ANALYTICAL METHODS FOR TEXTILE COMPOSITES
used to define a special element for a finite element calculation on a coarser scale. But
the coarser finite element calculation should not be categorized as a unit cell calculation.
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Fig. 5-4. Nonperiodic strains in a periodic structure under nonuniform external
loads. Schematic of two unit cells in a larger array.
Several other factors limit the usefulness of the unit cell approach. If the textile
architecture is periodic but complicated, the unit cell can become prohibitively large. In a
2D plain weave, a computational cell can be defined that contains just a handful of tow
segments (e.g. Fig. 5-3). In a 3D interlock weave, in contrast, the unit cell may contain
segments of ~100 tows. One example is the angle interlock woven panel of Fig. 5-5, in
which the phase of the warp weavers is staggered in the filler direction. This architecture
has a period of ten fillers in the stuffer direction and ten stuffers in the filler direction.
The unit cell also contains ten warp weavers; and, because of the absence of translational
invariance in the through-thickness direction, it must span from top to bottom of the
panel. In all, it contains segments of 140 different tows. Finite element calculations of
such a cell, with grids fine enough to represent the details of each tow's geometry, are not
viable. To proceed, further assumptions about the distributions of stresses must be made
to break the cell down into smaller constituents.
5.1.5 Macroscopic Length Scales
The analysis of a structure is greatly simplified if the structural material can be
treated as homogeneous over length scales comparable to any feature of the structure.
Identifying the minimum length scales for homogeneity defines the term macroscopic in
discussing elastic or other mechanical properties. A measured elastic constant is only a
