A collection of rubber bands. Some scraps of balloons. A tennis ball cut in half.

Such simple props are what mathematician Scott Spector uses to introduce novices to his complex field of research. Spector is internationally known for analyzing equations relating to elasticity, in order to help predict the behavior of materials and understand material failure. 

Nonmathematicians think of elasticity as a physical property. But it is also a mathematical theory used to explain what happens when certain polymers (like rubber) and crystals are subjected to stress. 

As a mathematician who’s also a specialist in solid mechanics, Spector is a rare bird. His contributions to his field won him SIUC’s Outstanding Scholar Award for 2000. The National Science Foundation has supported his research continuously since 1986, and he’s also had grants from the Air Force Office of Scientific Research. 

Spector’s colleagues consider him one of the world’s top theorists in nonlinear elasticity, a mathematical theory used to study material instability. Engineers draw on such findings in designing and manufacturing new materials. 

Nonlinear elasticity describes the behavior of materials that can bend in more than one way under stress, or that can take different configurations at rest and then return to their original form. Spector demonstrates the latter with the tennis ball half. He can turn it inside out and it still sits stably on his desk; then he can restore its original, right-side-out state. Modeling such a material requires equations to which there can be more than one answer.

Most of Spector’s work applies to elastomers—elastic polymers, such as silicone rubber. But, he says, "There are lots of materials you can model with nonlinear elasticity—and you can then make first predictions. If you know the material is going to fail elastically, there’s no reason for going further" with more complex models, such as those needed to deal with properties such as plasticity. 

 He adds, "A material isn’t intrinsically elastic or plastic; it all depends on what you’re trying to do with the material as well as the time scale involved." 

Stretch and release a rubber band and it will spring back to its original length. Do the same to, say, a band of plastic and it will stay permanently deformed to some extent. But stretch either one too hard, and it will break. Or keep a rubber band stretched out for a long time, and it will lose its ability to return to its original length. Elasticity and plasticity, like other properties of materials, have limits.

For more than a decade Spector has worked on elasticity equations relating to a type of material instability called cavitation: the formation of holes in a solid material when tensile (outward) forces are applied. 

When cavitation progresses too far, it causes cracking and material failure. But in elastic materials, cavitation can be helpful: engineers sometimes induce it to prevent fracture. For example, holes in an elastomer, which are able to expand and shrink, can allow the material to endure stresses that might otherwise tear it apart.

Cavitation in elastic materials has a lot to do with the material’s stored energy. (Force, or stress, imparts energy: when a rubber band is being stretched, it contains more energy than when it isn’t.) When an elastic material under tensile stress is trying to reach a lower energy state, will holes form? If so, how many, what shape, and in what locations? How does changing the temperature of the material affect the formation of holes?

"These are mathematically idealized problems relating to cavitation," says Spector. But, as he points out, if theorists can’t answer the simplified questions, they can’t hope to model the complex, real-world behavior of materials.

One of Spector’s major achievements, with fellow mathematician Stefan Müller, was proving it’s possible for an elastic material to reach equilibrium at its lowest possible energy state by cavitating. That finding had no immediate practical application. But by giving mathematicians a much better understanding of the equations involved, it enabled them to tackle other sorts of problems.

"What the best pure mathematicians in the world are doing is inventing new tools so that applied mathematicians like me can use them to solve particular problems," says Spector. By testing the ability of these new techniques to describe certain phenomena, he advances the state of the art in his field. For instance, he and his wife, SIUC mathematics professor Kathy Pericak-Spector, have published the first findings on "dynamic" cavitation in elastomers—the formation of holes over time.

Spector’s work, although theoretical, rests on a solid grounding in physics, and some of it does bear on matters of immediate interest to materials scientists. 

For example, he is attempting to apply elasticity models to a problem in fiber optics. The cores of optical fibers (which are made primarily of silica, a glass) are vulnerable to sudden failure caused by bullet-shaped holes cascading down the length of the fiber. Scientists have speculated that the laser light, focusing and refocusing down the length of the fiber, melts the silica at regular intervals or even heats it enough to form gas bubbles. 

But Spector thinks the holes might be caused by elastic cavitation. The manufacturing process for optical fibers introduces a lot of residual stresses into the silica as it is pulled into thin strands. Although glass is not ordinarily thought of as an elastic material, Spector thinks that the optical fibers might, in a sense, be behaving elastically. Elasticity equations might be able to explain the material failure involved and predict when it would occur. 

Spector decided he wanted to be a mathematician when, in the fifth or sixth grade, he read a Time-Life Series book on math. "Working math puzzles was like a game," he says. "I wanted to do that for a living rather than working."

He now knows what hard work math can be. But solving the problems, he says, remains a delight.

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