A homogenization model for concrete creep

roots of Eqs. (11). From these relations, the expressions 
of ( )homMTk t and ( )
hom
MT t are easily delivered by applying (10): 
( )hom 0
1
k
k
i
t
N M
MT i
i
k k k et

+ −
=
= +  , ( )hom 0
1
i
t
N M
MT i
i
et

  
+ −
=
= +  (13) 
with 
ik and i the moduli associated with the relaxation times 
k
i and i
 . As already 
reported (see e.g. [11,12]), we observe that the number of relaxation times defining the 
macroscale behavior is higher than that at microscale, indicating an enrichment of the 
relaxation spectra due to the homogenization procedure. 
It should be pointed out that the above exact analytical formulation of the inverse LC 
problem has been obtained in [40] in the case of two-phase matrix/inclusion 
microstructures and by applying the MT scheme. Thus, the proposed procedure may 
be viewed as a generalization to viscoelastic multiphase materials of the MT scheme. 
Note also that the rather simple expressions in Eqs. (12)-(13) relate mainly to the 
microstructure with spherical inclusions and to the use of the MT scheme. However, it 
is shown in [42] that exact results can also be obtained for the microstructure with 
ellipsoidal inclusions in the limiting cases of penny-like and needle-like shapes. The 
more general case of ellipsoidal shapes with variable aspect ratio is currently under 
investigation. 
3. RESULTS AND DISCUSSIONS 
3.1 Simulation results 
We detail in this section the procedure adopted for identifying the parameters of 
the viscodamage model. As explained above, for simplification the parameters of the 
damage evolution law are taken to be constant, which implies that they can be 
identified on a classical (static) compression test. This constitutes then the first step of 
the characterization procedure of the parameters. The following values are retained for 
the material as they provide a classical form of stress-strain curve in the case of a 
compression uniaxial test: 0 8cA .= and 439cB = [42]. The second step is to identify 
the microscopic viscoelastic creep parameters. These parameters are adjusted on the 
macroscopic experimental strains (both longitudinal and transversal) by applying the 
standard least-square method to minimize the differences between numerical results 
and test data. To reduce the domain of identification for the relaxation times 1τ
m and 
2τ
m , we have chosen to limit its values such that they correspond to the test duration, 
i.e. inferior to 40 days. Further, we propose to fix 1τ
m and to carry out the identification 
over the other parameters such as 2τ
m , and then to repeat this procedure for increasing 
values of 1τ
m . The first value of 1τ
m is set to 1 day, and the subsequent ones are 
incremented by 1 day. The parameters retained correspond to the minimal overall error 
calculated between the simulated and experimental results and in the sense of the least-
squared method. This identification process has been carried out with the aid of the 
Matlab software. 
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The resulting model is implemented in the finite element code Cast3M developed at 
CEA [49]. The Figure 13 shows the longitudinal and transversal total strains obtained 
numerically (curves) and experimentally (symbols, [48]). Since the parameters have 
been identified on these data, we observe a good concordance between both results. 
This also indicates that the number of chains in the microscopic Maxwell models is 
sufficient for characterizing the material on the duration of the tests. The Figure 14 
reports the evolution of the Poisson’s ratio obtained from Eq. (1) involving both 
longitudinal and transversal measured total strains. We observe that the experimental 
Poisson’s ratio rapidly decreases from about 0.24 to 0.19 the first 4-5 days, then much 
more slowly to reach the value of 0.175 at the end of the test (42 days). This evolution 
confirms the need of different time functions for the macroscopic bulk and shear 
moduli. 
Figure 13: numerical (curves) and experimental (symbols, [48]) evolutions of 
longitudinal and transversal total strains. 
This progressive damage growth is in fact a consequence of the time-dependency of 
the Poisson’s ratio or equivalently of the non-proportionality between the evolutions of 
longitudinal and transversal strains. According to Figure 13 and Figure 15, in this 
simulation case the creep strains originate essentially from the viscoelastic nature of 
the mechanical parameters, and to a lesser extent from the damage evolutions. Another 
remark is that, even for the moderate and widely used creep compressive loading level 
of nearly 30% of the material strength, a non-negligible damage appears 
instantaneously, i.e. in static condition. It seems that this damage is often disregarded 
in the identification procedure of most models. Finally, it appears that for such loading 
condition and generalized Maxwell models, damage stabilizes, as well as total and 
pseudo-strains, at constant values for times going to infinity. 
Time (days) 
T
o
ta
l 
st
ra
in
s 
(-
) 
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Figure 14: evolutions of the Poisson’s ratio calculated from the experimental total 
strains of Figure 13. 
In Figure 15 are plotted the damage evolutions calculated with the model during the 
creep test. The variations range from about 0.18 at 0t = to 0.22 at 42 days, indicating 
that damage initiates instantaneously due to the application of the loading, and then 
propagates moderately during the test. 
Figure 15: calculated evolutions of the damage variable during the creep test. 
3.2 Discussions 
We examine and detail in this section the response of the model for different 
configurations of the material microstructure, in terms of strains, damage and 
mechanical properties evolutions during a creep test. We choose to analyze the effects 
of a variation of pore volume fraction in the simulation of the creep test as previously 
presented. In order to simplify the analyses and due to a lack of experimental data, the 
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damage parameters are kept unchanged for all the configurations. This is clearly a 
strong hypothesis which would deserve further investigations, because each material is 
expected to have a specific damage evolution law, as a function of the pseudo-strains. 
In addition to the initial porosity of the material 0 42. = , two different and well 
separated values have been considered: 0 20. = and 0 60. = . The lower one can be 
viewed as typical of a high performance cement paste, whereas the greater one is 
characteristic of severely degraded (i.e. by leaching) cement pastes (e.g. [50]). It 
should be noticed that the value 0 60. = is retained mainly in a demonstrative and 
‘academic’ purpose, since in practice it is doubtful that such material composition 
would be realistic. Indeed even for a strongly degraded material, the remaining part of 
the solid phase (matrix) would probably have a behavior different from the initial one, 
due to the effects of the degradation phenomena. Moreover, the high particle volume 
fraction of 60% is out of the traditionally accepted validity range of the MT scheme 
(see e.g. [51]). Considering that all the other model parameters are conserved, the 
microstructures then differ only by the volume fraction of their constituents (i.e. matrix 
and pores), whose respective behavior is the same. Error! Reference source not 
found. shows the evolutions of both bulk and shear moduli for the 3 configurations 
when subjected to a creep loading (recall that the overall moduli are independent of the 
loading), and we clearly notice that these parameters are as expected strongly affected 
by  . For all cases, both moduli decrease rapidly the first 3-4 days, then much more 
progressively. 
Time (days) Time (days) 
S
h
ea
r 
m
o
d
u
lu
s 
(P
a
) 
B
u
lk
 m
o
d
u
lu
s 
(P
a
)  = 60% 
 = 42% 
 = 20% 
 = 60% 
 = 42% 
 = 20% 
Figure 16: numerical evolutions of bulk (left) and shear (right) moduli for different 
volume fractions of porosity . 
We examine now the damage response of the model to the creep tests while 
considering the same compressive loading for the 3 microstructures. 
Time (days) Time (days) 
T
ra
n
sv
er
sa
l 
st
ra
in
s 
(-
) 
 = 60% 
 = 42% 
 = 20% 
L
o
n
g
it
u
d
in
a
l 
st
ra
in
s 
(-
)  = 60% 
 = 42% 
 = 20% 
Figure 17 presents the evolutions of both longitudinal (left) and transversal (right) total 
strains simulated for the 3 porosities. We observe, as anticipated from Error! 
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Reference source not found., that the magnitude of the strains increases for higher 
porosity values. This feature results largely from the decrease of the mechanical 
properties, and also from the time-dependant damage propagation (see below). From 
this viewpoint it is instructive to examine the time evolutions of the pseudo-strains 
corresponding to the studied cases, as they govern the damage growth. 
Time (days) Time (days) 
T
ra
n
sv
er
sa
l 
st
ra
in
s 
(-
) 
 = 60% 
 = 42% 
 = 20% 
L
o
n
g
it
u
d
in
a
l 
st
ra
in
s 
(-
)  = 60% 
 = 42% 
 = 20% 
Figure 17: numerical evolutions of longitudinal (left) and transversal (right) total 
strains for different volume fractions of porosity . 
Time (days) Time (days) 
T
ra
n
sv
er
sa
l 
st
ra
in
s 
(-
) 
L
o
n
g
it
u
d
in
a
l 
st
ra
in
s 
(-
) 
Figure 18 presents the longitudinal (left) and transversal (right) total strains obtained 
for the 3 cases. We remark that, although the increase of applied loading is relatively 
small when comparing the different cases, the strains vary significantly. The rupture 
happens at about 41, 29 and 17 days for the loads equal to 83, 84 and 85% of 
cf , 
respectively. After a rapid increase in magnitude the first 5 days followed by a more 
progressive evolution, the strain curves shape exhibits an inflection point which marks 
the onset of a quick rise leading to the overall fracture. This rupture happens suddenly 
as an unstable process. 
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Time (days) Time (days) 
T
ra
n
sv
er
sa
l 
st
ra
in
s 
(-
) 
L
o
n
g
it
u
d
in
a
l 
st
ra
in
s 
(-
) 
Figure 18: numerical evolutions of longitudinal (left) and transversal (right) total 
strains for different values of compressive creep loading. 
4. CONCLUSION 
We have presented in this study a creep model for concrete. The viscoelastic 
model results from a homogenization procedure, where the MT scheme is applied in 
the LC space to estimate the evolutions of the linear viscoelastic bulk and shear moduli 
of the material. This procedure relies on the material representation as a composite 
consisting of spherical elastic inclusions (aggregates and/or voids) embedded in a 
linear viscoelastic matrix whose behavior is assumed to be ruled by a generalized 
Maxwell model. One key feature of the model is that analytical expressions of the 
time-dependant effective mechanical properties are derived in the time space as a 
function of the properties of the components, and that these expressions are exact in 
some simple cases. The model is then applied to the simulation of classical creep tests 
performed on cement paste specimens, for which accurate experimental data are 
available in the literature. For simplicity, the cement paste is represented as a 
viscoelastic matrix spherical voids, which is described by a 3-chain generalized 
Maxwell model. A good agreement is obtained between experimental and numerical 
results of creep strains upon an adequate identification of the model parameters. A 
study in progress takes advantage of the viscoelastic multi-scale model to investigate 
the effects of the shape and volume fraction of the aggregates and pores on the 
macroscopic properties of the material. 
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