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so, for the entire composite, the Young modulus
can be computed using the rule of mixtures:
= E ⋅ϕ
+ E ⋅(
1 − ϕ ). (18)
EC F F RM
F
If we denote P , the basic elastic property of
M
the matrix, P and P the basic elastic property
F f
of the fi bers respective the fi ller, then the upper
limit of the homogenized coeffi cients can be estimated
computing the arithmetic mean of these
basic elastic properties:
PM
⋅ϕ
M + PF
⋅ϕ
F + Pf
⋅ϕ
f
Aa
=
.
3
(19)
Th e lower limit of the homogenized elastic coeffi
cients can be estimated computing the harmonic
mean of the basic elasticity properties of
the isotropic compounds:
3
Ah
=
.
1 1 1
+ +
P ⋅ϕ
P ⋅ϕ
P ⋅ϕ
M
M
F
F
f
f
(20)
An intermediate limit between the arithmetic
and harmonic mean is given by the geometric
mean writt en below:
Ag = 3 PM
⋅ϕ
M ⋅ PF
⋅ϕ
F ⋅ Pf
⋅ϕ
f ,
(21)
where P and A can be the Young modulus respective
the shear modulus.
RESULTS
According to equations (13) and (16), the upper
and lower limits of the homogenized coeffi
cients for a 27% fi bers volume fraction SMC
material have been computed and presented
as follows. For 0° angular variation of the ellipsoidal
inclusion, the upper limit of the homogenized
coeffi cients is 2.52 and the lower limit
of the homogenized coeffi cients is 0.83. For
± 15° angular variation, the upper limit of the
homogenized coeffi cients is 2.37 and the lower
limit of the homogenized coeffi cients is 0.851.
For ± 30° angular variation, the upper limit of
the homogenized coeffi cients is 2.17 and the
lower limit of the homogenized coeffi cients is
0.886. Fig. 1 shows the Young moduli of the isotropic
SMC compounds, the upper and lower
limits of the homogenized elastic coeffi cients
as well as a comparison with the experimental
value. According to equations (19) – (21), the
averaging methods of Young modulus of various
SMCs with diff erent fi bers volume fractions
present following distributions in fi g. 2.
CONCLUSIONS
By increasing the fi bers volume fraction, the difference
has been equally divided between matrix
and fi ller volume fraction. Th e averaging methods
to compute the Young and shear moduli of
various SMCs with diff erent fi bers volume fractions
show that for a 27% fi bers volume fraction
SMC, the arithmetic means between matrix,
fi bers and fi ller Young moduli respective shear
moduli give close values to those determined
experimentally. Th e presented results suggest
that the environmental geometry given through
the angular variation of the ellipsoidal domains
can lead to diff erent results for same fi bers volume
fraction. Th e upper limits of the homogenized
elastic coeffi cients are very close to experi-
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Fig. 2 Averaging methods to compute the Young moduli of various SMCs
mental data, showing that this homogenization
method give bett er results than the computed
composite’s Young modulus determined by help
of the rule of mixtures.
REFERENCES:
[1] H.I. Ene, G.I. Pasa, Th e Homogenization Method.
Applications at Composite Materials Th eory, Romanian
Academy Publishing House, Bucharest, 1987 (in
Romanian).
[2] H.G. Science and
Technology, Hanser, 1993.
[3] C. Miehe, J. Schröder, M. Becker, Computational
homogenization analysis in fi nite elasticity: material and
structural instabilities on the micro- and macro-scales
of periodic composites and their interaction, Comput.
Methods Appl. Mech. Eng. 191 (2002) 4971-5005.
[4] N. Bakhvalov, G. Panasenko, Homogenization.
Averaging Processes in Periodic Media: Mathematical
Problems in the Mechanics of Composite Materials,
Kluwer Academic Publishers, 1989.
[5] J.M. Whitney, R.L. McCullough, Delaware
Composite Design Encyclopedia: Micromechanical
Materials Modelling, Vol. 2, University of Delaware,
L.A., 1990.
[6] T.J. Barth, T. Chan, R. Haimes, Multiscale and
Multiresolution Methods: Th eory and Applications,
Springer, 2001.
[7] A. Pavliotis, A. Stuart, Multiscale Methods:
Averaging and Homogenization, Springer, 2008.
[8] V.V. Jikov, S.M. Kozlov, O.A. Oleynik,
Homogenization of Diff erential Operators and Integral
Functions, Springer, 1994.
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