calculating the crack propagation of concrete initial defects, two critical points should be considered, viz. the initial defects cracking and the complete cracking of concrete cover depth. It is assumed that the crack stress intensity factors of these two critical points are equal to the double K fracture parameters, so, we can obtain the initial threshold expansion pressure pini and the threshold cracking pressure pun at the initial defect points A and B by substituting Equation (1) and Equation (2). Since the crack development of the initial defect at point B is along the reinforcement bar, which has little effect on the crack expansion of the initial defect at point A, considering the crack state of the initial defect at point A as the criterion for judging the crack penetration of the concrete cover depth.
The initial threshold pressure
pini is as follows:
Hence, the expression for the threshold cracking pressure
is given as:
where
* denotes the length of fracture zone and can be obtained by following equation [
19]:
where
is the tensile strength of concrete.
2.1.2. The Corrosion Rate of Steel Bar
For an initial unrestrained RC specimen with the bottom clear cover
C and original reinforcing bar
R, the thick-walled concrete cylinder is shown in
Figure 2. The original radius of the steel bar is
R, the radius of steel bar after corrosion is
a1, the radial loss of steel is
R-a1, and the combined radius of un-corroded steel plus free-expansive corrosion products is
a2. Based on the amount of theoretical analysis and experimental study, it is verified that there are a lot of porous zones around the interface between the reinforcement bar and concrete, which affects the cracking time of concrete cover depth. For the sake of simplicity, the porous zone is assumed to be uniform and its thickness is indicated by
d0, assumed to be 12.5 mm as adopted by Liu and Weyers [
6], Bhargava et al. [
45] and Kotes [
46]. The corrosion products must first fill this porous zone before their volume expansion starts to create uniform radial inner pressure
p around the surrounding concrete, due to which the concrete gets an internal radial displacement
σcon, therefore:
where
uc denotes the Poisson’s ratio;
Eef denotes the effective modulus of elasticity for concrete cover, which can be obtained as:
where
Ec denotes elastic modulus;
j denotes creep coefficient of the concrete cover. The value for
j as per the reference is 2.0 [
6].
Based on the geometric condition shown in
Figure 2, the corrosion-induced loss of volume per meter on the longitudinal direction of steel can be written as:
Therefore, the corrosion rate denoted as
ρ can be written as follows:
Additionally, the volume of the porous zone per meter on the longitudinal direction can be expressed as:
In addition, the volume of concrete per meter on the longitudinal direction caused by the radial displacement
σcon can be determined as:
For the sake of simplicity, the volume of crack per meter on the longitudinal direction after concrete cover cracking can be estimated by [
47,
48]:
where
n is the volume ratio between the corrosion products and the basic steel. The expression is given as follows:
Generally, when concrete cover begins to crack, corrosion-induced product penetration will occur in both the porous zone and cracks. Hence, the Δ
Vrust, which denotes the total corrosion volume per meter on the longitudinal direction, consists of four parts:
Combining Equations (19) to (25) obtains a relationship as follows:
Solving Equation (21), the corrosion rate
ρ can be obtained:
Hence, one can see that there is a strong correlation between the corrosion rate ρ and the shape of the initial defect.
2.1.4. Corrosion Current Density of Steel Bars in Concrete
During the cracking process for concrete cover, the corrosion current density of the reinforcing bar is regarded as one of the determinant factors on relative behaviors. In past decades, a number of researchers have conducted several in-depth researches on this topic. In this work, the corrosion current density utilized is determined by Lu et al. [
29], and the expression is given as:
where
icorr denotes corrosion current density (μA/cm
2);
Ct denotes concrete chloride content on the surface of reinforcement (kg/m
3);
T is the temperature (K);
RH is relative humidity;
t is corrosion duration (years);
ρ0 is concrete resistivity (kohm.cm).
The above formula is mainly used for steel corrosion in the natural corrosion environment, while the corrosion current density in the steel corrosion accelerated by electrification is artificially set and a known parameter.
Past studies [
3] have shown that concrete cover cracking time is closely related to cover depth, the diameter of steel reinforcement, the strength of concrete, concrete water–cement (
w/c) ratio and other factors. Concrete
w/c ratio has a great influence on concrete strength; the larger the concrete
w/c ratio is, the lower the concrete strength is. In this paper, the concrete
w/c ratio is selected as one of the main factors affecting cover cracking. Similarly, cover depth also has an important effect on cover cracking time; the thicker the cover, the longer it takes for the crack to reach the concrete surface, and the longer it takes for the corrosion products to fill the pores in the concrete [
17]. Based on the Vu and Stewart model [
27], the formula of impact factor is established, considering the influence of the
w/c ratio and cover depth on cover cracking time and can be expressed as:
Substituting Equation (32) into (31), the following relationship is obtained:
where
k is the adjustment coefficient.
By conducting regression analysis on the experimental results proposed by Liu and Weyers [
26], a modified formula of corrosion current density can be obtained:
2.1.5. Cracking Time Model
In the past, cover cracking time was mainly studied in the natural corrosion environment and the electrified acceleration environment; in the natural corrosion environment, the corrosion current density will change with time, and in the electrified acceleration environment, the corrosion current density is a constant value, so, this paper needs to consider cover cracking time in these environments.
For reinforced concrete structures in naturally corrosive environments, combining Equations (29) to (30) and Equation (34), the following relationship is obtained:
By integrating Equation (35), the formula of concrete cover cracking time is obtained:
where
H* can be expressed as:
For reinforced concrete structures in naturally corrosive environments, based on the Wang [
34] model, the cover cracking time can be expressed as follows:
where
a3, b1, c1 is the combination coefficient, which can be obtained by Equation (27).