JEC COMPOSITES MAGAZINE - Issue #102 - January/February 2016 - 44

trace
at 232 GPa, and the lowest one is C-Ply 55, at
139 GPa. Their pristine values shown in the
vertical axis are accurate within 1%. They could
be measured experimentally to determine the
as-built laminate that will contain manufacturing
defects and/or usage damage. Trace provides a
rational basis to assess the true performance of
laminates with or without imperfections. It is
equally easy to measure the effects of cold-dry
and hot-wet conditions through trace. Trace
makes scaling possible, i.e. if hot-wet conditions
reduce trace by 5%, the stiffness of all laminates
will decrease by the same amount. By the same
token, the relative stiffness of any laminate of
IM6/epoxy over IM7/977-3 is simply their trace
ratios, i.e. 232/218 = 1.064 or 6.4%, using the
data in Figure 4.
Fig. 4: Longitudinal stiffness of CFRP with and without trace

cannot be found, not to mention the nightmare in tape layup or
fibre placement. On the other hand, we will show that when trace is
recognized, there is a new order. This is also truly true.
With trace, the material/laminate optimization is drastically simplified. The right of Figure 4 shows the same longitudinal stiffness
components, but they are normalized by the respective trace of
each CFRP. For [π/4] of IM7/977-3 (the second CFRP after IM6/
epoxy), E1° is 73 GPa, and the trace-normalized value is 73/218 =
0.34. Suddenly, a universal ply/laminate emerges in the bottom row
that will cover all CFRP materials. The variation for this laminate
stiffness is 0.1%, far less than the common 3% experimental uncertainties. The optimization procedure is significantly simplified, and
now doable. It is now a 2-step operation, i.e. 1) selecting the best
universal laminate, and 2) scaling for the best material. Instead of a
mosaic of laminates for a structure, a homogenized structure with
a finite number of laminates can be selected to bring about weight
and cost savings that are not possible with a heterogeneous mosaic
structure.
With trace as the normalizing factor, laminate stiffness becomes independent of materials (see horizontal lines in Figure 5). Each circular
dot represents a CFRP, as listed in Figure 4; the highest one is iM6,

Fig. 5: Trace-based universal laminate constants for 4 laminates (trace values
are those listed in Figure 2)

Laminates simplified
The total stiffness components for an 8-ply thick and a 32-ply thick
[π/4] laminate, i.e. [0/±45/90]2 or 8 Total, are shown in Figure 6.
The highly heterogeneous laminate is on the left, the nearly homogeneous one on the right. In the upper part, their absolute dimensions
and the resulting trace for [A] and [D] are shown (185.7 and 15.5,
and 752 and 1002, respectively). Note that they are totally different
in not only their values but also their units. It is not possible to make
a judgment on these values. The lower part shows the power of trace.
When the stiffness components are divided by their respective trace,
their normalized (also universal and dimensionless) values are less
than 1%, i.e. 0.371 vs. 0.372. In fact, this is one measure of the degree
of homogenization. This is also an important design goal that can
strengthen a laminated structure and reduce cost at the same time.
This also confirms the assertion in Figure 1 that the traces for [Q],
[A*] and [D*] are equal for all laminates. Note 0.371 x (1 - 0.32) =
0.337, which is the E1* shown in Figure 4 for a homogenized [π/4],
where a Poisson's ratio of 0.3 is used. This is universal and covers all
CFRP materials.

Fig. 6: Stiffness of homogenized [π/4] in absolute and normalized units

44 jec composites magazine / No102 January - February 2016

Table of Contents for the Digital Edition of JEC COMPOSITES MAGAZINE - Issue #102 - January/February 2016