JEC COMPOSITES MAGAZINE - Issue #102 - January/February 2016 - 45

[π/4] would be 0.336 x 180 = 60, as seen in Figure 4. The accuracy
is a fraction of 1% among over a dozen carbon/epoxy materials. In
fact, this can be a good way to measure the manufacturing quality of a
given [π/4] laminate. Inhomogeneities like voids, laminate thickness,
degree of cure and ply wrinkling will all cause lower stiffness values
than the pristine value based on trace. Equally useful is the change
in trace due to temperature and moisture, which in turn will affect
laminate stiffness and buckling strength.

Fig. 7: Relative cost of composites as a linear function of trace found in multiple
products of Avanco

The upper stiffness is material dependent (IM7/8552 in this case).
The lower stiffness, that is trace normalized, becomes independent
of the material. Thus, it is universal and makes design simple, just one
solution for each laminate for all materials. Each actual material can
simply be scaled by its trace (the use of universal laminate constants
corresponds to the value of trace being unity, thus for each material
the scaling factor is simply the value of it trace). Only trace can do
this. It's a miracle, truly true.

Trace is simple and simplifies design
Trace is exact, invariant and rational. It is not an arbitrary collection
of stiffness components. Other invariants of the same stiffness matrix
cannot do what trace can do. It represents rightfully the total stiffness
capability of a material. Lamination changes the distribution of the
stiffness but not the total capacity that is fixed.
For carbon/epoxy composites, trace tells us that only the fibre
dominant components, like Qxx and Ex, need to be measured. They
are 88% of trace. Transverse and shear moduli, which are only 5 and
3% of trace, can be ignored without much impact on the stiffness of
the laminates. For testing to generate laminate stiffness, only Ex is
needed. Trace would simply be Ex/0.88. The less reliable and more
costly shear test is no longer needed.
Once the value of trace is known, the stiffness components of all
laminates will be known, e.g., if trace for IM7/8552 is 188 GPa, E1 of

The most profound change brought on by trace is the realization
that each laminate is a material system by itself. This is contrary to
the traditional view that each material, such asIM7/8552, is very
complicated because it can have multiple laminates. Design allowable
requires multiple materials with multiple laminates (as seen on the
left of Figure 4). In this new view embodied by trace, each laminate is
a material system that includes all CFRPs. The universal laminate has
a trace value of unity. So, boundary-value problems for plane stress
and plate bending need to be solved only once for each laminate.
With trace for each actual material, scaling can be applied to stress
solutions, such as open hole, natural frequency and buckling, by postprocessing, without recalculation. For isotropy, the stress concentration factor of OHT is 3. This factor is independent of material, metals
or quasi-isotropic laminates. Trace teaches us that this constant stress
concentration is true for all laminates, e.g. for [0/±30], it is 2.5 for all
CFRPs (exact solution from Lekhnitskii's Anisotropic Plates). With
trace, this simplification is truly true.
With 124 composites and their hybrids, Avanco found a correlation
between cost and trace, as seen in Figure 7. Only trace was found to
unify such data.

Training opportunities
A special tutorial session on trace, with a free book from Tsai and
Melo, and design and testing tools will be offered at JEC World in
Paris (9-11 March 2016) and during the online Composites Design
Workshop XI by Stanford University on 1-4 February 2016. These
are opportunities for engineers to learn and to become proficient in
using trace to make their composite materials and structures simple,
sensible, and competitive. Such a miracle is truly true. n
More information:
www.stanford.edu/group/composites

Composites Design Workshop XI
Online, Live, Truly World-class
February 1-5, 2016; Noon to 4 PM PST
20 intensive hours, plus homework problems
US$ 1,200: free hardcover and e-books,
Plus multiple Excel-based design tools
All sessions recorded/downloadable
No travel and subsistence expenses
Certificate for 3 CE credit hours: an extra $200
Cutting edge trace to revolutionize design/testing
Register: http://compositesdesign.stanford.edu

No102 January - February 2016 /

jec composites magazine 45

http://www.stanford.edu/group/composites http://compositesdesign.stanford.edu

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