March 1, 2024

Study unravels mathematical mystery of cable-like structures

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Rigid rod systems acquire rigidity through the random addition of additional rods and cables, as captured through graph theory. The research team’s main subject of study, shown here, are structures consisting of a large number of pores – arranged in columns and rows with randomly added cables and rods. Credit: Georgia Institute of Technology

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Rigid rod systems acquire rigidity through the random addition of additional rods and cables, as captured through graph theory. The research team’s main subject of study, shown here, are structures consisting of a large number of pores – arranged in columns and rows with randomly added cables and rods. Credit: Georgia Institute of Technology

Are our bodies solid or liquid? We all know the convention: solids maintain their shape, while liquids fill the containers they are in. But often in the real world, these lines are blurred. Imagine walking on a beach. Sometimes the sand gives way underfoot, deforming like a liquid, but when enough grains of sand come together, they can support the weight like a solid surface.

Modeling these types of systems is notoriously difficult — but Zeb Rocklin, an assistant professor in Georgia Tech’s School of Physics, has written a new paper doing just that.

Rocklin’s study, “Stiffness percolation in a random tensegrity via analytic graph theory,” was published in Proceedings of the National Academy of Sciences. The results have the potential to impact fields ranging from biology to engineering and nanotechnology, showing that these types of deformable solids offer a rare combination of durability and flexibility.

“I’m very proud of our team, especially Will and Vishal, the two Georgia Tech graduate students who co-led the study,” says Rocklin.

Lead author William Stephenson and co-author Vishal Sudhakar completed their undergraduate studies at the Institute during the period of this research. Stephenson is now a first-year undergraduate student at the University of Michigan, Ann Arbor, and Sudhakar has been admitted to Georgia Tech as a graduate student. Additionally, co-author Michael Czajkowski is a postdoctoral researcher in the School of Physics, and co-author James McInerney completed his graduate studies at the Rocklin School of Physics. McInerney is now a postdoctoral researcher at the University of Michigan.

Connecting the dots… with cables

Imagine building molecules in chemistry class – large wooden spheres connected with sticks or rods. Although many models use rods, including mathematical models, real-life biological systems are built with polymers, which function more like elastic strings.

Similarly, when creating mathematical or biological models, researchers often treat all elements as rods, rather than treating some of them as cables or ropes. But, “there are tradeoffs between how mathematically tractable a model is and how physically plausible it is,” says Rocklin.

“Physicists may have some beautiful mathematical theories, but they are not always realistic.” For example, a model that uses connective strings may not capture the dynamics that connective strings provide. “With a string you can stretch it and it will fight you, but when you compress it it collapses.”

“But in this study, we extend current theories,” he says, adding elements similar to cables. “And that turns out to be incredibly difficult, because these theories use mathematical equations. In contrast, the distance between the two ends of a cable is represented by an inequality, which is not an equation at all.

“So how do you create a mathematical theory when you don’t start with equations?” Although a rod is a certain length in a mathematical equation, the ends of the rope must be represented as less than or equal to a certain length.

In this situation, “all the usual analytical theories completely break down,” says Rocklin. “It becomes very difficult for physicists or mathematicians.”

“The trick was realizing that these physical systems were logically equivalent to something called a directed graph,” adds Rocklin, “where different modes of deformation are linked to each other in specific ways. This allows us to take a relatively complicated system and massively compress it.” into a much smaller system. And when we did that, we were able to turn it into something that was extremely easy for the computer to do.”

From biology to engineering

Rocklin’s team found that when modeling with cables and springs, the range of the target changed – becoming smoother, with a wider margin of error. “This can be very important for something like a biological system, because a biological system tries to stay close to this critical point,” says Rocklin. “Our model shows that the region around the critical point is actually much wider than models using only rods have previously shown.”

Rocklin also points out applications for engineers. For example, since Rocklin’s new theory suggests that even haphazard cable structures can be strong and flexible, it could help engineers harness cables as building materials to create safer, more durable bridges. The theory also provides a way to easily model these cable-based structures to ensure their safety before they are built, and provides a way for engineers to iterate on designs.

Rocklin also looks at potential applications in nanotechnology. “In nanotechnology, you have to accept an increasing amount of clutter, because you can’t just have a skilled worker come in and put segments there, and you can’t have a conventional factory machine put segments there,” says Rocklin.

But biology has known how to establish effective but disordered rod-and-cable structures for hundreds of millions of years. “This will tell us what kinds of machines we can make with these disordered structures when we get to the point where we’re able to do what biology can do. And that’s a possible future design principle for engineers to explore, at least very small scales.” , where we can’t choose exactly where each cable goes,” says Rocklin.

“Our theory shows that with cables we can maintain a combination of flexibility and strength with much less precision than would otherwise be necessary.”

More information:
William Stephenson et al, Stiffness percolation in a random tensegrity via analytic graph theory, Proceedings of the National Academy of Sciences (2023). DOI: 10.1073/pnas.2302536120

Diary information:
Proceedings of the National Academy of Sciences

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