On the effective viscosity of fresh concrete: A homogenization approach
Tóm tắt On the effective viscosity of fresh concrete: A homogenization approach: ...f interest to note that the first two terms on the right side of Eq. (4) corresponds to the Einstein’s model. Fig. 3 shows the evolution of the normalized effective viscosity µGSC/µ f estimated by the GSC (Eq. (2)) together with its second order series expansion (Eq. (4)) in comparison with Einst... 9 R O X P H I U D F W L R Q R I S D U W L F O H V 1 R U P D O L ] H G Y L V F R V L W \ ' D W D R I I U H V K F H P H Q W 6 W U X E O H D Q G 6 X Q 7 R X W R X H W D O ...Chemistry, 53, (7), (1949), pp. 1042–1056. https://doi.org/10.1021/j150472a007. [7] G. F. Eveson, S. G. Ward, and R. L. Whitmore. Classical colloids. Theory of size distri- bution; paints, coals, greases, etc. Discussions of the Faraday Society, 11, (1951), pp. 11–14. https://doi.org/10.1039/DF9...
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nding
separately in fluid but they locally pasted together, event at small or medium volume
fraction, to form alternative composite particles those block inside a part of the fluid
phase. Therefore, a suspension with cohesive particles of volume fraction φ can be con-
sidered as a suspension with alternative particles of volume fraction φ+ φb, where φb the
total volume of the blocked fluid. Theoretically, φb must be smaller or equal to 1− φmax
that is 0.36(φ+ φb) for a packing of mono size sphere.
294 Tuan Nguyen-Sy, Duong Nguyen-The
It is important to remark that such assumption allows reducing the volume fraction
of the EFZ to zero. Therefore the GSC can be employed for the mixture of alternative
particles and remaining (non blocked) fluid. Fig. 7 shows that the results obtained by
considering a minimum possible value of φb that is φb = 0 provides a lower bound of the
experimental data. A maximum value φb = 0.36(φ+φb), i.e. φb = 0.56φ, (with mono-size
assumption for cement particles) seems provides a good fit with the experimental data.
9 R O X P H I U D F W L R Q R I S D U W L F O H V
1
R U
P
D O
L ]
H G
Y
L V
F R
V L
W \
' D W D R I I U H V K F H P H Q W
6 W U X E O H D Q G 6 X Q
7 R X W R X H W D O
8 S S H U E R X Q G E O R F N H G
S R U R V L W \
8 S S H U E R X Q G E O R F N H G
S R U R V L W \
Fig. 7. Effective viscosity of cement: a comparison of the homogenization scheme
and experimental data of fresh cement
3.2. The second scale: fresh concrete
A comparison with experimental data shows that the improved GSC scheme (IGSC),
that was presented in Section 2.3, works very well for fresh concrete at the second scale
that is a mixture of fresh cement and gravels (Fig. 8). That method assumes a spherical
shape of the EFZ and employs the Eshelbys solution to estimate the local strain rate of
the EFZ. However analytical results are very cumbersome.
Let consider a more simple assumption such that the average localization strain rate
of the fluid phase located in the coated phase surrounding the gravel particles equal
to that of the EFZ. This assumption allows using the solution of the localization factor
obtained for the coated fluid phase of the composite sphere for the EFZ. We note by B1
the localization factor of the particles and B2 the localization factor of the coated fluid and
the EFZ. Using Eq. (1), to calculate the effective viscosity of the suspension, we obtain
µhom = lim
µp→∞
[
φµpB1 + (1− φ)µ f B2
]
. (5)
The closed-form solution to this problem is
µ
µcement
=
−b−√b2 − 4ac
a
, (6)
On the effective viscosity of fresh concrete: A homogenization approach 295
9 R O X P H I U D F W L R Q R I J U D Y H O V
5
H O
D W
L Y
H
Y L
V F
R V
L W \
R
I F
R Q
F U
H W
H
D Q
G
F H
P
H Q
W 7 R X W R X H W D O
, P S U R Y H G * 6 &
, P S U R Y H G * 6 & Z L W K D Q
D V V X P S W L R Q R I X Q L I R U P
V W U H V V I L H O G L Q W K H ( ) =
Fig. 8. Relative viscosity between concrete and cement versus the volume fraction of gravels:
a comparison between the present model and data of Toutou et al. [22]
where the parameters a, b and c are defined by
a = − 24
5
(
δ2 + δ+ 1
)
(δ− 1)4
(
4δ6 + 16δ5 + 40δ4 + 55δ3 + 40δ2 + 16δ+ 4
)
,
b = − 24δ3(δ− 1)
(
2δ8 + 4δ7 + 6δ6 + 13δ5 + 20δ4 + 13δ3 + 6δ2 + 4δ+ 2
)
φmax
+
6
5
(δ− 1)
(
32δ11 + 64δ10 + 96δ9 + 36δ8 − 24δ7 − 28δ6 + 168δ5
+ 339δ4 + 174δ3 + 9δ2 + 6δ+ 3),
c = 3δ3
[
56δ3(δ+ 1)
(
δ2 − 1)+ (16δ7 + 19) (δ2 + δ+ 1)] φmax
− 6
5
(δ− 1)
(
16δ11 + 32δ10 + 48δ9 + 88δ8 + 128δ7
+ 56δ6 + 84δ5 + 187δ4 + 122δ3 + 57δ2 + 38δ+ 19),
(7)
with δ3 = φ/φmax and φmax = 1− VEFZ. For the particular case with no EFZ zone, i.e.
VEFZ = 0, the solution (6) is simplified to the solution (2) of the GSC. Note that φmax =
0.64 for a packing of sphere. A comparison of the solution (6) with experimental data of
fresh concrete [22] as well as with the improved GSC presented in Section 2.3 is shown
on Fig. 8. In this figure, the viscosity of fresh concrete is normalized to the viscosity
of the fresh cement paste so the input parameters such as the viscosity of fluid and the
porosity of concrete are not required for the simulation. The only parameter needed is
the volume fraction of gravels. Of course, the viscosity of fluid is needed for determining
the absolute viscosity of the mixture. Even though a very simplified assumption of the
mechanical field in the EFZ is considered, the solution (6) fit very well with data and the
model of Section 2.3. More importantly, solution (6) is explicit and then very easy to use
in practice.
296 Tuan Nguyen-Sy, Duong Nguyen-The
4. CONCLUSIONS
Two homogenization concepts are proposed to model fresh concrete at two scales:
fresh cement scale and gravel scale. At the first scale, fresh cement is regarded as a sus-
pension of cohesive particles in viscous fluid. The cohesive particles are assumed to
locally pasted together to form alternative particles, each of them is a particle packing
containing blocked fluid. We consider a mono suspension and an ideal polydisperse sus-
pension (such as the volume fraction of the blocked zone tends to zero) and we observed
that the ideal polydisperse suspension provide an under bound while the mono size sus-
pension correspond to a upper bound. Most of experimental data fit with the mono size
assumption but some point lay in between the bounds.
At the second scale, the mixture between fresh cement and gravel is modeled by
an improved GSC. Analytical solutions are derived considering a simplified assumption
of uniform mechanical field in the extra fluid zone (EFZ). A mono size suspension for
gravels is a good approximation and a comparison with experimental data show that
such assumption allows the model to fit very well with data.
The workflow of the present homogenization model for the prediction of the effective
viscosity of fresh concrete is resumed as following:
(1) Input: volume fraction of gravels φg, volume fraction of cement particles φc0 and
viscosity of fluid µ f .
(2) First scale: compute the relative volume fraction of cement in cement-fluid mix-
ture (fresh cement) φc = φc0/(1 − φg) then compute the volume fraction of the local
cement particle parkings φcp = φc/0.64. Introducing the volume fraction φcp together
with the input viscosity of the fluid phase µ f in Eqs. (2) and (3) to compute the effective
viscosity of fresh cement.
(3) Second scale: Once the viscosity of fresh cement is obtained from the first scale
and the volume fraction of gravel particles is given by input data, we can compute the
final effective viscosity of fresh concrete using Eqs. (6) and (7).
The dependence of effective viscosity on the thixotropic and curing behavior of cement-
based materials are out of the scope of the present paper that focus on a homogenization
method and the effect of the microstructure of the suspension.
ACKNOWLEDGMENT
This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 107.01-2016.17. The authors express
deep thanks to Reviewers for their suggestions and comments.
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APPENDIX
The mathematical development of the proposed closed form solutions
The homogenization method consists to estimate the effective shear viscosity of a
suspension by an average over its representative elementary volume (REV) (see Eq. (1)).
To deal with a large range of particle concentration, it is preferred to adopt the concept
of the generalized self-consistent homogenization theory [19, 20]. Unfortunately, the an-
alytical solutions developed by Herve and Zaoui [20] are not straightforward for the
problem of suspension of solid particles in viscous fluid because of a fluid’s Poisson ratio
of ν f = 0.5 that leads to several undefined terms containing a division to (1− 2ν f ). Their
workflow can be followed but a major modification is necessary.
Indeed, following Herve and Zaoui [20], we consider a two phase concentric sphere,
made of a rigid core and a linear viscous fluid coated layer, surrounded by an infinite
homogeneous viscous matrix (Fig. 9). The rigid core and the coated phase of the coated
inclusion are numbered by 1 and 2, respectively. The infinite matrix is the phase 3.
Under a simple uniform macroscopic shear stress, the fluid velocity in the system
has following components in spherical coordinates
vr = Vr(r) sin2 θ cos 2φ, vθ = Vθ(r) sin θ cos θ cos 2φ,
vφ = Vφ(r) sin θ sin 2φ, Vφ(r) = −Vθ(r).
(8)
On the effective viscosity of fresh concrete: A homogenization approach 299
On the effective viscosity of fresh concrete: a homogenization approach 11
fraction φcp together with the input viscosity of the fluid phase µf in Eqs. (2)
and (3) to compute the effective viscosity of fresh cement.
(3) Second scale: One the viscosity of fresh cement is obtained from the first scale
and the volume fraction of gravel particles is given by input data, we can
compute the final effective viscosity of fresh concrete using equations (6) and
(7).
The dependence of effective viscosity on the thixotropic and curing behavior of
cement-based materials are out of the scope of the present paper that focus on a homog-
enization method and the effect of the microstructure of the suspension.
ACKNOWLEDGMENT
This research is funded by Vietnam National Foundation for Science and Technology
Development (NAFOSTED) under grant number 107.01-2016.17.
APPENDIX: THE MATHEMATICAL DEVELOPMENT OF THE
PROPOSED CLOSED FORM SOLUTIONS
The homogenization method consists to estimate the effective shear viscosity of
a suspension by an average over its representative elementary volume (REV) (see Eq.
(1)). To deal with a large range of particle concentration, it is preferred to adopt the
concept of the generalized self-consistent homogenization theory [18, 19]. Unfortunately,
the analytical solutions developed by Herve and Zaoui [19] are not straightforward for the
problem of suspension of solid particles in viscous fluid because of a fluid’s Poisson ratio
of νf=0.5 that leads to several undefined terms containing a division to (1-2νf ). Their
workflow can be followed but a major modification is necessary.
Indeed, following Herve and Zaoui [19], we consider a two phase concentric sphere,
made of a rigid core and a linear viscous fluid coated layer, surrounded by an infinite
homogeneous viscous matrix (Fig. 9). The rigid core and the coated phase of the coated
inclusion are numbered by 1 and 2, respectively. The infinite matrix is the phase 3.
Fig. 9. Coated spherical inclusion in an infinite reference matrix.Fig. 9. Coated spherical inclusion in an infinite reference matrix
The correspondent strain rates are
ε˙rr = V ′r sin2 θ cos 2φ,
ε˙θθ =
1
r
Vr sin2 θ cos 2φ+
1
r
Vθ cos 2θ cos 2φ,
ε˙φφ =
2
r
Vφ cos 2φ+
1
r
Vr sin2 θ cos 2φ+
1
r
Vθ cos2 θ cos 2φ,
ε˙rθ =
1
2
(
2
r
Vr +V ′θ −
1
r
Vθ
)
sin θ cos θ cos 2φ,
ε˙rφ = −12
(
2
r
Vr −V ′φ −
1
r
Vφ
)
sin θ sin 2φ,
ε˙θφ = −1r
(
2Vθ +Vφ
)
cos θ sin 2φ.
(9)
The volumetric strain rate is
tr ε˙ =
(
V ′r +
2
r
Vr − 3r Vθ
)
sin2 θ cos 2φ. (10)
The stresses components are: σjk = 2µ
(
ν
(1− 2ν) tr ε˙+ ε˙ jk
)
with j, k = r, θ, φ. The veloci-
ties Vr and Vθ , those are solutions of the combinations of the constitutive equation and the
equilibrium equation, and the correspondent radial and tangent stress:
[
Vr,Vθ , S
(i)
rr , S
(i)
rθ
]t
= L(i)r [A, B,C, D]t with i = 1 or 2 and
L(i)r =
r − 6νi
4νi − 7r
3 − 3
2r4
5− 4νi
2 (1− 2νi)
1
r2
r r3
1
r4
1
r2
2µi
6νi
4νi − 7µir
2 12
r5
µi
2 (νi − 5)
1− 2νi
µi
r3
2µi −27+ 2νi4νi − 7µir
2 − 8
r5
µi
2 (νi + 1)
1− 2νi
µi
r3
. (11)
300 Tuan Nguyen-Sy, Duong Nguyen-The
The strain rate concentration factors of the coated layer (see Eq. (1)) is
B˜1 =
A1
A3
− B1
A3
21R21
5 (4ν1 − 7) , (12)
and
B˜2 =
A2
A3
− B2
A3
21R22
5 (4ν2 − 7)
1− X5
1− X3 , (13)
where we noted the ratio between the radii of the inner core and the coated inclusion
by: X = R1/R2; ν1 and ν2 the Poissons ratios of the core and the coated phase of the
inclusion, respectively. They are related to the bulk and shear moduli by the classical
formula: νi = (3ki − 2µi)/(6ki + 2µi) with i = 1 or 2. The terms A1/A3, A2/A3, B1/A3
and B2/A3 are determined by
A1
A3
=
P(2)22
P(2)11 P
(2)
22 − P(2)12 P(2)21
,
B1
A3
=
−P(2)21
P(2)11 P
(2)
22 − P(2)12 P(2)21
, (14)
and
A2
A3
=
P(1)11 P
(2)
22 − P(1)12 P(2)21
P(2)11 P
(2)
22 − P(2)12 P(2)21
,
B2
A3
=
P(1)22 P
(2)
21
P(2)11 P
(2)
22 − P(2)12 P(2)21
. (15)
where the [4×4] matrix P(1) and P(2) are determined by
P(1) = M(1);P(2) = M(2)M(1), (16)
with
M(i) =
(
L(i+1)Ri
)−1
L(i)Ri , (17)
in which the matrix L(i)r is defined by Eq. (11).
The parameters B˜1 and B˜2 are functions of the elastic properties of the coated inclu-
sion and the matrix phase as well as the ratio X of the coated inclusion. It is of interest to
remark that for the case of solid with ν1 6= 0.5 and ν2 6= 0.5, the solutions (12) and (13)
are equivalent to the classical solution obtained by Herve and Zaoui [20]. However the
solutions obtained herein are also valid for fluid with ν2 = 0.5.
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