JEC COMPOSITES MAGAZINE - Issue #95 - Special JEC Europe - March 2015 - 71

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Composite materials:
unlocking mystery with invariants

A common, understandable invariant-based framework has been
discovered that can rate all CFRPs and their processes, simplify
design and testing, and lower the barrier of acceptance.

stePhen W. tsai, Professor researCh
eMeritus, dePartMent of aeronautiCs &
astronautiCs, stanford university
Jose daniel d. Melo, Phd Professor of Materials
engineering federal university of rio grande do norte

n a speech delivered at CompositesWorld's Carbon Fiber 2014
conference on December 10, 2014,
John Byrne, head of materials and
structures for Boeing Commercial
Airplanes, talked about the lessons
learned from their pioneering effort
in using CFRP for the 787 primary
structures, and cited explicitly, "Composites in general are poorly understood... therefore not employed
optimally." We hope that our effort in
the new book "Composite Materials
Design and Testing" can demystify
the often unnecessary complexity of
composite materials and offer more
orderly, practical solutions.
There are simpler and easier approaches that will lead to new opportunities in design and testing. In
traditional practice, ply properties are
the starting point. There are 4 elastic
constants (Ex, Ey, nx, and Es) and 5
anchors for strength (X, X', Y, Y' and
S). Ex and X and X' are the easiest to
measure; the others are more difficult. Shear modulus and strength can

hardly be measured directly. These
properties are treated like random
variables. What is worse, designers
demand their allowable values. It is
our plan to show that these properties are all related, and can be cast
into a general framework. With such a
framework, only the easiest test of [0]
be needed, from which all laminate
stiffness and strength can be inferred.
The unifying entity is the trace of a
material.

Trace is the one
Trace is the sum of the diagonal
components of the stiffness matrix of
metals and composites; i.e., Tr [A] =
A11 + A22 + 2A66. This is an exact
mathematical relation, not just a convenient collection of the elastic components. This invariant (that has no
directional preference or variation) is
the one and only number needed to
completely characterize the stiffness
of all composite laminates formed by
carbon fibre-reinforced plies. Instead
of the building block approach that
goes from the bottom of a pyramid
to the top, we see trace as the shining
light from the top of the pyramid that
unlocks the stiffness of all composite
materials. It is that simple and cor-

rect; not an empirical fit.
Testing of coupons in the bottom row
of the traditional strength pyramid
can be replaced by data from [0]
specimens; we can go directly to
as-built elements and components.
Trace defines the upper bound of
laminate performance and sets a
standard to rate the quality of design,
manufacturing and testing. It makes
composites as clear and straightforward as metals. Each material has a
value in trace, which is easy to estimate for unidirectional or fabric plies,
and their laminates.
In Table 1, the value of trace for 16
unidirectional CFRP are ranked
in descending order in the second
column from the left. With an order
of 1% accuracy, these 16 common
CFRP materials can be represented
by a master ply using their median
values. The trace-normalized effective
stiffnesses E1 and their coefficient
of variation for 7 laminates defined
based on the percentage of [0],
[±45] and [90] plies are shown in
the bottom 2 rows. For example, the
effective stiffness E1 of [π/4] is 0.34
x Trace. For IM7/9773 with a trace
value of 218 GPa, E1° = 0.34 x 218 =

No95 March 2015 / jec

composites magazine 71

JEC COMPOSITES MAGAZINE - Issue #95 - Special JEC Europe - March 2015

Table of Contents for the Digital Edition of JEC COMPOSITES MAGAZINE - Issue #95 - Special JEC Europe - March 2015