# Math solution has many applications

The solution to a mathematical problem MIT engineers originally tackled as a "theoretical curiosity" could lead to dental implants that won't crack and tank armor that's more resistant to missiles. It could also radically change the way automotive companies inspect the gears in car transmissions, saving time on the factory floor.

"It was so much fun playing with the problem from a scientific point of view," said Subra Suresh, the Richard P. Simmons Professor of Materials Science and Engineering. "Unbeknownst to us, its solution had many applications."

The researchers' achievement was to develop a mathematical theory to describe a ubiquitous class of materials. They are now using that theory to create new materials with exceptional properties. The work has also led to a more efficient way of testing the quality of components made of these materials.

In graded materials -- the focus of the MIT research -- two or more different materials are combined in such a way that the proportion of one is greater at the surface but is gradually replaced by another with depth.

"A samurai sword is a good example of a graded material," said Professor Suresh, who holds appointments in the Departments of Mechanical Engineering and Materials Science and Engineering. "It's made in such a way that the material at the surface is very sharp. But that material is also brittle and can break, so as you go inward, the material changes so it's not as brittle." The gear teeth in car transmissions, human teeth and the Earth itself are other graded materials.

About three years ago, Professor Suresh began a program to create novel graded materials. Once such a material was developed, the researchers planned to explore how its properties changed along the graded continuum. This proved to be more difficult than they expected.

There are standard ways for measuring the properties of a single material by "indenting" it. An indentor works by poking the material of interest with an object like a sharp probe, then measuring the force applied and how deeply the object penetrates the material. "This gives you a great deal of information about the material and its properties," Professor Suresh said.

But the engineers quickly found that this wouldn't work for graded materials. "There was no [mathematical] theory to interpret the measurements from such indentation instruments when they were applied to graded materials," he said.

So Professor Suresh and research scientist Antonios Giannakopoulos developed that theory. "The major impact of the theory is that it provides clean mathematical solutions involving quantities which can be accurately measured by experiments," Professor Suresh said.

The work has since led to patents on a new machine for testing graded materials and for the software that extracts information about the materials' properties. Both are licensed by Instron Corp., which was so interested in the work that it licensed it even before one of the patents was filed.

##### A NEW INSTRUMENT

After developing the theory, the researchers wanted to test it via computer simulations and experiments. The simulations confirmed their results, but experiments were impossible. The standard indentors for measuring materials' properties were available in two size ranges, but one range was too small and the other too large for the researchers' purposes.

With nanoindentors, for example, the probe used to poke the material is so tiny that it extends into only one section of a graded material, "so you can't generalize to cover the rest of the material," Professor Suresh explained. At the other extreme, the probe of a macroindentor is too big to capture the details of the gradient. "We needed something in between," Professor Suresh said.

So Professor Suresh and Jorge Alcala, now of Universidad de Castilla-La Mancha near Madrid, built an intermediate machine -- the microindentor -- from off-the-shelf parts for about \$5,000. (In contrast, nanoindentors sell for well over \$100,000.) The tests they conducted with the microindentor provided the experimental validation of their theory.

##### TRANSMISSION GEARS

The researchers have since found that the new machine coupled with the new theory has industrial applications. For example, it could speed the quality-control process for gears in car transmissions.

Automotive companies currently check the quality of these gears via indentation tests on the production floor. "But these gears are graded materials," Professor Suresh said. "So they indent the gear, get the properties of the first layer, then machine away some material from the surface and indent it again. And they have to keep doing that process. With our theory and the microindentor, they can just indent the gear once."

Professor Suresh noted that one of the patents for the work includes a flow chart of how to use the microindentor and interpret its readings. "We took out the equations so someone without a mathematical or engineering background could apply it on the shop floor," he said.

##### BETTER MATERIALS

The theory is also giving the researchers insights that could lead to materials with exceptional properties. For example, the theory predicted that the surface of a properly constructed graded material will be more resistant to cracking and wear.

That could have implications for, among other things, the materials used in dental and orthopedic implants. "These implants are currently made of a homogeneous material," Professor Suresh said. "By using a grad-ed material, the implant might last longer." For example, graded materials used for the armor on tanks might prevent the armor from shattering upon impact with a missile.

"What I find so exciting about this work is that what started as a pure theoretical curiosity has essentially led to a new field," Professor Suresh said.

The researchers have reported their work in the International Journal of Solids and Structures (IJSS) and Acta Materialia. A fourth IJSS article, with O. Jorgensen of Risï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ï¿½ National Laboratory in Denmark, is in press, as is an article in the Journal of the American Ceramic Society. The work was sponsored by the Department of Energy, the Office of Naval Research and the National Science Foundation.

A version of this article appeared in MIT Tech Talk on February 11, 1998.

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