Page 91 - 3D Fibre Reinforced Polymer Composites

80 30 Fibre Reinforced Polymer Composites
height of the warp yarn is a function of only x. Similar to the functions defined in both
y-z and z-x planes by Naik and Ganesh (1992), the following three different sinusoidal
functions may be utilised to define the shape and the yarn undulation:

z, (x, Y) =
2

ZZ (x. Y) =
2
h, +hw+h,
Z,(X,Y) = (4.35)
2 Qw +gw

where

(4.36)

In equation (4.39, zl(x,y) defines the upper surface of the warp yarn, z~(x,y) defines the
lower surface of the warp yarn and also the upper surface of the weft yam, and z3(x,y)
defines the lower surface of the weft yarn. The ranges of x and y are [0, 0.5~~;
0,
O.S(uft.gf)] for the warp yarn and [O, 0.5(aw+gw);- 0, 0.5ufl for the weft yarn,
respectively. The heights of the warp and weft yarns are given by

(4.37)

and equation (4.36) was obtained by setting the height of both yarns to be zero. The
undulation function is defined as the trajectory of the centre of an individual yam. The
undulation functions for both warp and weft yarns are given by:

z, (Y) = 0.5(z, (0, Y) + zz (0, y>) =
2

Differentiating the two undulation functions, we can readily obtain the expressions for
the fibre orientation angles, Ow&) and O&), of the warp and weft yarns in relation to
the global x and y coordinates. With the off-axis fibre orientation angle known for each
yarn, the reduced compliance constants, Sij( B), of the undulated yarns along the global