Fracture Mechanics of Concrete and Concrete Structures High Performance, Fiber Reinforced Concrete, Special Loadings and Structural Applications- B. H. Oh, et al. (eds)ⓒ 2010 Korea Concrete Institute, ISBN 978-89-5708-182-2

Study on size effect in bending behavior of ECC
K. Asano
University of Tsukuba, Tsukuba, Ibaraki, Japan

T. Kanakubo

University of Tsukuba, Tsukuba, Ibaraki, Japan

T. Matsushima
University of Tsukuba, Tsukuba, Ibaraki, Japan

ABSTRACT: In this paper, the bending test of Engineered Cementitious Composites (ECC) with PVA fiber
is conducted to investigate the size effect for bending behavior of ECC. The main parameter is the size of thespecimens. The test results show that the bending strength increases as the size of specimen decreases. To express the wall effect of fibers, probability of fibers which cross the surface of specimen is solved theoretically.
In the case of 40×40mm section specimen, 23% of fibers are influenced by specimen surface and considered
to be oriented toward two-dimension. The coefficient of contribution to bending strength by two-dimensional
orientation of fibers is proposed based on the probability.
1 INTRODUCTION

2 BENDING TEST

To use the feature of Engineered Cementitious
Composites (ECC) (Li 1993) well, it is important toevaluate the tensile performance of ECC appropriately. Although a lot of methods to evaluate the tensile performance are proposed, uniaxial tension test
is most proper. However, the shape, the size and theboundary condition of the specimens cause large influences on the test results. Therefore, it is difficult
to perform the uniaxial tension test as a standard
test. Although the tensile performance can be evaluated from the result of the bending test, the bending
test results give a structural performance which is
not a material performance. Therefore, it is necessary to understand a bending behavior and to clarify
the relation between tensile behavior and bending
one. Moreover, the size effect is caused by the difference of the size of the specimen. In addition, it isconsidered that the wall effect caused by twodimensional orientation of fibers along the specimen
surfaces affects the size effect on tensile and bending behavior. In this paper, the bending test of ECC
with PVA fiber is conducted to investigate the size
effect for bending behavior of ECC. The main parameter is the size of the specimens. After that,probability of fibers which cross the surface ofspecimen is solved theoretically to express the wall
effect of fibers, The coefficient of contribution tobending strength by two-dimensional orientation offibers is discussed based on the probability.

2.1 Employed material
The characteristics of PVA fiber are shown in Table
1. The compressive strength and the elastic modulus
of ECC at the testing age are shown in Table 2.
Moderate-heat Portland cement and fly ash (type II
by JIS A 6201) are used as binder. Crushed limestone which specific surface area is 2500cm2/g isused as the fine aggregate. The fiber volume fractionis 1.5% (PVA15) and 2.0% (PVA20). PVA15 and
PVA20 were mixed by two batches each using apractical mixer with volume of 1m3. The specimens
were uniformly and continuously cast into the mold.
After steam curing for 8 hours with 35 Celsius degree, the specimens were cured in atmospheric environment.
2.2 Specimens
Figure 1 shows the shape of the specimens. Table 3
shows the list of the specimens. The 4-point bending
test is applied. The number of specimens for thesame parameter is three.
2.3 Loading and measurements
The displacement controlled 2MN universal testingmachine was used. The loading speed was set to0.5mm/min. The applied load and the axial deformation at the test section were measured to calculateJ = − D ( h, T The)∇h position of displacement transducers
curvature.
(1)
was set to have the same dimensional ratio in eachsizeTheof specimens.
However,the gauge
lengththeproportionality
coefficientD(h,T)
is ofcalled
B40
was
25mm
due
to
the
loadingdevicerestricmoisture permeability and it is a nonlinear function
tion.
of the relative humidity h and temperature T (Bažant& Najjar 1972). The moisture mass balance requiresthat Test
the variation
2.4
results in time of the water mass per unit
volume of concrete (water content w) be equal to theTable
4 shows
themoisture
test results.
divergenceof thefluxTheJ bending stress isTable.1
Characteristics
PVAthefiber.evolution of hydration
explicitly
accountsof for
Tensilereactionand DiameterSF content. This sorption
isotherm
LengthElastic modulus
strength
reads
(mm)
(mm)
(MPa)
(GPa)
12
0.04
1690
40.6
⎡
⎤
1
⎢
⎥
) = G (performancewe (h, αCompressive
α c , α s )⎢1 − of ECC.
+
Table.2⎥c , α sFiber
1∞
10(g α− α c )h ⎥Elastic
1 c
Air ⎢⎣ e Compressive
⎦

Batch

(4)
volumecontent
strengthmodulusfraction⎡ 10(g α ∞ − α )h⎤(%) K (α (%)
(MPa)
c
c
1
⎢
,α ) e− 1⎥(GPa)

defined as a value of bending moment divided by
the∂wmodulus of section and the bending strain is thec5.4 s ⎢1⎥18.9
PVA15-1
55.3(2)− = ∇ • multiplied
J
⎣
⎦
1.5
curvaturebyspecimen
depth.
These
fac∂t
PVA15-2
5.6
56.9
19.9
tors are defined to consider the differences of the
PVA20-1
45.0
16.0 thewhere the 2.0first term11.0(gel isotherm)
represents
sizeTheof water
specimens.
Figureshows
the relationshipcontent
w can2 be
expressed
as the sum PVA20-2
bound (adsorbed) watersecond47.1 and the16.4
betweenthe bending
versus thebending
water,
water physicallyof the evaporable
waterstresswe (capillaryterm(capillary
isotherm)
represents
thecapillarystrain.
In
spite
of
the
differenceoffiber
volumevapor, and adsorbed water) and the non-evaporable water. This expression is valid only for low content
fractionand thesize ofwater
specimen,deflection(Mills1966, of SF. The coefficient G represents the amount of(chemicallybound)
wn thehardening
behavior
in
which
the
load
increases
afterto1
Pantazopoulo & Mills 1995). It is reasonable
water
per
unit
volumeheld
the gel pores at 100%
first
crack
can
be
recognized.
The
averagevalue
andassume that the evaporable water is a function of relative humidity, and it canin be
expressed (Norlingthecoefficientofvariation
(COV)
of
the
maximum
relative humidity, h, degree of hydration, αc, and Mjornell 1997) as
bendingbendingα strain
at the maxidegree ofstresssilicaandfumethereaction,s, i.e. we=we(h,αc,αs)
mum
are
shown
in
Table
5
and
Table
6.
are a
= age-dependent sorption/desorptionThere
isotherm
1. The shapecof the specimens(*:25mm in B40).
lot
of
the
specimens
which
COV
is
lower
than
(Norling Mjonell 1997). Under this assumption10%
and Figure
G1 (α c , α s ) = k vg
α c+ ks α s
(5)
c
vg
s
for
the
maximum
bending
stress.
However,theby substituting Equation 1 into Equation 2 one Table 3. The list of the specimens.
range
obtainsof the COV is from 20% to 30% for the bendTestcsa parameters. From theing strain. A clear relationship can not be confirmed
Namewhereof kspecivg and kBatch
vg are material
sectionmens
between the size of the specimens and the COV.
maximum amount of water per
unit volume
that can
(mm)
(mm)
−

∂w ∂h
e + ∇ • ( D ∇h) = ∂we α
c
h
∂α
∂h ∂t

∂w

& + ∂α e α& + w&

(3)

fill all pores (both capillary pores and gel pores), onee
3.1
Probability
ofcalled
fibers whichcross capacity).
the surface Theofisotherm(alsomoisture
specimengoverning
equation (Equation 3) must be completed

B40-1550
can calculate K1 as one obtains40
PVA15-1
B100-15100
100
B200-15200⎡⎛∞ 200⎞ ⎤10⎜ g α− α ⎟h ⎥
⎢
1 c
c⎠ ⎥
⎝
B400-15
PVA15-2
400
w0 − 0.188
α s + 0.22α s − G400
⎢1 − ecs 1⎢
B40-2040⎣50 ⎥⎦K1(α c ,α s ) =⎛B100-20
PVA20-1100∞ 100⎞
10⎜ g α− α ⎟h1 cc⎝⎠e
B200-20200 − 1200
B400-20
PVA20-2400c400setc.). In the literature various formulations can befound to describe the sorption isotherm of normal
3.2 Theoretical probability
concrete (Xi et al. 1994). However, in the present
papersizetheof semi-empiricalexpression
byThe
the specimen andthe lengthproposedof the fiber
Norling
Mjornellis adopted
because it
aretreatedas the (1997)
parameters.
The probability

that the fiber crosses the boundary is constructed. (7)To
simplify the construction, the diameter of the fiber iswhere q isAtthe
flux, T is theabsolute
disregarded.
first,heat
the complementary
eventof thetemperature,
and
λ
is
the
heat
conductivity;in thistarget probability is considered. The left figure
in

c

s

s

n

3 QUANTIFICATION OF THE SIZE EFFECT

where ∂w /∂h is the slope of the sorption/desorption(6)

Bending stress (MPa)by thisappropriate
boundary
and"Buffon's
initial conditions.
In
paper, theidea ofNeedle" (RoThe&relation
betweenthe amountof evaporable
zanov
Sobor 1976)
enhancedto three-dimension
water
and relative
humiditycalledthe‘‘adsorptionisdiscussed.
"Buffon's
Needle"is treatsneedle ofif measured
with increasing
relativity
aisotherm”
certain lengththat is distributedat randomin theThe material parameters k vg and k vg and g1 canhumidityplane
and area,‘‘desorption
isotherm”
the opposite
limitedand expresses
the incalculationof
be calibrated by fitting experimental data relevant toFiber fraction : 1.5%
case.probability
Neglectingoftheirdifference(Xi et al.in 1994),
thecrossing
the boundary
the areain
free (evaporable)
water content
concrete
Fiberinfraction: 2.0% at
10thethefollowing,‘‘sorption
isotherm”
be usedbyalgebraic
formulation.
In thiswill
paper,
it is withenvarious
ages (Di Luzio & Cusatis
2009b).
10referencetotothree-dimension,both sorption andanddesorption
conditions.
hancedthe probabilityof
By fiberthe crossing
way, if thethesurface
hysteresisof the frommoisture
theis calculatedthe2.2 5Temperature evolution
isotherm would
takenarea.
into account,
twofibers
different
boundaryin the bespatialWhen theare5
B40
relation, evaporable
water vs relative
humidity,
must Note that, at early age,B40
distributed
in a three-dimensional
space,
the fibers
since the chemical reactions
B100
B100
B200
be usedareaccording
to the signbyofthe
themold
variation
the associated with cement
which
actually restrained
and aoffree
hydration and SF reaction
B200
B400
B400
relativity (castinghumidity.surface)
The shapeof theto sorption
boundary
are thought
be oriare exothermic, the temperaturefield is not uniform
0
1
2
3
4
0
1
2
3
isotherm
for two-dimension
HPC is influenced
by manyinparameters,
ented
along
as shown
Figure 3.
for non-adiabatic systems even if the environmental4
Bending strain (%)
Bending strain (%)
especiallythosethat crossinfluenceextent andof theThese
fibersdo notthe surface
andrateexistsintemperature
is constant. Heat conduction
can bechemical
reactionsand, in turn, isdeterminetheneighbor.
This phenomenon
defined asporeadescribed
in
concrete,
at
least
for
temperature
not
Figure 2. The relation between bending stress and bending strain.
structure andevent
pore insizewhichdistribution(water-to-cement
probability
the originallyrandomexceeding 100°C (Bažant & Kaplan 1996), by
ratio, cement
chemicalcomposition,
SF content,orientedfibers are
influenced
by the boundary(wall
Fourier’s law, which reads
curing time and method, temperature, mix additives,effect).

Proceedings of FraMCoS-7, May 23-28, 2010

q = − λ ∇TTable 4. Test results.
Size of thecross section

At the maximum load
Bending moment Curvature(mm)

Mmax(kNm)

φmax(1/m)

0.100

0.215

0.095

0.185

B40-15-3

0.105

0.201

B100-15-1

1.525

0.548

1.670

0.539

B100-15-3

1.780

0.847

B200-15-1

7.911

0.032

6.465

0.030

6.803

0.015

52.7400.005

40.2760.005

34.168

0.005

0.124

0.487

Name of specimens
B40-15-1

40×40

B40-15-2

B100-15-2

100×100

B200-15-2

200×200

B200-15-3
B400-15-1
B400-15-2

400×400

B400-15-3
B40-20-1
40×40

B40-20-2
B40-20-3
B100-20-1
B100-20-2

0.110

0.247

0.109

0.367

1.863

0.677

1.728

0.726

B100-20-3

2.160

1.170

B200-20-1

7.591

0.027

100×100

B200-20-2

8.048

0.014

B200-20-3

200×200

6.763

0.025

B400-20-1

67.0020.012

B400-20-2

400×400

B400-20-3

52.3580.009

54.202

0.004

Average of the maximumCoefficient ofvariation
bending stress
(%)
(MPa)

Specimen

B40-15

9.47

3.1

B40-20B100-15

9.83

5.8

B100-20

B200-15

5.27

8.9

B200-20

B400-15

3.94

18.6

B400-20

Table 6. Bending strain at the maximum load.
Specimen

Average ofbending strain at the
max.
(%)

Bending stress

Bending strain
The
proportionality
coefficient D(h,T)σ (MPa)ε (%)
moisturepermeability
and it is a nonlinea
0.85
of9.58the relative humidity
h and temperature
9.06
0.73
& Najjar 1972). The moisture mass balanc
9.76
0.80
that
the variation intime of the water mas
9.08
volume of concrete0.55(water content w) be eq
9.97
0.54
divergenceof the moisture
flux J
max

max

10.45

0.85

∂ = ∇•J− 4.82
∂
5.06
5.92
w
t

0.64
0.60
0.30

0.18
The water content
w can be expressed aof the evaporable0.21
water we (capillary wa
3.18
0.19
vapor,
and adsorbed
water) and the non-e
11.49
1.96
(chemicallybound)
water wn (Mil10.30
Pantazopoulo& 0.99Mills 1995). It is reas
10.47 that the evaporable
1.45
assumewater is a fu
11.20
0.68
relative humidity, h, degree of hydration
10.44 of silica fume0.72 reaction, α , i.e. w =w
degree
s
e
12.85
1.18
= age-dependent sorption/desorption
5.58
0.55
(Norling
Mjonell 1997).
Under this assum
5.98
0.28
by substituting Equation 1 into Equati
5.08
0.49
obtains
4.93
3.72

6.27

0.49

4.91
0.38
∂w
∂w ∂h
eα
) =
+ ∇ • ( D ∇0.14− 5.03e
h
c
h
∂α
∂h ∂t

c

Table 5. Maximum bending stress.
Specimen

J = − D ( h , T ) ∇h

Coefficient ofvariation(%)

Specimen

B40-150.79

6.2

B40-20B100-15

0.65

22.5

B100-20

B200-15

0.52

29.2

B200-20

B400-150.19

6.0

B400-20∂w& + ∂α e α& + wssslope ofofthe sorption/
where∂wmaximume/∂h is the
Averageof theCoefficient
isotherm
bending
stress (also called
variation moisture capac
governing equation(%)(Equation 3) must be(MPa)by appropriate boundary and initial conditi
the amount of e
10.75 The relation between
4.9
water and relative humidity is called ‘‘
11.50isotherm” if measured
8.7
with increasing
humidity
and‘‘desorption
isotherm” in th
5.54
6.7
case. Neglecting their difference (Xi et al.
isotherm” will be
5.40 the following, ‘‘sorption
11.4
reference to both sorption and desorption c
By the way, if the hysteresis of theisotherm would be taken into account, tworelation,
Averageof evaporable water vs relative humi
Coefficientof of the varia
beusedto the sign
bending strain according
at thevariation
max.relativity humidity. The shape of the(%)(%) isotherm for HPC is influenced by many pespecially those that influence extent and
1.47
27.2
chemical reactions
and, in turn, determ
structure
and
pore
size
distribution (water0.86
26.2
ratio, cement chemical composition, SF
temperature, mix
0.44 curing time and method,
26.5etc.). In the literature various formulatio
0.34 found to describe43.3
the sorption isotherm
concrete (Xi et al. 1994). However, in th
paper the semi-empirical expression pro
Norling Mjornell (1997) is adopted b
Proceedings of FraMCoS-7, May 23-28, 2010

byappropriate boundary and initial conditions. (1)(b − Thel f cos θ )(D − l f sin θ )relation between the amount of evaporable
water and relative humidity is called ‘‘adsorption
where,
cosθifis measured
expressed as
the increasing
absolute value.
This
isotherm”
withrelativity
isbecauseandthe ‘‘desorption
range of θ is isotherm”
from 0 to π
the prohumidity
in, andthe opposite
jectionof the fibertheirlengthon a horizontal
can becase. Neglecting
difference(Xi et al.axis
1994),
inexpressed.
the following, ‘‘sorption isotherm” will be used with
2) Central
and right
figurereference
to bothsorptionand desorption conditions.
to Formula
ByThethecosφway,is multiplied
if the hysteresisof (1)theincluding
moisturelf
when
the
fiber
inclines
by
the
axial
directionwithisotherm would be taken into account, two different
the
angle
of
φ
as
shown
in
the
centralfigure.
Whenrelation, evaporable water vs relative humidity, must
assumingthat the surface
existsthe axial
dibe used according
to the sign
oftoward
the variationof therectiondistance
it shallrelativityat certain
humidity.
The L,shapeof betheconsideredsorptionthat
the lf for
sinφHPC
expresses
the influencedof fiisotherm
is influenced
by manyregion
parameters,
bers
in
the
axial
direction.
The
axial
lengthofespecially those that influence extent and rate of thethenon-influenced
solid and,can beindescribedas L－lf sinφ.
chemical reactions
turn, determine
pore
Therefore,
theporevolumeof the solid(water-to-cement
is given by the
structure andsize distribution
following
formula.
ratio, cement chemical composition, SF content,
curing time and method, temperature, mix additives,(b − l cosInθ cos
D − l sinθ cosvarious
φ )(L − l sin
φ)etc.).
theφ )(literatureformulations
can(2)befound to describe the sorption isotherm of normal
concrete(Xi et al.that
1994).
However,the present
The probability
the fiber
is not in
influencedbypaper
the semi-empirical
proposed by
theboundaryis given by theexpression
following formula.
Norling Mjornell (1997) is adopted because itfff

Proceedings of FraMCoS-7, May 23-28, 2010

explicitlyof hydration(b − l cosθ cosaccounts
φ )(D − l sinfor
θ costheφ )(L evolution− l sin φ )(3)reaction and SFbDLcontent. This sorption isotherm
reads
f

f

f

Formula (3) gives the probability in case of anarbitrary orientation angle,to⎡ θ and φ. It is necessary⎤1calculate the probability ⎢in all the orientation⎥ angles.
we (hprobability, α , α ) = G (α , α ) 1 −
+ dθ
⎢
⎥
c s
The
thatc θ svaries
withrandomangle
1∞
10(g α− α c )h ⎥⎢from 0 to π is given by thee 1 c formula.
⎣ following
⎦
(4)
1

π

⎡ 10(g α ∞ − α )h
⎤
c
1 c
⎢
K1 (α c , α s ) e
− 1⎥
⎢
⎥
⎣
⎦

⋅ dθ

(4)

Similarly, for the case of φ,where the first term (gel isotherm) represents the1physicallybound (adsorbed) water and the second
⋅ dφ
(5)
π
term
(capillary isotherm) represents the capillarywater. This expression is valid only for low content
expressedGby
formulas (3),and (5)the(4),
amountofof TheSF. events
The coefficient
1 represents
aretheperindependent
events.
thewaterunit volume
held inThetheprobability
gel pores atthat100%
fiber
is notinfluenced
the boundaries( p (Norling) can berelative
humidity,and byit canbe expressedcalculatedby theMjornell 1997)
as double integral of these product.
However, it is not necessary to treat axial direction
when only thesection is considered.
c pure kbendings
G1 (α c , α s ) =a kpart
Therefore,vg α cofc +formulavg α s s (3) is given as follows.(5)(L − l sinkφcvg) =andksvg are material parameters. From (6)
the
where
1
L
maximum amount of water per unit volume that canfill all pores (both capillary pores and gel pores), one
Finally,p is K
given
as follows.
can calculate
1 as one obtainsf⎡0

K (α c α s ) =
,

1

c

s

⎛
10⎜
⎝

e

1

1

⎝

g α c∞ − α c ⎞⎟h ⎤⎥1⎢⎣

g αc − αc h
∞

⎛

10⎜

⎢
Surfaceofα specimen
w − 0.188
s + 0.22α s − G ⎢1 − e

⎞
⎟
⎠

⎠

⎥
⎥
⎦

(6)

−1

The material parameters kcvg and ksvg and g1 canbe calibrated by fitting experimental data relevant tofree (evaporable) water content in concrete atvarious ages (Di Luzio & Cusatis 2009b).
Mold

J = − D (4h, Tshows) ∇hFigurethe cross-section of the specimen(1)with shows the cross-section of the specimen withb(mm)×D(mm).
Four coefficient
fibers are D(h,T)
expressedby
The proportionality
is called
straight
lines
in
the
figure.
Asfortwofibers,themoisture permeability and it is a nonlinear function
bothof thefiber touchboundariesTwith
theof theends
relative
humidity
h andthetemperature(Bažant
orientation
angle
of
θ.
The
other
two
fiberstouchthe& Najjar 1972). The moisture mass balance requires
corner
at
the
end
also
with
the
orientation
angle
of
θ.
that the variation in time of the water mass per unit
These
fibersindicatethemaximum
area
in
which
the
volume of concrete (water content w) be equal to thefiber
has possibility
to crossdivergenceof the moisture
fluxtheJ boundary, in otherwords, possibility to be influenced by the surface ofthe∂wcross section. The grayed rectangle in the figure− = ∇ • Jthe area in which when the center of (2)expresses
the∂tfiber exists, the fiber is not influenced by the boundary.The
Thewatercentralfigurewincan
Figure4 shows the
expancontent
be expressedas the
sum
sion
the
above
mentionedconsideration
3of the evaporable water we (capillary water, towater
dimensional
fiber
existence
towardtheaxial
direcvapor, and adsorbed water) and the non-evaporable
tionof the specimen.
fiber inclines
with thedi1966,(chemicallybound) Thewater
wn (Mills
rection
of
φ
by
the
axial
direction
keepingtheangle
Pantazopoulo & Mills 1995). It is reasonable toofθ to thedirection.
the grayedassumethatcross-sectional
the evaporable
water isThen
a functionofrectangle
in
the
left
figureexpandsbyconsidering
3relative humidity, h, degree of hydration, αc, anddimensional
orientation
as shown
i.e.the
we=right
we(h,figure.
αc,αs)
degree of silica
fume reaction,
αs, inIf
thecenterof
thefiber
existsinthegrayedrectan= age-dependent sorption/desorption isotherm
gle,that fiber
does1997).
not crossthe boundary.
(Norling
MjonellUnder
this assumption and
Next,
the
above-mentioned
probabilityisby substituting Equation 1 into Equationevent
2 one
formulated.
It is described step by step from the left
obtains
figure in Figure 4.
1) Left figure∂w∂w ∂he α& + ∂we αis&given
e probability
two-dimension
by cal&) =+ ∇ • ( D ∇in
− The
h
(3)
c
s + wn
h
∂
α
∂
α
∂
∂
h
t
culation of the ratio of thec grayeds inner rectangle
area to cross section area of the specimen with varying
of ∂w
angle
θ. When the length of the fiber is dewheree/∂h is the slope of the sorption/desorption
fined
as
l
,
inner
rectangle
area is givenby the The
folf
isotherm(alsocalledmoisture
capacity).
lowingformula.
governing
equation (Equation 3) must be completed

2.2 Temperature evolution
Fiber
Restrictedorientation
Note that,Freeat early age,since thechemical reactions
Figure3.
Fiberinfluencedbythesurface.
associated with cement hydration and SF reaction
are exothermic, the temperature field is not uniform
for non-adiabatic systems even if the environmental
temperature
is constant. Heat conduction can beFiberdescribed
in
concrete, at leastφ for temperature notθexceeding 100°C (Bažant
& Kaplan 1996), byD
D
Fourier’s law, which reads
q = − λ ∇T
b

b

(7)

Figureof fiwhere4. qTheisnon-influenced
the heat area
flux,for Tcenteris position
the absoluteber.
temperature, and λ is the heat conductivity; in this(b − l cosθ cosφ )(D − l sinθ cosφ )(L − l sin φ ) dθdφ

=

f

f

bDL

π π

∫0 ∫0
π

(b − l cosθ cosφ )(D − l sinθ cosφ ) dθdφfπ

∫0 ∫02ππ2(b − l cosθ cosφ )(D − l sinθ cosφ ) dθdφ
f

f

bD

π2

2

=1 −

4l f ⎛ 1 1 ⎞ l f⎜ + ⎟+
π 2 ⎝ b D ⎠ 2πbD

Because p is a complementary event of the target
probability that the fiber crosses the boundaries, theprobability p that the fiber crosses the boundaries isgiven as the following equations.

2

1
0
3

(7)

In case of this study, for B40 specimen, 23% offibers are influenced by specimen surface and considered to be oriented toward two-dimension.

Normalized stress

4l f ⎛ 1 1 ⎞ l f 2⎜ + ⎟−
π 2 ⎝ b D ⎠ 2πbD

21

3.3 Evaluation of Bending Test Results
The bending strength (maximum of bending stress)
observed in bending test is evaluated using previously
discussed probability. The fiber distributed along thesurface is considered to be in two-dimensional andhas a big influence to the bending strength. In the literature (Li 1990), the strength of pull out test of thesingle fiber is expressed according to the orientation
angle. It is assumed that the probability given by Eq.
(7) has the certain influence to the bending strength.
In the bending test, it is assumed that bending behavior has the same tendency with the behavior of pullout test. Eq. (8) is introduced referring to the proposed model by exponential formula in order to express the size effect considering the probability.
Where, αb is introduced as a contribution coefficientthat expresses the influence of two-dimensional oriented fiber to bending strength. Figure 5 shows the relation between the bending strength of each size
specimen normalized by that observed by B400
specimen and the probability given by Equation (7).
The curve in the figure indicates Equation (8). The
value of αb is obtained from the least squares method.
αb is 4.68 in case of fiber volume fraction of 1.5%,
and 3.59 in case of 2.0%. The size effect becomes
smaller as the fiber fraction increases.
σ max= exp(α b ⋅ p )σ0(8)

σmax : bending strength of each size specimen
σ0 : bending strength of the largest size specimen
αb : contribution coefficient (αb>1)p: probability that fiber crosses boundaries

The proportionality coefficient D(h,T)
moisture permeability and it is a nonlineaof the relative humidity h and temperature& Najjar 1972). The moisture mass balancthat the variation inαtimeof the water mas
b=4.68volume ofFiberconcrete(watercontent w) be eq
fraction : 1.5%
divergence of the
moisture flux J
Experimental value
Calculation value0.2− ∂ 0.1= ∇ • J∂ Probability
w

0.3

t

p =1 − p=

J = − D ( h , T ) ∇h

2

f

bD

=4 2

3

f

Normalized stress

π π

∫0 ∫0

0

The water content w can be expressed aof the evaporable water we (capillary wa
vapor, and adsorbed water) and the non-e(chemically bound) water wn (MilPantazopoulo & Mills 1995). It is reas
assume that the evaporable water is a fu
=3.59 of hydration
relative humidity, hα, bdegree
degree ofFibersilicafraction
fume :reaction,
αs, i.e. we=w
2.0%
= age-dependent
sorption/desorption
Experimental value(Norling Mjonell
1997).
Under this assum
Calculation
value
by 0.1
substituting 0.2
Equation 10.3into Equati
obtainsProbability∂w strength.
∂w ∂hprobability and bending
Figure 5. The relation between
e α ∂we α+ ∇ • ( D ∇h ) =
− e
c
s
h
∂α
∂α
∂h ∂t

4 CONCLUSION

c

&+

s

&+w

where ∂we/∂h is the slope of the sorption/

isotherm
calledbehavior
moistureof capac
To quantify the size
effect (alsoin bending
governing
equation(Equation3) must be
ECC, the bending test is carried out using the speciby
appropriate
boundary
andinitialmen with various sizes as the main parameter. The conditi
the amountprobability that theThe
fiberrelation
crossesbetween
the boundary
is of eand relative
humidityinisthis
called ‘‘constructed. Thewater
followings
are concluded
isotherm” if measured with increasing
study.
humidityand show‘‘desorption(1) The bending
test resultsthat the isotherm”
bending in th
their difference
strength increasescase.
as theNeglecting
size of specimen
decreases.(Xi et al.
(2) To expressthe
thefollowing,
wall effect‘‘sorptionof fibers,isotherm”
probabil- will bereferencetobothsorptionanddesorptioncity of fibers which cross the surface of specimen
is
By
the
way,
if
the
hysteresisofthesolved theoretically. In the case of 40×40mm section
would
be takenaccount, two
specimen, 23% ofisotherm
fibers are
influenced
by intospecimen
relation,toevaporable
water
vs relative
surface and considered
be oriented
towardtwo- humi
beusedaccording
to
the
sign
of
the varia
dimension.
relativity
humidity.
Theshape(3) The coefficients of contribution to bending of theisotherm fororientation
HPC is influenced
strength by two-dimensionalof fibers byaremany p
especiallythose that influence extent andproposed based on
the probability.

chemical reactions and, in turn, determ
structure and pore size distribution (waterratio, cement chemical composition, SF
ACKNOWLEDGEMENT
curing time and method, temperature, mixetc.). InbytheGrand-in-Aid
literature various
formulatio
This study is supported
for Scienfoundtodescribethesorption
isothermtific Research (B: 20360247) from MEXT, Japan. The
concrete(Xi cooperation
et al. 1994).ofHowever,authors acknowledge
the kind
Koken in th
Sangyo Co. Ltd. paper the semi-empirical expression pro
Norling Mjornell (1997) is adopted b
Proceedings of FraMCoS-7, May 23-28, 2010

J = − D ( h , T ) ∇h
REFERENCE(1)

Li, V.C.
From Micromechanics
to Structural
The 1993.
proportionality
coefficientD(h,T)Engineering
is called–The Design of Cementitious Composites for Civil Engimoisturepermeability
and
it
is
a
nonlinearfunction
neering Applications, Journal of Structural Mechanics
andof Earthquake
the relativeEng.,
humidityhandtemperature
T
JSCE, Vol.10, No.2, pp.37-48 (Bažant
Li,
V. C., Wang,
and Backer,
S.: mass
Effect balanceof Inclining
Angle,& Najjar1972).Y. The
moisturerequiresand Surface
Treatment
Synthetic
thatBundling,
the variationin timeof theonwater
massFiber
per Pullunitout fromaconcrete
Cement Matrix,
Composites,Vol.
21, No.2,
pp.
volumeof(watercontent
w
)
be
equal
to
the
132-140, 1990
divergence of the moisture flux J
Rozanov, IU.A. & Sobor, I.M. 1976. Theory of Probability andMonte Carlo Method, Tokyo, Sougoukagaku publishingw∂(2)− company= ∇•J∂

explicitly accounts for the evolution of hydration
reaction and SF content. This sorption isotherm
reads
⎡
⎤
1
⎢
⎥
we (h, α c , α s ) = G1 (α c , α s )⎢1 −+⎥∞10(g α− α c )h ⎥
⎢
e 1 c
⎣
⎦
⎡ 10(g α ∞ − α )h
⎤
c
1 c
⎢
K1 (α c , α s ) e
− 1⎥
⎢
⎥
⎣
⎦

t

The water content w can be expressed as the sumof the evaporable water we (capillary water, water
vapor, and adsorbed water) and the non-evaporable(chemically bound) water wn (Mills 1966,
Pantazopoulo & Mills 1995). It is reasonable toassume that the evaporable water is a function ofrelative humidity, h, degree of hydration, αc, anddegree of silica fume reaction, αs, i.e. we=we(h,αc,αs)= age-dependent sorption/desorption isotherm(Norling Mjonell 1997). Under this assumption andby substituting Equation 1 into Equation 2 one
obtains−∂w ∂he + ∇ • ( D ∇h) = ∂we α
c
h
∂α
∂h ∂t

c

∂w

& + ∂α e α& + w&
s

s

n

(3)

where ∂we/∂h is the slope of the sorption/desorption
isotherm (also called moisture capacity). The
governing equation (Equation 3) must be completed
by appropriate boundary and initial conditions.
The relation between the amount of evaporable
water and relative humidity is called ‘‘adsorption
isotherm” if measured with increasing relativity
humidity and ‘‘desorption isotherm” in the opposite
case. Neglecting their difference (Xi et al. 1994), in
the following, ‘‘sorption isotherm” will be used withreference to both sorption and desorption conditions.
By the way, if the hysteresis of the moistureisotherm would be taken into account, two different
relation, evaporable water vs relative humidity, must
be used according to the sign of the variation of therelativity humidity. The shape of the sorption
isotherm for HPC is influenced by many parameters,
especially those that influence extent and rate of thechemical reactions and, in turn, determine pore
structure and pore size distribution (water-to-cement
ratio, cement chemical composition, SF content,
curing time and method, temperature, mix additives,etc.). In the literature various formulations can befound to describe the sorption isotherm of normal
concrete (Xi et al. 1994). However, in the present
paper the semi-empirical expression proposed by
Norling Mjornell (1997) is adopted because it
Proceedings of FraMCoS-7, May 23-28, 2010(4)

where the first term (gel isotherm) represents thephysically bound (adsorbed) water and the second
term (capillary isotherm) represents the capillarywater. This expression is valid only for low content
of SF. The coefficient G1 represents the amount ofwater per unit volume held in the gel pores at 100%
relative humidity, and it can be expressed (Norling
Mjornell 1997) asc α c+ ks α sG (α c α s ) = k vg
c vg s(5),1where kcvg and ksvg are material parameters. From the
maximum amount of water per unit volume that canfill all pores (both capillary pores and gel pores), one
can calculate K1 as one obtainsw −0K (α c α s ) =
,

1

⎡
⎢
−
⎢1 −
1
⎢
⎣

α s + 0.22α s G

0.188

c

s

⎛
10⎜
⎝

e

⎛

e

g αc − αc h
∞

1

⎞
⎟
⎠

10⎜

⎝

g α c∞ − α c ⎞⎟h ⎤⎥
1

⎠

⎥
⎥
⎦

(6)

−1

The material parameters kcvg and ksvg and g1 canbe calibrated by fitting experimental data relevant tofree (evaporable) water content in concrete atvarious ages (Di Luzio & Cusatis 2009b).
2.2 Temperature evolution
Note that, at early age, since the chemical reactions
associated with cement hydration and SF reaction
are exothermic, the temperature field is not uniform
for non-adiabatic systems even if the environmental
temperature is constant. Heat conduction can be
described in concrete, at least for temperature notexceeding 100°C (Bažant & Kaplan 1996), by
Fourier’s law, which reads
q = − λ ∇T

(7)where q is the heat flux, T is the absolutetemperature, and λ is the heat conductivity; in this