Simulation of Coupled Corrosive Product Formation ...concrete.t.u-tokyo.ac.jp/research/intro/material/ACT2005-Vol3-No2... · K. Toongoenthong and K. Maekawa / Journal of Advanced

Journal of Advanced Concrete Technology Vol. 3, No. 2, 253-265, June 2005 / Copyright © 2005 Japan Concrete Institute 253

Scientific paper

Simulation of Coupled Corrosive Product Formation, Migration into Crack and Propagation in Reinforced Concrete Sections Kukrit Toongoenthong1 and Koichi Maekawa2

Received 13 September 2004, accepted 22 December 2004

Abstract This paper aims to numerically simulate corrosion induced cracking, its propagation over sections of reinforced con-crete members and the penetration of corrosive gel product into crack gaps. A coupled steel core and surrounding corro-sion product are mechanically represented by a fictitious growing composite, with which the corrosive cracking initia-tion and subsequent propagation are simulated by 2D nonlinear crack analysis. The injection of corrosive gels into evolving cracks is substantiated in cases where corrosive cracks stably propagate such as large covers and/or compara-tively small diameters of steel, and the coupled system of gel formation, migration and crack propagation is newly pre-sented. The simulation scheme was verified through RC sections subjected to accelerated corrosion by electric current with regard to crack patterns and critical corrosion rates when cracks reach the outer surface of members.

1. Introduction

Steel corrosion in reinforced concrete structures has been recognized as one of the major threats for build-ings and infrastructures in service. The safety and ser-viceability of corroded RC structures has been widely addressed by practicing engineers. In some durability designs, initiation of corrosion is chosen as a limit state of durability performance with regard to carbonation and chloride penetration. This is due to the current diffi-culty to predict the lifetime of mechanical damage of the concrete cover after corrosion initiation. Since the 1980s, theoretical proposals and experimental investiga-tions on steel corrosion in concrete have been intensely carried out and reported in the literatures (e.g., Bazant 1979; Browne 1980; Tuutti 1982; Ravindrarajah and Ong 1987). Here, not only experimental investigation but also a reliable numerical approach for predicting critical corrosion mass loss for concrete cover cracking is indispensable as an important tool for rational dura-bility assessment.

Morikawa et al. (1988) and Tsunomoto et al. (1990) conducted the analysis of critical weight loss for cover cracking due to reinforcement corrosion. In their analy-ses, the target specimens have smaller covers compared to the reinforcing bar diameters (C/D ratio 1.2 to 2.8). The analytical result of weight loss for cracking was close to the experimental one at the high corrosion rate. However, in the case of long-term corrosion, the analy-sis underestimated the critical weight loss compared to the reality. Molina et al. (1993) proposed the numerical

model and verified it against their own experimental results. In their analysis, not only the critical mass loss but also the numerically estimated crack width was computed, and the analytical results showed overestima-tion of corrosion weight loss of steel. The analysis of corroded RC specimens having a relatively large C/D ratio was conducted by Tsutsumi et al. (1998). Their model was verified by RC specimens having C/D of 2.31, 3.85 and 5.26. It was found that the weight loss was much underestimated, especially in the case of large C/D ratios. Pantazopoulou and Papoulia (2001) investi-gated simple analytical modelling for cover cracking. The verification of the model was made using many past experimental results. While a qualitative tendency was seen, it was quantitatively found that the computed weight loss was greatly underestimated compared to the experimental results. In their conclusions, an important comment was made on a more reliable approach and the need for experimental evidence on the penetration of corrosion products into cracks. As a matter of fact, this point raised by Pantazopoulou (2001) is the main target of this paper.

The non-linear analysis with multi-mechanics of cor-rosive substances and structural concrete has been pro-posed for simulation of post-corrosion cracked members (Toongoenthong and Maekawa 2004). Here, a core of non-corroded steel and its surrounding rust substances are mechanically represented by a comparable growing material, and the corrosion degree is dynamically asso-ciated with the numerical simulator based on thermo-dynamic multi-chemo physics (Maekawa et al. 1999), or non-destructive testing and inspection data in reality.

This approach has been verified chiefly in terms of structural safety assessment of RC members after the formation of section-penetrating cracks caused by cor-rosive volumetric expansion of steel (Toongoenthong and Maekawa 2004). However, it has not been dis-cussed nor verified on predicted duration from the ini-

1Design Engineer, Civil Engineering Division, Taisei Corporation, Japan. E-mail: [email protected] 2Professor, Department of Civil Engineering, TheUniversity of Tokyo, Japan.

254 K. Toongoenthong and K. Maekawa / Journal of Advanced Concrete Technology Vol. 3, No. 2, 253-265, 2005

tiation of corrosion to spalling of the cover concrete (internal corrosion crack reaches the surface of mem-bers). In this paper, the concept of the multi-mechanical approach with two phases of corroded reinforcement system and concrete will be extended to the limit state evaluation of cover spalling. The authors employ the concept of multi-mechanics composed of corrosion product, remaining non-corroded steel and surrounding structural concrete. Examined as stated above is its ca-pability for predicting the limit state of section damage, which is associated with spall-off of cover concrete for a wide range of cover dimensions relative to reinforcing bar diameters. The proposed computational approach is also extended to the nonlinear post-cracking mechanics of the concrete cover in RC sections. The effect of creep in concrete is also taken into account in the analysis via the reduction of effective concrete stiffness.

The migration of corrosive gel products into stable crack gaps is quantitatively highlighted in the coupled computational system of chemo-physics and mechanics with substances formation. In this paper, the authors will concentrate on this gel product migration into crack gaps as the original focus of this study.

2. Multi-mechanics of corrosive substances and concrete

For simulation of corrosion occurring in RC sections, the authors assume a two-phase mechanical system composed of corrosion product with non-corroded steel as one integrated phase and nearby concrete as another. After initiation of corrosion, the volumetric expansion of rust is constrained by the surrounding concrete. Ac-cordingly, compressive stress is introduced to the cor-roded reinforcement system, while tensile stress devel-ops in the surrounding area. Internal equilibrium and deformational compatibility may lead to pre- and post-cracking fields of stress and strain. Here, the combined corrosion product and non-corroded steel are modelled together as a growing compatible composite. For sur-rounding structural concrete, it was represented by smeared crack elements having coherent tension soften-ing after cracking with a finite element size and fracture energy of concrete (Okamura and Maekawa 1991; Maekawa et al. 2003).

Figure 1 shows a schematic diagram for the ex-panded geometry of corroded steel and concrete. The uniform corrosion accumulation around the reinforce-ment is simply assumed in this computation. Although actual corrosion in RC structures may not be uniformly distributed due to interfacial zones and bleeding beneath the bars, the hypothesis of this uniformity is thought to represent the critical index for corrosion estimation of cover cracking. When considering the possibility of the migration of the corrosion product into crack spaces or adjacent concrete voids, it is not only volume loss of corroded steel (denoted V loss) that causes the build-up of stress in the nearby concrete. Some amount of corrosion

product may possibly occupy interior spaces bordered by a pair of crack planes or bleeding gaps beneath steel bars before cracks reach the concrete surface. Let Veffec-

tive denote the effective volume loss per unit length from the parent steel that really causes the stress buildup, and Qcr, the rest amount of corrosion product that may be injected inside concrete cracks and micro-voids. Using the analogy of plastic theory, the effective volume loss that causes stresses and cracking to cover concrete can be expressed as,

crlosseffective QVV −= (1)

In order to estimate Vloss, we need some non-destructive testing or other means to identify this value for existing structures. For durability design, this total amount of volume loss, Vloss, can be obtained from ac-cumulated mass loss per unit surface area of steel (kg/m2), which is obtained from the simulation (Maekawa et al. 1999, 2003).

If there is no restraint to the corrosion product from the surrounding concrete, all effective corrosion sub-stances can expand freely with no confining stress. First, the corrosive volume loss from the mother steel is cal-culated by,

42DAV steeleffective πγγ ⋅=⋅= (2)

where Asteel = original steel area, D = original steel di-ameter (m) and γ = effective volume fraction loss (non-dimensional effective mass loss) of steel per unit length ranging from 0 to 1; 0 means no corrosion, while unity means complete loss of the reinforcement section.

Consequently, the mother steel exhibits a smaller di-ameter due to corrosion loss. The remaining diameter of un-corroded steel (denoted by Dcorroded-steel) yields,

γ−=− 1DD steelcorroded (3)

As the volume loss results in expansive corroded prod-uct, we have the final volume of corroded product (Vcorr) as,

42DVcorr αγπ= (4)

where α is defined as the coefficient of expansion of corroded substance (assumed α=2). Morikawa et al. (1988) conducted the X-ray diffraction of corrosion product in the acceleration test, and found that Fe3O4 is the main component of the corrosion product. Moreover Liu and Weyers (1998) provided information on the relative volumes of iron and its corrosion products in many forms. Among various iron oxides, Fe3O4 has vol-ume expansion of approximately 2 with regard to α. Consequently, α=2 is used in this paper as the first as-sumption. This coefficient will also be verified through the benchmark experiments of structural concrete in the following sections.

The corroded product, which effectively creates in-

K. Toongoenthong and K. Maekawa / Journal of Advanced Concrete Technology Vol. 3, No. 2, 253-265, 2005 255

ternal stress, consequently forms around the mother steel. The mean diameter of the corroded reinforcement system, which consists of mother steel and its corrosive product, is obtained in the case of stress-free and geo-metrically uniform expansion as,

( )11 −+= αγDD yercorrodedla (5)

Then, the reinforcement free-expansion strain (εs, free) of isotropy yields,

( ) 111, −−+= αγε frees (6)

The value stated above is the mean strain of the cor-roded system with respect to the initial diameter of rein-forcing bars, and it is a function of the specific level of corrosion ‘γ’. The stress induced by restraint of the sur-rounding concrete is computed by multiplying the total strain of the corroded system with its average stiffness representing multi-mechanical properties. The average stiffness of the corroded system, which is denoted by Es,

eq, is computed based upon the steel and corroded layers. As the corroded system is assumed to have uniform characteristics, the average stiffness of the corroded system can be expressed based upon the volume fraction as,

( )( )GE

E

s

eqsγαγ

αγ

+

−

−+=

111

, (7)

where Es = stiffness of steel (~200 GPa) and G = stiff-ness of corroded substance (first assumed as 7 GPa). Note that the linear constitutive law of the corroded substance is assumed in this formulation. Moreover, a wide range of corrosion substance stiffness has been proposed in the literature. A stiffness value as low as 0.02 GPa was suggested by Yoshioka and Yonezawa

(1982). Molina et al. (1993) used the value of 2 to 4 GPa in

their modelling and concluded insignificant influence on computed critical mass loss. Lundgren (2002) proposed the stress-level dependent stiffness of corrosion sub-stance, which can be up to 14 GPa under high confine-ment. Then, the assumed value of G is also the target of experimental verification in the following section.

By employing the computed stress-free expansion strain and equivalent stiffness of corroded reinforcement system, self-equilibrated stress is computed by the fol-lowing equivalent elasticity of the steel-rust composite elements as,

( ))()( ,, γεδεσ freesklkleqsijklij ED ⋅−⋅= (8)

where, Dijkl = constitutive model function for corroded steel continuum and δkl = Kroneker’s delta, which is defined on X-Y coordinates including the RC cross sec-tion concerned. In this paper, the isotropic linear matrix of Dijkl was assumed with Poisson’s ratio of 0.3 for the analyses of RC sections.

Morikawa et al (1988) pointed out the substantial ef-fect of concrete creep on corrosive cracking initiation. Consequently, in the two-dimensional crack analysis of this study, the effect of concrete creep is simply taken into account by factoring the stiffness of concrete with the creep coefficient. Comparisons with non-creep analysis will also be made. 3. Verification of computed critical corrosion for cover cracking

3.1 Equivalent growing material of rust-steel composite The galvanostatic corrosion experiments of 13 speci-mens were selected for experimental verification from the existing literature, as listed in Table 1. Expansion of

D

Free expansionof rust

Steel w/o corrosion

Confined expansionof RC system

where D: initial steel diameter, γ: mass fraction loss, α: rust expansion coefficient

L

L

D

Concrete

( )11 −+ αγD

D+δ

D+δ

Steel

x

y

Before corrosion After corrosion

Plain concrete

Corroded steel system

Fig. 1 Schematic representation for expansion of corroded steel in RC.


corrosion product around corroded steel is thought to exhibit uniformity under the accelerated corrosion with galvanostatic process. Thus, this electrically accelerated condition, which actually differs from the mode of real-time scale corrosion, is preferable for experimental veri-fication of the numerical modelling. Although there is a wide variety of specimen dimensions and reinforcement arrangement, the experimental common objective is to investigate the critical mass loss of steel during cover cracking and spalling. The experimentally measured corrosive mass loss is directly used for computing the equivalent stiffness of the corroded reinforcement sys-tem and stress-free expansion strain as explained in the previous section. By applying Eq. (8) for finite elements to represent steel bars, varying stiffness and free expan-sion, which are derived from the measured corrosion mass loss, are incorporated into 2D finite element analysis. The experimentally measured mass loss of steel at failure, computed critical mass loss and speci-men details are also given in Table 1. Table 2 summa-rizes the concrete material properties used in FEM analysis including experimentally obtained compressive strength of concrete, estimated tensile strength and ten-sile fracture energy of concrete.

An example of the finite element mesh used in the analysis is shown in Fig. 2. The FEM mesh is composed of three types of elements, i.e. expansive material ele-ment, plain concrete and interface elements. The elastic

element with the equivalent expansive material model is used to represent the corroded reinforcement system, as discussed in Section 2. The concrete domain is modelled with the smeared crack plain concrete elements having tension softening that is consistent with the finite ele-ment size and the fracture energy of concrete (Okamura and Maekawa 1991; Maekawa et al. 2003). The interfa-cial zone between corroded steel elements and contact-ing concrete is modelled by the joint interface elements to represent the stress release at separation and stiffness recovery at re-contact. For the normal direction, large stiffness in the closure mode and negligibly small stiff-ness in the opening mode are assumed. For the shear direction, the interface element has zero stiffness both in the closure and opening modes. Here, the authors con-ducted trial analysis to clarify the importance of interfa-cial zone modelling.

As shown in Fig. 3a, when the steel surface and con-crete were unrealistically assumed to be perfectly con-nected without joint elements, overestimation of crack-ing damage around reinforcing bars was observed, which is obviously different from the reality as observed in the experiment. On the other hand, a reasonable cracking damage condition can be obtained by more realistic modelling with interfacial zone as shown in Fig. 3b. To avoid the freedom of rigid body motions of the entire analysis domain, the central nodes of reinforcing bars were numerically fixed by displacement.

Table 1 Experimentally observed critical mass loss of steel and FEM prediction for concrete cover cracking.

Reference Min. Covers

(SpecimenLength)

in mm.

Steel Diameter

(mm)

C/D ratioExperimental Critical Mass

Loss(%)

Computed Critical Mass

Loss (%)

(Exp./FEM) Buffer

Capacity',

Q (%)

Creep Effect(Long Period

Corrosion)

Andrade et al. 20x20 (380) 20x30 (380) 30x30 (380)

16 16 16

1.25 1.25 1.875

0.37% 0.448% 0.45%

0.30% 0.42% 0.44%

1.23 1.07 1.02

- - -

× × ×

Al-Sulaimani

et al.

70x70 (150) 65x65 (150) 68x68 (150)

10 20 14

7.00 3.25 4.86

4.50% 1.80% 2.89%

4.45% 1.34% 2.18%

1.01 1.34 1.33

3.62% 0.88% 1.35%

○ ○ ○

Cabrera and

Ghoddoussi 69x69 (150) 12 5.75 2.25% 2.16% 1.04 1.34% ○

Morikawa et al.

30x30 (200) 30x30 (200) 50x50 (200) 50x50 200) 70x70 (200) 70x70 (200)

25 25 25 25 25 25

1.2 1.2 2 2

2.8 2.8

0.08% 0.20% 0.12% 0.30% 0.15% 0.30%

0.10% 0.18% 0.158% 0.28% 0.21% 0.29%

0.80 1.11 0.76 1.07 0.71 1.03

- - - - - -

× ○ × ○ × ○

Table 2 Material properties for FEM simulation analysis.

References

Concrete

Compressive strength from Experiment

Concrete Tensile

strength used in FEM (*)

Calculated

fracture energy (**)

Andrade et al. 30 MPa 2.58 MPa 0.066 Nmm/mm2

Al-Sulaimani et al. 30 MPa 2.58 MPa 0.066 Nmm/mm2

Cabrera and Ghoddoussi 56 MPa 3.91 MPa 0.101 Nmm/mm2

Morikawa et al. 35.6 MPa 2.89 MPa 0.074 Nmm/mm2

Tachibana et al. 35.6 MPa 2.89 MPa 0.074 Nmm/mm2

Cabrera 35 MPa 2.86 MPa 0.073 Nmm/mm2

Toongoenthong and Maekawa 34 MPa 2.80 MPa 0.072 Nmm/mm2

Matsuo et al. 47.8 MPa 3.52 MPa 0.091 Nmm/mm2

3/2)(58.0(*) ct ff ′=

( ) 7.00(**) cmcmf ffG α=

where ft = concrete tensile strength (kg/cm2)f’c = concrete compressive strength (kg/cm2)

where Gf = fracture energyα = a coefficient based on maximum

aggregate size (N⋅mm/mm2)fcm = cylindrical compressive strength (kg/cm2)fcm0 = 100


First, let us concentrate on the simulation of small cover cases of RC specimens. For the specimens with relatively small cover to the bar diameter, occurrence of cover cracking tends to be very rapid, hence provides no time for corrosion gel penetration into cracks or con-crete voids, i.e. Qcr ≈ 0 due to an insignificant amount of corrosion product injected into cracks, and consequently, we have Veffective=Vloss where Qcr =0 in Eq. (1). As a re-sult, the computed effective critical mass loss for small C/D cases corresponding to experimentally observed cover cracking reasonably matches the experimental measurement as shown in Table 1. This successful simulation for small cover cases confirmed the numeri-cal approach concept similarly to past research, where the amount of corrosion penetration was so small as to be numerically negligible. Figure 4(a) shows the nu-merical crack pattern of the fair simulation. The experi-mental and computed values of critical mass loss are close to each other without considering the amount of corrosion gel penetration. Here, the creep effect of con-crete is taken into account. Especially, it is somehow influential for the cases where crack initiation takes a long time, for example when there is a large cover depth relative to the bar diameter (Al-Sulaimani et al. 1990, Cabrera and Ghoddoussi 1992), or in the case of a long-term accelerated corrosion test (Morikawa et al. 1988). For the sake of simplicity, concrete mean elasticity was uniformly reduced using a creep coefficient of two. As Morikawa et al. (1988) pointed out, creep of concrete results in higher weigh loss associated with cover con-

crete cracking in experiments when a slow corrosion rate develops. By analysing the specimens of Morikawa et al. (1988), this creep effect is also captured in this simulation as discussed later (Fig. 5a).

However, in the case of large cover specimen (large C/D ratio), the only computed effective critical mass loss gives the underestimated result of critical total mass loss in general. Some finite element discretized meshes of large cover specimens are shown in Fig. 4. In these cases, finer sized finite elements are appropriate for discussing the stability of crack propagation in detail. A relatively smaller computed mass loss of 0.57% is ob-tained compared with the experimental measurement of critical total mass loss of 4.5%. The large discrepancy between the predictive and experimental values is thought to be caused by the migration of corrosive sub-stance into crack gaps (buffer effect), as discussed in the following section. 3.2 Penetration of corrosive product into cracks (Qcr ≠0) Penetration of corrosive product into crack gaps of con-crete is thought to occur because the corroded substance is a form of gel with some liquid characteristics. If this penetration is substantial, internal pressure around the corroded steel may be reduced and an internal gap formed by a pair of crack planes may work as a buffer or a reservoir. In the case of a large cover thickness rela-tive to the bar diameters, for example, crack propagation is slow and stable enough to migrate into crack gaps and

Typical FEM mesh for RC sectionin the analysis

③ Plain concrete element

σ

ε

Nonlinear Tension Modelof Crack Concrete

Tension Stiffening ‘c’ = f (mesh size, fracture energy)(Okamura and Maekawa 1991; Maekawa et al. 2003)

② Universal interface element① Expansive material elementWith very low resistance in shear direction

and high contact stiffness in normal closure

Normal direction

OpenLow stiffness

ClosureHigh stiffness

δ

σ

Shear direction

OpenLow stiffness

ClosureLow stiffness

γ

τ

)1(1exp −+=− αγε ansionfree

Fig. 2 FEM mesh and elements used in analysis of corrosive crack propagation.

b) Analysis with interface jointa) Analysis without interface joint

Does not match with reality! Consistent with reality

(by Cabreara 1996)

Experimental Crack Pattern

Fig. 3 Importance of proper interface zone modelling in FEM.


the crack gap spaces can be large (the assumption of Qcr ≈ 0 is no longer valid). This buffer capacity is defined as the volume surrounded by crack planes, which can be occupied by corrosion product at most. Hence, the total critical corrosion mass loss should include this buffer capacity. The created spaces surrounded by crack planes can be simply computed by integrating the computed crack width with the increment of crack ligament. The crack width is obtained by the relative displacements of nodes crossing a strain localization zone as a crack band. In this case, the volume change of the concrete contin-uum is negligibly small compared with the newly cre-ated spaces by cracking. Thus, we can simply estimate the total buffer capacity defined by Qcr in Eq. (1) as,

( )cr domainQ u n ds≈ ⋅∫ (9)

where u and n denote the displacement vector and the unit directional vector normal to the boundary surface of

analysis domain. The simultaneous coupling of bound-ary integral equation (9) with the finite element field ones is the first original challenge to compute corrosion crack propagation associated with rust gel injection into the analysis domain.

Figure 5 shows the computed crack ligament in the analysis with the effective corrosion mass loss. It can be seen that a relatively large cover thickness to bar diame-ter specimen of Alsulaimani et al. (1990) and Cabrera and Ghoddoussi (1992) exhibits stable crack growth before cracking reaches the surface of the cover con-crete. It takes time until complete crack growth to the surface occurs. This means that the rust gel may possi-bly migrate into cracks if given sufficient time. As a result, the corrosion product may occupy the cracking spaces around the corroded reinforcement before unsta-ble crack growth driven by a small fraction of consecu-tive corrosion. Here, this capacity of corrosion product accumulated in the crack gaps is denoted as buffer ca-

Ref.: Andrade et al.Cross-sectional 150x150DB 16Minimum Cover 30x30Exp. Critical Mass Loss 0.45%

Numerical Critical Mass loss = 0.44% Numerical Critical Mass loss = 0.3%



Numerical Critical Mass loss = 0.42% (a) Small cover to bar diameter cases.

ε > 1%ε > 0.5%ε > 0.35%ε > 0.2%ε > 0%ε > -5%

ε > 1%ε > 0.5%ε > 0.35%ε > 0.2%ε > 0%ε > -5%

Mass loss = 0.57%

Ref.: Al-Sulaimani et al.Cross-sectional 150x150DB 10Minimum Cover 70x70Exp. Critical Mass Loss 4.5%

Mass loss = 0.55%

Mass loss= 0.42%

With buffer capacity; Numerical Critical Mass Loss 4.45%(Calculated Buffer Capacity= 3.62%)

Without buffer capacity; Numerical critical mass loss 0.57%

(b) Large cover to bar diameter cases.

Fig. 4 Numerical crack propagation by FEM with and without buffer capacity of crack gaps.


pacity and its computation by 2D analysis is shown in Table 1. By incorporating the buffer capacity by Eq. (9) with the effective one into the total computed mass loss (see Eq. (1)), the analysis can give a fair estimation of critical mass loss at cover cracking that compares well with the experimental value as shown in Fig. 4.

In order to confirm the existence of the phenomenon of corrosive gel penetration into crack gaps, the authors also conducted a simple experiment on a corroded RC specimen with a relatively large cover. Figure 6a shows the specimen and experimental setup for the accelerated corrosion process. In the experiment, to verify the buffer phenomenon, the cathodic accelerated corrosion process was terminated before the corrosive crack plane could reach the surface of the concrete block. After that, the concrete was carefully chipped off to reveal the corro-sive gel penetration near the corroded reinforcement. The specimen was a cube measuring 20 cm on each side.

A single reinforcement 16 mm in diameter was embed-ded at the middle of the specimen (C/D ratio = 5.75). The migration of corrosion product away from the par-ent steel was experimentally confirmed as shown in Fig. 6b and Fig. 6c showing the propagation front of corro-sive gel inside the RC specimen. The experimental mass loss of corroded steel was measured as 0.77%.

Figure 6d shows the corrosive crack propagation by FEM based on the experimental measured mass loss as input taking into consideration the buffer capacity. The tested compressive strength of concrete was 35.1 MPa. In this case, computationally obtained effective mass loss was 0.65%. Thus, the mass loss of 0.12 (= 0.77 - 0.65%) stored in the crack gaps cannot be ignored com-pared with the total mass loss of steel. The propagation front of cracking is similar to the experimental one. The ratio of buffer to the effective mass loss is not constant, instead varying over a wide range according to the ge-

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.1 0.2 0.3 0.4

Corrosion % mass loss

Norm

aliz

ed

cra

ck

ligam

ent

Cover 3 cm without creep Cover 3 cm with creepCover 5 cm without creep Cover 5 cm with creepCover 7 cm without creep Cover 7 cm with creep

Increase of corrosion mass loss with

creep consideration

(a) Effect of creep.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0 0.2 0.4 0.6 0.8 1

Corrosion % mass loss

Norm

aliz

ed

cra

ck

ligam

ent

Andrade et al. Al-Sulaimani et al.(1)

Morikawa et al. (1) Al-Sulaimani et al.(2)

Morikawa et al. (2) Al-Sulaimani et al. (3)

Morikawa et al. (3) Cabrera and Ghoddoussi

Stable crack growthin large cover specimen

Buffer phenomenon

Concrete

Reinforcement

CrackSteel

Corrosion gel

Migration of corrosion productinto crack space

Surrounding concrete

Mig

ratio

n fr

ont o

f co

rros

ion

gel i

n re

al R

C s

peci

men

(b) Effect of crack gap volume. Fig. 5 Corrosive cracks propagation with effective corrosion degree.


ometry of the sections and the crack ligament length. On the other hand, instantaneous crack propagation to

the concrete cover is numerically observed in the case of relatively small cover depth to bar diameter speci-mens of Andrade et al. (1993) and Morikawa et al. (1988) as shown in Fig. 5a. Here, the corrosion product is hardly accumulated inside the crack spaces due to the immediate propagation of corrosive cracks to the sur-face and insufficient time for gel migration. For these cases (small C/D ratio), computation can give fair

agreement of critical mass loss only by considering the effective mass loss around the mother steel. In this case, the creep effect (linear creep coefficient 2) is seen to be substantial with regard to the corrosion rate and the cracking time. Figure 7a and Figure 7b summarize the comparison of the experimental and the computed total critical mass losses with and without consideration of the amount of corrosion gel penetration by nonlinear FE analysis. Consideration of the buffer effect is indispen-sable for mechanical damage simulation of cover con-

a. Specimen and experimental setup b. Partial corrosion gel penetration(cross section view)

c. Corrosion gel penetration front inside specimen d. Corrosive crack propagation in FEM Fig. 6 Observed corrosive gel penetration into crack gaps before failure of cover concrete in experiment and analysis.

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

0.00% 1.00% 2.00% 3.00% 4.00% 5.00%

Experiment Critical Mass Loss

FEM

Critica

l M

ass

Loss

0≅crQ

Without considering amount ofcorrosion gel penetration

0.00%

1.00%

2.00%

3.00%

4.00%

5.00%

0.00% 1.00% 2.00% 3.00% 4.00% 5.00%

Experiment Critical Mass Loss

FEM

Critical

Mas

s Los

s

0 4.1. =

av erageF EME xp

%83.18=CV

( )∫ ≠⋅≈domaincr dsnuQ 0

a. Underestimation of critical mass loss when amount of b. Reasonable estimation of critical mass loss when corrosion penetration is neglected. amount of corrosion penetration is considered. .

Fig. 7 Comparison of critical mass loss: experiment and FEM computation.


crete. As the capacity of Qcr is so large compared with the total corrosion mass, the effect of bleeding and inter-facial zones on the lifetime of cracking may tend to be less for thick concrete covers.

3.3 Analysis of RC section with multiple rein-forcements In section 3.2, the numerical critical mass loss at cover cracking was verified with the experimental results of RC specimens with embedded single bars. In this sec-tion, the results of further analysis performed on RC sections with multiple arranged reinforcing steels are discussed. This section also aims to verify simulation with regard to the corrosive crack propagation pattern when RC sections contain more than single steel. Here, the experimental results for RC beams obtained by Ta-chibana et al. (1990) are used for verification of corro-sive crack propagation. The targeted RC beam section is 150 mm by 200 mm with two D16 mm reinforcing bars. The beam length is 2 m. The minimum cover at the bot-tom is 30 mm. The compressive strength of concrete is reported as 35.6 MPa. The RC beam was corroded with a galvanostatic corrosion system employing the electric current connected to the reinforcement. The experimen-tal growth of cracks after a 15-day current application period is shown in Fig. 8.

In order to verify the applicability of the corrosive cracking propagation analysis, finite element mesh of the same dimensions is used and input with the experi-mental material properties of concrete and reinforce-

ment. The reinforcement is represented by the evolving equivalent elastic elements with isotropic expansion in each time step (Section 2). The computation terminates when the numerical crack propagation reaches the sur-face of the member as experimentally observed. Due to the relatively long corrosion period, the effect of creep is also considered in the analysis through the reduction of concrete stiffness by a creep coefficient of 2. Figure 8 shows the comparison of the experimental corrosive crack patterns with the numerical ones. The computed mass loss of reinforcement was also given by consider-ing the possible buffer capacity as Qcr, which reflects the possibility that corrosion product can occupy corro-sive cracking spaces. A little smaller analytical value of estimated reinforcement mass loss might be attributed to the fact that some of the corrosion product may be dis-charged outside just before or after cover cracking. This leakage of rust gel is not taken into account in the nu-merical simulation until cracking reaches the surface of concrete members.

Next, RC sections with multiple bars are discussed by employing the detailed characters on corrosive cracking by Cabrera (1996). Three selected specimens in total have the same cross sectional dimension of 100 mm by 300 mm with a length of 200 mm along steel bars. Each specimen has three steel bars embedded inside as shown in Fig. 9. The three different sizes of reinforcement di-ameters are the points of discussion, i.e. DB12, DB16 and DB20. The three specimens were corroded by the galvanostatic corrosion system with electric current. At the end of a 28-day current application period, the cor-rosive cracking patterns were recorded along with measurement of the reinforcement weight loss and maximum surface crack width, as shown in Fig. 10.

The numerical analysis of crack propagation was car-ried out for all experimental cases with consideration of creep. The computation is terminated when the com-puted numerical crack width reaches the experimentally observed value, under the assumption that the same cor-rosion damage level is thus reached. The reinforcement mass loss with corrosion product injected inside the surrounding crack spaces is estimated to correspond to the observed crack propagation. The comparison of the final crack pattern is shown in Fig. 10 along with the estimated numerical reinforcement mass loss. Here, the numerical estimated mass loss is slightly underestimated, which might be explained by the possible leakage of

20

44

300

100

Concrete Reinforcement

20

16

12

Steel Diameter (mm)

0.56mm6.76%A-20

0.56mm12.8%A-16

0.38mm17.73%A-12

Max. surface crack width

Experimental Mass Loss

Specimen

20

16

12

Steel Diameter (mm)

0.56mm6.76%A-20

0.56mm12.8%A-16

0.38mm17.73%A-12

Max. surface crack width

Experimental Mass Loss

Specimen

Fig. 9 RC specimens with multiple reinforcements by Cabrera (1996).

Growth of cracks after15-day Current application

period with 5% loss of reinforcement

Computed mass loss of 4.63%with creep & buffer capacity

150

200

DB16

Experiment FEManalysis

Fig. 8 Growth of corrosive cracking in experiment and FEM analysis.


corrosion product from inside the RC specimen just before or after cracking during the long-term corrosion process. The present numerical computation does not include this leakage of corrosion gel, as stated before.

Figure 11 shows the sequential process of corrosive crack evolution for case A-16 at different corrosion lev-els. The initial crack was observed to form at the bottom cover side and then propagate further to the main body of the specimen. At the beginning stage of crack propa-gation, the computational crack pattern is almost sym-metric. However, this symmetry is then lost, similarly to the experiment according to the evolution of corrosion. In the analysis, the predictor-corrector method was ap-plied to search for the unique stable equilibrium when cracking proceeds. Thus, the higher energy solution of perfect symmetry of crack damage was not computa-tionally selected but the convergence of solution reached the asymmetric solution of the lowest strain energy (Maekawa, et al. 2003).

The corroded RC beams tested by Toongoenthong and Maekawa (2004) were also analysed for verification. All three sections of the RC beams were corroded by the accelerated galvanostatic method. After a period of sev-eral weeks of corrosion, the damage in terms of corro-sive crack width was measured with measurement of mass loss of corroded steel after the structural loading test. Table 3 shows the recorded corrosive crack width information and the measured mass loss from the ex-

periment for all three beams. The RC sections were de-signed to be a single layer arrangement of reinforcement with relatively small cover depth and clearance between bars, as shown in Fig. 12. With this dense single-layer reinforcement arrangement, the corrosive horizontal crack then quickly propagated and resulted in a local-ized horizontal crack plane being observed in both the experimental and the numerical crack patterns, as shown in Fig. 12. The buffer capacity at the crack propagation to the surface of structural concrete is thought insignifi-cant since sudden propagation is numerically predicted within a short period of time. The numerically computed crack width for all three cases is given in Table 3 with the assumed creep coefficient of 2.0 to take into account the long-term corrosion period. The computed crack width fairly agrees with the experimental observation.

Finally, verification of the analytical approach was at-tempted by using the experimental data of Matsuo et al. (2004) on corroded RC beam experiments. The three RC beams with sectional dimension of 20 cm by 40 cm with two 22 mm reinforcing bars were used. The length of all beams was 3.0 m. The beams were corroded with the chemo-electrical process of galvanostatic by using electric current. Different corrosion periods of 18, 36 and 72 hours were used for the three beams. The loca-tion of corrosive cracking was recorded for all beams and is shown in Fig. 13. Most of the corroded beams show corrosive cracking running along two longitudinal

Case A-12

Experiment mass loss 17.73%Max. exp. surface crack width 0.38 mm

Computed mass loss 13.45%Max. computed crack width ~ 0.38 mm

Case A-16



Case A-20



Fig. 10 Experimental (Cabrera 1996) and numerical corrosive crack patterns with computed mass loss.

at 1.45% estimated mass loss

at 10.87% estimated mass loss

at 2.3% estimated mass loss at 4.37% estimated mass loss

at 6.45% estimated mass loss at 8.56% estimated mass loss Fig. 11 Crack evolution with increasing mass loss (case A-16).


bars on the bottom faces. The information of mass loss of the corroded bars was also reported along with the maximum crack width, as shown in Table 4. Analysis of the RC section was then carried out and forced to stop when the numerical surface crack width reached the experimentally observed one. After this, the numerical mass loss was estimated with consideration of the pos-sible migration of corrosive product inside the corrosive crack spaces as listed in Table 4.

In the experiment, maximum crack width was re-ported over the domain of specimens because of the scatter of crack widths place by place as shown in Fig. 13. However, in the analysis of RC beam sections with unit breadth, non-uniformity of crack width along the beam span was not considered. Thus, strictly speaking, experimental and analytical values cannot be directly compared, but the analytical crack width is thought to be a mean value over the longitudinal span under the assumption that no leakage of corrosion product outside occurs. Thus, it would be reasonable to have relatively larger values of computed mass loss corresponding to

the mean crack widths of analysis. As a matter of fact, this length of reinforcement (3000 mm) is larger than the other cases discussed in the previous sections (150-380 mm for the data in Fig. 7, 2000 mm in Fig. 8 and 200 mm in Fig. 10). It seems that in the cases of N-36h and N-144h, uniformity of surface crack width over the bottom surface is comparatively less than in the case of N-72h. The analytically obtained mass loss is closer to the reality for apparently uniform crack width develop-ment of N-72h.

4. Conclusions

A multi-mechanical model of corrosive substance and concrete was formulated in this study and a two-phase system to take into account the reinforcement corrosion inside RC was proposed. The corrosion product and corroded reinforcement are idealized as an integrated phase and the surrounding concrete as another contin-uum. The equivalent stiffness of the corroded rein-forcement system derives from the degree of reinforce-

250

350

Tension mid-span Tension near support Compression mid-span

Crack width (analysis)

Crack width (experiment) 0~0.3, 0.5 mm

0.1 mm

0.2~0.7 mm

0.41 mm

0.3~0.5 mm0.31 mm

Fig. 12 Experimental and numerically created horizontal corrosive crack plane of RC beams.

Table 3 Experimentally obtained corrosion and computed corrosion crack width.

Compression mid-span

Tension mid-span

Tension near support

Experimental crack width (mm) 0*)~0.3~0.5 mm 0.2~0.7 mm 0.3~0.5 mm

Experimental mass loss (%) 1.89% 8.76% 6.7%

Computed crack width (mm) 0.1mm 0.41mm 0.31mm

*) Visible cracking (0.3~0.5 mm) in one side but cracking of the other side was hardly detected

Table 4 Experimental and numerical crack width and corroded mass loss.

Specimen No. N-36h N-72h N-144h

Experimental max. crack width (mm) 0.3 mm 0.4 mm 1.4 mm

Experimental mass loss (%) 2.2% 6.9% 17.7%

Computed crack width (mm) 0.3 mm 0.4 mm 1.4 mm

Computed max. mass loss (%) 6.2% 8.7% 30%


ment corrosion. The free expansion strain of corroded reinforcement is also computed through the assumption of uniform corrosion product formed around the parent steel solid. In order to investigate the corrosive cracking propagation into structural concrete, a non-linear finite element with multi-directional fixed crack modelling was employed by representing the corroded reinforce-ment system by equivalent elastic element with the av-eraged free expansion of corroded steel. The concrete cracking driven by the expansion of corroded steel was then consistently traced. The following conclusions are obtained. (1) Two-dimensional analysis of corrosive crack initia-

tion becomes possible by means of non-linear finite element analysis with the proposed multi-mechanics of corrosive substance and concrete. Verification of critical mass loss for cover cracking of corroded RC specimens with single reinforcing steel was performed based on previously reported experiments. Numerically estimated critical mass loss with consideration of buffer capacity of rust gels and concrete creep shows good agreement with the experimental facts.

(2) The relationship between crack ligament and corro-sion mass loss shows stable crack propagation prior to cover cracking in the specimens with a large cover depth relative to the bar diameter, i.e. speci-mens with a large buffer space where corrosive gel products can be stored. The buffer capacity was as-sumed based on the fact that corrosion product might occupy cracking spaces around the corroded steel. The importance of considering the buffer ca-pacity was computationally demonstrated.

(3) The assumption of corrosion product migration was verified by the experiments of corroded RC speci-mens before occurrence of cover cracking. The in-

vestigation of the corrosion gel penetration front in-side the tested specimen revealed the existence of a buffer phenomenon, i.e. corrosion gel migration into crack spaces around the corroded parent steel.

(4) Creep deformation under long term of corrosion was included in the analysis by reducing concrete elastic stiffness with regard to creep coefficient. In-crease in necessary corrosion mass loss for concrete cover cracking was numerically reproduced in the case of a long-term corrosion period compared with short-term corrosion period.

(5) Further numerical analysis of RC sections with multiple reinforcing bars was conducted to verify the applicability of the analytical approach to the corrosive crack propagation. The results show fair prediction in terms of the cracking propagation pat-tern and the corrosion mass loss and/or surface crack width compared with the experiments for the RC sections containing multi reinforcement.

Here, it must be mentioned that the assumption of uniform corrosion around steel bars is not directly ap-plicable to all cases of corroded reinforced concrete, especially for long-term slow rate corrosion in nature and unsaturated states of moisture. The applied experi-mental condition may represent the idealized condition of corroded reinforcement and be a standard index, which should be modified by safety factors related to moisture states, weathering conditions, local quality of the concrete beneath reinforcing bars, and cracking.

Acknowledgments The authors appreciate their fruitful and valuable dis-cussions with Dr. T. Ishida and Dr. T. Kishi of The Uni-versity of Tokyo. This study was financially supported by Grant-in-Aid for Scientific Research (S) No.15106008.

Beam N-36h

Beam N-72h

Beam N-144h

Side Face

Bottom Face

Side Face

Bottom Face

Side Face

Bottom Face

Corrosive crack in experiment

200

400D22

Beam section

Numerical crack propagationin analysis

Corrosive crackingat bottom in analysis

and experiment

3000

Unit in mm. Fig. 13 Experimental and numerical corrosive cracking damage for corroded beams.


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Simulation of Coupled Corrosive Product Formation ...concrete.t.u-tokyo.ac.jp/research/intro/material/ACT2005-Vol3-No2... · K. Toongoenthong and K. Maekawa / Journal of Advanced