ANALYTICAL METHODS FOR TEXTILE COMPOSITES
6-8
τ
k
= (
τ
c
−
τ
xy
)/
φ
(6.3)
which leads to the sloping boundaries for negative axial stress in Fig. 6-2(a). The
maximum sustainable axial shear stress,
τ
xy
, is
τ
c
, which imposes the horizontal boundary
in Fig. 6-2(a). Fiber rupture under aligned tension is approximately unaffected by
simultaneous shear, so that the right hand boundary in Fig. 6-2(a) is vertical.
Figure 6-2(b) shows the situation for combined axial shear and transverse tension
or compression. Once again, the failure locus is bounded on the shear axis by
τ
c
, the axial
shear flow stress. The right hand boundary is limited by matrix cracking under transverse
tension, imposing a critical stress σ
y
(c)
in the absence of axial shear. Transverse
compression will probably lead to transverse shear failure in tows at some stress σ
ts
,
although direct observation of the failure mechanism has not been reported. The failure
locus in the space (σ
y
,σ
xy
) has been completed as an ellipse, following the quadratic
strength rules popularized for unidirectional composites [6.19]. Fleck and Jelf defined an
effective stress consisting of a quadratic combination of σ
xy
and σ
y
in developing a model
of plasticity in polymer composites under combined transverse tension and axial shear
[6.20]. They also suggested an elliptical failure locus, reasoning that failure under such
biaxial loading is a plastic instability (kinking), but their experimental confirmation is
sparse and indecisive.
Swanson et al. propose the simple failure criterion in 2D braids under multiaxial
loads that the
strain in any tow system should exceed a critical value. This leads to
failure loci that are parallelograms, with encouraging but imperfect correspondence with
experiments [6.21]. Their proposal, which is similar to a first ply failure condition, is
reasonable as long as one set of approximately straight tows with significant volume
fraction is aligned with the major stress axis. In aerospace applications, where structures
must be strong and stiff, this situation will usually be the preferred design. However,
their approach will be of dubious value for any textile architecture that admits failure by
matrix-mediated shear.
6.4 Codes for Predicting Nonlinear Stress-Strain Behaviour and Ultimate
Strength
The capabilities of the codes collected in the handbook for predicting nonlinear
stress-strain behaviour and ultimate strength are summarized in Table 6.2.
6.4.1 Nonlinearity
Different approaches have been used for modeling nonlinearity due to plastic shear
and tensile microcracking. Nonlinearity due to shear plasticity is generally accompanied
by material hardening (e.g. Fig. 4-2), which is modeled by replacing the shear terms of the
local stiffness matrix with a strain-dependent function that approximates the experimentally
measured response. Computation proceeds incrementally, following some algorithm to
insure convergence to a self-consistent, equilibrium state. TEXCAD implements this
approach. Nonlinearity due to tensile microcracking is usually handled by monitoring each
