Page 92 - 3D Fibre Reinforced Polymer Composites

Micromechanics Models for Mechanical Properties 81
axes can be determined using transformed equations similar to those given in (4.30) and
(4.3 1).
As proposed by Naik and Ganesh (1992), with reference to Figure 4.7(b) and (c), for
the warp yam in one slice the off-axis angle at the midpoint of that slice is used to
calculate the off-axis compliance constants for the idealised warp layer, while for the
weft yarn in one slice the off-axis effective compliance constants for the idealised weft
layer are set to equal to the average values of Sij( 6J over the integration interval from 0
to the maximum values of I%&). With the compliance constants known for each layer
in one idealised slice in Figure 4.7(c), the effective properties for that slice can be
determined using the classical laminate theory. Similar to the case of one-dimensional
fibre undulation model, the overall effective properties for the unit cell shown in Figure
4.7(a) can be determined by applying iso-stress conditions to all slices in y direction.
Evidently, determination of the off-axis compliance constants in the weft yarn in
terms of average values of Sij( 8, introduces approximations. Another model, proposed
by Naik and Ganesh (1992) and referred to as the element array model, was to enhance
the approximation in the weft direction. In the element array model, the slices of the
unit cell shown in Figure 4.7(b) were further divided into elements along the x direction
prior to idealisation. This can be better illustrated in Figure 4.8, in which the unit cell is
discretised along both warp and weft directions into elements. For each element the off-
axis angles at the centre of the element are chosen to determine the reduced properties
of the idealised layers, which can be further used to calculate the properties of that
element using the classical laminate theory. The overall effective properties of the unit
cell can be obtained by assembling all elements in two combinations, i.e., series-parallel
combination and parallel-series combination. In the series-parallel combination,
elements are assembled in series into slices first along the loading direction under iso-
stress condition and then the slices are considered in parallel under iso-strain condition.
In the parallel-series combination, elements are grouped in parallel into slices first
across the loading direction under an iso-strain condition and then the slices are
considered in series under an iso-stress condition. It is expected that the parallel-series
combination predict a higher value of stiffness compared to the series-parallel
combination.
Naik and his colleagues have performed an extensive research both numerically and
experimentally to verify the slices array model and element array model (Naik, 1994).
Shembekar and Naik (1992) also investigated the effect of fibre undulation shifts
between individual weave lamina in a laminated plate. It is beyond the scope and limit
of this chapter. Readers who require further details on the models themselves and
experimental verificatiofare referred to the book by Naik (1994).

4.3.3 Three-Dimensional (3D) Models
Both 1D and 2D models discussed above were developed based on the classical
laminate theory. Although accounting for yarn undulation, yarn shape and spacing,
these models predict the in-plane elastic properties only. 3D models have been
developed to evaluate the out-of-plane elastic properties in addition to the in-plane
properties. It is not possible to present all models. In the following we choose to
present the models proposed by Hahn and Pandey (1994) and Vandeurzen et al. (1996a,
1996b, 1998) for the case of plain weave composites.