PREDICTION OF ELASTIC CONSTANTS AND THERMAL EXPANSION
where the a
ij
are the direction cosines between the local tow coordinate system and the
global system; and
T T
T
σ ε
−
=
1
, (5.25)
which follows from orthogonality. (These transformations are well known results of
tensor algebra [5.20,5.21].) For a tow whose axis is orientated at angles
θ
and
β
with
respect to the global system, as shown in Fig. 5-7, the a
ij
are given by
a
ij
=
cos
θ
cos
β
sin
θ
cos
β
sin
β
-sin
θ
cos
θ
0
-cos
θ
sin
β
sin
θ
sin
β
cos
β
. (5.26)
The tow is assumed to be transversely isotropic. Therefore, there is a degree of freedom
in the definition of the local coordinate system. Without loss of generality, Eq. (5.26)
refers to coordinate systems in which the 2 axis of the tow is perpendicular to the global
z-axis.
Figure 5-7. Coordinates for transformation of tow properties.
Orientation averaging preserves the symmetry inherent in the tessellation of
grains over large gauge lengths. Thus, using an example common in flat textile sheets, if
three orthogonal planes of symmetry exist in the pattern of grains, then the macroscopic
stiffness and compliance tensors derived by orientation averaging will exhibit orthotropic
symmetry (e.g., Fig. 5-8).
Whether either Eq. (5.22a) or (5.22b) is a good approximation depends on the
textile architecture. Isostrain conditions apply when translational invariance obtains in the
direction of the applied load. Isostress conditions apply when translational invariance
