Relation between crack width and corrosion degree in corroding

Relation between crack width and corrosion degree in corroding

Fracture Mechanics of Concrete and Concrete Structures Assessment, Durability, Monitoring and Retrofitting of Concrete Structures- B. H. Oh, et al. (eds)
ⓒ 2010 Korea Concrete Institute, Seoul, ISBN 978-89-5708-181-5
Relation between crack width and corrosion degree in corroding
elements exposed to the natural atmosphere
C. Andrade
IETcc- CSIC, Madrid, Spain
A. Muñoz
Universidad Marista de Querétaro
A. Torres-Acosta
Instituto Mexicano del Transporte, Querétaro, México
ABSTRACT: Reinforcement corrosion leads into several damage types which influence the structural loadbearing capacity, among which can be mentioned the cracking of concrete cover. This work presents results
of crack width generated in concrete elements fabricated in 1990 with chlorides added in the mix and exposed
to the natural atmosphere of Madrid-Spain climate. These elements, one T beam and one column. Two
expressions have been fitted to the results: w = k Px / (c/φ) and w = k Px / Ro, where w is a crack width in the
time, k is a proportional factor, Px is the corrosion penetration in the time, c/φ is the concrete cover/diameter
relation and Ro is the original radius of the bar. The expressions were also fitted to results taken from the
literature made applying a current. The beam shows larger crack widths than the beam and the accelerated
tests give intermediate results. Based in all the results, although the scatter is important, it has been calculated
the k and k’ slopes which resulted respectively in values of 9.5 and 35.5.
KEY WORDS: Corrosion, crack width, concrete cover, formulas
1 INTRODUCTION
Corrosion of reinforcement produces iron oxides that
induce cracking of concrete cover due the traction
efforts induced by the generation of the rust. The
cracks generate parallel to the direction of the bars
(Beeby 1983, Reinhardt 1984). Three steps have been
identified: 1) Period of initiation of cracking, during
when the cracks are developed from the bar up to
reaching the surface of the concrete, 2) Period of
spread of the cracking, during which the width of
crack grows, and 3) this spread progress coalescence
several cracks may give place to the detachment of
entire chunks of the concrete cover ending up in
delaminations and spalling.
The relation between the crack opening and the
quantity of oxide generated by the corrosion,
expressed as penetration of the corrosion or loss of
diameter of the bars, has been the subject of
previous works by the authors by means of
different tests (both accelerated and not accelerated)
(Andrade 1993, Torres-Acosta 1999). Furthermore,
some models (Pantazopoulou 2001, Liu 1995) analyze
cracking time as function of concrete cover, concrete
and rust properties controlled by the rate of rust
accumulation, while that other models (Andrade
1993, Martín-Perez 1998) assume constant rate of
rust production. Another model (Leung 2001)
obtain one upper and lower bound assuming the
steel / concrete interface to be perfectly smooth or
perfectly bonded. Various numerical approaches
use a finite element method analyzing cracking
with the fixed smear crack model, assuming linear
softening of the concrete (Andrade 1993), assuming
linear elastic fracture mechanics and movable
mesh placed around the crack tip to capture the
local stress concentration (Padovan 1997) and
with boundary element approach (Ohtsu 1997).
Another papers develop models based on a critical
corrosion attack penetration to initiate cracking
and they relate it to the rebar radius (TorresAcosta 1998), steel cross section loss due to
corrosion (Vidal 2004) and cover / diameter ratio
and concrete characteristics (Andrade 1995,
Rasheeduzzufar 1992). In all investigations the
calculations an the simulated cracking patterns are
compared to experimental observations. One has
concluded that the beginning of the cracking
depends principally on the relation concrete cover
/ diameter of the bars, the quality of the concrete
and his tensile strength.
J
=Also
− D (hthe
, T )∇
h
porosity
plays a role, as has been reported
(1)
(Alonso 1998) that the voids around the bars can
allocate
the corrosioncoefficient
products D(h,T)
after corrosion
The proportionality
is called
initiation.
Then,
the
porosity
around
the
barsfunction
initially
moisture permeability and it is a nonlinear
may
the pressure
The oxides
may
of thedelay
relative
humiditytransmission.
h and temperature
T (Bažant
later
diffuse
out
of
the
steel/concrete
interface
& Najjar 1972). The moisture mass balance requires
throughout
the poreinnetwork.
that the variation
time of the water mass per unit
The
majority
of the
mentioned
studies
by
volume of concrete
(water
content
w) bewere
equalmade
to the
applying
an of
external
current.flux
ThisJ procedure needs the
divergence
the moisture
gravimetric calibration because the electrochemical loss
results
in general smaller than the real ones (Alonso
(2)
− ∂w = Also
∇ • J the crack width opening is smaller when
1998).
∂t
the test is accelerated (Alonso 1998). Low corrosion
rates
produce
cracks.
The
waterwider
content
w can be expressed as the sum
In
present
paper
the study
is made water,
on elements
water
of the evaporable water
we (capillary
exposed
from
1990
to
the
action
of
a
natural
vapor, and adsorbed water) and the non-evaporable
atmosphere
Madrid water
climatew (Rodríguez
1993,
(Mills 1966,
(chemically inbound)
n
Rodriguez
1996).
These
elements
are
a
beam
and
Pantazopoulo & Mills 1995). It is reasonable toa
column
with chlorides
the mix of
in
assume fabricated
that the evaporable
wateradded
is a tofunction
which
the
corrosion
rate
has
been
monitored
together
relative humidity, h, degree of hydration, αc, and
with
theofcrack
The authors
=we(h,αwork
degree
silicaopening.
fume reaction,
αs, i.e.ofwepresent
c,αs)
have
used
the
simplified
formula
used
Torres= age-dependent sorption/desorption byisotherm
Acosta
only and
the
(Norling(Torres-Acosta
Mjonell 1997).1999)
Underconsidering
this assumption
bar
diameter
and
not
the
concrete
cover
depth
and
by substituting Equation 1 into Equation 2 one
compare
obtains it to a similar one but including the cover
depth. The comparison of previous formula for the
crack with growing developed using accelerated
∂w
∂w ∂h
e α& + ∂weinα&natural
corrosion,
to• the
values registered
condi&
( D ∇h ) =
+∇
− e
(3)
c
s + wn
h
∂
α
∂
α
∂
∂
h
t
tions during almost 15 years
has enabled
to calculate
c
s
closer to reality fitting factors.
where ∂we/∂h is the slope of the sorption/desorption
isotherm
(also called moisture capacity). The
2 EXPERIMENTAL
governing equation (Equation 3) must be completed
by appropriate boundary and initial conditions.
relationelements
between the amount of evaporable
2.1 The
Concrete
water
and
relative
humidity is called ‘‘adsorption
The studied elements are a column and a T beam that
isotherm”
if
measured
with increasing
relativity
were
made
to‘‘desorption
study the evolution
ofinthe
corrosion
humidity
and
isotherm”
the
opposite
rate and the corrosion induce cracks when exposed to
case. Neglecting their difference (Xi et al. 1994), in
the atmosphere non protected from the sun and rain.
theThe
following,
‘‘sorption
isotherm”
will be used
with
column
was
of 200 and
x 200
mm in area
and 2 m
reference
to
both
sorption
desorption
conditions.
in length (Fig. 1). The reinforcement consists of two
By the way, if the hysteresis of the moisture
sections with different quantities. The first section
isotherm
would be takentwointolow
account,
two differentin
has
two evaporable
top bars and
oneshumidity,
of 12 mm
relation,
water
vs
relative
must
diameter and stirrups of 6 mm in diameter each 200
be
used
according
to
the
sign
of
the
variation
of
the
mm in length. The second section has six bars of 12
relativity
humidity.
The
shape
of
the
sorption
mm
in diameter
stirrups of by
6 mm
100 mm
isotherm
for HPCand
is influenced
manyeach
parameters,
in length. The cover is the same for the whole
especially those that influence extent and rate of the
element being 30 mm.
chemical
reactions
and,
in turn, determine
pore
The section
of the
Tdistribution
beam consists
of a base of
structure
and
pore
size
(water-to-cement
300 x 100 mm a web of 200 x 100 mm. The
ratio, cement chemical composition, SF content,
reinforcement of the web are two bars of 16 mm in
curing time
and
method,part
temperature,
mix additives,
diameter
and
in
the top various
of the
base there are
etc.).
In
the
literature
formulations
canfour
be
bars of 12 mm in diameter and in the low part, in the
found
to
describe
the
sorption
isotherm
of
normal
central part, two bars of 16 mm in diameter and, in
concrete
(Xi bars
et al.of1994).
However,
in The
the present
the
ends,
two
12 mmexpression
in diameter.proposed
stirrups
paper
the
semi-empirical
by
are 6 mm of diameter to each 200 mm. The cover of
Norling
Mjornell
(1997)
is
adopted
because
it
the element is also of 30 mm.
Proceedings of FraMCoS-7, May 23-28, 2010
explicitly accounts for the evolution of hydration
reaction and SF content. This sorption isotherm
reads
we (h α c α s ) =
,
,
⎡
G1 (α c , α s )⎢⎢1 −
⎢
⎣
1
⎤
⎥
− α c )h ⎥⎥
⎦
∞
(g α
c
e
∞ − α )h
⎡
(g α
c
c −
⎢
K (α c α s ) e
⎢
10
,
1
10
1
1
⎣
+
(4)
⎤
1⎥
⎥
⎦
where the first term (gel isotherm) represents the
physically
bound (adsorbed) water and the second
Figure 1. Detail of the sections of the studied elements.
term
(capillary isotherm) represents the capillary
water.Both
Thiselements
expression
is fabricated
valid only infor1990
low and
content
were
the
represents
amount
of
ofnominal
SF. Thedosage
coefficient
of theG1concrete
usedthewas
of: 360
3 unit volume held in the
3 gel pores at 100%
water
per
kg/m of cement, 1080 kg/m of aggregate 5-12
relative
humidity,
and3 itofcan
be 0-5
expressed
(Norling
mm and
840 kg/m
sand
mm. The
w/c
Mjornell
1997)
as
relation was of 0.7. In order to induce corrosion
3% of CaCl *2H O was added to the concrete
c 2α c +2kwas
s made during 7 days (5)
by
(α α The
) = k curing
G mix.
c s vg c vg α s s
,
means of the placement of plastic cover to avoid
the desiccation
with intermittent watering.
c
ksvgset
areofmaterial
parameters.
From the
whereAkview
vg and
of the
beams and
columns fabricated
maximum
amount ofofwater
per unitisvolume
and the disposition
the elements
so that athat
facecan
of
filltheallelements
pores (both
capillary
poresand
andthe
gelother
pores),
one
remains
in shade
facing
as one
obtains
canthecalculate
K12).
sun (Fig.
Madrid
has a continental weather
reaching temperatures around 0-5ºC in winter and 35⎡
⎛
⎞ ⎤
∞
40ºC in summer. The RH evolves
from
− α ⎟h ⎥ 10%
⎜ g α around
⎢
c
c
⎝
⎠
w −to 60-70%
α s + αins winter
in summer
⎥
c
s − G ⎢⎢ − e as average values.
Raining may appear around 75-100
days per⎥⎦ year
(6)
⎣
(α α ) =
K with
of
c saround 3155 hours/year
⎛
⎞ TOW and around
∞
⎜ g α − α ⎟h
c c ⎠ −Snow may occur
600-700ml/m2 collected
e ⎝ annually.
one to two times per year (Andrade, 2002).
1
10
0
0.188
0.22
1
1
1
,
1
10
1
1
The material parameters kcvg and ksvg and g1 can
be calibrated by fitting experimental data relevant to
free (evaporable) water content in concrete at
various 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 not
exceeding 100°C (Bažant & Kaplan 1996), by
Fourier’s law, which reads
q
= − λ ∇T
(7)
where q is the heat flux, T is the absolute
temperature, and λ is the heat conductivity; in this
Figure 2. Exposition of elements to the atmosphere.
2.2 Measurement of corrosion rate
Since elements fabrication until now, temperature
and humidity, inside and outside of the concrete,
have been measured by means of a hygrometer
Vaisala (see Fig. 3). The relative humidity, RH, and
temperature, T, are measured at the open atmosphere
and inside a hole made in the beam (Fig. 3). The
readings are taken after few minutes when the values
show stability.
The measures are taken with the portable corrosion rate meter Gecor in its version 6 and 8 (Feliú
1990), which have modulated confinement of the
current. That is the current is confined in a certain
area and the corrosion rate is referred to that area.
The corrosion rate values are calculated from the
ratio between current applied and shift in potential
removing mathematically the ohmic drop.
J =measurements
− D ( h , T ) ∇h
From all the
they have been
obtained about 170 values for the T beam and 87
for the column. The proportionality coefficient D(h,T)
In order to make
a statistical
analysis
moisture
permeability
andtheit ismeasa nonlinea
urements was always
in the
same position.
All
of the taken
relative
humidity
h and temperature
the data where &
considered:
33
points
in
the
T
beam
Najjar 1972). The moisture mass balanc
and 24 in the that
column
that are sown
in of
Figure
4. mas
the variation
in time
the water
Regarding the C/φ
parameter,
for
the
case
of
the
volume of concrete (water contentTw) be eq
beam, there aredivergence
different diameters
of barsflux
thatJinof the moisture
fluence the cracks and for each crack width detected,
according to the zone
the form of the crack, there
∂w =and
∇of• Jthe diameter of the bar,
was assigned a−value
∂t
existing three different ones: 6 (stirrups), 12 (bottom
bars) and 16 mm (top
Forcontent
the casewofcan
the column,
Thebars).
water
be expressed a
the diameters were
6
and
12
mm,
and
also
were
of the evaporable water wthey
e (capillary wa
considered in thevapor,
calculation
of
k.
In
the
formula,
and adsorbed water) and the
the non-e
cover was taken(chemically
the actual inbound)
each case.water
The time
wn (Mil
used in the calculations,
was that&corresponding
Pantazopoulo
Mills 1995).to the
It is reas
moment of measurement
of
the
crack
width. water is a fu
assume that the evaporable
relative humidity, h, degree of hydration
fume reaction, αs, i.e. we=w
2.4 Formulas degree
used in of
thesilica
analysis
= age-dependent sorption/desorption
The main goal(Norling
of present
paper1997).
is to verify
Mjonell
Under the
this assum
accuracy of simple
formulae
to
predict
the
average
by substituting Equation 1 into Equati
crack width produced
obtains in elements corroding in
natural conditions, in function of the accumulated
corrosion, Px. In previous papers they have been
∂w
∂w ∂h
e α& + ∂we α& + w
proposed (Andrade
+ ∇ • ( D ∇h ) =
− e1993):
∂h ∂t
Figure 3. Corrosion rate meter measuring the internal relative
humidity in the beam.
2.3 Measurement of cracking
From the manufacture of the elements the crack
widths were measured in 11 dates for the T beam and
7 measurements for the column. The dates are given
in the following table.
Table 1. Dates of measurements of the crack width.
T Beam
Column
Measure
Date
Measure
Date
0
15/02/1990
0
15/02/1990
1
1/07/1993
1
25/08/1995
2
17/11/1993
2
28/05/1997
3
21/06/1994
3
26/03/1998
4
4/05/1995
4
1/07/1998
5
9/07/1996
5
18/03/2004
6
9/07/1997
6
13/09/2004
7
27/03/1998
7
14/09/2005
8
22/06/1998
8
16.09.2006
9
18/03/2004
10
13/09/2004
11
14/09/2005
12
16.09.2006
h
⎛ Px ⎞
⎟
φ ⎟⎠the
∂we/∂h
⎝ C is
The first formula is: w = k ⎜⎜
∂α
c
c
∂α
s
(1)
s
slope of the sorption/
where
isotherm (also called moisture capac
governing
(Equation
3) must be
where w is the crack
openingequation
in the time
t, k is a factor
by
appropriate
boundary
and
initial
of proportionality without dimensions, c/φ is the relationconditi
The
between
the amount
penetration
of the of e
cover/diameter of the
bar relation
and Px is the
water
and
relative
humidity
is called ‘‘
corrosion in time t, where Px is equal to:
isotherm” if measured with increasing
humidity
and ‘‘desorption isotherm” in th
rep
Px = 0.0115 I corr t
case. Neglecting their difference (Xi et al.
the following, ‘‘sorption isotherm” will be
and rearranging
Replacing Preference
x in the equation
to both sorption
and desorption c
the terms:
By the way, if the hysteresis of the
isotherm would be taken into account, two
w
relation,
water vs relative humi
( mm )C ( mmevaporable
)
k=
(2)
repused according to the sign of the varia
be
0.0115 I corr (mm / a? )t (a? )φ (mm )
relativity humidity. The shape of the
isotherm for HPC is influenced by many p
For the second
formula
procedure
especially
thosethethatsame
influence
extent and
followed, onlychemical
that usedreactions
Ro that isand,
the inoriginal
turn, determ
radius of the structure
bar instead
c/φsize
(Torres-Acosta
andofpore
distribution (water1999, Torres-Acosta
1999,
Torres-Acosta
1998).
ratio, cement chemical composition,
SF
For what ultimately
stays:
curing time and method, temperature, mix
etc.). In the literature various formulatio
wfound
( mm ) Roto
( mm )describe the sorption isotherm
k' =
(3) in th
concrete
(Xi tet al. 1994). However,
rep
0.0115 Ipaper
(the
)
mm / a?
(
)
a?
corr
semi-empirical expression pro
Norling
Mjornell
(1997) is adopted b
k’ has the dimensions of millimeters.
Proceedings of FraMCoS-7, May 23-28, 2010
2
4
6
8
Ancho de fisura (mm)
0.5
y = 0.0177x - 0.0416
0
0
10 12 14 16 18 20
2
4
6
8
1.5
1
y = 0.0369x + 0.0526
0.5
0
10 12 14 16 18 20
0
Tiempo (a? s)
2
4
Ancho de fisura (mm)
Ancho de fisura (mm)
2
y = 0.0512x - 0.113
⎡
G1 (α c , α s )⎢⎢1 −
⎢
⎣
0.5
0
6
8
we (h α c α s ) =
,
,
Ancho de fisura (mm)
2
1
y = 0.0344x - 0.0669
0
0
2
4
6
8
1
y = 0.0198x - 0.0206
0
4
6
8
y = 0.0234x - 0.0347
0
2
4
6
8
10
1
,
0
0
2
4
6
8
1
0
2
4
6
8
10
12
1
y = 0.0409x - 0.0266
1
0.5
1
18
20
⎣
0
2
4
6
8
10
12 14 16
18
y = 0.0295x - 0.0846
0.5
0
0
0
20
2
4
6
8
10 12 14 16 18 20
Tiempo (a? s)
Tiempo (a? s)
where the first term (gel isotherm) represents the
physically bound (adsorbed) water and the second
term (capillary isotherm) represents the capillary
water. This expression is valid only for low content
of SF. The coefficient G1 represents the amount of
water per unit volume held in the gel pores at 100%
relative humidity, and it can be expressed (Norling
Mjornell 1997) as
0
1.5
1
y = 0.0 202 x - 0.00 46
0.5
1.5
1
y = 0.0 274 x + 0 .01
0.5
2
4
6
8
1 .5
1
10 1 2 1 4 1 6 18 2 0
0
2
4
6
Ti em p o (a ? s)
8
10
12
14
16
18
0
0
20
y = 0 .0 12 x + 0 .07 45
0 .5
0
0
0
2
Ancho de fis ura (mm )
Ancho de fisura (mm)
y = 0 .0 16 3 x + 0.12 5 1
0.5
P un to d e m e d id a 78
2
2
Ancho de fis ura (mm)
Ancho de fisura (mm)
2
1.5
1
Pu nto de m edida 87
Pu nto d e m edid a 79
Punto d e m e did a 6 5
2
4
6
8
10
12
14
16
18
0
20
2
4
6
8
10
12
14
16
18
20
T ie m p o ( a? s )
Tiem po (a? s )
Tiem po (a? s )
Punto de m edida 73
Punto de m ed ida 6 6
2
Ancho de fisura (mm)
2
1.5
1
y = 0.02 16x - 0.055 1
0.5
1 .5
y = 0.0 076x + 0. 4447
1
0 .5
0
0
0
2
4
6
8
1 0 1 2 14 16 18 20
0
2
4
6
8
10
12
14
16
18
20
T iempo (a? s )
Tie mp o (a? s)
P unto de m edida 69
Punto de m edida 72
2
Ancho de fisura (mm)
2
1.5
1
y = 0.0252x + 0.047
0.5
0
0
2
4
6
8
10
12
14
16
18
1.5
1
y = 0.0378x + 0.1518
0.5
0
0
20
2
4
6
8
Pu n to d e m ed id a 82
Pun to de m e did a 85
2
y = 0.0213x + 0.1533
1 .5
1
c α c+ ks α s
G (α c α s ) = k vg
c vg s
0.5
0
2
4
6
8
10
12
14
16
18
y = 0.0 286x + 0 .05 25
0 .5
1 .5
1
0
2
4
6
8
10
12
14
16
18
y = 0 .0 04 4x + 0 .0 8 47
0 .5
14
16
18
20
16
18
20
2
1 .5
1
0 .5
0
0
20
Tiem po (a? s)
,
2
Ancho de fis ura (m m)
Ancho de fisura (mm)
2
1.5
0
12
P unto de m ed ida 8 0
Ancho de fisura (mm)
P unto de m edida 64
1
10
Tiem po (a? s)
Tiem po (a? s)
y = 0 .01 18x - 0.0 074
(5)
0
0
20
2
4
6
8
10
12
14
16
18
20
0
2
4
6
8
T ie m po (a ? s )
T ie m po ( a? s )
10
12
14
Tie m po (a? s )
1
Punto de medida 63
5
5
4
y = 0.3013x - 1.0705
4
Punto de medida 64
15
12
Punto de medida 66
5
Anc ho de fis ur a (mm)
Punto de medida 6
Punto de medida 11
y = 0.2589x - 0.5336
A ncho de fisur a (mm)
4
A ncho de fis ura (mm)
Punto de medida 16
5
4
5
4
where kcvg and ksvg are material parameters. From the
maximum amount of water per unit volume that can
fill all pores (both capillary pores and gel pores), one
can calculate K1 as one obtains
3
2
1
2
4
6
8
10
12
14 16
3
2
1
0
0
18 20
2
4
6
8
10 12
14
16
18
3
2
y = 0.0421x - 0.0608
1
6
3
0
2
4
6
Tiempo (a? s )
8
10
12
14
16
18
0
20
y = 0.5416x - 0.5041
3
2
1
0
0
0
20
y = 0.6588x - 1.1452
9
2
4
6
8
10 12
2
4
6
y = 0.5654x - 0.8838
2
8
10
12
14
16
18
1
20
0
2
4
6
8
Tiempo (a? s)
Tiempo ( a? s)
Tiempo (a? s)
3
0
0
14 16 18 20
10 12 14 16
18 20
Tiempo (a? s )
Punto de medida 62
15
Ancho de fisura (m m)
Ancho de fisura (mm)
10
9
8
7
6
5
4
3
2
1
0
y = 0.3545x - 0.8276
0
2
4
6
8
12
9
6
y = 0.5796x - 0.7499
3
0
10 12 14 16 18 20
0
Tiempo (a? s )
2
4
6
8
10
12
14
16 18
20
16
20
Tiempo (a? s)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
P unto de medida 60
y = 0.2997x - 0.9032
0
2
4
6
8
10 12 14
10
9
8
7
6
5
4
3
2
1
0
Ancho de fisura (m m)
An cho de fis ura (m m)
Punto de m edida 23
16 18 20
10 12 14 16 18
α s + 0.22α s G
− 0.188
20
0
c
Punto de medida 37
s
Anc ho de fisura ( mm)
K (α c α s ) =
,
1
10
9
8
7
6
5
4
3
2
1
0
⎛
y = 0.3687x + 0.1086
0
2
4
6
8
10⎜
10 12 14 16 18 20
Tiempo (a? s )
Punto de medida 25
Punto de medida 43
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
Anc ho de fisura (mm)
Ancho de fisura (mm)
5
4.5
4
3.5
3
2.5
2
1.5
1
0.5
0
e
⎝
e
y = 0.07x - 0.1194
4
y = 0.1001x + 0.0306
3
2
8
10 12 14
18
g αc − αc h
⎞ ⎤
⎟ ⎥
⎠
⎥
⎥
⎦
P unto de medida 56
1
4
y = 0.1063x - 0.0016
3
2
1
0
0
2
4
6
8
10
12 14
16
18
16
18
Tie mpo (a? s)
5
4
(6)
20
3
y = 0.117x - 0.117
2
1
0
0
⎠
Pu nto de medida 45
5
6
Punto de medida 54
g α c∞ − α c ⎞⎟h
1
4
5
Ancho de fis ura (m m)
8
Tiempo (a? s)
∞
A nc ho de fisur a ( mm)
6
⎛
10⎜
⎝
2
4
6
8
10
12
14
20
T ie mpo ( a? s)
−1
Punto de medida 53
Punto de medida 48
5
Ancho de fisura (m m)
4
Ancho de fisura (mm)
2
Anc ho de fisura (m m)
Ancho de fisura (mm)
w
y = 0.3712x - 1.197
0
2
Tiempo (a? s)
⎡
⎢
−
⎢1 −
1
⎢
⎣
Punto de medida 27
10
9
8
7
6
5
4
3
2
1
0
y = 0.4205x - 0.952
0
Tiempo (a? s )
4
3
y = 0.0334x - 0.0234
2
5
4
y = 0.06x - 0.0364
3
2
The material parameters kcvg and ksvg and g1 can
be calibrated by fitting experimental data relevant to
free (evaporable) water content in concrete at
various ages (Di Luzio & Cusatis 2009b).
6
8
10
12
14
16
18
0
20
2
4
6
8
2
1
Ancho de fisura (mm)
Ancho de fisura (mm)
y = 0.1868x - 0.6826
0
0
2
4
6
8
0
2
4
6
8
10 12
14
4
2
1
y = 0.0268x + 0.0773
0
3
2
y = 0.0762x - 0.2597
1
0
0
2
4
6
8
4
6
8
10 12 14 16 18 20
10
12
14
16
18
20
2
4
6
8
0
0
2
4
6
10 12 14
16
18
20
Punto de medida 121
5
y = 0.2878x - 0.8517
3
2
1
10 12 14 16 18 20
8
Tiempo (a? s)
4
3
0
2
4
6
Tiempo (a? s)
8
y = 0.0225x - 0.0169
2
1
0
0
0
Tiempo (a? s)
1
Punto de medida 124
5
4
10 12 14 16 18 20
2
Punto de medida 86
5
4
Tiempo (a? s)
0
Tiempo (a? s)
5
3
1
0
16 18 20
Tiempo (a? s )
Punto de medida 81
Punto de medida 114
5
3
0
Tiempo (a? s)
Tiempo (a? s)
4
1
10 12 14 16 18 20
Ancho de fisura (mm)
4
Ancho de fisura (mm)
2
Ancho de fisura (mm)
y = 0.0284x - 0.0239
0
0
10 12 14 16 18 20
2
4
6
8
10 12 14 16 18 20
Tiempo (a? s)
Tiempo (a? s)
Punto de medida 107
Punto de medida 111
10
9
8
7
6
5
4
3
2
1
0
5
Ancho de fisura (mm)
Ancho de fisura (mm)
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 not
Figure 4. Position
the points&of Kaplan
measurement
of crack
exceeding
100°C of(Bažant
1996),
by
width
and
slopes
with
time.
Fourier’s law, which reads
y = 0.3811x - 1.1112
0
2
4
6
8
10 12 14 16 18 20
4
3
2
1
y = 0.0218x - 0.0272
0
0
2
4
6
Tiempo (a? s)
10 12 14 16 18 20
5
10
9
8
7
6
5
4
3
2
1
0
Ancho de fisura (mm)
Ancho de fisura (mm)
8
Tiempo (a? s)
Punto de medida 69
Punto de medida 102
y = 0.3475x - 0.5485
0
2
4
6
8
4
3
2
y = 0.0198x + 0.0222
1
0
10 12 14 16 18 20
0
2
4
6
Tiempo (a? s)
0
2
4
6
8
10 12 14 16 18 20
Tiempo (a? s)
10 12 14 16 18 20
5
4
y = 0.2329x - 0.5372
3
2
1
2
4
6
8
10 12 14 16 18 20
Tiempo (a? s)
5
4
y = 0.1874x - 0.0416
3
2
1
0
0
0
Ancho de fisura (mm)
5
y = 0.455x - 0.8503
Punto de medida 135
0
2
4
6
8
10 12 14 16 18 20
Tiempo (a? s)
0
2
4
6
8
10 12 14 16 18 20
Tiempo (a? s)
Ancho de fisura (mm)
10
9
8
7
6
5
4
3
2
1
0
Ancho de fisura (mm)
y = 0.6268x - 1.6796
Ancho de fisura (mm)
10
9
8
7
6
5
4
3
2
1
0
8
Tiempo (a? s)
Punto de medida 137
Punto de medida 145
Punto de medida 93
Punto de medida 89
Ancho de fisura (mm)
where ∂we/∂h is the slope of the sorption/desorption
isotherm
(alsothe called
moistureTable
capacity).
Regarding
crack widths,
3 showsThe
the
governing
equation
(Equation
3)
must
be
completed
averaged values obtained in the tested elements at
by appropriate
boundary and initial conditions.
each
date
The relation between the amount of evaporable
Table
Mean
and Standard
deviation
of the ‘‘adsorption
average crack
water3.and
relative
humidity
is called
width
in the beam
and the pillarwith
shownincreasing
in Figure 4. relativity
isotherm”
if measured
(mm)
Waverisotherm”
humidity and ‘‘desorption
inDesv.
the Est.
opposite
1-Jul-93
0.31
0.34
case. Neglecting their difference (Xi et al. 1994), in
17-Nov-93
the following,
‘‘sorption0.43isotherm” will0.42
be used with
21-Jun-94
0.48and desorption
0.46conditions.
reference to
both
sorption
4-May-95
0.63
0.62
By the way,
if the 1.07
hysteresis of 1.02
the moisture
9-Jul-96
isotherm would
two different
9-Jul-97be taken
1.49into account,1.26
T Beam evaporable water vs relative humidity, must
relation,
27-Mar-98
0.86
1.11
be used according
sign of the variation
of the
22-Jun-98 to the
0.87
1.00
2.38 shape of 1.86
relativity 18-Mar-04
humidity. The
the sorption
13-Sep-04
2.68
2.33parameters,
isotherm for
HPC is influenced
by many
14-Sep-05
3.41
2.78 rate of the
especially16-Oct-06
those that influence
extent and
3.94
3.14
chemical 25-Aug-95
reactions and,
in turn, determine
pore
0.20
0.21
structure and
pore
size
distribution
(water-to-cement
28-May-97
0.26
0.28
ratio, cement
chemical0.28composition,0.26
SF content,
26-Mar-98
curing
time
and
method,
temperature,
mix
additives,
1-Jul-98
0.30
0.31
Column
etc.).
In the
literature 0.36
various formulations
can be
18-Mar-04
0.16
13-Sep-04 the 0.39
0.12 of normal
found to describe
sorption isotherm
14-Sep-05
0.43 However, 0.14
concrete (Xi
et al. 1994).
in the present
16-Oct-06
0.51 expression 0.15
paper the semi-empirical
proposed by
Norling Mjornell (1997) is adopted because it
16
⎤
1⎥
⎥
⎦
2
1.5
Punto de medida 20
(3)
14
Punto de medida 62
2
1.5
10 12 14 16 18 20
Tiempo (a? s)
(4)
y = 0.0404x - 0.1166
0.5
Tiempo (a? s)
y = 0.0348x - 0.0935
0.5
1
10 12 14 16 18 20
Tiempo (a? s)
1
+
Punto de medida 56
2
0
10
1.5
10 12 14 16 18 20
1.5
Punto de medida 61
2
Ancho de fisura (mm)
Ancho de fisura (mm)
2
1.5
2
1
Tiempo (a? s)
1
Punto de medida 50
Punto de medida 42
0
2
0.5
0
10 12 14 16 18 20
Tiempo (a? s)
0.5
20
1.5
∞
(g α
c
e
∞ − α )h
⎡
(g α
c
c −
⎢
K (α c α s ) e
⎢
Punto de medida 40
1.5
0.5
10 12 14 16 18
⎤
⎥
− α c )h ⎥⎥
⎦
10 12 14 16 18 20
Tiempo (a? s)
Ancho de fisura (mm)
4
Ancho de fisura (mm)
2
Ancho de fisura (mm)
0
8
Punto de medida 63
1.5
1
6
Tiempo (a? s)
Punto de medida 44
Figure 3. Icorr evolution measured in the beam during test time.
Proceedings of FraMCoS-7, May 23-28, 2010
1
0
0
Tiempo (a? s)
∂w
α&c + e α&s + w&n
∂α
c
s
y = 0.0229x - 0.0409
1.5
Tiempo (a? s)
0
∂w ∂h
e + ∇ • ( D ∇h) = ∂we
h
∂α
∂h ∂t
1
0.5
2
Ancho de fisura (mm)
10 12 14 16 18 20
Punto de medida 57
2
Ancho de fisura (mm)
8
0
−
Ancho de fisura (mm)
6
Ancho de fisura (mm)
Pantazopoulo & Mills 1995). It is reasonable to
assume that the evaporable water is a function of
relative humidity, h, degree of hydration, αc, and
degree of silica fume reaction, αs, i.e. we=we(h,αc,αs)
= age-dependent sorption/desorption isotherm
(Norling Mjonell 1997). Under this assumption and
by substituting Equation 1 into Equation 2 one
obtains
4
Anc ho de fisura (mm)
n
y = 0.0339x - 0.051
0
2
2
1.5
Tiempo (a? s)
Ancho de fisura (mm)
Figure
4 shows the results of corrosion rate Icorr
− ∂w = ∇ • in
J the beam throughout the time. (2)
obtained
The
∂t
2
average value is 0.128 µA/cm (1.5 µm/year). On the
pillar
were as
made.
The
Theonly
waterisolated
contentmeasurements
w can be expressed
the sum
results
indicated
in
it
a
higher
value
of
the
corrosion
of the evaporable water we (capillary water, water
rate,
of 2 water)
µm/year
was
considered in
vapor,then
anda value
adsorbed
and
thealso
non-evaporable
the
calculations.
(chemically
bound) water w (Mills 1966,
1
0.5
Punto de medida 58
Punto de medida 55
2
1.5
0
Ancho de fisura (mm)
that the variation in time of the water mass per unit
volume of concrete (water content w) be equal to the
3divergence
RESULTS
of the moisture flux J
Punto de medida 47
Ancho de fisura (mm)
they have been taken as average annual values of
corrosion
rate Icorrrep =coefficient
1,5 as said
is the
mean
The proportionality
D(h,T)
is called
recorded
in
the
T
beam
throughout
14
years
of
life
moisture permeability and it is a nonlinear functionof
the
elements,
thetemperature
sake of a T
sensitivity
of the
relative however
humidity for
h and
(Bažant
analysis
also
a
value
of
2,0
µm/year
will
be
tried.
& Najjar 1972). The moisture mass balance requires
explicitly accounts for the evolution of hydration
reaction and SF content. This sorption isotherm
reads
A ncho de fis ur a ( mm)
J =For
− D (the
h, T calculation
) ∇h
of the factor of proportionality
(1)
4
y = 0.134x + 0.1269
3
2
1
0
0
2
4
6
8
10 12 14 16 18 20
Tiempo (a? s)
= −The
λ ∇T Figure 4 shows the evolution of crack
(7)
width in the different locations selected in the
beam and in the column. The cracks grow at
where
q is
flux,
T trends)
is thedepending
absolute
different
ratesthe(theheat
slopes
of the
temperature,
and
λ
is
the
heat
conductivity;
thisa
of the place. It has to be stressed that there isinnot
q
unique value of the crack width The reasons for this
behaviour can be several: a) the bars have different
sizes and the stirrups different cover depths, b) there
are noticed pre-existing drying shrinkage cracks and
c) the corrosion rate has likely been uneven along the
bars in spite that the chloride was added in the mix.
The statistical distribution found is given for each
element in Figure 5. The distribution is log normal
and the crack widths are larger in the beam than in
the pillar perhaps due to the reinforcement detailing,
− D (h, Tdetailed
) ∇h
2006). AlthoughJ a= more
analysis has to be
made the slope values here calculated seems to
reflect better an averaged
behaviour. coefficient D(h,T)
The proportionality
moisture permeability and it is a nonlinea
of the relative humidity h and temperature
& Najjar 1972). The moisture mass balanc
that the variation in time of the water mas
volume of concrete (water content w) be eq
divergence of the moisture flux J
∂
a) Ec.− 2∂t = ∇ • J
w
b) Ec. 3
Figure 6. Representation of equations (2) and (3) plotting
crack width versus corrosion penetration divided by c/φ or
by the bar radius Ro. The water content w can be expressed
a) Beam T
b) Pillar
Figure 5. Statistical distribution of crack widths found under
natural corrosion.
4 DISCUSSION
In order to make later a comparison, the equations
proposed were first fitted to the experimental data in
existing literature (Rodriguez 1996, Andrade 2002,
Feliú 1990, Andrade 1993, Torres-Acosta 1999
Torres-Acosta 1998, Cabrera 1996). Table 2 shows
the results obtained using the values of crack width
and corrosion given by the authors mentioned in it.
In Figure 6 it has been analyzed present results
plotting the crack width versus the C/φ (equation 2)
or versus the bar radius Ro (equation 3). From the
figure it can be deduced the linear trend in both cases
but the relative value depends on the type of element
in the case of the natural corrosion being higher the
values of crack widths found in the beam than in the
pillar.
With respect to the values obtained in accelerated
tests in the laboratory, they show intermediate values
between those of the beam and the pillar. This is an
interesting fact as it enables to use accelerated tests to
predict evolutions in real conditions.
Table 2. Results of the value of k for different authors calculated
with the formulas (2) and (3).
k = w C / Px 
k' = w Ro / Px
Average of all
4.79
28.44
Torres
2.10
27.99
Castro
11.60
26.22
Andrade
3.41
30.52
Cabrera
0.61
18.67
Rodríguez 1
8.53
24.70
Rodríguez 2
8.58
41.23
In Figure 7 an averaged trend has been fitted to all
results which enables in an averaged value for the
slopes k y k’ of 9.5 y 35.5 respectively which are 50
y 40% higher than those proposed before (Muñoz
a
of the evaporable water we (capillary wa
vapor, and adsorbed water) and the non-e
(chemically bound) water wn (Mil
Pantazopoulo & Mills 1995). It is reas
assume that the evaporable water is a fu
relative humidity, h, degree of hydration
degree of silica fume reaction, αs, i.e. we=w
= age-dependent sorption/desorption
a) Ec.(Norling
2
Ec. 3 Under this assum
Mjonellb)1997).
Figure 7. Fitting oby
fan substituting
averaged trend Equation
in the data of
1 natural
into Equati
and accelerated corrosion
tests.
obtains
∂w
∂w ∂h
e α& + ∂we α& + w
5 CONCLUSIONS
+ ∇ • ( D ∇h ) =
− e
c
s
∂h ∂t
h
∂α
c
∂α
s
There has been studied the evolution of the cracks
generated by where
the corrosion
reinforced
slope
of the sorption/
∂we/∂h is ofthetwo
elements, a T beam
and
a
column
that
contained
isotherm
(also
called
moisture
capac
chlorides from governing
the mixingequation
and have(Equation
been exposed
3)
must
be
from 1990 to bytheappropriate
atmosphere
of Madrid.
The conditi
boundary
and
initial
conclusions that have
drawn
up are:the amount of e
The been
relation
between
water and relative humidity is called ‘‘
1. The crack width
evolution
has been first
isotherm”
if measured
withfitted
increasing
to two simplehumidity
equations
and ‘‘desorption isotherm” in th
(2)
case. Neglecting their difference (Xi et al.
(3)
⎛ P ⎞ following,
⎛ P ⎞ isotherm” will be
‘‘sorption
⎟⎟
w = k ⎜⎜ xthe
w = k ⎜⎜ x ⎟⎟
to both⎝sorption
and desorption c
Ro ⎠
⎝ C φreference
⎠
By the way, if the hysteresis of the
2. These expressions
bothbetotaken
represent
well
isothermseem
would
into account,
two
the evolution
of
the
crack
width
in
time
in humi
relation, evaporable water vs relative
natural conditions.
be used according to the sign of the varia
3. The values of
crack widths
have been
relativity
humidity.
The larger
shapein of the
the T beam than
in theforpillar.
values of by
themany p
isotherm
HPC The
is influenced
accelerated especially
tests published
in
the
literature
those that influence extent and
show intermediate
whichand,
indicate
that determ
chemicalvalues
reactions
in turn,
the accelerated
tests
may
be
used
for
prestructure and pore size distribution (waterdictions in real
applied
ratio,conditions
cement provided
chemical hecomposition,
SF
currents are limited.
curing time and method, temperature, mix
4. As for the values
of the
factors
etc.). In
the proportionality
literature various
formulatio
k and k’, thatfound
appears
in
the
formula,
they
to describe the sorptionhave
isotherm
been averaged
with
results
obtained
in
accelerated
concrete (Xi et al. 1994). However, in th
tests. The values
have been expression
9.5 for
paper obtained
the semi-empirical
pro
equation 2 and
35.5
for
equation
3.
Norling Mjornell (1997) is adopted b
Proceedings of FraMCoS-7, May 23-28, 2010
J = − D ( h , T ) ∇h
REFERENCES
(1)
Alonso
Andrade C., Rodriguez
J. and D(h,T)
Diez J.M.:
TheC.,proportionality
coefficient
is “Factors
called
controlling cracking of concrete affected by reinforcement
moisture
permeability
and
it
is
a
nonlinear
function
corrosion”. Materials and Structures, 31, August-September,
of pp.
the435-441,
relative1998.
humidity h and temperature T (Bažant
& NajjarC.,1972).
balance
requires
Andrade
AlonsoThe
C. ymoisture
Molina F.mass
J., “Cover
cracking
as a
of rebar in
corrosion:
I – Experimental
test”.
thatfunction
the variation
time of Part
the water
mass per unit
Materials
and Structures,
26 (1993),
volume
of concrete
(water
contentpp.w453-464.
) be equal to the
Andrade C., Alonso C., Molina F. J., “Cover cracking as a
divergence
of
the
moisture
flux
J
function of rebar corrosion: Part II – Numerical model”.
Materials and Structures, 26 (1993), pp. 532-548.
w
∂
Andrade
C., Molina F.J., “Cover cracking as a
(2)
− function
= ∇C.,
• JofAlonso
rebar
corrosion: Part I – Experimental test”.
∂t
Materials and Structures, 26 (1993), pp. 453-464.
Andrade C., Alonso C., Rodríguez J., Casal J.: Relation between
The waterand
content
w can
be expressed
the sum
corrosion
concrete
cracking.
Reporte asinterno
del
(capillary
water,
of proyecto
the evaporable
water
w
e
Brite/Euram BE-4062.
DG XII, C.E.C., water
Agosto
vapor,
1995.and adsorbed water) and the non-evaporable
Andrade
C., Alonso
C., Sarría
J.: “Corrosion
rate evolution
1966,in
(chemically
bound)
water
wn (Mills
concrete
structures
exposed
to
the
atmosphere”,
Cement
Pantazopoulo
& Mills 1995). It is reasonable to&
Concrete Composites, 24 (2002), pp. 55-64.
assume that the evaporable water is a function of
Beeby A. W., “Cracking, cover and corrosion of reinforcerelative
h, degree5, of
hydration,
αc, and
ment”. humidity,
Concrete International,
2 (1983),
Pg. 35-40.
we(h,αc,αs)
degree J.ofG.,
silica
fume reaction,
αs, i.e.
Cabrera
“Deterioration
of concrete
duewto
e=reinforcement
corrosion,” Cement
& Concrete Composites,isotherm
18 (1996),
= steel
age-dependent
sorption/desorption
pp. 47-59.
(Norling
Mjonell 1997). Under this assumption and
Feliú,
S., González,Equation
J.A., Feliú,
S.Jr., Equation
and Andrade,
C.,
by "Confinement
substituting
1 into
2 one
of the electrical signal or in-situ measurement
obtains
of Polarization Resistance in Reinforced concrete," ACI
Mater. J., 87, pp 457. (1990).
Leung
Y.: “Modeling of ∂concrete
cracking
induced by steel
∂w
∂w K.
we
e αEngineering, Maye ∂h + ∇ •Journal
in
Civil
α
( D ∇hof
) =Materials
− expansion”,
(3)
c
s
n
h
∂α
∂α
∂h ∂2001.
t
June,
c
s
Liu Y., Weyers R.E.: “Modelling the time to corrosion cracking
in chloride contaminated reinforcement concrete structures”,
the slope
the675-681.
sorption/desorption
where
∂we/∂h isJournal,
ACI Materials
95 (6),ofpp.
isotherm (also
called lifemoisture
The
Martín-Perez
B.: “Service
modelingcapacity).
of RC highway
structures equation
exposed to(Equation
chlorides”. 3)
PhD
dissertation,
Dept. of
governing
must
be completed
Engineering,
University
Toronto,
1998.
by Civil
appropriate
boundary
andofinitial
conditions.
Muñoz A., Andrade C., Torres-Acosta A., Rodríguez J.:
The relation between the amount of evaporable
“Relation Between Crack Width and Diameter of Rebar
water
relative
humidity
is called
‘‘adsorption
Loss and
due to
Corrosion
of Reinforced
Concrete
Members”
isotherm”
if measured
with increasing
relativity
Electrochemical
Society Symposium,
Cancun, Mexico,
Oct
29th – Nov
2006.
humidity
and3rd,‘‘desorption
isotherm” in the opposite
&+
& + w&
case. Neglecting their difference (Xi et al. 1994), in
the following, ‘‘sorption isotherm” will be used with
reference to both sorption and desorption conditions.
By the way, if the hysteresis of the moisture
isotherm would be taken into account, two different
relation, evaporable water vs relative humidity, must
be used according to the sign of the variation of the
relativity humidity. The shape of the sorption
isotherm for HPC is influenced by many parameters,
especially those that influence extent and rate of the
chemical 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 be
found 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
Ohtsu M.,accounts
Yoshimurafor
S.: the
“Analysis
of crack
explicitly
evolution
of propagation
hydration
and
crack
initiation
due
to
corrosion
reinforcement”,
reaction and SF content. This sorption isotherm
Const. And Build. Mat., 11 (7-8), pp 437-442, 1997.
reads
Padovan J., Jae J.: “FE modeling of expansive oxide induced
fracture of rebar reinforced concrete”, Engineering
Fracture Mechanics, 56⎡(6), pp. 797-812, 1997.⎤
Pantazopoulou S. J., Papoulia
K. D.:1 “Modeling
cover
⎢
⎥
we (h,cracking
α c , α s )due
= G1to(αreinforcement
+
in RC structures”,
⎥
c , α s )⎢1 − 10corrosion
∞
(g αApril
h⎥
− α2001.
Journal of Engineering ⎢Mechanics,
c )A.
1 c
e Al-Gahtani
⎣
⎦S.: “Cor(4)
Rasheeduzzufar, S. S. Al-Saadoun,
rosion cracking in relation
cover
and
∞ −diameter,
⎡ 10(to
⎤
)h
g1αbar
αCivil
concrete quality”,
Journal
of
Material
in
Engineering,
c
c
⎢
⎥
K1 (α1992.
−1
Vol 4 (4), Nov.
c , α s )⎢e
⎥
⎣
⎦ elastic
Reinhardt H. W.: “Fracture
mechanics of an
softening material like concrete”, Heron Vol. 29 No. 2,
1984.
where
the first term (gel isotherm) represents the
Rodríguez J., Ortega L. M., Casal J., “Load bearing capacity
physically
bound
(adsorbed)
water reinforcement,”
and the second
of concrete
columns
with corroded
en
term Corrosion
(capillaryof Reinforcement
isotherm) represents
capillary
in Concretethe
(1996),
C. L.
water.Page,
ThisJ.W.
expression
is valid
only(Eds.).
for low content
Figg, and P.B.
Bamforth
Rodríguez
J.,
Ortega
L.
M.,
Vidal
Ma.
Casal J.,
of
of SF. The coefficient G1 represents theA.,amount
“Disminución
de
la
adherencia
entre
hormigón
y 100%
barras
watercorrugadas,
per unit volume
held
in
the
gel
pores
at
debida a la corrosión,” Hormigón y Acero,
relative
humidity,
and it can be expressed (Norling
4º. Trimestre (1993), p. 49.
Mjornell
1997)
as
Torres Acosta A. A., Castro P., Sagüés A. A., “Effect of
corrosion rate in the cracking process of concrete”, Proc.
National
c α c +Congress
s α s of the Mexican ElectroG1 (α cXIVth
,α ) = k
k vg
(5)
chemical
Society
Spanish),
August 1999, pp. 24-28,
s vg
c (in
s
Mérida, México
Torres c Acosta A.
A., “Cracking induced by localized
are materialinparameters.
From the
wherecorrosion
k vg andofksvgreinforcement
chloride contaminated
concrete”.
Ph.
D.
Thesis,
University
of
South
Florida,
maximum amount of water per unit volume that
can
Florida,
USA,
1999.
fill all pores (both capillary pores and gel pores), one
AcostaKA.as
A.,one
Sagüés
A. A., “Concrete cover crackobtains
canTorres
calculate
1
ing and corrosion
expansion of embedded reinforcing
steel”, P. Castro, O. Troconis y C. Andrade (Eds.),
⎡ Latin
⎛ American
⎞ ⎤
∞
Proceedings of the third NACE
10⎜ g α
− α ⎟h ⎥region
⎢
1 c
c
⎝
⎠
corrosion
on
of corrosion
188α s + 0.22
w0 − 0.congress
α s −rehabilitation
G1 ⎢1 − e
⎥
c
s
⎥
damaged infrastructures (1998),⎢ pp. 215-229.
⎦ (6)
⎣
(α , α T.,
) = Castel A., Francois R.: “Analyzing crack width to
K1Vidal
c predict
s
⎛
⎞
∞
corrosion in10reinforced
Cement and
h
⎜ g α − α ⎟concrete”,
1 c
⎠ −1
e ⎝2004,
Concrete Research 34,
pp.c 165-174.
The material parameters kcvg and ksvg and g1 can
be calibrated by fitting experimental data relevant to
free (evaporable) water content in concrete at
various 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 not
exceeding 100°C (Bažant & Kaplan 1996), by
Fourier’s law, which reads
q
= − λ ∇T
(7)
where q is the heat flux, T is the absolute
temperature, and λ is the heat conductivity; in this