Page 96 - 3D Fibre Reinforced Polymer Composites

Micromechanics Models for Mechanical Properties 85
Vandeurzen et a1 (1996b) also considered the iso-stress condition in addition to the iso-
strain condition to evaluate the effective properties of the unit cell. For the case of non-
mixed yam systems, the effective properties of the unit cell can be approximated as:

(4.42)

where w is the volume fraction of the corresponding yarn system or matrix, Ny is the
total number of yam systems and i is the i" yarn system. Subscript m refers to the
matrix. The above two equations can be regarded as approximations of equations (4.40)
and (4.41) because the transformation matrix [TJ is set to be constant in a yam system.
For the mixed yarn system, above equations can also be used except for wm=O.
Vandeurzen et al. (1996b) presented the above equations in a convenient form for
implementation in their custom design tool TEXCOMP. They also presented a new
model, referred to as the combi-cell model, for mixing up the yarn and matrix as shown
in Figure 4.9(b). A combi-cell consists of a yarn layer (Y) and a matrix layer (M) as
shown in Figure 4.10(b), which simplifies the micro-cell model in Figure 4.10(a). By
minimising complementary strain energy, the effective properties of the combi-cell can
be written as

where k is the volume fraction and Y and M refer to the yam and matrix respectively,
and [Ai] is the relation matrix, which defines a linear relationship between the externally
applied stress { 5) and the layer internal stresses (a, } , namely,
(0, [A, IIW (i=Y, M) (4.44)
1
=
The compliance matrix of each micro-cell is then calculated by transforming the
compliance matrix of the combi-cell given in equation (4.43) to the unit cell coordinate
system.

matrix
I I I I

Figure 4.10 Combi-cell model (Vandeurzen et al, 1996b)