Multi-sale modeling to link observed behavior, characterization and
analysis
H. Mihashi
Tohoku University, Sendai, Japan
K. Maekawa & T. Ishida
The University of Tokyo, Tokyo, Japan
S. Asamoto
Saitama University, Saitama, Japan
I. Maruyama
Nagoya University, Nagoya, Japan
ABSTRACT: This paper provides an overview of the mathematical modeling for time-dependent behaviour
of cementitious materials. Since the cementitious materials have complex and multi-scale structures, the
modeling need to bridge a very wide range of the order from nano-scale for material science level to meterscale for structural analysis. Furthermore the structure of the cementitious materials is formed through
chemical reaction and changes with time as a function of the environmental condition. Thus analytical models
for observed behaviour in the cementitious materials need to be sufficiently comprehensive and precise
characterization for material structure and measurement of the environmental conditions are essential for
building up the constitutive models.
1 INTRODUCTION
For durability design and/or prediction of life time
of concrete structures, nowadays it is essential to
apply some relevant mathematical models to
simulate the time-dependent behaviour of concrete.
Cementitious materials such as concrete have
complex multi-component and multi-scale structures
that first form through chemical reaction and then
continue to change with time (e.g. Jennings et al.
2008). As a result, various aspects of the behaviour
have been observed but sometimes the real
mechanisms are not clarified. If the observed
behaviour is just simplified as a material law, the
potential of modern numerical methods for
computerized structural design is not fully used.
It could be noticed that a lot of information on the
cementitious materials exists dealing with the
various aspects of formation and development of the
micro-structure and strength. In the field of cement
chemistry, chemical reaction process, hydration
products and micro-pore systems were deeply
studied. Material structure of the heterogeneous
composite system, mechanism of crack initiation
and propagation, and mass transportation through
the internal network of the porous micro-structure
and cracks were studied in the field of material
engineering.
For the analysis and design of
buildings and infrastuctures, however, those
scientific knowledges are limited to be transferred
but averaged material laws have been obtained by a
large amount of experimental studies.
One way to come to more realistic and more
reliable material laws is to link the observed timedependent behaviour with details of the material
structure and with real mechanisms (Wittmann,
1983). However, the observed behaviour of concrete
cannot be linked directly each other and a hierarchic
system of at least three different levels needs to be
introduced. Table 1 shows an example proposed
by Wittmann (1983). For combining all of these
observed behaviour into a unified system,
conceptural modelings are essential to integrate
carefully all of the important findings on these
different levels into a general computational
framework based on mathematical modelling.
Here the modelling of concrete should cover
microstructure, hydration, temperature, moisture
content, state of equilibrium, and mechanical and
volumetric changes. The integration is to be
sought in terms of a computational tool that can take
into consideration the development of microstructure and various changes of concrete exposed to
any arbitrary environment. Furthermore, for
predicting the life time of concrete structures, it
needs to predict the structural performance of
concrete together with the transport of various
external agents into the concrete microstructure
(Maekawa et al. 1999).
In this paper, some examples of such
mathematical modelling and computational tools
together with some experimental technologies to
characterize the material structure are overviewed.
Table 1 Three structural levels of concrete and corresponding
characteristic features, mechanisms and models. (Wittmann 1983)
Structural
Levels
Micro-level
Characteristic
features
Structure of
Hardend Cement
Paste, Xerogel
Meso-level
Pores, Cracks,
Inclusions
Macro-level
Quasi-homogeneous
Structural Elements
2 MATHEMATICAL MODELING FOR TIMEDEPENDENT BEHAVIOR OF
CEMENTITIOUS MATERIALS BASED ON
MICRO-MESOSCOPIC APPROACH
2.1 Microstructural modeling of concrete
As a starting point for predicting the time-dependent
behavior of concrete or concrete structures, i.e. heat
production and resultant thermal deformation,
autogenous shrinkage, drying shrinkage and creep
behavior, microstructural modeling of concrete is
discussed. For the initial state of concrete, which is
determined by the properties and performance of
raw materials, mixture proportion, work operation,
and hydration process, a couple of models have been
proposed with which the evolution of the initial state
of the concrete can be predicted as a function of the
mixture composition and curing condition. The five
most frequently discussed micro-structural models
are the HYMOSTRUC (van Breugel 1991),
CEMHYD3D (Bentz & Garboczi 1993), Navi’s
model (Navi & Pignat 1999), DuCOM (Maekawa et
al. 1999, Maekawa et al. 2003) and CCBM
(Maruyama et al. 2007). Regarding, especially
deformation
phenomenon,
HYMOSTRUC,
CEMHYD3D, CCBM, and DuCOM are dealing
with, and consequently, these models are
overviewed here.
HYMOSTRUC predicts the degree of hydration
from the data of water to cement ratio, mineral
components of cement, particle size distribution, and
curing condition (i.e. humidity and temperature).
The particle size distribution is represented by
Rosin-Rammler function and the particles are
uniformly placed in the space in statistical way.
Shape of cement particles is assumed to be
spherical, and it keeps the geometry during
hydration process. Inter-particle contact with each
other occurs when the cement particles grow. The
effect of inter-particle contact on the rate of
hydration is modeled with the concept of ‘cell
density’.
Mechanisms
Particle Displacement,
Capillary Tension,
Disjoining Pressure,
Surface Free Energy
Crack Formation and
Extension, Differential
Stresses
Apparent Mechanisms
Type of Models
Materials Science
Models
Materials Engineering
Models
Structural Engineering
Models
Material laws
The rate of hydration is described as a function of
basic rate of hydration and coefficients of diffusion
process in hydrates with Arrhenius’ law for curing
temperature effect. The basic rate of hydration is
determined by the amount of C3S and C2S. The
coefficients of diffusion process in hydrates take
into account of density of hydrates, residual water
and relative humidity in pore structure.
Using HYMOSTRUC, van Breugel & Lokhorst
(2001) explained the creep behavior of early age
concrete based on the concept proposed by Ghosh
(1973). Assuming inherent creep of cement hydrates
and strain conservation among cement hydrates,
redistribution of stress between old hydrates and
new hydration products was mathematically solved
in his study.
Koenders & van Breugel (1997) developed a
model for autogenous shrinkage based on the
HYMOSTRUC. In this model, a simple pore size
distribution curve was used, and this pore size
distribution, residual water, and adsorption curve
gave the thermodynamic equilibrium, and relative
humidity in cement paste matrix. Thermodynamic
equilibrium and a similar approach to Munich model
(Wittmann 1982), which is based on the surface
tension concept, can produce the deformation of
cement paste in early age.
CEMHYD3D is using the method of Cell
Automaton, and accumulation of reaction in local
cells represents the total reaction of cement paste
and gives the discrete (or voxel) 3-dimensional
cement paste structure. The accumulation is
determined by the local chemical reaction rule
which is a function of density of substances in target
cell and neighborhood cells. This unique model can
take into account many chemical reactions at once.
But discrete system in size and time, may produce
some other problems, such as evaluation of time
with changing curing temperature and reaction of
fine particles like silica fume. In spite of these
problems, combining with statistical approach,
CEMHYD3D has been successfully applied to many
engineering problems.
Regarding shrinkage problem, combining
CEMHYD3D and geometrical algorithm (Bentz et
al. 1995) can take into account the thermodynamic
equilibrium and relative humidity in the cement
paste matrix. Using these parameters, and
considering specific surface free energy and
Bangham equation (Bangham & Fakhoury 1931) it
can simulate the deformation of porous material
(Bentz et al. 1998).
Recently, CCBM was proposed by Maruyama et
al. in Japan. This model is based on the fundamental
kinetic model for Portland cement developed by
Tomosawa (1974 &1997). The kinetic model is
expressed as a single equation composed of four rate
determining coefficients which represent the rate of
surface solution, formation and destruction of initial
impermeable layer (for dormant period), and the
following diffusion controlled process. This model
also assumes that cement is spherical and it keeps its
shape during hydration. Aiming for the universality,
CCBM introduced the particle size distribution,
reaction of each mineral compositions, coefficients
for precipitation rate, and difference of density of
cement gel which is affected by the temperature
history. Model parameters were determined
according to the cement reaction data obtained by
X-ray/Rietvelt method (Hoshino et al. 2006,
Matsushita et al. 2007). Phase composition problem
was also modeled according to experimental data
(Maruyama et al. 2007). Fig.1 shows the comparison
of degree of hydration between the experimental
results and calculated results by CCBM. This model
focused on the problem of early age cracking in RC
members. Heat capacity and coupling of the heat
and moisture transport were modeled precisely and
succeeded in predicting the temperature history and
distribution (Maruyama et al. 2006). Using capillary
tension and disjoining pressure approach for
autogenous shrinkage, self-induced stress in RC
members coupled with thermal deformation and
autogenous shrinkage was evaluated with FE
analysis (Maruyama & Sato 2005).
Another Japanese micro-structural simulation
model named DuCOM was also developed by
Maekawa, Ishida, Kishi and others in the University
of Tokyo (Maekawa et al. 1999, Maekawa et al.
2003). The analytical system can simulate hydration
reaction, micro-pore structure formation, moisture
transport/equilibrium and their system dynamics
based on thermodynamics. DuCOM is comprised of
mainly the multi-component hydration heat model,
microstructure development model and moisture
transport and equilibrium model. Each model is
briefly explained below.
In the multi-component hydration heat model of
DuCOM, the chemical compounds of Portland
cement are aluminate (C3A), alite (C3S), belite
(C2S), ferrite (C4AF), and gypsum. Other blending
materials, such as blast furnace slag, fly ash or
limestone powder, are incorporated into the model
as individual components. The total heat generation
rate per unit volume of such a blended cement is
represented as the sum of the specific heat
generation rate of the individual clinker component.
The temperature-dependent heat generation ratio of
each clinker component is based on the Arrhenius
law. The heat generation rate can be idealized by
two material property parameters, i.e. the reference
heat rate and activation energy, when temperature is
constant at 293K. In the heat generation, the
following factors are taken into account as mutual
interaction: retardation of the cement-slag reaction
at an early stage caused by the presence of fly ash
and organic admixtures, free water consumption due
to hydration progress, heat generation rate change of
slag and fly ash based on the amount of calcium
hydroxide in the liquid phase, the interdependence
between C3S and C2S and relative fineness of each
mineral component. Finally, the temperature
distribution and the degree of hydration of the
concrete
can
be
obtained
by
applying
thermodynamic energy conservation to the space
and time domain of interest. The hydration degree of
hardened cement paste in this model is assumed to
be the ratio of accumulated heat Q of hydrated
products to final heat generation Q∞. According to
the hydration model, the hydration reaction process
for an arbitrary cement powder can be simulated
rationally.
Next, the microstructure development model is
summarized. The model can simulate the
microstructure formation process of the hardening
cement paste matrix for any mix proportion under
arbitrary environmental conditions based on the
information from the above multi-component
hydration model. In the microstructure model,
micro-pores in the concrete are classified into three
types: capillary pores, gel pores, and interlayer pores.
The capillary pores are located in the relatively large
interparticle spaces between the powder particles;
they act as free space for the precipitation of hydrate
products. The gel pores are thermo-dynamically
defined as the spaces where new C-S-H gel grains
conditionally precipitates only when the temperature
rises (Nakarai et al. 2007). The interlayer pores are
located between the layer structures of the C-S-H
gel grains; the layer thickness is defined as 2.8 Å
corresponding to water molecular size. From the
average degree of hydration and the weight of
chemically combined water computed in the
hydration model, the volume of gel products and
porosity of capillary pores, gel pores and interlayer
pores are calculated in the model. The entire micropore structure is idealized as the total porosity of the
above-mentioned capillary pores, gel pores and
interlayer pores. The pore distribution of each pore
is represented by a simple Raleigh Ritz function,
postulating that the pores are cylindrical in shape.
C3S
1
C 2S
EC 0.35-10o C
0.8
C 3A
EC 0.50-10o C
C 4AF
LC 0.35-10o C
LC 0.50-10o C
0.6
0.4
Degree of hydration
0.2
0
o
EC 0.35-20 C
0.8
o
o
o
EC 0.50-20 C
LC 0.50-20 C
LC 0.35-20 C
0.6
0.4
0.2
0
0.8
o
EC 0.35-40 C
0.6
0.4
0.2
0
1
LC 0.35-40oC
EC 0.50-313K
10
100
1
10
100
1
10
100
1
LC 0.50-40o C
10
100
1000
Age (hour)
Fig.1 Experimental results of degree of hydration of C3S, C2S, C3A, and C4AF
(dots), and calculated ones by CCBM using fitted parameters (lines).
Moisture in the cementitious microstructure can
be presented in both liquid and vapor phases in the
moisture transport and equilibrium model. The
vapor and liquid phase transports are formulated
based on mass conservation laws. The pore moisture
is classified as adsorbed water, condensed water in
capillary pores and gel pores, and interlayer water in
the interlayer pores in the model.
Firstly, the moisture equilibrium model is
explained. Considering the surface equilibrium
between the vapor and liquid phases in fine pores,
the relationship between the radius rs of pore where
the meniscus is formed and pore humidity can be
obtained from Laplace equation and Kelvin’s
equation, when the interface is a part of an ideal
spherical surface. The amount of water in the
microstructure at a given ambient relative humidity
can be calculated because the porosity distribution
of the microstructure is known from the above
mentioned microstructure development model. Pores
of radius less than rs would be completely filled,
while the other pores would be empty. Thus, the
model gives the saturation of pores by integrating
the total pores volume from zero to rs. The pores
include physically adsorbed water in their solid
walls. In the model, the adsorbed layer is idealized
based on the modified BET theory (Hillerborg
1985). The degree of moisture saturation in concrete
is then given as the total of saturation in micro-pores
consisting of capillary pores, gel pores and
interlayer pores, and the volume of adsorbed layer.
The adsorption-desorption hysteresis is explained as
the inkbottle effect in the model. It is assumed that
moisture can be trapped in inkbottle-shape pores
owing to the complex geometrical characteristics of
the microstructure. During the wetting stage, as
relative humidity increases, liquid water can
gradually condense in the smaller pores. On the
other hand, during the drying stage, evaporation of
moisture occurs from larger pores. During virgin
drying, the evaporation of trapped water in larger
inkbottle-shaped pores can be inhibited due to
connection to smaller pores. The result is an
apparent adsorption-desorption hysteresis. In the
model, the saturation in monotonic drying is
idealized based on the inkbottle effect. Since the
stability of the condensed water in inkbottle-shaped
pores is strongly dependent on the ambient
temperature according to experimental findings, the
temperature sensitivity in saturation-humidity paths
is implemented (Ishida et al. 2007). The hysteresis
of interlayer water is also modeled based on the
experiments that revealed the hysteresis has a
temperature dependence, too (Ishida et al. 2007).
Next, the moisture transport model is described.
Moisture transport in a porous body is driven by the
pore pressure gradient and the temperature gradient.
Ishida et al. (2007) proposed a formulation of vapor
flux that can be applied to arbitrary temperature
conditions. The model takes into account of factors
reducing the apparent diffusivity of vapor, such as
the complexity of the pore network. The vapor flux
is driven by the gradient of absolute vapor density ρv
of the system, instead of by relative humidity.
2.2 Mesoscopic modeling for time-dependent
behaviour of concrete
Recently, a project has been in progress, which
couples
the
microscopic
thermo-physical
information obtained from DuCOM with
macroscopic structural information such as stress,
strain, deformation, cracking and others computed
by a 3D FEM structural system named COM3
(Maekawa et al. 2003). One of the targets of this
project is the development of a multi-scale
constitutive model that can simulate the timedependent deformation of concrete such as
shrinkage and creep based on the hydration reaction,
pore structure and moisture state in the pores
(Maekawa et al. 2003, Zhu et al. 2004, Asamoto et
al. 2006). Outline of the multi-scale constitutive
model is described below.
In the model, concrete is idealized as a two-phase
composite with aggregate and hardening cement
paste. The aggregates are modeled as elastic
particles with a stiffness determined by their density,
while the hardening of cement paste is expressed by
the progressive formation of finite fictitious clusters
as the hydration proceeds based on solidification
theory (Bazant & Prasannan 1989). The number of
clusters is dependent on the degree of hydration as
obtained from DuCOM. The stress in the hardening
paste is given as the summation of stresses applied
to all clusters. Both volumetric terms and deviation
of stress and strain are computed, taking into
account of interactions between aggregate particles
and the cement paste matrix.
According to the model, the mechanical behavior
of a fictitious cluster is associated with the
thermodynamic state of moisture in micro-pores
such as capillary pores, gel pores and interlayer
pores. Recently, moisture migration in gel pores was
divided into moisture transport through the internal
pores of C-S-H gel grains and water in motion
within the inter-particle spaces of hydrate microproducts (Asamoto et al. 2006). The motion of
moisture in capillary and gel pores is idealized based
on seepage theory, which describes moisture in the
pores migrating gradually under sustained stress.
DuCOM
thermo-hygro-physical system
Multi-component hydration heat model
Hydration
degree,
temperature
Outer
product
Inner
product
Microstructure development model
Outer product
density ρr=ρr(φr)
ζ
rr
r0
Multi-scale constitutive model
Pore distribution for
capillary, gel and
interlayer pores
Concrete
σ0=Vagσag+Vcpσcp
ε0=Vagεag+Vcpεcp
dV/d log r Vapor
εe
Saturation,
pore humidity
Condensed
water
Liquid transport
rc
log r
Trapped water
in inkbottleshaped pore
Time
σcp, σag, εcp, εag: Volumetric stress
and strain of aggregate, cement
paste, respectively
Kag: Volumetric stiffness of aggregate
Vcp, Vag: Volume fractions of
aggregate, cement paste
CSH gel
Vapor transport
Cc
Capillary water
10-8~10-6m
kslL
CgL
Gel water L
3.0x10-9~10-7m
kslS
CgS
εgL
εgS
εly=εe+εc+εgS+ε
σgSly +ε
= lEe ⋅ ε e
Instantaneous
stiffness
Ee
Ec
ki
Cc
kslL
CgL
kslS
CgS
Solidifying
cluster of
cement
paste
ki
Free space for precipitation
of hydrate products
Gel water S
10-9~3.0x10-9m
Interlayer water
10-10m
σly
σly, εly: Volumetric stress and
strain in the cluster, respectively
dε c
dt
Ee
Ec
Rheological model
of solidifying cluster
σly
εc
εl
σ ly = Ec ⋅ ε c + EcCc
Microstructure at
matured age
Aggregate
εag=σag/3Kag
Vag
δm
δmax
Microstructure at
an early age
Cement paste
εcp =f(σcp)
Moisture transfer and equilibrium model
gel
Cement paste model based
on solidification theory
Concrete model
(Two-phase composite)
Vcp
Unhydrated
core
dr
Since the rate of motion depends on pore size, it
takes longer for moisture in smaller pores to reach
the equilibrium. The moisture in interlayer pores
diffuses and causes the volumetric change only at
high temperature. Each component is composed of
an elastic spring, a dashpot and a slider whose
parameters are determined by temperature, moisture
saturation, pore size distribution and other factors of
the hardening cement paste.
The driving force behind volumetric change in the
model is assumed to be related to capillary pressure
and solid surface energy. These phenomena are
computed, coupling with the thermodynamic state of
moisture in the micro-pores. The capillary force is
calculated based on the Laplace equation, the Kelvin
equation and the degree of saturation of capillary
and gel pores subjected to pore pressure. Shrinkage
under a relatively high humidity condition is caused
mainly by the capillary force in the model. On the
other hand, in the case of a relatively low humidity
condition, the increase in the solid surface energy of
gel particles due to desorption of water is idealized
as the main mechanism to drive the shrinkage. By
coupling the above volumetric stress depending on
moisture states in pores with the C-S-H skeleton
stress, the equilibrium between skeleton stress and
external loading can be satisfied. Thus, the timedependent deformation can be computed according
to the boundary conditions without conventional
classification such as autogenous/drying and
basic/drying creep. Currently, the model is able to
reasonably simulate the time-dependent deformation
of cementitious composites under arbitrary boundary
conditions (Zhu et al. 2004, Asamoto et al. 2006).
Fig.2 summarizes DuCOM and multi-scale
constitutive model concept.
Number of cluster is
based on hydration
degree
1
Interparticle space
among gel grains
2
3
1: Capillary pores
2: Outer products
3: Inner products
4: Unhydrated core
4
C-S-H gel
grain
10-5~6m
Ee, Ec, Cc, CgL, CgS, kslL, kslS, ki : Function of
thermo-dynamics state (hydration ,
temperature, saturation etc)
Fig.2 DuCOM and multi-scale constitutive model
2.3 Experiments for verifying models
Some of model components, especially related
with the phenomena in micro-pores, are dependent
on various macroscopic assumptions arising from
experiments. Regarding CCBM, model parameters
of hydration process are determined according to the
experimental data of X-ray/Rietvelt analysis, DSC,
and MIP with several assumptions for amorphous
phase component. And the accuracy of the model is
discussed with experimental data of ig. loss,
calorimeter, and X-ray/internal standard method. In
the case of DuCOM, the modeling of the
temperature-dependent hydration process and
microstructure formation of gel pores is based on
indirect information obtained from the adiabatic
temperature test and the thermogravimetric analysis.
The hysteresis and the temperature sensitivity of
moisture such as trapped water in inkbottle-shaped
pores and interlayer water are simply idealized
according to macroscopic experimental findings.
The ultimate recoverable and irrecoverable timedependent deformation in the model is also based on
the various experimental data of shrinkage and
creep. The hydration, microstructure, moisture
states, and time-dependent deformation in fine pores
have been extensively studied over several decades,
but it has been difficult to directly observe the
phenomena and verify the theories, particularly at
nanometer scale. Thus, the modeling must be
postulated from macroscopic information and
remain empirical. According to the model, the
computational prediction can give a good agreement
with the phenomena that have been experienced,
Experimental
Characterization
analysis
of structure
Phenomena
Capillary
pores
Mesopores
10nm
Gel
pores
1nm
Interlayer
pores
1Åm
z Neutron
Radiography
(1mm~10mm)
Microcracking
drying
Saturation
Surface
tension
Pore
Pore
distribution
0
Pore
distribution
dV
d(lnr)
Liquid
water
Pore radius
0
lnr
0.2 0.4 0.6 0.8 1.0
RH
Hysterisis
Trapped water
in inkbottle shaped pores
vapor
Sc Sink
rc
Pore radius
Seepage of water
Disjoining
pressure
z Digital & 3D laser
microscope
(1mm~1µm)
[20?C]
1.0
0.8 W/C=25%
0.6
0.4
0.2
W/C=55%
Pore pressure
100nm
Modeling
Sustained load
Microcracking
1µm
while the unusual extending conditions and
phenomena in the far future are hardly predicted at
convincing level. In order to predict the long-term
durability of RC structures and to utilize a variety of
cementitious materials for various structures, the
intrinsic micro information from experimental
analysis is necessary.
Recent remarkable development of the
experimental technique is expected to reveal the
above unclear microphysical phenomena. It may
enable us to investigate the microstructure and
moisture behavior at micrometer and nanometer
scale. Directly observing the time-dependent
behavior of cement hydrates in fine pores, each
phenomenon can be clarified and the various
theories assuming from macroscopic information
can be verified. Since the molecule behavior is
dominant at nanometer scale and should be modeled
not only by classical thermodynamics but also by
molecular dynamics, statistical theories and others,
the experimental studies are useful for idealizing the
phenomena at nanometer level. Identifying the
dominant phenomenon and physics in each scale for
the time-dependent behavior at macro-scale, the
modeling of the time-dependency of cement hydrate
at the multi-scale is possible and enables to predict
the deterioration process of RC structures under
arbitrary boundary conditions. Thus, as shown in
Fig.3, the linkage between the phenomenon,
experimental analysis and modeling based on
theories at each scale is quite important. In the next
chapter, the state-of-the-art of experimental
techniques to verify the theory at each scale will be
described.
Adsorption &
desorption
Micro diffusion of water molecule
Dispersion of interlayer water
at elevated temperature
Pc = −
Pore pressure, Kelvin equation
ta =
 − 2γMw 
k1 ×
h
÷
h = exp
÷
(1− h / hm )(1− h / hm + k2 ×
h) m
 RTρl rl 
z Internal pore
humidity sensor
intrusion
porosimetry
Understanding
µ
of phenomena (1 m~10nm)
BET theory
∆l / l = λ∆γ
Munich model
z Mercury
φ(r )= φ cp ×
Vcp (r )+ φ gl ×
Vgl (r )+ φlr
Linking
with model
Vi (r ) = 1 − exp(− Bi ×
r)
dVi (r )= Bi ×
r×
exp(− Bi ×
r )×
d ln r
z Gas
adsorption
isotherm (1µm~10nm)
z SAS
Pore distribution model
(100µm~10nm)
z SEM
(1µm~1nm)
z NMR
(10nm~1nm)
ρ RT
2 γM w
2γ
=− l
ln h rs = −
RTρl ln h
rs
Mw
Ec
Cc
Spring and dashpot
for visco-elasticity
Fig.3 Schematic illustration of the linkage of phenomena,
characterization of structures and modeling
kg
Cg
Dashpot and slider
for visco-plasticity
3 EXPERIMENTAL TECHNOLOGY AND
METHODOLOGY TO MEASURE
MICROSCOPIC INFORMATION OF
CEMENTITIOUS MATERIAL
3.1 Propagation and distribution of micro-crack
and moisture transport in micro-crack
(1mm~1µm level)
3.1.1 Time-dependent behavior in micro-crack
The behavior of microcracks in cementitious
materials has been extensively studied because the
microcracks can affect the diffusivities of chloride
ion, CO2 and other harmful agents in concrete and
they lead to the reduction of the durability. Gran
(1995) reported an impregnation technique based on
fluorescent ethanol that can be slowly replaced with
the pore water. This technique is called as
Fluorescent Liquid Replacement(FLR) technique
and does not need traditional preparation procedures
such as drying and evacuation that may induce
microcracks. The FLR technique gives impregnation
depths that are several orders of magnitude larger
than the measurable impregnation obtained with
traditional procedure using epoxy. According to
him, the FLR technique can be used to observe the
crack development due to loading, freezing/thawing,
corrosion etc., even in the case of low W/C concrete
with dense porosity (Gran 1995). Ammouche et al.
(2000 & 2001) proposed an image analysis
technique for the quantitative assessment of
microcraks in cement-based materials. The
highlighted microcracks and other microdefects of
the polished samples were observed by using an
Section 1
Profile1
Total
Section 1
Section 2
Section 3
optical microscope. The various micro information
such as porosity, air bubble and microcracks was
extracted from the observed image conducting a
pretreatment of the color image and an automatic
thresholding on the gray level histogram. The
characteristics of the crack network were quantified
using classical stereological methods. The technique
enabled to determine the specific crack length, the
degree of orientation and other crack pattern
characteristics and seemed to lead to the basis of the
micromechanical discussion.
Recently, the development of laser microscopes
and digital microscopes has been progressive. The
advantage of these technologies is to observe the
phenomenon in micro-pores with high resolution
and color under arbitrary conditions such as wetting
conditions, while SEM, X-ray diffraction and others
need some preprocessing of samples, for example,
vacuum drying and evaporation coating. In the case
of laser microscopes, 3D laser scanning technology
has been advanced and enables us to examine the
surface roughness of specimens at high resolution of
nm level (http://www.keyence.co.jp/microscope/lase
r /vk_9700/index.jsp?). The observable 2D range of
3D laser microscopes and latest digital microscopes
is about from 1 µm to 1 mm. Although these
technologies are not applicable for studying
phenomena at nanometer scale, it is expected that
the propagation of micro-crack under sustained
stress and the meniscus formation in micro-meter
pores are observable because they are able to
continuously measure such behavior. An example of
observation on the surface of hardened cement paste
by using the above mentioned 3D laser microscopes
is shown in Fig.4.
Section 2 Section 3
Horizontal Height
Average
Length of
Cross-sectional
Angle
distance difference height
cross-section
area
82.537um 16.696um 21.949um 11.436°
188.183um
1813.561um2
8.010um 5.032um 25.912um 32.140°
16.855um
209.862um2
6.965um 1.046um 23.975um 8.540°
11.905um
167.993um2
8.358um 5.862um 20.813um 35.042°
19.360um
174.249um2
23.333um 11.940um 70.700um 75.722°
Total
8.358um 5.862um 25.912um 35.042°
Maximum
6.965um 1.046um 20.813um 8.540°
Minimum
7.778um 3.980um 23.567um 25.241°
Average
Standard deviation 0.592um 2.102um 2.102um 11.868°
1.776um 6.306um 6.305um 35.605°
3σ
48.120um
19.360um
11.905um
16.040um
3.098um
9.293um
552.104um2
209.862um2
167.993um2
184.035um2
18.441um2
55.322um2
Fig.4 Observation of cement paste surface by 3D laser microscopes
3.1.2 Moisture transport around crack and between
aggregate and cement paste matrix
Recently, Neutron Radiography (NR) technique
was adopted for the observation of moisture
transport behavior around crack. Kanematsu et al.
(2007) observed moisture behavior in different crack
width ranging from 0.05 to 0.3mm in the specimen
whose size was 100 x 100 x 20mm. Relative water
content in the concrete was also the one of
parameters. According to the experiment, it was
confirmed that the speed of penetration of moisture
in the crack is affected by the relative water content
in the concrete. The typical figure of NR is shown in
Fig.5.
Fig.5 W/C=50%, crack width 0.05mm, relative water content 30%(Kanematsu et al. 2007)
For mitigation of autogenous shrinkage, recently
saturated porous aggregate was applied to the
concrete of low water to binder ratio. For the
performance-based design of these kinds of material
and evaluation of concrete performance, effect of
water in porous aggregate on the reduction of
shrinkage of concrete should be evaluated.
Regarding this aspect, Lura (2003) used ink for the
observation of the effective distance from the
aggregate surface, and this experiment revealed that
the range within 1mm from the cement paste matrix
to the aggregate surface was colored, and, at least,
1mm distance from the aggregate was supplied
water from the aggregate. Bentz et al. (2006) used
the X-ray tomography technique for getting the
information of water distribution around saturated
light weight aggregate. From this experiment, water
was supplied to the position of 2mm away from the
aggregate surface. Maruyama et al. (2007) also
carried out a similar experiment using NR, which is
shown in Fig.6. In this experiment, porous aggregate
was limitated by hardened cement paste with
W/C=0.55, and W/C of cement paste matrix was
0.30. Subtraction of water strength obtained by NR
test, that the range of supplied water in hardening
cement paste was 4mm from the surface of imitated
aggregate was detected. Fig.6 shows the distribution
of water strength of NR around cement paste matrix
- aggregate interface.
Fig.6 Moisture transfer between different porous media during hydration process
adsorption isotherm relationship, the accumulated
pore capacity curve similar to the mercury intrusion
porosimetry is gained. The adsorption volume is
obtained with the volumetric method, while the
mercury intrusion volume in the mercury intrusion
porosimetry directly corresponds to the volume of
pores.
The BET theory is used for evaluating the
specific surface area based on the adsorption
isotherm. The theory is expanded from the Langmuir
theory and multilayer adsorption state is considered,
assuming the adsorption site on the adsorbent. The
adsorption isotherm expressed with the BET theory
nearly corresponds to the II type adsorption
isotherm. The range of relative humidity in which
the BET theory is accepted is about from h=0.05 to
0.35 in the case of hardened cement paste. The
monolayer adsorption volume and interaction
constant are determined from the results of the BET
plot within this range. In Fig.7, the solid line in the
graph is the fitting results of the relative humidity
range of h=0.05 to 0.35 in a straight line. The
monolayer adsorption volume, specific surface area
and interaction constant are calculated based on the
fitting results and are marked in each graph.
Davis (1984) showed the correlation of the BET
specific surface area using nitrogen adsorption
isotherm and the mercury intrusion porosimetry by
targeting a variety of samples. Davis pointed out that
the mercury intrusion porosimetry gives a larger
specific surface area than the BET methods does.
However, the results in both methods are not so
different and they can be directly compared for
practical purposes. Fig.8 shows the comparison of
the specific surface of hardened cement paste
measured with these two different methods. In these
results, the mercury intrusion methods tend to give
larger specific surface area than the BET method
does.
3.2 Microstructure and moisture state in micropores (1µm~10nm level)
3.2.1 Pore distribution and volume
The mercury intrusion porosimetry has been widely
used to study the microstructure of hardened cement
paste. It measures the volume of mercury which
infiltrates into sample pores according to injection
pressure. By converting the injection pressure of the
test results into the cylindrical pore radius, and
assuming the mercury intrusion volume as volume
of the pores which corresponds to the cylindrical
shaped pores, the relationship between the pore
radius and volume can be obtained. This relationship
is said to be the cumulative pore capacity curve and
is the most basic relationship which describes the
distribution characteristics of pores in the
microstructure. The specific surface area of sample
pores is also evaluated from the mercury intrusion
porosimetry results. Based on the modeling which
assumes the geometry as the cylindrical shape,
specific surface area is evaluated as a total sum of
pore surface of cylindrical shape.
A gas adsorption test is also an accepted
methodology to examine the pore structure. It
measures the specified volume of gas adsorption to
adsorbent which it balances in relative humidity
under an isothermal environment, that is, the test
measures the adsorption isotherm. In the gas
adsorption test, as well as the mercury intrusion
porosimetry, the volume of gas adsorption
corresponds to sample pore volume. There are
various gases that can be used in the gas adsorption
tests. From a viewpoint of microstructure evaluation,
nitrogen is often used because it is an inert gas.
Kelvin equation is generally used to analyze the pore
size distribution from the adsorption isotherm.
Applying the Kelvin equation to the results of the
0.5
0.4
0.3
0.2
0.1
0.6
W/C=0.4
3
Monolayer volume (cm /g) : 4.311
BET surface area (m2/g) : 18.765
C value of BET equation : 32.721
0.5
h/(V(1-h))
h/(V(1-h))
0.6
W/C=0.3
Monolayer volume (cm3/g) : 2.014
BET surface area (m2/g) : 8.768
C value of BET equation : 14.684
0.4
0.3
0.2
0.1
0
0.2
0.4
0.6
h
0.8
1
0.4
0.3
0.2
0.1
0
0
W/C=0.5
Monolayer volume (cm3/g) : 5.893
2
BET surface area (m /g) : 25.653
C value of BET equation : 36.758
0.5
h/(V(1-h))
0.6
0
0
0.2
0.4
0.6
h
0.8
1
0
0.2
0.4
0.6
0.8
1
h
Fig.7 BET plot of hardened cement paste; N2 adsorption (Chiba Institute of Technology, Utsumi Lab.)
300
2
2
BETBET
surface
Surface
area(N
area(N
Surface area(N
2 )[m /g]
2 )[m /g]
40
250
BET surface area(N2)[m2/g]
35
30
200
25
150
20
100
15
10
50
5
0
0
0
0
5
10
15
20
25
30
35
40
Mercury surface area[m2/g]
Fig.8 Comparison of surface area of hardened
cement pastes with W/C=0.3, 0.4 and 0.5
obtained from N2 adsorption isotherms and
mercury intrusion porosimetry (Chiba Institute
of Technology, Utsumi Lab.)
Samples to be used in the mercury intrusion
porosimetrys and gas adsorption tests need to be
dried before the tests. Gallé (2001) studied porosity
obtained in the mercury intrusion porosimetry
results at the maximum injection pressure of 413
MPa by targeting the dried sample using each drying
method. It was found that samples executed in an
oven-drying at 105 ºC have the highest porosity in
the mercury intrusion porosimetry. Juenger et al.
(2001) deliberated the differences of the BET
specific surface area and the pore volume based on
the drying conditions targeting the nitrogen
adsorption test. According to their experimental
results, the oven drying at 105 ºC has the lowest
BET specific surface area compared to that of D-dry
and a combination of methanol elution and D-dry in
the nitrogen adsorption. The measured porosity is
dependent on not only test method but also predrying conditions.
Odler (2003) deliberated on the difference of the
BET specific surface area depending on each type of
adsorbate. Fig.9 shows the summarized results of the
BET specific surface of H2O adsorption and N2
adsorption described in references (Odler 2003,
Mikhail & Abo-El-Enel 1972, Mikhail & Selim
1966). From these results, it is found that water
vapor adsorption evaluates the higher specific
surface area than that of the nitrogen adsorption.
Feldman and Sereda (1968) explained the difference
in the specific surface area between water vapor
adsorption and nitrogen adsorption in terms of the
filling of interlayer pores only by water. Daimon et
al. (1977) pointed out that the only water could
absorb into intercrystallite pore of C-S-H gel and
interlayer pores, while nitrogen cannot enter such
small pores. Recently, Jennings and his co-workers
50
100
150
200
250
300
2
2/g]
BETBET
surface
Surface
Surface area(H
area(H
area(H
/g]
2 0)[m
2O)[m
Fig.9 Comparison of BET surface area obtained
from H2O adsorption and N2 adsorption
isotherms (∆: Odler 2003, ○: Mikhail & Abo-ElEnel 1972, µ: Mikhail & Selim 1966)
(Thomas et al. 1999, Jennings 2000, Tennis &
Jennings 2000) suggested that the calcium silicate
hydrate (C-S-H) in cement paste is classified into
two types: a high density C-S-H (HD C-S-H) and a
low density C-S-H (LD C-S-H). HD C-S-H is made
of densely packed particles into which nitrogen
cannot penetrate. Particles of LD C-S-H are not
packed as tightly as those of HD C-S-H, and
nitrogen can penetrate partially into this structure.
On the other hand, water can infiltrate into both
types of C-S-H and as a result the difference in the
specific surface area by vapor adsorption from
nitrogen adsorption arises. There is, however, no
unified explanation about the variation of surface
area due to adsorbents.
Due to the characteristic and theory of each test
method, the range of pore size of the analysis target
is specified. In the case of mercury intrusion
porosimetry, the pore radius in which mercury can
be infiltrated shall be specified depending on the
injection pressure. Currently, equipment which can
pressurize up to 400 MPa is widely used, and the
minimum pore radius which the mercury can
infiltrate during pressurization is about r=1.8 x 10-9
[m] according to Washburn equation. That is, based
on the maximum injection pressure of the equipment
to be used, the minimum limit value of the
theoretically evaluable pore radius in mercury
intrusion porosimetrys is specified.
On the other hand, the evaluable minimum pore
diameter in the gas adsorption test is the minimum
value of pore diameter whose molecule diameter to
be used can be detected from the gas adsorption
tests. Generally, since adsorption potential of micro
pores with less than 2.0 x 10-9 [m] is extremely
large, it is thought that Kelvin equation is not
approved in such fine pores. Dubinin equation
(Dubinin 1960) is also used for evaluating micro
pore size. In gas adsorption test, the maximum value
of pore diameter is theoretically infinitely large. At
the area close to saturated vapor pressure, however,
the experimental error is perceptively sensitive to
temperature control and pressure control, and
practically, pore radius range which can be
evaluated with Kelvin equation is assumed to be
about from 1 to 100 nm (see Fig.10).
micropore
mesopore
macropore
Gas adsorption(Dubinin eq.)
Gas adsorption(Kelvin eq.)
Mercury porosimetry(pmax=400MPa, Washburn eq.)
○：H2O Molecular radius
□：N2 Molecular radius
1x10
-10
1x10
-9
1x10
-8
-7
1x10
r [m]
1x10
-6
-5
1x10
1x10
-4
Fig.10 Measurement range of mercury porosimetry and gas adsorption
3.2.2 Internal humidity
Grasley et al. (2007 & 2006) developed a new
internal relative humidity measurement system and
proposed a model to predict drying stress gradient in
concrete. The system involved embedding of a
small, and digital, RH and temperature sensor in a
small plastic tube with Gore-Tex caps into fresh
concrete. The sensors were a capacitive type RH
sensor and used a thermistor to measure
temperature. In the system, the internal humidity
could be measured at the accuracy of ±1.8%(when
RH is less than 80%) and the temperature sensor had
an accuracy of ±0.3°C (at 20°C). Using the system,
the potential free shrinkage strain based on KelvinLaplace equation was calculated and the drying
stress induced by internal and external restraint at
any point across the cross-section was determined.
The model revealed that the tensile stress in the
surface layer routinely exceeded the tensile strength
of concrete in drying concrete, even in the case of
free shrinkage.
Ceramic humidity sensor is also used to measure
internal pore humidity in concrete. When water
vapor within a porous ceramic material diffuses and
is adsorbed on its crystal surface, moisture
dissociates into hydroxyl and proton on the surface,
reducing the electric resistance of the ceramic. The
humidity-sensing mechanism of a humidity sensor
utilizes this phenomenon, ionic conduction by water
adsorption. Among the wide variety of ceramic
humidity sensors available with various materials,
sensors made with NASICON ceramic materials
have the potential to be used as humidity sensors for
concrete, as they show a high ionic conductivity
while having relatively high durability and longterm stability (Yagi & Saiki 1991, Ozawa et al.
2007). Fig.11 shows the configuration of a sensor
element, which consists of an alumina substrate on
which a ceramic moisture-sensitive film is deposited
between two electrodes. With its body being small
and thin (5 x 13 x 0.8 mm) as shown in Fig.12, this
type of sensor embedded in concrete allows
measurement of humidity in concrete without
significantly disturbing the humidity profile of
concrete.
The measurement range and accuracy of this
sensor are 20 to 90% RH and ± 3 to 5%,
respectively, within the effective temperature range
of 0 to 50°C. When embedding this sensor in
concrete, it is necessary to calibrate its sensing
characteristics beforehand by conducting controlled
humidity testing, as various ions contained in the
moisture in concrete can affect the ion conduction.
Fig.13 and Fig.14 show typical sensing
characteristics of this sensor in air and in concrete,
respectively, obtained from controlled humidity
testing for 400 days. Fig.13 reveals that the
measurement error tends to increase as the humidity
decreases in air. In concrete, however, the
measurement error tends to increase both in the high
and low humidity ranges as shown in Fig.14. When
using this sensor, the measured data should be
corrected accordingly based on the recognized
sensing characteristics.
Fig.15 shows typical humidity measurements at
depths of 5, 10, and 50 mm from the surface of a
concrete specimen with a cross-sectional size of 10
by 10 mm.
100
80
Reference RH　[％]
Alumina substrate
Sensor layer
Upper electrode
Lower electrode
60
40
20
0
Electrode
0
50
100
Meas ured RH　[％ ]
Fig.11 Configuration of sensor
element
Fig.12 Sensor element
100
Relative Humidity
（％）
Reference RH　[％]
100
Fig.13 Characteristics of sensor in air
80
60
40
20
0
0
50
Measured RH　[％]
100
Fig.14 Characteristics of sensor in concrete
3.3 Measurable information in nano-pores
The microphysical information in nano-pores is
considerable to evaluate the durability of concrete
structures because it is known that the moisture in
fine pores greatly affects long-term creep, shrinkage
under low humidity and others. It has been,
however, very difficult to measure or observe the
structure and the moisture behavior in nano-pores
because the preprocess, for example, vacuum drying
50mm
90
5mm
80
10mm
70
Ambient
60
50
0
1
2
3
4
Time(Days)
5
6
Fig.15 Typical humidity measurements
and evaporation coating for experimental
technologies such as SEM, X-ray diffraction and
others can cause the destruction of the nano-pores
structure and dispersion of moisture in nano-pores.
Recently, the study focusing on the microstructure
and the moisture behavior in pores of nanomater
scale has been progressive using latest experimental
technologies.
Nuclear Magnetic Resonance (NMR) is one of
interesting technologies to be able to investigate the
time-dependent behavior in fine unsaturated and
saturated pores. Valckenborg et al. (2001) examined
the drying process and the pore water distribution of
saturated mortar by means of NMR. It was found
that the moisture cannot be extracted from gel pores
(<10nm) until the water in capillary pores (101000nm) is drained as shown in Fig.16. Hazrati et al.
(2002) measured water content profiles using proton
nuclear magnetic resonance imaging (NMRI)
technique during the capillary absorption of water in
unsaturated mortar samples. The water diffusivity
coefficient obtained in the experiment is strongly
dependent on the water content and can be
approximated by the sum of two exponential
functions. Heijden et al. (2007) built a dedicated
NMR setup in order to study the moisture transport
in heated concrete. One-dimensional moisture
profiles in fire-clay brick, calcium-silicate brick and
concrete during heating were measured and
compared with their model. Aono et al. (2007)
investigated the change of microstructure of
hardened cement paste due to drying and dryingwetting cycle using MIP, Archimedian method and
Si29-NMR. It was found that the pore distribution is
coarser due to drying because the condensation
polymerization of silicate anion is progressive
during drying.
Small-angle scattering techniques (SAS) using
neutrons (SANS) or X-rays (SAXS) are also
powerful method to study the microstructure of
hardening cement paste. The angular profile of the
Fig.16 Pore water distribution during drying
(Valck enborg et al. 2001)
small-angle scattered intensity from the sample is
effectively a Fourier transform of the microstructure
and can in principle be analyzed to determine the
size distribution, volume fraction, and shape of the
scattering features (Jennings et al. 2008). The
advantage of the technique is also to examine the
microstructure in saturated state. Thomas et al.
(1998) reported the variation on the surface area of
OPC with aging by SANS (Allen & Thomas 2007).
They found that the surface area increases slightly
with long-term curing and is greater in the case of
higher water-to-cement ratio. In addition, it was
suggested that the dense and inner product of C-S-H
cannot be observed by SANS or SAXS and they
interpreted the experimental findings as evidence for
two-different morphologies of C-S-H. Allen et al.
(2007) investigated the mean formula and mass
density of the nanoscale C-S-H gel particles in
hydrating cement by combining SANS and SAXS
data and by exploiting the hydrogen/deuterium
neutron isotope effect both in water and methanol.
They have established a C-S-H formula
(CaO)1.70(SiO2)(H2O)1.80 with a mass of density
2.604Mgm-3 and a water mass fraction of 0.174.
Studies by NMR and SAS techniques on
microstructure and moisture behavior in micro-pores
have been in progress recently. The nanoscale
information may lead to further modeling of timedependent behavior of cementitious materials under
arbitrary conditions.
Fig.17 surface area (solid circles) and heat evolved
(line) for OPC/H2O at 30 °C and W/C=0.4 (Thomas et
al. 1998)
3.4 Summary of current technologies and future
prospects
4.1 Effect of aggregates and interaction with
cement paste phase (1µm ~ 1-10mm level)
As described above, recent development of
experimental techniques to study microphysical
information has been remarkable and progressive. It
is expected that the development enables us to
reveal the classical assumption about timedependent behavior of cementitious materials and to
enhance the numerical models. Some of techniques,
however, show different results even in the same
measuring target. Since these differences appear to
be caused by the characteristic and limitation of
each test method, it is indispensable to understand
the technique correctly and to analyze the varying
test results comprehensively.
In order to verify various theories about
micromechanics of cementitious materials, study on
the microstructure related with moisture behavior in
micro-pores is essential because hydration process,
microstructure formation, transportation of ion,
shrinkage and creep are strongly affected by
moisture state in pores. Currently, NMR or SAS
seems to be one of the effective techniques to
examine such time-dependent behavior under
various conditions. For further investigation for
microphysical information, the application of
various experimental techniques should be taken
into account. Thus, the collaborative studies with
researchers in other scientific fields such as
chemical engineering, biotechnical engineering,
nano-technology and others are necessary.
Aggregates are used to reduce the shrinkage of
concrete by restraining the shrinkage of the matrix
paste. The roles of the type and amount of aggregate
to reduce shrinkage were addressed by Pickett
(1956) ， Roper (1959), and Hansen & Nielsen
(1965) as shown in Fig.18. They also indicated that
some types of aggregates often exhibit a large
degree of shrinkage. Consequently, the selection of
the type of aggregate can have a significant effect on
the long-term behavior of a concrete structure. The
water absorption of aggregates is directly related to
the shrinkage strain of concrete in the same types of
rocks. However, this correlation decreases when
different types of rocks are used (Snowdon &
Edwards 1962). It is well known that artificial
lightweight aggregate does not exhibit so much large
shrinkage strain in spite of its large water absorption
ratio. Fujiwara (1984) found that internal specific
surface areas (SSAs) of coarse aggregates measured
by the BET method using nitrogen were in close
agreement with the shrinkage strains, as shown in
Fig.19. Larger SSA will be associated with a greater
surface energy or disjoining pressure (Powers 1965),
resulting in a higher shrinkage (or expansion) strain.
On the other hands, a previous research showed that
the measured SSAs vary with the adsorbates used
(Brunauer et al. 1938). Hence, BET surface area of
the shrinking aggregate using water vapor will be
reasonable information. Fig.20 shows that the SSAs
using water vapor have a good agreement with
concrete shrinkage strains with various kinds of
aggregates (Imamoto & Arai 2007). Furthermore,
the evaluation of the shrinkage behavior from the
viewpoint of the SSA will provide a perspective for
new mixture design methodologies for concrete.
4 MATHEMATICAL MODELING FOR TIMEDEPENDENT BEHAVIOR OF CONCRETE
COMPOSITES BASED ON MESOMACROSCOPIC APPROACH
Fig.18 Volume of aggregates versus shrinkage strain
ratios (Hansen & Nielsen 1965)
Fig.19 BET(N2) versus aggregate shrinkage strain
(Fujiwara 1984)
-6
Drying shrinkage strain(x10 )
1200
1100
1000
900
800
R2=0.84
700
600
H1
H2
L1
500
L2
G
ALA
400
0
2
4
6
8
10
Specific surface area of coarse aggregates (m 2/g)
Fig.20 BET(H2O) versus concrete shrinkage strain
(constant aggregate volume) (Imamoto & Arai 2007)
500
400
300
200
Expt. (Mix A)
100
Expt. (Mix B)
Expt. (Mix C)
0
0
20
40
60
Age (Day)
80
100
property on mechanical behavior of concrete should
be studied and taken into account in the design.
Asamoto et al. (2008) investigated quantitatively
the influence of various aggregate properties on
concrete shrinkage behavior using the multi-scale
constitutive model described in Section 2.2. One of
the aggregate properties could be possibly
responsible for the large shrinkage of the bridge
concrete. For example, the targeted aggregate had
relatively softer elastic modulus in comparison with
normal aggregate, (JSCE concrete committee 2005).
According to the numerical simulation, however,
such large shrinkage due to the soft stiffness of
aggregate takes place only in the case of extremely
low Young’s modulus (1.0GPa) as shown in Fig.22.
Since aggregate shrinkage appeared to be dominant
effect on concrete shrinkage as mentioned above, an
aggregate shrinkage model was proposed based on
an earlier pioneering research (Fujiwara 1984) and
implemented in the multi-scale constitutive model in
order to analytically examine the influence of
aggregate shrinkage on concrete shrinkage. This
demonstrated that the significant increase in
concrete shrinkage could be calculated when
aggregate shrinkage related to aggregate moisture
states was taken into account as shown in Fig.23.
According to the numerical simulation, it was also
suggested that aggregate shrinkage was likely to
cause the remarkable variation in concrete
shrinkage.
Drying shrinkage（µ）
Autogeneous shrinkage (µ)
Recently, it was reported that a prestressed
reinforced concrete (PRC) bridge in Japan had
suffered severe cracking of RC member surfaces
even though only three years had passed after the
construction. A number of cracks were found on the
surface of the RC members and vertical cracks were
observed in the web even though it was horizontally
prestressed. According to the emergency committee
set up by JSCE, one of the reasons for this serious
damage might have been excessive shrinkage of the
concrete (JSCE Concrete Committee 2005). Since
such serious concrete shrinkage was thought to be
associated with aggregate properties, the committee
studied the influence of aggregate properties on
concrete shrinkage. Shrinkage experiments of three
types of concretes with different types of aggregates
were conducted. The aggregates for mix A were
from the same source as used for the bridge
concrete, while the aggregates for mix B were
standard ones extracted from a different site. The
sand in mix C was standard but the gravel was the
same as that in mix A. The test results are shown in
Fig.21. The concrete with the same type of
aggregates as that of the PRC bridge exhibits
extremely large shrinkage, about twice larger than
that of concrete containing standard aggregates.
Although little attention has been paid to the
characteristics of aggregates themselves in the
design process considering concrete mix proportion,
structure size and boundary conditions, the above
mentioned findings suggested that the effect of each
1000
800
600
400
Expt. (Mix A)
200
Expt. (Mix B)
Expt. (Mix C)
0
0
20
40
60
80
Drying time (Day)
100
Fig.21 Autogeneous shrinkage and drying shrinkage of concretes with different aggregates
(JSCE Concrete Committee 2005)
1000
Dryinjg shrinkage（µ）
Autogeneous shrinkage (µ)
Cement paste
Young's modulus
Young's modulus
Young's modulus
Young's modulus
Young's modulus
Expt. (Mix A)
1500
of aggregate=1.0GPa
of aggregate=3.16GPa
of aggregate=6.32GPa
of aggregate=9.48GPa
of aggregate=28.4GPa
500
0
0
20
40
60
Age (Day)
80
Cement paste
Young's modulus
Young's modulus
Young's modulus
Young's modulus
Young's modulus
Expt. (Mix A)
2000
1500
1000
of aggregate=1.0GPa
of aggregate=3.16GPa
of aggregate=6.32GPa
of aggregate=9.48GPa
of aggregate=28.4GPa
500
0
0
100
20
40
60
80
Drying time (Day)
100
1500
1000
Analysis (with aggregate shrinkage)
Analysis (without aggregate shrinkage)
Expt. (Mix A)
800
600
Drying shrinkage (µ)
Autogenous shrinkage (µ)
Fig.22 Computed results of shrinkage with aggregates of varying Young’s modulus
400
200
0
0
20
40
60
80
100
Analysis (with aggregate shrinkage)
Analysis (without aggregate shrinkage)
Expt. (mix A)
1000
500
0
0
20
Age (Day)
40
60
Drying time (Day)
80
100
Fig.23 Computed results considering aggregate shrinkage
(maximum aggregate shrinkage is assumed to be 1400 µ)
I cr (t ) ≥ γ cr
4.2 Thermal stress and crack risk assessment of
massive concrete (1cm ~ 1-10m level)
Technology for estimating the temperature cracking
of massive concrete structures due to the hydration
heat of cement has been dramatically developed in
Japan over the past two decades, beginning with the
publication of the 1986 edition of the Standard
Specifications for Design and Construction of
Concrete Structures by the Japan Society of Civil
Engineers (JSCE 1986). These specifications
introduced a quantitative index to the risk of thermal
cracking, i.e. “thermal cracking index”. In current
standard specification (JSCE 2002), the index
including effect of drying shrinkage and
autogeneous shrinkage, represented by the following
expression is used.
f t (t )
(1)
σ t (t )
where, ft(t): tensile strength of concrete and σt(t):
maximum thermal stress in tension. The thermal
cracking index should satisfy the safety factor γcr for
the required functions and durability of the concrete
structures as shown in Eq.(2).
Thermal cracking index:
I cr (t ) =
(2)
Fig.24 represents the relationship between
probability of cracking Pf and safety factor γcr. For
the ordinary reinforced concrete structures, the
reference values of safety factors are provided
according to the level of cracking risk as described
below.
1) -1.75 and higher: when cracking is prevented.
2) -1.45 and higher: number of cracks is
controlled as much as possible.
3) -1.00 and higher: when the crack width is
controlled while allowing cracking.
In the Fig.24, the probability is larger than 50%
when the safety value is 1.0, because the cracking
index is evaluated from the tensile strength of the
test cylinder that can be different form that of the
real concrete structure. To calculate the cracking
index, a specific procedure for determining the
probability of cracking is also provided in the
specifications. As a simple method of analyzing
thermal stress, CP method is recommended in the
specification, which was proposed by Japan
Concrete Institute in 1985 (JCI 1985). CP method is
a practically useful method allowing incorporation
of the concrete placing process.
Fig.25 shows the relative proportion of analytical
methods for thermal stress in Japan. The CP method
Probability of cracking (%) Pf
is used as the most practically method in Japan,
because the thermal stress can be calculated easily
and two-dimensionally for wall structures. In the
case of FEM, three dimensional analysis is better for
simulating such type of structures as well.
Fig.26 shows the comparative ratio of thermal
crack control methods in Japan. As shown in Fig.26,
selection of low-heat type cement and reduction of
cement content are often adapted for preventing
thermal cracking. The crack-inducing joint is also
often applied to wall structures. The effective
method according to the structure type is examined
based on thermal crack index determined by JSCE
specifications.
Next, a recent analytical system named JCMAC3
in Japan is described. The program can deal with not
only thermal strain but also other initial strain at an
early age comprehensively and three-dimensionally.
Thermal strain, autogeneous shrinkage and drying
shrinkage can be taken into account as initial strain
at an early age (Fig.27). The initial strain is
computed based on the recent research progress in
Japan.
A discrete FEM solving three-dimensional heat
conduction equation gives temperature distribution
at each time-step and thermal strain is obtained.
Drying shrinkage is computed by moisture transport
analysis in the system. The moisture transport is
idealized in Eq.(3) (Kagohashi et al. 2000). Vapor
permeability and vapor capacity are dependent on
relative humidity and lead to non-linear analysis.
100
90
80
70
60
50
40
30
20
10
0
0.5
1.0
1.5
Safety Factor γcr
2.0
 dq  ∂P
 ∂q 
= ∇ ( λ p ∇P ) −  
 
 dP  ∂t
 ∂t 
dP
J = −λ p
dn
dP
− λp
= α( P − P0 )
dn
(3)
Fig.24 Relationship between Probability
of cracking and safety factor
Expansive cement 3％
3D FEM 16%
Crack control reinforcements 8％
2D FEM 19%
Temperature preserving
curing 5%
Setting of crack
inducing joints 16%
Selection of cement kinds 38％
Pipe cooling 8%
CP 56%
Pre-cooling of materials 11％
Fig.25 Relative proportion of analytical methods
for thermal stress in Japan (Tanabe 2004)
Reduction of cement contents 11％
Fig.26 Comparative ratio of thermal crack
control methods in Japan (Tanabe 2004)
Fig.27 Outline of JCMAC3
where, J: moisture flux(g/h), λp: vapor
permeability(g/h⋅m⋅mmHg), dp/dn: gradient of
vapor, (dq/dn): vapor capacity(g/h⋅m3⋅mmHg), q:
vapor density(g/m3), n: direction cosine, (∂q/∂t):
varying vapor transfer density due to hydration
reaction(g/h⋅m3), P: vapor pressure at the
surface(mmHg), P0: surrounding vapor pressure, α:
evaporation ratio(g/m2⋅h⋅mmHg) ． Autogeneous
shrinkage is determined from Eq.(4) (JSCE 2002)
and distributed uniformly in each element.
[
{
ε′as (t ) = γε′as∞ 1 − exp − a (t − t s )b
}]
Dirichlet series creep function is used for the
computation as shown in Eq.(5).
 t − τ 
1 

1 − exp
µ
i =1 Ci ( τ) 
 i 
N
φ(t , τ) = ∑
(5)
where, Ci(τ): function of time, µi: delaying time.
Fig.28 represents the physical meaning. N Kelvin
type creep models are connected in series and each
dashpot works behind µi. Then, total visco-elastic
strain is regarded as summation of creep strain in
each dashpot. The increment of creep strain in three
dimension ∆{εcr} is idealized using stress increment
∆{σr}.
(4)
ε’as(t): autogeneous shrinkage strain of concrete
from start of setting ts to age t[x 10-6], ε’as∞: final
value of shrinkage strain[x 10-6], γ: coefficient
representing the influence of cement type, a and b:
coefficients representing the characteristics of
autogeneous shrinkage progress.
JCMAC3 can comprehensively simulate initial
strain at an early age including thermal strain,
autogeneous shrinkage and drying shrinkage and
lead to stress analysis at an early age. The addition
of elements and change of heat transfer interface
properties due to casting are possible. In JCMAC3,
creep strain is calculated based on rate type theory
(Ando et al. 1996), while the influence of creep is
taken into account by using effective elastic modulus
determined from the increment of effective strain at
each step in the case of CP method. In JCMAC3,
creep strain is determined by the strain at previous
step.
∆{ε cr } = [ L1 ]∆{σ r } + {L2 }
(6)
0
0
0 
1 − ν cr − ν cr

−
ν
1
0
0
0 
cr

1
0
0
0 
1 
[ L1 ] =


1 + 2ν cr
0
0 
E' ' 
 Sym.
1 + 2ν cr
0 


1 + 2ν cr 

N
{L2 } = ∑ (1 − exp( − ∆yi ) ){qi (tr )}
(8)
i =1
1 − e−t
1
Creep function
クリープ関数
0.1
1.0
φ ( t ,τ )
 t −τ
1 − exp
 µ5
10.0



1 / C 5 (τ )
1 / C 4 (τ )
Fig.28 Schematic representation of physical meaning of Dirichlet series creep function
{qi (tr +1 )} = {qi (tr )} exp(− ∆yi ) +
∆yi = (t r +1 − t r ) / τ i ,
λi
∆{σ}
Ci (tr +1 / 2 )
N
1− λi
1
=∑
E ' ' i =1 Ci (t r +1 / 2 )
(7)
,
(9)
λi =
1 − exp[− ∆yi ] ,
∆yi
(10)
where, νcr is creep Poisson ratio. At step t=tr+1, {L2}
can be obtained from {qi(tr)} because [L1] in Eq.(6)
is known. Based on {qi(tr)} at previous step, the
relationship between stress and strain considering
visco-elastic effect can be computed. Dashpot
parameters, Ci(tr+1/2) and µi can be determined by
curve-fitting the creep function with Eq.(7).
In JCMAC3, computer memory can be
effectively used by adapting the incremental creep
model, so the analysis of massive structure having
many elements is possible. In addition, there are
several mathematical solvers that can solve
simultaneous linear equations very fast. JCAMC3 is
expected to be a useful tool for simulation of
cracking in concrete, since it implements recent
research results and has very practical aspect.
5 MATHEMATICAL MODELING FOR TIMEDEPENDENT BEHAVIOR OF CRACKED
STRUCTURAL CONCRETE
5.1 Thermal crack control (1mm ~ 1cm level)
Maximum Crack width(mm)
In the section 4.2, the evaluation for the crack risk of
massive concrete with the thermal cracking index
was described. In this section, the thermal crack
control design and its analytical method after
cracked are introduced from a practical point of
view. The relationship between the thermal cracking
index and the maximum crack width is given in
JSCE guidelines for concrete (JSCE 2002),
according to the reinforcement ratio, as shown in
Fig.29. This relationship was proposed based upon
the experimental results. The relationship enables us
to determine the appropriate reinforcement ratio to
control the maximum crack width based on the
crack index.
On the other hand, Japan Concrete Institute
proposed the following three kinds of analytical
methods that extend the CP method.
1) Extended CP method (For calculating
temperature, stress, and crack width).
0.5
0.4
2) Surface crack CP method (For tracing the
crack propagation on the concrete surface).
3) Hybrid CP/FEM method (Focusing on the
part where detailed information required).
According to the methods, it is possible to
calculate crack propagation and crack width history.
JCI implemented the above methods in the program
named ‘JCMAC2’ that was released in the market
for concrete engineers. The three analytical methods
are summarized below.
5.1.1 Extended CP method (JCI 1992)
To directly calculate the crack width, the Extended
CP method was developed by advancing the CP
method. In the method, restraining body and
restrained body are focused on and only the crack
which penetrates over the restrained body is dealt
with. It is postulated that the crack occurs when the
thermal stress exceeds the tensile strength of
concrete.
The procedure of Extended CP method is shown
in Fig.30. The calculation is carried out in two steps.
In the first step, the stress distribution in the
restrained body is calculated ignoring the
deformation of the restraining body. Here,
Bernoulli-Euler theory is used. In other words, the
stress transfer between the reinforcement and the
concrete in the restrained body is calculated by the
CP method, ignoring the deformation of the
restraining body. The total stress balance in the
whole section including the restraining body is taken
into account in the second step. Bernoulli-Euler
theory is also applied to the section of the entire
system. The stress is computed by using CP method
again. Then, the redistribution of the stress to
balance the total stress in the whole system of the
restrained and restraining bodies is calculated. As a
result, the discontinuous plane is formed between
the restrained body and the restraining body.
Reinforcement ratio
0.25~0.30%
0.55~0.65%
0.85~0.95%
0.3
0.2
0.1
0.2 0.4 0.6 0.8 1.0 1.2 1.4
Crack Index
Fig.29 Relationship between cracking index
and maximum crack width
1. Calculate stress distribution by CP Method
Stress release zone l cu
l su
Crack Rebar
Debonding zone
Crack Rebar
Restrained
body
Restrained
body
Restraining
body
MR
NR
3. Second step: Resolv e the imbalance of
forces in the whole section
l sb Restraining
l cb
body
2. First step: Exchange of forces within the
restrained body after cracking
4. Calculate the crack width at the position of rebar
Fig.30 Flow of Extended CP Method
5.1.2 Surface crack CP method (JCI 2003)
5.1.3 Hybrid CP/FEM method (JCI 2003)
Extended CP method is based on the assumption that
once the crack occurs it penetrates over the cross
section. Thus, it cannot simulate the gradual crack
propagation such as surface cracking. The surface
crack CP method (JCI 2003) was developed in 2003,
in order to calculate the crack depth from the surface
of the concrete in each age. This method was also
developed based on the CP method. Whereas
Extended CP method assumes that the height (or
depth) of stress release zone is always constant, the
height of the zone of the surface crack CP method
can increase according to the crack propagation. To
determine the height, the calculation can be repeated
as changing the stress release zone until the thermal
stress is less than the tensile strength in the whole
zone.
The hybrid CP/FEM method combining the CP
method and finite element method (FEM) was
developed and commercialized with a program to
analyze the temperature rise due to hydration
reaction developed by Suzuki et al. (1990). In this
hybrid CP/FEM method, the detailed structural
responses such as stress and strain distribution in
each element, cracking pattern and others can be
simulated by nonlinear 3D FEM analysis, while the
simple information can be obtained by the CP
method based on the Bernoulli-Euler theory as
shown in Fig.31. The crack width can be computed
considering not only internal penetration of crack
but also the propagation at the surface and threedimensional effect of steel confinement.
Cracking by initial strain
Restrained body
Restraining body
N R = RN EA( ∆ε 0 − ∆ε r )
M R = RM EI ( ∆φ 0 − ∆φ r )
RN,RM: Coefficients of axial and flexural
restraint of restraining body
Plane section remains plane
Joint element
M
N
M
R
R
External restraining
bending moment
N
R
R External restraining
axial force
3D-FE Model Parts include the rebar arrangement exactly
Fig.31 Concept of Hybrid CP/FEM Method
5.2 Time-dependent structural damage under
sustained and repeated loads (1cm ~ 1m level)
The meso-scale modeling of concrete brings about
volumetric strain related to hydration and drying,
and the composite material based strain is forwarded
to post-cracking structural analyses as shown in
Fig.32. Here, the time-dependency of tension
stiffening and softening after cracking is
indispensable for most macroscopic analysis. As
shown in Fig.33, space-averaged tensile strain under
sustained forces increases with the progress of time.
This nonlinearity is due to both time-dependent local
bond and tensile creep of concrete continuum
between cracks (Maekawa et al. 2006). Some
research efforts have been addressed to the timedependent modeling of local bond or macroscopic
tension stiffening.
Fig.34 shows an example of post-cracking creep
deflection of RC beams in flexure and the timedependent creep analysis with a time-dependent
tension stiffening model. The free-stress shrinkage
strain is computed in space according to the loss of
moisture under the drying state. The moisture
migration is also computed based upon the micropore structural analysis. In flexure, the timedependent curvature is governed by the flexural
tension creep since the neutral axis is closer to the
compression fiber. Generally speaking, the creep
deflection is so large compared to the one of precracking creep responses. Thus, the multi-scale
structural analysis can be a versatile tool for
prediction of long-term performance of RC members
under ambient conditions.
thermo-hydro physical model
This multi-scale analysis can be applied to the
performance assessment of existing structures
including initial defects (see Fig.35). It is reported
that the time-dependent deflection is accelerated by
the presence of initial damages around the anchorage
zones and the increasing risk of creep failure is
pointed out (Toongoenthong & Maekawa 2005). The
potential of the multi-scale analysis is thought to be
high owing to its versatility. The multi-scale
approach is being applied to the nonlinear analysis
of structures under high cycle fatigue loads.
It must be noted that the time-dependency is not
negligible for fatigue as shown in Fig.36. As a
matter of fact, the apparent fatigue life of materials
and structures is also dependent on the rate of stress
and/or strain. Under large amplitude of stresses,
concrete is subjected to higher rate of creep. Then,
the time-dependent damage is also included as well
as the effect of stress repetition. On the other hand,
the fatigue life corresponding to the lower amplitude
of stress is dominated by the damage induced by the
stress repetition. Thus, the fatigue constitutive
modeling has to include the time-factor (Maekawa &
Toongoenthong 2006).
Fig.37 shows a high cyclic load response of RC
beams in combined stress condition of flexure and
shear. Recently, the direct path-integral scheme has
been available for fatigue simulation similar to the
nonlinear seismic response analysis. The simulation
can be applied to the case under moving loads as
well as the pulsation of a fixed point loading
(Gebreyouhannes et al. 2007).
Moisture & Free shrinkage
Micro-structure,
Moisture equilibrium,
Capillary tension and
Gel disjoining pressure
dr
Moisture state
micro-structure
hydration
Detailed method to use local free shrinkage strain in space
Apparent tensile strength
equivalently reduced
Local free shrinkage
Strain profile
Sectional
Input
strain
shrinkage
tension
Local stress profile tension
ex) In case of
free sectional
Stress = 0
cracking zone
cracking zone
Assumed stress
Real local stress profile
compression
reinforcement
Simplified method to use section-averaged shrinkage strain + equivalent tensile strength
Fig.32 Effect of shrinkage strain on local and averaged stress profile
Specimen
Temp.：20±5℃、RH：60±20%
15 cm
297.8cm
cm
15
1#D19
2500
2500
2000
experiment
Strain (µ)
Strain (µ)
2000
1500
1000
analysis
experiment
+2 crack
1000
500
+1 crack
+1 crack
1500
6 crack
500
analysis
0
20
60
0
Elapsed time (days)
5
20
40
60
80
100
Elapsed time (days)
5
experiment
4
Stress （MPa）
Stress （MPa）
40
experiment
3
2
analysis
1
4
analysis
3
2
1
0
500
1000
1500
2000
Strain （µ）
0
500
1000
1500
Strain （µ）
2000
Fig.33 Computed and experimental post-cracking tensile creep of RC
50
21
14
2 days
Uniform Load: 4.86 kN/m
400 mm
21
14 2 days
-0.0008 -0.0006 -0.0004 -0.0002
0
Mid-span deflection (mm)
50
35
0
Intrinsic free shrinkage A
100
350 cm
divided into 7 layers
- 0 . 0 0 0 8 - 0 . 0 0 0 6 - 0 . 0 0 0 4 - 0 .0 0 0 2
200
163mm
3@D12
100
200 days
130
analysis : non-uniform shrinkage profile -A
30
25
Sensitivity analysis : uniform shrinkage profile -B
20
15
Sensitivity analysis : no shrinkage + 0.75ft
10
experiment
5
0
Intrinsic free shrinkage
Assumed uniform B
0
50
100
150
200
250
Elapsed time (days)
Fig.34 Creep deflection of slab and the effect of shrinkage
300
Load (kN)
Induced cracking damage (1st step)
150
no pre-damage
125
100
75
analysis
experiment
pre-cracking
50
sustained load level
25
0
1
Deflection (mm)
0
Final failure crack pattern (50kN)
2
3
1.5
4
Deflection (mm)
5
pre-cracking
analysis
experiment
1
no pre-damage
0.5
0
Experimental failure crack pattern
0.01
0.1
1
10
100
1000
Elapsed time (day)
Fig.35 Analytical result of beam with inherent crack-like defect until anchorage zone
30
300
1
design
Compressive Stress (MPa)
250
200
20
0.01Hz
150
100
10
analysis rate of
loading = 1.0Hz
50
0
0
0.0005
0.001
0.001
0.0015
0.002
0.002
0.0025
0.003
0.003
Stress Rate / fc’ (/min)
specifie
d S-N
diagram
for
1
0.9
0.8
0.7
0.6
0.5
0.4
0.3
0.2
0.1
0
∆σ/ fc’ = 0.5
∆σ/ fc’ (amplitude)= 0.9
0.1
data by
Award and
Hilsdorf
0.01
∆σ/ fc’ = 0.1
analysis
σmax/ fc’ = 0.9
Compressive Strain
0.001
0
1
2
3
4
5
Log N
6
1
10
100
1000
10000
100000
Cycles to Failure
(a) Stress amplitude effect
(b) Stress-rate effect
Fig.36 Computed S-N diagram and stress-rate effect of fatigue compressive strength of concrete
5
90%
80%
4
70%
75%
1.0
60%
3
2
1
0
-1
10
10
0
10
1
2
10
10
3
10
4
Number of cycles, N
5
10
10
6
Stress ratio normalized by
fixed static capacity
Mid span Deflection, mm
M ax. stress/Com p.Strength
F
0.001
Okamura and Ueda relation
0.9
log(Vc/ Vco) =-0.036*(1-r|r|).logN
r = Vmin/ Vmax case, r = 0
0.8
0.7
≈3 orders
FE analysis
experiment
0.6
FE analysis
0.5
0.4
0
10
Fixed pulsating load
Moving load constant load
10
1
10
2
10
3
10
4
10
5
10
6
10
Number of loading cycles/passages
Fig.37 Shear fatigue response of RC beam and S-N curves for fixed pulsation and moving loads
7
6 CONCLUDING REMARKS
This paper provided an overview of mathematical
models
for
time-dependent
behaviour
of
cementitious materials and the following concluding
remarks were obtained:
1) Since the cementitious materials have complex
and multi-scale structures, a hierarchic system
of at least three different levels needs to be
introduced for linking the observed behaviour
with the material structure and real
mechanisms.
2) For linking the various aspects of the observed
behaviour on different levels, conceptual
modeling to integrate them into a general
computational
framework
based
on
mathematical models is important.
3) By linking the observed behaviour of
cementitious materials with details of the
material structure and with real mechanisms,
experimental results can be interpreted on a
more general basis and development of
various advanced technologies can contribute
to the improvement of the mathematical
models.
4) Mathematical models based on the material
structure and real mechanisms are very useful
to predict time-dependent behaviour and longterm performance of concrete structures.
ACKNOWLEDGEMENT
This paper is the summary of the activities report of
the working group on mechanisms and modeling of
microstructure (WG1) in the JCI technical
committee on time-dependent behaviour of
cementitious materials. The authors would like to
thank all of the other members of this WG1: T. Endo
(Tohoku Gakuin Univ.), K. Imamoto (Tokyo Univ.
of Science), M. Ishikawa (Tohoku Gakuin Univ.),
M. Kunieda (Nagoya Univ.), H. Morimoto (Gifu
Univ.), Y. Sato (Oh-ita Univ.) and H. Ustumi (Chiba
Institute of Technology) for their valuable
contribution to this WG activities.
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