Composite Materials Research Progress

6
Jacquemin Frédéric and Fréour Sylvain
In statistics, given a set of data, X = { x1, x2, ..., xn} and corresponding weights, W = {
w1, w2, ..., wn}, the weighted geometric (respectively, arithmetic) mean
i
X
GA
i=
1,2,..., n
(respectively,
AA
i
X ) is calculated as:
i=
1,2,..., n
⎛ n ⎞
1/
⎜ ∑ w ⎟
GA ⎛ n ⎞ ⎜ i ⎟
i ⎜ w
X = ⎟ ⎝i=
1 ⎠
= ⎜∏
x i
i 1,2,..., n
i ⎟
⎝ i=
1 ⎠
∑
∑ =
i
X
AA 1 n
= xi
wi
i=
1,2,..., n n
w i 1
i
i=
1
(5)
Both averages have been extensively used in the field of materials science, in order to
achieve various scale transition modelling over a wide range of materials. The interested
reader can refer to: (Morawiec, 1989; Matthies and Humbert, 1993; Matthies et al., 1994) that
can be considered as typical illustrations of works taking advantage of the geometric average
for estimating the properties and mechanical states of polycrystals (nevertheless, Eshelby-
Kröner self-consistent model was not involved in any of these articles), whereas the
previously cited references (Kocks et al., 1998; Gloaguen et al., 2002; Fréour et al., 2003;
Jacquemin et al., 2005) show applications of arithmetic averages for studying of polycrystals
or composite structures.
According to equations (4) and (5), the explicit writing of a volume weighted average
directly depend on the averaging method chosen to perform this operation. Since the present
work aims to express analytical forms involving such volume averages, it is necessary to
select one average type in order to ensure a better understanding for the reader. Usually, in
this field of research, the arithmetic and not the geometric volume weighted average is used.
Moreover, in a recent work, the alternative geometric averages were also used for estimating
the effective properties of carbon-epoxy composites (Fréour et al., to be published).
Nevertheless, the obtained results were not found as satisfactory than in the previously
studied cases of metallic polycrystals or metal ceramic assemblies. Actually, the very strongly
heterogeneous properties presented by the constituents of carbon reinforced polymer matrix
composites yields a strong underestimation of the effective properties of the composite ply
predicted according to Eshelby-Kröner model involving the geometric average, by
comparison to the expected (measured) properties. Thus, the geometric average should not be
considered as a reliable alternate solution to the classical arithmetic average for achieving
scale transition modelling of composite structures. Consequently, arithmetic averages
satisfying to relation (4) only will be used in the following of this manuscript.
(4)