DACIA ELECTRA - ingineria-automobilului.ro

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P , homologous through periodicity in neigh-
2
bour periods, it can be notice that the dependence
in x
η
is the same and the dependence in x is almost
the same since the distance P P is small. Th e
1 2
function uη depends on the periodic coeffi cients
a , on the function f(x) and on the boundary ij ∂Ω
. Using the development (6), the expressions
∂u ∂xi
η and η
p are [1]:
η
∂u
⎛ ∂ 1 ∂ ⎞ 0 1
=
⎜ + ⋅
⎟ ⋅(
u + η ⋅ u + ... ) =
∂xi
⎝ ∂xi
η ∂yi
⎠
(8)
0 1
1 2
∂u
∂u
⎛ ∂u
∂u
⎞
= + + η ⋅ ⎜ ⎟
⎜
+
⎟
+ ...,
∂xi
∂yi
⎝ ∂xi
∂yi
⎠
η
p ( x ) p0(
x,
y ) p1
i = i + η ⋅ i ( x,
y ) (9)
p ( x,
y ) ...,
2 ⎛ 0 1 ⎞
0
( , ) ( ) ⎜
∂u
∂u
1
p ⎟
i x y = aij
y ⋅ + , p ( , ) =
⎜ ⎟ i x y (10)
⎝
∂x
j ∂y
j ⎠
⎛ 1 2 ⎞
) ( ) ⎜
∂u
∂u
= a
⎟
ij y ⋅ + , .
⎜ ⎟
⎝
∂x
j ∂y
j ⎠
represent the homogenized coeffi cients.
APPLICATION FOR A 27% FIBERS
VOLUME FRA CTION SHEET MOLDING
COMPOUND
For a SMC material is preferable to estimate
these homogenized coeffi cients between an upper
and a lower limit. Since the fi bers volume
fraction of common SMCs is 27%, to lighten
the calculus, an ellipsoidal inclusion of area 0.27
situated in a square of side 1 is considered. In
the structure’s periodicity cell (SPC) of a SMC
+η ⋅ i +
composite material, the fi bers bundle is seen
where:
as an ellipsoidal inclusion. Let us consider the
function f(x , x ) = 10 in inclusion and 1 in ma-
1 2
10
Fig. 1 Th e upper (E+) and lower limits (E-)
of the homogenized elastic coeffi cients
trix. To determine the upper and the lower limit
of the homogenized coeffi cients, fi rst the arithmetic
mean as a function of x -axis followed by
2
the harmonic mean as a function of x -axis must
1
be computed. Th e lower limit is obtained computing
fi rst the harmonic mean as a function of
x -axis and then the arithmetic mean as a func-
1
tion of x -axis. If we denote with φ(x ) the arith-
2 1
metic mean against x -axis of the function f(x ,
2 1
x ), it follows:
2
(11)
(12)
Th e upper limit is obtained computing the harmonic
mean of the function φ(x ): 1
(13)
To compute the lower limit, we consider ψ(x ) 2
the harmonic mean of the function f(x , x ) 1 2
against x : 1
(14)
(15)
Th e lower limit will be given by the arithmetic
mean of the function ψ(x ): 2
(16)
Th e Young modulus of the replacement matrix
(E ) can be estimated computing the harmon-
RM
ic mean of the basic elastic properties of the isotropic
compounds, as follows:
(17)