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Next, longitudinal cracks appear from the reinforcement to the concrete surface, which may finally lead to a general failure of the RC structure. Therefore, by taking into account the above assertions, studies on RC structures incorporating concrete cover cracking in chloride-laden environments are endowed with great engineering significance and such efforts are capable of prolonging the service life and reducing the maintenance cost of RC structures [5].

As a fundamental problem with respect to the chloride-induced corrosion of reinforcing bars, concrete cover cracking time is regarded as one of the most essential characteristics in evaluating the service life of RC structures. Specifically, the threshold expansion pressure and corrosion rate of concrete reaches the maximum at the moment of cover cracking, which has been widely studied by researchers [1,6,7,8,9]. In the past thirty years, much effort focused on experimental studies has been directed towards the exploration of cover cracking time induced by the corrosion of reinforcing bars [10,11,12,13,14], along with a variety of corrosion models that were developed and implemented to predict concrete cover cracking time [15,16,17,18,19]. Precisely, among the above models, thick or thin-walled cylinder and elasticity theory are usually utilized for modeling the concrete with embedded steel bars and evaluating the cover cracking time, respectively [6,20,21]. It is worth noting that in those approaches, concrete is succinctly assumed to be homogeneous and isotropic without initial defects. In fact, concrete is a heterogeneous material, which mainly consists of cement, aggregates and chemical admixture. It inevitably contains assorted initial defects such as random cracks. Noticing this fact, Zhang et al. [22] developed a dynamic model based on the fracture mechanics approach by taking into account the initial defects. This model is capable of predicting the initiation time of initial defects, cover cracking time, threshold expansion pressure, and critical corrosion rate of reinforcing bar, which provided a more reasonable prediction associated with the serviceability of RC structures. However, in this model, two essential factors were ignored: the shape of the initial defects and the corrosion current density with time. These two factors are verified to play significant roles on the stresses induced by corrosion products residing in concrete, and subsequently affect the cover cracking time [23,24,25,26,27,28,29]. Therefore, in order to obtain a certain model with high accuracy with regard to the cover cracking problem, the coupling effects of the shape of initial defects and modified corrosion current density should be taken into account. Furthermore, although the aforementioned models [15,16,17,18,19,30,31] are capable of addressing and modelling several problems with regard to concrete cover cracking time, it is worth stressing that the approaches adopted were generally developed within a deterministic framework and in deficiency of delineating the significant uncertainties during the deterioration process of RC structures, which means the applicability is inevitably limited. Hence, it is appropriate to adopt a probabilistic approach for the characterization of the cracking process and predicting the relevant remaining service life of RC structures.

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]:

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]:

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]:

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/2c 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.

Generally, when concrete cover begins to crack, corrosion-induced product penetration will occur in both the porous zone and cracks. Hence, the ΔVrust, which denotes the total corrosion volume per meter on the longitudinal direction, consists of four parts:

As shown in Figure 9g,h, the probability curve of the elastic modulus Ec and cracking fracture toughness KIc is similar to each other, and the results of the effects of two factors on the cover cracking probability are negligible.

It can be seen from Figure 10 that the effects of COV for different factors on the failure probability of cover cracking shows a similar trend, and the failure probability of cover cracking also increases with the growth of time.

Specifically, it is concluded that the COV of cove depth and chloride content are the factors that affect cracking probability the most (Figure 10a,c). Moreover, the w/c ratio, relative humidity and cracking fracture toughness also have a great influence on the cracking probability (Figure 10d,f,h). In Figure 10a, an increase in the coefficient of variation (C) from 0.1 to 0.6 has no effect on cover cracking time. In Figure 10c, it shows that the cover cracking time, which emerges at a 0.2-year increase, increases as Cl changes from 0.1 to 0.7. In Figure 10d,f,h, an increase in the coefficient of variation of d, w/c ratio, KIc from 0.1 to 0.6, 0.05 to 0.3, 0.1 to 0.4 can lead to a increment in cover cracking time between 1.8 and 1.9 years, respectively.

As shown in Figure 10b,e,g, the probability curve of the COV for the diameter of the steel bar d, along with temperature T and the elastic modulus Ec on the cover cracking probability are similar to each other, and the results show that the influence of these three factors on the cover cracking probability is negligible. 2b1af7f3a8