Page 102 - Mechanics Analysis Composite Materials

Chapter 3. Mechanics of a unidirectional p1.v 87

Only two of the foregoing expressions, namely Eq. (3.76) for E, and Eq. (3.80)
for vz1, both following from the rule of mixtures, demonstrate good agreement with
experimental results. Moreover, expressions analogous to Eqs. (3.76) and (3.80)
follow practically from all numerous studies based on different micromechanical
models. Comparison of predicted and experimental results is presented in
Figs. 3.35-3.37, where theoretical dependencies of normalized moduli on the fiber
volume fraction are shown with lines. The circles correspond to the test data for
epoxy composites reinforced with different fibers that were obtained by the authors
or taken from publications of Tarnopol'skii and Roze (1969), Kondo and Aoki
(1982), Lee et al. (1995). As can be seen in Fig. 3.35, not only first-order model,
Eq. (3.76), but zero-order model, Eqs. (3.61), as well, provide fair prediction for E,,
while Figs. 3.36 and 3.37 for E2 and Gl2 call for the improvement of the first-order
model (the corresponding results are shown with solid lines).
Second-order models allow for the fiber shape and distribution, but in contrast to
higher-order models ignore complicated stressed state of fibers and matrix under the
ply loading shown in Fig. 3.29. To demonstrate this approach, consider a layer-wise
fiber distribution (see Fig. 3.5) and assume that the fibers are absolutely rigid and
the matrix is in the simplest uniaxial stressed state under transverse tension. The
typical element of this model is shown in Fig. 3.38 from which we can obtain the
following equation

nR2 nR
q=-=-. (3.82)
2Ra 2a
Because 2R < a, vf < n/4 = 0.785. Equilibrium condition yields
R
2R02 = / crndx3 , (3.83)
-R
where x3 =Rcosa and 02 is some average transverse stress that induces average
strain

0.8
0.6

0.4

0.2

0
0 0.2 0.4 0.6 0.8
Fig. 3.35. Dependence of the normalized longitudinal modulus on fiber volume fraction: (- -- -) zero-
order model, Eqs. (3.61); (-) first-order model, Eqs. (3.76); (*) experimental data.