ANALYTICAL METHODS FOR TEXTILE COMPOSITES
the macroscopic scale. Under the isostrain assumption, the determination of the effective
or macroscopic stiffness tensor becomes a simple volume averaging of the constituent
stiffnesses, transformed from the local material coordinate system to the global
coordinate system (Section 5.1.7); i.e. the method of orientation averaging. Following the
formal mechanics reviewed in Section 5.1.6, this is equivalent to coupling homogeneous
displacement boundary conditions (Eq. (5.4)) with an assumption of linearly varying
displacements throughout the enclosed volume and finding the relation between average
stress and average strain.
In most of the codes in Table 5.3, the textile is treated as being periodic and the
averaging is performed over the volume, V, of the unit cell. In simple models that include
only straight yarn segments (e.g., the diagonal brick model), orientation averaging then
leads to a closed-form summation of volume weighted stiffnesses. In the more general
codes, such as TEXCAD, numerical integration is performed to account for the curvature
of tow paths.
When quasi-laminar textiles are approximated as translationally invariant in-
plane, orientation averaging leads to a simple sum over layers. Indeed, standard laminate
analysis applied to in-plane loads can be regarded as an instance of orientation averaging.
Orientation averaging can be applied just as well to nonperiodic textiles. This may
be done by substituting a periodic approximation for nonperiodic elements of the textile
architecture, defining a unit cell, and proceeding as above. Alternatively, some other
representative volume element can be defined and analyzed, large enough that spatial
variations associated with nonperiodic elements of the reinforcement have a small effect
on the calculated macroscopic (spatially averaged) stiffness tensor.
The isostrain assumption that leads to stiffness averaging also yields simple
expressions for thermal expansion coefficients. For example, the effective thermal
expansion coefficients,
, of the unit cell of a periodic textile follow according to [5.27]
α α
ij ijkl ijkl
V
kl
S
V
C dV=
∫
1
* *
(5.28)
where an asterisk again indicates transformation of local properties to the global
coordinate system and
denotes the spatially averaged compliance (the inverse of
).
