Toronto Conference

8/2/2019 Toronto Conference

1/20

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


13/20

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



Total chloride

Free chloride

Diffusion +

Advective transport


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



Diffusion only

Total chloride

Free chloride


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



Total chloride

Free chloride

Diffusion +

Advective transport


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


14/20

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]


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%


Time[days]

Markers Lines

ExperimentComputation

0 10 20 30 40 50 600

5

10

15

20

W/C65%

W/C55%

RH :50%

CO2:10%


Time[days]

Markers Lines

ExperimentComputation


15/20

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%


16/20

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



Single calculation


12.9 days dried

1.0

0 30 [cm]


tt f

0 5 10 15 20 25 300.04

0.05

0.06

0.07

0.08

0.09

0.10



Single calculation



35.0 days dried

1.0

0

tt f

30 [cm]


17/20

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


18/20

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




Load


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




Load


Cracked elements

due to bendingc


19/20

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.

REFERENCES

1K. Maekawa, R. P. Chaube, and T. Kishi, Modeling of Concrete Performance, E&FN

SPON, 1999.2K. Maekawa, P. Irawan and H. Okamura, Path-dependent Three Dimensional Constitutive

Laws of Reinforced Concrete Formation and Experimental Verifications, Structural

Engineering and Mechanics, Vol.15, No.6, pp.743-754, 1997.3H. Okamura and K. Maekawa, Nonlinear Analysis and Constitutive Models of Reinforced

Concrete, Gihodo, Tokyo 1991.4R. Mabrouk, T. Ishida, and K. Maekawa, Solidification model of hardening concrete

composite for predicting creep and shrinkage of concrete, Proceedings of the JCI, Vol.20, No.2,


20/20

pp.691-696 1998.5http://concrete.t.u-tokyo.ac.jp/en/demos/ducom/index.html, Concrete Laboratory, University

of Tokyo, 1996-1999.6Kishi, T. and Maekawa, K. Multi-component model for hydration heating of portland

cement, Concrete Library of JSCE, No.28, pp. 97-115, 1996.7

Kishi, T. and Maekawa, K., Multi-component model for hydration heating of blended cementwith blast furnace slag and fly ash, Concrete Library of JSCE, No.30, pp. 125-139, 1997.

8Chaube, R.P. and Maekawa, K. A study of the moisture transport process in concrete as a

composite material, Proc. of the JCI, Vol. 16, No.1, pp.895-900, 1994.9Chaube, R.P. and Maekawa, K. A permeability model of concrete considering its

microstructual characteristics, Proc. of the JCI, Vol. 18, No.1, pp. 927-932, 1996.10

Ishida, T., Chaube, R.P., Kishi, T. and Maekawa, K., Modeling of pore water content in

concrete under generic drying wetting conditions, Concrete Library of JSCE, No.31, pp. 275-287,

1998.11

T. Maruya, S. Tangtermsirikul, and Y. Matsuoka, Modeling of Chloride Ion Movement in

the Surface Layer of Hardened Concrete, Concrete Library of JSCE, No.32, pp.69-84, 1998.12

O. E. Gjrv and K. Sakai, Testing of Chloride Diffusivity for Concrete, Proceedings ofthe International Conference on Concrete under Severe Conditions, CONSEC95, pp.645-654,

1995.13

J. R. Welty, C. E. Wicks and R.E. Wilson, Fundamentals of Momentum, Heat, and Mass

transfer, John Wiley & Sons, Inc., 1969.14

H. Freiser and Q. Fernando, Ionic Equilibria in Analytical Chemistry, John Wiley & Sons,

Inc., (1963).15

T. Saeki, H. Ohga and S. Nagataki, Mechanism of Carbonation and Prediction of

Carbonation Process of Concrete, Concrete Library of JSCE, No.17, pp.23-36, 1991.16

West, J.M., Corrosion and oxidation, Sangyo-tosyo, 1983.17

Ishida, T., An integrated computational system of mass/energy generation, transport and

mechanics of materials and structures, PhD thesis submitted to University of Tokyo, 1999 (In

Japanese).18

T. Uomoto and Y. Takada, Factors Affecting Concrete Carbonation Ratio, Concrete

Library of JSCE, No.21, pp.31-44, 1993.19

K. Yokozeki, K. Motohashi, K. Okada and T. Tsutsumi, A Rational Model to Predict the

Service Life of RC Structures in Marine Environment, Forth CANMET/ACI International

Conference on Durability of Concrete, SP170-40, pp.777-798, 1997.20

T. Shimomura, Modelling of Initial Defect of Concrete due to Drying Shrinkage,

Concrete Under Severe Conditions 2, CONSEC 98, Vol.3, pp.2074-2083, 1998.21

T. Nishi, T. Shimomura and H. Sato, Modeling of Diffusion of Vapor Within Cracked

Concrete, Proceedings of the JCI, Vol. 21, No. 2, pp.859-864, 1999 (In Japanese).22

X. An, K. Maekawa and H. Okamura, Numerical Simulation of Size Effect in Shear

Strength of RC Beams, Proceedings of JSCE, No.564, V-35, pp.297-316, 1997.

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