ANALYTICAL METHODS FOR TEXTILE COMPOSITES
6-2
increases in the applied load cannot then increase the average axial stress in the cracked ply.
In textile composites, transverse matrix cracks form most easily
tows aligned
normal to the load. Indeed, in many textiles, cracks may not form within the tows at all.
In such cases, the crack spacing is simply determined by the tow diameter. Transverse
cracking of this kind is seen between tows in 2D weaves and 3D interlock weaves under in-
plane tension [6.1]. A credible approach to predicting the associated softening of a cracked
layer of transversely oriented tows would be to embed the same mechanics used in
analyzing matrix cracking in 0/90° laminates [6.2] in a quasi-laminar model of the textile.
However, none of the codes available to date performs this operation.
In other textiles, transverse matrix cracks do form within tows. The effect of such
internal crack systems has been modeled with some success by simply knocking down the
transverse stiffness of the affected tows to some arbitrary, small value [6.3].
6.1.2 Shear Deformation
Figure 4-2(a) showed shear deformation data acquired in a test of a ±45° tape
laminate in uniaxial tension. Such approximately elastic/perfectly-plastic behaviour is
common to most polymer composites loaded in shear, including textile composites (Fig. 4-
2(b)). The ±45° laminated test specimens yield reasonable constitutive laws for tow
segments in textile composites made of the same fibers and matrix and with the same fiber
volume fraction. For modeling textiles, the data could be fitted with a simple
elastic/perfectly plastic law with flow stress τ
c
(Fig. 4-2(a));
2
or a numerical, parametric
curve such as a Ramberg-Osgood strain hardening law [6.4],
γ
=
τ
/G
xy
[1 + 3(
τ
/
τ
c
)
n-1
/7] (6.1)
where
τ
is the axial shear stress,
γ
the axial shear strain, G
xy
the shear modulus,
and n is a
hardening exponent.
6.1.3 Plastic Tow Straightening
Waviness in nominally straight tows allows nonlinear axial strain when the tow
straightens (Section 4.3). Constitutive laws for axial plasticity due to tow straightening can
be based on the constitutive law for axial shear discussed above. The resolved axial shear
stress,
τ
s
, due to the tensile stress,
σ
x
, in the nominal tow direction in the presence of
misalignment,
φ
, is approximately
τ
s
=
σ
x
φ
(6.2)
where
φ
is the local misalignment angle. When
τ
s
exceeds the flow stress,
τ
c
, the tow
segment will begin to straighten. The evolution of the straightening depends on the
distribution of
φ
. An analytic constitutive law is readily developed for the common case
that
φ
is normally distributed. The law is completed by specification of the width,
σ
φ
, of
the distribution, which usually requires an experiment.
2
“Shear flow” here refers to the regime of nearly perfectly-plastic shear deformation visible over large
strains in Fig. 4-2. The deformation is mediated by microcracking, crazing, and frictional slip, as already
noted.
