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Getting More Stiffness with Less

Mike May

Everyone understands intuitively the mechanical concept of stiffness. A baseball bat is stiff; a garden hose is not. Mechanical engineers often pursue enhanced stiffness through the use of composites—materials composed of more than one component, such as the fiberglass used in pole vaulting or the carbon fiber used in some bicycle frames. To make a stiffer composite, it makes perfect sense to use stiffer components. In fact, all mechanical engineers followed that rule, until Roderic Lakes at the University of Wisconsin at Madison started experimenting with so-called negative stiffness.

Flexible ruler displays negative stiffness . . .Click to Enlarge Image

In general, a material's stiffness creates a force when something tries to deform it. We associate the term with positive stiffness, or things that push back when you try to change their shape. Something with negative stiffness, on the other hand, actually creates a force that amplifies the deformation. As shown in the illustration, you can take a flexible plastic ruler and stand it on end. Push down on the upper end until the ruler bends out to one side. While continuing the downward pressure, use one finger to push against the bulging side of the ruler. As you push harder, the ruler takes an S shape. Keep pushing and, eventually, the ruler pops through and bulges out on the opposite side. That is negative stiffness.

This instability bestowed by negative stiffness makes it seem like a poor feature to add to anything, especially to something that should be stiff, or resist external forces. In fact, when one mechanical engineer, who happens to work with composites, heard about the possibility of using negative stiffness to make something stiffer, he said, "It does not sound very realistic." Such a reaction might be found often, because, as Lakes says, "Nobody has ever really thought that you could do a composite with negative stiffness. So people have had a narrow window on this sort of thing."

Lakes, on the other hand, seeks out the unexpected. "Part of our general theme of research," he said, "is to try to expand the space of possibilities and make materials do what normally people think they shouldn't be able to do." With a twist here and there, he changes how materials perform. Imagine the ruler example again, but glue your pushing finger to it. Now, when the ruler tries to pop through to the other side, it pulls on your finger. But what if your finger pulled back, like a spring? In other words, maybe you can stabilize materials with negative stiffness by attaching them to materials with positive stiffness. That's just what Lakes thought. In describing this, he said, "There's kind of a dance or balance between the positive and negative."

He started attacking this problem from the theoretical side. A variety of mathematical equations calculate the stiffness of a composite, but no one had ever tried using a component with negative stiffness. In the March 26 issue of Physical Review Letters, Lakes published calculations that show that a composite created from parts with positive and negative stiffness could surpass the stiffness of one made entirely with positive stiffness. In fact, Lakes said, "If we were able to stabilize things perfectly, the theory has no limit to the stiffness."

But what about real-world experiments? In the February issue of Philosophical Magazine Letters, Lakes described a series of experiments with silicone tubes that had an outside diameter of 10 millimeters. When he compressed one of these tubes lengthwise, it pushed back, showing positive stiffness. Then he buckled the side of one tube, making it look like a partially crushed soda can. The buckled silicone tube displayed negative stiffness. He made composites from these materials by taking a tube that was 40 millimeters long and then gluing short, non-buckled pieces across the top and bottom, which created an I-shaped composite of three tubes. He applied sinusoidal compressions to the top of the I, and recorded how the structure damped the vibrations. In experiments that included a buckled tube for the upright portion of the I-shaped composite, its maximum damping was orders of magnitude higher than that of composites composed of all positive-stiffness components. The experiments suggested that a composite made of materials with positive and negative stiffness could be very useful in anything subjected to vibration, such as airplane wings.

Lakes also looked at a microscopic system. As described in the March 29 issue of Nature, Lakes and his colleagues created a composite of tin, which has positive stiffness, and vanadium oxide, which has negative stiffness over a range of temperatures. Crystals of vanadium oxide switch between two shapes, much like a flexible ruler. In a composite composed of tin and just one percent vanadium oxide, the peak stiffness increased by about eight percent over pure tin. "That's what's really so counterintuitive," Lakes said. "You really don't need as much negative stiffness as you think."

A little pliancy in the mind of an investigator can't hurt either.

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