ANALYTICAL METHODS FOR TEXTILE COMPOSITES
z
c
= ±
t
2
. (2.7)
where x
c
is measured from the cross-over. The yarn cross-section shown in Fig. 2-16
consists of a straight central portion, with sinusoidal, lenticular ends. If the area of this
shape is known (from eqs. 2.3 or 2.4) then the shape parameter L
u
(defined in Fig. 2-16)
can be determined by
L
u
=
(2.8)
Textile composites are often characterized by the “crimp angle”,
θ
c
. This is the
maximum angle the yarn makes with respect to the x-y plane (Fig. 2-16). From the simple
geometric description just given, the crimp angle must be given by
θ
c
= Tan
−1
t π
2L
u
(2.9)
The constraint w ≥ L
u
imposes a minimum value for
θ
c
.
The yarn cross-sections and crossover points are similar for satin weave fabrics.
The distinguishing characteristic is the longer straight segments of yarn.
2.3.2.2 2D Braids
A simplified description of the unit cell geometry for a bias and a triaxial braid is
given in Ref. [2.10]. The formulation is similar to that given for the plain weave in the
previous section. However, because the yarns interlace at nonorthogonal angles, the
equations are more complex. They are not reproduced here.
More advanced process models attempt to predict the yarn cross section from the
mechanical interactions between yarns as the textile is formed. One such model includes the
twisting that must occur when yarns cross at non-orthogonal angles [2.14]. However, even
advanced models assume relatively simple forms for cross sections.
2.3.2.3 3D Weaves
Even though unit cells can be quite large because of the complex patterns preferred
for the phases of warp weavers, geometric idealizations for 3D interlock weaves are quite
