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

Crack formation and tensile stress-crack opening behavior of fiber

reinforced cementitious composites (FRCC)
E.B. Pereira & G. Fischer
Technical University of Denmark, Lyngby, Denmark
J.A.O. Barros
University of Minho, Guimaraes, Portugal
M. Lepech
Stanford University, Palo Alto, U.S.A.

ABSTRACT: The formation and further development of cracking in strain hardening cementitious compos-
ites under tensile loading strongly influences their mechanical behavior. The work presented in this paper de-
scribes the crack formation in fiber reinforced cement composites (FRCC). The experimental results are ana-
lyzed using a digital image analysis technique to gain detailed insight in the cracking process during the
propagation and the opening phases under tensile loading. The data and observations obtained from these tests
are used to derive the tensile stress-crack opening behavior of different types of FRCC and to analyze and
compare the effect of various composite parameters including fiber reinforcement, cementitious matrix, and
interfacial bond properties. In the experimental program, the FRCC specimens are notched and tested in direct
tension with the purpose of inducing a single crack during testing. Subsequently, the corresponding crack be-
havior in terms of stress-crack opening for all specimens is derived, allowing a quantitative evaluation of the
performance of the different fiber reinforcements. The monitoring of the crack formation with the digital im-
age acquisition allows the identification of the stages of crack development. The comparison of the stage se-
quences among different tensioned specimens contributes to the qualitative assessment of the bridging effect
obtained in each composite system and to the description of the features associated with the singular crack
generated during testing.

1 INTRODUCTION ness material. Engineered Cement Composites

(ECC) represent a class of cementitious based mate-
In the perspective of the structural design, strain- rials, typically reinforced with Polyvinyl Alcohol
hardening ability in tension is often referred to as the (PVA) fibers, which was objectively proved to be
most important property of Strain-hardening Cement able to gather these requisites in an efficient manner
Composites (SHCC). The spectrum of structural (Li 2003, Fischer & Li 2007).
problems, of quasi-static or dynamic nature, which For the full use of all the SHCC material potenti-
can be effectively solved by using these materials is alities, in the recent past some efforts have been
wide. They also open new fields of innovative struc- made to develop numerical modeling strategies es-
tural systems and strengthening techniques. pecially suited to simulate the behavior of SHCC
Under certain circumstances, strain-hardening re- structures (Kabele 2007). These models may be of a
sults in the ability of the material to develop multiple continuum nature or not, but in general their per-
cracks in tension. This relevant material property formance and accuracy has always to rely on a deep
has, in a simple view, a dual advantage in engineer- and precise assessment of the material behavior.
ing applications: while much more cracks develop This usually requires the study of the material at a
for the same deformation level, the crack opening is smaller scale than the structural.
much smaller. The resulting benefits in terms of du- The material behavior of SHCC is a delicate bal-
rability and the preservation of functional properties ance of a wide multiplicity of factors. Among others,
of the structural elements are evident. The higher the main ones are the interfacial bonding and fiber
energy dissipation ability, per se, at the level of a pull-out properties, the relation between the material
single crack is even multiplied by the high number parameters of the isolated fibers and of the matrix,
of cracks developed, resulting in a very high tough- the variation of the flaw size, the fiber orientation
and their distribution in the matrix. The randomness In the present work − D (tensile
J = the h, T )∇h behavior of six fiber
associated to all these parameters also plays an im- reinforced cementitious composites is assessed. The
portant role in the composite behavior. Adding more The proportionality
matrix used was essentially the same coefficient
for all the D(h,T)
complexity to the problem, all these factors interact composites, while moisture permeability and it is was
the fiber reinforcement a nonlinea
with each other in a highly coupled fashion. As a re- changed in type, of the relative
content humidity
and nature. h and
This work temperature
fo-
sult, it is not easy, if not impracticable, to study them cuses mainly on & theNajjar
study1972).
of the The moisture mass balanc
test-setup and the
separately or in an uncoupled fashion. In this con- conditions neededthatto the variation
assure that thein time of theofwater
behavior a mas
text, the assessment of the behavior of the SHCC at volume of concrete (water content w) be eq
single crack is accessed as close as possible. Since
the level of a single planar crack may represent a divergence
these materials are desirably of the moisture
designed to flux J
develop
valuable strategy to extract the necessary parameters multiple cracks in tension, the characterization of the
for the mechanical characterization of SHCC, in a mechanical behavior ∂w in terms of its tensile stress-
modeling or a design perspective. This strategy does crack opening at −the∂t
= ∇ • Jof a single crack is not
level
not imply the need for a full understanding of all the straightforward and can be troublesome.
micro mechanisms happening at the level of the fi- The experimentalThe work water content wofcan
is composed twobeparts.
expressed a
bers, matrix or interfaces. Consequently, it can pro- of the evaporable water we tensile
First, the experimental characterization of the (capillary wa
vide the relevant information about the material vapor,
stress-crack opening and adsorbed
behavior of eachwater) and the
composite is non-e
properties of the composite for a precise and correct (chemically
carried out. Secondly, giving more bound)
focuswater
to the ob- wn (Mil
structural modeling and design with SHCC materials, servation of the Pantazopoulo
stages of crack & Mills 1995). and
appearance It is reas
in an easy to handle fashion. Alternatively, it can also assumecracking
propagation in tension, that the isevaporable watera is a fu
assessed with
configure an essential tool for the material design relative humidity, h, degree oftwo
digital imaging system. For this last purpose, hydration
process. Some difficulties may emerge, however, different variantsdegree
of theofproposed
silica fumegeometry , i.e. we=w
reaction,forαsthe
when trying to assess the behavior of a single crack tensile tests were=developed.
age-dependent sorption/desorption
The intention is to al-
experimentally, mainly if the material is specifically (Norling Mjonell 1997). Underfield
low the visual access and further digital strain this assum
designed to develop multiple cracks and demonstrates by substituting Equation 1 into Equati
interpolation of the cracking region, which is not
high tolerance to damage, as is often the case. possible with the obtains
standard four-notched specimens.
In order to produce the adequate mechanical con-
ditions for the emergence and evolution of a single
∂w ∂w
crack, the stress distribution on test specimens needs 2.1 Tensile tests in e ∂h + ∇specimens
− notched • ( D ∇h ) = e α& + ∂we α& + w
to be locally changed and intensified. This may be ∂h ∂t h ∂α c
∂α s
c s
typically achieved by introducing constrictions or As reported in previous works (Fischer et al. 2007),
notches in some sections of the test specimen (Shah the execution of awhere
notch∂w eachisofthetheslope
in /∂h fouroffaces
the of
sorption/
e
1996, Fischer et al. 2007). At this scale level, it is the prismatic ECC isotherm
specimens
tory in assuring governing
(also
the properequation
seems called
to be
shielding(Equation
moisture
satisfac- capac
important to gather more information about the ef- 3)
and develop- must be
fect of various composite parameters. The fiber rein- by
ment of a crack inside appropriate
the notched boundary
region.andThisinitial
con- conditi
forcement, the cementitious matrix and interfacial
the first crackingwater
The relation
dition is naturally dependant
stressand
on between
and relative
the ratiothe
the peakhumidity
amount of e
between
stress. Inis the
bond properties can be assessed by observing the
present work theisotherm”
geometry if
called ‘‘
stress-crack opening behavior of what may be con-
tensile tests with humidity
different and
measured
adopted with
to conduct
fiber‘‘desorption
increasing
the
sidered as a single planar crack, at a meso-scale
tious compositescase.
isotherm”
reinforced cementi- in th
level. Digital imaging systems may represent a valu- Neglecting
is represented their
in difference
Figure 1. (Xi et al.
The
able contribution for the assessment of these mate- adopted dimensions the following,
specimen’s cross reference
allow the
section toto40%
‘‘sorption
reduction isotherm”
of the will be
rial features. These systems allow the representation bothofsorption andvalue,.
its initial desorption c
of the generated strain fields during testing, with By the way, if the hysteresis of the
high precision and at small scales. This is typically isotherm would be taken into account, two
done by interpolation of the observed displacements relation, evaporable water vs relative humi
from a matrix of dots superimposed to the digital be used according to the sign of the varia
image. The allowable visual magnification deter- relativity humidity. The shape of the
mines the scale at which the strengthening mecha- isotherm for HPC is influenced by many p
nisms are studied. The crack appearance and devel- especially those that influence extent and
opment during tension in notched SHCC specimens chemical reactions and, in turn, determ
can be assessed, particularly in the early stages of structure and pore size distribution (water-
the test sequence, while the crack plane fully devel- ratio, cement chemical composition, SF
ops and the tensile stresses acting at the matrix curing time and method, temperature, mix
gradually transfer into the fibers bridging the crack. etc.). In the literature various formulatio
found to describe the sorption isotherm
Figure 1. Schematicconcrete (Xi ofet the
representation al.specimen
1994). However,
geometry in th
2 EXPERIMENTAL PROGRAM paper the assessment
A, adopted for the experimental semi-empirical expression
of the tensile be- pro
havior of FRCC. TheNorling
specimen’sMjornell
length was (1997)
120.0 mm.is adopted b

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

 J = − D (ah,total
within T )∇h extension of 0.5 mm, the notch thick- (1) explicitly accounts for the evolution of hydration
ness. The very small notch thickness was adopted reaction and SF content. This sorption isotherm
withThetheproportionality coefficient
intention of isolating, D(h,T)
as much is called
as possible, reads
moisture permeability and it is a nonlinear function
one single planar crack
of Elongated
the relativeplates
humidity h and
of 60 cmtemperature
long, 10.5 Tcm(Bažant
wide ⎡ ⎤
& Najjar 1972). The moisture mass balance requires
and 1.6 cm thick were cast and left to dry cure for 28
we (h α c α s ) = G1 (α c , α s )⎢⎢1 − 1 ⎥
+
that the variation in time of the water mass per unit ∞
, ,
days. From each of these plates the specimens were (g α
c 10 − α c )h ⎥⎥
volume
cut in theoflongitudinal
concrete (water content
direction andw)the
be casting
equal tofree
the e
⎢
⎣ 1
⎦ (4)
divergence
surface wasofrectified.
the moisture
The flux J composition was
matrix
⎡ (g α
∞ − α )h ⎤
similar in all of them, and the weight proportions of
K (α c α s ) e
, ⎢ c 10
c − 1
1⎥
the∂wingredients are represented in table 1. ⎢ ⎥
(2)
1
− = ∇•J ⎣ ⎦
∂t
Table 1. Weight proportions of the matrix constituents.
The waterFly
contentw can
Finebe expressed
Quartzaspowder
the sum where the first term (gel isotherm) represents the
Cement ash sand
of the evaporable water we (capillary water, water physically bound (adsorbed) water and the second
vapor,
1 and adsorbed
2 water)0.35 and the 0.35
non-evaporable term 2.(capillary
Figure isotherm)
Picture of the test-setuprepresents the capillary
used for the tensile behavior

(chemically bound) water wn (Mills 1966, water. This by expression

the hydraulic is valid
andonly forclip
lowgages
assessment of the FRCC, showing the specimen fixed in the
extremities grips the two content
used
Pantazopoulo
Fibers of two & different
Mills 1995). natures It were
is reasonable
used: PVAto ofmonitor
to SF. The the crack opening.G1 represents the amount of
coefficient
assume thatalcohol)
(polyvinyl the evaporable
based and water is a function of
PAN (polyacrylnitrile) water per unit volume held in the gel pores at 100%
relativeThe
based. humidity, h, degreeand
main geometrical of mechanical c, and
hydration, αproper- relative
these humidity, atanda meso-scale
mechanisms it can be expressed
level may(Norling
provide
degree of silica fume reaction, αs, i.e.
ties of these fibers are presented in table =we(h,αc,αs)
we2. Mjornell 1997) as
important information about the interaction between
= age-dependent sorption/desorption isotherm the fibers and the matrix, with a special focus on the
(Norling
Table Mjonell
2. Main 1997).
properties of theUnder this assumption and
fibers studied. c mechanisms
G1 (α c , α s ) = k vg
strengthening α c + k s α s and on the cracking (5)
by substituting EquationTensile 1 intoLength
EquationDiameter
2 one c vg s
stages and crack development.
obtains
Fiber Abbrev. strength The monitoring of crack appearance and progres-
µm
whereduring
sion kcvg and ksvg are
testing withmaterial parameters.
the optical system From
requiresthe
∂w ∂h
MPa
∂w
mm
∂w maximum amount of water per unit volume that can
the existence of a smooth and visually accessible
−PVAe Rec +15∇ • (PVA eα
Dh ∇h) = 1600 &+ 8e α & + w&40.0 (3) fill all pores
surface in one (both capillary
of the pores faces,
specimen’s and gelwhich
pores),is one
in-
∂h ∂t ∂α c
∂α s n
can calculatewith K1 astheone obtains
PAN D10 1.5 PAN 1.5 826 c 6s 12.7 compatible slim grooves used in geometry
A (Fig.1). The small scale at which the observations
PAN D10 3.0 PAN 3.0 767 6 18.0
where ∂we/∂h is the6.7slope413of the sorption/desorption are supposed to be carried out ⎡implies
⎢
also the
⎞ ⎤ need
⎜ g α − α ⎟h ⎥
⎛ ∞
c c
10

isotherm
PAN D18 6.7
(alsoPANcalled moisture6 capacity). 26.8
The of adoptingw very α s + apertures
small α s − G ⎢ −ine the optical⎠ lenses
⎝ 1

to maximize the depth ofs field,

−
0.188
c 0.22 1 ⎥

governing ⎦ (6)
PAN D18 30equation (Equation
295 3) must be completed
0 1
PAN 30 6 57.0 ⎢ which in turn⎥ de-
K (α c α sfor ⎣
by appropriate boundary and initial conditions. mands
1
, ) = extra lighting. These requirements impose
⎜ g α − α ⎟h
⎛ ∞ ⎞

Thetherelation between the mixes

amountcontained
of evaporable e ⎝ c c ⎠shape,
10

For PAN fibers all the 2% of a delicate balance of specimen’s 1

−
lighting con-
1

water inandvolume,
fibers relativewhile
humidity
for PVA is called
fibers 1%‘‘adsorption
and 2% ditions and the precision of the measurements. For
isotherm”
volume if measured
amounts were used. with increasing relativity thisThe material
purpose, in parameters
the present kwork c
andtwo
vg and gspeci
ks distinct vg can 1
humidity and ‘‘desorption
The test sequence consisted isotherm” in the opposite
of submitting the be calibrated by fitting experimental data relevant to
case. Neglecting
specimens their difference
to a displacement (Xi et constant
controlled al. 1994),ax-in free (evaporable) water content in concrete at
the extension
ial following,rate‘‘sorption
of 5 µisotherm”
m/s. This will be used with
deformation rate various ages (Di Luzio & Cusatis 2009b).
reference
was to bothfrom
transmitted sorption
the and desorption
hydraulic conditions.
actuator to the
By the way, if the hysteresis
specimen by means of two hydraulic grips. Theseof the moisture 2.2 Temperature evolution
isotherm
grips wouldfixed
conferred be taken intotoaccount,
support two different
the specimen’s ends
relation, evaporable
(rotations water vs relative
and transverse humidity, were
displacements must Note that, at early age, since the chemical reactions
be used according
blocked). to the sign
While testing, of the variation
the opening of the of the
notch associated with cement hydration and SF reaction
relativity humidity. The shape
was evaluated by means of two clip gages, posi-of the sorption are exothermic, the temperature field is not uniform
isotherm
tioned for HPC sides
in opposite is influenced
as shownbyinmanyFigure parameters,
2. for non-adiabatic systems even if the environmental
especially those that influence extent and rate of the temperature is constant. Heat conduction can be
chemical reactions and, in turn, determine pore described in concrete, at least for temperature not
structure
2.2 andmonitoring
Optical pore size distribution (water-to-cement
of crack development exceeding 100°C (Bažant & Kaplan 1996), by
ratio, cement chemical composition, SF content, Fourier’s law, which reads
curing
The time and
cracking method, temperature,
mechanisms taking place mixinside
additives,
fiber
etc.). In thecement
reinforced literature various manifest
composites formulations can be
themselves q = − λ3.∇Geometries
Figure T (7)
adopted for the visual inspection and op-
found towhen
visually describe
thesethe sorption
cracks eitherisotherm
reach or of normal
initiate at tical monitoring of the FRCC specimens. Geometry B provides
concrete
the surface(Xiofet the
al. specimens.
1994). However, in the present
The monitoring of
a flat surface and the geometry C tries to replicate the same
where qobtained
conditions is thewithheat flux, used
the shape is tensile
T for the absolute
tests, the
paper the semi-empirical expression proposed by temperature,
geometry A. Theand λ is the
specimens’ heatwasconductivity;
length 120.0 mm. in this
Norling Mjornell (1997) is adopted because it

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

men shapes were conceived specifically for the 7
J = − D ( h , T ) ∇h
study of crack development with the optical system 2% PVA
(see Fig. 3). One of the proposed solutions consists 6 The proportionality coefficient D(h,T)
on simply suppressing one of the notches, keeping moisture permeability and it is a nonlinea
the same shape and dimensions of the notched cross- 5 of the relative humidity h and temperature
& Najjar 1972). The moisture mass balanc

T ensile stress [M Pa]

section area. This is achieved by executing the two
10 mm depth notches, one in each of the opposite 4 that the variation
1% PVA in time of the water mas
smaller sides, and one 4 mm depth slit along one of volume of concrete (water content w) be eq
the longer sides, as represented in Figure 3 (geome- 3 divergence of the moisture flux J
try B). In this fashion, one of the specimen’s faces is
− ∂ = ∇•J
kept flat in the notched region. The original pre- 2 w
cracked stress field in the notched area is, neverthe- ∂ t
less, changed when compared with the one induced 1
by geometry A. However, the flat surface is the most The water content w can be expressed a
favorable for the optical monitoring. 0 of the evaporable water we (capillary wa
The other geometry proposed consisted on creat- 0 vapor,
0.5 and1 adsorbed
1.5 water)
2 and the non-e
ing a round cylindrical surface replacing one of the (chemically
CMOD [mm]bound) water wn (Mil
slits (geometry C, see Figs. 3 and 4). This procedure 7 Pantazopoulo & Mills 1995). It is reas
tries to replicate, as close as possible, the stress assume that the evaporable water is a fu
fields generated near the notched area by replacing 6 relative humidity, h, degree of hydration
the discrete notch with a cylindrical smooth surface degree of silica fume reaction, αs, i.e. we=w
with 22 mm of diameter. This cylindrical surface 5 = age-dependent sorption/desorption
PAN 6.7
reaches a depth of 2 mm, the same as the original (Norling Mjonell 1997). Under this assum
T ensile stress [M Pa]
discrete notch. In this case the intention is to achieve 4 by PANsubstituting
30 Equation 1 into Equati
the best compromise between the flatness and obtains
smoothness required for a precise digital interpola- 3
tion of the deformations, while keeping the local
∂w ∂h
strong constriction necessary to intensify the stress − e + ∇ • ( D ∇h) = ∂we ∂w
α&c + e α&s + w
field adequately. At the same time, while allowing
2
∂h ∂t h ∂α ∂α
c s
the visual access of the crack propagation, the pro- 1
posed shape approximately preserves the axial cen- where ∂we/∂h is the slope of the sorption/
tering of the tensile loading (Fig. 4). isotherm (also called moisture capac
0
0 governing
0.5 1equation 1.5 (Equation
2 3) must be
by appropriate
CMOD [mm]boundary and initial conditi
7 The relation between the amount of e
water and relative humidity is called ‘‘
6 isotherm” if measured with increasing
humidity and ‘‘desorption isotherm” in th
PAN 3.0

5 case. Neglecting their difference (Xi et al.

the following,
PAN 1.5
‘‘sorption isotherm” will be
T ensile stess [M Pa]

4 reference to both sorption and desorption c

By the way, if the hysteresis of the
3 isotherm would be taken into account, two
relation, evaporable water vs relative humi
2 be used according to the sign of the varia
Figure 4. Picture of one of the specimens conceived with the relativity humidity. The shape of the
geometry C, allowing the visual inspection of crack develop-
1 isotherm for HPC is influenced by many p
ment and the preservation, as close as possible, of the notched
cross section. especially those that influence extent and
0
chemical reactions and, in turn, determ
structure and pore1.5size distribution (water-
ratio, cement
CMOD [mm]chemical composition, SF
0 0.5 1 2
3 RESULTS AND DISCUSSION
curing time and method, temperature, mix
3.1 Tensile tests etc.). In the
CMOD curves for the six tested FRCC. literature various formulatio
Figure 5. Graphics showing the obtained tensile stress versus
found to describe the sorption isotherm
The obtained results in terms of tensile stress versus concrete
The tensile stress values(Xi et al.
result from 1994). However, in th
the normali-
crack mouth opening displacement (CMOD) for the paper the semi-empirical
zation of the tensile load with respect to the notched expression pro
six different composites are shown in Figure 5. Norling Mjornell (1997)
cross sectional area. For clarity, only the results is adopted
of b

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

 J = −D
three ( h , T ) ∇h
specimens are shown for each composite. The (1) explicitly accounts
summarize and condense for thein evolution
a rational of andhydration
easy to
anomalous results were also excluded. reactionapproach
handle and SFthecontent. This sorption
tensile behavior of FRCC, isotherm
very
Thegeneral,
In proportionality
the first coefficient D(h,T) is called
pertinent observation about reads in a design perspective.
useful
moisture permeability and it is a nonlinear function
the obtained results is that, for each composite, there Observing Figure 5, it becomes clear that a strong
of athegood
is relative humidity
agreement h and temperature
between curves obtained T (Bažant
from correlation can be established ⎡ between the geometri- ⎤
& Najjar 1972). The moisture mass balance requires
different specimens. The scatter of results may cal and mechanical properties
w (h, α c , α s ) = of
G1 (αthe ⎢
) 1−
of the1 used fibers ⎥
+ and
that the variation in time of the water mass per unit
change with the procedure used to cast the speci- thee features c , α sstages
⎢
10(g α
∞
previously − α c )h ⎥⎥
described.
1 c
volumedue
mens, of concrete
to differently (waterinduced
content fiber
w) bedistributions
equal to the Without detailing too much ⎢
⎣ e micromechanical
the ⎦ (4)
as-
divergence
and predominant of the orientations.
moisture fluxExcludingJ this factor, pects of each fiber reinforcement type, what is rele-
⎡ 10(g α ∞ − α )h ⎤
one may say that the true character of the material vant in the present context
K1 (α c ,magnitude
α )e ⎢ c
is to1 stress c
the importance
− 1⎥
s⎢
− ∂w = It∇ •isJ clear that parameters like the nature, (2)
behavior in tension is captured by this type of test- of the shape, size, and significance ⎥ of
⎣ ⎦
∂t
setup. the each of the identified stages in the overall assessed
geometrical and the mechanical properties of the tensile behavior. These will reveal themselves in a
usedThefibers
waterplay content w can be expressed
an important role in theasobserved the sum where theway
particular firstwhen
termthe (gelmaterial
isotherm) is usedrepresents
in a struc- the
of the evaporable water we (capillary
features of the tensile behavior of eachwater, water
composite. physically
ture. One brief bound (adsorbed)
example wouldwater be the andability
the second
of the
vapor, and adsorbed water) and the non-evaporable
The direct correlation between them and the final term (capillary
material to developisotherm) multiplerepresents
cracks the in capillary
tension,
(chemicallytensile
composite bound)behavior water is notw easily (Mills 1966,
accessible, roughly dependant on the ratio between first content
water. This expression is valid only for low crack-
Pantazopoulo & Mills 1995). It is reasonable
as generally recognized. However,
n
each of the pre- to of SF.
ing stressThe(first
coefficient
stage) and G1 represents
the peak bridgingthe amount of
stress
assumestress
sented that vs theCMODevaporable curves water is a function
is apparently able toof water stage).
(third per unitAnther
volumeexample held in would
the gelbepores the at 100%
durabil-
relativethehumidity,
reveal role that hthe , degree of hydration,
geometrical αc, and
and mechanical relative
ity humidity, design
and functional and it aspects,
can be expressed
very dependent (Norling on
degree of silica
properties of thefume fibersreaction,
have in αs, the
i.e. w e=we(h,αc,αs)
overall tensile Mjornell 1997) as
crack opening, which is most likely correlated with
= age-dependent sorption/desorption isotherm
behavior of the composite. the inclination of the second stage hardening branch.
(Norling Mjonellresults 1997).seem Under this assumption and c of distinctive
s
G1The
(α , α ) = k α c + k α s
vg s identifiable in all (5)
The obtained to reveal some few dis- sequence stages above men-
by substituting Equation 1 into Equation 2 one
tinct generic stages of a typical mechanical behavior. tioned c seems
s vgto c be clearly the
obtains
At the onset of tensile testing, the evolution starts tested composites. The adopted fiber reinforcements
with what can be assumed as a predominantly elastic reveal kcvg and kmechanical
where different s
vg are material parameters.inFrom
performances tension,the
behavior, with a very steep increase of load bearing
∂w ∂h ∂w maximum amount of water per unit volume that can
but still these stages remain clearly distinctive and
− e e α& + ∂we αincreases.
& &
s + wn
capacity ∇ • ( Dtensile
while
+ ∇h) = deformation This
(3) fill allthepores
keep same(both
formalcapillary
details.pores
Thisand gel pores),
suggests that aone ra-
∂ h ∂ t
first stage may be assumed h ∂ α c
∂ α can calculate K1 as one can obtains
c to end swhen the matrix tional design approach be based on the assump-
cracking strength is achieved, with a very quick tion that, in general, the stress-crack opening behav-
where ∂w
transition
e/∂h is the slope of the sorption/desorption
of the tensile stresses from the cracked ior of FRCC follows a standard ⎡ sequence
⎢ 10
of⎞ stages
⎜ g α − α ⎟h ⎥
⎛ ∞
c c
⎤

isotherm
matrix to (also
the called
fibers. moisture
Preliminary capacity).
stages of micro-The assuming wthe− shape 0.188 α s + α s −G ⎢ −e
of a
c curves (Fig.6).
0.22
generic
1
⎝ 1design ⎠ tensile
⎥

governing equation (Equation 3) must be completed ⎦ (6)

0 1
cracking and micro-defect propagation may affect stress-crack opening ⎢ ⎥
K (α c α s ) = ⎣
by appropriate
the shape of this boundary
transition andeventually
initial conditions.
stage, ruled mainly by 1
,

⎜ g α − α ⎟h
⎛ ∞ ⎞

The relation between the amount of evaporable e ⎝ c c⎠ −

fracture mechanisms and dependent of 10
1
1

water and relative

micro-defect shape andhumidity
size. is readjustment
called ‘‘adsorption
isotherm”
In a second if measured
stage, with
the internal increasing relativity
of the
The material parameters kcvg and ksvg and g1 can
humidity and
composite ‘‘desorption
microstructure isotherm”
to the new insubsequently
the opposite
damaged con-
be calibrated by fitting experimental data relevant to
case. Neglecting
figuration typically their difference
reveals itself by(Xilike
theet al. 1994), in free (evaporable) water content in concrete at
the following,
observed sharp‘‘sorption
load decay. isotherm”
Factors will be used with
fiber-matrix
various ages (Di Luzio & Cusatis 2009b).
reference
bonding, to
fiber both sorption
diameter, and
number desorption
of fibersconditions.
bridging
By the
the crackway, and ifthe the ratio
hysteresis
between of thethe moisture
Young’s
2.2 Temperature evolution
isotherm would
modulus of fibers be taken into account, two different
and the bulk matrix may play an
relation,
important evaporable
role. water by vs relative humidity,stage, must Note that, at early age, since the chemical reactions
be used
The according
former to
is followed the sign aof
new the variation
hardening of the associated with cement hydration and SF reaction
relativity
most likelyhumidity.
supported by The theshape of
full mobilizationthe sorption
of the
are exothermic, the temperature field is not uniform
isothermThe
fiber-matrix forpeakHPCstress
bonding is influenced
mechanisms by inmany
and while parameters,
fibers
for non-adiabatic systems even if the environmental
especially
stretch. those thateffect influence
is reached extent and
this rate of the
third stage,
temperature is constant. Heat conduction can be
chemical
when reactions
the stiffening and, in
of turn,
the fiberdetermine
reinforcement pore described in concrete, at least for temperature not
structure
is exhausted and pore
either size
duedistribution (water-to-cement
to fiber debonding or fiber
exceeding 100°C (Bažant & Kaplan 1996), by
ratio,
rupture cement
predominance. chemical composition, SF last
content, Fourier’s law, which reads
curing
The time
fourth andand method,
last stage temperature,
consists ofmix theadditives,sof- Figure 6. Diagram representing the generic bridging stress
( σ B ) versus crack opening (w) relationship, typical of FRCC.
etc.). the
tening Inbranch
the literature
observedvarious in all formulations
curves. It coincides can be q = − λ ∇T (7)
found
with to describe
gradual the sorption
neutralization isotherm of
of the remaining normal As previously reported (Fischer et al. 2007), the
concrete
bonds (Xi
between ettheal.opposite
1994). However,
crack faces.in the present
where q is of
simplification thethisheat flux, by
behavior T aistri-linear
the absolute
curve
paper
This the
is semi-empirical
obviously a expression
simplistic proposed
interpretation of thebya temperature,
may andthe
enclose all λ isnecessary
the heat and
conductivity;
fundamental in this
de-
Norling
complex Mjornell
mechanisms (1997)
takingisplace
crack in the bulk composite, but somehow they
adoptedat thebecause
level of it
tails for the overall composite tensile behavior. One

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

plausible way to represent it would be as shown in J = − D ( h , T ) ∇h
Figure 6, where the first branch may be suppressed
for further simplification. The proportionality coefficient D(h,T)
The identification of the parameters that define moisture permeability and it is a nonlinea
the shape of the bridging stress, σ B , versus the of the relative humidity h and temperature
crack opening, w relationship may be done experi- & Najjar 1972). The moisture mass balanc
mentally. For that purpose the above suggested setup that the variation in time of the water mas
conditions may be adopted. The first cracking volume of concrete (water content w) be eq
strength, σ fc , the initial bridging stress, σ B ,initial , the divergence of the moisture flux J
peak bridging stress, σ B , peak , and the corresponding
− ∂ = ∇•J
crack opening at peak bridging stress, w0 , represent w
the most significant parameters to be identified. ∂ t
Subsequently there is the residual bridging stress,
σ 1 , the corresponding crack opening, w1 , and finally The water content w can be expressed a
the ultimate cohesive crack opening, w2 . All these of the evaporable water we (capillary wa
data, and the scatter associated to each parameter, vapor, and adsorbed water) and the non-e
may be used to support the structural design and to (chemically bound) water wn (Mil
define safety factors for each mechanical parameter. Pantazopoulo & Mills 1995). It is reas
Alternatively, in some cases the material requisites assume that the evaporable water is a fu
may be imposed by the structural concept. These relative humidity, h, degree of hydration
requisites may assume the shape of a predetermined degree of silica fume reaction, αs, i.e. we=w
bridging stress-crack opening curve, which will con- = age-dependent sorption/desorption
stitute the support of the material design task with a (Norling Mjonell 1997). Under this assum
FRCC based solution. In this fashion, material and by substituting Equation 1 into Equati
structural design may be integrated. obtains
∂w ∂h
3.2 Crack development under tension − e + ∇ • ( D ∇h) = ∂we ∂w
α&c + e α&s + w
∂h ∂t h ∂α ∂α
c s
The collection of photos taken during tensile testing
of the specimens with geometry C was used to digi- where ∂we/∂h is the slope of the sorption/
tally interpolate the strain fields near the notched re- isotherm (also called moisture capac
gion. The point matrix used to interpolate the strain governing equation (Equation 3) must be
fields was discretized with enough accuracy to make by appropriate boundary and initial conditi
possible the identification of even invisible cracks, The relation between the amount of e
with openings smaller than 0.1 mm. The test proce- water and relative humidity is called ‘‘
dure was the same as previously referred. isotherm” if measured with increasing
The sequence of pictures shown in Figure 7 humidity and ‘‘desorption isotherm” in th
represents, in gray scale, the strain fields obtained in case. Neglecting their difference (Xi et al.
five specific stages of the tensile testing. Stage zero the following, ‘‘sorption isotherm” will be
refers to the onset of the test sequence, with the reference to both sorption and desorption c
specimen still unloaded. The white dots reflect uni- By the way, if the hysteresis of the
dentifiable regions of the specimen’s surface, mainly isotherm would be taken into account, two
constituted by pores with deficient illumination and relation, evaporable water vs relative humi
excessive geometric gradients. The represented be used according to the sign of the varia
strain fields are useful to support the interpretation relativity humidity. The shape of the
of the cracking process observed in the composite, isotherm for HPC is influenced by many p
but only in a qualitative perspective. They represent especially those that influence extent and
a convenient way to visualize and interpret the dis- chemical reactions and, in turn, determ
crete displacements observed in all the sampling structure and pore size distribution (water-
points, using a reference length of 15 pixels. ratio, cement chemical composition, SF
Among all the tests preformed, the evolution of curing time and method, temperature, mix
cracking of the composite reinforced with 2% of etc.). In the literature various formulatio
PVA fibers was selected (Fig. 7), due to its greater found to describe the sorption isotherm
potential to develop multiple cracks. concrete (Xi et al. 1994). However, in th
Figure 7. Pictures of the composite containing 2% of PVA and
The selected sequence of five pictures tries to paper Stage
the semi-empirical
zero corresponds toexpression
adopting geometry C, with the corresponding interpolated
strain pattern superimposed. the ref- pro
summarize the most important stages of the over
all cracking process evolution observed in the
Norling
erence stage, unloaded Mjornell (1997) is adopted b
and undeformed.

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

 J = − D ( hspecimens.
notched , T ) ∇h First of all, it is important to (1) re- explicitly
This is the accounts for the that
kind of situation evolution
is not of hydration
desirable for
fer that the CMOD measured during all testing in the reaction
the and SF content.
characterization This sorption
of the stress-crack isotherm
opening be-
The proportionality
opposite notches was approximately coefficient D(h,T) the same is called
(less reads of FRCC. The pronounced crack branching
havior
moisture permeability and it is a nonlinear function
than 0.001 mm of difference). This was expectable, and crack multiplication increases the resulting axial
of the relative
nonetheless humidity h and
a confirmation temperature
is essential to assureT (Bažantthat deformation for the same⎡ load level, when compared ⎤
& Najjar 1972). The moisture mass balance requires
the prescribed boundary conditions are in fact being with the situation where ⎢a single crack
w (h, α c , αtime, ⎥
is present.
+ At
s ) = Gthe(α , α ) 1 −
1

that the variation

effectively in time ofgiving
accomplished, the water mass per unit
the sensitivity of thee same 1 cbehavior
s ⎢ of10several
(g α
∞ − α c )h ⎥
small ⎥ parallel
1 c likely
volume of concrete (water content w) be equal to the
the test. It may be assumed then that the displace- cracks opening simultaneously, ⎢
⎣ e most in (4)
⎦ differ-
divergence
ment is taking of the
place moisture
exclusivelyflux Jin the direction per- ent stages of its stress-crack opening diagram, masks
∞ − α )hopening
10(g α- crack
pendicular to the notched plane. the true character of the⎡⎢stress 1 c c
⎤
be-
K1 (α c , α ssingle
) e − 1⎥features
∂Observing the sequence of pictures in Figure 7, it
(2)
havior of the assumed ⎢ crack. These ⎥
is− apparent
= ∇ • J that the presence of the notches induces
w

∂t are relevant in a material⎣design perspective,⎦ and for

the appearance and propagation of what may be con- their clear and unmasked experimental characteriza-
The water
sidered as a single content w canLooking
crack. be expressed at Stage as 9,
theclose
sum where
tion thethe first term
isolation (gel isotherm)
of a single crack should represents the
be attained
of the evaporable water we (capillary water,visible
to the mouth of the left and right notches it is water physically
as close as bound
possible. (adsorbed) water and
For that purpose, thethe second
execution
vapor,
the and adsorbed
appearance of small water) and the
regions wherenon-evaporable
high strain termthe(capillary
of four notches isotherm) represents
assuming the capillary
the configuration
(chemically
gradients bound) already
are detected, water taking wn (Millsthe shape1966, of a water. This expression is valid only
adopted in geometry A (see Fig. 1) is decisive. for low content
Pantazopoulo & Mills 1995). It is reasonableany
preliminary crack. It is not possible to visualize to of SF. The coefficient G1 represents the amount of
assume that
discrete crackthewith evaporable
naked eye water or is evena function
with high of water per unit volume held in the gel pores at 100%
relative humidity,
magnifications (up to h, 10 degree
times),ofsohydration,
these preliminaryαc, and relative humidity, and it can be expressed (Norling
degreestrain
small of silica fume reaction,
concentration regions αs, assume
i.e. we=weventually
e(h,αc,αs)
Mjornell 1997) as
= shape
the age-dependent
of a damaged sorption/desorption
region of material which isotherm in
(Norling Mjonell 1997). Under this assumption and
further stages will originate the discrete crack. c α c+ ks α s
G (α c α s ) = k vg (5)
by Assubstituting
shown in Stage Equation 10, only 1 into Equationafter
0.2 seconds 2 onethe 1
,
c vg s
obtains9, the visible areas of high gradients in the
Stage
strain field are now spread through the entire speci- where kcvg and ksvg are material parameters. From the
men’s width, closely aligned and assuming the shape
∂w ∂h ∂w ∂w maximum amount of water per unit volume that can
− a efuture
of • ( D ∇h)crack.
+ ∇tension = e Itα&is+ still e αnot
& s + wn
&
possible to
(3) fill all pores (both capillary pores and gel pores), one
∂ h ∂ t h ∂
visualize a discrete crack cin the specimen’sα c
∂ α surface, can calculate K1 as one obtains
s
but the strain gradients are very localized and con-
wheretoto∂wathe
fined
e/∂h is the slope of the sorption/desorption
thin aligned region. Stages 9 and 10 corre- ⎡
⎢
⎛
10⎜ g αc αc h
∞
−
⎞ ⎤
⎟ ⎥
isotherm
spond (also
right called
after and moisture
right before capacity).
peak crack- The w − 0.188α s
c + 0.22 α s G
−
s ⎢1 − e ⎝ 1 ⎠
⎥

governing equation (Equation 3) must be completed (6)

0 1
ing stress reaching, respectively. Figure 8. Picture of the composite ⎢ containing 2% of⎥ PVA
K αc α s
adopting )geometry
⎦
B, at the instant ⎣ when the peak bridging
by At appropriate
Stage 15boundary the crackand initial conditions.
is apparently fully devel- 1
( , =
stress is reached. g αc αc h
⎛ ∞
−
⎞
oped.Thecarrying
From relationcapacity
here, betweenofthetheamount
there is a constant of evaporable
increase of the 10⎜ ⎟
e ⎝ 1 ⎠ −1

water
load and relative humidity is called
specimen ‘‘adsorption
with the
The results obtained withc geometry B revealed
isotherm”
gradual increase if faces.
measured
of the spacing with continuous
increasing
between the relativity
opposite
The material parameters k and
themselves very useful in thevgunderstanding k s
vg and g1 can
of the
humidityboth
crack mouth and notches
‘‘desorptionThe isotherm”
dark inregion
the opposite
line con-
be calibrated by fitting experimental data
cracking behavior of the studied composites. The relevant to
case. Neglecting
necting their difference
represents (Xi most
the et al. likely
1994),the
where in free (evaporable)
appearance water
of parallel crackscontent in concrete
and their at
interaction,
the
strong following,
strain gradients‘‘sorption areisotherm”
observed,will be used with various
their ages (Di Luzio
chronology and & theCusatis 2009b).observed in
differences
reference
tensile crack, tostage
both
alreadysorption
fully and desorption 148,
developed. conditions. these features between different composites inputs
By Thethe last way, if the
represented hysteresis of
is the Stage the moisture
which
2.2 Temperature
also evolution of great value. All this
qualitative information
isotherm
corresponds would beThe
to the taken
instant intoline
when account,
the two different
peak bridging
data is, though, extensive and falls out of the scope
relation,
stress evaporable
is reached. water
white vs relative humidity,
separating themusttwo
Note
of thethat, at early
present work.age,
Thesince the chemical
geometry B allowsreactions
confin-
be used
regions ofaccording
the specimen toarethealready
sign oftoo
reveals thehigh
that variation
the ofcom-
strain gradi- the associated with cement hydration and
ing the crack processes to a much smaller area, mak-SF reaction
relativity
ents in that humidity.
region The shape of theto be sorption are exothermic,
ing easier their the temperature
observation withfield
smallis not uniform
scale high
isotherm
puted, so for oneHPC may is influenced
assume that aby many parameters,
completely discrete
for non-adiabatic
precision systems even if the environmental
optical systems.
especially
crack is those
opening. that influence extent tensile
and ratetest of for
the temperature
Going backistoconstant.
geometryHeat conduction
C, with increased can be
magni-
chemical
In Figure reactions
8 the same and, in
stage turn,
of thedetermine pore described in concrete, at least for temperature
fication it is possible to observe with more detail not
structure
the and
same composite poreinterpolated
size distribution
is shown, this time(water-to-cement
adopting the
exceeding
what happens 100°C
close (Bažant & Kaplan
to the lateral notches. 1996), by
The small
ratio, cement tochemical
geometry B. The composition,
strain fieldsSF were content,not
Fourier’swhere
regions law, which reads gradients where first de-
high strain
curingattained
superimposed time andwhen method,
the photo,temperature,
since the mixstress
cracking additives,
pat-
tected, at Stage 9 (see Fig. 7), may be studied with
etc.).
tern In the literature thevarious formulations
peak bridging canwas be q = − detail.
more λ ∇T At the same time, a closer insight to (7) the
found
reached to is describe
clear the
enough. sorption
It may isotherm
be observed of normal
that the
crack initiation and propagation in the notched re-
concrete (Xi
geometry B doeset propagation
al.not1994).createHowever,the conditionsin the present
for the
whereis also
gion q is allowed,
the heatnecessary
flux, Tto isdiscern
the absolute
the dis-
paper
appearance the semi-empirical
and expressionforproposed
of a single crack, by
and
temperature, and λ is the heat conductivity; in this
crete character of the formed crack.
Norling
the Mjornell
high potential of(1997)
the composite
tion of multiple cracks is consequently revealed.
is adopted because
the genera-it
In Figure 9 the sequence of the most relevant

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

stages of cracking are identified on the stress- J = − D ( h , T ) ∇h
CMOD diagram obtained from the tensile test. For
clarity, the results are shown only up to a measured The proportionality coefficient D(h,T)
value of the CMOD of 1 mm. The series of stages moisture permeability and it is a nonlinea
shown was selected in order to represent the most of the relative humidity h and temperature
important features detected during the hole testing & Najjar 1972). The moisture mass balanc
sequence. that the variation in time of the water mas
volume of concrete (water content w) be eq
7 divergence of the moisture flux J

− ∂ = ∇•J
stage 15
stage 63 w
6
stage 148 ∂ t

5
The water content w can be expressed a
Tensile stress [MPa]

of the evaporable water we (capillary wa

4 vapor, and adsorbed water) and the non-e
stage 10 (chemically bound) water wn (Mil
3 Pantazopoulo & Mills 1995). It is reas
stage 9 assume that the evaporable water is a fu
2 relative humidity, h, degree of hydration
stage 8
degree of silica fume reaction, αs, i.e. we=w
1 stage 7 = age-dependent sorption/desorption
(Norling Mjonell 1997). Under this assum
stage 6
by substituting Equation 1 into Equati
0
obtains
0 0.25 0.5 0.75 1
∂w ∂h
e + ∇ • ( D ∇h) = ∂we ∂w
CMOD [mm]
Figure 9. Identification of the relevant stages on the stress – −
h α&c + e α&s + w
∂h ∂t ∂α ∂α
CMOD diagram obtained with geometry C specimen. c s
The represented stages (Fig. 9) are revealed in where ∂we/∂h is the slope of the sorption/
Figure 10, where the notch in the right hand part of isotherm (also called moisture capac
the specimen was magnified. For each stage, the true governing equation (Equation 3) must be
photo is presented in the left hand side and the inter- by appropriate boundary and initial conditi
polated strain field is shown in the right hand side. The relation between the amount of e
water and relative humidity is called ‘‘
isotherm” if measured with increasing
humidity and ‘‘desorption isotherm” in th
case. Neglecting their difference (Xi et al.
the following, ‘‘sorption isotherm” will be
reference to both sorption and desorption c
By the way, if the hysteresis of the
isotherm would be taken into account, two
relation, evaporable water vs relative humi
be used according to the sign of the varia
relativity humidity. The shape of the
isotherm for HPC is influenced by many p
especially those that influence extent and
chemical reactions and, in turn, determ
structure and pore size distribution (water-
ratio, cement chemical composition, SF
curing time and method, temperature, mix
etc.). In the literature various formulatio
found to describe the sorption isotherm
concrete (Xi et al. 1994). However, in th
paperof the
Figure 10. Identification therelevant
semi-empirical
stages on theexpression
stress – pro
Norling Mjornell (1997) is adopted b
CMOD diagram obtained with geometry C specimen.

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

 J − Dobserving
=By ( h , T ) ∇h Figures 9 and 10 it is possible(1)
to explicitly accounts for the evolution of hydration
follow the initiation and further propagation of the reaction and SF content. This sorption isotherm
The crack.
tensile proportionality
It is also coefficient
possible to D(h,T) observe isthat calledthe reads
moisture permeability and it is a nonlinear function
eventually formed crack is rather invisible up to the
of the relative
Stage 15. However, humidity and stage
in hthis temperature T (Bažant
the interpolated ⎡ ⎤
& Najjar 1972). The moisture mass balanceand
strain field reveals already a quite clear requires
pro-
we (h α c α s ) = G1 (α c , α s )⎢⎢1 − 1 ⎥
+
that the variation in time of the water mass per unit ∞
, ,
nounced region of intense strain gradient concentra- (g α
c 10 − α c )h ⎥⎥
volume
tion, withof theconcrete
shape(water content w) thin
of a continuous be equal
line.toThis the e
⎢
⎣ 1
⎦ (4)
divergence
line of the moisture
tracks already the future flux J The interpolated
crack.
⎡ (g α
∞ − α )h ⎤
strain field reveals itself very useful on identifying
K (α c α s ) e
, ⎢ c 10
c − 1
1⎥
the∂wpreliminary stages of the crack formation. The ⎢ ⎥
(2)
1
− =scale ∇ • J is not small enough to state something ⎣ ⎦
used∂t
conclusive about the cracking processing zone, but
the The water content
sequence of stages w can
from be6expressed
to 10 (see as theFig.sum 9) where the first term (gel isotherm) represents the
of the evaporable water we (capillary water, water
seems to indicate that, in the present situation, it may physically bound (adsorbed) water and the second
vapor, and
assume adsorbed
a sharp, thin and water) and theshape.
elongated non-evaporable
This was term (capillary isotherm) represents the capillary
(chemicallyandbound)
expectable, agrees with water wn (Mills of 1966,
the observations other water. This expression is valid only for low content
Pantazopoulo & Mills 1995). It is reasonable to
researchers (Bolander Jr. & Hikosaka 1995, Otsuka of SF. The coefficient G1 represents the amount of
assume
& Date 2000).that the evaporable water is a function of water per unit volume held in the gel pores at 100%
relative humidity,
In Figure 11 the hcrack , degree of hydration,
patterns obtained from αc, and the relative humidity, and it can be expressed (Norling
degree oftesting
tensile silica fume
of dogbone reaction,shaped αs, i.e. wspecimens
e=we(h,αc,αs)
is Mjornell 1997) as
a) b)
= age-dependent sorption/desorption isotherm
shown. In Figure 11.a the presented photograph re-
(Norling Mjonell 1997).observable
Under thisin assumption and G1 tensile ctwo
k vg + k s α shaped
Figure 11. Pictures showing the cracked pattern obtained from
α cdogbone (5)
vg s s cracksECC
veals the crack pattern a dogbone sur- (α , α test
) = of
by substituting Equation 1 into Equation 2 one
face right before localization and rupture. In Figure
the
c s c
ing a system of closely parallel
specimens, exhibit-
along the longitudinal
obtains
11.b the interpolated strain field obtained using the axis. a) Photo of the specimen right before the peak strength is
digital imaging system is superimposed to the original where kand
reached vg and k vg field
c b) strain s
are material
obtained parameters. From
with the digital the
imaging
specimen’s photo, allowing the observation of the
∂w ∂h ∂w ∂w maximum amount of water per unit volume that can
system (Lárusson et al. 2009).
− e
cracking + ∇ • ( D ∇at
processes h muche earlier
) = α&c +
e α& +(Lárusson
stages w&n et
(3) fill all pores (both capillary pores and gel pores), one
∂ h ∂t h
al. 2009). The results obtained ∂α ∂α s
calculate K1 as one obtains
4canCONCLUSIONS
c from sthe tensile testing
of several ECC specimens reveal that, in general, fol-
In the present work the cracking process of notched
where ∂w
lowing
e/∂h is the slope of the sorption/desorption
a preliminary stage of elastic behavior a pro-
tensile specimens of FRCC was
⎡
⎢
⎛
⎜ g α ∞
c − α ⎟h ⎥
experimentally
10
c
⎞ ⎤
as-
isotherm
nounced (also
cracking called
stage moisture
with capacity).
strain hardening The
takes w − α s + α s −G −e
sessed. The distinctc fiberss used⎢⎢ as reinforcement
0.188 0.22
⎝ 1
⎠1
⎥
of
governing ⎦ (6)
place. This equation
stage typically (Equationends 3) withmust be completed
localization for a 0 1
⎥
K (α c α s ) =
the composites revealed very ⎣ distinct stress-crack
by appropriate
limit strain of around boundary 4%, andafterinitial
the full conditions.
development of 1
,

opening behaviors. It is⎛⎜ g αpossible, ∞

− α ⎟h however, to as-
⎞

The
a system relation between
of closely parallel the amount oftheevaporable
cracks along longitudi-
e
10
⎝ c 1 c ⎠ −
sume that a typical sequence of characteristic 1
stages
water
nal axis and relativemodeling
(Lárusson et humiditycarried
al. 2009). is called ‘‘adsorption took place in all the performed tests. The particular
isotherm”
The numerical if measured with increasing
out in relativity
previous
The material
features of each parameters
stage, suchkas
c
andmagnitude,
vg its ksvg and gdura- 1 can
humidity andbehavior
works (Fischer ‘‘desorption
et al. 2007) isotherm”
showed in the
that the opposite
tensile
be calibrated by fitting experimental
tion and significance are the reflex of a complex col- data relevant to
case. Neglecting
stress-strain their difference
of these (Xi etbehavior
dogbones al. 1994),
may be ob- ina free (evaporable)
lection water content
of micromechanical in concrete
mechanisms taking at
the following,
tained from the‘‘sorption
stress-crack isotherm”
opening will be used with
of
various ages (Di Luzio & Cusatis
place. These are determined by the matrix, the fi- 2009b).
reference
single crack. to both
The sorption
structuraland desorption
behavior will conditions.
be mainly
bers’ properties and their interaction. They may con-
By the way,
the result of the iflevel
sum theofhysteresis
of the mechanical ofcrack.
the The moisture
behavior en-
2.2 Temperature
stitute very important evolution
information for supporting the
isotherm
countered would be
at the takenstress-CMOD
into account, two
each single different
uni-
design, either of the material itself or of the structure
relation,
axial evaporable
character of the water vs relative humidity,
law derived must Note that,
where the at early age,
material since the
is meant to be chemical
used. The reactions
pro-
be
from used according
experimental to the
testing sign
wasofconsidered
the variation as aofnon- the associated with cement hydration
posed curve to describe the tensile behavior of and SF reaction
a sin-
relativity
linear spring. humidity.
The overall The shape
behavior ofof the sorption
the dogbone
are crack
gle exothermic,in FRCC theassumes
temperature a simplefieldandis not uniform
convenient
isotherm
consisted, for
in HPC isrepresenting
this fashion, influenced
of the by many
association parameters,
in series
for non-adiabatic
format, consequently systems even if an
representing theeventually
environmental use-
especiallyofthose
of spring elements that influence extentelastic
either and rate
the of the
non-linear
temperature is constant. Heat
ful instrument to be included in a design strategyconduction can for be
chemical
behavior reactions
each crack and,
or thein turn,
linear determine behavior pore described in concrete,
this type of materials. at least for temperature not
structure
of the intact and pore
material size distribution
between cracks.(water-to-cement
Aditionally, a
exceeding
The isolation 100°C of a(Bažant
single crack& Kaplan 1996),test-
during tensile by
ratio,randomness
statisticalcement approachchemical composition,
was considered to SFaccountcontent, for
Fourier’s
ing law, whichpremise
is an important reads for the truth assessment
curing time
the and method,
associated temperature,
with the mix additives,
material ad me-
of the composite mechanical behavior. The high po-
etc.).
chanical In the literature
properties, in various
particularformulations
the matrix can tensile be q = − λ∇
tential ofT multiple crack development revealed (7) by
found to describe the sorption isotherm of normal
cracking stress.
SHCC is deleterious to that purpose. The local inten-
concrete (Xi et al. 1994). However, in the present where q ofis thethestress
sification heatfieldflux,achieved
T is by the executing
absolute
paper the semi-empirical expression proposed by temperature,
the four notches and λinisthe the specimens,
heat conductivity;following in this
the
Norling Mjornell (1997) is adopted because it aforementioned geometry A, revealed to be a valid

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

strategy to gather the conditions for the formation − D (h, T )∇hCementitious Composites
Li, V.C., 2003. OnJ =Engineered
and propagation of a meso-scopic single crack dur- (ECC) - A Review of the Material and Its Applications, J.
Adv. Conc. Tech., 1 (3): 215-230.
ing tensile tests. The Fracture
Otsuka, K. & Date, H., 2000. proportionality coefficient
process zone in concrete D(h,T)
The geometry A did not allow the visual access to tension specimen,moisture permeability
J. Eng. Frac. and it is a nonlinea
Mech., 65: 111-131.
the inner cracking region, consequently the geome- of the
Shah, S.P. et al., 1996. relativecharacterization
Toughness humidity h and temperature
and tough-
tries B and C were developed. The observations ening mechanisms. & Najjar 1972). The moisture mass
In Naaman, A.E. & Reinhardt, H.W. balanc
made with the specimen’s geometry C allowed to in- that the
E &variation in time of the water mas
(eds.). High performance fiber reinforced cement compos-
ites 2; Proc. Int. symp., FN Spon, London.
fer that the development of a single crack is, in a volume of concrete (water content w) be eq
meso-scopic perspective, efficiently achieved for the divergence of the moisture flux J
level of the bridging stresses in study. The tensile
− ∂ = ∇•J
stress – crack opening curves obtained by using w
specimen’s geometry A may constitute an important ∂ t
element in both the material and the structural de-
sign activities. The behavior of a single crack is cap- The water content w can be expressed a
tured in a simple and appealing way, providing the of the evaporable water we (capillary wa
designer with the most relevant features of the me- vapor, and adsorbed water) and the non-e
chanical behavior of the material. (chemically bound) water wn (Mil
The digital interpolation of the strain fields in Pantazopoulo & Mills 1995). It is reas
specimens’ geometries B and C allowed the identifi- assume that the evaporable water is a fu
cation of the features of the most relevant cracking relative humidity, h, degree of hydration
stages. The initiation of the crack at the notches and degree of silica fume reaction, αs, i.e. we=w
its subsequent development always in symmetric = age-dependent sorption/desorption
conditions in terms of deformations sustains the fact (Norling Mjonell 1997). Under this assum
that the test setup, particularly the support condi- by substituting Equation 1 into Equati
tions, was adequate in assuring the initiation and obtains
propagation of one planar crack.
∂w ∂h
− e + ∇ • ( D ∇h) = ∂we ∂w
α&c + e α&s + w
ACKNOWLEDGMENTS ∂h ∂t h ∂α ∂α
c s
The first author wishes to thank the Portuguese Na- where ∂we/∂h is the slope of the sorption/
tional Science Foundation for the financial support, isotherm (also called moisture capac
through grant SFRH / BD / 36515 / 2007, funded by governing equation (Equation 3) must be
POPH - QREN, the Social European Fund and the by appropriate boundary and initial conditi
MCTES. The relation between the amount of e
water and relative humidity is called ‘‘
isotherm” if measured with increasing
REFERENCES humidity and ‘‘desorption isotherm” in th
Bolander Jr, J. & Hikosaka, H., 1995. Simulation of Fracture in
case. Neglecting their difference (Xi et al.
Cement-based Composites, Cem. & Conc. Comp., 17: 135- the following, ‘‘sorption isotherm” will be
145. reference to both sorption and desorption c
Fischer, G. & Li, V.C., 2007. Effect of fiber reinforcement on By the way, if the hysteresis of the
the response of structural members, Eng. Frac. Mech., 74: isotherm would be taken into account, two
258-272. relation, evaporable water vs relative humi
Fischer, G., Stang, H. & Dick-Nielsen, L. 2007. Initiation and
development of cracking in ECC materials: Experimental be used according to the sign of the varia
observations and modeling. In G. F. A. Carpinteri, P. Gam- relativity humidity. The shape of the
barova & G. P. (eds.), High Performance Concrete, Brick- isotherm for HPC is influenced by many p
Masonry and Environmental Aspects; Proc. int. symp., Vol. especially those that influence extent and
3, Ia-FraMCos. Taylor & Francis, pp. 1517–1522. chemical reactions and, in turn, determ
Kabele, P., 2007. Multiscale framework for modeling of frac-
ture in high performance fiber reinforced cementitious
structure and pore size distribution (water-
composites, Eng. Frac. Mech., 74: 194-209. ratio, cement chemical composition, SF
Lárusson, L.H., Fischer, G. & Jönsson, J., 2010. Mechanical curing time and method, temperature, mix
interaction of Engineered Cementitious Composite (ECC) etc.). In the literature various formulatio
reinforced with Fiber Reinforced Polymer (FRP) rebar in found to describe the sorption isotherm
tensile loading. In G. P. A. G. van Zijl & W. P. Boshoff concrete (Xi et al. 1994). However, in th
(eds.) Advances in Cement-based Materials (ACM2009);
Proc. int. syp., South Africa, CRC Press/ Balkema. paper the semi-empirical expression pro
Norling Mjornell (1997) is adopted b

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