Incorporating frustration: How mismatched shapes can increase material strength and toughness

Anyone who has ever tried to tile flooring, backsplash, or even art and craft projects will know the emotional frustration of working with pieces whose shapes don’t completely complement each other. However, we find that some creatures can actually rely on similar inconsistencies to create geometric frustration that results in complex natural structures with outstanding properties, such as protective shells and sturdy yet flexible bones.

Currently, researchers at the University of Michigan are developing mathematical models that show one way nature can achieve this. These models help design advanced materials such as medical devices, sustainable construction, and more.

“The frustration of using these discrepancies of building blocks leads to excellent complexity, which can help provide excellent material properties,” said Xiaoming Mao, professor at UM Physics and senior author of the new study.

Mao’s research group has always been interested in the relationship between the structure of a material and its properties. However, this particular project was inspired by a photograph in which she saw a single-celled organism called Kokkorisophore, which makes a beautiful geometric shell from calcium carbonate.

It made me wonder how these simple organisms use basic components to create complex structures. When researchers delved into the problem, they found that this strategy was not confined to the Kokukorisophore.

“There are essentially many examples,” Mao Zedong said. “And geometric frustration is one of the paths leading to these incredible structures.”

In addition to providing a stunning aesthetic, frustration also helped to supply the materials with desirable properties from an engineering perspective. They may be flexible, but they were strong, tough and characterized by gaps within the structure.

“If we can learn to build with these properties, we can use less materials and get better mechanical properties,” Mao said. “It gives us a new way to think about sustainable manufacturing.”

To take a step towards it, the team turned to a branch of mathematics called graph theory to explore how structures can be a reasonable level of frustration. Their approach allowed researchers to create phase diagrams that would help predict material properties for various frustrating structures based on how they packed the building blocks.

They did this because of two types of frustration, called cumulative and cumulative. Non-active cases prevent some irregularity in the shape of the building block from properly arranging the final pattern.

In cumulative frustration, building blocks can be perfectly shaped, but their geometry prevents them from cramming straight into the space they are trying to cover. For example, try tile a flat pentagonal surface.

The team discovered distinct geometric patterns that emerged from these two types of frustration. In both cases, organizations in the block and void structure were required to fully utilize the less synergistic effects that the frustrating material provides. However, the structure is too organized and there are also points in which its material properties pay for the price.

“So, something like a simple crystal won’t give you what you want, but it doesn’t work either,” Moo said.

There’s Goldilocks level somewhere in between, she said.

Mao and colleagues at UM reported their work in the Journal Physical Review Letters. The team also included Nicholas Kotov, a professor of chemical engineering, and graduate student researchers Jose Ortiz Tavarez and Zen Yang.

The team’s work will continue at Compass, a research center of the National Science Foundation, headquartered in UM. The next step involves working with collaborators to study real-world examples through the lens of new models and extract new lessons in advanced materials in engineering.

“What we’re studying now is what we call the “toy model” of theoretical physics,” Mao said. “We really want to see real materials. That’s where compasses come in. There are a lot of people who are researching real things like this, generating real data, and using AI to analyze the complexity and internal patterns of these materials. This is how we connect complexity with functionality.”