Modeling the cracking of cover concrete due to non-uniform corrosion of reinforcement

Corrosion Science 89 (2014) 189–202

Contents lists available at ScienceDirect

Corrosion Science journal homepage: www.elsevier.com/locate/corsci

Modeling the cracking of cover concrete due to non-uniform corrosion of reinforcement Xiuli Du a, Liu Jin a,b,⇑, Renbo Zhang a a b

Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing, China Department of Civil Engineering, Tsinghua University, Beijing, China

a r t i c l e

i n f o

Article history: Received 9 March 2014 Accepted 18 August 2014 Available online 27 August 2014 Keywords: A. Concrete A. Steel B. Modelling studies C. Stress corrosion

a b s t r a c t In situations when external chloride penetration is the cause of depassivation, the corrosion process may start from the outer region of a rebar, which might expand non-uniformly. Therefore, the main objective of the present work is to explore the effect of non-uniform corrosion on cracking behavior of cover concrete. The influences of concrete heterogeneities and the porous layer generated at the rebar/concrete interface on the failure patterns and the corrosion level of cover concrete are considered. The random aggregate structures of concrete are built, and the concrete is regarded as a composite composed of three phases, i.e. the aggregate, mortar matrix, and the interfacial transition zones (ITZs). The plasticity damaged model is employed to describe the mechanical properties of the mortar matrix and the ITZs, and it is assumed that the aggregate is elastic. Non-uniform radial displacement with a half ellipse shape is adopted to describe the expansion distribution of the corrosion products. The failure pattern and the corrosion pressure of cover concrete, and the critical corrosion level when the cover concrete cracks due to non-uniform corrosion expansion are studied based on the meso-scale numerical method. The comparison of the simulation results with the available test results on the failure pattern of cover concrete shows fairly good agreement. Moreover, the influence of meso-structural heterogeneities is explored, and the cracking behavior obtained under non-uniform and uniform expansion conditions are compared. Finally, the influences of cover thickness, rebar diameter and the location of rebar (namely side-located rebar and corner-located rebar), on the failure pattern and the corrosion level are examined. Ó 2014 Elsevier Ltd. All rights reserved.

1. Introduction Corrosion of steel reinforcement is one of the main pathologies of reinforced concrete structures. It involves generation of an oxide layer on the bar surface, which results in a decrease of the net cross-sectional area, thus, reducing its strength and decreasing the overall safety of the structure [1]. When a significant reduction in area is reached, the volumetric expansion of the oxide induces internal pressure on the surrounding concrete, causing the cracking of cover concrete and, eventually, the full spalling of the cover [2,3]. Accordingly, corrosion-induced cracking of cover concrete is an important and essential issue in concrete structures because it directly affects not only durability, but also the service life of these engineering structures.

⇑ Corresponding author at: Key Laboratory of Urban Security and Disaster Engineering of Ministry of Education, Beijing University of Technology, Beijing, China. Tel.: +86 15811171018. E-mail address: [email protected] (L. Jin). http://dx.doi.org/10.1016/j.corsci.2014.08.025 0010-938X/Ó 2014 Elsevier Ltd. All rights reserved.

Much effort, involving experimental, analytical and numerical effort, has been carried out to investigate the failure pattern of cover concrete and determine the critical corrosion amount which causes the cracking of cover concrete. Experimental effort has been carried out to investigate the cracking behavior of cover concrete and the critical corrosion amount when the cover concrete cracks due to steel reinforcement corrosion, e.g. based on the electric corrosion test method [2,4–7] and the mechanical dilatation method [8,9]. These experimental studies are mainly focused on testing the critical amount of the rebar corrosion required for the cracking of cover concrete and developing corresponding empirical models. Therefore, the cracking mechanism and the cracking process of cover concrete cannot be captured by these experimental approaches. In the analytical studies, a number of analytical models [10–13] have also been developed for the study of the cracking of cover concrete induced by corrosion. These models have been widely used to evaluate the critical corrosion amount and predict the time to crack initiation in the concrete cover. However, most of these analytical models were based on the theory of elasticity and the

190

X. Du et al. / Corrosion Science 89 (2014) 189–202

effect of the residual strength of cracked concrete was not considered. Furthermore, all these analytical methods have assumed that the corrosion of reinforcing bar was uniform and thus the corresponding expansion pressure was uniform around a rebar. In situations when external chloride penetration is the cause of depassivation, the corrosion process may start from the outer region of a rebar, which might expand non-uniformly [14–16]. In such a case, a model assuming uniform expansion may be used only as a very rough approximation [17]. Except for these experimental and analytical efforts, a lot of studies have been carried out by using nonlinear fracture mechanics or finite element methods [1,15,18–22]. For instance, Jang and Oh [15] used the macro-scale finite element method to explore the effect of non-uniform corrosion on cracking behavior of concrete cover, and they have found that the pressures to cause cracking of concrete cover under non-uniform corrosion conditions are much smaller than those under uniform corrosion cases. Tran et al. [6] adopted a three-dimensional Rigid-Body-Spring (RBSM) method combined with a three-phase material corrosion–expansion model to simulate the crack propagation in cover concrete due to rebar corrosion. Recently, Ozbolt et al. [23] have employed the micro-plane model to study the damage in concrete caused by corrosion of reinforcement, and they also explored the influence of the anode–cathode regions on the corrosion behavior. Sanz et al. [1] developed a model – called expansive joint element to simulate the expansion of the oxide, and they used finite elements with an embedded adaptable cohesive crack to describe concrete cracking. All these simulation studies have made great contributions in analyzing the corrosion behavior of steel reinforcement and the relevant cover cracking. Some of the effort assumed that the corrosion process was uniform. Furthermore, in these investigations, the concrete was often assumed as homogeneous, without considering the influence of heterogeneities of concrete on the cracking behavior. As known, the failure behavior of concrete is closely associated with the heterogeneities of concrete meso- or micro-structure [16,24–26], and the concrete heterogeneities should be accounted for in the simulations. At the micro- or meso-scale, the macro-nonlinearity and the size effect of concrete can be described well. And therefore, a lot of micro-mechanical and meso-mechanical methods have been developed to study the failure behavior of concrete. For more details about these micro- or meso-mechanical methods, refers to [25]. In our previous work [27], considering the influence of concrete heterogeneities, a meso-scale mechanical method was developed to explore the cracking behavior of cover concrete due to corrosion of steel reinforcement. However in the work, the assumption of uniform corrosion was made, and the effect of the porous layer existed at the interface of reinforcing bar and surrounding concrete on corrosion level was ignored. As mentioned previously, the corrosion process should be non-uniform, and the uniform corrosion assumption should be a rough approximation. Therefore, the present work concentrates on the cracking behavior of the heterogeneous cover concrete induced by non-uniform corrosion expansion. Non-uniform corrosion behavior leads to a non-uniform distribution of expansion pressures. And the non-uniform distribution of expansion pressure may cause adverse effect for the cracking of concrete cover because higher pressures are concentrated at the outer region of rebar toward concrete cover [15,22]. This may cause higher tensile stress development and fast occurrence of cracks in concrete cover, which reduces time-to-cracking and eventually service life of concrete structures [15,28]. In light of this, both the heterogeneities and the porous layer existed at the steel/concrete interface were considered, and concrete was assumed to be a three-phase composite composed of aggregate, mortar matrix and the interfacial transition zones (i.e. ITZs) in the present study. A meso-scale mechanical model was

established for the investigation on the cracking of cover concrete. In the model, non-uniform radial displacement with a half ellipse shape was utilized to describe the expansion distribution of the corrosion products. A series of simulations have been carried out, and some influencing factors have also been studied. After this introduction, the paper is decomposed as follows. The next Section describes the expansion pattern of corrosion products (i.e. the half ellipse-shape distribution) and presents the corresponding corrosion level considering the effect of the porous layer generated at the steel/concrete interface. Section 3 presents the concrete meso-mechanical model for simulation of the cracking behavior of cover concrete due to non-uniform corrosion-induced expansion of steel reinforcement. In Section 4, the cracking process of cover concrete was simulated, and the present meso-mechanical model was verified with the available test observations. In Section 5, the numerical results obtained by macro-scale homogeneous and meso-scale heterogeneous models were compared. Uniform and non-uniform corrosion expansion behavior was also studied. Moreover, some influencing factors, including the thickness of cover, the diameter of reinforcing bar, and the location of steel bar on the failure patterns of cover concrete and the critical corrosion level when cover cracks were explored. Finally, Section 6 presents the conclusions.

2. Non-uniform corrosion-induced cover cracking analysis Steel in concrete is protected from corrosion by surface film of ferric oxide. The corrosion will start when the film is broken or depassivated [29]. The depassivation can be caused by reaching a threshold concentration of chloride ions in concrete near the steel surface [30]. The chloride-induced reinforcement corrosion process and cracking patterns are illustrated in Fig. 1 [19]. Once the chloride content at the surface of the steel rebar reaches a threshold value, the corrosion of steel reinforcement will be initiated (see Fig. 1(a)). The corrosion products expand into the porous transition zone between the steel rebar and the surrounding concrete, as presented in Fig. 1(b). As the corrosion progresses, the corrosion products accumulate in the steel/concrete interface and generate expansive pressure on the surrounding concrete, as shown in Fig. 1(c). The pressure arising from corrosion-induced expansion of the steel bar in concrete generally induces tensile stresses and strains in the surrounding concrete. The tensile strains in concrete increase as the corrosion of the steel bar progresses. Further increase of the tensile strains will cause cracking in the surrounding concrete and the cracking will also occur at the surface of the concrete cover during the expansion process [19], as depicted in Fig. 1(d). Therefore, to study the cracking behavior of concrete cover due to corrosion-induced expansion, one should analyze the quantitative relationship between the corrosion amount of the steel rebar and the internal pressure generated from corrosion. In the previous work [27], the cracking of cover concrete due to uniform corrosion-induced expansion was modeled and explored. And as mentioned in the Introduction part, the corrosion process may start from the outer region of a rebar, which might expand non-uniformly [14–16]. The assumption of uniform expansion should be an approximation. The non-uniform corrosion behavior of a reinforcing bar was therefore to be discussed in the present work. To develop the corrosion cracking model of cover concrete, the following four assumptions are made in this work: 1) Failure behavior of concrete is closely associated with mesostructure of concrete. Meso-scale heterogeneous concrete models were employed.

191

X. Du et al. / Corrosion Science 89 (2014) 189–202

(d)

(a)

(b)

(c)

Corrosion initiation

Free expansion of rust

Stress initiation

Concrete cracking

Fig. 1. Chloride-induced reinforcement corrosion and cracking patterns (Chen and Mahadevan [19]).

2) Corrosion process is spatially non-uniform around steel reinforcement, which causes a non-uniform radial expansive pressure at the interface of steel and concrete; a half ellipses-shape corrosion distribution curve was employed to describe the expansion behavior of corrosion products. 3) The stress and deformation generated within the surrounding concrete were induced only by the corrosion of steel reinforcement. Other influencing factors, such as freezing, thawing and dynamic loadings, were excluded in this work, though they may affect the processes of cover cracking. 4) There was a porous layer around the steel/concrete interface caused by the transition from cement paste to steel, entrapped/entrained air voids, and corrosion products diffusing into the capillary voids in the cement paste, etc. It should be noted that, corrosion products may penetrate into the corrosion-induced cracks [31], which was not considered in the present simulation model. 2.1. Expansion pattern of corrosion products Yuan and Ji’s [32] measurements indicate that the thickness distribution of the corrosion products layer has the shape of a half ellipse, and the measured data are presented in Fig. 2. Based on this assumption and the test results, the model for the distribution configuration of the corroded steel section before the cover cracking is described in Fig. 3. The expression of the model under the polar coordinate is given as follows [32]:

( uh ¼

ðRþu1 ÞðRþu2 Þ pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ffi  R 0 6 h 6 180 2 2 2 2 ðRþu1 Þ cos hþðRþu2 Þ sin h

180 6 h 6 360

u2

ð1Þ

in which R is the original radius of the steel bar; uh is the corroded thickness at h under polar coordinate; u1 is the maximum corroded thickness nearest the concrete cover (for the side located bar, i.e. the red line shown in Fig. 3(b)) or the maximum corroded thickness at the corner of the concrete specimen (for the corner located bar,

220.1 121.2

146.5 46.3

55.2 R

7.8

0

0

9.9 8.9

6.6

Measured surface Ellipse

Unit: µm

6.1

Fig. 2. The observations of steel surface corrosion features [32].

i.e. the two green lines shown in Fig. 3(b)); u2 is the corroded thickness at the side far away from concrete cover, u2 = (1/30–1/20)u1 [30]. The ratio of u2/u1 determines the shape of the half elliptical curve, that is to say, the elliptical shape varies with u2/u1 ratio. And the effect of u2/u1 ratio was to be studied in Section 4. 2.2. Corrosion level The corrosion products of a reinforcing bar in concrete induce pressure to the surrounding concrete. This expansion pressure induces tensile stresses in concrete around the reinforcing bar and the continuous increase of expansion pressure eventually causes cracking through concrete cover [9,11,15,20]. As known, not all corrosion products contribute to the build-up of stresses and initial cracking of the cover. Despite the apparent insolubility of the final corrosion products, the process of corrosion involves soluble species that can dissolve in the concrete pore solution and subsequently migrate or diffuse through the cement paste matrix away from the corroding steel. In other words, there is a porous transition zone between a reinforcing bar and the surrounding concrete [11,33]. The thickness d0 of this layer depends mainly on the porosity of concrete and compaction degree, and is assed to be in the range of 10–20 lm [13,34]. Herein this study, similar with that in Liu and Weyers’s work [11], the porous zone was assumed to be uniform and its thickness was set as 12.5 lm. Once the voids in the porous zone are occupied completely with rust, further rust accumulation will trigger expansion stress and then lead to the cracking in the surrounding concrete cover. When the thickness d0 of the interfacial layer is known, the volume Vil of the interfacial layer per unit length should be equal to 2pd0. At the stage of free expansion (see Fig. 1(b)) of corrosion products, the volume of the corroded part of steel bar per unit length is set as Vs1, and the volume Vr1 of the corrosion rust per unit length should be the sum of the volumes of the interfacial layer and the corroded part of steel bar, i.e.,

V r1 ¼ V il þ V s1 ¼ 2pd0 þ V s1

ð2Þ

After the voids in the porous zone are occupied completely with rust, tensile stresses initiate in the surrounding concrete (see Fig. 1(c) and (d)). At this stage, the corroded status is shown in Fig. 3. In Fig. 3(b), S1 denotes the expansion section area of the surrounding concrete, and S2 is the section area of the corroded part of the steel bar. The expansion volume Vc of the surrounding concrete of the steel bar per unit length can be evaluated as

V c ¼ S1  1 ¼

1 pRðu1 þ 3u2 Þ 2

ð3Þ

The volume of the corroded part of the steel bar per unit length is assumed to be Vs2. At this stage, the volume of the corrosion products Vr2, therefore, can be calculated as

V r2 ¼ V c þ V s2

ð4Þ

192

X. Du et al. / Corrosion Science 89 (2014) 189–202

Cover (side located bar)

u1

Products surface Original surface Corroded surface

u

u1

S2 The side near cover

R

Cover (corner located bar)

900

S1 R

r

The side far away from cover

Steel bar

Steel bar

u2

u2

(a)

(b)

Fig. 3. (a) Contour line model of rebar of non-uniform corrosion, and (b) the section of the reinforcement bar of non-uniform corrosion. (For interpretation of the references to colour in this figure legend, the reader is referred to the web version of this article.)

The total volume of the corrosion products per unit length should be Vr = Vr1 + Vr2, and the total volume of the corroded part of the steel bar becomes Vs = Vs1 + Vs2. The ratio of volume of expansive corrosion products to the volume of iron consumed in the corrosion process, i.e. the ratio of volume expansion of corrosion products, is set as n. Accordingly, the relationship between Vr and Vs can be expressed as the following form

V r ¼ nV s

ð5Þ

Substituting Eqs. (3) and (4) into Eq. (5), one can get the total corrosion volume Vs of steel bar per unit length as

Vs ¼

4pRd0 þ pRðu1 þ 3u2 Þ 2ðn  1Þ

ð6Þ

Let the rebar corrosion level q denote the percentage of steel mass loss Mloss to the original steel mass Ms per unit length, namely [13]

q¼

M loss  100% Ms

ð7Þ

Therefore, the corrosion level q can be drawn as follows

q¼

Vs

pR

2

 100% ¼

4d0 þ u1 þ 3u2  100% 2ðn  1Þ  R

ð8Þ

It is well known that corrosion of reinforcement results in the transformation of metallic iron to the corrosion products due to the process of oxidation resulting in an increase in volume which, depending on the level of oxidation, may be up to about 6.5 times the original iron volume [11,12]. This volume increase is believed to be the principal cause of the concrete expansion and the cover cracking. The volume expansion caused by corrosion creates strain only in concrete (i.e. strain in steel is neglected). This assumption was typically used by other researcher to predict the internal pressure caused by corrosion [11,34]. In addition, Lu et al.’s [13] studies indicate that the influence of material properties of corrosion products on critical corrosion level is negligible, and the corrosion products can be regarded as rigid. Accordingly, for the sake of simplicity, the deformation of corrosion products is neglected in the present analysis. Generally speaking, the magnitude of the ratio of volume expansion of corrosion products n varies from 2 to 4, similar with that used in [13,35,36], n is taken to be 3, i.e. n = 3. Hence, Eq. (8) becomes

q¼

Vs

p R2

 100% ¼

4d0 þ u1 þ 3u2  100% 4R

ð9Þ

3. Meso-scale computational model The global mechanical performances of concrete, involving the macroscopic mechanical properties and the failure patterns are closely associated with the meso- or micro-structure of concrete. Due to the complexity of concrete, the failure phenomena can be studied using numerical approaches. In the simulations of the failure process of concrete, the heterogeneity should be accounted for, and thus it calls for the adoption of a meso-scale or micro-scale mechanical model in which heterogeneity can be explicitly simulated [16,21,24,26,27,37]. At the meso-scale, three different phases can be distinguished—coarse aggregates particles, mortar matrix and the interfacial transition zones (i.e., the ITZs) between the former two phases. Each of the components has its own heterogeneous character. At the micro-scale, the mortar matrix, for instance, mainly consists of hardened cement paste, air voids and fine aggregate particles which are not explicitly considered as an individual phase in the model. If the micro-structure model of concrete is adopted, then the simulation will require an enormous amount of computation. For the consideration of computational amount, the meso-scale was thus often selected [24]. Lots of meso-mechanical methods, including the random particle model developed by Bazˇant et al. [38], the lattice model [21], the random aggregate model [22,24,27], and the meso-element equivalent method [25], etc, have been utilized to study the failure behavior of concrete. Each of these approaches has its own advantages and disadvantages, see more details in Kim and Abu Al-Rub [39] and Unger and Eckwardt [24]. The concrete random aggregate structure that can well describe the meso-structure of concrete has been successfully used to model the fracture of concrete at the meso-scale. Similar with the previous work [27], the random aggregate model was therefore adopted in the following simulations. 3.1. Random aggregate structure of concrete Similar with the work in [16,27,37], the aggregate particles were modeled as spheres with random spatial distribution enfolded by an interface layer of the ITZ. Due to great computational expense only two-dimensional simulations were performed in the present study. The popular ‘‘takeand-place’’ method [16,25–27,37] was employed to generate the coarse circular aggregate particles. The fine ones and other components (e.g. voids and micro-cracks) were assumed to be mixed up with mortar matrix establishing the matrix phase. Therefore this phase was considered as a homogeneous component at the

193

X. Du et al. / Corrosion Science 89 (2014) 189–202

c

Aggregate

150 mm

Interface

d

Mortar matrix y x

150 mm Fig. 4. Meso-scopic mechanical model for study of the interaction of reinforcement and heterogeneous concrete (Note: the radial displacement is applied on the hole to describe the corrosion-induced expansion behavior).

meso-scale. And the distribution of circular aggregate particles with different sizes employed in the concrete specimens was determined by the Fuller curve [27,38]. The aggregate particles were placed into the mortar matrix one by one at randomly determined positions in such a way that no overlapping with particles already placed. The aggregate fraction in concrete, if only particles with a diameter above 2 mm are considered, is around 30% and can rise up above 70%, if smaller inclusions are considered too. Herein the present study, the matrix was assumed to include all the particles smaller than 2 mm, which is similar with that of Unger and Eckwardt [24]. Fig. 4 depicts a typical schematic diagram of concrete mesostructure sized of 150 mm  150 mm, which consists of coarse aggregate particles, mortar matrix and the ITZs. The steel reinforcement was substituted by a ‘‘hole’’ where a reinforcing bar should occupy. The cover thickness is 30 mm and the circular ‘‘hole’’ is 16 mm (i.e. the diameter of the reinforcing bar). It should be noted that the mechanical behavior at the steel–concrete interface is very complicated [40,41], therefore a simplified model was utilized in this work. In the present study, to describe the interaction between reinforcing steel and the surrounding concrete, the prescribed radial displacement generated by the corrosion-induced expansion was applied on the hole. The size of the aggregate particles within concrete has a large range, and herein three equivalent aggregate sizes were utilized, i.e. 10 mm, 15 mm and 25 mm. According to the Fuller’s curve, 56 circular equivalent aggregate particles in the class of 10– 25 mm are placed into the specimen. The aggregate content is about 45%. In Fig. 4, the green areas mean the aggregate phase, the red1 thin-layer zones are the ITZ phase and the grey zones denote the matrix phase. It is in this manner that different mechanical properties are assigned to different phases. Because the ITZs’ thickness in concrete surrounding aggregates is tiny (i.e. the real ITZ thickness is between 30 lm and 50 lm), it is difficult to take them into account explicitly in the modeling of concrete [42]. Similar with those in Šavija et al.’s work [16] and the previous work [27], the thickness of the ITZ phase was set as 1 mm. The meshing of the computation model was done with four node solid elements with a linear strain field. To have a better understanding of cover concrete fracture, the average mesh size was 1 mm. This is consistent with that in the work from Hentz and co-workers [43]. 3.2. Material model for concrete meso-components Because of the heterogeneity of concrete, under static loadings, the cracks generated within concrete prefer to propagate along the weaker zones (e.g. the ITZs and mortar matrix zones), and no 1 For interpretation of color in Fig. 4, the reader is referred to the web version of this article.

damage generates in the aggregate phase [24,27,37] for normal concrete. This is because that the tensile and compressive strengths of aggregate are much larger than those of mortar matrix and the ITZs. Therefore, the aggregate was set as elastic in the present simulations. So far, many theoretical approaches have been proposed to describe the mechanical behavior of concrete, including the linear elastic fracture mechanics, the plasticity theory, the elastic-damaged theory and the plasticity-damaged theory [27,39]. Linear elastic fracture mechanics approaches can hardly be used on concrete, since fracture pattern usually consists of a main crack with branches, bridges, secondly cracks and micro-cracks. The plasticity theory failed to address the degradation of the material stiffness due to micro-cracking, and the continuum damage mechanics with elasticity also could not describe the irreversible deformations and the inelastic volumetric expansion in compression. Since both the stiffness degradation behavior and irreversible plastic deformations are contributing to the nonlinear behavior of concrete, a reasonable constitutive model should characterize the two distinct mechanisms. In recent years, many coupled plasticity-damage models have been proposed to describe the mechanical behavior of concrete, such as in the work [39,44,45]. The concrete damaged plasticity model proposed first by Lubliner et al. [44] and then developed later by Lee and Fenves [45] can well describe the mechanical behavior of concrete subjected to arbitrary loading conditions, including cyclic and dynamic loadings, and therefore it has been widely used in the study of the mechanical behavior of concrete, e.g. in the simulations of the dynamic failure behavior [26] and static failure behavior of concrete [27]. The model takes into consideration the degradation of the elastic stiffness induced by plastic straining both in tension and compression, as well as the triaxial effect. The model is a continuum, plasticity based model for concrete and it assumes that the main failure mechanisms of the concrete are cracking in tension and crushing in compression. For more details about the constitutive model, refers to the work [26,45]. Here, the mechanical model was employed to describe the mechanical properties of the mortar matrix and the ITZ in the present work. Similar with the previous work [27], to avoid or release unreasonable mesh sensitive results, the tensile post-failure behavior was given in terms of a fracture energy cracking criterion by specifying a stress–displacement curve instead of a stress– strain curve. It is to be noted that, for high performance concrete, aggregates and mortar matrix strengths are no significantly different, and cracks can propagate through aggregates in this concrete. For this case, the damaged model can be used to describe the mechanical behavior of aggregate. The major mechanical parameters of the three meso-components used in the simulations, namely the aggregate phase, the matrix phase and the ITZ phase, are given in Table 1 [27]. It is to

194

X. Du et al. / Corrosion Science 89 (2014) 189–202

Table 1 Mechanical parameters for the meso-constituents [27]. Constituents

Elastic modulus (GPa)

Poisson’s ratio

Tensile strength (MPa)

Aggregate Mortar matrix ITZ

70 30 25

0.16 0.2 0.22

– 1.43 1.2

be noted that, according to these given parameters, one can get the macroscopic uniaxial tensile strength ft of concrete, i.e. ft = 1.50 MPa (see the details in Appendix).

3.3. Loading and boundary conditions In general, the internal uniform pressure due to corrosioninduced expansion was applied to the radial direction at the outer

(a) u1= u2

(b) u1= 2u2

(c) u1= 5u2

(d) u1= 10u2

(e) u1= 20u2

(f) u1= 30u2

Fig. 5. Failure modes of the concrete cover under different u1/u2 (Note: the red areas in the contours mean the heavily damaged regions). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

(a) u1=8.60µm

(b) u1=12.74µm

(c) u1=20.15µm

(d) u1=31.26µm

(e) u1=53.55µm

(f) u1=149.90µm

Fig. 6. Cracking process of the concrete specimen with a cover thickness of 30 mm and a rebar diameter of 16 mm.

195

X. Du et al. / Corrosion Science 89 (2014) 189–202

(a)

test result [6]

Simulation result [6]

test result [46]

Simulation result [47]

(b)

(c)

test result [33]

Simulation result [35] Fig. 7. Comparison of the present meso-scale simulation results and the available test observations as well as others’ macro-scale simulation results, (a) Tran et al.’s [6] test results and simulation results (for side located rebar), (b) Fischer’s [46] test observations and simulation results (for corner located rebar), and (c) Vu et al.’s [33] test results and simulation results.

Homogeneous model

Middle side bar

Uniform corrosion

Non-uniform corrosion

Heterogeneous model

Uniform corrosion

Non-uniform corrosion

Corner side bar

Note: c = 30 mm, d = 16 mm Fig. 8. Failure patterns of the four concrete samples (Note: the upper row shows the failure patterns of homogeneous cover concrete, and the lower row presents the failure patterns of heterogeneous cover concrete).

surface line of the reinforcing bars, e.g. in [13,16]. To capture the softening behavior of concrete, the displacement-control method was adopted in this work. The prescribed radial displacement of

Eq. (1) was applied around the ‘‘hole’’, which was positioned where the steel reinforcement should be. An incremental iterative algorithm was selected in the present analyses, and the

196

X. Du et al. / Corrosion Science 89 (2014) 189–202

d = 16 mm

d = 20 mm

d = 25 mm

(a)

c = 30 mm

(b)

c = 40 mm

(c) c = 50 mm

Damage dt 0

0.95

Fig. 9. Failure patterns of the cover concrete with different cover thicknesses (i.e. 30 mm, 40 mm and 50 mm) and diameters of reinforcement (i.e. 16 mm, 20 mm and 25 mm) (for side located rebar).

Newton–Raphson method was utilized for the solution of nonlinear equations. The detailed description of the simulations is illustrated in the following Sections. 4. Simulation on cracking behavior of cover concrete 4.1. Non-uniform corrosion distribution As mentioned in Section 2.1, the ratio of u1/u2 determines the shape of the half ellipse. Therefore, the effect of u1/u2 ratio was examined. The cracking behavior of a concrete specimen subjected to non-uniform corrosion-induced expansion was studied. Under different u1/u2 ratios, the final cracking patterns of the concrete sample with a cover thickness of 30 mm and a rebar diameter of 16 mm are shown in Fig. 5. It can be seen from Fig. 5 that, the failure patterns of the specimen with the u1/u2 ratio less than 5 are quite distinct from those with the u1/u2 ratio larger than 5. When u1 is equal to u2, i.e. u1/u2 = 1 (Fig. 5(a)), the corrosion process is uniform, and the obtained distribution of cracking is different from the others obtained from non-uniform corrosion. Moreover, from Fig. 5(d)–(f) one can find that the three failure patterns of the cover concrete are very close to each other. In other words, the effect of u1/u2 ratio on the failure patterns can be ignored when u1/u2 > 10. In the following simulations on the non-uniform corrosion behavior, the u1/u2 = 30 was thus used. It is to be noted that, it requires about 50 min for the computations on a standard PC (Intel Core2 Duo CPU, 2.93 GHz, memory 2G) running 32-Bit Windows 7. 4.2. Cracking process of cover concrete The failure (cracking) process of the concrete specimen with the cover thickness of 30 mm and rebar diameter of 16 mm, subjected

to non-uniform expansion induced by steel corrosion, was presented in Fig. 6. It is obvious that the cracking of concrete cover is the direct consequence of an increasing radial expansion of corroded reinforcement. As can be seen in Fig. 6, when the tensile stress just exceeds the tensile strength of the meso-components, the damage initiates at the left and right sides of the rebar where the stress concentration behavior occurs, for example when u1 gets to 8.60 lm (see Fig. 6a). As the radial displacement increases, the corrosion pressure increases and the damaged areas expand and propagate gradually, and micro-cracks propagate from the rebar and the surface deformation occurs due to expansion pressure. When the radial displacement u1 gets to 12.74 lm (see Fig. 6b), the damage generates at the cover surface, and two major cracks propagate along the left and right sides. Finally, the cracks almost propagate through the concrete protective layer and the cover concrete flakes when the radial displacement u1 reaches to 149.9 lm, as indicated in Fig. 6(f).

4.3. Comparison of numerical and experimental results Fig. 7 compares the present meso-scale simulation results with the available test observations and the macro-scale simulation results obtained from others. Fig. 7(a) compares the meso-scale failure pattern of a concrete sample with a side located rebar with Tran et al.’s [6] test observations and their numerical results, and Fig. 7(b) shows the present meso-scale failure pattern of a concrete sample with a corner located rebar and Fischer’s [46] test results as well as Ozbolt et al.’s [47] simulation results. In addition, Fig. 7(c) illustrates the comparison of the present simulation results, Vu et al.’s [33] test observations and Chernin and Val’s [35] macroscale simulation results.

197

X. Du et al. / Corrosion Science 89 (2014) 189–202

Pressure [MPa]

3 0 -3

16 mm

25 mm

20 mm

0 3 6

u1= 1.66µm

u1=20.15µm

u1= 1.66µm

u1=23.65µm

u1= 6.44µm

u1= 35.15µm

u1=12.74µm

u1=31.26µm

u1=12.87µm

u1=43.93µm

u1= 19.25µm

u1= 59.65µm

(a) Concrete cover thickness c =30 mm 6

Pressure [MPa]

3 0 -3

20 mm

16 mm

25 mm

0 3 6

u1= 3.79µm

u1= 38.94µm

u1= 6.44µm

u1= 38.31µm

u1= 6.44µm

u1= 66.75µm

u1= 19.44µm

u1=36.36µm

u1= 18.41µm

u1= 65.75µm

u1=19.28µm

u1=60.25µm

(b) Concrete cover thickness c =40mm 9

Pressure [MPa]

6 3 0 -3

20 mm

16 mm

25 mm

0 3 6 9

u1= 7.16µm

u1=38.73µm

u1= 6.44µm

u1=36.51µm

u1= 6.74µm

u1=36.53µm

u1=20.20µm

u1=60.36µm

u1=18.81µm

u1=63.40µm

u1=19.92µm

u1=63.70µm

(c) Concrete cover thickness c =50mm Fig. 10. Distribution of the pressure induced by rebar corrosion with different cover thicknesses and different rebar diameters (for side located rebar).

As can be seen from Fig. 7, the failure patterns obtained by the present meso-scale numerical method have good agreements with the available experimental results. Moreover, compared with the macro-scale simulation results obtained from Tran et al’s [6], Ozbolt et al.’s [47] and Chernin and Val’s [35] work, the meso-scale failure patterns in this study can well describe the tortuosity of crack propagation within concrete. This illustrates the reliability and rationality of the present numerical approach. The verified meso-scale mechanical analysis method was employed to explore the effects of some influencing parameters in the next Section. It should be noted here that, an error analysis was not carried out to quantify the agreement between the results obtained from the present model and from experimental studies. This is because, the mechanical parameters of concrete, especially the tensile/

compressive strengths of concrete used in the present study which are exactly the same as those in the previous work [27], are distinct with those in the available experimental work. In reality, if the selected global mechanical parameters of concrete in the simulations are exactly the same as those in the experiments, then one can easily have an error analysis. This should be a deficiency, which should be improved in future work.

5. Parametric study and discussion Eight groups of simulation results (i.e. the failure patterns), involving the results obtained from the macro-scale model with uniform and non-uniform corrosion expansions, as well as the meso-scale model with uniform and non-uniform expansions, are

198

X. Du et al. / Corrosion Science 89 (2014) 189–202

d = 16 mm

(a)

d = 20 mm

d = 25 mm

c1 c2 c1 = c2 = 30 mm

(b)

c1 = c2 = 40 mm

(c)

c1 = c2 = 50 mm Damage dt 0

0.95

Fig. 11. Failure patterns of the cover concrete with different cover thicknesses (i.e. 30 mm, 40 mm and 50 mm) and diameters of reinforcement (i.e. 16 mm, 20 mm and 25 mm) (for corner located rebar).

presented in Fig. 8. Herein, the macro-scale model means that the concrete used is homogeneous, and the meso-scale model means that the concrete is assumed to be a composite composed of aggregate particles, mortar matrix and the ITZs. In the macro-scale model, the global mechanical properties of concrete were used, including the elastic modulus and tensile strength, etc., as presented in Appendix. And in the meso-scale heterogeneous model of concrete, the mechanical properties of concrete meso-components were adopted. As known, the cover thickness, rebar diameter, the concrete strength and the location of rebar, are the four major influencing parameters that affect the cracking behavior and the corrosion behavior of cover concrete. The effect of concrete strength has been studied and discussed in the work [15,27,36], and consistent results have been obtained. It is found that the cracking pressure of concrete cover increases as concrete strength increases, and the effect of concrete strength on the failure pattern of cover concrete can be ignored. Herein the present work, therefore, the effects of cover thickness, rebar diameter and rebar location were modeled and explored. Figs. 9–12 show the simulation results on the cracking patterns of cover concrete and the distribution of the corrosion pressures generated on the surrounding concrete. Figs. 9 and 11 present the final cracking patterns of the cover concrete with different cover thicknesses c and rebar diameters d, for side located bar and corner located bar, respectively. In the simulations, the cover thicknesses of 30 mm, 40 mm and 50 mm were selected, and the rebar diameters of 16 mm, 20 mm and 25 mm were utilized. The obtained corrosion expansion pressures for side located rebar and corner located rebar are plotted in Figs. 10 and 12, respectively. It should be noted that the ‘‘corrosion pressure’’ (in MPa) in the whole context is the

averaged pressure acted on the corrosion products. Table 2 presents the critical corrosion level of the rebar when cover concrete cracks. The detailed analyses and discussion on these simulation results are shown below. 5.1. Comparison of macro-scale and meso-scale models In Fig. 8, the upper row of the contours shows the failure patterns of the homogeneous cover concrete, and the lower row presents the failure patterns of the heterogeneous cover concrete with uniform and non-uniform corrosion expansions. Moreover, the two different locations of the steel bar were considered, i.e. the side located bar and the corner located bar. One can see from Fig. 8 that the cracking patterns of the concrete cover with the side located rebar are significantly different from those with the corner located rebar. Moreover, one can see from Fig. 8 that, compared with the homogeneous model, the present meso-scale heterogeneous concrete model can describe the cracking behavior and failure pattern of cover concrete more realistically and can characterize the tortuosity of crack propagation more vividly. This illustrates the importance of concrete heterogeneities in the simulation of concrete fracture again. In the macro-scale computational models [15,18,20,35], the obtained critical corrosion levels, i.e. the corrosion levels of rebar when cover concrete cracks, are exactly the same when the ratios of cover thickness c and rebar diameter d are similar. That is to say, when a macro-mechanical model is employed, the obtained critical corrosion level will be only related to c/d ratio, and it is independent on pure c or d. However, some experimental results [9,48] shows that the critical corrosion level and the critical corrosion

199

X. Du et al. / Corrosion Science 89 (2014) 189–202

6

Pressure [MPa]

3 0 -3

25 mm

20 mm

16 mm

0 3 6

u1=14.01µm

u1=40.94µm

u1= 9.14µm

u1=42.10µm

u1= 9.14µm

u1=41.84µm

u1=21.99µm

u1=79.45µm

u1=21.43µm

u1=75.15µm

u1=20.36µm

u1=73.35µm

(a) Concrete cover thickness c =30mm 6

Pressure [MPa]

3 0 -3

20 mm

16 mm

25 mm

0 3 6

u1= 7.70µm

u1=39.55µm

u1=20.04µm

u1=71.25µm

u1= 9.14µm

u1=39.93µm

u1= 9.14µm

u1=41.07µm

u1=20.15µm

u1=67.60µm

u1=22.24µm

u1=68.55µm

(b) Concrete cover thickness c =40mm 9

Pressure [MPa]

6 3 0 -3

25 mm

20 mm

16 mm

0 3 6 9

u1=10.28µm

u1=40.47µm

u1=10.52µm

u1=40.03µm

u1=11.23µm

u1=40.12µm

u1=20.17µm

u1=65.05µm

u1=20.05µm

u1=61.00µm

u1=20.00µm

u1=64.30µm

(c) Concrete cover thickness c =50mm Fig. 12. Distribution of the pressure induced by rebar corrosion with different cover thicknesses and different rebar diameters (for corner located rebar).

pressure may also be dependent on pure c, c/d, d or else. Therefore, the macro-scale homogeneous model cannot accurately describe the effects of cover thickness and rebar diameter, because it does not consider the effect of the meso-/micro-structural heterogeneities of concrete. Unlike the homogeneous model, the heterogeneous model considering the non-uniform characteristics should be able to model the complex cracking behavior of cover concrete. From Table 2, it can be found that under a same c/d ratio (e.g. c/d = 2, 2.5) but different magnitudes of c and d (e.g. c1 = 40 mm, d1 = 20 mm and c2 = 50 mm, d2 = 25 mm), the obtained critical displacements are distinct, and the critical corrosion levels of the rebar when cover concrete cracks are also different. This indicates that the present heterogeneous concrete model can effectively describe the cracking behavior of concrete cover due to the corrosion-induced expansion.

5.2. Comparison of uniform and non-uniform corrosion expansions A numerical study of corrosion behavior was conducted by Jang and Oh [15], who concluded that the assumption of uniform corrosion may not lead to a conservative estimate for time to cracking. Herein, the comparison of the failure patterns obtained from uniform and non-uniform expansions was made. It can be found from Fig. 8 that the failure patterns obtained from the non-uniform corrosion-induced expansion are quite different from those from the uniform one. At large amounts of corrosion, there are inside cracks propagating from the rebar conforming to the behavior of uniform corrosion. When the corrosion process is assumed to be uniform, the corrosion expansion pressure acts in all direction around the rebar. Then, the expansion pressure under the rebar leads to some inside cracks. This situation is different from the Tran et al.’s [6]

200

X. Du et al. / Corrosion Science 89 (2014) 189–202

Table 2 Critical corrosion level of the rebar when cover concrete cracks. Cover thickness c (mm)

Rebar diameter d (mm)

c/d ratio

Side located rebar Critical displacement u1 (lm)

Corner located rebar Corrosion level q (%) Non-uniform

Uniform

Critical displacement u1 (lm)

Corrosion level q (%) Non-uniform

Uniform

30

16 20 25

1.875 1.5 1.2

17.05 17.33 15.79

10.74 8.63 6.74

11.61*

14.26 12.87 12.38

10.26 8.02 06.36

11.08*

40

16 20 25

2.5 2 1.6

19.02 17.71 17.43

11.08 8.69 6.92

14.57*

18.41 15.63 16.85

10.98 8.40 6.85

13.45*

50

16 20 25

3.125 2.5 2

22.67 23.27 21.55

11.71 9.45 7.37

22.87 21.89 17.49

11.74 9.26 6.92

Note: data with ‘‘*’’ are quoted from the previous work [27].

experiments for local corrosion (see Fig. 7(a)). The failure patterns of the concrete specimen subjected to non-uniform corrosioninduced expansion are very close to Tran et al.’s [6] test observations. In addition, the present simulation results obtained from the non-uniform corrosion-induced expansion are very similar to those from Šavija et al. [16]. In Table 2, a comparison of the critical corrosion levels obtained from the present non-uniform corrosion model and the previous uniform corrosion model [27] is made. The data with ‘‘*’’ in Table 2 are quoted from Du and Jin [27]. It can be found from Table 2 that the critical corrosion level obtained from the non-uniform corrosion is less than that obtained from the uniform corrosion. This means that the cracking of concrete cover due to corrosion of steel bar occurs much earlier when corrosion is localized at the outer region of rebar. This is the case of usual pitting corrosion occurring in actual concrete under sea environments [15]. In summary, the non-uniform corrosion model can better simulate crack patterns in comparison with the experimental results. 5.3. Effect of rebar diameter From Figs. 9 and 11, it can be clearly seen that the obtained failure patterns of cover concrete with different rebar diameters are very similar as the cover thickness keeps the same. However, with the increase of rebar diameter, the maximum corrosion pressure generated on the corrosion products decreases obviously, as can be concluded from the corrosion pressure distributions presented in Figs. 10 and 12. Furthermore, it can be seen from Table 2 that the critical corrosion level decreases as the steel bar diameter increases, for both of the cases with side located bar or corner located bar. This is because, for the same concrete cover thickness, the same steel loss of the larger steel bar diameter produces larger strain energy in the concrete. Consequently, the larger the rebar diameter is, the more easily the concrete cover cracks. In a word, compared with the rebar with a smaller diameter, the steel bar of larger diameter makes concrete cover crack more easily. 5.4. Effect of cover thickness To examine the effect of concrete cover thickness on the cracking behavior and the critical corrosion level, three cover thicknesses were selected, i.e. 30 mm, 40 mm and 50 mm. It can be seen from Figs. 9 and 11 that the failure patterns become more and more complicated as cover thickness increases. The number of the cracking paths increases obviously with increasing the cover

thickness. And the larger the cover thickness is, the larger the spalling region is. From the corrosion pressure curves plotted in Figs. 10 and 12, one can see that the larger the cover thickness is, the larger the expansive pressure is. And it can be noted from Table 2 that the larger critical radial displacement leads to a larger critical corrosion level. This is because that the thicker the concrete cover is, the more strain energy will be dissipated in the cracking process of concrete. And thus the critical corrosion level increases as cover thickness increases. This is consistent with that in [34].

5.5. Effect of rebar location The cracking in cover concrete due to non-uniform corrosioninduced expansion of steel reinforcement placed at the corner region of concrete was studied and presented in Fig. 11. In Fig. 11, the cover thickness c2 means the distance from the surface of reinforcing steel to the right surface of concrete, and c1 means the distance from the surface of reinforcing steel to the top surface of concrete. Comparing Fig. 11 with Fig. 9, one can conclude that the failure pattern of cover concrete with corner located rebar has a great difference with that of cover concrete with side located rebar. Comparing Fig. 12 with Fig. 10, one can find that the two corrosion pressure distributions are obviously distinct. One can see from Table 2 that the value of the critical corrosion levels for the specimens with corner located rebar is less than that with side located rebar. This means that the cracking of concrete cover with corner located rebar occurs much earlier than that with side located rebar. The obtained results are consistent with those of Jang and Oh [15].

6. Conclusions Considering the concrete heterogeneities and assuming the non-uniform corrosion-induced expansion, the cracking behavior of cover concrete due to corrosion-induced expansion of steel reinforcement was studied by using the meso-scale numerical method. In the two-dimensional simulations, concrete was assumed to be a three-phase composite composed of aggregate, mortar matrix and the ITZs. Non-uniform radial displacement with a half ellipse shape was adopted to describe the expansion distribution of the corrosion products. The present numerical method was verified with the available test observations. And some influencing parameters were investigated. Some conclusions can be drawn from the simulation results, as given below.

X. Du et al. / Corrosion Science 89 (2014) 189–202

1) Compared with macro-scale homogeneous model, the present meso-scale heterogeneous model can describe the cracking and failure pattern of cover concrete more realistically. Moreover, it can reflect the corrosion-induced expansion behavior more rationality. Therefore, in the simulation of failure of cover concrete, the heterogeneities should be considered. 2) The failure patterns obtained from the non-uniform corrosion-induced expansion are quite different from those obtained from the uniform one. And the cracking of concrete cover due to rebar corrosion-induced expansion occurs much earlier when corrosion is localized at the outer region of rebar. Simple assumption of uniform corrosion may lead to unconservative estimation for service life. 3) The critical corrosion level decreases as the steel bar diameter increases. The larger the rebar diameter is, the more easily the concrete cover cracks. However, the effect of bar diameter on the cracking pattern of cover concrete is found to be rather small. Compared with the rebar with a smaller diameter, the steel bar of larger diameter makes concrete cover crack more easily. 4) The larger the cover thickness is, the larger the spalling region is. The larger the cover thickness is, the larger the expansive pressure is. Larger cover thickness leads to a larger critical corrosion level. 5) The cracking of concrete cover with corner located rebar occurs much earlier than that with side located rebar. 6) Good agreement between the present meso-scale numerical results and the available test results is obtained.

201

computation amount. In addition, for high performance concrete, the tensile/compressive strengths of aggregates are no significantly different from those of mortar matrix. For this case, cracks can run through paste and aggregates in the concrete. And therefore, other constitutive model that can describe the damage or cracking behavior should be used for aggregates. Acknowledgments This research was supported by the National Key Basic Research and Development Program of China (No. 2011CB013600), the major program of the National Natural Science Foundation of China (No. 91215301), the Municipal Natural Science Foundation of Beijing (No. 8100001) and the Doctoral Fund of Innovation (No. YB201207). The support is gratefully acknowledged. Appendix A The failure processes of the concrete random-aggregate structure subjected to uniaxial tensile loadings (at X- and Y-directions, respectively) were studied. In the simulations, the main mechanical parameters used were listed in Table 1. Fig. 13(a) and (b) present the two failure patterns of the concrete specimen at X and Y directions, respectively. And the corresponding macroscopic stress–strain relationship is plotted in Fig. 13(c). From Fig. 13(c) it can be noticed that the obtained macroscopic tensile strength of concrete is about 1.50 MPa. References

It is noted that, the present meso-scale model should have some limitations, because there are some assumptions existed in the developed model. For instance, the aggregate shape was set to be circular, the ITZ thickness used was larger than the actual size of interface due to the limits of computational efficiency, and the analysis model was two dimensional for the consideration of

hole

(a) Load at X-direction

(b) Load at Y-direction

Global tensile strength (MPa)

2.5 2.0 X 1.50 MPa

1.5 Y

1.0

Uniaxial tension

0.5

Loading at X direction Loading at Y direction

0.0 0.0

0.5

1.0

1.5

2.0

-4

Normal strain (x10 )

(c) Global tensile strength of the concrete specimen Fig. 13. The failure modes and the macroscopic stress–strain relationships of the concrete specimen under uniaxial tension.

[1] B. Sanz, J. Planas, J.M. Sancho, An experimental and numerical study of the pattern of cracking of concrete due to steel reinforcement corrosion, Eng. Fract. Mech. 114 (2013) 26–41. [2] C. Andrade, M. Alonso, F. Molina, Cover cracking as a function of bar corrosion: Part I – Experimental test, Mater. Struct. 26 (1993) 453–464. [3] M. Alonso, C. Andrade, J. Rodriguez, J. Diez, Factors controlling cracking of concrete affected by reinforcement corrosion, Mater. Struct. 31 (1997) 435– 441. [4] J.G. Cabrera, Deterioration of concrete due to reinforcement steel corrosion, Cem. Concr. Compos. 18 (1996) 47–59. [5] A.S. Al-Harthy, M.G. Stewart, J. Mullard, Concrete cover cracking caused by steel reinforcement corrosion, Mag. Concr. Res. 63 (2011) 655–667. [6] K.K. Tran, H. Nakamura, K. Kawamura, M. Kunieda, Analysis of crack propagation due to rebar corrosion using RBSM, Cem. Concr. Compos. 33 (2011) 906–917. [7] Y. Zhao, J. Dong, Y. Wu, H. Wang, X. Li, Q. Xu, Steel corrosion and corrosioninduced cracking recycled aggregate concrete, Corros. Sci. 85 (2014) 241–250. [8] M.L. Allan, Probability of corrosion induced cracking in reinforced concrete, Cem. Concr. Res. 25 (1995) 1179–1190. [9] S.J. Williamson, L.A. Clark, Pressure required to cause cover cracking of concrete due to reinforcement corrosion, Mag. Concr. Res. 52 (2000) 455–467. [10] Z.P. Bazˇant, Physical model for steel corrosion in sea structures, ASCE J. Struct. Div. 105 (1979) 1137–1153. [11] Y. Liu, R.E. Weyers, Modeling the time-to-corrosion cracking in chloride contaminated reinforced concrete structures, ACI Mater. J. 95 (1998) 675–681. [12] K. Bhargava, A.K. Ghosh, Y. Mori, et al., Modeling of time to corrosion-induced cover cracking in reinforced concrete structures, Cem. Concr. Res. 35 (2005) 2203–2218. [13] C.H. Lu, W.L. Jin, R.G. Liu, Reinforcement corrosion-induced cover cracking and its time prediction for reinforced concrete structures, Corros. Sci. 53 (2011) 1337–1347. [14] H. Wong, Y.X. Zhao, A.R. Karimi, N.R. Buenfeld, W.L. Jin, On the penetration of corrosion products from reinforcing steel into concrete due to chlorideinduced corrosion, Corros. Sci. 52 (2010) 2469–2480. [15] B.S. Jang, B.H. Oh, Effects of non-uniform corrosion on the cracking and service life of reinforced concrete structures, Cem. Concr. Res. 40 (2010) 1441–1450. [16] B. Šavija, M. Lukovic´, J. Pacheco, E. Schlangen, Cracking of the concrete cover due to reinforcement corrosion: a two dimensional lattice model study, Const. Build. Mater. 44 (2013) 626–638. [17] U. Angst, B. Elsener, A. Jamali, B. Adey, Concrete cover cracking owing to reinforcement corrosion – theoretical considerations and practical experience, Mater. Corros. 63 (2012) 1069–1077. [18] Y.G. Du, A.H.C. Chan, L.A. Clark, Finite element analysis of the effects of radial expansion of corroded reinforcement, Comp. Struct. 84 (2006) 917–929. [19] D. Chen, S. Mahadevan, Chloride-induced reinforcement corrosion and concrete cracking simulation, Cem. Concr. Compos. 30 (2008) 227–238.

202

X. Du et al. / Corrosion Science 89 (2014) 189–202

[20] K.H. Kim, S.Y. Jang, B.H. Oh, Modeling mechanical behavior of reinforced concrete due to corrosion of steel bar, ACI Mater. J. 107 (2010) 106–113. [21] P. Grassl, T. Davies, Lattice modelling of corrosion induced cracking and bond in reinforced concrete, Cem. Concr. Compos. 33 (2011) 918–924. [22] T. Pan, Y. Lu, Stochastic modeling of reinforced concrete cracking due to nonuniform corrosion: FEM-based cross-scale analysis, ASCE J. Mater. Civil Eng. 24 (2012) 698–706. [23] J. Ozbolt, F. Oršanic, G. Balabanic, M. Kušter, Modeling damage in concrete caused by corrosion of reinforcement: coupled 3D FE model, Int. J. Fract. 178 (2012) 233–244. [24] J.F. Unger, S. Eckardt, Multiscale modeling of concrete, Arch. Comput. Methods Eng. 18 (2011) 341–393. [25] X.L. Du, L. Jin, G.W. Ma, Meso-element equivalent method for the simulation of macro mechanical properties of concrete, Int. J. Damage Mech. 22 (2013) 617– 642. [26] X.L. Du, L. Jin, G.W. Ma, Numerical simulation of dynamic tensile-failure of concrete at meso-scale, Int. J. Impact Eng. 66 (2014) 5–17. [27] X.L. Du, L. Jin, Meso-scale numerical investigation on cracking of cover concrete induced by corrosion of reinforcing steel, Eng. Fail. Anal. 39 (2014) 21–33. [28] M. Maage, S. Helland, E. Poulsen, O. Vennesland, J.E. Carlsen, Service life prediction of existing concrete structures exposed to marine environment, ACI Mater. J. 93 (1996) 602–608. [29] J. Ozbolt, G. Balabanic, M. Kušter, 3D numerical modelling of steel corrosion in concrete structures, Corros. Sci. 53 (2011) 4166–4177. [30] G. Glass, N. Buenfeld, Chloride threshold levels for corrosion induced deterioration of steel in concrete, in: L.-O. Nilsson, J.O. Uivier (Eds.), Chloride Penetration into Concrete, RILEM, Saint-Remy-Chevreuse, 1995, pp. 429–440. [31] Y. Zhao, J. Lu, B. Hu, W. Jin, Crack shape and rust distribution in corrosioninduced cracking concrete, Corros. Sci. 55 (2012) 385–393. [32] Y.S. Yuan, Y.S. Ji, Modeling corroded section configuration of steel bar in concrete structure, Const. Build. Mater. 23 (2009) 2461–2466. [33] K. Vu, M.G. Stewart, J. Mullard, Corrosion-induced cracking: experimental data and predictive models, ACI Struct. J. 102 (2005) 719–726. [34] T.E. Maaddawy, K. Soudki, A model for prediction of time from corrosion initiation to corrosion cracking, Cem. Concr. Compos. 29 (2007) 168–175.

[35] L. Chernin, D.V. Val, K.Y. Volokh, Analytical modelling of concrete cover cracking caused by corrosion of reinforcement, Mater. Struct. 43 (2010) 543– 556. [36] Y.X. Zhao, J. Yu, W.L. Jin, Damage analysis and cracking model of reinforced concrete structures with rebar corrosion, Corros. Sci. 53 (2011) 3388–3397. [37] P. Wriggers, S.O. Moftah, Mesoscale models for concrete: homogenisation and damage behavior, Finite Elements Anal. Des. 42 (2006) 623–636. [38] Z.P. Bazˇant, M.R. Tabbara, M.T. Kazemi, et al., Random particle model for fracture of aggregate or fiber composites, ASCE J. Eng. Mech. 116 (1990) 1686– 1705. [39] S.M. Kim, R.K. Abu Al-Rub, Meso-scale computational modeling of the plasticdamage response of cementitious composites, Cem. Concr. Res. 41 (2011) 339– 358. [40] M.R. Ben Romdhane, F.-J. Ulm, Computational mechanics of the steel–concrete interface, Int. J. Numer. Anal. Meth. Geomech. 26 (2002) 99–120. [41] B. Richard, F. Ragueneau, C. Cremona, et al., A three-dimension steel/concrete interface model including corrosion effects, Eng. Fract. Mech. 77 (2010) 951– 973. [42] F. Grondin, M. Matallah, How to consider the interfacial transition zones in the finite element modelling of concrete?, Cem Concr. Res. 58 (2014) 67–75. [43] S. Hentz, F.V. Donz´e, L. Daudeville, Discrete element modeling of concrete submitted to dynamic loading at high strain rates, Comp. Struct. 82 (2004) 2509–2524. [44] J. Lubliner, J. Oliver, S. Oller, E. On´ate, A plastic-damage model for concrete, Int. J. Solids Struct. 25 (1989) 299–326. [45] J. Lee, G.L. Fenves, Plastic-damage model for cyclic loading of concrete structures, ASCE J. Eng. Mech. 124 (1998) 892–900. [46] C. Fischer, Beitrag zu den Auswirkungen der Bewehrungsstahlkorrosion auf den Verbund zwischen Stahl und Beton, Dissertation, Stuttgart, 2013. [47] J. Ozbolt, F. Orsanic, G. Balabanic, Modeling pull-out resistance of corroded reinforcement in concrete: coupled three-dimensional finite element model, Cem. Concr. Compos. 46 (2014) 41–55. [48] D.V. Val, L. Chernin, M.G. Stewart, Experimental and numerical investigation of corrosion-induced cover cracking in reinforced concrete structures, ASCE J. Struct. Eng. 135 (2009) 376–385.