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78 30 Fibre Reinforced Polymer Composites

plane shear modulus decreases almost linearly with the fibre volume fraction which
decreases with ng.

4.3.2 Two-Dimensional (2D) Models
The fibre undulation model (Ishikawa and Chou, 1982b) considered fibre continuity and
fibre undulation in one direction only, and is thus deemed as a ID model. 2D models
should take into account fibre undulation and continuity in both the warp and weft
directions. In 1992, Naik and colleagues (Naik and Shembekar, 1992a,b; Shembekar
and Naik, 1992 and Naik and Ganesh, 1992) extended the fibre undulation model and
developed 2D models, which includes the fibre undulation and continuity in both warp
and weft directions, the possible presence of a gap between adjacent yarns, and the
actual cross-sectional geometry of fibre yams. To present the fundamental concepts of
2D models, let us consider a representative cell of a plain wave lamina, as shown in
Figure 4.6, for a plain weave shown in Figure 4.1.

Figure 4.6 The repetitive unit cell of plain weave lamina

The unit cell consists of two fibre yarns, warp and weft, and pure matrix regions. It is
desirable to obtain accurate geometrical descriptions of individual fibres or even the
warp and weft yarns themselves in the space. Due to the nature of manufacturing
process, the geometry of fibres or fibre yarns inevitably vary from one cell to another,
thus assumptions must be introduced based on experimental observations to simplify the
geometrical visualisation problem of fibres or fibre yarns. One assumption is to assume
that the repetitive unit cell possesses two planes of symmetry in the interlacing region.
By virtue of the symmetry, Naik and Ganesh (1992) considered only one quarter of the
repetitive unit cell as shown in Figure 4.7(a) and proposed two models based on the
classical laminate theory, one is referred to as slice array model and the other element
array model.
In the slice array model, the unit cell is discretised into slices, for example three
slices as shown in Figure 4.7(b), along the loading direction 0, direction in this case).
Each slice is then transformed into a four-layered laminate, i.e., an asymmetrical cross
ply sandwiched between two pure matrix layers as shown in Figure 4.7(c). The
effective elastic constants of the plain weave lamina are evaluated from the properties of