Page 84 - 3D Fibre Reinforced Polymer Composites

Micromechanics Models for Mechanical Properties 73

(4.24)

where A;, Eli and Di; are the stretching, stretching-bending coupling and bending
stiffness matrices of a two-ply cross-ply asymmetrical laminate.
The constitutive equations (4.21) can also be written in an inverted form as follows:

(4.25)

Application of iso-stress field to the middle plane yields the following equations for the
effective compliance constants:

(4.26)

The above equations provide upper bounds of the compliance constants and lower
bounds of the stiffness constants when inverted. For non-hybrid weaves, the averages
can be simplified as:
- .- 2 . -

a, =aii, b, =(1--)6.., V d.. =d* (4.27)
11
11
nb
where al;, bi and dli are the stretching, stretching-bending coupling and bending
compliance matrices of a two-ply cross-ply asymmetrical laminate.
Mosaic model provides upper and lower bounds for the effective stiffness and
compliance constants for a unit cell of woven composite. However, fibre continuity and
non-uniform stresses and strains in the interlaced region are not considered although a
good agreement between predictions and experimental results was reported. It is clear
that fibre continuity and undulation are not taken into account in the idealisation
process. Consequently, a one-dimensional crimp model named as “fibre undulation
model” was proposed that takes into account the fibre continuity and undulation omitted
in the “mosaic model”.
Figure 4.3 depicts the concept of the fibre undulation model. In this model, it is
assumed that the geometry of fibre undulation in the weft yarn can be expressed in the
form of the following sinusoidal function within the length of a,:

(4.28)

and the sectional shape of the warp yarn is assumed to take the following form: