fahmi_dry_cyc_impr.eps

Acta Polytechnica Vol. 52 No. 2/2012

Modeling of Concrete Creep Based on
Microprestress-solidification Theory

P. Havlásek, M. Jirásek

Abstract

Creep of concrete is strongly affected by the evolution of pore humidity and temperature, which in turn depend on
the environmental conditions and on the size and shape of the concrete member. Current codes of practice take that
into account only approximately, in a very simplified way. A more realistic description can be achieved by advanced
models, such as model B3 and its improved version that uses the concept of microprestress. The value of microprestress
is influenced by the evolution of pore humidity and temperature. In this paper, values of parameters used by the
microprestress-solidification theory (MPS) are recommended and their influence on the creep compliance function is
evaluated and checked against experimental data from the literature. Certain deficiencies of MPS are pointed out, and
a modified version of MPS is proposed.

Keywords: creep, concrete, compliance function, Kelvin chain, solidification, microprestress, finite elements.

1 Introduction
In contrast tometals, concrete exhibits creep already
at room temperature. This phenomenon results in a
gradual but considerable increase in deformation at
sustained loads, and needs to be taken into account
in thedesignandanalysis of concrete structures. The
present paper examines an advanced concrete creep
model, which extends the original B3 model [1] and
uses the concepts of solidification [5,6] and micro-
prestress [3,4,2]. The main objective of the paper
is to clarify the role of non-traditionalmodel param-
eters and provide hints on their identification. The
creep tests performed byKommendant, Polivka, and
Pirtz [8], Nasser and Neville [9], and Fahmi, Polivka
and Bresler [7] are used as a source of experimental
data, which are comparedwith the results of numer-
ical simulations. References [8] and [9] were focused
mainly on creep of sealed concrete specimens sub-
jected to elevated but constant temperatures. Refer-
ence [7] studied creep under variable temperature for
both sealed and drying specimens. The same refer-
ences were used in [2] to demonstrate the functional-
ity of theMicroprestress-SolidificationTheory,which
is the constitutive model described in Section 2. All

numerical computations have been performed using
thefinite elementpackageOOFEM[10–12]developed
mainly at the CTU in Prague by Bořek Patzák.

2 Description of the material
model

The complete constitutive model for creep and
shrinkage of concrete can be represented by the rhe-
ological scheme shown inFigure 1. It consists of (i) a
non-aging elastic spring, representing instantaneous
elastic deformation, (ii) a solidifying Kelvin chain,
representing short-term creep, (iii) an aging dashpot
with viscosity dependent on the microprestress, S,
representing long-term creep, (iv) a shrinkage unit,
representingvolume changes due to drying, and (v) a
unit representing thermal expansion. All these units
areconnected in series, and thus the total strain is the
sum of the individual contributions, while the stress
transmitted by all units is the same. Attention is fo-
cusedhere on themechanical strain, composed of the
first three contributions to the total strain, which are
stress-dependent. In the experiments, shrinkage and
thermal strains were measured separately on load-

σσ

E0
E1

η1

η2

ηM
SS

S
S

CS
ksh αT

εa εv εf εsh εT

solidification

Fig. 1: Rheological scheme of the complete hygro-thermo-mechanical model

Acta Polytechnica Vol. 52 No. 2/2012

free specimens and subtracted from the strain of the
loaded specimen under the same environmental con-
ditions. It should be noted that even after subtrac-
tion of shrinkage and thermal strain, the evolution of
mechanical strain is affected by humidity and tem-
perature. The reference case is so-called basic creep,
i.e. creep in sealed conditions and at room temper-
ature. Dry concrete creeps less than wet concrete,
but the process of drying accelerates creep. Elevated
temperature leads to faster cement hydration and
thus to faster reduction of compliance due to aging,
but it also accelerates the viscous processes that are
at the origin of creep and the process of micropre-
stress relaxation.
Solidification theory [5] reflects theprocessof con-

crete aging due to cement hydration, which leads to
the deposition of new layers of solidified hydration
products (mainly calcium-silicate-hydrate gels, C-S-
H). It is assumed that the creep ofC-S-H is described
by non-aging viscoelasticity, and aging is caused by
the growth in volumeof the solidifiedmaterial, which
leads to a special structure of the compliance func-
tion, reflectedbymodelB3. According to thismodel,
basic creep is described by a compliance function of
the form

Jb(t, t
′) = q1 + q2

∫ t
t′

ns−m

s − t′ +(s − t′)1−n
ds + (1)

q3 ln
[
1+(t − t′)n

]
+ q4 ln

t′
, t ≥ t′,

where t is the current time (measured as the age of
the concrete, expressed in days), t′ is the age at load
application, n = 0.1, m = 0.5, and q1, q2, q3 and
q4 are parameters determined by fitting of experi-
mental results or estimated from concrete compo-
sition and strength using empirical formulae. The
first (constant) term corresponds to the compliance
q1 = 1/E0 of the elastic spring in Figure 1, the sec-
ond and third terms to the solidifying viscoelastic
material (in numerical simulations approximated by
a solidifyingKelvin chain), and the fourth term to an
aging viscous dashpot with viscosity ηf(t)= t/q4.
Microprestress-solidification theory is an exten-

sion of the above model to variable humidity and
temperature. It replaces the explicit dependence of
viscosity ηf on time by its dependence on the so-
called microprestress, S, which is governed by a sep-
arate evolution equation. The microprestress is un-
derstoodas the stress in themicrostructuregenerated
due to large localized volume changes during the hy-
dration process. It builds up at very early stages of
microstructure formation and then is gradually re-
duced by relaxation processes. The microprestress is
considered to be much bigger than any stress acting
on themacroscopic level, and therefore it is not influ-
enced by the macroscopic stress. Additional micro-
prestress is generated by changes in internal relative

humidity and temperature. This is described by the
non-linear differential equation

dS
dt
+ ψS(T, h)c0S

2 = k1

∣∣∣∣d(T lnh)dt
∣∣∣∣ (2)

in which T denotes the absolute temperature, h is
the relative pore humidity (partial pressure of water
vapor divided by the saturation pressure), c0 and k1
are constant parameters, and ψS is a variable fac-
tor that reflects the acceleration ofmicroprestress re-
laxation at higher temperature and its deceleration
at lower humidity (compared to the standard condi-
tions). Owing to the presence of the absolute value
operator on the right-hand side of (2), additionalmi-
croprestress is generatedbybothdrying andwetting,
and by both heating and cooling, as suggested in [2].
The dependence of factor ψS on temperature and

humidity is assumed in the form

ψS(T, h) = exp

[
QS
R

(
1
T0

−
1
T

)]
· (3)

[
αS +(1 − αS)h2

]
where QS is the activation energy, R is the Boltz-
mann constant, T0 is the reference temperature
(room temperature) in absolute scale and αS is a pa-
rameter. Thedefault parametervalues recommended
in [2] are QS /R =3000 K and αS ≈ 0.1.
As discussed in [3], highmicroprestress facilitates

sliding in the microstructure and thus accelerates
creep. Therefore, the viscosity of the dashpot that
represents long-term viscous flow is assumed to be
inversely proportional to the microprestress. This
viscosity acts as a proportionality factor between the
flow rate and the stress. Themodel is thus described
by the equations

σ = ηf
dεf
dt

(4)

ηf =
1
cS

(5)

with a constant parameter c, which is not indepen-
dent and can be linked to the already introduced pa-
rameters. It suffices to impose the requirement that,
under standard conditions (T = T0 and h = 1) and
constant stress, the evolution of flow strain should
be logarithmic and should exactly correspond to the
last termof the compliance function (1) ofmodel B3.
A simple comparison reveals that c = c0q4. At the
same time, we obtain the appropriate initial condi-
tion for microprestress, which must supplement dif-
ferential equation (2). The initial condition reads
S0 = 1/(c0t0), where t0 is a suitably selected time
that precedes the onset of drying and temperature
variations.
As already mentioned, parameters q1, q2, q3 and

q4 are related to basic creep and can be predicted

Acta Polytechnica Vol. 52 No. 2/2012

from the composition of the concretemixture and its
average 28-day compressive strength using empirical
formulae [1]. The part of the compliance function
that contains q2 and q3 is related to viscoelastic ef-
fects in the solidifying part of the model. In numer-
ical simulations, this part of compliance is approx-
imated by Dirichlet series corresponding to a solid-
ifying Kelvin chain. The stiffnesses and viscosities
of individual Kelvin units can be conveniently deter-
mined from the continuous retardation spectrum of
the non-aging compliance function that describes the
solidifying constituent. The flow term representing
long-term creep is handled separately. For general
applications with variable environmental conditions,
it is necessary todetermineparameters c0 and k1 that
appear in the microprestress evolution equation (2)
and indirectly affect the flow viscosity. The model
response is also influenced by parameters QS /R and
αS, forwhichdefault valueshavebeen recommended.
As will be shown later, a better agreement with ex-
perimental results can be obtained if the default val-
ues are adjusted. Also, the assumption that changes
of T lnh contribute to the build-up ofmicroprestress
independently of their sign will be shown to be too
simplistic.

3 Numerical simulations
In this section, experimental data are compared to
results obtained with MPS theory, which reduces to
the standard B3 model in the special case of basic
creep. All examples concerning drying and thermally
induced creep have been run as a staggered problem,
with the heat and moisture transport analyses pre-
ceding themechanical analysis. The available experi-
mental data contained themechanical strains (due to
elasticity and creep), with the thermal and shrinkage
strains subtracted.

3.1 Experiments of Kommendant,
Polivka and Pirtz (1976)

At the time of writing, the original report was not
available to the authors of the present paper; there-
fore the experimental data as well as the recom-
mendedbasic creepparameters (q1 =20.0, q2 =70.0,
q3 = 5.6 and q4 = 7.0, all in 10

−6/MPa) were taken
from [2]. Under constant uniaxial load and constant
temperature, it is assumed that there are similar con-
ditions in the whole specimen. This allowed all com-
putations to be carriedout on just one finite element.
Figure 2 shows the experimental (points) and cal-

culated (curves) compliance functions for two dif-
ferent ages at loading and three different levels of
temperature. For the younger age at loading, t′ =
28 days, the computed curves provide an excellent
fit of the measured data for temperatures 23◦C and

43◦C. For the highest temperature, T = 71◦C, the
compliance is somewhat overpredicted for load dura-
tions from 1 week to 2 years. For the higher age at
loading, t′ = 90 days, the measured data are under-
predicted for all temperatures, but starting from load
durations of 10 days all the creep rates are predicted
very well. The reduced accuracy can be attributed
to the general tendency of the B3 model to overem-
phasize the effect of aging.

100

120

0.1 1 10 100 1000

co
m

p
lia

n
ce

J
[
1
0

-6
/M

P
a
]

duration of loading t-t’ [days]

ex. data T=23°C
T=23°C

ex. data T=43°C
T=43°C

ex. data T=71°C
T=71°C

100

120

0.1 1 10 100 1000

co
m

p
lia

n
ce

J
[
1
0

-6
/M

P
a
]

duration of loading t-t’ [days]

ex. data T=23°C
T=23°C

ex. data T=43°C
T=43°C

ex. data T=71°C
T=71°C

Fig. 2: Experimental data (Kommendant, Polivka and
Pirtz)andcomputedcompliance functions for ageat load-
ing t′ =28 days (top) and t′ =90 days (bottom)

In this example, the present results agree with
those presented in the original work [2], which veri-
fies the correct implementation. The calculated data
are independent of parameters c0 and k1. One can
obtain exactly the same curves as T = 23◦C in Fi-
gure 2 just by substituting parameters q1–q4, age at
loading t′ and the duration of loading t − t′ into the
full version of the B3 model; see equation 1.
TheoriginalB3model containsa simple extension

to basic creep at constant elevated temperatures; see
section 1.7.2 in [1]. The actual age at loading and
the load duration are replaced by the equivalent age

Acta Polytechnica Vol. 52 No. 2/2012

and the equivalent load duration, which evolve faster
at elevated temperatures. The calculated compliance
functions for default values of activation energies, as-
sumed water content w = 200 kg/m3 and average
28-day compressive strength f̄c =35 MPa are shown
in Figure 3. For loading at age t′ =28 days, the ini-
tial compliance is overestimated and for the highest
temperature 71◦C the rate of creep for longer load-
ing durations is too low. The compliance functions
for loading at age t′ = 90 days fit the experimental
data nicely except for the highest temperature. In
all cases the rate of creep is captured better by MPS
theory.

100

120

0.1 1 10 100 1000

co
m

p
lia

n
ce

J
[
1
0

-6
/M

P
a
]

duration of loading t-t’ [days]

ex. data T=23°C
B3 T=23°C

ex. data T=43°C
B3 T=43°C

ex. data T=71°C
B3 T=71°C

100

120

0.1 1 10 100 1000

co
m

p
lia

n
ce

J
[
1
0

-6
/M

P
a
]

duration of loading t-t’ [days]

ex. data T=23°C
B3 T=23°C

ex. data T=43°C
B3 T=43°C

ex. data T=71°C
B3 T=71°C

Fig. 3: Experimental data (Kommendant, Polivka and
Pirtz) andcomputedcompliance functions for ageat load-
ing t′ = 28 days (top) and t′ = 90 days (bottom) using
the original model B3

3.2 Experiments of Nasser and
Neville (1965)

Nasser and Neville studied the creep of cylindrical
concrete specimens subjected to three different lev-
els of temperature. In their experiments, all speci-

mens were sealed in water-tight jackets and placed
in a water bath in order to guarantee constant tem-
perature. At the age of 14 days the specimens were
loaded to 35%, 60% or 69% of the average compres-
sive strength at the time of loading; unfortunately,
just the lowest load level is in the range in which
concrete creep can be considered as linear. Paper [9]
does not contain enough information to allow the pa-
rameters of MPS theory to be predicted, but the
values q1 = 15, q2 = 80, q3 = 24 and q4 = 5 (all
in 10−6/MPa) published in [2] again provide good
agreement at room temperature, see the first graph
in Figure 4.

100

1 10 100 1000

co
m

p
lia

n
ce

J
[
1
0

-6
/M

P
a
]

duration of loading t-t’ [days]

ex. data T=21°C
standard

100

120

1 10 100 1000

co
m

p
lia

n
ce

J
[
1
0

-6
/M

P
a
]

duration of loading t-t’ [days]

ex. data T=71°C
standard
modified

100

120

1 10 100 1000

co
m

p
lia

n
ce

J
[
1
0

-6
/M

P
a
]

duration of loading t-t’ [days]

ex. data T=96°C
standard
modified

Fig. 4: Experimental data (Nasser andNeville) and com-
pliance functions for temperatures 21◦C, 71◦Cand96◦C

Acta Polytechnica Vol. 52 No. 2/2012

100

120

1 10 100 1000

co
m

p
lia

n
ce

J
[
1
0

-6
/M

P
a
]

duration of loading t-t’ [days]

ex. data T=21°C
B3 T=21°C

ex. data T=71°C
B3 T=71°C

ex. data T=96°C
B3 T=96°C

Fig. 5: Experimental data (Nasser andNeville) and com-
pliance functions obtained with the original model B3 for
temperatures 21◦C, 71◦C and 96◦C

For thehigher temperature, T =71◦C, the agree-
ment is goodup to 20days at loading, but afterwards
the computed rate of creep is too low. A remedy can
be sought in modifying the activation energy. Re-
duction of QS /R from the default value 3000 K to
the adjusted value of 2200K leads to an excellent fit;
see the curve labeled in Figure 4 as modified. Unfor-
tunately, the prediction for the highest temperature
(T =96◦C) is improved only partially.
Changes in activation energy have no influence

on the results when the temperature is close to the
room temperature. Before loading, the specimens
had been subjected to an environment at the given
temperature, which accelerated the hydration pro-
cesses in concrete, i.e. the maturity of concrete. In
other words, the higher the temperature, the lower
the initial compliance. On the other hand, for longer
periods of loading the higher temperature accelerates
the rate of bond breakages, which accelerates creep.
This justifies the shape of the obtained curve for the
medium temperature, which is different from the one
published in [2], where the initial compliance for this
temperature was higher than for the room tempera-
ture.
The compliance functions obtained with the B3

model are shown for all tested temperatures in Fi-
gure5. Again, defaultvalueswereused for theactiva-
tion energies, assumedwater content w =200 kg/m3

and compressive strength f̄c =35 MPa. Experimen-
tal data for the room temperature and for the high-
est temperature are captured nicely, but the compli-
ance function for T = 71◦C is overestimated (until
100 days of loadduration), and the final rate of creep
seems to be too low.

3.3 Experiments of Fahmi, Polivka
and Bresler (1972)

In these experiments, all specimens had the shape
of a hollow cylinder with inner diameter 12.7 cm,

outer diameter 15.24 cm and height 101.6 cm. The
weight ratio of the components of the concrete mix-
ture was water:cement:aggregates= 0.58 : 1 : 2.
From this we can estimate that the concretemixture
contained approximately 520 kg of cement per cu-
bic meter. The average 21-day compressive strength
was 40.3 MPa. Using CEB-FIP recommendations,
the 28-day strength can be estimated as 42.2 MPa.
The experiment was performed for four different his-
tories of loading, temperature and relative humidity.
The loading programs of the first two specimens are
summarized in Tables 1 and 2, the other two loading
programswith cyclic thermal loading are specified in
Tables 3 and 4.

Table 1: Testingprogramof the sealed specimenwith one
temperature cycle (Data set #1)

time duration T RH σ
[day] [◦C] [%] [MPa]

21 23 100 0
37 23 98 −6.27
26 47 98 −6.27
82 60 98 −6.27
10 23 98 −6.27
25 23 98 0

Table 2: Testing program of the drying specimen with
one temperature cycle (Data set #2)

time duration T RH σ
[day] [◦C] [%] [MPa]

18 23 100 0
14 23 50 0
37 23 50 −6.27
108 60 50 −6.27
10 23 50 −6.27
25 23 50 0

Table 3: Testing program of the sealed specimen sub-
jected to several temperature cycles (Data set #3). As-
terisks denote a section which is repeated 4×

time duration T RH σ
[day] [◦C] [%] [MPa]

21 23 100 0
35 23 98 −6.27
9 40 98 −6.27
5 60 98 −6.27
14 23 98 −6.27
7∗ 60 98 −6.27
7∗ 23 98 −6.27
7 60 98 −6.27
12 23 98 −6.27
40 23 98 0

Acta Polytechnica Vol. 52 No. 2/2012

Table 4: Testing program of the drying specimen sub-
jected to several temperature cycles (Data set #4). As-
terisks denote a section which is repeated 4×

time duration T RH σ
[day] [◦C] [%] [MPa]

18 23 100 0
14 23 50 0
33 23 50 −6.27
15 60 50 −6.27
14 23 50 −6.27
7∗ 60 50 −6.27
7∗ 23 50 −6.27
7 60 50 −6.27
13 23 50 −6.27
14 23 50 0

The four parameters of the B3 model describing
the basic creep, q1, q2, q3 and q4, were determined
from the composition of the concrete mixture and
from the compressive strength using empirical for-
mulae according to [1]. The result of this prediction
exceeded expectations; only minor adjustments were
necessary to get the optimal fit (see the first part
of the strain evolution in Figure 6). The following
values were used: q1 = 19.5, q2 = 160, q3 = 5.25
and q4 =12.5 (all in 10

−6/MPa). They differ signifi-
cantly from the values recommended in [2], q1 =25,
q2 =100, q3 =1.5 and q4 =6, which do not provide
satisfactory agreement with experimental data.
MPS theory uses three additional parameters, c0,

k1 and c, but parameter c should be equal to c0q4.
It has been found that the remaining parameters c0
and k1 are not independent. What matters for creep
is only their product. For different combinations of
c0 and k1 giving the same product, the evolution of
microprestress is different but the evolution of creep
strain is exactly the same. Sincemicroprestress is not
directlymeasurable, c0 and k1 cannot (andneed not)
be determined separately. In practical computations,
k1 can be set to a fixed value (eg. 1 MPa/K), and
c0 can be varied until the best fit with experimen-
tal data is obtained; in all the following figures c0 is
specified inMPa−1day−1. All other parameterswere
used according to standard recommendations.
A really good fit of the first experimental data

set (98% relative humidity, i.e., h = 0.98) was ob-
tained for c0 =0.235MPa

−1day−1; seeFigure6. The
agreement is satisfactory except for the last interval,
which corresponds to unloading. It is worth noting
that the thermally induced part of creep accounts for
more than a half of the total creep (compare the ex-
perimental data with the solid curve labeled basic in
Figure 6). Unfortunately, with default values of the
other parameters, the same value of c0 could not be
used to fit experimental data set number 2, because

it would have led to overestimation of the creep (see
the dashed curve in Figure 7). In the first loading in-
terval of 37 days, creep takes place at room tempera-
ture and the best agreementwould be obtained with
parameter c0 set to 0.940MPa

−1day−1; see the dash-
dotted curve in Figure 7. However, at the later stage
when the temperature rises to 60◦C, the creepwould
be grossly overestimated. A reasonable agreement
during this stage of loading is obtained with c0 re-
duced to0.067MPa−1day−1 (solid curve inFigure7),
but then the creep is underestimated in the first in-
terval in Figure 6 left. Raising parameter αS from
its recommendedvalue 0.1 to 0.3 (short-dashed curve
in Figure 7 right) has approximately the same effect
as decreasing c0 from 0.235 to 0.067 MPa

−1day−1.
Parameter αS controls the effect of reduced humid-
ity on the rate of microprestress relaxation, and its
modification has no effect on the response of sealed
specimens.

200

400

600

800

1000

1200

1400

1600

1800

0 50 100 150 200

m
e

ch
a

n
ic

a
l s

tr
a

in
[

1
0

-6
]

age of concrete [day]

experimental data
basic

c0 = 0.067
c0 = 0.235
c0 = 0.671

Fig. 6: Mechanical strain evolution for sealed specimens,
with relative pore humidity assumed to be 98%, loaded
by compressive stress 6.27 MPa at time t′ =21 days

200

400

600

800

1000

1200

1400

1600

1800

0 50 100 150 200

m
e

ch
a

n
ic

a
l s

tr
a

in
[

1
0

-6
]

age of concrete [day]

experimental data
c0 = 0.067
c0 = 0.235

c0 = 0.235 modified
c0 = 0.940

Fig. 7: Mechanical strain evolution for drying specimens
at 50% relative environmental humidity, loaded by com-
pressive stress 6.27 MPa at time t′ =32 days

Acta Polytechnica Vol. 52 No. 2/2012

500

1000

1500

2000

2500

3000

0 20 40 60 80 100 120 140 160 180

m
e

ch
a

n
ic

a
l s

tr
a

in
[

1
0

-6
]

age of concrete [day]

experimental data
c0 = 0.235

basic

Fig. 8: Mechanical strain evolution for sealed speci-
men, loaded by compressive stress 6.27 MPa at time
t
′ = 21 days and subjected to cyclic variations of tem-
perature

1000

2000

3000

4000

5000

6000

0 20 40 60 80 100 120 140 160 180

m
e

ch
a

n
ic

a
l s

tr
a

in
[

1
0

-6
]

age of concrete [day]

experimental data
c0 = 0.235

Fig. 9: Mechanical strain evolution for drying speci-
mens, loaded by compressive stress 6.27 MPa at time
t
′ = 32 days and subjected to cyclic variations of tem-
perature

For the last two testing programsdescribed inTa-
bles 3and4, the agreementbetween the experimental
and computed data is reasonable only until the end
of the second heating cycle (solid curves in Figure 8
and Figure 9). For data set 3, the final predicted
compliance exceeds the measured value almost twice
(Figure 8), for data set 4 almost five times (Figure 9).
In order to obtain a better agreement, parameter c0
would have to be reduced, but this would result in an
underestimation of the creep in the first two testing
programs. The experimental data showthat the tem-
perature cycles significantly increase the creep only
in the first cycle; during subsequent thermal cycling
their effect on creepdiminishes. Therefore it couldbe
beneficial to enhance the material model by adding
internal memory, which would improve the behavior
under cyclic thermal loading, while the response to
sustained loading would remain unchanged.

Another deficiency of the model is illustrated by
the graphs in Figure 10. They refer to the first set
of experiments. As documented by the solid curve
in Figure 6, a good fit was obtained by setting pa-
rameter c0 =0.235 MPa

−1day−1, assuming that the
relative pore humidity is 98%. The pores are ini-
tially completely filled with water; however, even if
the specimen isperfectly sealed, the relativehumidity
decreases slightly due to the water deficiency caused
by the hydration reaction. This phenomenon is re-
ferred to as self-desiccation.

200

400

600

800

1000

1200

1400

1600

1800

0 50 100 150 200

m
e

ch
a

n
ic

a
l s

tr
a

in
[

1
0

-6
]

age of concrete [day]

experimental data
RH = 95%
RH = 96%
RH = 97%
RH = 98%
RH = 99%

RH = 100%
basic

Fig. 10: Mechanical strain evolution for sealed speci-
mens, loaded by compressive stress 6.27 MPa from age
21 days, with the assumed relative humidity of the pores
varied from 95% to 100%. Parameters of MPS theory:
k1 =1 MPa/K, c0 =0.235 MPa

−1 day−1

The problem is that the exact value of pore hu-
midity in a sealed specimen and its evolution in time
are difficult to determine. In simple engineering cal-
culations, a constant value of 98% is often used. Un-
fortunately, the response of the model is quite sensi-
tive to this choice, and the creepcurvesobtainedwith
other assumed values of pore humidity in the range
from95% to 100%would be different; see Figure 10.
The source this strong sensitivity is the assumption
that the instantaneously generated microprestress is
proportional to the absolute value of the change of
T ln(h); see the right-hand side of (2). Rewriting (2)
as

dS
dt
+ ψS(T, h)c0S

2 = k1

∣∣∣∣lnhdTdt +
T

dh
dt

∣∣∣∣ (6)
we can see that at (almost) constant humidity close
to 100%, the right-hand side is proportional to the
magnitude of the temperature rate, with proportion-
ality factor k1| ln(h)| ≈ k1(1− h). For instance, if the
assumedhumidity is changed from99% to 98%, this
proportionality factor is doubled.

Acta Polytechnica Vol. 52 No. 2/2012

200

400

600

800

1000

1200

1400

0 50 100 150 200

m
e
ch

a
n
ic

a
l s

tr
a
in

[
1
0

-6
]

age of concrete [day]

experimental data
original MPS

improved MPS

Fig. 11: Mechanical strain evolution for sealed speci-
mens loaded by compressive stress 6.27 MPa at time
t
′ =21 days

500

1000

1500

2000

0 50 100 150 200

m
e
ch

a
n
ic

a
l s

tr
a
in

[
1
0

-6
]

age of concrete [day]

experimental data
original MPS
κT = -ln(0.98)

κT adjusted
improved

Fig. 12: Mechanical strain evolution for drying speci-
mens loaded by compressive stress 6.27 MPa at time
t
′ =32 days

4 Improved material model
and its validation

As a simple remedy to overcome these problems, the
microprestress relaxation equation (2) is replaced by

dS
dt
+ ψS(T, h)c0S

2 = k1

∣∣∣∣ Th
dh
dt

− κT kT(T)
dT
dt

∣∣∣∣ (7)
with kT(T) = e

−cT (Tmax−T) (8)

in which κT and cT are new parameters and Tmax is
themaximumreached temperature. With κT =0.02,
the creep curves in Figure 10 plotted for different
assumed pore humidities would be almost identical
with the solid curve, which nicely fits the experimen-
tal results. The introduction of a newparameter pro-
videsmore flexibility, which is needed to improve the
fit of the second testing program in Figure 7, with
combined effects of drying and temperature varia-

tion. For sealed specimens and monotonous thermal
loading, only the product c0k1κT matters, and so the
good fit in Figure 7 could be obtained with different
combinations of κT and c0.
The resultsare shown inFigures11and12 for sus-

tained thermal loading (data sets 1 and2)and inFig-
ures 13 and 14 for cyclic thermal loading (data sets 3
and4). Default values of parameters αS, αR, αE and
activation energies are used. In these plots, data se-
ries labeled original MPS show results obtainedwith
standard MPS.
The data series κT = − ln(0.98) were obtained

with c0 =0.235 MPa
−1day−1, k1 =1 MPa/K, κT =

0.020203 and cT = 0. The data series κTadjusted
correspond to parameters c0 = 0.235 MPa

−1day−1,
k1 = 4 MPa/K, κT = 0.005051 and cT = 0. Note
that in the case of constant relative humidity (Fig-
ures 11 and 13) these series coincide with the data
series original MPS.

200

400

600

800

1000

1200

1400

0 20 40 60 80 100 120 140 160 180

m
e
ch

a
n
ic

a
l s

tr
a
in

[
1
0

-6
]

age of concrete [day]

experimental data
original MPS

improved

Fig. 13: Mechanical strain evolution for sealed speci-
men, loaded by compressive stress 6.27 MPa at time
t
′ = 21 days and subjected to cyclic variations of tem-
perature

500

1000

1500

2000

2500

0 50 100 150 200

m
e
ch

a
n
ic

a
l s

tr
a
in

[
1
0

-6
]

age of concrete [day]

experimental data
original MPS
κT = -ln(0.98)

κT adjusted
improved

Fig. 14: Mechanical strain evolution for drying speci-
mens, loaded by compressive stress 6.27 MPa at time
t
′ = 32 days and subjected to cyclic variations of tem-
perature

Acta Polytechnica Vol. 52 No. 2/2012

Thebest agreementwith experimental data is ob-
tainedwith c0 =0.235MPa

−1day−1, k1 =4MPa/K,
κT =0.005051and cT =0.3K

−1; these series are la-
beled improved. In Figure 11, only a small change
can be observed compared to data series original
MPS; these differences arise when the temperature
ceases to be monotonous. For the sealed specimen
(Figure 11), this change is detrimental, but looking
at Figures 13 and 14, this deterioration is negligible
compared to the substantial improvement in the case
of cyclic thermal loading.

5 Conclusions
The material model based on MPS theory has been
successfully implemented into the OOFEM finite el-
ement package, and has been used in simulations of
concrete creep at variable temperature andhumidity.
MPS theory performswell for standard sustained

levels of temperature and load levelswithin the linear
range of creep, provided that the activation energy
is properly adjusted. For higher sustained tempera-
tures (above 70◦C) the experimental data are repro-
duced with somewhat lower accuracy.
For sealed specimens subjected to variable tem-

perature, the results predicted by MPS theory are
very sensitive to the assumed value of relative pore
humidity (which is slightly below 100% due to self-
desiccation). In order to overcome this deficiency, a
modified version of themodel has been proposed and
successfully validated. Excessive sensitivity to the
specific choice of relative humidity has been elimi-
nated. Also, it has become easier to calibrate the
model because thermal andmoisture effects on creep
are partially separated.
The original MPS theory grossly overestimates

creep when the specimen is subjected to cyclic tem-
perature. A new variable kT has been introduced in
order to reduce the influence of subsequent thermal
cycles on creep. This modification does not affect
creep tests in which the evolution of temperature is
monotonous.

Acknowledgement

Financial support for this work was provided by
projects 103/09/H078 and P105/10/2400 of the
Czech Science Foundation. The financial support is
gratefully acknowledged.

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Petr Havlásek
E-mail: petr.havlasek@fsv.cvut.cz
Milan Jirásek
E-mail: milan.jirasek@fsv.cvut.cz
Department of Mechanics
Faculty of Civil Engineering
Czech Technical University in Prague
Thákurova 7, 166 29 Prague 6