On the respect to the Hashin-Shtrikman bounds of some analytical methods applying to porous media for estimating elastic moduli

 PA is closely under the upp r bounds as shown in Fig. 6. 
(a) Buck modulus (b) Bulk modulus (closer look of (a)) 
Figure 6. Comparison of elastic modulus estimated by PA and MTA of a 2D 3-
component unit cell: the matrix ( MK , MP ) = (40, 20) kN/mm2, ellipse inclusions 
( I1K , I1P ) =(1, 0.4) kN/mm2, circular inclusions ( I2K , I2P )= (0, 0) kN/mm2. 
3.3. 3D 3-component examples 
In this part, we apply MTA and PA for some 3D porous media with several types 
of inclusion, including platelet, needle, and sphere. In the following examples, the 
properties of the matrix ( MK , MP ) are constant at (40, 20) kN/mm2. The bulk modulus 
is taken into consideration in different cases of volume fraction from low to high. 
Figs. 7(a-c) plot the estimation of MTA and PA for the case of ellipsoid 
inclusions ( I1K , I1P ) = (10, 0.4) kN/mm2 and sphere voids ( I2K , I2P ) = (0, 0) kN/mm2. 
We can observe that PA, MTA, and HSU nearly coincide when the volume fraction of 
needles is small I1X = 5% and 15%. Whereas, as can be seen in Fig.7c when the 
volume fraction of needles is 75%, the MTA estimation start to exceed the HSU and 
PA estimation still respect the upper of H-S bounds. 
Fig. 8(a-c) plot the estimation of MTA and PA for the case when inclusions are 
platelets ( I1K , I1P ) = (10, 0.4) kN/mm2 and sphere ( I2K , I2P ) = (0, 0) kN/mm2. In this 
case, the violation of MTA is first observed in Fig. 8(b) when the platelet phase has a 
volume fraction of I1X = 15%. This is more obvious in Fig. 8(c) when the sample 
contains a high proportion of platelet I1X = 75%. Again, the violation to H-S bounds of 
PA is acknowledged. 
Similarly, we consider the case when inclusions are platelets and ellipsoids 
(voids). Figs. 9 (a-c) plot the estimation of MTA and PA in the three cases of platelet 
inclusion volume fraction 5%, 15%, 75% respectively. The trend is not different from 
(b) Bulk modulus (closer look of (a))
Figure 6. Comparison of elastic modulus estimated by PA and MTA of a 2D 3-component unit cell: the matrix
(KM , µM) = (40, 20) kN/mm2, ellipse inclusions (KI1, µI1) = (1, 0.4) kN/mm2, circular inclusions
(KI2, µI2) = (0, 0) kN/mm2
3.3. 3D 3-component examples
the case of platelet and sphere inclusion. The MTA may invade but PA always 
respects the H-S bounds. 
Note that, in these examples, I2X varies and I I1 I2X X X  . 
 (a) I1 5%X (b) I1 15%X 
 (c) I1 75%X 
Figure 7. Comparison of Bulk modulus estimated by PA and MTA of a 3D material 
with the matrix ( MK , MP ) = (40, 20) kN/mm2, ellipsoid inclusions ( I1K , I1P ) = (10, 0.4) 
kN/mm2 and sphere inclusions ( I2K , I2P ) = (0, 0) kN/mm2, I I1 I2X X X  . 
(a) I1 5%X (b) I1 15%X 
(a) υI1 = 5%
the case of platelet and sphere inclusion. The MTA may invade but PA lways 
respects the H-S bounds. 
Note that, in these examples, I2X varies nd I I1 I2X X X  . 
 ( I (b) I1 15%X 
 (c) I1 75%X 
Figure 7. Comparis of Bulk modulus estimated by PA and MTA of a 3D material 
with the matrix ( MK , MP ) = (40, 20) kN/mm2, ellipsoid inclusions ( I1K , I1P ) = (10, 0.4) 
kN/mm2 and sphere inclusions ( I2K , I2P ) = (0, 0) kN/mm2, I I1 I2X X X  . 
(a) I1 5%X (b) I1 15%X 
(b) υI1 = 15%
the cas of pl telet and sphere inclusion. The invade but PA always 
respects the H-S bounds. 
Note that, in these examples, I2X varies and I I1 I2X X X  . 
 (a) I1 5%X (b) I1 15%X 
 (c) I1 75%X 
Figure 7. Comparison of Bulk modulus estimated by PA and MTA of a 3D material 
with the matrix ( MK , MP ) = (40, 20) kN/mm2, ellipsoid inclusions ( I1K , I1P ) = (10, 0.4) 
kN/mm2 and sphere inclusions ( I2K , I2P ) = (0, 0) kN/mm2, I I1 I2X X X  . 
(a) I1 5%X (b) I1 15%X 
(c) υI1 75%
Figure 7. Comparison of Bulk modulus estimated by PA and MTA of a 3D material with the matrix
(KM , µM) = (40, 20) kN/mm2, ellipsoid inclusions (KI1, µI1) = (10, 0.4) kN/mm2 and sphere inclusions
(KI2, µI2) = (0, 0) kN/mm2, υI = υI1 + υI2
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Nguyen, N., et al. / Journal of Science and Technology in Civil Engineering
In this part, we apply MTA and PA for some 3D porous media with several types of inclusion, in-
cluding platelet, needle, and sphere. In the following examples, the properties of the matrix (KM, µM)
are constant at (40, 20) kN/mm2. The bulk modulus is taken into consideration in different cases of
volume fraction from low to high.
Fig. 7 plots the estimation of MTA and PA for the case of ellipsoid inclusions (KI1, µI1) = (10, 0.4)
kN/mm2 and sphere voids (KI2, µI2) = (0, 0) kN/mm2. We can observe that PA, MTA, and HSU nearly
coincide when the volume fraction of needles is small υI1 = 5% and 15%. Whereas, as can be seen in
Fig. 7(c) when the volume fraction of needles is 75%, the MTA estimation start to exceed the HSU
and PA estimation still respect the upper of H-S bounds.
Fig. 8 plots the estimation of MTA and PA for the case when inclusions are platelets (KI1, µI1) =
(10, 0.4) kN/mm2 and sphere (KI2, µI2) = (0, 0) kN/mm2. In this case, the violation of MTA is first
observed in Fig. 8(b) when the platelet phase has a volume fraction of υI1 = 15%. This is more
obvious in Fig. 8(c) when the sample contains a high proportion of platelet υI1 = 75%. Again, the
violation to H-S bounds of PA is acknowledged.
the case of platelet and sphere inclusion. The MTA may invade but PA always 
respects the H-S bounds. 
Note that, in these examples, I2X varies and I I1 I2X X X  . 
 (a) I1 5%X (b) I1 15%X 
 (c) I1 75%X 
Figure 7. Comparison of Bulk modulus esti ated by PA and MTA of a 3D material 
with the matrix ( MK , MP ) = (40, 20) kN/mm2, ellipsoid inclusions ( I1K , I1P ) = (10, 0.4) 
kN/mm2 a sphere inclusions ( I2K , I2P ) = (0, 0) kN/mm2, I I1 I2X X X  . 
(a) I1 5%X (b) I1 15%X (a) υI1 = 5%
the case of platelet and sphere inclusion. The MTA may invade but PA always 
respects th H-S bounds. 
Note that, in these examples, I2X varies and I I1 I2X X X  . 
 (a) I1 5%X (b) I1 15%X 
 (c) I1 75%X 
Figure 7. Co parison of Bulk modulus estimated by PA nd MTA of a 3D material 
with the matrix ( MK , MP ) = (40, 20) kN/m 2, ellipsoid inclusions ( I1K , I1P ) = (10, 0.4) 
kN/m 2 and sphere inclusi ns ( I2K , I2P ) = (0, 0) kN/mm2, I I1 I2X X . 
(a) I1 5%X (b) I1 15%X 
(b) υI1 = 15%
(c) I1 75%X 
Figure 8. Comparison of elastic modulus estimated by PA and MTA of a 3D material 
with the matrix ( MK , MP )= (40, 20) kN/mm2, platelet inclusions ( I1K , I1P ) = (10, 0.4) 
kN/mm2, spherical inclusions ( I2K , I2P ) = (0, 0) kN/mm2. 
(a) I1 5%X (b) I1 15%X 
(c) I1 75%X 
(c) υI1 = 75%
Figure 8. Comparison of elastic modulus estimated by PA and MTA of a 3D material with the matrix
(KM , µM) = (40, 20) kN/mm2, platelet inclusions (KI1, µI1) = (10, 0.4) kN/mm2, spherical inclusions
(KI2, µI2) = (0, 0) kN/mm2
Similarly, we consider the case when inclusions are platelets and ellipsoids (voids). Fig. 9 plots
the estimation of MTA and PA in the three cases of platelet inclusion volume fraction 5%, 15%, 75%
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Nguyen, N., et al. / Journal of Science and Technology in Civil Engineering
respectively. The trend is not different from the case of platelet and sphere inclusion. The MTA may
invade but PA always respects the H-S bounds.
(c) I1 75%X 
Figure 8. Compariso of elastic modulus estimated by PA and MTA of a 3D material 
with the matrix ( MK , MP )= (40, 20) kN/mm2, platelet inclusions ( I1K , I1P ) = (10, 0.4) 
kN/mm2, spherical inclusions ( I2K , I2P ) = (0, 0) kN/mm2. 
(a) I1 5%X (b) I1 15%X 
(c) I1 75%X 
(a) υI1 = 5%
(c) I1 75%X 
 . parison of elastic modulus estimat d by PA and MTA of a 3D material 
it t atrix ( M , MP )= (40, 20) kN/mm2, platelet inclusions ( I1K , I1P ) = (10, 0.4) 
k / 2, spherical inclusions ( I2K , I2P ) = (0, 0) kN/mm2. 
(a) I1 5X (b) I1 15%X 
(c) I1 75%X 
(b) υI1 = 15%
(c) I 
Figure 8. Comparison of elastic modulus esti ated by PA and TA of a 3D material 
with the matrix ( MK , MP )= (40, 20) kN/mm2, platelet inclusions ( I1K , I1P ) = (10, 0.4) 
kN/mm2, spherical inclusions ( I2K , I2P ) = (0, 0) kN/mm2. 
(a) I1 5%X (b) I1 15%X 
(c) I1 75%X 
(c) υI1 = 75%
Figure 9. Comparison of elastic modulus estimated by PA and MTA of a 3D material with the matrix
(KM , µM) = (40, 20) kN/mm2, platelet inclusions (KI1, µI1) = (10, 0.4) kN/mm2, ellipsoid inclusions
(KI2, µI2) = (0, 0) kN/mm2
Note that, in these examples, υI2 varies and υI = υI1 + υI2.
4. Conclusions
This work has investigated the respect to H-S bounds of the MTA and PA0 of some porous media.
The results reveal that: (i) in a 2D porous medium, in which the void inclusion is ideally circular, the
estimation of MA, MTA, and PA coincide and agree with that of FEM while the deviation is clearly
when the shape of inclusions changes to ellipses; (ii) The MTA estimation can violate the H-S bounds
when the sample is highly porous in the case of 2D-3 phase composite; (iii) The influence of platelet
inclusion is significant in the sense of the violation HS- bounds when estimating by MTA. In all the
investigated example, the polarization approximation respects the H-S bounds which suggests that PA
is a reliable method.
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Nguyen, N., et al. / Journal of Science and Technology in Civil Engineering
Acknowledgments
This research is funded by Vietnam National Foundation for Science and Technology Develop-
ment (NAFOSTED) under Grant Number 107.02-2017.309.
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