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SERVICE-LIFE EVALUATION OF REINFORCED CONCRETE UNDER
COUPLED FORCES AND ENVIRONMENTAL ACTIONS
Koichi MAEKAWA and Tetsuya ISHIDA
University of Tokyo
7-3-1 Hongo, Bunkyo-ku, Tokyo, Japan
ABSTRACT
The authors propose a so-called life-span simulator that can predict concrete structural
behaviors under arbitrary external forces and environmental conditions. In order to realize this
kind of technology, two computational systems have been developed; one is a thermo-hygro
system that covers microscopic phenomena in C-S-H gel and capillary pores, and the other is astructural analysis system, which deal with macroscopic stress and deformational field. In this
paper, the unification of mechanics and thermo-dynamics of materials and structures has been
made with the ion transport of chloride, CO2 and O2 dissolution. This proposed integrated system
can be used for the simultaneous overall evaluation of structural and material performances
without distinguishing between structure and durability.
INTRODUCTION
For sustainable development in the coming century, it is necessary that the infrastructures
retain their required performances over the long term. In order to construct a durable and reliable
structure, it is necessary to evaluate the life cycle cost and its benefits as well as the initial cost of
construction. On the other hand, for an already deteriorated structure, a rational maintenance and
repair plan should be implemented in accordance with the condition of the structure. Considering
these points, it is therefore indispensable to grasp the structural performances under the expected
environmental and load conditions during the service life.
The objective of our research is to develop a so-called lifespan simulator that enables us to
predict the structural behavior for arbitrary conditions. Fig.1 shows the schematic representation
of the lifespan simulator of material science and mechanics of structures. Our research group has
been developing two numerical simulation tools. One is a thermo-hygro system named DuCOM
[1], which covers the micro-scale phenomena governed by thermodynamics. This computational
system is capable of evaluating the early age development of cementitious materials and
deterioration processes of hydrated products under long-term environmental actions. In the
following section, the overall scheme of this system and each material modeling will be
introduced. The other one is a nonlinear path-dependent structural analytical system named
COM3 [2][3]. For arbitrary mechanical actions including temperature and shrinkage effect, the
structural response as well as mechanical states of constituent elements can be predicted. The
solidification model of hardening concrete composite has been also installed in this system for
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predicting time-dependent behavior depending on the temperature, moisture profile, and
micropore structure of materials [4].
It has to be also noted that the structural deformation and capacity are really linked with both
micro-pore based deterioration and large-scale mechanical defects represented by cracking, yielding
and damaging of materials with respect to control volume. In turn, the progress in macro-scale
material damage and defects are also dependent on both the structural deformation and
environmental boundary conditions. Here, nonlinearly accelerated change of material and structural
performances takes place simultaneously. For example, corrosion and associated volume expansion
induces additional cracks and defects which also accelerate the migration of moisture and ions. In
this paper, the unification of mechanics and thermo-dynamics of materials and structures will be
tackled for showing the possibility and future direction of research and development. The authors
understand that the unified approach of mechanics which governs stress and strain fields and
thermo-hygro physics ruling mass and energy transport associated with thermo-dynamic state
equilibrium would serve as a technicality of ensuring total performances of concrete structures as
well as structural concrete performance over the life span of concrete structures.
THERMO-HYGRO PHYSICS FOR CONCRETE PERFORMANCE --DuCOM --
The development of young aged concrete is intimately associated with the thermodynamic
processes, such as hydration of powders, moisture transport and micro structure development,
which show dynamic progress from 10-1
to 101[days]. It has to be noted that these phenomena
exhibit strong mutual link. For example, the development of micro structures can be achieved by
the precipitation of hydrated products, and the moisture profile in cementitious materials
influences the rate of hydration. Furthermore, properties of pore structure determine the moisture
conductivity. Our research group has been developing 3D finite element analysis program,
10-1
Scale
100-103[m]
Scale
10-6-10-9[m]
100
104
103
102
101
Macroscopic cracking
Stress, Strain, Accelerations, Degree of
damage, Plasticity, Crack density etc.
Continuum Mechanics
Hydration
startsHeat. Initial
defects
Time
(Days)
Deformational compatibility
Momentum conservation
Thermo-hygro system
State lawsMass/energy balance
Output:
Oxidation, Carbonation,
deterioration
Unified
evaluation
Environmental ActionsDrying-wetting. Wind. Sunlight.
ions/salts etc.
Mechanical ActionsGround accelerationGravityTemperature and shrinkage effects
Output: Hydration degree, Microstructure,Distributions of Moisture /Salt /Oxygen /CO2,
pH in pore water, corrosion rate etc.
Fig.1 Lifespan simulation for materials and structures
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allowing to simulate these interactive processes (Fig.1) [1][5]. This section simply summarizes the
overall schemes and the core points, since the details were already resented in other literatures.
The hydration of both constituent minerals of cement and pozzolans is traced by
simultaneous differential equations based on the Arrhenius law of chemical reaction [6][7]. The
rate of hydration is mathematically specified in terms of temperature, free water content in
capillary pores, degree of hydration and associated cluster thickness of C-S-H gel layers
precipitated around non-reacted cement particles (Fig.2). Then, the chemical process and its
Fig.2 Modeling of the hydration of cement and pozzolans
Fig.3 Schematic representation of moisture transport modeling in concrete
TimeHydrationHeatRateCa + SiO4
4
CaCa
Ca saltwith Sp
adsorption of Sp
Sp
Consumption of Ca ion
Delaying ofCa(OH)2 nucleation
100%%1 25%
stage1 stage2 stage3
( 30%)
1%
C3S
C2S
C3AC4AF
SG
FA
( )H s H QE
R T T i i i i T i
i=
, exp0
1 1
0
Ca2+
Ca +
Ca2+
Ca2+
+
( )
i
ii
ip
S
P
P
t div K P S
tW
t
+ = 0
Particle growthMOISTURECONDUCTIVITY
Liquid + vapor Computed from porestructure directly
Random pore model
gel
Vaportransport
Liquid transport
Knudsenfactor
History dependent liquidviscosity
PORE STRUCTUREDEVELOPMENT
Based on cementparticle expansion
Growth dependent onthe average degreeof hydration
saturation
RH
slope
MOISTURECAPACITY Obtained from the
pore structures(B.E.T theory)
Hysteresis isothermmodel consideringinkbottle effect
MOISTURE LOSSDUE TO HYDRATION
Obtained directly from hydrationmodel. Based on reaction patternof each clinker component
Cement composition dependent
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interaction among minerals and additive pozzolans are considered by sharing common variables
associated with pore solution, water and temperature.
During the hydration process, mass and heat energy conservations have to be satisfied with
respect to moisture and temperature. At the same time, moisture migration in terms of vapor and
liquid water and heat flux are incorporated in the conservation conditions of the second law of
thermo-dynamics. The equilibrium conditions are simultaneously to be solved together, and the mass
and energy transport resistance denoted by permeability and conductivity has to be formulated.
The permeability of vapor and liquid water is mathematically formulated based on the
micro-pore size distribution as demonstrated in Fig.3 [8][9]. The path of moisture in cement paste
is thought to be assembly of small sized fictitious pipes and its integration results in the
macroscopic permeability. Tortuosity on percolation and the thermo-dynamic activation of surface
energy onto the micro-scale viscosity of pore water are taken into account. It is to be noted that the
simple micro-mechanical modeling is applied without any variable fitting.
As a natural way, the pore structure formation model, as illustrated in Fig.4, is added in the
system dynamics of transient concrete performance modeling [1]. The statistical approach to the
Fig.5 Framework of DuCOM thermo-hygro physics
Conservation
lawssatisfied?
Hydration
computation
Microstructure
computation
Pore pressurecomputation
Next
Iteration
START
yes
no
Chloride
transport and
equilibrium
Incrementtime,continue
Carbon dioxidetransport and
equilibrium
Corrosion model Ion equilibriummodel
( )( ) ( ) 0, =+
iiiiii Qdiv
t
SJ
Governing
equations
Size, shape, mix proportions,
initial and boundary conditions
Temperature,
hydration level of
each component
Bi-modal porosity
distribution,
interlayer porosity
Pore pressures,
RH and moisture
distribution
Dissolved and
bound chloride
concentration
Carbon dioxidetransport and
equilibrium
Gas and dissolved
CO2 concentrationpH in pore waterGas and dissolved
O2 concentration
Corrosion rate,
amount of O2consumption
Fig.4 Outline of the pore structure development computation
Matrix micropore structure
HydrationDeg
reeofMatrix
Total surface area (/m3)
Capillaries, gel, and interlayer
dr
r
ro
( ) r r r=
RepresentativeCSH grain
Outer productsdensity at r
particle
radius
Meanseparationmax
m
The particle growthVolume and weight ofinter and outer products
Bulk porosity ofCapillaries,gel and interlayer
porosity
0.0
1.0
outerproduct
inner product
unhydratedcore
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micro pore structural geometry of hardened cement paste having interlayer, C-S-H gel and
capillary pores is used. The porosity distribution of hydrated and non-hydrated compounds around
referential cement particles is calculated and the surface area of micro-pores is estimated
mathematically. By assuming statistical distribution function with regard to the pore sizes, the
authors extend the geometrical description of micro pores. The connective mode of each pore
volume is also defined with simple probability on the basis of which the path-dependency ofisotherm of moisture is successfully described [10].
Recently, in addition to the above modeling related to early age development phenomenon,
the authors have been extending the scope of DuCOM in order to cover the deterioration and
resolution of cementitious materials and steel corrosion. Here, concentrations of chloride ion,
oxide, and carbon dioxide were added to the thermo-hygro system, as additional degrees of
freedom to be solved (Fig.5). Each physical variable should satisfy the law of mass conservation
shown in Fig.5, same as the story in terms of the temperature and moisture profile computation in
the previous discussions. Potential term S(), flux term J(), and sink term Q() constituting thegoverning equations, are formulated as a nonlinear function of variables i based onthermodynamic theory. The obtained material properties are shared through common variables
beyond each sub-system, therefore interactive problem, such as corrosion due to simultaneousattack of chloride ions and carbon dioxide, can be simulated in a natural way. Coupling these
materials modeling, an early age development process and deterioration phenomenon during the
service period can be evaluated for arbitrary materials, curing and environmental conditions in a
unified manner. In the following sections, the authors will introduce the general ideas of each
material modeling and its coupling system.
Formulation of Chloride Ion Transport
It is a well-known fact that chlorides in cementitious materials have free and bound
components. The bound components exist in the form of chloro aluminates and adsorbed phase on
the pore walls, making them unavailable for free transport. It has been reported that the amount of
bounded chlorides would be dependent on the binders, electric potential of pore wall, and pH in
pore solutions. However, their exact mechanisms are still not clear. In this paper, the free and
bound components of chlorides under equilibrium conditions are tentatively expressed by the
following empirical equations proposed by Maruya et al. as [11],
( )
tot
tot
tot
totfixed
C
C
C
C
=
0.3
0.31.0
1.0
543.0
1.035.01
125.0
(1)
where, Ctot; total amount of chloride [wt% of cement] (=Cfree+ Cbound, amount of free chloride and
bound chloride, respectively), fixed= Cfree + Cbound; equilibrium ratio of fixed chloride componentto the total chloride ion component.
Considering the advective transport due to the bulk movement of pore solution phase as well
as the ionic diffusion due to concentration difference, the flux of free chlorides in pore water can
be expressed as,
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S
PKCSCD
SClClClCl
=+
= uuJ (2)
where, JClT
= [JxJyJz] ; flux vector of the ions [mol/m2.s], ; porosity of the porous media, S; degree
of saturation of the porous medium,DCl; diffusion coefficient of the chloride ions in pore solution
phase [m2
/s], =(/2)2
accounts for the average tortuosity of a single pore as a fictitious pipe formass transfer, and this parameter considers the tortuosity of hardened cement paste matrix, which is
uniformly and randomly connected in 3-D system [1][9], T = [/x/y/z] : the gradient operator,CCl : concentration of ions in the pore solution phase [mol/l], : density of water, and u
T= [ux uy uz]
is the advective velocity of ions due to the bulk movement of pore solution phase [m/s]. The
advective velocity u is directly obtained from the pore pressure gradient P and moistureconductivity K, which depends on water content, micro pore structures, and moisture history as
shown in Fig. 3. In the case of chloride ion transport in concrete, S represents the degree of saturation
in terms of the free water only, as adsorbed and interlayer components of water are also present. Here,
it has to be noted that diffusion coefficientDCl would be a function of ion concentration, since ionic
interaction effects will be significant in the fine micro structures at increased concentrations, thereby
reducing the apparent diffusive movement driven by the gradient of ion concentration [12]. Thismechanism, however, is not clearly understood, therefore we neglect the dependency of the ionic
concentration on the diffusion process in the modeling. From the several numerical sensitivity
analysis, a constant value of 3.010-11 [m2/s] is given forDCl.The first term on the right-hand side of Eq. (2) expresses the diffusion of ions, whereas the
second term describes the advective transport due to the bulk movement of condensed pore water.
The advective velocity of free chloride ions might be also dependent on the ion concentration,
similarly to the diffusion coefficient. In this paper, however, we assumed that the velocity vector
of ions would be equal to that of pore liquid water, since there is not enough experimental data to
establish a model for this aspect.
Material parameters shown in the Eq.(2), such as porosity, saturation and advective velocity,
are obtained directly by the thermo-hygro physics. Therefore, the flux of chloride ions can be
obtained without any empirical equations and/or intentional fittings, once mix proportions, powder
materials, curing and environmental conditions are given to the analytical system. Same story can
be applied for other modeling, say, formulation of CO2 and O2 transport, steel corrosion and ion
equilibrium.
The mass balance condition for free chloride can be expressed as,
( ) 0=+
ClClcl QdivJSCt
(3)
where, QCl; the rate of binding or the change of free chloride to bound chloride per unit volume of
concrete [mol/m3.s], which can be computed by assuming local equilibrium conditions shown in
the eq.(1) . From the above discussions and formulations, distribution of bounded and free
chloride ions can be obtained at arbitrary stage.
Modeling of Carbonation
For simulating carbonation phenomena in concrete, equilibrium of gas and dissolved carbon
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dioxide, their transport, ionic equilibriums, and carbonation reaction process are formulated based
on thermodynamics and chemical equilibrium theory. Mass balance condition for dissolved and
gaseous carbon dioxide in porous medium can be expressed as,
( ) 0]}1[{2222
=++
COCOdCOgCO QdivJSSt
(4)
where, gCO2; density of CO2 gas [kg/m3], dCO2; density of dissolved CO2 in pore water [kg/m
3],
JCO2; total flux of dissolved and gaseous CO2 [kg/m2.s], QCO2; sink term that represents the rate of
CO2 consumption due to carbonation [kg/m3.s]. For representing local equilibrium between
gaseous and dissolved CO2, we use Henrys law, which states the relationship between the
solubility of gas in pore water and the partial pressure of the gas [13].
The transfer of the carbon dioxide is considered in both phases of dissolved and gaseous
carbon dioxide. The flux of carbon dioxide can be formulated based on Ficks first law of
diffusion. However, factors such as complicated pore network, Knudsen diffusion etc, reduce the
apparent diffusivity of carbon dioxide. Considering the effect of Knudsen diffusion, tortuosity, and
connectivity of pores on diffusivity, the flux of CO2JCO2 can be expressed as,
( )
+=
=+=
c
c
r k
g
gCO
r
d
dCOgCOgCOdCOdCOCON
dVDDdVDDDDJ1
0
0
0
2222222(5)
where, DgCO2; diffusion coefficient of gaseous CO2 in porous medium[m2/s], DdCO2; diffusion
coefficient of dissolved CO2 in porous medium[m2/s], D0
g; diffusivity of CO2 gas in a free
atmosphere[m2/s],D0
d; diffusivity of dissolved CO2 in pore water [m
2/s], V; pore volume, rc; pore
radius in which the equilibrated interface of liquid and vapor is created, which is determined by
thermodynamic conditions,Nk; Knudsen number, which is the ratio of the mean free path length
of a molecule of CO2 gas to the pore diameter. Knudsen effect on the gaseous CO2 transport is not
negligible in low RH condition, since porous medium for gas transport becomes finer as relative
humidity decreases. As shown in eq.(5), diffusion coefficientDdCO2 is obtained by integrating the
diffusivity of saturated pores over the entire porosity distribution, whereas DdCO2 is obtained by
summing up the diffusivity of gaseous CO2 through unsaturated pores.
The carbonation reaction in cementitious materials is simply described by the following
chemical reaction.
3
-2
3
2 CaCOCOCa ++ (6)
The calcium ion decomposed from the dissolution of calcium hydroxide is assumed to react with
carbonate ion, whereas the reaction of silicic acid calcium hydrate (C-S-H) is not considered, since
its solubility is quite low compared with calcium hydroxide. The rate of the reaction can be
expressed by the following differential equation, assuming that the reaction is of the first order
with respect to Ca2+
and CO32-
concentrations as,
]CO][Ca[ 232CaCO3
2+==
kt
CQCO (7)
where, CCaCO3; concentration of calcium carbonate, kis a reaction rate coefficient, which shows
the temperature dependence. In the current stage, we focus on the carbonation phenomenon under
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constant temperature, and coefficient kis assumed to be constant (k=2.08 [l/mol.sec]) determined
from several sensitivity analyses. The authors understand that the formulation based on the
Arrhenius law of chemical reaction should be considered for more generic treatment.
In order to calculate the rate of reaction with eq.(7), it is necessary to obtain the concentration
of calcium ion and carbonic acid in the pore water at arbitrary stage. In this study, we consider the
following ion equilibriums; the dissociation of water and carbonic acid, and the dissolution andthe dissociation of calcium hydroxide and calcium carbonate. Here, the presence of chlorides is
not considered, although we understand that chloride ions are likely to affect the equilibrium
conditions. The formulation including chlorides remains for future study.
++
+
++
+2
3332
2
CO2HHCOHCOH
OHHOH
( )-2
3
2
3
2
2
COCaCaCO
2OHCaOHCa
+
++
+
(8)
Although the hydronium ion H3O+
is present in water and confers acidic properties upon aqueous
solutions, it is customary to use the symbol H+
in place of H3O+. As shown in eq.(8), carbonation
is an acid-base reaction, where cation and anion act as Brnsted acid and base respectively.
Furthermore, the solubility of precipitations is dependent on pH in pore solutions. Therefore,
according to the basic principles on ion equilibrium, the authors aim to formulate the carbonatereaction in concrete [14].
First of all, let us consider the equilibrium reaction of carbonic acid. From the law of mass
action, the corresponding equilibrium expression is,
]OH][H[ +=wK ]HCO[
]CO][H[
]COH[
]HCO][H[
3
23
32
3
++
== ba KK (9)
where, Ki is the equilibrium constant of concentration for each dissociation, we give these values
as, Kw=1.0010-14
, Ka=1.0010-14
, Kb=4.7910-14
at 25 respectively [13]. Next, the mass
conservation law is applied for the ions from dissolution of carbon dioxide and re-dissolution of
calcium carbonate.
[ ]-2
3-33210 COHCOCOH ++=+ SC (10)
where, C0 is the concentration of dissolved carbon dioxide [mol/l], which can be obtained from
dCO2 in eq.(4). S1 is the solubility of calcium carbonate, which can be calculated by thesolubility-product constant. Using eq.(9) and (10), concentrations of H2CO3, HCO3
-and CO3
2-can
be obtained as,
( )
( )
( )baa
ba
baa
a
baa
KKK
KKSC
KKK
KSC
KKKSC
++=+=
++=+=
++=+=
++
++
+
++
+
]H[]H[][CO
]H[]H[
]H[][HCO
]H[]H[
]H[]CO[H
22102
-2
3
21101
-
3
2
2
010032
(11)
The solubility of calcium carbonate can be obtained by the following relationship as,
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]CO][Ca[2
3
21 +=spK (12)
where, 1spK is the solubility-product constant of the calcium carbonate (=4.710-9, at 25).
Similarly, the solubility of calcium hydroxide can be calculated as,
2-22 ]OH][Ca[ +=sp
K (13)
where, 2spK is the solubility-product constant of the calcium hydroxide (=5.510-6, at 25) [13].
Considering the common ion effect on the each solubility, eq. (12) and eq. (13) can be replaced
with the solubility of calcium carbonate S1 and that of calcium hydroxide S2 as,
( ) ( ) ( ) [ ]2-212102211 OH +=++= SSKSCSSK spsp (14)
From the mass conservation conditions, concentration of ions should satisfy the following
relationships.
dddC ]CO[]HCO[]COH[-2
3
-
3320 ++= (15)
ssssS ]Ca[]CO[]HCO[]COH[
2-2
3
-
3321
+
=++= cS ]Ca[2
2
+
= (16)where, [i]d, [i]s, and [i]c are the concentration of ion from the dissolution of CO2 gas, calcium
carbonate and calcium hydroxide, respectively. For example, the total concentration of carbonic
acid [H2CO3] shown in eq. (9) becomes the summation of [H2CO3]d from CO2 gas and [H2CO3]s
from CaCO3.
In addition, the above ions should satisfy the law of proton balance, in which the amount of
donor is equal to that of accepter in terms of proton in the Brnsted-Lowry theory. The equation
deduced by the law of proton balance is obtained as,
ccssc ]CO[2]HCO[]OH[]HCO[]COH[2]Ca[2]H[-2
3
-
3332
2 ++=+++ ++ (17)
From the above equations describing the conditions of ion equilibrium, finally we obtain as,
[ ] 020111012 2H22]H[ CCKSSS w ++=+++ ++ (18)
Using eq.(18), the concentration of proton in pore solutions can be calculated at arbitrary
stage, once the concentration of calcium hydroxide and that of carbonic acid before dissociation
are given.
It has been reported that micro-pore structure in cementitious materials would be changed
due to carbonation. In this paper, the authors use an empirical set of equations that are proposed by
Saeki et al. as [15],
6.00.5
0.10.6
2
22
Ca(OH)
Ca(OH)Ca(OH)
=
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In this section, we introduce the general scheme of micro-cell corrosion model based on
thermodynamics and electro-chemistry. In our modeling, it has been assumed that the steel
corrosion would occur uniformly over the surface areas of the reinforcing bars in a finite volume,whereas the formation of pits due to localized attack of chlorides and the corrosion with macro
cell remains for future study. For making it possible to treat the formation of macro cell, we
understand that it is necessary to consider magneto-electrical field governed by Maxwells
principle as well as the mass, momentum and energy conservations. Fig.6 shows the flow of the
computation of corrosion rate. When we consider the micro-cell based corrosion, it can be
assumed that the area of anode is equal to that of cathode and they are not separated from each
other. Therefore, we do not treat the electrical conductivity of concrete, which governs the
macroscopic transfer of ions in pore water.
First of all, electric potential of corrosion cell is obtained from the ambient temperature, pH
in pore solution and partial pressure of oxide, which are calculated by other subroutine in the
system. The potential of half-cell can be expressed with the Nernst equation as [16],
( ) ( ) ( )
( ) ++=
+
+
2FeFe FeFe
2
ln
PtaqFesFe
hFzRTEE
e
( ) ( ) ( ) ( )
( ) ( ) pHPPFzRTEEe
06.0ln
aqOH4Pt4lOH2gO
2O22O2OO
-
22
+=
=++ (20)
where, EFe; standard cell potential of Fe, anode (V, SHE), EO2; standard cell potential of O2,
cathode (V, SHE), EFe; standard cell potential of Fe at 25 (=-0.44V,SHE), E
O2; standard cell
potential of O2 at 25 (=0.40V,SHE),zFe; the number of charge of Fe ions (=2), zO2; the number
of charge of O2 (=2), P; atmospheric pressure. Strictly speaking, the solution of other ions in pore
water might affect the electric potential of cell. However, it is difficult to consider the effect of ion
solutions on the half-cell potentials, therefore we adopt the above equations, assuming the ideal
conditions.
Next, based on the thermo-dynamical conditions, the condition of passive layers is evaluated
by the Pourbaix diagram, which shows that there are conditions where steel corrodes, areas where
protective oxides form, and an area of immunity to corrosion depending upon the pH and the
potential of the steel. From the electric potential and the formation of passive layers, electric
current that involves chemical reaction can be calculated so that conservation law of electric
Computation ofelectric potential of
corrosion cell
TemperaturepH in pore solution
Partial pressure of O2
Evaluation of the conditionof the passivity
Computation of thecorrosion rate
pH in pore solutionConcentration of Cl- ions
Amount of dissolved O2in pore water
TemperatureAmount of steel
corrosion
Amount ofconsumed O2
Output
Fig.6 Overall scheme of corrosion
computation
logia log|ic|
EO2
EFeLogi0of Fe
logicorr
zFRT 303.2
[V]
Ecorr
Tafel gradient
Logi0of O2
Fig.7 The relationship between electric current
and voltage for anode and cathode
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charge should be satisfied in a local area (Fig.7). The relationship between electric current and
voltage for anode and cathode can be expressed by the following Nernst equation as,
( ) ( ) ( ) ( )0Oc
0Fe
alog5.0303.2log5.0303.2
2iiFzRTiiFzRT
ca== (21)
where, a; overvoltage at anode [V], c; overvoltage at cathode [V], F; Faradays constant, ia;
electric current density at anode [A/m2
], ic; electric current density at cathode [A/m2
]. CorrosioncurrentIcorrcan be obtained as the point of intersection of two lines. The existence of passive layer
reduces the corrosion progress. In this model, this phenomena is described by changing the Tafel
gradient.
When the amount of oxygen supplied to the reaction is not enough, the rate of corrosion
would be controlled by the diffusion process of oxygen. In this paper, coupling with oxygen
transport model, this phenomenon can be simulated. The detailed discussion on the formulations
of the oxygen is omitted for lack of space, since they are almost same as those of carbon dioxide
[17]. Finally, using the Faradays law, electric current of corrosion is converted to the rate of steel
corrosion.
It has to be noted that these models are only derived from the thermodynamics and
electrochemistry, and the authors understand that further development and improvement are stillneeded thorough various verification of corrosion phenomena in real concrete structures.
CONTINUUM MECHANICS OF MATERIALS AND STRUCTURES -- COM3 --
For simulating structural behaviors
expressed by displacement, deformation,
stresses and macro-defects of materials in
view of continuum plasticity, fracturing and
cracking, well established continuum
mechanics can be used as illustrated in Fig.8.
The compatibility condition, equilibrium and
constitutive modeling of material mechanics
are the basis and the spatial averaging of
overall defects in control volume of finite
element is incorporated into the constitutive
model of quasi-continuum. The authors
adopted a 3D finite element computer code
named COM3 for structural dynamics,
which has been also developed at the
University of Tokyo for static as well as
dynamic ultimate limit states [2][3].
This frame of structural mechanics has
an inter-link with thermo-hygro physics in
terms of mechanical performances of
materials through the constitutive modeling
in both space and time. In this study, the
instantaneous stiffness, short-term strengths
Time
(days)10 10 10 10 10 10
- 1 0 1 2 3 4Service Starts
ExternalLoads
Environment (weather)
effects
Reinforcements
MacroCracks
1010 -1,-210 -6
Shear stress transfer
across crack
YieldStress of steel
Strainof steel
Stress
Strain
Crack
Comp.
Tension
Fig.8 Macro-scale defects and micro-scale pore
structures
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of concrete in tension and compression, free volumetric contraction rooted in coupled water loss
and self-desiccation caused by varying pore sizes are considered in the creep constitutive
modeling of liner convolution integral (Fig.8). The volumetric change provoked by the hydration
in progress and water loss is physically tied with surface tension force developing inside the
micro-capillary pores. Of course, the micro-pore size distribution and moisture balance of
thermo-dynamic equilibrium are given from the code DuCOM at each time step.The cracking is the most important damage index associated with mass transport inside the
targeted structures. Cracks are assumed to be induced normal to the maximum principal stress
direction in 3D extent when the tensile principal stress exceeds the tensile strength of concrete. As
stated before, the strength is numerically evaluated from the degree of micro structural formation.
In reality, the explicit relation of the specific strength and formed porosity with intrinsic sizes is
adopted in this study. After crack initiation, the tension softening on progressive crack planes is
taken into account in the form of fracture mechanics. In the reinforced concrete zone, in which
bond stress transfer is expected being effective, the tension stiffness model is brought together.
Since the external load level, with which the environmental action be coupled in design, is rather
lower than the ultimate limit states, compression induced damage accompanying dispersed
micro-cracking is disregarded in this study.
UNIFICATION OF THERMO-PHYSICS OF MATERIALS AND MECHANICS OF
STRUCTURES
For numerical evaluation of the total
structural and material performances, we
propose the dual parallel processing of coupling
two sub-systems shown in Fig. 9 [17]. This
system can be embodied on the multitask
operation system. In this framework, constituent
sub-systems, which have different schemes to
solve the different governing equations, dont
need to be combined into a single process. The
operation system manages the job of each system,
and two sub-systems are connected by
high-speed signal bus or networks so as to
mutually share the common data information.
First, material properties are calculated by
DuCOM. After one step of execution, calculated
results, such as temperature, water content, pore
pressure, pore structure, stiffness, and strength,
are stored in the common data area. After that, a
signal is sent to the sleeping process (COM3) to
start execution. COM3 that becomes active reads the information from the common data area and
performs the stress computation. In this analysis, the damage level of RC member is obtained, and
calculated results are written in the common area after its execution. These steps are continued till
DuCOM
Standby
Write
Read
Calculation
COM3
Calculationconsideringcrack damage
Standby
Standby
Write
Read
Repeat until final step
Calculationconsideringdifferentproperties
Common
storage area
Strength,stiffness,temperature,watercontent,and pore
pressure,etc..
Degree of
damage
Shape, sizerestraintcondition
Initial andboundaryconditions
Fig. 9 Parallel processing of DuCOM and
COM3
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one of the processes completes its computation. Following these procedures, each FE program can
share the computational results between two systems at each gauss point in each finite element.
The chief advantage of unifying material and structural analysis in this manner is numerical
stability of explicit scheme. Furthermore, this coupling method under multi-task operation enables
engineers to easily link independently developed computer codes even if being written by
different computer languages and algorithms. As a matter of fact, slight modification for data
exchange with the common memory space through high-speed bus is needed with a short system
manager program alone.
NUMERICAL SIMULATIONS
Chloride Transport into Concrete Under Cyclic Drying-Wetting Condition
Using the proposed method, transport of chloride ion under alternate drying wetting
conditions were simulated. It has been confirmed in the past research that the concentration of
chloride near the surface layer is higher than that of the solution when a concrete specimen is
submerged in it. This phenomenon cannot be explained by the diffusion theory alone. In order to
consider this behavior, we use the ion adsorption model in the surface layer proposed by Maruya
et al. This model expresses the flux of chloride ions driven by the gradient of electrical force; the
positive charge at the pore surface draws chloride ions that have negative electric charges.
0 0.01 0.02 0.03 0.04 0.050.0
1.0
2.0
3.0
4.0
5.0After 28days
Distance from the surface [m]
Chloride content [wt% of cement]
Total chloride
Free chlorideDiffusion only
Markers : Test data (Maruya et al.)
Drying 7days
Cl ion:0.51[mol/l]Wetting 7days
Lines : Computation
0 0.01 0.02 0.03 0.04 0.050.0
1.0
2.0
3.0
4.0
5.0After 28days
Distance from the surface [m]
Chloride content [wt% of cement]
Total chloride
Free chloride
Diffusion +
Advective transport
Markers : Test data (Maruya et al.)
Drying 7days
Cl ion:0.51[mol/l]
Wetting 7days
Lines : Computation
0 0.01 0.02 0.03 0.04 0.050.0
1.0
2.0
3.0
4.0
5.0After 182days
Distance from the surface [m]
Chloride content [wt% of cement]
Diffusion only
Total chloride
Free chloride
Markers : Test data (Maruya et al.)
Drying 7days
Cl ion:0.51[mol/l]
Wetting 7days
Lines : Computation
0 0.01 0.02 0.03 0.04 0.050.0
1.0
2.0
3.0
4.0
5.0After 182days
Distance from the surface [m]
Chloride content [wt% of cement]
Total chloride
Free chloride
Diffusion +
Advective transport
Markers : Test data (Maruya et al.)
Drying 7days
Cl ion:0.51[mol/l]
Wetting 7days
Lines : Computation
Fig.10 Chloride content profile in concrete exposed to cyclic wetting and drying
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For verification, the experimental data by Maruya et al. were used [11]. The size of mortar
specimens were 5510 [cm] and the water to powder ratio was 50%. After 28 days of sealedcuring, the specimens were exposed to cyclic alternate drying (7 days) and wetting (7 days) cycles.
The drying condition was 60%RH, whereas the wetting was exposed to a chloride solution of 0.51
[mol/l] at 20. In the FEM analysis, mix proportions and the chemical composition of the
cements (C3A, C4AF, C3S, C2S, and gypsum) were given. The curing conditions and exposure
conditions were also given as boundary conditions for the target structures. All of these input
values corresponded to the experimental conditions. Fig.10 shows the distribution of free and
bound chlorides from the boundary surface. For comparison, we analyzed two cases; one
considering only diffusive movement and the other including the advective transport due to the
bulk movement of pore water as well as the diffusion process. As shown in the analytical results,
the distribution of bound and free chlorides can be reasonably simulated with advective transport
due to the rapid suction of pore water under wetting phase.
Fig.11 Carbonation phenomena for different CO2 concentrations and W/C
Fig.12 Carbonation phenomena for different CO2 concentrations and W/C
0 100 200 300 4000
10
20
30
40
Time[Days]
Depth of carbonation[mm]
CO2=1%
RH=55%
W/C50% W/C60% W/C70%
Markers Lines
Experiment [9]Computation
0 100 200 300 4000
20
40
60
80
100
Time[days]
Depth of carbonation[mm]
CO2=10%
RH=55%W/C50% W/C60% W/C70%
Markers Lines
Experiment [9]Computation
0 10 20 30 40 50 600
5
10
15
20
W/C65%
W/C55%
RH :80%
CO2:10%
Depth of carbonation[mm]
Time[days]
Markers Lines
ExperimentComputation
0 10 20 30 40 50 600
5
10
15
20
W/C65%
W/C55%
RH :50%
CO2:10%
Depth of carbonation[mm]
Time[days]
Markers Lines
ExperimentComputation
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0 20 40 60 80 100 1200
5
10
15
20
25
30
35
W/C50%
CO2:3%
Structural age until cracking due to corrosion [year]
Cover depth [mm]
W/C60%
W/C40%
60%RH 10days
99%RH 10days
Cl ion:0.51[mol/l]
Carbonation Phenomena in concrete
In this section, computations were performed to predict the progress of carbonation for
different CO2 concentrations, relative humidity, and water to cement ratio. The amount ofCa(OH)2 existing in cementitious materials can be obtained by multi-component hydration model
as [6][7],
( ) ( )
( ) 6324
2323223233
AHC10HOH2CaAFC
OHCaHSC4HS2COH3CaHSC6HS2C
++++++
(22)
When blast furnace slag and fly ash are used, Ca(OH)2 will be consumed during hydration. The
consumption ratios of slag and fly ash reactions are assumed to be 22% and 100% of reacted mass,
respectively, in this analysis [6][7].
First, the accelerated carbonation tests were studied. For verification, the experimental data
done by Uomoto et al were used [18]. Fig.11 shows the comparison of analytical results and
empirical formula that was regressed with
the square root t equation. Similar to the
previous case, all of the input values in the
analysis corresponded to the experimental
conditions. Analytical results show the
relationship between the depth of concrete in
which pH in pore water becomes less than
10.0 and exposed time. The simulations can
roughly predict the progress of carbonation
for different CO2 concentration and water to
powder ratio.
Next, we studied the influence of the
ambient relative humidity on the progress of
carbonation. In the acceleration test,
specimens were exposed to 50%RH and Fig.14 Time till first signs of cracking due to
corrosion for concrete
Fig.13 Distribution of pH, calcium hydroxide and calcium carbonate under the action of carbonic acid.
0 2 4 6 8 10 127
8
9
10
11
12
13
14
0.00
0.05
0.10
0.15
0.20
Distance from the surface [cm]
pH CO2 [mol/l]
After 1800days
W/C=25%
W/C=55%
pH
CO2
0 2 4 6 8 10 120
20
40
60
80
100
120
140
160
0.00
0.20
0.40
0.60
0.80
1.00
1.20
Distance from the surface [cm]
a(OH)2 [kg/m3] CaCO3 [mol/l]
Ca(OH)2
CaCO3
After 1800days
W/C=25%W/C=55%
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80%RH with CO2 concentrations of 10%. As shown in Fig.12, analysis can reasonably follow the
experimental data for different W/C and environmental conditions.
Fig.13 shows the distribution of pH in pore water, CO2, calcium hydroxide, and calcium
carbonate inside concrete, exposed to the CO2 concentration of 3%. Two different water to powder
ratio, W/C=25% and 50%, were analyzed. It can be shown that higher resistance for the carbonic
acid action is achieved in the case of low W/C.
Numerical Simulation of Coupled Carbonation and Chloride Induced Corrosion
Corrosion of steel in concrete due to simultaneous attack of chloride ions and carbon dioxide
were simulated. One-dimensional concrete members that have three different water to powder
ratio, W/C=40, 50, 60%, with only one face exposed to the environment were considered. In this
analysis, the stage where concrete cracking occurs was defined as a limit state with respect to the
steel corrosion. The progressive period until the initiation of longitudinal cracking were estimated
by the equation proposed by Yokozeki et al [19] . which is a function of cover depth. Fig.14 shows
the relationships between cover depth and structural age until cracking due to corrosion obtained
by the proposed thermo-hygro system. It can be seen that the concrete nearer to the exposure
surface would show early sign of corrosion induced cracking, and low W/C concrete has a higher
Fig.15 Moisture and internal stress distribution in concrete exposed to drying condition
Restrained x and
y displacements
Restrained in
all directions
Mass/energy
transfer from
surface element
2.04.0
6.0
10.0
Unit[cm]
8.0
1.0
1.0
x
yz
60cm
0 5 10 15 20 25 300.05
0.06
0.07
0.08
0.09
0.10
Distance from the surface[cm]
Water content[kg/m3]
Single calculation
Parallel calculation
1.0
0
tt f
30 [cm]
Cracked element(Softening zone)
2.3 days dried
0 5 10 15 20 25 300.04
0.05
0.06
0.07
0.08
0.09
0.10
Distance from the surface[cm]
Water content[kg/m3]
Single calculation
Parallel calculation
12.9 days dried
1.0
0 30 [cm]
Cracked element(Softening zone)
tt f
0 5 10 15 20 25 300.04
0.05
0.06
0.07
0.08
0.09
0.10
Distance from the surface[cm]
Water content[kg/m3]
Single calculation
Parallel calculation
Cracked element(Softening zone)
35.0 days dried
1.0
0
tt f
30 [cm]
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resistance against corrosion.
Moisture Distribution in Cracked Concrete
In the following sections, in order to show the possibility of the unification of structure and
durability design, several primitive simulations were conducted by using the proposed parallel
computational system. First case study is moisture loss behavior in cracked concrete. It has beenreported that there would be close relationship between the moisture conductivity and the damage
level of cracked concrete, that is, moisture conductivity would be dependent on the crack width, or
the continuity of each cracking. The proposed system, in which the information can be shared
between thermo-hygro and structural mechanics system, can describe this aspect quantitatively by
considering the inter-relationship between the moisture conductivity and properties of cracking.
For representing the acceleration of drying out of concrete due to cracking, the following model
proposed by Shimomura were used in this analysis [20].
+++
+=
kingafter cracJJJJ
ckingbefore craJJJ
cr
L
cr
VLV
LV
w
(23)
where,Jw is the total mass flux of water in concrete, JVandJL are mass flux of vapor and liquid innon-damaged concrete respectively, andJV
crandJL
crare mass flux of the vapor and liquid water
through cracks. In this simulation, onlyJVcr
is taken into account for the first approximation, since
diffusion of vapor would be predominant when concrete are exposed to drying conditions. From
the experimental study done by Shimomura et al., it has been confirmed that the flux JVcr
can be
expressed as [21],
hDJ aVcr
V = (24)
where, ; average strain of cracked concrete, which can be computed by COM3, V; density ofvapor,Da; vapor diffusivity in free atmosphere, h; relative humidity. This formulation assumes
elastic deformation of uncracked region in tension to be small compared with crack opening.
The target structure in this analysis is a concrete slab, which has 30% water to powder ratio
using medium heat cement. The volume of aggregate was 70%. After 3 days of sealed curing, the
specimen was exposed to 50%RH. Fig. 15 shows the mesh layout and the restraint condition used
in this analysis.
Fig.15 shows the cracked elements, the distribution of moisture, and normalized tensile stress
at each point from the boundary surface exposed to drying condition. Moisture distribution
calculated without stress analysis is also shown in Fig.15. As shown in the results, the crack
occurs from the element near the surface, and the crack progresses internally with the progress of
drying. It is also shown that the amount of moisture loss becomes large due to cracking.
Ingress of Chloride Ion in RC Beam Damaged by External Load
The second case is the numerical simulation about the ingress of chloride ion in RC beam
damaged by external load. Fig 16 shows the size of the beam, layout of FE mesh and load
condition used in this analysis. The reinforcement ratio is 0.96%. For FE analysis of RC structures,
AN proposed the model which combines the nonlinearity of cracked concrete in RC zone and
plain concrete zone (PL zone) [22]. In this analysis also, we considered two different zones in RC
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beams to take into account the difference of concrete mechanics near or far from reinforcing bars
(Fig.16). As for the mix proportion given to DuCOM, water to cement ratio is 45%, and the
volume of aggregate is 65%. After 7 days of sealed curing, load is applied with displacement
control. Fig.17 shows the load-deflection relationship and cracked elements due to bending.
After loading, behaviors of chloride transport into damaged RC beam were simulated. The
bottom surface of the beam is exposed to the concentration of chloride ion 1.4 [mol/l] under
Fig.16 Mesh layout and load condition used in FE analysis
Fig.17 Distribution of chloride ion in damaged RC beam due to external load
90
20
15
10
13
p=0.96%
Unit [cm]
RC Zone
PL Zone
Load
0.0 0.1 0.2 0.3 0.4 0.50.0
0.5
1.0
1.5
2.0
2.5
3.0
Deflection at the center section [mm]
Load [tf]
a b c
Crack occurs
at the bottom
Cracked elements
due to bending
0 2 4 6 8 10 12 14 160.0
0.20.4
0.6
0.8
1.0
1.2
1.4
1.6
Distance from the bottom [cm]
Chloride content [Wt% of cement]
Ingress of chloride ion
Load
Without considerationof cracks and masstransport coupling
Cracked elements
due to bendinga
0 2 4 6 8 10 12 14 160.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Distance from the bottom [cm]
Chloride content [Wt% of cement]
Ingress of chloride ion
Load
Without considerationof cracks and masstransport coupling
Cracked elements
due to bendingb
0 2 4 6 8 10 12 14 160.0
0.2
0.4
0.6
0.8
1.0
1.2
1.4
1.6
Distance from the bottom [cm]
Chloride content [Wt% of cement]
Ingress of chloride ion
Load
Without considerationof cracks and masstransport coupling
Cracked elements
due to bendingc
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alternate drying (7days) and wetting (7days) cycles. Wetting is simulated by an environmental
relative humidity of 99.9%, whereas drying condition is given as 50%RH. During wetting stage,
the moisture flux through cracked area cannot be negligible, since cracks would cause the rapid
suction of pore water. However, there has not been enough knowledge to quantify this aspect yet.
Therefore, liquid conductivity after cracking was roughly assumed becoming to 10 times before
cracking. Fig.17 shows each distribution of chloride ion at point a, b and c. The parallel simulationclearly shows deeper ingress of chloride ion within 100 days, compared to the results without
considering cracks and mass transport coupling. It can be also seen that the amount of ingress of
chloride ion increases near the center section, since in cracked element the bulk movement of
chloride ion in pore water can easily take place.
CONCLUSIONS
The numerical simulation system that can evaluate structural behaviors under coupled forces
and environmental actions was proposed in this paper. This system consists of two computational
system, that is, one is a thermo-hygro system that covers microscopic phenomena in C-S-H gel
and capillary pores, and the other is structural analysis system, which deal with macroscopic stress
and deformational field.In thermo-hygro system, generation and transfer of heat, moisture, gas and ions in micro-pore
structures were formulated based on thermodynamics and electrochemistry. Coupling these
materials modeling, an early age development process and deterioration phenomenon during the
service period can be evaluated for arbitrary materials, curing and environmental conditions in a
unified manner. Numerical verifications show that this method can roughly predict ingress of ion,
carbonation and corrosion phenomena for different materials, curing and environmental
conditions.
The macroscopic structural behaviors were linked with both the microphysical phenomenon
and external load and restraint conditions. In this paper, the unification of mechanics and
thermo-dynamics of materials and structures has been made. Though each component in this
system are crudely simplified and further progress and development is still needed for
accomplishing entire system, the system dynamics of micro-scale pore structure formation and
macro-scale defects and deformation of structures can be shown as a possible approach in this
study.
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