Composite Materials Research Progress

22
Jacquemin Frédéric and Fréour Sylvain
Moreover, it was established in (Hill, 1967), that the self-consistent model was
compatible with the following volume averages on both pseudo-macroscopic stresses and
strains:
σ
ε
i
i
i=
r, m
i=
r, m
= σ
For a given applied macroscopic thermo-hygro-elastic load {σ I , ΔC I, ΔT} one can easily
determine ε I through (33), provided that the effective elastic behaviour L I of the ply has been
calculated using either the homogenization procedure corresponding to Eshelby-Kröner
model or the corresponding Mori-Tanaka alternate solution (see previous developments
provided in section 2 above). Then, the pseudo-macroscopic strains are determined through
(35).
4.3. Analytical Expression for Calculating the Mechanical States Experienced
by the Constituents of Fiber-Reinforced Composites According to
Eshelby-Kröner Model
The main impediment requiring to be overcome in order to achieve closed-forms from
relation (35) is the determination of Morris’ tensor E I . Actually, according to the integrals
appearing in relation (8), this tensor will admit only numerical solutions in most cases.
However, some analytical forms for Morris’ tensor are actually available in the literature;
the interested reader can for instance refer to (of Mura, 1982; Kocks et al., 1998; or Qiu and
Weng 1991). Nevertheless, these forms were established considering either spherical, discshaped
of fiber-shaped inclusions embedded in an ideally isotropic macroscopic medium, that
is incompatible with the strong elastic anisotropy exhibited by fiber-reinforced composites at
macroscopic scale (Tsai and Hahn, 1987).
In the case of carbon-epoxy composites, a transversely isotropic macroscopic behaviour
being coherent with fiber shape is actually expected (and predicted by the numerical
computations). Assuming that the longitudinal (subscripted 1) axis is parallel to fiber axis, one
obtains the following conditions for the semi-lengths of the microstructure representative
ellipsoid: a1→∞, a2=a3. Moreover, the macroscopic elastic stiffness should satisfy :
I I I I I I
L11 ≠ L12
≠ L22
≠ L23
≠ L44
≠ L55
. Now, it is obvious, that these additional
hypotheses lead to drastic simplifications of Morris’ tensor (8), in the case that fiber
morphology is considered for the reinforcements. The line of reasoning required to achieve
the writing of analytical expressions for Morris’ tensor is extensively presented in (Welzel et
= ε
al., 2005; Fréour et al., 2005). Actually, one obtains (in contracted notation i.e,
components are given here):
I
I
(36)
I
E ij