Composite Materials Research Progress

Multi-scale Analysis of Fiber-Reinforced Composite Parts… 5
Therefore, the formalism was extent so that homogenisation relations were established for
estimating the macroscopic CME from those of the constituents (Jacquemin et al., 2005).
Many previously published documents have been dedicated to the determination of (at
least some of) the effective thermo-hygro-elastic properties of heterogeneous materials
through Kröner-Eshelby self-consistent approach (Kocks et al., 1998; Gloaguen et al., 2002;
Fréour et al., 2003a-b; Jacquemin et al., 2005). The main involved equations are:
( I + E
I
: [ L
i
− L
I ] )
L
I
= L
i
:
−1
(1)
i=
r, m
1
−1
i I I − I
i I I
( + L : R ) : L : ( L + L : R )
I 1
−1
i i i
β = L : L : β ΔC (2)
I
ΔC
i=
r, m
i=
r, m
1
−1
i I I − I
i I I
[ L + L : R ] : L : [ L + L : R ]
I
M =
−1
i i
: L : M
(3)
i=
r, m
i=
r, m
Where ΔC i is the moisture content of the studied i element of the composite structure.
The superscripts r and m are considered as replacement rule for the general superscript i, in
the cases that the properties of the reinforcements or those of the matrix have to be
considered, respectively. Actually, the pseudo-macroscopic moisture contents ΔC r and ΔC m
can be expressed as a function of the macroscopic hygroscopic load ΔC I (Loos and Springer,
1981), so that the hygro-mechanical states cancels in relation (2) that can finally be rewritten
as a function of the materials properties only, but at the exclusion of the ΔC i that are
unexpected to appear in such an expression (Jacquemin et al., 2005). Relation (2), that is
provided in the present work for predicting the macroscopic CME, is given for its enhanced
readability, compared to the more rigorous state exclusive relation.
In relations (1-3), the brackets < > stand for volume weighted averages (that in fact
replace volume integrals that would require Finite Elements Methods instead). Empirically, as
stated by Hill (Hill, 1952), arithmetic or geometric averages suggest themselves as good
approximations. On the one hand, the geometric mean of a set of positive data is defined as
the n th root of the product of all the members of the set, where n is the number of members.
On the other hand, in mathematics and statistics, the arithmetic mean (or simply the mean) of
a list of numbers is the sum of all the members of the list divided by the number of items in
the list. For Young’s modulus, as an example, the Geometric Average Y GA of the moduli
GA
according to the Reuss (YR) and Voigt (YV) models is defined as Y = YR
YV
,
whereas the corresponding Arithmetic Average Y AA AA Y Y
is: Y R +
= V
.
2