183
Nuclear Engineering and Design 76 (1983) 183-191
183-191
North-Holland, Amsterdam
MATHEMATICAL
MODEL
MATHEMATICAL
M O D E L FOR
F O R CREEP
C R E E P AND
A N D THERMAL
T H E R M A L SHRINKAGE
S H R I N K A G E OF
O F CONCRETE
C O N C R E T E AT
AT
HIGH
H I G H TEMPERATURE
TEMPERATURE
Zdenek
Z d e n ~ k P. BAZANT
BAZANT
Director. Center for Concrete and Geomaterials.
Institute. Northwestern
Professor of Civil Engineering and Director,
Geomaterials, Technological Institute,
University.
Evanston. IL 60201.
University. Evanston,
60201, USA
Received
Received 2 November 1982
possible to set up an approximate constitutive model for creep and shrinkage at
Based on the existing limited test data, it is possible
temperatures above 100°C,
100°C, up to about 400°C.
400°C. The model presented here describes
describes the effect of various constant temperatures
on the creep rate and the rate of aging, similar effects of the specific water content, the creep increase
increase caused by simultaneous
changes
changes in moisture content, the thermal volume changes
changes as well as the volume changes
changes caused
caused by changes in moisture content
pressure produced by heating.
(drying shrinkage
shrinkage or thermal shrinkage),
shrinkage), and the effect of pore pressure
heating. Generalizations to time-variable
stresses and multiaxial stresses are also given. The model should allow more realistic
realistic analysis
analysis of nuclear reactor vessels and
containments for accident situations,
situations, of concrete structures
structures subjected to fire, of vessels for coal gassification
gassification or liquefaction,
etc.
1. Introduction
Mathematical
Mathematical modeling of creep and
and hygral volume
changes
changes of concrete at temperatures
temperatures over 100°C
100°C is
needed
needed for finite element analysis of hypothetical
hypothetical accident
dent situations
situations in nuclear
nuclear reactor
reactor structures,
structures, as well as
assessments
assessments of fire resistance. Another
Another area
area of need is
the design of concrete vessels for coal gasification and
and
liquefaction. Although
Although considerable
considerable relevant experimental information
been obtained
information has been
obtained in recent years [1),
[1],
numerous
research remains
numerous gaps
gaps remain
remain and
and much
much more
more research
remains
to be done. Yet, structural
structural analyses
analyses have to be carried
carried
up at this time
out.
out. Therefore, it seems worthwhile to set up
an approximate
based on
approximate and
and rather
rather simplified model based
what is known
what
known now. This is chosen
chosen as the objective of
this work, even though
though one should
should be aware
aware that
that revisions are likely when
when further test data
data become available.
Analysis of deformations
deformations of concrete at high temperatures
atures cannot
cannot be carried
carried out
out without
without the knowledge of
the changes
changes in the specific content
content of water
water and
and its
migration, as well as the pressure
pore water. We will
pressure of pore
not
pore
not address
address here the question of calculating
calculating the pore
pressure, not
because of necessary
pressure,
not only because
necessary scope limitabut also because
because various approximate
tions but
approximate mathematimathematical models have been
been developed in other
other works [2-10).
[2-10].
For
purpose, the pore
pore pressure
pressure is assumed
For our
our purpose,
assumed to be
given.
The
The total, macroscopically measurable
measurable strain,
strain, (,
c, can
be subdivided
parts:
be
subdivided into three
three parts:
= £M q_ {T q_ ~H,
(1)
(1)
in which (M
~M = mechanical strain,
strain, i.e., the strain
strain caused
caused
by stress, (T
~T = thermal
thermal dilatation,
dilatation, and
and (H
~H = hygral strain,
strain,
i.e., the strain
strain caused
caused by humidity
humidity changes
changes (shrinkage
(shrinkage or
swelling). We will analyze these three
three components
components separately. The analysis
analysis will be restricted to the state
state of
possible multi
axial generalization
uniaxial stress and
and possible
multiaxial
generalization
will be indicated
indicated at the end.
2. Mechanical
Mechanical strain
For
For stresses
stresses less than
than about
about 0.5 of the strength
strength limit,
the strain
by stress (M,
proporstrain caused
caused by
c M, is approximately
approximately proportional to the stress, a,
o, which is considered
considered for the time
being to be constant
being
constant in time. Thus,
Thus,
,M = aJ(t,
o J ( t , t'),
/'),
(M
(2)
(2)
in which J((t,
t , t '')) == compliance function (also called the
creep function), which represents
represents the strain
strain at age t
caused
been acting since age t'
caused by a unit stress that
that has been
of concrete. At constant
constant specific moisture
moisture content,
content, w,
and
and constant
constant temperature
temperature T, the compliance function
0029-5493/83/$03.00
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184
184
z.P.
Z.P. Bazant
BaY,ant / Model for creep
creep and
and thermal
thermal shrinkaKe
shrinkage of
q[ COllerele
concrete
well approximated by the double
may be reasonably well
[11-13]. For an arbitrary time-constant
power law [11-13].
specific water content wand
w and an arbitrary time-constant
specific
temperature
temperature T.
T, the double power law may be written in
the form
1 +~rfw
l(t.t')=
<l>jw(t;-m+IX)(t_t')'"
J(t,t')=~~o + To(t~
.... +~)(,-c)",
u
T
/,_-----
To
(3)
(3)
E 0 = asymptotic modulus = the elastic modwhich Eo
ulus extrapolated to an infinitely
infinitely fast application of
n, m.
m, IXa == parameters for the given concrete.
concrete,
stress; n.
roughly independent of temperature (for their prediction, see
see refs. [12 and 13]); <l>T
~T == function of temperature.
temperature,
ion.
f,,, == function of water content.
content, and t~
t~ == equivalent hyfw
[I 1,14,15] == age
age of concrete at
dration period (maturity) [11.14.15]
temperature T for which the creep is the same as age t'
temperature
temperature To
TO== 25°C. The typical values
at reference temperature
parameters are n""
n = 1/8.
1 / 8 , m == 1/3.
1/3, IXc~== 0.3;
of material parameters
and, for reference temperature
temperature To
To = 25°C, <l>Tfw
q~Tf,. = <1>1
q~l = 3
to 6 (the times must be given in days); approximately,
approximately,
Eo
E 0 == 1.5 E(
E ( tn.
' ) . The conventional (static) elastic modulus
modulus
obtained from eq. (3) as E(I')=
E ( t ' ) = l / J1/1(1.1').
(t,
t'), where
is obtained
t' plus 0.1 day.
t == I'
known, the temperature
temperature effect on creep is
As is well known.
temperature the hydration
hydration
two-fold. First, at a higher temperature
manifested in aging, progresses faster, and since
process, manifested
for loads applied
applied at the higher age the creep is smaller,
tends to decrease creep. Second, at the higher
this effect tends
temperature the rate of bond
ruptures which are rebond ruptures
temperature
sponsible for creep and
and are manifested
manifested in viscosity
sponsible
parameters
parameters (e.g., those for the Maxwell chain model), is
tends to increase
increase creep and
and usually
larger. This effect tends
prevails over the first one [11,14,15].
[11,14,15].
The effect of temperature
temperature on the rate of hydration
hydration
The
(aging) may be described
described [15,11,14]
[15,11,14] by expressing
expressing the
equivalent hydration
period (maturity)
hydration period
(maturity) as
,
t;
JJc,,,')
(II t')
in
III
t; = Lt/?T [ T ( t " ) ] d t " .
(c)
(0)
(o)
J (t, t')
(4)
(4)
For
For a two-step
two-step temperature
temperature history
history in which
which the tempertemperature
Too until
until age
ature is equal
equal to the
the reference temperature
temperature T
which the
the temperature
temperature is raised
raised to a new constant
constant
t I1 at which
t; = ttoo +
+ #f3Tr ,I((t'
- tl);
t 1); see
ttemperature,
e m p e r a t u r e , T1,
t ' -1, eq. (4) yields t;
fig. lc.
o . Coefficient /3
lc. Also tt;~ = t too if T
T1I ==T To.
f3Tr is a
function
function of
of temperature,
temperature, and
and since the chemical
chemical reaction
reaction
of
of hydration
hydration is a thermally
thermally activated
activated process,
process, this coefficient may
may be
be expressed
expressed with
with the help
help of the rate-process
rate-process
cient
theory [16,15,11,14]
[16,15,11,14] and:
and:
theory
(5)
'i
t;
t
tO
Ib)~
(b)
t' ..
/3h I
(d)
~ / ~ t
h
t~
O
/
t
loglt t)
log (I - t')
O·C
O°C
100·C
IOO°C
Fig. 1.
1.
in which R = gas constant,
constant, U hh = the activation energy
of hydration;
hydration; and T, To
To must be given as the absolute
f3T=
temperatures.
temperatures. By definition.
definition, /3
T = 1 when T=
T = 7;).
T0. The
value of Uhh depends
depends on the type of cement and concrete.
crete, and actually eq. (5) is itself a simplification since
hydration
hydration involves a number
number of different <.:hemi<.:al
chemical
pro<.:esses
processes with different activation energies. Nevertheless.
less, as a good approximation
approximation up to about
about 1000e
100°C one
<.:an
can use U hh// RR == 4000 K. Above 100°C,
100°C, the hydration
hydration
process does not proceed, and above approximately
approximately
o
e, the reverse process,
process, dehydration,
pla<.:e.
200
200°C,
dehydration, takes place.
Dehydration,
Dehydration, however.
however, is not too significant below
0e and
400
400°C
and in this range may be neglected, as an approximation [1].
proximation
For
For nuclear
nuclear accident analysis, one needs to det~ just
just before the postulated
postulated ac<.:itermine
termine the value of t~
accident
dent happens,
happens, i.e., for operating
operating temperature
temperature wnditions.
conditions.
The
The variations
variations of t~ during
during a short-time
short-time aC<.:ident
accident may
probably be neglected, and
probably
and the same is true for calculations of response
response to fire.
The change
change of viscosity aspect of the temperature
temperature
modeled according
a<.:cording to
on creep
creep is logically also modeled
effect on
the rate-process
rate-process theory
theory since the mechanism
mechanism of
of creep
creep no
doubt consists
consists in thermally
thermally activated
activated ruptures
ruptures of
of bonds
bonds
doubt
within the solid microstructure.
microstructure. Therefore,
Therefore,
within
t;
(6)
which ~0
<1>0 = constant
constant parameter
parameter for"
fof the given type of
in which
concrete [12,13],
[12,13], and
and U=
activation energy
energy for the creep
concrete
U = activation
rate (i.e., for the bond
bond ruptures).
ruptures). While
While below
below 1000e
rate
100°C the
dependence expressed
expressed by
by eq. (6) is obscured
obscured by the
dependence
simultaneous effect of
of temperature
temperature on
on the
the hydration
hydration
simultaneous
rate, from
from 100°C
100°C to
to about
about 400°C
400°C this
this dependence
dependence can
can
rate,
be directly
directly measured,
measured, as the
the hydration
hydration effects are
are neglibe
z.P.
Z.P. Bazant
Ba~ant// Model
Modelfor
for creep
creepand
and thermal
thermalshrinkage
shrinkageof
of concrete
concrete
€
..
(0)
/
(o)
/
wet
,~we,/
~o,/ ~o,~///
qbw
II____
3
/I/// /
I
.o'~/ 4o7
Wo
ried
dried
1 atm
T
00
200" C
200·C
w,
."
"~'~
2
00
(b)
~
400"C
400·C
~
~
Wo
wo
iw
W,
wI
2.
Fig. 2.
gible, in comparison, and anyhow small. Pertinent
Pertinent test
data were reported, e.g., by Man~chal
Mar6chal [17-20]. To dedata
Mardchal's data, we note that i:~Tr =
termine U from Marechal's
oaf/at
(T, exp[T
/ R] or U // RR = [In(i:T/
oOJ/Ot =
=CTt
exp[T1I- I] --- rl)U
T-X)U/R]
[ln(~r/
i:~r,)](T1-1
)](TI- I --- T-I)-I.
T - I ) -1. Using Marechal's
Mar6chal's [17] figs. 5 and
and 4
T
(p. 113), we find from his curve for 0o == 100 bars and
(p:
100°C and
and 300°C
300°C that U// RR == [In(3.3/0.5)]
[ln(3.3/0.5)]
points at 100°C
l -1=
(373 - al -- 57r
573-1)
(37r
)-1 = 2017 K, and from his curve for
o = 50 bars, and
and points at 100°C
100°C and 400°C,
400°C, that U/R
U/R
0=
=
-I) = [In(3.2/0.2S)](373
[ln(3.2/0.28)](373 -- ]I -- 673
673-1)
-1I == 203S
2038 K. Thus,
Thus, it
consider (fig. 2):
seems one may consider
U / R -- == 2000 K.
U/R""
(7)
(7)
cement concrete, same
This applies for dried portland
portland cement
as Mar6chal's
Marechal's data.
data.
Mardchal's
Marechal's points
points for 20°C
20°C and
and 50°C
50°C give U// RR =
=
[1n(2.8/1.07)](323
3 - 1 )I) -1
0 3 0 K, a higher
[In(2.S/1.07)](32r-1I -- 2 929r
- I = 33030
higher
value. In an
an analysis
analysis of
of many
many test data
data in the literature
literature
[15,
275], the
the values of
of U/ RR in the
the range
range 0°C
O°C to 80°C
SO°C
[15, p. 275],
are
10000 K, with
are found
found to
to range
range from 3000
3000 K
K to 10000
with a
median
median roughly
roughly 4000
4000 K. One
One must
must remember,
remember, however,
however,
that
that for temperatures
temperatures below
below 100°C,
100°C, which
which are
are not
not of
of
direct
direct interest
interest here,
here, the
the value
value of
of U can
can be
be different.
different.
Since
Since the
the solids
solids forming
forming hardened
hardened cement
cement paste
paste are
are
strongly
strongly hydrophylic,
hydrophylic, and
and an
an ingress
ingress of
of water
water into
into the
the
pores
pores reduces
reduces the
the surface
surface energy
energy and
and thus
thus weakens
weakens the
the
bonds
bonds (as
(as the
the water
water molucules
molucules are
are trying
trying to
to enter
enter and
and
split
split the
the bonds),
bonds), the
the value
value of
of U no
no doubt
doubt depends
depends on
on
the
the specific water
water content,
content, w
W (kg
(kg of
of water
water per
per m
m33 of
of
concrete).
concrete). Comparing
Comparing the
the value
value 2000
2000 K,
K, which
which applies
applies
for
for dried
dried concrete
concrete above
above 100°C,
100°C, with
with the
the value
value roughly
roughly
4000
4000 K
K found
found in
in ref.
ref. [15]
[15] for
for temperatures
temperatures below
below 100°C,
100°C,
and
and assuming
assuming the
the same
same value
value above
above 100°C,
100°C, we
we may
may
extrapolate
extrapolate eq.
eq. (7)
(7) as
as
w-w° )
W-W)
--U
_0 2000
U- 2"" ( 11 +
+_
2000 K,
K,
R
WI
R
w] -- Wwo0
185
185
(S)
(8)
where
where aa linear
linear dependence
dependence on
on Ww is
is assumed
assumed due
due to
to lack
lack
of
of more
more precise
precise information;
information; Wwo0 == water
water content
content of
of dried
dried
concrete
concrete (i.e.,
(i.e., at
at 105°C
105°C and
and 11 atm),
atm), and
and WI
w] == water
water
content
content of
of saturated
saturated concrete
concrete at
at 25°C
25°C and
and 11 atm
atm (1
(1
atm
atm == 101325
101 325 Pa).
Pa).
3. Creep
C r e e p at various constant
constant water contents
contents
Aside from the direct effect of water content on
activation energy U,
U, as just explained, there exists, no
doubt, an additional direct effect of water content Ww on
the creep rate. This is because eq. (S)
(8) does not give a
sufficiently large difference between the creep rates of
wet and dried concretes, which is an order of magnitude
difference. Apparently, the presence of water not only
reduces the activation energy but also generally increases
the mobility of the solid particles that are displaced
during creep. Again, from Marechal's
Mar~chal's data
data [17, his figs.
4 and
and 5], the ratio of strain ratei:
rate ~ values for wet and
dried concretes is about
about 2.S/0.3
2.8/0.3 ::::
= 9 (from his fig. 5), or
560/70
560/70 == S
8 (from his fig.
fig. 4). As another
another experimental
evidence, Bazant,
Ba~ant, et al. [4] find in their fig.
fig. 6 for 200°C
the ratio of i:~ values at two hours after loading for
predried and
predried
and drying concrete to be about
about 1/5,
1 / 5 , and
and that
that
for drying concrete and
and sealed concrete to be about
about
1/2.
predried
1 / 2 . This gives for the ratio of i:~ values for predried
concrete and
and sealed concrete approximately
approximately 1/10,
1/10, which
is about
about the same result. Other
Other data,
data, e.g., those by
Wittmann [21,22,23],
[21,22,23], give also about
about the same
same result.
Wittmann
Thus, assuming
assuming again
again a linear
linear dependence
dependence due
due to lack of
of
Thus,
better evidence, we may
may propose
propose that
that (see fig. 2b):
better
W|
fw=l-kw~--~0,
--
W
with kw - v ~.
((9)
9)
Obviously, the
the effect of
of w
W is a very important
important one
one
Obviously,
and can
can never
never be
be neglected.
neglected.
and
One should
should also
also realize that
that the
the pore
pore pressures
pressures in
in
One
concrete can
can far
far exceed
exceed the
the saturation
saturation pressure
pressure
concrete
[2,3,10,4,5,6], and
and concrete
concrete then
then becomes
becomes significantly
significantly
[2,3,10,4,5,6],
oversaturated, i.e., w
W --w lWI
Eq. (9)
(9) would
would certainly
certainly
oversaturated,
( t(t).
). Eq.
too much
much variation
variation for
for oversaturated
oversaturated concrete,
concrete, and
and
give too
not applicable
applicable for
for w
W>
> %.
WI' Actually,
Actually, according
according to
to the
the
is not
test results
results of
of Ba~ant,
Bazant, Kim
Kim and
and Meiri
Meiri [24],
[24], itit appears
appears
test
that oversaturation
oversaturation significantly
significantly decreases
decreases rather
rather than
than
that
and roughly
roughly fw
fw ="" 0.5
0.5 which
which
increases the
the creep
creep rate,
rate, and
increases
ensues from
from measurements
measurements on
on cement
cement paste
paste exposed
exposed to
to
ensues
liquid pressurized
pressurized water
water at
at 200°C.
200°C. However,
However, more
more
liquid
extensive testing
testing is
is required.
required.
extensive
186
z.P. Badant
Bazant I/ Model for
jor creep and
shrinhage ,III
Z.P.
and thermal
thermal shrinkage
/ co/lerele
con¢rew
4. Creep at variable moisture content and temperature
As clearly demonstrated
demonstrated first by Pickett [25)
[25] and
and
studied
studied by many
many others
others [11.14).
[11,14], the creep of concrete in
a transient
transient moisture
moisture state
state can be much
much higher than
than at
constant
both
constant moisture
moisture content.
content. This appears
appears to be true both
for decreasing
decreasing and
and increasing moisture
moisture contents.
contents, as well
temperatures, most likely also above
as for elevated temperatures.
100°('.
phenomenon may be explained
100°C. The
The phenomenon
explained by the hypothesis that
pothesis
that movement of water
water molecules may facilitate changes
bonding of the solid microstructure
changes in the bonding
microstructure
[11).
[11].
Above 100°C
100°C the measurements
measurements on specimens
exposed
exposed to dry environments
environments show much
much less creep than
than
those which are sealed [26.24).
[26,24], contrary
contrary to what
what is seen
probably due to the fact
at room temperature.
temperature. This is probably
that
that the rate of moisture
moisture loss above 100°C
100°C is much
faster (about
(about 200-times)
200-times) than
than at room temperature
temperature [2.3).
[2,3],
so that
that the existing measurements
measurements at high temperature
temperature
pertain to specimens
pertain
specimens that
that have already
already dried and
and
achieved equilibrium before the measurement.
measurement. To model
power law.
this effect.
effect, we may generalize the double
double power
law, eq.
(3).
(3), as follows
1
;,)/w4,T
,,,+ ~)(t-t')".
J(t.I')= ~ +g(w/:T(t'c-m+o:)(t-t')'"
J(.t')=~+g(
~-0(t7
o
()
(10)
(10)
in which g is a function of W.
~, and
and w
~' represents
represents some
suitable
proper measure
suitable proper
measure of the rate of the moisture
moisture
be. however.
content.
content. It would be,
however, unrealistic to assume
assume
that
that g == g(
g ( #w).
) , since the relative increase
increase of the creep
rate seems to be of the same
same order
order of magnitude
magnitude for
both the short-time
both
short-time creep (say.
(say, several hours
hours after load
application)
application) and
and the long-time creep (say.
(say, many
many years
after load application.
application, at room temperature).
temperature), even
though
though the strain
strain rates (~ differ by orders
orders of magnitude.
magnitude.
Therefore.
Therefore, we must
must assume
assume that
that g depends
depends on the ratio
w
If. rather
just w.
~'/~
rather than
than just
#. Furthermore.
Furthermore, we should
should note
that
that the effect seems to be relatively the same for weak
concretes of high moisture
moisture contents
contents (high WI)
w~) and
and for
dense
dense concretes of small water
water contents.
contents. Also.
Also, the relative effect seems to be the same for concretes which
creep much (large ()
~) and
and those which creep little (small
f).
c). Therefore.
Therefore, insted
insted of w
A' one should
should use fI~/WI'
v / w l, and
and
instead
instead of f.~ one should
should use VL
~/~. This leads us to postulate
tulate that
that w
~i, may be defined as
= 6 , ' ~ / i w 1,
(11
(11))
in which WI
w~ represents
represents the total saturation
saturation water
water content
content
at room temperature.
temperature. We may call w
~ the relative drying
rate.
w). We should
N ow we need to specify functions g(
Now
g(~').
should
realize that
that for extremely high drying rates.
rates, such as
those experienced by the thin surface layer of a drying
specimen at the start
start of drying.
drying, or by cylinders at high
temperature.
temperature, the increase of creep caused
caused hy
by simultasimultaneous
neous drying is not much larger than
than it is for moderate
moderate
drying
drying rates.
rates, which are orders
orders of magnitude
magnitude smaller [27).
[27].
A similar comparison
comparison could he
be made
made between moderate
moderate
and
and very low drying rates. Therefore.
Therefore, it seems that
that the
function g(
g ( ~It·)
' ) should
should have the form sketched
sketched in fig. 2c.
consisting
consisting of a smooth
smooth transition
transition from an asymptotic
asymptotic
value for very low drying rates to an asymptotic
asymptotic value
for very high drying rates. The simplest function to
describe such a transition
transition is
g ( Ii· ) == 1 +
k w [[11 ++ tanh
w ( Ii'
- hw )].
g(~;,')
+/~w
tanh tg~(
~,'-/~w)],
(12)
(12)
k w === ½,
~. tw
in which one may set
set/~,~.
~,~ ==
= 1.
1, and
and hw
i~w is the value
It·
Ii') is highest. Ap~, for which the slope of curve g(
g(~¢,)
== I.
proximately take hw
proximately
/,,,,-1, which means
means that
that the drying
creep effect becomes significant when the value of w
# /Iw
w
becomes comparable
comparable to VL
,~/~. This should
should of course be
hecomes
checked
checked carefully against
against test data
data after detailed studies
studies
are completed.
completed.
axial stress
5. Generalization to multi
multiaxial
While important
important nonlinear
nonlinear effects certainly exist
probably use a linear formulation
above 100°C,
100°C, we may probably
just like at room temperature.
for many
purposes. just
many purposes,
temperature. Then
Then
the multiaxial generalization
generalization is completely specified by
the values of the Poisson's
Poisson's ratio v~, for instantaneous
instantaneous
deformations
deformations and
and for creep. Both are approximately
approximately
equal below 100°C (v ==
--- 0.18).
There is a question.
question, however.
however, as to the value of v
above 100°C.
100°C, since no direct measurements
measurements seem to
present for concrete. We will therefore try to
exist at the present
make a logical deduction
deduction from the measurements
measurements on
small cement
paste specimens
cement paste
specimens [24).
[24], which showed
showed that
that
pastes at 25°C
the value v == 0.25 applicable
applicable for cement
cement pastes
25°C
increases
increases to about
about 0.46 at 200°C,
200°C. This means
means that
that a
large increase of temperature
temperature intensifies the deviatoric
creep much
much more than
than the volumetric creep.
creep, for cement
cement
pastes. For
pastes.
For concretes.
concretes, however.
however, the relative magnitude
magnitude
of this effect should
should be much weaker.
weaker, since the Poisson's
ratio of concrete is dominated
dominated by the aggregates rather
rather
than
binder. At 25°C.
both short-time
than the binder.
25°C, v ==
= 0.18 for both
short-time
and
and long-time deformations.
deformations.
For
For approximate
approximate inferences.
inferences, one may use the simple
laws of mixtures known
known from the theory of composites.
composites,
which were shown
shown to be applicable to concrete as a
satisfactory
satisfactory approximation;
approximation: see DougiIJ
Dougill [28).
[28]. There
There are
z.P. Badant
Baiant // Model
Model for
for creep and
and thermal
thermal shrinkage
shrinkage of
of concrete
concrete
Z.P.
two simple
simple laws
laws of
of mixture,
mixture, corresponding
corresponding to
to the
the sosotwo
called series
series and
and parallel
parallel couplings
couplings of
of the
the deformable
deformable
called
one and
and the
the other
other ccomponent
elements representing
representing the
the one
elements
omponent
of the
the composite
composite ssolid-cement
paste (c)
(c) and
and aggregate
aggregate
of
o l i d - - c e m e n t paste
(a). The
The reality
reality lies between
between these
these two
two laws,
laws, which
which
(a).
much
closer to
to the
the mixture
mixture law
law
represent bounds,
bounds, aand
represent
n d is m
u c h closer
for the
the parallel
parallel coupling
coupling [28]. Based
Based on
on this, the
the Poisson's
Poisson's
for
of the
the composite
composite solid
solid may
may be
be approximated
approximated as
as
ratio of
ratio
Ecv ¢ v~ + Eav a Va
(13)
in which
which E~,
Ec ' E~,
Ea , uvc,
and
the elastic
elastic moduli
moduli and
and
in
n d vVaa = the
c' a
Poisson's ratios
ratios for cement
cement paste
paste aand
for aggregate,
aggregate, aand
Poisson's
n d for
nd
Vc ' v,
va = volume
volume fractions
fractions of
of hhardened
cement paste
paste and
and
v~,
a r d e n e d cement
aggregate in
in concrete.
concrete. For
For the
the aggregate
aggregate skeleton
skeleton without
without
aggregate
may assume
assume ~a
Va = 0.175.
0.175. Then,
Then, for a typical
typical
ccement
e m e n t we may
Eava/ Ecvc ='" 20, we get
get vV = (0.25 ++ 20 x
concrete with
with E~vJE~.o~
concrete
0.175)/(1 ++ 20) = 0.18.
0.18. This
This agrees with
with m
many
observa0.175)/(1
a n y observaand thus
thus supports
supports our
our way
way of
of argument.
argument. Now,
Now, for
tions, and
concrete at 200°C,
200°C, we
we may
may estimate
estimate in the same
same m
manner
concrete
anner
V = (0.46
(0.46 ++ 20 xX 0.175)/(1
0.175)/(1 ++ 20) = 0.19,
0.19, which
which is only
only a
different from
from the previous
previous value
value 0.18.
0.18. Our
Our theory
theory
little different
rather crude
crude anyway,
anyway, and
and so it appears
appears that
that we
we may
may
is rather
use for concrete
concrete at all temperatures
temperatures
use
0.18.
vv'"
~ 0.18.
(14)
(14)
Using the approximate
approximate effective modulus
modulus approach,
approach,
the same
same value of Vv is obtained
obtained for creep of concrete
concrete at
all temperatures.
temperatures. Clearly, it is the dominating
d o m i n a t i n g influence
of the aggregate skeleton which causes the Poisson's
ratio of concrete
concrete to be insensitive to the value for
cement
cement paste.
The
The value of Vv being known,
known, the volumetric and
and
deviatoric compliance
compliance functions may
may be expressed
expressed as
[11,14]:
the total
total stress,
stress, representing
representing aa resultant
resultant (over
(over aa unit
unit area
area
the
of concrete)
concrete) of
of the
the microstresses
microstresses in
in solid
solid as
as well
well as
as
of
water,
water, and
and the
the effective
effective stress,
stress, representing
representing the
the resultant
resultant
of the
the microstresses
micros tresses from
from the
the solid
solid part
part of
of the
the micromicroof
structure only.
only. Further,
Further, there
there is aa fine point
point whether
whether the
the
structure
unit
or microscopically
microscopically
unit area
area should
should be
be perfectly
perfectly plane
plane or
tortuous so
so as to
to pass
pass through
through the
the weakest
weakest connections
connections
tortuous
in the
the solid
solid microstructure.
microstructure. The
The second
second possibility
possibility is
in
more relevant
relevant for the
the assessments
assessments of
of fracture,
fracture, aand
in
more
n d in
n p to
to be
be used
used with
with pressure
pressure pp
that case
case the
the porosity
porosity np
that
should
should not
not be
be the
the actual
actual porosity
porosity n (i.e., the
the volume
volume
of pores)
pores) bbut
the so-called
so-called bboundary
porosity,
fraction of
u t the
o u n d a r y porosity,
as
as used
used in soil mechanics,
mechanics, which
which represents
represents the
the area
area
but
fraction of
of water
water over
over the
the macroscopically
macroscopically plane
plane but
fraction
microscopically
microscopically tortuous
tortuous weakest
weakest section
section through
through the
the
material [29]. The
The value
value of
of np
n p is always
always _<
s 1 and
and always
always
material
~ the actual
actual porosity.
porosity. FFrom
studies of
of the
the uplift prespres_>
r o m studies
seems that
that n vp = 0.9 is a reasonable
reasonable
sure in dams
dams [29], it seems
sure
estimate
e c o m m e n d for dams
estimate (some
(some investigators
investigators rrecommend
dams n pp '"
1, but
but for our
our purposes
purposes the difference
difference is not
not significant).
The effective volumetric
volumetric stress may
may be
be calculated
calculated as
The
o v ' = ov + n p p
JD(t,
J ° ( t , t')
t ' ) == 2(1 +
+ v)J(t,
v ) J ( t , t').
t').
(15)
(15)
6. Effect of pore pressure
At
A t high temperatures,
temperatures, the pressure of pore
pore water, p,
p,
can attain significant values and must be taken into
account in evaluating tensile fracture. A similar phenomenon
n o m e n o n has been studied in depth
d e p t h in soil mechanics,
and
a n d for concrete the effect of pore
pore pressure was examined
a m i n e d in many
m a n y works on
o n the problem
p r o b l e m of uplift pressure in dams (for review see ref. [29]). From
F r o m these works
it transpires that a distinction must be made between
( n p = 0.9),
(16)
(16)
which o(J vv = volumetric
in which
e a n total stress.
volumetric total stress = m
mean
stress.
As a simplification,
simplification, one
one may
may use this expression
expression bboth
As
oth
relation and
and the
the fracture criterion.
criterion.
for the sstress-strain
t r e s s - s t r a i n relation
7. Hygral and thermal volume changes
The
The thermal and
a n d hygral strains, representing the
volumetric (mean) strains caused
by changes of tempercaused by
ature T and
and water
water content
content w, may be
calculated as
be calculated
ature
T
ev=f
aTdT '
To
ret,
t'),
J V ( t , t ' t')
) = 6=( ½6(1- v ) J ( t v, t )J(t,
'),
187
cvl = f "'k s d w ,
(17)
(17)
wo
in which the thermal dilatation coefficient aT
a r and
a n d the
shrinkage coefficient ks
k s depend, strictly viewed, on T
and
a n d w, and To,
T0, W
wo0 == initial values of T and
a n d w.
For
For hardened
h a r d e n e d cement paste, a depends
depends significantly
on w, and
a n d the average value is about
a b o u t aT
a y '"
= 25 X
x 1O1 0 - 66/ °0CC.
.
For siliceous or calcareous
calcareous aggregates, aT'"
a r - - 7 to 8 X
x
For
10
1 0 - 66 //0C.
° C . For
F o r concrete, the aT
a r values are generally
around
10x
a r o u n d aT'"
a T = 10
× 1O1 0 - 66 / °oCC.
. From
F r o m this it is clear that
d o m i n a t e the value for concrete.
again the aggregates dominate
Thus, the variations of aT
a r due to changes in w should be
manisfested only very weakly for concrete. Therefore,
Therefore,
one may normally assume for concrete, as an approxic o n s t a n t value, aT'"
a r - - 10 X
X 1O1 0 - 66// °oCC.. The scatter is
mate constant
1 0 0 ° C - 4 3 0 ° C , the secant
however large. For the range 100°C-430°C,
value for Harmarthy's
H a r m a r t h y ' s data [30] for calcareous concrete
188
Z.P.
z.P. Ba~ant
Baiant / Model
Nfodel for creep
creep and thermal shrinkage
shrinkaKe oj
o/crmcrete
cof/creTe
is 14 X
× 1O10 66// °0CC.. For
For Chicago concretes with Elgin
sand
sand and
and dolomite gravel, Cruz
Cruz and
and Gillen [31] find for
X 11O.47SoC
475°C the values 8.4
8 . 4 ×x1100 - 6 6/ °/ CoC and
and 11 ×
0 - 6 6/ /° C0 e.
.
For
For quartzite
quartzite gravel concretes up to SOO°c,
500°C, Zoldners'
X 1Obasalt and
[32,33] data
data indicates
indicates 16 ×
1 0 - 66 // °0Ce.. For
For basalt
and
calcareous
calcareous aggregate concretes, Sullivan's [34] data
data indicate
X 10 - 6/oC
X 10cate S.3
5.3 ×
6 / ° C and
and 8.3 ×
1 0 - 66/0C.
/°C.
While drying shrinkage
shrinkage depends
depends nonlinearly on pore
proportional to the loss in
humidity,
humidity, h, it is roughly proportional
water
water content
content [3S].
[35]. Thus,
Thus, the shrinkage
shrinkage coefficient k
may
may be considered
considered approximately
approximately as constant.
constant.
Many
Many measurements
measurements of length and
and volume changes
changes
at high temperatures
temperatures have been made. It is, however,
difficult to separate
separate the changes
changes caused
caused directly by
thermal
thermal expansion
expansion of solids and
and those caused
caused indirectly
by the water
water loss due to temperature
temperature increase. We will
therefore try to supplement
reported observations
supplement reported
observations by
logical arguments.
arguments.
We try to reconcile what
what is known
known about
about cement
cement
paste, aggregates, and
paste,
and concrete. First, we assume,
assume, as
before, the parallel coupling model. The normal
normal strain
strain
must
must then be the same for the aggregate and
and cement
cement
paste constituents,
paste
constituents, and
and may be expressed,
expressed, for cement
cement
paste,
A~=a~AT+A~H+AoJE~,
and for the ag= lX e L1 T + L1(~ + L1ac/ Ee, and
paste, as L1(
L1T
gregate constituent
constituent as L1(
Ac == lX%aA
T ++ L1a
A%/E~.
a/E a. Equilibrium between the normal
normal stresses resulting from the
L1ac +
cement
paste and
cement paste
and the aggregate requires that
that veA%
+
vaL1aa
%A% == O.
0. Eliminating L1a
A%a and
and L1a
Ao~e from the last three
equations,
equations, one obtains
obtains
,~
E~. v,: a~
I+-----E~ % a~
-
E~,~
a, AT
I+----
of eqs. (18) and
and (19).
(19), which is closer to the parallel
coupling model than
than to the series coupling model.
Consider
Consider now a typical concrete for which T',,1I',
q,/v~ ""
= 10
and
Ea/ Ec~ ""
and Ea/E
= 2 (for short-time
short-time deformation).
deformation). For the
cement
paste we may assume
cement paste
assume as the ultimate
ultimate shrinkage
shrinkage
strain
X 10 - I>~ [26]. The
strain L1(~
Ae s = 4000 ×
The shrinkage
shrinkage terms of
eqs. (18) and
and (19) then give.
give, for concrete.
concrete, the shrinkage
shrinkage
strain
predictions L1(
strain predictions
Ae 5s ""
= 200 X
× 10 - I>6 for the parallel coupling model, and
and L1(
Ac ss ""
= 400 xX 10 - I>6 for the series coupling model. The geometric mean of these two values is
roughly 300
1>. which is a reasonable
prediction
3 0 0 ×X 1010 -6,
reasonable prediction
for structural
Note that
structural concrete. Note
that we got a value about
about
ten-times less than
paste.
than that for cement paste.
Let us now apply this same argument
argument to estimate
estimate the
paste
drying shrinkage
shrinkage at SOO°c,
500°C. For
For hardened
hardened cement
cement paste
we have roughly .1(:
A~s == 0.01 [24]. We should
should also consider that, due to a large short-time
short-time creep, the effective
paste is less.
value of the elastic modulus
modulus for cement
cement paste
less, and
we take E
Ea/Ee
J E c ""
= 4. Then,
Then, eqs. (18) and
and (19) yield L1(s
& s ""
=
2S0
250 X
x 10 --66 for the parallel coupling, and
and L1(
Ac s =
--- 1000 X
×
10 - I>6 for the series coupling. The
The geometric mean of
these two values is L1(s
A~s == SOO
500 X
× 1010 66• which is about
about
paste.
twenty-times less than
than the value for cement
cement paste.
The
The foregoing two estimates
estimates may now be used to
separate the observed deformations,
deformations, reported
reported in the
separate
literature,
literature, into the thermal
thermal dilatation
dilatation and
and drying shrinkage contributions.
contributions. We demonstrate
demonstrate it in fig. 3 for the
test data
data by Cruz
Cruz and
and Gillen [31]. The
The theoretical thermal dilatations
dilatations without
without simultaneous
simultaneous drying shrinkage
shrinkage
(called also thermal
thermal shrinkage
shrinkage when it occurs due to
driving the water
water out by heating) are shown
shown as the
dashed
dashed lines. We see that
that these lines are much above the
observed
paste. but only slightly
observed curves for cement
cement paste,
E~ v~
+
1
AcSc = OtTAT+ A¢ s,
0.01
0.OI
(18)
(18)
r-----------------/~------__.
//
/
~__../. ¢.,
....------/"c..
1+----
thermal
thermol "",I
~el
/,0
Assuming,
Assuming, in turn,
turn, a series coupling model, the
both constituents
stresses
stresses in both
constituents are the same and
and their
strains
strains are added.
added. Then
Then one simply obtains
obtains
(OCaUa +
+ a~v~)/tT=vd~¢
s.
(lX.Va
lXcvJL1T=vcL1(~·
~
( f)
0
-6
E
0
o
6
0
too 2oo ~,,,~oo
6oo°c
600 0 e
a
~
A , ==%
aaAT +
,S)
L1(
V.lX.L1
+vVc~ ( alX ccL1A T + AL1(~)
=
=
~~'\-
.c2
c
0o
-
s I]
thermal .-'~"1
0
Z
(19)
(19)
total
The
between those given by eqs.
The correct values lie between
(18) and
and (19), and
and may be expected to be closer to those
given by eg.
eq. (18) which are always less in magnitude
magnitude
than
than those given by eq. (19). By experience, reasonable
reasonable
results
results seem to be obtained
obtained by taking a geometric mean
mean
-0.01
-O.OI
Fig. 3.
/
189
189
z.P.
Z.P. Baiant
Ba£ant / Model for creep
creep and thermal
thermal shrinkage of
of concrete
concrete
above the observed curves for concrete. Note that the
dashed-line extrapolations represent roughly straightline extrapolations of the initial trend of the measured
curve, which seems rea~,onable
reasonable from the physical point
of view.
Further
Further deductions can be made with regard to the
s . Assume that the
value of shrinkage coefficient k s'
water-cement ratio (by weight) is 0.50 and that the final
water loss due to drying at zero humidity is about 0.2 c
where c == weight of cement in unit volume of concrete.
v J v a :::
= 0.1 and the concrete weights 2300 kg/m),
k g / m 3, we
If vjva
then get the total water loss Llw
Aw == 0.2 X
X 0.1 X
× 2300 == 46
k g / m 3. Then, if the corresponding
corresponding total shrinkage is
kg/m).
6 , we obtain ks
Ac s == 300 Xx 1010 -6,
k s :::
-- 1010 -55 m)
m3/kg.
Ll£s
/kg. On the
other hand, for cement paste taken alone, we have
Llw
Aw = 0.2 X
× 2000 kg/m)
k g / m 3 = 400 kg/m),
k g / m 3, and then, if the
:::
total shrinkage
shrinkage is Ll£
Ac s == 4000 X
X 10 - 6,
6 we obtain
obtain k ss-3
10 - 55 m
m3/kg.
10/kg.
Let us now estimate the values for high temperature,
temperature,
500°C. The water loss is about
about 0.8 of 0.5c or O.4c,
0.4c,
say 500°e.
and the total drying shrinkage
shrinkage is roughly 0.01. This
and
3
k s == 1010 .55 m
m3/kg
yields ks
/kg for cement paste. For concrete
Aw is about
about the double of the previous value 46 kg/m
k g / m 33,,
Llw
and
and if Ll£s
A~s is then about
about double the value at room
temperature,
temperature, then we obtain
obtain again about
about the same value
arguments, it seems
of k ss . In consequence of all these arguments,
reasonable
assume
reasonable to assume
Ac s = k s A w ,
with k s = 10 -~ m3/kg.
and refer the interested reader to the preceding works.
Assuming, as in previous works, the Maxwell chain
representation, we may express the stress, or more precisely the effective
effective stress, for the porous material as a
sum of partial stresses corresponding
corresponding to Maxwell units
of various relaxation times, i.e.,
ov +,,pp = ~ov,
stress
8. Generalization to time-variable stress
applications, the stresses are not
In most practical
practical applications,
constant but
but vary in time;
time, not
not just
just because
because of a possible
possible
constant
of applied
applied loads
loads but
but also as a consequence
consequence of
of
variation of
variation
the
the creep
creep itself. To
To generalize the preceding
preceding formulation
formulation
to
to time-variable
time-variable stresses,
stresses, one
one can
can proceed,
proceed, under
under the
the
assumption
of linearity stated
stated at the outset,
outset, basically
basically in
assumption of
two different
different ways. Either
Either one
one can
can apply
apply directly the
two
of superposition,
superposition, which
which leads
leads to
to integral
integral equaequaprinciple of
principle
or one
one can
can determine
determine an
an equivalent
tions in time, or
tions
rate-type formulation.
formulation. The
The latter
latter is much
much more
more convenirate-type
ent
ent for finite element
element computations,
computations, particularly
particularly since
since
no previous
previous history
history needs
needs to be stored.
stored. The
The determinadeterminano
of a rate-type
rate-type formulation
formulation is exactly
exactly the
the same
same as for
tion of
tion
moderate temperatures,
temperatures, as
as described
described before
before in
in detail
detail
moderate
[11,14,15].
[11,14,15]. Therefore,
Therefore, we give here
here only
only a brief
brief outline
outline
E
,/,,
O D
(21)
in which i, j == subscripts
subscripts referring to cartesian coordix, (i = 1, 2, 3), aoVv,, ai~
o,jD= volumetric and deviatoric
nates X,
component
component of total stress tensor aij'
osj, and p.~ refers to the
number
number of the Maxwell unit in the chain model (p.
(/~ =
1, 2 ....
. . NN).
) . The volumetric and deviatoric stress-strain
stress-strain
relation then has the form
~v
i~
°v
+
°v
~ ar~r + ks~V,
3K~(t~)
3~V(te)
• D
Otj
OijD
2G.(te) ~ 3 n ° ( t e ~ '
(22)
(22)
(23)
(23)
in which £v,
cv, £~
c~ == volumetric and
and deviatoric components
components
c;j; K",
K;,, G"
G;, = bulk and shear moduli
of the strain tensor £ij;
for the individual units of the Maxwell chain; and
and 1/~
nv
1/~
age-dependent viscosities. Due to the condition of
rh,D =~ age-dependent
isotropy,
(20)
(20)
measurements should
should be carried
carried
However, further measurements
out
out to ascertain
ascertain that
that ks
k s is indeed approximately
approximately constant.
stant.
%D =
E~(t)
K,(t)
6(½- v~'
E~,(t)
G,(t)
2(1 + v ) '
(24)
(24)
in which E~,
E" = Young's
units of
Young's elastic moduli for the units
the Maxwell chain
chain model for uniaxial stress. FurtherFurthermore, the viscosities may be expressed
expressed as
1
3(1 - 2 , )
,v (,o) - ~wlw*T ~ - ( ~ ,
~°(te)-
6(½-v)
2(1 + v)
(25)
(25)
G~, v ;
~V(te)= "~'0p. ~./e),
(26)
(26)
which ~'
T"~ represent
represent the
the relaxation
relaxation times of
of the individin which
units at
at reference
reference temperature,
temperature, determined
determined
ual Maxwell units
[11,14,15). Together
Together with
with the equaequadescribed before
before [11,14,15].
as described
tions defining t~,
Ie' q~r,
<PT' fw,
/w, and
and gw (te == fflr
ff3Td/,
(4»,
tions
dt, eq. (4)),
(20-26) define the
the constitutive
constitutive relation.
relation. Efficient
eqs. (20-26)
of these
numerical algorithms
algorithms for a step-by-step
step-by-step solution
solution of
numerical
equations have
have been
been presented
presented before
before [36,11,14,15].
[36,11,14,15).
equations
possible generalization,
generalization, which
which is easy
easy in the
the case
As a possible
of the
the rate-type
rate-type formulation,
formulation, function
function gw could
could depend
depend
of
on #,
p., i.e., have
have different
different values
values for various
various units
units of
of the
the
on
Maxwell chain
chain model
model (with
(with different
different coefficients in
in eq.
Maxwell
190
z.P. Ba£ant
Bazant / Model for creep and thermal shrinkage of"
Z.P.
~)/ ("(}l1crele
com'rele
(12».
(12)). Test data would be needed.
needed, however.
however, to decide
that.
9. Conclusions
Based on the existing limited test data.
data, an approximate constitutive model for creep and hygral volume
D
changes of concrete at temperatures over 100
C can be
100°C
set up and its material parameters estimated. The model
covers constant as well as variable temperature and
specific water content, defines the effect of these variables on the creep rate and the rate of aging, and
describes the volume changes due to changes of temperature and of moisture content. The effect of pore pressure is taken into account through the effective stress
concept. Generalizations to multi
axial stresses and time
multiaxial
variable stresses can be carried out. The model should
allow more realistic analysis of nuclear reactor structures under hypothetical accident conditions, concrete
structures subjected to fire.
fire, concrete vessels for coal
gassification or liquefaction, etc.
Acknowledgement
Acknowledgement
Most of the present work has been carried out under
Subcontract 31-109-38-6473
National
31-109-38-6473 with Argonne National
Laboratory, Argonne, Illinois, USA, and was supervised
by Dr. J. Kennedy, Manager of Computational Mechanics Section in Reactor Analysis and Safety Division.
The underlying mathematical concepts have been developed before under previous National
National Science Foundation Grant
Grant ENG75-14848.
ENG75-14848.
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