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When materials are forced into new shapes, a tipping point can shift them from flexibility and resilience to failing or breaking. Understanding that tipping point is at the core of Jani Onninen’s research. He has received a three-year grant from the National Science Foundation (NSF) to explore challenging mathematical problems of predicting how materials change under stress.

, a professor in the , is drawing on two fields of mathematics—geometric function theory and non-linear elasticity—to understand how and why materials fail under certain conditions.

“Imagine a blacksmith shaping hot metal,” Onninen says. “Each hammer strike creates a small deformation. Early on, each deformation is reversible. You can undo it and return to the original shape. But as the blacksmith continues hammering, the sequence of deformations approaches a limit where this reversibility breaks down. This signal tells us something critical. The blacksmith should stop—before the material reaches conditions conducive to forming a crack.”

Materials in the Real World

Traditional mathematical models use “Sobolev homeomorphisms” to describe a material when it deforms and collapses. These models assume two things. One, the material can return to its original shape (it’s “invertible”). Two, the deformation follows the path that uses the least energy. When these models show that a deformation can’t do these two things, it’s a warning signal that the material could fail.

In real life, however, materials don’t always behave according to these ideal mathematical models.

Materials tend to use the least amount of energy possible when they change shape. But sometimes the most efficient or “energy-saving” ways a material might deform don’t fit current math equations. So, researchers are trying to learn the most energy-efficient ways for a material to go from one shape to another.

Warning Signs Before Failure

At the heart of this research is the challenge of understanding and modeling more complex elastic deformations, as well as identifying warning signals in mathematics before materials reach their breaking point.

Onninen, in collaboration with former University postdoctoral researcher Ilmari Kangasniemi, has developed a new framework—the theory of quasiregular values—and achieved breakthroughs, including solving the Astala–Iwaniec–Martin uniqueness problem and providing fresh insights into Picard’s theorem, a foundational result in mathematics from the 1870s.

Onninen’s work is theoretical, studying what happens beyond the boundaries of current mathematical models. But basic research can lead to practical advances years or decades later. Eventually it could have applications in engineering, manufacturing and other fields to learn how much stress a material can handle. This could have implications for understanding wear and tear in infrastructure, like roads and bridges, clothing materials, such as cloth and plastics, and vehicle materials, like metals and plastics.

Building the Next Generation

The NSF grant will also support the training and mentorship of graduate students and early-career researchers, ensuring the continuation of this cutting-edge research.

“Some of the most exciting progress I’ve made has come from working closely with colleagues—sharing ideas, challenging each other and building something new together,” Onninen says, emphasizing the collaborative nature of mathematical discovery.

This latest grant marks Onninen’s seventh standard NSF award since joining the University.

“The mathematics department is thrilled that Professor Jani Onninen has received this prestigious NSF award, recognizing his groundbreaking work,” says , professor and department chair. “His research continues to elevate the department’s profile and provides outstanding opportunities for our graduate students to participate in cutting-edge research, fostering their development as the next generation of scholars.”

Story by John H. Tibbetts