ANALYTICAL METHODS FOR TEXTILE COMPOSITES
of the tow to composite stiffness is nearly independent of V
t
. The critical characteristic is
simply the total number of fibers in the tow, not their density of packing [5.8].
In the looping geometry of knitted fabrics made of yarns of approximately equal
weight
1
, on the other hand, all yarns have significant curvature over most of their lengths.
Macroscopic properties are not fiber dominated, but depend more strongly on the
response of tow segments to local transverse and shear stresses. Because transverse and
shear tow moduli depend nonlinearly on the assumed value of V
t
(Fig. 5-2), the
composite elastic constants can be influenced by the assignment of V
t
to a modest degree
[5.10].
5.1.4 Unit Cells and Periodic Boundary Conditions
Many textile processes yield patterns of interlaced tows or yarns that repeat in one
or two directions; they are periodic. A large volume of such a structure can be generated
in a model by stacking together unit cells, each of which represents the tows in one
period or cycle. (The term unit cell has been borrowed from crystallography, e.g. [5.11].)
The response of the textile composite to external loads can then be computed by
analyzing the behavior of a single unit cell with suitable boundary conditions.
For any periodic structure, there are infinitely many ways of choosing a unit cell.
For example, if a cuboid aligned with the axis system (x
1
, x
2
, x
3
) is one possible unit cell,
then so too is the rhombohedron obtained by shearing the faces normal to any one of
these axes in the direction of either of the other two axes through a displacement equal to
a multiple of its original length (Fig. 5-3). Equally, either unit cell outlined in Fig. 5-3
will remain a unit cell if it is displaced to the right or left by any distance.
Under uniform external loads, the stress and strain distribution in a periodic
textile composite must also be periodic. The analysis of a single unit cell should therefore
be subject to periodic boundary conditions for stresses or strains. The solutions obtained
for different unit cells chosen to represent the same periodic structure should in principle
be the same, although small differences may arise in practice because of inconsistent
approximations in the numerical methods used, especially in the definition of
computational grids. The choice of unit cell is usually guided by other symmetry
properties of the textile. For example, choosing the unit cell to be symmetric about a
1
When warp knitting is used to tie together heavy yarns (e.g., of carbon fibers) with a light thread (e.g., of
polyester), the heavier yarns are often kept as straight as possible by design. Generalizations intended for
fabrics with highly curved yarns are then inapplicable.
