Page 102 - 3D Fibre Reinforced Polymer Composites

Micromechanics Models for Mechanical Properties 91
4.4.1 Orientation Averaging Models
Simple orientation averaging models were originally developed for calculating
macroscopically averaged elastic properties of fibre reinforced composites
(Tarnopol'skii et a1 1973; Kregers and Melbardis, 1978). In these models, the
composite is treated as an assemblage of small volumes. In each individual volume, all
fibres are aligned and orientated depending on the reinforcement architecture. Each
volume can be modelled as unidirectional composites with transversely isotropic
properties. The overall effective properties of the composites can then be determined by
averaging the response of a representative body to the externally applied loads under the
assumption of either uniform stresses or uniform strains. It is clear that the assumption
of uniform stresses or strains is identical to that used in the 3D model for 2D woven
composites. It may be viewed, to certain extent, as rules of mixtures in the three
dimensional case, similar to those in Section 4.2.3.
Cox and Dadkhah (1995) applied the orientation averaging method to 3D woven
interlock composite, i.e., layer-to-layer and through-the-thickness angle interlock and
orthogonal interlock weaves. For orientation averaging, each composite is divided into
stuffer, filler, and two warp weavers volumes with fraction ci of the total composite
volume (i=s,f, wI and wz for stuffer, filler, and two warp weavers volume) with a sum
of ci being unity. Similar to equation (4.42), the following approximate expression for
the stiffness matrix of 3D woven composites with ideal geometry is obtained:

(4.48)

It was found that the ideal geometry is far different from the true geometry. Both
stuffer and filler yams are not straight and there exists significant out-of-plane
waviness, which varies along the stuffer and filler directions. To take into account the
most important effect of tow waviness on elastic properties, a symmetrical normal
distribution is formed for the out-of-plane alignment angle, 5. as follows:

with the density function given by

where o5 represents the width of the distributions. A waviness knockdown factor is
defined as

for crl 510' (4.49)

The waviness knockdown factor is used to reduce the values of Young's modulus in the
tow direction and Poisson ratio v12 for the stuffer and filler yarns.