NONLINEAR STRESS-STRAIN BEHAVIOUR AND STRENGTH
6-3
Tow straightening can arise in nominally straight tows with irregularity; or in tows
possessing regular oscillations, such as in plain weaves. If wavy but nominally straight
tows are represented as ideally straight in a code (the most common case - see Section 5.2),
tow straightening should be considered independently of axial shear due to far-field shear
loading. It must be introduced as a modification of the response of the tow to axial tension,
in which the tow stiffens with increasing tensile strain. Similarly, if the geometry of regular
oscillations is not modeled explicitly (as in the code WEAVE), but introduced in the elastic
regime via in-plane stiffness knockdowns, then tow straightening will again be represented
as a modification of the axial properties of the tow. If, on the other hand, the paths of
regularly oscillating tows are represented more or less literally in a code, the axial shear
strains associated with tow straightening will usually be computed as part of the general
analysis of internal stresses and strains. The constitutive laws assigned to the tows need
not then be modified.
Because a tow segment stiffens when it straightens, tow straightening under tension
will progressively encompass all segments of a tow. Under axial compression, in contrast,
a wavy tow becomes increasingly wavy and softens. Softening will occur first in the most
misaligned segments, increasing the misalignment and thus the preference for softening in
the same locations. This mechanism of localization will lead to kink band formation.
6.2 Tessellation Models
To predict nonlinear composite behaviour and strength, the partitioning of stress
amongst tow segments with different orientations must be calculated and local stresses
compared with the known yield and failure characteristics of tows under general states of
stress. As recounted in Section 5, macroscopic elastic properties are often approached by
modeling a textile composite as a tessellation of grains within each of which the fiber
orientation is approximately constant. (The grains may be finite or infinite. For example,
each ply in a tape laminate or infinitely long, nominally straight stuffers in a 3D interlock
weave would be described in a tessellation model as single grains.) The elastic constants of
a unidirectional composite are assigned to each grain. Using the same tessellation model in
predicting nonlinearity and strength, the nonlinear constitutive laws and failure criteria
appropriate to each grain might also be guessed to be those of a unidirectional composite,
modified to account for the higher degree of irregularity endemic to textiles. The status of
our knowledge of the details of these constitutive laws and failure criteria will be
summarized below.
In most of the models of nonlinearity in the codes collected here, matrix plasticity or
damage is assumed to be uniform within any tow or set of equivalent tow segments, i.e.,
within equivalent grains. This reduces the number of degrees of freedom to a manageable
level. If isostress or isostrain conditions are also assumed (more reasonably the latter when
predicting nonlinearity in fiber-dominated strain components), the resulting models are
straightforward generalizations of the orientation averaging models used with frequent in
success in the elastic regime, with incremental stiffnesses replacing constant stiffnesses.
Assuming isostrain conditions implies displacement continuity across grain
boundaries. Even when matrix cracks appear between neighbouring tows, friction inhibits
sliding, especially in 3D architectures, where tow separation is opposed by their
interlacing. Neglecting sliding between tows, i.e. assuming coherent tow interfaces,
probably has a negligible effect on macroscopic nonlinearity, at least until high strains
(typically > 0.05) bring severe damage.
