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IBRACON Structures and Materials Journal • 2012 • vol. 5 • nº 3
W. R. L. da Silva | J. Němeček | P. Štemberk
symmetrical indenter shape (β=1.034 for Berkovich tip). The effect
of non-rigid indenter can be taken into consideration by the equality
of compliances shown in Eq.(3),
(3)
,
1
1 1
2
2
i
i
r
E E E
n
n -
+
-
=
where E corresponds to the elastic modulus and n the Poisson’s
ratio of the tested materials, while E
i
and
n
i
are the parameters of
the indenter (E
i
=1141 GPa and
n
i
=0.07 for a diamond).
2.4.2 Deconvolution procedure
The individual phase properties of the analyzed mixtures were de-
termined by applying the statistical deconvolution to elastic modu-
lus histograms, [19,4]. The results on the elastic moduli in the form
of frequency plots are analyzed so that the minimization proce-
dure seeks parameters of
n
Gauss distributions in an experimental
probability density function, PDF.
In the deconvolution algorithm, a random seed and minimizing criteria
based on quadratic deviations between the experimental and theoreti-
cally computed overall PDFs are calculated to find the best fit. A com-
prehensive review about the deconvolution procedure is described in
[4]. The number of the searched phases was fixed to four based on
the number of characteristic peaks in the PDF. The dominant phas-
es in the PDF need not necessarily to correlate with the chemically
distinct phases and are further considered as mechanically distinct
phases which will be used in the homogenization procedure.
2.5 Analytical elastic homogenization
The homogenization corresponds to a technique used to upscale
the mechanical properties from the microscale to the upper lev-
el to find the effective properties of the RVE. In this paper, the
elastic homogenization was performed by means of the analytical
Mori-Tanaka scheme, [20]. This scheme describes a composite
by a morphologically prevailing matrix (the reference medium)
reinforced by distinct non-continuous spherical inclusions. In the
Mori-Tanaka method, the homogenized bulk and shear moduli of a
r
-phase composite are assessed as indicated in Eq.4 to 7.
(4)
1
0
0
1
0
0
hom
))1
(
1(
))1
(
1(
-
-
-
×
+
-
×
+ ×
=
å
å
k k
f
k k
k f
k
r
r r
r
r
r r
a
a
(5)
1
0
0
1
0
0
hom
))1
(
1(
))1
(
1(
-
-
-
×
+
-
×
+ ×
=
å
å
mm b
mm b m
m
r
r r
r
r
r r
f
f
(6)
0
0
0
0
4 3
3
m
a
× + ×
×
=
k
k
(7)
0
0
0
0
0
20
15
12
6
m
m
b
×
+ ×
× + ×
=
k
k
where the subscript
0
corresponds to the reference medium
and
r
corresponds to a particular inclusion. Thus,
k
0
and
µ
0
are
the bulk and shear moduli of the reference medium, while
k
r
and
µ
r
refer to the inclusion phases. These values are the input
to the equations that define the homogenized elastic modu-
lus. After defining these values, the bulk and shear moduli can
be recomputed to the elastic modulus and the Poisson’s ratio
though Eq.8 and 9.
(8)
hom
hom
hom
hom
hom
3
9
m
m
+ ×
×
×
=
k
k
E
(9)
hom
hom
hom
hom
hom
2
6
2
3
m
m
n
× + ×
× - ×
=
k
k
3. Results
3.1 Experimental results in hardened
state – Macroscale analysis
3.1.1 Elastic modulus and Compressive strength
The results from elastic modulus and uniaxial compression tests
are summarized in Table 5.
Table 5 – Elastic modulus and Compressive
strength - Macroscale analysis
Sample
Elastic
modulus [GPa]
Compressive
strength
[MPa]
1
41.2
61.4
2
38.1
58.2
3
37.5
49.4
4
39.1
57.0
5
38.8
55.7
6
39.5
54.7
Average
result [GPa]
39.0
–
–
Standard
deviation [GPa]
1.28