ANALYTICAL METHODS FOR TEXTILE COMPOSITES
5.1.7 Orientation Averaging
"Orientation Averaging" is based on two tenets: that the textile composite can be
represented geometrically as a tessellation of grains of unidirectional composite; and that
either isostress or isostrain conditions apply [5.15-5.19].
Grains are usually defined according to the ideal fabric geometry implied by
geometrical models of the textile process. Irregularities such as pinching, waviness, or
crimp are not modeled. Curved tow segments are usually divided into just a few grains in
each of which the fiber orientation takes a single, spatially averaged value. Because
curved tows are ubiquitous in textiles, the definition of grains is obviously not unique.
Each grain is assigned the elastic properties of a unidirectional composite using a
model of the kind discussed in Section 5.1.2. Let C
(α)
denote the stiffness matrix for
grain
α
; and S
(α)
the corresponding compliance matrix. The components of C
(α)
and S
(α)
refer to an axis system aligned with the local fiber direction. Denote the composite
stiffness matrix C and the compliance S, both defined relative to a global coordinate
system common to all grains. Then
(isostrain conditions) (5.22a)
or
(isostress conditions) (5.22b)
where
and
denote C
(α)
and S
(α)
respectively transformed into the global
coordinate system:
(5.23)
where T
σ
and T
ε
are the stress and strain transformation matrices, respectively, given by
( )
T
a a a a a a a a a
a a a a a a a a a
a a a a a a a a a
a a a a a a a a a a a a a a a a a a
a a a a a a a a a a a a a a
ε
=
+ + +
+ +
11
2
12
2
13
2
12 13 11 13 11 12
21
2
22
2
23
2
22 23 23 21 21 22
31
2
32
2
33
2
32 33 33 31 31 32
21 31 32 22 23 33 22 33 23 32 23 31 21 33 21 32 22 31
11 31 12 32 13 33 32 13 33 12 11 33 13 31
2 2 2
2 2 2
( ) (
)
( ) ( ) ( )
( ) ( ) ( )
a a a a
a a a a a a a a a a a a a a a a a a
31 12 32 11
11 21 12 22 13 23 12 23 13 22 13 21 11 23 11 22 12 21
2 2 2
+
+ + +
(5.24)
