コンクリート工学年次論文集，Vol.32，No.1，2010- Technical Paper -QUANTITATIVELY EVALUATION OF CRACK PROPAGATIONDUE TO REBAR CORROSION
Khoa K. TRAN*1, Hikaru NAKAMURA*2, Keisuke KAWAMURA*3 and Minoru KUNIEDA*4

ABSTRACT
Cracking behavior due to rebar corrosion for concrete specimen with single rebar was evaluated analytically. The
evaluation was performed by comparing with results of the electric corrosion test. Three-dimensional RigidBody-Spring-Method (RBSM) with three phase material corrosion expansion model was applied to simulate internalcracking patterns and surface crack width propagation. The effect of rebar local corrosion and penetration of
corrosion products into cracks during the corrosion process were investigated and cracking behavior due to rebar
corrosion was evaluated quantitatively.
Keywords: Rigid Body Spring Model, Local Corrosion, Penetration of corrosion products, Expansion Model, Crack
Width, Internal Crack

1

INTRODUCTION

Cracking of concrete due to rebar corrosion is
one of the major deterioration behaviors and it causes
the spalling of concrete cover or acceleration ofdeterioration. It is necessary to predict the internal
damage from the observable surface condition during
maintenance process. Therefore, it is desirable toestablish a prediction method to quantitatively assess
the internal crack propagation behavior and rebar
corrosion rate from surface cracks. The internal crack
patterns due to rebar corrosion considering thedifference of cover thickness or diameter of reinforcing
bar has been clarified [1] and [2]. The propagation of
surface crack width against corrosion rate was alsostudied [3], [4] and [5]. Kawamura [3] could simulate
reasonably qualitative behavior of the cracking of
single rebar specimen but the quantitative behavior wasoverestimate when it was compared with the testing
results. Moreover, the effect of local corrosion and local
penetration of corrosion products into cracks after thecracking initiation has not been clarified in theliterature.
In this study, the crack propagation behavior isanalyzed quantitatively. The crack propagation behavior
is simulated using Rigid-Body-Spring-Method withthree-dimensional Voronoi particles. In the analysis, thethree phase material model which consists of rebar,
corrosion products layer and concrete is applied. For
the corrosion products layer, the effect of corrosion
products is considered by initial strain problem due tocorrosion expansion. Moreover, the local corrosion ofrebar and the effect of penetration of corrosion products
into cracks after the cracking initiation are simulated.
The applicability of the model is verified by comparing
*1
*2
*3
*4

with the testing results. As the results, the progress of
surface crack width, the propagation of internal cracks,
the effect of local corrosion and the local penetration of
corrosion products into cracks and so on are clarified.
2

EXPERIMENTAL STUDY

2.1 Testing method
Crack width measurement
Cutting section
C=30

150

85

30

D19
D19

80

85

250

300

150

Unit=mmFigure 1. Specimen setup
Figure 1 shows the specimen set up of theelectric corrosion test [3] and [6]. The corrosion process
was accelerated by using external direct current asshown in the Figure 2.

+-

reinforcement
concrete

Wooden packer Copper plate

NaCl 3%FigureSchematicof corrosion
test
Figure2.2.
Schematicof corrosion
test

2.2 Experimental crack patterns

Graduate School of Engineering, Nagoya University, JCI Member
Professor, Dept. of Civil Engineering, Nagoya University, Dr.E., JCI Member
Graduate School of Engineering, Nagoya University, JCI Member
Associate Prof., Dept. of Civil Engineering, Nagoya University, Dr.E., JCI Member-1043-

Figure 3 indicates the defined internal crack
types shown in crack patterns.
Internal crack patterns at several corrosion rates
are shown in Figure 4 which correspond to the marked
points in Figure 5. Initiation of visible crack occurs on
the surface of concrete cover (vertical crack). Whencorrosion products increase, the crack propagates inside
the concrete cover. After that, lateral cracks appear andtheir lengths increase.
Vertical crack

RBSM is applied [7]. Each element has six degrees of
freedom at center points. Boundary between twoelements is divided into triangles formed by the centerand vertices of the boundary. At each center point of
triangle, three springs, one normal and two shear
springs are set. The analytical model is divided intoelements using Voronoi random polygons. In RBSM
model, crack widths can be automatically measured
during the analysis. The three-dimensional model ispossible to simulate complicated problems.
Zero springs

Lateral crack
Inside crack

Figure 3. Internal crack types
Triangle center point Nuclear

Voronoi particles

Figure 6. RBSM Voronoi polygonsσ

162 mg/cm2

445 mg/cm2

915 mg/cm2

Surface crack
width= 0.54mm

Surface crack
width= 0.94mm

Surface crack
width= 1.52mm

f c’

E

2.3 Surface crack width propagation
Propagation of surface crack width against
corrosion rate is shown in Figure 5. It can be seen that
opening of surface crack initiates after a significant
amount of corrosion products are formed. After the
initiation, surface crack width propagates rapidly up to
the value about 0.4mm. After that, the speed ofpropagation reduces with occurrence of lateral cracks
and effect of penetration of corrosion products intocracks to be discussed in the later part of this study.
Finally, the speed of propagation increases again withthe further propagation of the lateral crack.
Surface crack width (mm)

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
200
400
600
Corrosion rate (mg/cm 2 )800

Figure 5. Surface crack width propagation3

f tl = 0.25 f t
ε tl = 0.75GF / f t h
ε tu = 5.0GF / f t h

G fc / h

ANALYTICAL MODEL

3.1 Three- dimensional RBSM
Rigid- Body-Spring-Method (RBSM) is one ofthe discrete approaches and it is easy to deal with crack
propagation of concrete directly. The method represents
a continuum material as an assemblage of rigid particle
elements interconnected by zero springs along theirboundaries (Figure 6). In this study, three-dimensional

GFt/h
ft1

Ec E
c
ε0 / 2 ε0Figure 4. Internal crack patterns

0

σ
ft

ε tu

ε

Concrete compressive model

εt0

εt1

εtu ε

Concrete tensile model
c= 0.138f’c
φ= 37 o’’Concrete shear model- Morh-Coulomb criterionwith tension and compression ca ps

Figure 7. Concrete material model
3.2 Material models
Figure 7 shows concrete material models used inanalysis.
In the compressive model, f’c is compressive
strength of concrete, Gfc compressive fracture energy,
Ec is Young modulus of concrete.
The tensile behavior of concrete up to thestrength is modeled by using linear elastic. Whilebilinear softening branch of 1/4 model is assumed after
cracking as shown, in which ft is tensile strength, Gft is
tensile fracture energy and h is distance between
centers of Voronoi elements.
Tangential springs represent the shear
transferring mechanism of concrete. The shear strength
is assumed to follow the Morh- Coulomb type criterionwith the tension and compression caps. The shear
fracture criterion is expressed as follows [8]:-1044-

τ2
≥1
τf2

(1)

where c − σ tan φ , forσ ≥ 0.5 f ' c
c − 0.5 f c tan φ , forσ < 0.5 f ' c

τf =

(2)

Rebar is modeled as linear elastic.

3.3 Corrosion expansion model
Modeling expansion of corrosion products isshown in Figure 8. Three phase material model
including rebar, corrosion products and concrete isapplied. The merit of the model is that the properties ofcorrosion products such as thickness (H) and elastic
modulus (Er) are assumed directly. The model isefficient to investigate the effect of corrosion products.
In the model, initial strain is gradually increased innormal springs located on boundary of the corrosion
products layer based on initial strain problem.
Concrete
Corrosion
products
Rebar
Expansion
strainassumed as 2.5 [10], ρs is rebar density (=7.85x103
mg/cm3).
It is not easy to measure properties of corrosion
products such as thickness and elastic modulus byexperiment. In this study, the initial thickness of 1.0mm
and the elastic modulus of 500MPa are recommended
because they can simulate the cracking behavior
reasonably in comparison with the experimental results.

4

UNIFORM CORROSION MODEL

In this model, it is assumed that corrosion
products are uniformly formed around rebar before andafter the cracking initiation.

4.1 Internal crack patterns
Analytical crack patterns at several values of thesurface crack widths are obtained as shown in Figure
10. The analytical internal crack patterns appear to be
similar to the experimental ones. However, at the highlevels of corrosion products amounts, the analyticalones show inside cracks propagating from the rebar
clearly but the testing results do not show this crack.
The difference may be due to the assumption of
uniform corrosion in the analysis.

Figure 8. RBSM corrosion expansion model
U
Ucor

Surface cra ck
width 0.54mmr

Initial rebar
radius(3)

In RBSM model, initial thickness of corrosion
products layer is modeled constantly as H soU
(4)
ε free =HThis parameter is input data in the analyticalprogram by increasing the corrosion rate. U iscomputed from the corrosion rate as the following
equation [10]Wr (dV − 1)

ρs

(5)

where Wr is corrosion rate (mg/cm2), dV is volume
expansion ratio of corrosion products which depends oncorrosion product that forms [4]. In this study, it is

4.2 Surface crack width propagation
Analytical surface crack width propagation in
the case of uniform corrosion model is compared withthe experimental results in Figure 11 (black line).
1.6
1.4
Surface crack width (mm)FigureDeformation
aroundrebar
Figure 10.9.
Deformation a round rebar
due to corrosion
products products
due to corrosion
Lundgren [4] and Oh [9] suggested a model of
deformation around rebar due to corrosion products
which can be described in Figure 9, in which r, x, U and
Ucor are initial radius of rebar, corrosion penetration
depth, free increase of the radius and the real increaseof the radius respectively. The strain in the corrosion
products is

U=

Surfa ce cra ck
width 1.52mm

Figure 10. Internal crack patternsx

U −U
ε cor = cor= ε real − ε free
x +U

Surfa ce cra ck
width 0.94mm

1.2
1
0.8
0.6

Uniform Corrosion Model
Local corrosion HALF Model
Local corrosion QUATER Model
Local Corrosion- Local Penetration
HALF Model
Experiment

0.4
0.2
0

200

400
600
Corrosion rate (mg/cm2)

800

Figure 11. Surface crack width propagation
The analytical results appear similarity with theexperimental results, i.e. crack width initiating after acertain amount of corrosion products and then crack
opening propagating speedily up. However, analyticalcrack width values are larger than the experimental
results, especially when corrosion rates increase.
Reason of the difference is corrosion products tending
to penetrate into cracks after cracking initiation which-1045-

reduces expansive pressure onto concrete. This effectwill be discussed in detail later in this study.

5

EFFECT OF LOCAL CORROSION

5.1 Local corrosion model
Chloride diffusivity coefficient through cracked
concrete is much more than through the sound concrete[11]. As the results, the rebar is much more corroded
around the crack than the other part surrounded by
un-cracked or minor cracked concrete.
Figure 12 shows experimental results of local
corrosion of the rebar accordingly to the internal crack
patterns shown in Figure 4. Local corrosion can beobserved near the vertical crack.

162 mg/cm2
Surface crack
width= 0.54mm

445 mg/cm2

915 mg/cm2

Surface crack
width= 0.94mm

Surface crack
width= 1.52mm

Figure 12. Local corrosion
In this study, local corrosion of rebar after
cracking initiation is simply assumed as follows:
Rebar is locally corroded when vertical cracknear rebar exceeds 0.1mm width.
Local corrosion model is QUARTER
corrosion model or HALF corrosion model
as shown in Figure 13.
At the same corrosion rate, total amount ofcorrosion products in the local corrosion
model (SL) is the same as the one in theuniform model (Su).
SL

Surf ace crack
width 0.54mm

QUATER MODEL
Figure 14.Local corrosion- int. crack patterns
It can be seen that the analytical crack patterns are
similar in the two cases of the local corrosion and they
are reasonably agreement with the testing results in
Figure 4.
5.3 Surface crack width propagation
Analytical surface crack width propagations arealso shown in Figure 11 where red line for HALF
model and green line for QUARTER model. The
surface crack width propagations in two local corrosion
models are similar. When rebar is corroded, theconcrete cover is bended due to internal expansion
pressure and the tensile stress is maximum on the coversurface just above the rebar [3] and [5]. With the sameamount of corrosion products formed around rebar onlyin the concrete cover thickness, the surface crack
opening will propagate similarly.
The surface crack width in the local corrosion
models is larger than the one in the uniform corrosion
model at the same corrosion rate. It is due to moreamount of corrosion products formed in the concrete
cover thickness in the local corrosion models than in
the uniform model.

QUATERUNIFORM

Surf ace crack
width 1.52mm

HALF MODEL

6

Su

Surf ace crack
width 0.94mm

EFFECTOFPENETRATIONOF
CORROSION PRODUCTS INTO CRACKS

6.1 Penetration of corrosion products
During the corrosion process, it is known that
corrosion products tend to penetrate into cracks after
the cracking initiation (Figure 15). This effect may
reduce corrosion expansion pressure on concrete
accordingly [12] and [13].

S u= SLSL

HALF

Figure 13. Local corrosion models
Based on the above assumption, the free increase
U of corrosion products layer is re-calculated for each
local corrosion model. That is the strain in the corrosion
products are 2 times or 4 times of the uniform corrosion
model for the HALF and QUARTER corrosion models.
5.2 Internal crack patterns
Figure 14 shows internal crack patterns of the
HALF corrosion model and the QUARTER corrosion
model.

-1046-

Verticalcrack
Initialrebar
radiusLateral crack

Figure 15. Penetration of
corrosion products into cracks

Corrosion products

Top surf ace

Section throughcenter of rebar

Figure 16. Local penetration of corrosion
products into cracks
In order to simulate this effect in the analysis,
some assumptions are made as follows: Corrosion products can only penetrate intocracks if crack width is larger than 0.1mm
 Corrosion products fully fill in the cracks
 Only cracks above the plane at center of rebar
are taken into account. That is, only thepenetrations to vertical cracks are taken into
account.
Volume of cracks can be directly calculated in
the analysis as shown in Figure 17.
When the penetration of corrosion products intocracks is considered, the effective corrosion products
volume Vcor,eff is:(6)
Vcor , eff = Vcor − Vcrk

The analytical surface crack width propagationin the case of HALF local corrosion model combinedwith local penetration model is also shown in the
Figure 11 (blue line). It can be seen that when thepenetration of corrosion products into cracks are taken
into account, it can be simulate the crack propagation
closely to the testing results in both of the values of
surface crack width and the tendency of opening speedof the surface crack width.
It is noted that analytical internal crack patterns
in this case are the same as the ones of HALF local
corrosion model shown in Figure 14.

7

7.1 Propagation of crack width of vertical cracknear rebar
Figure 18 shows the propagation of verticalcrack width near rebar in the cases of local corrosion
HALF model (black line) and local corrosion model
combined with local penetration model (red line).
When the effect of penetration of corrosion into cracks
is taken into account, the analytical results appear to beclosely to the testing results.

where Vcor is volume of corrosion products in the
corrosion model and Vcrk is volume cracks as computedin Figure 17.
Normal spring i
εn,i : strain of normal spring i

hi
NSPG

Vcrk =∑ε .h.Sn,i i i

1.6
1.4
1.2
1
0.8
0.6
0.4
0.2
0

Local Corrosion
Local Corrosion-Local Penetration

200

400

600

800

1000

Figure 18. Propagation of vertical crack
width near rebar
7.2 Propagation of lateral crack length
The analytical lateral crack length propagation in
the cases of local corrosion HALF model (black line)
and local corrosion model combined with local
penetration model (red line) are shown in Figure 19.
The tendency of analytical result is similar to thetesting results. The propagation of lateral crack lengthis also closer to the experiment when the penetration
effect is considered.

Only cracks in
the hatchedarea areaccounted1

where NSPG is numbers of springs
wi =εn,i .h i is crack width at interfa ce surface i

Figure 17. Computation volumeof cracks to simulate penetrationof corrosion products into crack
The reduction of the free increase of thecorrosion products layer is calculated from the effective
corrosion products volume.
In the analysis, the local penetration model is
combined with HALF local corrosion.

6.2 Surface crack width propagation

ExperimentCorrosion rate (mg/cm 2 )Interfa ce surface i
Si is area of surface i

Vcrk is total volume of cracks

PROPAGATION OF INTERNAL CRACKSCra ck width (mm)Experimental results also show that corrosion
products mainly penetrate into cracks in the concrete
cover thickness rather than the lateral cracks since local
corrosion occurs due to the effect of vertical crack.
Moreover, part of corrosion products also move out thesurface of specimen (Figure 16).

7.3 Propagation of lateral crack width
Figure 20 shows propagation of lateral crack
width at several positions from rebar in the cases of
local corrosion and local corrosion combined with local
penetration respectively. Red and black lines represent
positions at 10mm and 40mm from rebar. With thesame color, solid lines represent the combination caseand dashed lines do for the local corrosion case. Again,
when the effect of penetration of corrosion products istaken into account, lateral crack widths can also be-1047-

simulated reasonably against the experimental results.

REFERENCES

La tera l crack length(mm)

60
50
40

Experiment

30
20
10
0

Local Corrosion-Penetration HALF Model
Local Corrosion HALF

200

400

600

800

1000

Corrosion ra te (mg/cm 2 )Figure 19. Propagation of lateral crack length1

Crack width (mm)

0.8
0.6

10mm( Local Corrosion+Local Penetration)
10mm( Local Corrosion)
40mm( Local Corrosion+Local Penetration)
40mm( Local Corrosion)
10mm f rom rebar ( Exp.)
40mm f rom rebar ( Exp.)

0.4
0.2
0

200

400

600

800

1000

Corrosion rate (mg/cm2)

Figure 20. Propagation of lateral crack width
8

CONCLUSIONS(1)

3D- RBSM analytical model was applied toevaluate cracking behavior of the concrete
specimen with single rebar. Internal crack patterns,
surface crack width propagation and internal crack
propagation were simulated and compared with theexperimental results. The model of local corrosion
after cracking initiation and local penetration of
corrosion products into cracks during corrosion
process was proposed.
When the local corrosion model is taken into(2)
account, it can simulate the internal crack patterns
reasonably.
(3)
When the local corrosion model is combinedwith the local penetration model of corrosion
products into cracks, they can simulate reasonablyinternal crack patterns and quantitatively surface
crack width propagation as well as internal crack
propagation in comparison with the testing results.

ACKNOWLEDGEMENT
This study was financially supported by
Grant-in-Aid for Scientific Research (C) No. 20560432.
The authors would like express appreciation to this
support.

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