Proof expands limits of composite materials

n
an advance that could lead to composite materials with virtually limitless
performance capabilities, a University of Wisconsin-Madison scientist
has dispelled a 50-year-old theoretical notion that composite materials
must be made only of “stable” individual materials to be
stable overall. Writing in the Feb. 2 issue of the journal *Physical
Review Letters,* Professor Walter
Drugan proves that a composite material can be stable overall even
if it contains a material having a negative stiffness, or one unstable
by itself—as long as it is contained within another material that
is sufficiently stable. “It’s saying you’re allowed
to use a much wider range of properties for one of the two materials,”
he says.

Comprising everything from
golf clubs and bicycle frames to bridge beams and airplane wings, composite
materials, or materials made by combining multiple distinct materials,
deliver such unattainable single-material advantages as optimal combinations
of high stiffness, strength, lightness, hardness, fracture resistance
or economy. “The idea is that you have one material with some
great properties, but it also has some disadvantages, so you combine
it with another material to try to ameliorate the disadvantages and
get the best of both,” says Drugan.

Until now, materials engineers
adhered to proven mathematical limits on composite performance, he says.
“For example, if you give me two materials and one has one stiffness
and the other has another stiffness, there are rigorous mathematical
bounds that show that with these two materials, you cannot make a material
that has a stiffness greater than this upper bound,” says Drugan.
“However, all these theoretical limits are based on the assumption
that every material in the composite has a positive stiffness—in
other words, that every material is stable by itself.”

When slightly disturbed,
stable materials, like those with positive stiffness, return easily
to their original state. A slightly compressed spring, for example,
bounces back after the compression force is removed. Unstable materials,
like those with negative stiffness, quickly collapse or undergo a large,
rapid deformation at the slightest perturbation. In an example from
the structures field, if a vertical column supports a load that becomes
too great, even a slight disturbance can cause the column to buckle.

The idea of incorporating
a material with negative stiffness into a composite designed to be highly
stiff originated with UW-Madison Wisconsin Distinguished Professor of
Engineering Physics Roderic Lakes, says Drugan. Some six years ago,
Lakes noticed that, in the mathematical formulas that predict how a
composite will perform based on its component material properties, employing
a material with a suitably chosen negative stiffness theoretically would
yield an infinitely stiff composite.

Lakes took his ideas into
the lab, where he created such a composite by embedding a material that
behaved like one with negative stiffness in a matrix of a material with
positive stiffness—somewhat like the shell of a golf ball surrounds
its core. Through dynamic experiments, conducted under oscillatory loading,
he showed that the composite stiffness was greater than the mathematical
bounds indicated it could be, given the combination of materials.

Since Lakes’ experiments
were dynamic, and since dynamics often has a stabilizing effect, it
remained unknown whether such material response could be obtained in
the static loading case, which is practically important since many structural
components are designed to support static loads.

Lakes and Drugan, who have
had a continuing research collaboration on this topic, published a 2002
paper in the *Journal of the Mechanics and Physics of Solids*
in which they showed that if a composite material containing a negative-stiffness
phase could be stable, and if they tuned the negative stiffness the
right way, the predicted composite property could be infinite stiffness
for a broad range of composite materials.

Then Drugan set out to prove
theoretically that such a material can be stable under static loading.
“In general this is a very challenging problem, but I finally
found a clean way to analyze it,” he says.

He applied a stability theory
criterion that says that a sample of material will be stable under all
possible small perturbations, or changes in size and shape, if the total
internal energy produced by such a perturbation always exceeds work
done by the applied loads. His analysis
also determines minimal requirements for each material of the composite,
so that the overall composite still will be stable. Roughly, the material
with positive stiffness must be stiff enough to stabilize the material
with negative stiffness—yet there are limits on how large the
negative stiffness can be.

Although Drugan analyzed
composites made of only two materials, he believes researchers easily
could build upon his proof to determine the stability of composites
made of even more materials. And though his proof examines materials
in which negative stiffness is the cause of instability, he says it
may be possible to make stable composites from a stable material and
an unstable material whose instability is produced by another phenomenon.
“Once you’ve shown this is possible in one case, then you
start to believe that it’s probably going to be true in these
other cases, too,” he says.

He hopes his proof will awaken
materials engineers to a new, broad range of possibilities for making
composite materials. “If you’re going to make a composite
material from two different materials, you think about all the possible
properties that each of the individual materials can have in order to
obtain an outstanding overall performance,” he says.

“If you’re suddenly
able to greatly expand the range of properties that one of these materials
can have, then you have a much wider range of possibilities for the
overall composite. And that’s what this research does. It says,
‘You don’t need to limit yourself to two stable materials
anymore.’ And in fact, when you do search for composites with
the best possible properties, you should look at all the possibilities,
including this allowable range of negative stiffness.”