Modelling of Corrosion-Induced Concrete Cover Cracking Due to Chloride Attacking

materials

Article Modelling of Corrosion-Induced Concrete Cover Cracking Due to Chloride Attacking

Pei-Yuan Lun 1,2, Xiao-Gang Zhang 1,*, Ce Jiang 1, Yi-Fei Ma 3 and Lei Fu 4

1 Guangdong Provincial Key Laboratory of Durability for Marine Civil Engineering, College of Civil and Transportation Engineering, Shenzhen University, Shenzhen 518060, China; [email protected] (P.-Y.L.); [email protected] (C.J.) 2 School of Civil Engineering, Central South University, Changsha 410000, China 3 The Construction Quality and Safety Supervision Center of Ningxia Hui Autonomous Region, Yinchuan 750001, China; [email protected] 4 Kaifeng Credit Information Center, Kaifeng 475000, China; [email protected] * Correspondence: [email protected]

Abstract: The premature failure of reinforced concrete (RC) structures is significantly affected by chloride-induced corrosion of reinforcing steel. Although researchers have achieved many outstand- ing results in the structural capacity of RC structures in the past few decades, the topic of service life has gradually attracted researchers’ attention. In this work, based on the stress intensity, two models are developed to predict the threshold expansive pressure, corrosion rate and cover cracking time of the corrosion-induced cracking process for RC structures. Specifically, in the proposed models, both the influence of initial defects and modified corrosion current density are taken into account. The results given by these models are in a good agreement with practical experience and laboratory

studies, and the inﬂuence of each parameter on cover cracking is analyzed. In addition, considering the uncertainty existing in the deterioration process of RC structures, a methodology based on the

Citation: Lun, P.-Y.; Zhang, X.-G.; third-moment method in regard to the stochastic process is proposed, which is able to evaluate the Jiang, C.; Ma, Y.-F.; Fu, L. Modelling cracking risk of RC structures quantitatively and predict their service life. This method provides a of Corrosion-Induced Concrete Cover good means to solve relevant problems and can prolong the service life of concrete infrastructures Cracking Due to Chloride Attacking. subjected to corrosion by applying timely inspection and repairs. Materials 2021, 14, 1440. https:// doi.org/10.3390/ma14061440 Keywords: concrete cover cracking; modiﬁed corrosion current density; semi-elliptical defect; stress intensity factor; probabilistic analysis Academic Editor: Dolores Eliche Quesada

Received: 10 January 2021 1. Introduction Accepted: 7 March 2021 Published: 16 March 2021 The chloride-induced corrosion of reinforcement bars is considered as one of the dominant causes of the deterioration of reinforced concrete (RC) structures [1–3]. In view of

Publisher’s Note: MDPI stays neutral its close relation to the service performance and residual life of in-service RC structure [4], with regard to jurisdictional claims in it has drawn intensified study from a number of researchers. The corrosion of reinforcing published maps and institutional affil- bars caused by chloride is mainly the presence of free chloride ions. iations. In detail, chloride ions are normally widespread in structures undergoing de-icing programs such as infrastructure buildings, cross-sea bridges and industrial structures, and all structures undergoing de-icing programs, which firstly results in the corrosion of reinforcement and further leads to the appearance and accumulation of corrosion products between steel and concrete. Meanwhile, the rusty strains between steel and concrete Copyright: © 2021 by the authors. Licensee MDPI, Basel, Switzerland. are produced. This article is an open access article Next, longitudinal cracks appear from the reinforcement to the concrete surface, which distributed under the terms and may finally lead to a general failure of the RC structure. Therefore, by taking into account conditions of the Creative Commons the above assertions, studies on RC structures incorporating concrete cover cracking in Attribution (CC BY) license (https:// chloride-laden environments are endowed with great engineering significance and such creativecommons.org/licenses/by/ efforts are capable of prolonging the service life and reducing the maintenance cost of RC 4.0/). structures [5].

Materials 2021, 14, 1440. https://doi.org/10.3390/ma14061440 https://www.mdpi.com/journal/materials Materials 2021, 14, 1440 2 of 22

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 [1,6–9]. In the past thirty years, much ef- fort focused on experimental studies has been directed towards the exploration of cover cracking time induced by the corrosion of reinforcing bars [10–14], along with a variety of corrosion models that were developed and implemented to predict concrete cover cracking time [15–19]. 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 [23–29]. 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 [15–19,30,31] 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. In the current paper, the main objective is to propose a corrosion-induced cracking model considering both the natural environment corrosion and acceleration to power corrosion. In the proposed model, the initial defects in three-dimensional and the modified corrosion current density formula of reinforcement bars are taken into account [32,33]. Formulas for predicting the cover cracking time, threshold expansion pressure and corrosion rate are also developed, respectively. In order to validate its accuracy and validity, comparisons of the predictions are conducted with experimental results obtained in the relevant literature [6,34,35]. In addition, the influences of relevant parameters on cover cracking time, the threshold expansive pressure and corrosion rate are also investigated. Lastly, by adopting the third-moment method, a cover cracking time probabilistic analysis is conducted.

2. Corrosion-Induced Cracking Model and Its Veriﬁcation Concrete is a heterogeneous material that inevitably contains randomly distributed initial defects with assorted mechanical behaviors. Those defects are attributable to the coactions of a number of factors such as overloading, shrinkage caused by restraint, weather (extremely dry, cold, or hot), poor workmanship, and settlement of concrete [36]. Precisely, the randomness associated to the size and the location of the initial defects will cause the emergence of variability for relative structures’ behaviors during service life, which are directly related to the safety of structures [20,37]. Therefore, in order to characterize the random responses of the relevant structure, several approaches were developed in the preceding decades, aiming to identify and evaluate the size and location of the initial defects. Materials 2021, 14, x FOR PEER REVIEW 3 of 23

which are directly related to the safety of structures [20,37]. Therefore, in order to characterize the random responses of the relevant structure, several approaches were developed in the preceding decades, aiming to identify and evaluate the size and location of the initial defects. To date, a number of approaches have been successfully developed and implemented based on different methods, including infrared thermography, ground pene- Materials 2021, 14, 1440 trating radar and radiographic methods, etc. [38]. 3 of 22

2.1. Corrosion-Induced Cracking Model In this section, the corrosion-induced cracking model is developed. In detail, this To date, a number of approaches have been successfully developed and implemented based model is mainly involved in four aspects, including the determination of threshold pres- on different methods, including infrared thermography, ground penetrating radar and sure, the corrosion rate, weight loss and corrosion current density for steel bars in con- radiographic methods, etc. [38]. crete. The development of the corrosion-induced cracking model is described as follows: 2.1. Corrosion-Induced Cracking Model 2.1.1. The Threshold Pressure of the Initial Crack and Cover Crack In this section, the corrosion-induced cracking model is developed. In detail, this modelAt is the mainly interface involved between in four the aspects, reinforced including bar and the concrete, determination the emanation of threshold of the pressure, initial thedefect corrosion is mainly rate, attributed weight loss to andthe corrosionconcrete settlement current density [22]. forIn t steelhis work, bars infor concrete. these initial The developmentdefects, a semi of-elliptical the corrosion-induced crack existed in cracking a thick- modelwalled is cylinder described is adopted as follows: for modeling such an initial defect [32,33]. Additionally, in order to ensure the accuracy of relative anal- 2.1.1.ysis, the The effects Threshold caused Pressure by these of the initial Initial defects Crack on and the Cover growth Crack process of the corrosion- inducedAt the crack interface are also between considered. the reinforced In detail, barFigure and 1 concrete, shows an the idealized emanation section of the of initiala con- defectcrete cylinder is mainly with attributed an initial to defect the concrete which is settlement assumed [to22 ].be Ina three this work,-dimensional for these semi initial-el- defects,lipsoid and a semi-elliptical can be simplified crack f existedor plane in problems. a thick-walled Specifically, cylinder in is Figure adopted 1, forc denotes modeling the suchhalf length an initial of the defect semi [32-elliptical,33]. Additionally, crack; a denotes in order the to radial ensure depth the accuracy of the initial of relative defect analysis,through the the concrete effects caused cover; by R denotes these initial the inner defects steel on theradius; growth C denotes process the of concrete the corrosion- cover induceddepth; the crack angle are β is also utilized considered. to describe In detail, the position Figure around1 shows the an semi idealized-elliptical section crack of var- a concreteies between cylinder 0 ≤ β with≤ π, and an initial A and defect B denote which the is deepest assumed points to be of a three-dimensionalthe crack in vertical semi- and ellipsoidhorizontal and directions, can be simpliﬁedrespectively. for plane problems. Speciﬁcally, in Figure1, c denotes the halfThe length theory of of the fracture semi-elliptical mechanics crack; is firsta denotes introduced the s radialuccinctly depth here of in the order initial to defectinves- throughtigate the the corrosion concrete-induced cover; R crackingdenotes process the inner insteel RC structures radius; C denotesincorporating the concrete the effects cover of depth;initial defects. the angle Inβ orderis utilized to develop to describe the model the position for predicting around thethe semi-ellipticaltime of corrosion crack-induced varies betweencracking 0on≤ concreteβ ≤ π, andcover,A anda modelB denote proposed the deepest by Liu and points Weyers of the [6] crack is utilized, in vertical and rela- and horizontaltive parameters directions, such as respectively. stress intensity factors are obtained from the literature [32,33]:

FigureFigure 1.1. Schematic diagramdiagram ofof semi-ellipticsemi-elliptic initialinitial defect:defect: cc—half—half lengthlength ofof thethe semi-ellipticalsemi-elliptical crack; acrack;—radial a— depthradial d ofepth the of initial the initial defect; defect;R—inner R—inner steel steel radius; radius;C—concrete C—concrete cover cover depth; depth;β—position β— angle;positionA—deepest angle; A— pointsdeepest of thepoints crack of inthe vertical crack direction;in verticalB direction;—deepest B points—deepest of the points crack inof horizontalthe crack in horizontal directions. directions.

TheFor the theory deepest of fracture of A: mechanics is ﬁrst introduced succinctly here in order to investi- gate the corrosion-induced cracking process in RC structures incorporating the effects of K pF a/ Q (1) initial defects. In order to develop the modelIA for A predicting the time of corrosion-induced crackingFor onthe concrete deepest cover,of B: 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 A: K pF a/ Q (2) IBp B KIA = pFA πa/Q (1) For the deepest of B: p KIB = pFB πa/Q (2)

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: √ 2Q F = (G M + G M + G M + G ) (3) A π 0 1A 1 2A 2 3A 3 Materials 2021, 14, 1440 4 of 22

√ 2 Q F = (G M + G M + G M + G ) (4) B π 0 1B 4 2B 5 3B 6 where = F + G0 (R/C)u E F ln u F ln(R/C) F 2 G1 = − + + + E 2(a/C)3/2u1/2 2(a/C)3/2u1/2 (a/C)(R/C) 3 F ln u F F ln(R/C) 1 (5) G2 = − 2 + + 2 + E (a/C) (a/C)(R/C) (a/C) 2 Fw F F ln(R/C) G3 = − + − + 2E 2(a/C)1/2u3/2 u(R/C) 2(a/C)1/2u3/2

2 = + (R/C) E 1 2(R/C)+1 2 2 = (R/C) (R/C+1) F 2(R/C)+1 h i (6) w = ln u + (a/C) + 2u1/2(a/C)1/2 u = (R/C) + (a/C)

The parameters MiA were obtained by

M = √2π (−Y + 3Y ) − 24 1A 2Q 0 1 5 M2A = 3 (7) M = √6π (Y − 2Y ) + 8 3A 2Q 0 1 5

The geometry correction factors, Y0 and Y1 are expressed as

a a 2 a 4 Y0 = A0 + A1 C + A2 C + A3 C a a 2 a 4 (8) Y1 = B0 + B1 C + B2 C + B3 C

where Ai and Bi are gained by

[−5.051( a )] A0 = 0.07e c + 1.044 [−3.393( a )] A1 = 0.665e c − 0.433 [−3.386( a )] (9) A2 = 1.16e c + 0.711 [−4.165( a )] A3 = 1.46e c − 0.179

[−0.035( a )] B0 = −2.16e c + 2.825 [−5.574( a )] B1 = 0.265e c − 0.225 [−4.0251( a )] (10) B2 = 0.753e c + 0.307 [0.0791( a )] B3 = −1.284e c + 1.398 Q is the elliptical crack shape factor:

a 1.65 Q = 1 + 1.464 (11) c 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]:

ini KI < KIc no crack propagation; ini un KIc < KI < KIc steady crack propagation; (12) un KI > KIc unsteady crack propagation;

ini un where KI denotes the stress intensity factor appeared in Equation (1); KIc and KIc 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 Materials 2021, 14, 1440 5 of 22

to experimental observations and analytical methods attempting to determine these parameters can be traced [41–43]. 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]:

r 0 C V 1/α K = K (h ≤ 2m) (13) Ics Ic h 2R · C

where h, V, and KIc are the height, volume and fracture toughness of a three-point bending beam, respectively; α0 is the Weibull modulus related to the variation of experimental results, which can be obtained by

0 π α = √ (0.013 ≤ CV ≤ 0.230) (14) 6.5CV 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. The initial threshold pressure pini is as follows:

Kini Pini = √ Ics (15) FA πa/Q Hence, the expression for the threshold cracking pressure Pun is given as:

Kun Pun = Ics (16) p ∗ FA · π(a + ∆a )/Q

where ∆a* denotes the length of fracture zone and can be obtained by following equation [19]: 1 Kun 2 ∆a∗ = Ics (17) 2π ft

where ft is the tensile strength of concrete. Materials 2021, 14, 1440 6 of 22

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 Figure2. 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 Materials 2021, 14, x FOR PEER REVIEWd0, assumed to be 12.5 mm as adopted by Liu and Weyers [6], Bhargava et al. [45]7 and of 23 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: 2 3 p RRCR2 11  3  p R(R+ C) (1 + υc) +ccR (1 − υc)  σ  con= · 2 (18)(18) con E E2RC2 RC+ C2 C ef ef

wherewhereu uc cdenotes denotes thethe Poisson’sPoisson’s ratio;ratio;E Eefef denotesdenotes thethe effectiveeffective modulusmodulus ofof elasticity elasticity for for concreteconcrete cover, cover, which which can can be be obtained obtained as: as: E cEc Eef = (19) Eef 1.0= + ϕ (19) 1.0  where Ec denotes elastic modulus; j denotes creep coefﬁcient of the concrete cover. The 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]. value for j as per the reference is 2.0 [6].

(a) Before corrosion (b) After corrosion

FigureFigure 2. 2.Calculation Calculation modelmodel ofof corrosioncorrosion rate:rate: ((aa)) beforebefore corrosion;corrosion; ((b)) afterafter corrosion; R—original—original radius of steel bar; a1—radius of steel bar after corrosion; a2—combined radius of un-corroded steel radius of steel bar; a1—radius of steel bar after corrosion; a2—combined radius of un-corroded steel plus free-expansive corrosion product; d0—the thickness of porous zone. plus free-expansive corrosion product; d0—the thickness of porous zone.

BasedBased onon the the geometric geometric condition condition shownshown inin Figure Figure2 ,2, the the corrosion-induced corrosion-induced loss loss of of volumevolume per per meter meter on on the the longitudinal longitudinal direction direction of of steel steel can can be be written written as: as: 22 V  ()2 R  2 a (20) ∆Vsteelsteel= π(R − a1 ) 1 (20) Therefore, the corrosion rate denoted as ρ can be written as follows: Therefore, the corrosion rate denoted as ρ can be written as follows:

Vsteel ∆=Vsteel (21) ρ = V (21) Vsteelsteel Additionally, the volume of the porous zone per meter on the longitudinal direction can be expressed as:

V2 Rd (22) p0 In addition, the volume of concrete per meter on the longitudinal direction caused by the radial displacement con can be determined as:

V 2 ( R  d ) (23) con 0 con 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]:

Materials 2021, 14, 1440 7 of 22

Additionally, the volume of the porous zone per meter on the longitudinal direction can be expressed as: ∆Vp = 2πRd0 (22) In addition, the volume of concrete per meter on the longitudinal direction caused by the radial displacement σcon can be determined as:

∆Vcon = 2π(R + d0)σcon (23)

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]:

1 ∆Vcrack = ∑ ω ac 3 i p (24) ∑ ωi = 2πR(n − 1)(1 − 1 − ρ)

where n is the volume ratio between the corrosion products and the basic steel. The expression is given as follows: n = ∆Vrust/∆Vsteel (25) Generally, when concrete cover begins to crack, corrosion-induced product penetration will occur in both the porous zone and cracks. Hence, the ∆Vrust, which denotes the total corrosion volume per meter on the longitudinal direction, consists of four parts:

∆Vrust = ∆Vp + ∆Vcon + ∆Vcrack + ∆Vsteel (26)

Combining Equations (19) to (25) obtains a relationship as follows:

2 p πηρR2 = π(n − 1)acR(1 − 1 − ρ) + 2πRd + 2π(R + d )σ + πρR2 (27) 3 0 0 con Solving Equation (21), the corrosion rate ρ can be obtained: q 2 −b1 ± b1 − 4a3c1 ρ = (28) 2a3 Hence, one can see that there is a strong correlation between the corrosion rate ρ and the shape of the initial defect.

2.1.3. Steel Bar Weight Loss Calculation When concrete cover cracks, the loss of steel weight can be expressed as follows:

2 Mloss = ρcr Ms = 0.0616ρcrD (29)

where Ms is the original steel weight (g); ρcr is the corrosion rate of steel bar when cover cracks (%); D is the diameter of steel bar (mm). The relationship among time, corrosion rate of the reinforcing bar and the loss of reinforcement weight is established based on Faraday’s law of corrosion and can be expressed as follows: MI M = corr t (30) loss zF

where Mloss is the loss of steel wight (g); z is lon valence; Icorr is corrosion current (A); M is the molecular mass of iron (g); t is corrosion time (s); F is Faraday’s constant (96500C).

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. Materials 2021, 14, 1440 8 of 22

In this work, the corrosion current density utilized is determined by Lu et al. [29], and the expression is given as: 1 3034 −3 0 icorr1(t) = √ exp 1.23 + 0.618 ln Ct − − 5 × 10 ρ (31) 3 1 + t T · (2.5 + RH)

2 where icorr denotes corrosion current density (µA/cm ); Ct denotes concrete chloride content on the surface of reinforcement (kg/m3); 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:

(1 − w/c)2 f = k (32) conc C Substituting Equation (32) into (31), the following relationship is obtained: 2 (1 − w/c) 1 3034 −3 0 icorr1(t) = k · √ exp 1.23 + 0.618 ln Ct − − 5 × 10 ρ (33) C 3 1 + t T · (2.5 + RH) 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 − w/c) 3034 −3 0 icorr1(t) = 62.69 √ · exp 1.23 + 0.618 ln Ct − − 5 × 10 ρ (34) 3 1 + t · C T · (2.5 + RH)

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:

Z t 2 M Mloss = 0.0616ρcrD = πD icorr1dt (35) zF 0 By integrating Equation (35), the formula of concrete cover cracking time is obtained:

" #1.5 0.285ρ DC(1 − w/c)1.64 t = cr + 1 − 1 (36) cr H∗ Materials 2021, 14, 1440 9 of 22

where H* can be expressed as:

3034 H∗ = exp 1.23 + 0.618 ln C − − 5 × 10−3ρ0 (37) t T · (2.5 + RH)

For reinforced concrete structures in naturally corrosive environments, based on the Wang [34] model, the cover cracking time can be expressed as follows: q D −b ± b2 − 4a c ρcrD 1 1 3 1 tcr = 78.3 = 78.3 (38) icorr icorr · 2a3

where a3, b1, c1 is the combination coefﬁcient, which can be obtained by Equation (27).

2.2. Experimental Verification of Cracking Model In order to verify the rationality and applicability of the above model in this study, five-year naturally exposed experiments, short time indoor tests and corroded concrete members in service on the prediction of time to cracking conducted by Liu and Weyers [6], Wang [34] and Shi [35] were used. The determination process of the relative parameters involved with the present model are explained as below: Initially, according to the experimental results conducted by Wu et al. [49], for concrete, ini un 1/2 1/2 the stable values of KIc and KIc are selected as 1.034 MPa·m and 2.072 MPa·m . Secondly, the value of the corresponding coefficients variation are determined as 0.061 and 0.073, respectively. In addition, other parameters for the proposed model are chosen as the same as the results adopted by Liu and Weyers [6], Wang [34] and Shi [35].

2.2.1. Compared to the Experimental Data of Liu and Weyers The experimental data of slabs in reference [6] is listed in Table1, and the comparison results of different models with the experimental data are listed in Table2.

Table 1. The value of basic parameters. Ct: chloride content; w/c: water–cement ratio; T: temperature; Ec: elastic modulus; vc: Poisson’s ratio; f t: tensile strength.

Specimen w/c Exposure 2R/mm C/mm C /kg/m3 T/K E /MPa ν ϕ f /MPa Designation Ratio t c c t Period/yr S1 16 48 0.43 4.92 295 27,000 0.18 2.0 3.3 1.84 S2 16 70 0.43 4.92 295 27,000 0.18 2.0 3.3 3.54 S3 16 27 0.45 6.02 295 27,000 0.18 2.0 3.3 0.72 S4 12.7 52 0.43 4.92 293 27,000 0.18 2.0 3.3 2.38

Table 2. Results comparison of present model with experimental prediction and other models.

Time to Cover Cracking/yr Specimen The Threshold Corrosion Error Value of Present Designation Experimental Present Pressure/MPa Rate/% Ref. [6] Ref. [50] Model to Experimental Value Model Results/% S1 4.81 0.46 1.84 1.53–2.06 1.79 2.00 8.69 S2 5.17 0.48 3.54 3.34–4.49 3.10 3.51 0.85 S3 4.11 0.44 0.72 0.56–0.75 0.80 0.76 5.56 S4 5.06 0.55 2.38 1.79–2.40 2.25 2.16 9.24

It was shown that in Table2, there is a relatively good coincidence between the prediction by the proposed model and the experimental results on the cover cracking time. Meanwhile, the predictions by the proposed model are comparatively improved when compared to that yielded by the models of Liu and Weyers [6] and Zhang et al. [50]. Moreover, the proposed model is also capable of evaluating the threshold pressure and Materials 2021, 14, 1440 10 of 22

corrosion rate, therefore, it is veriﬁed that the proposed model is able to predict the cover cracking time for both indoor and outdoor specimens.

2.2.2. Compared to the Acceleration Experimental Data of Wang The acceleration experimental data of specimens in reference [34] are listed in Table3, and the comparison results of different models with experimental data are listed in Table4.

Table 3. The value of basic parameters. d0: thickness of porous zone; icorr: corrosion current densitiy.

Specimen Cracking Date 2R/mm C/mm d /m fc/MPa Ec/MPa ft/MPa i /mA/cm2 Designation 0 corr Time/h Source A1 16 20 10 25 30,000 3.3 100 96 Andrade B1 16 50 20 25 30,000 3.3 100 208 Alonso B2 16 70 25 25 30,000 3.3 100 264 C1 16 33 15 30 30,000 3.3 150 95 Maaddawy D1 50 16 10 13.4 22,000 3.3 100 194.7 Vu D2 25 16 10 28.8 30,000 3.3 100 116

Table 4. Results comparison of present model with experimental prediction and other models.

Time to Cover Cracking/h Specimen The threshold Corrosion Error Value of Present Designation Experimental Present Pressure/MPa Rate/% Ref. [16] Ref. [34] Model to Experimental Value Model Results/% A1 3.68 0.35 96 84–119 82.08 106.55 10.99 B1 4.85 0.64 208 156–191 197.84 192.70 7.36 B2 5.17 0.79 264 203–237 281.35 239.18 9.40 C1 4.38 0.49 95 78–101 78.38 99.28 4.51 D1 1.95 0.21 194.7 - 172.68 194.37 1.70 D2 2.80 0.25 116 - 94.07 119.67 3.16

It can be seen from Table4, there is a good matching between the predictions of the proposed model and the experimental results achieved on the cover cracking time of accelerate corrosion specimens. In addition, when compared to the predictions generated by the models of Wang [34] and Maaddawy et al. [16], it is shown that the accuracy of the predictions by the proposed model is improved.

2.2.3. Compared to the Actual Measuring Data of Shi The actual measuring data of members in reference [35] and the comparison results are listed in Table5.

Table 5. Results comparison of the present model with measuring prediction.

Actual Time to Calculated Specimen Designation 2R/mm C/mm w/c Ratio C /kg/m3 fc/MPa t Cracking/yr Value/yr Concrete column 10 20.35 0.40 11 20 0.26 0.33 Concrete walkway slab 12 25.0 0.40 11 20 0.44 0.42

Table5 indicates the comparison results between the predictions by the proposed model and measuring results on the cover cracking time. It is revealed that the prediction is in good agreement with the actual measuring results. Therefore, it is proved that the proposed model can well predict the time to cover cracking of concrete structures in service. Materials 2021, 14, 1440 11 of 22

3. Influences of Various Parameters on Threshold Pressure, Corrosion Rate and Cracking Time In order to reflect the effectiveness of parameter influence, the parameters in the model need to be standardized. Based on engineering experience, the settlement height of concrete is about 1% of the concrete height; when the specimen height is about 200 mm, the length of initial defects is defined as 2 mm [20]. According to the selection standard of critical chloride concentration [51], the critical chloride ion concentration is 0.2%, assuming the mass of concrete is 2400 kg/m3, and the total chloride content (reinforcement surface) is calculated to be 4.8 kg/m3. In addition, the standard values for other variables need to be determined, and the value of other basic variables is listed in Table6.

Table 6. The value of basic parameters to determine.

Basic Variables Abbreviations Value Unit ini 1/2 Initial fracture toughness KIc 1.034 MPa·m un 1/2 Cracking fracture toughness KIc 2.072 MPa·m The length of initial defects a 2 mm 3 Chloride content Ct 4.8 kg/m Temperature T 293 K Relative humidity RH 80 % Concrete resistivity r 10,000 Ohm·cm Cover depth C 30 mm Diameter of steel bar D 16 mm Water-to-cement ratio w/c ratio 0.45 - 4 Elastic Modulus Ec 2.7 × 10 MPa Tensile strength f t 3.3 MPa Poisson’s ratio vc 0.18 - Creep coefﬁcient φcr 2.0 - Expansion rates of corrosion products n 2.0 - Thickness of porous zone d0 12.5 µm Initial defect cracking size a/c 0.20 - Cover cracking size a/c 0.72 -

3.1. Inﬂuence of the Initial Defect Length The initial defect length (a) is set to increase from 2 to 6 mm at 1 mm, and the corresponding initial defect size (a/c) is 0.1, 0.15, 0.2, 0.25, and 0.3, respectively. When the concrete cover cracks, the initial crack growth limits (a/c) are 0.64, 0.72 and 0.8. The calculation results of threshold pressure, corrosion rate of rebar and cover cracking time obtained by substituting the calculation parameters into the model are shown in Figure3. Figure3 shows that the initial defect length ( a) has a similar variation trend to the threshold pressure, corrosion rate of rebar and the cover cracking time, all of which gradually decrease with the increase in the initial defect length. This is because the increase in the initial defect length indicates that the on-site pouring and curing concrete are not in place, or that the concrete construction technology needs to be improved, and concrete density becomes worse, leading to the advance of the cover cracking time. When a/c = 0.72, the initial defect length increases from 2 mm to 7 mm, and the corresponding reduction rates of the threshold cracking pressure, corrosion rate and cover cracking time are 19.66%, 3.54% and 4.12%, respectively, indicating that the initial defect length has a greater impact on the threshold cracking pressure, followed by the cover cracking time. The initial defects cracks, or concrete cover cracks, the initial threshold pressure, the threshold cracking pressure, corrosion rate of rebar and cover cracking time all increase with the increase in the initial defect size (a/c). This is because the greater the ratio of the initial defect size, the smaller the initial defects area of the semi-ellipse is, indicating that the inﬂuence of the initial defect size on concrete cover cracking is reduced. In general, the initial defect size will change the cracking behavior and the failure mechanics mode of concrete cover cracking and affect the critical indexes such as concrete cover cracking time. At present, Materials 2021, 14, x FOR PEER REVIEW 12 of 23

the length of initial defects is defined as 2 mm [20]. According to the selection standard of critical chloride concentration [51], the critical chloride ion concentration is 0.2%, assuming the mass of concrete is 2400 kg/m3, and the total chloride content (reinforcement surface) is calculated to be 4.8 kg/m3. In addition, the standard values for other variables need to be determined, and the value of other basic variables is listed in Table 6.

Table 6. The value of basic parameters to determine.

Basic Variables Abbreviations Value Unit Initial fracture toughness KIcini 1.034 MPa·m1/2 Cracking fracture toughness KIcun 2.072 MPa·m1/2 The length of initial defects a 2 mm Chloride content Ct 4.8 kg/m3 Temperature T 293 K Relative humidity RH 80 % Concrete resistivity r 10,000 Ohm·cm Cover depth C 30 mm Diameter of steel bar D 16 mm Water-to-cement ratio w/c ratio 0.45 - Elastic Modulus Ec 2.7 × 104 MPa Tensile strength ft 3.3 MPa Poisson’s ratio vc 0.18 - Creep coefficient ϕcr 2.0 - Expansion rates of corrosion prod- n 2.0 - ucts Thickness of porous zone d0 12.5 μm Initial defect cracking size a/c 0.20 - Cover cracking size a/c 0.72 -

3.1. Influence of the Initial Defect Length Materials 2021, 14, 1440 12 of 22 The initial defect length (a) is set to increase from 2 to 6 mm at 1 mm, and the corresponding initial defect size (a/c) is 0.1, 0.15, 0.2, 0.25, and 0.3, respectively. When the concrete cover cracks, the initial crack growth limits (a/c) are 0.64, 0.72 and 0.8. The calculation resultsthe relevant of threshold experimental pressure, and corrosion theoretical rate research of rebar of and the initialcover cracking defect size time on obtained the concrete by scoverubstituting cracking the process calculation is still parameters blank. into the model are shown in Figure 3.

1.40 4.60 a/c=0.1 a/c=0.64 a/c=0.15 a/c=0.72 1.30 a/c=0.2 4.40 a/c=0.8 a/c=0.25 1.20 a/c=0.3 4.20 1.10 4.00 1.00 3.80 0.90 3.60 0.80

The initial threshold pressure threshold pressure The(MPa) initial 0.70 (MPa) pressure The cracking threshold 3.40

0.60 3.20 Materials 2021, 14, x FOR PEER2 REVIEW3 4 5 6 7 1 2 3 4 5 6 7 8 13 of 23

The length of initial defects (mm) The length of initial defects (mm) (a) The initial threshold pressure (b) The threshold cracking pressure

0.450 1.06 a/c=0.64 a/c=0.64 a/c=0.72 0.445 a/c=0.72 a/c=0.8 a/c=0.8 1.04 0.440

1.02 0.435

0.430 1.00

Cracking time time (year) Cracking The corrosion rate (%) rate The corrosion 0.425 0.98 0.420

0.415 0.96 1 2 3 4 5 6 7 8 1 2 3 4 5 6 7 8 The length of initial defects (mm) The length of initial defects (mm) (c) The corrosion rate (d) The cover cracking time

Figure 3. InfluenceInﬂuence of of fine ﬁne crack on threshol thresholdd pressure, corrosion rate and cracking time.

Figure 3 shows that the initial defect length (a) has a similar variation trend to the 3.2. Inﬂuence of the Expansion Rates of Corrosion Products threshold pressure, corrosion rate of rebar and the cover cracking time, all of which grad- uallyBased decrease on with previous the increase research inresults, the initial the defect range length. of the This expansion is because rates the of increase corrosion in theproducts initial isdefect 2–4times, length which indicates is set that to increasethe on-site from pouring 2 to 4 and at 0.5. curing The concrete calculation are resultsnot in place,of the or corrosion that the rateconcrete and coverconstruction cracking technology time obtained needs by to substitutingbe improved, the and calculation concrete densityparameters becomes into theworse, model leading are shown to the inadvance Figure of4. the cover cracking time. When a/c = 0.72, Figure4 shows that the expansion rates of corrosion products have a greater impact the initial defect length increases from 2 mm to 7 mm, and the corresponding reduction on the corrosion rate of rebar and cover cracking time, as the expansion rates of corrosion rates of the threshold cracking pressure, corrosion rate and cover cracking time are products increase from 2.0 to 4.0, the reduction rates of the corrosion rate and cover cracking 19.66%, 3.54% and 4.12%, respectively, indicating that the initial defect length has a greater time is 199.33% and 226.32%, respectively, and the difference between the two is not big; impact on the threshold cracking pressure, followed by the cover cracking time. The initial there is a one-to-one correspondence. This is because the limited pore area inside the defects cracks, or concrete cover cracks, the initial threshold pressure, the threshold crack- concrete cannot satisfy the expansion of the corrosion product volume greatly, in the case ing pressure, corrosion rate of rebar and cover cracking time all increase with the increase of a low rate of reinforcement corrosion rust had a great expansion pressure caused by in the initial defect size (a/c). This is because the greater the ratio of the initial defect size, coating concrete cover cracking in advance, so the selection of reasonable expansion rates the smaller the initial defects area of the semi-ellipse is, indicating that the influence of the of corrosion products for predicting the cover cracking time is crucial. initial defect size on concrete cover cracking is reduced. In general, the initial defect size will change the cracking behavior and the failure mechanics mode of concrete cover cracking and affect the critical indexes such as concrete cover cracking time. At present, the relevant experimental and theoretical research of the initial defect size on the concrete cover cracking process is still blank.

3.2. Influence of the Expansion Rates of Corrosion Products Based on previous research results, the range of the expansion rates of corrosion products is 2–4 times, which is set to increase from 2 to 4 at 0.5. The calculation results of the corrosion rate and cover cracking time obtained by substituting the calculation parameters into the model are shown in Figure 4. Figure 4 shows that the expansion rates of corrosion products have a greater impact on the corrosion rate of rebar and cover cracking time, as the expansion rates of corrosion products increase from 2.0 to 4.0, the reduction rates of the corrosion rate and cover cracking time is 199.33% and 226.32%, respectively, and the difference between the two is not big; there is a one-to-one correspondence. This is because the limited pore area inside the concrete cannot satisfy the expansion of the corrosion product volume greatly, in the case of a low rate of reinforcement corrosion rust had a great expansion pressure caused by coating concrete cover cracking in advance, so the selection of reasonable expansion rates of corrosion products for predicting the cover cracking time is crucial.

Materials 2021, 14, x FOR PEER REVIEW 14 of 23

MaterialsMaterials2021 2021,,14 14,, 1440 x FOR PEER REVIEW 1314of of 2223

0.5 1.1 Cracking time (year) 0.40.5 1.01.1 The corrosion rate (%) Cracking time (year) 0.9

1.0 timeCracking (year) 0.40.4 The corrosion rate (%) 0.80.9 0.30.4 timeCracking (year) 0.70.8 0.30.3 0.60.7

The corrosionThe rate (%) 0.20.3 0.50.6

The corrosionThe rate (%) 0.20.2 0.40.5

0.10.2 0.30.4 2.0 2.5 3.0 3.5 4.0 0.1 The expansion rates of corrosion products 0.3 2.0 2.5 3.0 3.5 4.0 Figure 4. RelationshipThe expansion betweenrates of corrosion corrosion products products expansion rates and corrosion rate as well as cover cracking time. FigureFigure 4.4. Relationship between between corrosion corrosion products products expansion expansion rates rates and and corrosion corrosion rate rate as aswell well as as covercover crackingcracking time.time. 3.3. Influence of the Thickness of the Porous Zone 3.3.3.3. InﬂuenceInfluenceCurrentl ofy,of thethe Thickness Thicknessthickness ofof of thethe the PorousPorous porous ZoneZo zonene at the interface was selected by referring to theCurrently,Currentl data providedy, thethe thicknessthickness by other ofof researchers, thethe porousporous zonewhichzone atat is thethe set interfaceinterface to increase waswas from selectedselected 5 to by by25 referring referringμm at 5 μm.toto thethe The datadata calculation provided results by other other of the researchers, researchers, threshold pressure,which which is is corrosionset set to to increase increase rate of from fromthe rebar5 5to to 25 and 25 μmµ cmover at at 5 cracking5μm.µm. The The calculation time calculation obtained results results by of substituting the of thethreshold threshold the pressure, calculation pressure, corrosion parameters corrosion rate of rate into the of rebar the the model rebarand c overand are showncovercracking cracking in Figure time time obtained 5. obtained by substituting by substituting the the calculation calculation parameters parameters into into the the model are are shownshownThe inin results FigureFigure 5of 5.. Figure 5 show as the thickness of the porous zone increases from 5 to 25 μm,TheThe the resultsresults growth ofof FigureFigurerates of5 5 show theshow corresponding as as the the thickness thickness threshold of of the the porous porous pressur zone zonee, corrosion increases increases rate from from of 5 5the to to µ rebar2525 μm,m, and the the cover growth growth cracking rates rates of timeof the the correspondingare corresponding 0%, 203.61% threshold and threshold 226.32%, pressure, pressur respectively, corrosione, corrosion which rate of rate indicates the of rebar the and cover cracking time are 0%, 203.61% and 226.32%, respectively, which indicates that thatrebar the and thickness cover cracking of the porous time are zone 0%, has 203.61% a relatively and 226.32%, large effect respectively, on the corrosion which indicates rate of the thickness of the porous zone has a relatively large effect on the corrosion rate of the thethat rebar the thickness and cover of cracking the porous time zone but hashas littlea relatively effect on large the initialeffect onthreshold the corrosion pressure rate and of rebar and cover cracking time but has little effect on the initial threshold pressure and the thethe thresholdrebar and covercracking cracking pressure. time This but hasis because little effect the onpressure the initial is mainly threshold sensitive pressure to andthe threshold cracking pressure. This is because the pressure is mainly sensitive to the initial initialthe threshold defect size, cracking which pressure.will affect This the fracture is because expansion the pressure trend, iswhile mainly the thicknesssensitive ofto thethe defect size, which will affect the fracture expansion trend, while the thickness of the porous porousinitial defect zone willsize, not which affec willt the affect fracture the fracture development. expansion However, trend, whilethe increase the thickness in the thick- of the zone will not affect the fracture development. However, the increase in the thickness of nessporous of thezone porous will not zone affec improvest the fracture the porositydevelopment. of concrete However, and the provides increase more in the storage thick- the porous zone improves the porosity of concrete and provides more storage space for spaceness of for the corrosion porous zone products, improves which the leads porosity to the of increase concrete in and rust provides expansion more and storage cover corrosion products, which leads to the increase in rust expansion and cover cracking time. crackingspace for time. corrosion products, which leads to the increase in rust expansion and cover cracking time. 5.0 0.8 2.0 Cracking time (year) 5.0 0.8 2.0 0.7 The corrosion rate (%) 4.0 Cracking time (year) The threshold cracking pressure 0.7 The corrosion rate (%) time Cracking (year) 0.6 1.5 4.0 The threshold cracking pressure

Cracking time time Cracking (year) 3.0 0.6 1.5 0.5 3.0 0.5 2.0 0.4 1.0

The corrosionrateThe (%)

The threshold pressrue (MPa) The threshold 2.0 The initial threshold pressure 0.4 1.0

The corrosionrateThe (%) 0.3

The threshold pressrue (MPa) The threshold 1.0 The initial threshold pressure 0.3 1.0 0.2 0.5 5.0 10.0 15.0 20.0 25.0 5.0 10.0 15.0 20.0 25.0 The thickness of porous zone ( m) 0.2 The thickness of porous zone (m) 0.5 5.0 10.0 15.0 20.0 25.0 5.0 10.0 15.0 20.0 25.0 (a) The threshold pressure (b) The corrosion rate and cracking time The thickness of porous zone ( m) The thickness of porous zone (m) (a) The thresholdFigure pressure 5. InfluenceInﬂuence of the thickness(b) The of the corrosion porous zone.zone. rate and cracking time Figure 5. Influence of the thickness of the porous zone. 3.4. Influence Inﬂuence of Cover Dept Depthh 3.4. InfluenceThe cover of depth Cover Deptis set h to increase from 20 to 65 mm at 5 mm;mm; thethe calculationcalculation results of the The threshold cover depth p pressure,ressure, is set corrosion to increase rate from of rebar 20 to and 65 covermm at cracking 5 mm; the time calculation are obtained results by substitutingof the threshold the calculationpressure, corrosion parameters rate into of rebar the model, and cover as shown cracking in Figure time are6 6.. obtained by substituting the calculation parameters into the model, as shown in Figure 6.

Materials 20212021,,, 1414,,, 1440xx FORFOR PEERPEER REVIEWREVIEW 1415 of 2223

6.0 0.48 3.0 6.0 0.48 3.0 Cracking time (year) Cracking time (year) 5.0 0.47 The corrosion rate (%) 5.0 0.47 The corrosion rate (%) 2.5 2.5

Cracking time time Cracking (year) 4.0 The threshold cracking pressure 0.46 time Cracking (year) 4.0 The threshold cracking pressure 0.46 2.0 2.0 3.0 0.45 3.0 0.45 1.5 1.5 2.0 0.44

2.0 corrosionrateThe (%) 0.44

The corrosionrateThe (%)

The threshold pressrue (MPa) The pressrue threshold The threshold pressrue (MPa) The pressrue threshold 1.0 1.0 The initial threshold pressure 0.43 1.0 1.0 The initial threshold pressure 0.43

0.0 0.42 0.5 0.0 0.42 0.5 20.0 30.0 40.0 50.0 60.0 70.0 20.0 30.0 40.0 50.0 60.0 70.0 20.0 30.0 40.0 50.0 60.0 70.0 20.0 30.0 40.0 50.0 60.0 70.0 Cover depth(mm) Cover depth(mm) Cover depth(mm) Cover depth(mm) (a) The threshold pressure (b) The corrosion rate and cracking time Figure 6. Influence of the cover depth on threshold pressure, corrosion rate and cracking time. Figure 6. InfluenceInﬂuence of the cove coverr depth on threshold pressure, corrosion raterate andand crackingcracking time.time.

Results of Figure 66 showshow thatthat asas covercover depthdepth increasesincreases fromfrom 2020 mmmm toto 6565 mm, mm, thethe growth rates of the threshold cracking pressure, corrosion rate of the rebar and cover cracking time are 38.65%,38.65%, 10.54% and 326.9%, respectively, which which indicates that cover depth has the largest impact on cover cracking time, and a relatively small impact on the threshold pressure pressure and and corrosion corrosion rate rate of of the the rebar. rebar. Therefore, Therefore, for for some some structures structures with with a longa long design design life life and and commemorative commemorative significance, signiﬁcance, such such as as bridges bridges in in coastal coastal areas and museums inin variousvarious regions,regions, the the service service life life of of the the structures structures can can be be improved improved by by increasing increas- ingthe coverthe cover depth. depth.

3.5. Influence Inﬂuence of w/c Ratio and Chloride Con Contenttent The concreteconcrete water–cementwater–cement ( w/c(w/c)) ratio ratio is is set set to to increase increase from from 0.3 to0.3 0.7 to at0.7 0.1, at and0.1, chlo-and 3 3 3 chlorideride content content (reinforcement (reinforcement surface) surface) is set isto setincrease to increase from 4.8 from to12.0 4.8 kg/m to12.0 atkg/m 1.23 kg/m at 1.2. 3 kg/mThe calculation3. The calculation results results of cover of cover cracking cracking time time obtained obtained by substituting by substituting the the calculation calcula- tionparameters parameters into theinto model the model are shown are shown in Figure in Figure7. 7.

0.7 12.0 0.7 12.0 3 Chloride content (kg/m3) Chloride content (kg/m ) 11.0 w/c ratio 11.0 0.6 w/c ratio 0.6

10.0 content (kg/m Chloride

9.0 0.5 9.0 0.5

ratio 8.0 ratio 8.0

w/c w/c 0.4 0.4 7.0 7.0

)

3 6.0 ) 0.3 6.0 0.3 5.0 5.0

0.2 4.0 0.20.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.84.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 1.6 1.8 Cracking time (year) Cracking time (year) Figure 7. InfluenceInfluence of w/c ratio and chloride content on cover cracking time. Results of Figure7 show that as the w/c ratio increases from 0.3 to 0.7, the cover Results of Figure 7 show that as the w/c ratio increases from 0.3 to 0.7, the cover crack- cracking time decreases by 244.7% significantly, indicating that the w/c ratio has a great ing time decreases by 244.7% significantly, indicating that the w/c ratio has a great influ- influence on cover cracking time. This is because the w/c ratio will affect the tensile strength ence on cover cracking time. This is because the w/c ratio will affect the tensile strength of of concrete, so the corrosion rate of the reinforcing bar increases, and the corresponding concrete, so the corrosion rate of the reinforcing bar increases, and the corresponding cover cracking time will be shortened. Moreover, the w/c ratio has an indirect effect cover cracking time will be shortened. Moreover, the w/c ratio has an indirect effect on the thresholdon the threshold pressure pressure and corrosion and corrosion rate of ratethe rebar of the by rebar affecting by affecting the porosity the porosity and tensile and threshold pressure and corrosion rate of the rebar by affecting the porosity and tensile3 strengthtensile strength of concrete. of concrete. As the Asconcrete the concrete chloride chloride content content increased increased from 4.8 from kg/m 4.8 kg/m3 to 12 kgto strength 3of concrete. As the concrete chloride content increased from 4.8 kg/m3 to 12 kg 12 kg/m3 , the corresponding growth rate of cover cracking time was 85.48% significantly. /m3, the corresponding growth rate of cover cracking time was 85.48% significantly. This

Materials 2021, 14, 1440 15 of 22

This is because in the steel bar corrosion in the event of a reaction, chloride ions have played a catalytic role but are not consumed themselves; as concrete chloride content increased, the process of steel corrosion was accelerated virtually, resulting in cover cracking time greatly in advance.

4. The Cover Cracking Time Probabilistic Analysis 4.1. Probability Model Based on the Third Moment (TM) Method In fact, the randomness is one of the essential characteristics for concrete material and is closely related to structure safety. For the problem of estimating the cover cracking time in actual projects, from a practical point of view, approaches based on the deterministic method are limited. Hence, in order to reflect the real response on the cover cracking time, it is necessary to adopt the random probability theory on relevant studies. In this work, the TM method proposed by Stanish et al. [52], Zhang et al. [53] and Zhao et al. [54,55] is adopted to calculate the reliability and probability of failure for reinforced concrete structures. For a reinforced concrete structures, the service life (lifetime) is defined as the period of time that meets or exceeds the initial requirements. Normally, a performance-based assessment criterion needs to be established firstly for determination of the service life of reinforced concrete structure. The general form for such criterion of limit state function is given as follows: G(L, S, t) = L(t) − S(t) (39) where S(t) is the action (load) or its effect at time t and L(t) is the acceptable limit (resistance) for the action or its effect. Based on Equation (47), the probability of concrete (structural) failure, Pf can be determined by Pf (t) = P[G(L, S, t) ≤ 0] = P[S(t) ≥ L(t)] (40)

It is noted that for concrete, once the KI (stress intensity factor) is larger than KIc (fracture toughness), concrete cover is not able to sustain the crack growth. Therefore, the tcr implies the critical time for cracks appearing in the concrete cover (gained by the value of KIc of concrete cover), t represents the changed time when corrosion products gradually accumulate. Based on the analytical solution, it can be found that the governing parameters (C, d, T, RH, Ec, Cl, w/c, KIc) are considered as random variables. Therefore, the probability of concrete failure can be determined from Equation (41) directly, where t replaces S and tcr replaces L, as follows: p f ,c(t) = P[t ≥ tcr] (41)

4.2. Verification of the Proposed Model In this section, the efficiency and accuracy of the proposed model on the probability of corrosion-induced failure is verified with different cover cracking. Specifically, comparisons between the results of the proposed model and those obtained by traditional Monte Carlo (MC) simulations are conducted. In addition, the standard, mean, coefficient of variation (COV) and distribution type of several fundamental variables are obtained and listed in Tables6 and7. In detail, results yielded by the proposed model and MC simulations for cover cracking failure probability with sample sizes N = 10, 100, 1000 and 10,000 are listed in Figure8. It is found that, when the sample size N is less than 1000, the prediction of failure probability of the MC simulation is unstable. Additionally, for the case of the sample size N = 10,000, the predictions generated by both the proposed model and the MC simulation are in good agreement. However, the TM method used in the proposed model is simpler than the MC simulation when dealing with the analysis of the probability of cover cracking. Materials 2021, 14, x FOR PEER REVIEW 17 of 23

Materials 2021, 14, 1440 in good agreement. However, the TM method used in the proposed model is simpler16 than of 22 the MC simulation when dealing with the analysis of the probability of cover cracking.

Table 7. The parameter values used in model verification of rust expansion cracking time. Table 7. The parameter values used in model veriﬁcation of rust expansion cracking time. Coefficient of Distribution Number Basic Variables Variable Name Mean References Coefﬁcient of Distribution Number Basic Variables Variable Name Mean Variation Type References Cracking fracture tough- Variation Type 1 KIcun 2.072 0.177 Normal [40] Cracking fracture 1 ness K un 2.072 0.177 Normal [40] toughness Ic t 3 2 Chloride content C 4.8 kg/m3 0.5 Normal [52] 2 Chloride content Ct 4.8 kg/m 0.5 Normal [52] 33 Temperature Temperature TT 293293 K K 0.20.2 Lognormal Lognormal [ [5353]] 44 Relative Relative humidity humidity RHRH 75%75% 0.050.05 Normal Normal [ [5656]] 55 Cover Cover depth depth CC 4040 mm mm 0.20.2 Normal Normal [ [5757]] 66 Diameter Diameter of of steel steel bar dd 1616 mm mm 0.20.2 Lognormal Lognormal [ [5858]] 7 Water-to-cement ratio w/c ratio 0.45 0.1 Lognormal [52] 7 Water-to-cement ratio w/c ratio 0.454 0.1 Lognormal [52] 8 Elastic Modulus Ec 2.7 × 10 MPa 0.12 Lognormal [58] 8 Elastic Modulus Ec 2.7 × 104 MPa 0.12 Lognormal [58]

1.0

0.8 MC(N=10) MC(N=100) MC(N=1000) 0.6 MC(N=10000) TM 0.4

Failure probabilityFailure

0.2

0.0 0 2 4 6 8 10 Time (year) Figure 8. Comparison of TM and MC on cover cracking probability.probability.

4.3. Time-DependentTime-Dependent ProbabilityProbability AssessmentAssessment ofof thethe ProposedProposed ModelModel 4.3.1. Mean Value of Various Factors 4.3.1. Mean Value of Various Factors By adopting the value of various factors (C, d, T, RH, Ec, Cl, w/c, KIc) in Table7, the By adopting the value of various factors (C, d, T, RH, Ec, Cl, w/c, KIc) in Table 7, the corresponding failure probability of cover cracking influenced by their mean values of corresponding failure probability of cover cracking influenced by their mean values of these factors are illustrated in Figure9. It can be seen from in Figure9 that the influence of these factors are illustrated in Figure 9. the mean values of these factors shows a similar trend, and the failure probability of cover It can be seen from in Figure 9 that the influence of the mean values of these factors cracking also increases with the growth of time. shows a similar trend, and the failure probability of cover cracking also increases with the Cove depth, the diameter of steel bar and the water–cement ratio are the most im- portantgrowth of factors time. affecting cracking probability (Figure9a,b,d). Moreover, it is found that 1.0 1.0 there are also great effects on the probability of cracking for chloride content, temperature and relative humidity (Figure9c,e,f). Specifically, in Figure9a, it shows that as the 0.8 MC,C=30mm 0.8 TM,d=16mm value of C increasesMC,C=40mm from 30 mm to 60 mm, the surface crackingTM,d=20mm time, which is defined MC,C=50mm TM,d=22mm as the time to reachMC,C=60mm a 50% cracking probability, decreases from approximatelyTM,d=25mm 0.8 years 0.6 to 2.7 years. In FigureTM,C=30mm9b, it shows that0.6 when d varies from 16 mm to 25 mm, it results in TM,C=40mm an approximatelyTM,C=50mm 1.2-year increase in the surface cracking time. In Figure9d, the surface TM,C=60mm 0.4 cracking time increases from 0.5 years0.4 to around 2.7 years as w/c changes from 0.35 to 0.65.

Failure probability Failure probability Failure 3 In Figure9 c,e,f, an increase in Ct, T, RH from 4.8 to 9.6 kg/m , 283 to 313 K, 0.65 to 0.95 can 0.2 lead to a reduction in the surface cracking0.2 time between 1.0 and 2.0 years, respectively. As shown in Figure9g,h, the probability curve of the elastic modulus Ec and cracking

0.0 fracture toughness KIc is similar to0.0 each other, and the results of the effects of two factors 0 2 on the4 cover6 cracking8 probability10 are negligible.0 2 4 6 8 10 TheTime (year) comparison results of effects of cover depth,Time chloride (year) content and temperature (a) Coverare summarized depth (C) in Figure9a,c,e. As indicated(b) Diameter in these ﬁgures,of steel thebar calculation(d) results of the two methods are very close, which shows that the third-order method can calculate the inﬂuence of these three factors on the cover failure probability more accurately.

Materials 2021, 14, x FOR PEER REVIEW 17 of 23

in good agreement. However, the TM method used in the proposed model is simpler than the MC simulation when dealing with the analysis of the probability of cover cracking.

Table 7. The parameter values used in model verification of rust expansion cracking time.

Coefficient of Distribution Number Basic Variables Variable Name Mean References Variation Type Cracking fracture tough- 1 KIcun 2.072 0.177 Normal [40] ness 2 Chloride content Ct 4.8 kg/m3 0.5 Normal [52] 3 Temperature T 293 K 0.2 Lognormal [53] 4 Relative humidity RH 75% 0.05 Normal [56] 5 Cover depth C 40 mm 0.2 Normal [57] 6 Diameter of steel bar d 16 mm 0.2 Lognormal [58] 7 Water-to-cement ratio w/c ratio 0.45 0.1 Lognormal [52] 8 Elastic Modulus Ec 2.7 × 104 MPa 0.12 Lognormal [58]

1.0

0.8 MC(N=10) MC(N=100) MC(N=1000) 0.6 MC(N=10000) TM 0.4

Failure probabilityFailure

0.2

0.0 0 2 4 6 8 10 Time (year) Figure 8. Comparison of TM and MC on cover cracking probability.

4.3. Time-Dependent Probability Assessment of the Proposed Model 4.3.1. Mean Value of Various Factors

By adopting the value of various factors (C, d, T, RH, Ec, Cl, w/c, KIc) in Table 7, the corresponding failure probability of cover cracking influenced by their mean values of Materials 2021, 14, 1440 these factors are illustrated in Figure 9. 17 of 22 It can be seen from in Figure 9 that the influence of the mean values of these factors shows a similar trend, and the failure probability of cover cracking also increases with the growth of time. 1.0 1.0

0.8 MC,C=30mm 0.8 TM,d=16mm MC,C=40mm TM,d=20mm MC,C=50mm TM,d=22mm MC,C=60mm TM,d=25mm 0.6 TM,C=30mm 0.6 TM,C=40mm TM,C=50mm TM,C=60mm 0.4 0.4

Failure probability Failure probability Failure

0.2 0.2

0.0 0.0 Materials 2021, 14, x FOR0 PEER REVIEW2 4 6 8 10 0 2 4 6 8 10 18 of 23

Time (year) Time (year) (a) Cover depth (C) (b) Diameter of steel bar (d)

1.0 1.0

MC,Cl=4.8kg/m^3 TM,w/c=0.35 MC,Cl=6.0kg/m^3 0.8 0.8 TM,w/c=0.45 MC,Cl=7.2kg/m^3 MC,Cl=9.6kg/m^3 TM,w/c=0.55 TM,Cl=4.8kg/m^3 TM,w/c=0.65 0.6 TM,Cl=6.0kg/m^3 0.6 TM,Cl=7.2kg/m^3 TM,Cl=9.6kg/m^3

0.4 0.4

Failure probability Failure

Failure probabilityFailure

0.2 0.2

0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 Time (year) Time (year) (c) Chloride content (Ct) (d) Water–cement ratio (w/c) 1.0 1.0

MC,T=283K TM,RH=0.65 0.8 MC,T=293K 0.8 TM,RH=0.75 MC,T=303K TM,RH=0.85 MC,T=313K TM,RH=0.95 TM,T=283K 0.6 TM,T=293K 0.6 TM,T=303K TM,T=313K

0.4 0.4

Failure probability Failure

Failure probabilityFailure

0.2 0.2

0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 Time (year) Time (year) (e) Temperature (T) (f) Relative humidity (RH) 1.0 1.0

TM,Ec=25GPa 0.8 0.8 TM,Ec=27GPa TM,K=2 TM,Ec=30GPa TM,K=2.8 TM,Ec=35GPa TM,K=3.2 0.6 0.6

0.4 0.4

Failure probability Failure probability Failure

0.2 0.2

0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 Time (year) Time (year) (g) Elastic Modulus (Ec) (h) Cracking fracture toughness (KIc)

FigureFigure 9. Inﬂuence 9. Influence of theof the mean mean values values of of different different factors on on the the failure failure probability probability of cover of cover cracking. cracking.

Cove depth, the diameter of steel bar and the water–cement ratio are the most important factors affecting cracking probability (Figure 9a,b,d). Moreover, it is found that there are also great effects on the probability of cracking for chloride content, temperature and relative humidity (Figure 9c,e,f). Specifically, in Figure 9a, it shows that as the value of C increases from 30 mm to 60 mm, the surface cracking time, which is defined as the time to reach a 50% cracking probability, decreases from approximately 0.8 years to 2.7 years. In Figure 9b, it shows that when d varies from 16 mm to 25 mm, it results in an approximately 1.2-year increase in the surface cracking time. In Figure 9d, the surface cracking time increases from 0.5 years to around 2.7 years as w/c changes from 0.35 to 0.65.

Materials 2021, 14, x FOR PEER REVIEW 19 of 23

In Figure 9c,e,f, an increase in Ct, T, RH from 4.8 to 9.6 kg/m3, 283 to 313 K, 0.65 to 0.95 can lead to a reduction in the surface cracking time between 1.0 and 2.0 years, respectively. 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. The comparison results of effects of cover depth, chloride content and temperature are summarized in Figure 9a,c,e. As indicated in these figures, the calculation results of the two methods are very close, which shows that the third-order method can calculate the influence of these three factors on the cover failure probability more accurately.

4.3.2. Coefficient of Variation Due to the variation of the values for the above influencing factors, relative effects by coefficient of variation (COV) on the probability of cracking need to be examined. In detail, the relevant results are represented in Figure 10. 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 Materials 2021, 14, 1440 cracking also increases with the growth of time. 18 of 22 Specifically, it is concluded that the COV of cove depth and chloride content are the factors that affect cracking probability the most (Figure10a,c). Moreover, the w/c ratio, relative humidity and cracking fracture toughness also have a great influence on the cracking 4.3.2.probability Coefficient (Figure of Variation 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 crackingDue to thetime, variation which emerges of the at values a 0.2-year for theincrease, above increases influencing as Cl changes factors, from relative 0.1 to effects by coefficient0.7. In Figure10d,f,h, of variation an (COV) increase on thein the probability coefficient of variation cracking of need d, w/c to ratio be examined., KIc from 0.1 In detail, the relevantto 0.6, 0.05 results to 0.3, 0.1 are to represented 0.4 can lead to in a Figureincrement 10 in. cover cracking time between 1.8 and 1.9 years, respectively. 1.0 1.0

MC,cov=0.1 MC,cov=0.1 0.8 MC,cov=0.2 0.8 MC,cov=0.2 MC,cov=0.4 MC,cov=0.4 MC,cov=0.6 MC,cov=0.6 TM,cov=0.1 TM,cov=0.1 0.6 TM,cov=0.2 0.6 TM,cov=0.2 TM,cov=0.4 TM,cov=0.4 TM,cov=0.6 TM,cov=0.6 0.4 0.4

Failure probabilityFailure

Failure probability Failure

0.2 0.2

0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 Time (year) Time (year) (a) Cover depth (C) (b) Diameter of steel bar (d) 1.0 1.0

MC,cov=0.1 0.8 MC,cov=0.3 0.8 TM,cov=0.05 MC,cov=0.7 TM,cov=0.1 MC,cov=0.5 TM,cov=0.2 TM,cov=0.1 TM,cov=0.3 0.6 TM,cov=0.3 0.6 TM,cov=0.5 TM,cov=0.7

0.4 0.4

Failure probability Failure probability Failure

0.2 0.2

0.0 0.0 Materials 2021, 14, x FOR0 PEER REVIEW2 4 6 8 10 0 2 4 6 8 10 20 of 23

Time (year) Time (year) (c) Chloride content (Ct) (d) Water–cement ratio (w/c)

1.0 1.0

MC,cov=0.1 0.8 MC,cov=0.2 0.8 TM,cov=0.05 MC,cov=0.4 TM,cov=0.1 MC,cov=0.6 TM,cov=0.2 TM,cov=0.1 TM,cov=0.4 0.6 TM,cov=0.2 0.6 TM,cov=0.4 TM,cov=0.6

0.4 0.4

Failure probability Failure probability Failure

0.2 0.2

0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 Time (year) Time (year) (e) Temperature (T) (f) Relative humidity (RH) 1.0 1.0

TM,cov=0.04 TM,cov=0.1 0.8 TM,cov=0.1 0.8 TM,cov=0.177 TM,cov=0.2 TM,cov=0.4 TM,cov=0.3

0.6 0.6

0.4 0.4

Failure probability Failure

Failure probabilityFailure

0.2 0.2

0.0 0.0 0 2 4 6 8 10 0 2 4 6 8 10 Time (year) Time (year) (g) Elastic Modulus (Ec) (h) Cracking fracture toughness (KIc) Figure 10. Influence of the coefficient of variation of different factors on failure probability of cover cracking. Figure 10. Inﬂuence of the coefﬁcient of variation of different factors on failure probability of cover cracking. 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. The comparison results of the effects of the coefficient of variation of cover depth, chloride content, diameter of steel bar and temperature are summarized in Figure 10a,b,c,e. The results show the calculation results of the two methods are very close, which shows that the third-order method can calculate the influence of these factors on the cover failure probability more accurately.

5. Conclusions In this study, a corrosion-induced cracking model based on the theory of fracture mechanics was proposed. Specifically, in this model, the three-dimensional initial defect, modified corrosion rate of the reinforcing bar in a natural environment and the acceleration to power corrosion were taken into account. In addition, the influence of relative parameters on several critical corrosion-induced crack indices were also analyzed. The proposed model is capable of quantitatively evaluating the risk of cracking for RC structures and predicting their service life. The following conclusions can be drawn: (1) The predictions yielded by the proposed model are in close agreement with the experimental results obtained from published data and achieve better precision when compared with other existing models.

Materials 2021, 14, 1440 19 of 22

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. The comparison results of the effects of the coefficient of variation of cover depth, chloride content, diameter of steel bar and temperature are summarized in Figure 10a,b,c,e. The results show the calculation results of the two methods are very close, which shows that the third-order method can calculate the influence of these factors on the cover failure probability more accurately.

5. Conclusions In this study, a corrosion-induced cracking model based on the theory of fracture mechanics was proposed. Specifically, in this model, the three-dimensional initial defect, modified corrosion rate of the reinforcing bar in a natural environment and the acceleration to power corrosion were taken into account. In addition, the influence of relative parameters on several critical corrosion-induced crack indices were also analyzed. The proposed model is capable of quantitatively evaluating the risk of cracking for RC structures and predicting their service life. The following conclusions can be drawn: (1) The predictions yielded by the proposed model are in close agreement with the experimental results obtained from published data and achieve better precision when compared with other existing models. (2) Through sensitive analysis of relative parameters in the cracking model, it was found that critical indices including threshold pressure, corrosion rate and cracking time decrease slowly when the initial defect size increases from 2 mm to 8 mm. Besides, both the volume expansion ratio of corrosion products and the thickness of the porous zone had great effects on the critical indices of corrosion rate and cracking time. It is found that the values of these two critical indices decline sharply with the increase in volume expansion ration from 2.0 to 4.0 and the decrease in the thickness of the porous zone from 25 µm to 5 µm, respectively. It is shown that other parameters including cover depth, w/c ratio and chloride content were only influential to the cracking time. (3) The results of probabilistic analysis for cover cracking time revealed that there are certain effects on the failure probability of cracking caused by both the mean values and COV of basic variables. The cracking risk of RC structures was found to grow with the increase in the mean value for chloride content, temperature, relative humidity and w/c ratio, and decreased with the increase in cover depth and the steel bar diameter. Among the w/c ratio, cover depth and steel bar dimeter, the cracking time was more sensitive to the w/c ratio than the other two variables. (4) Compared with traditional methods such as the MC simulation, the proposed TM method was able to obtain higher efficiency and greater simplicity in the stochastic analysis of corrosion-induced cracking problems. Materials 2021, 14, 1440 20 of 22

Author Contributions: Conceptualization, P.-Y.L.; methodology, C.J.; software, L.F. and Y.-F.M.; formal analysis, P.-Y.L.; investigation, C.J.; resources, L.F.; writing—original draft preparation, P.-Y.L.; writing—review and editing, P.-Y.L.; project administration, X.-G.Z.; funding acquisition, X.-G.Z. and Y.-F.M. All authors have read and agreed to the published version of the manuscript. Funding: The study was supported by the National Natural Science Foundation of China (Grant No. 51878413), the Science and Technology Plan Project of Shenzhen (Grant No. JCYJ20190808112019066) and Ningxia Youth Talents Supporting Program (TJGC2019030). Institutional Review Board Statement: Not applicable. Informed Consent Statement: Not applicable. Data Availability Statement: Data available in a publicly accessible repository that does not is- sue DOIs. Conﬂicts of Interest: The authors declare no conﬂict of interest.

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