A concrete theory
UWM mathematician
Concrete is a universal building material, but it’s prone to wear, tear, and cracking over time. In addition to the main loading patterns, in large structures, deformations caused by temperature fluctuations - if not properly compensated for - can be a driving force of premature failure. To counteract that deformation, engineers usually design large structures to be assembled from lay it in large slabs. Those slabs are joined together with connection elements shielded from aggressive environments by a rubberized membrane, almost like caulk. But, said Burns Healy, that connection element is expensive and concrete often fails at those joins. “It would be much better if we could make larger monolithic pieces of concrete without joints rather than having to connect each of them,” he said. That’s why Healy is teaming up with researchers from civil and environmental engineering professor Konstantin Sobolev’s group in Burns Healy the UWM College of Engineering and Applied Science. They’re trying to find new ways of shaping concrete through 3D printing that will increase its durability – and hopefully decrease its cost. Healy is a visiting assistant professor in UWM’s Mathematical Sciences Department. He specializes in geometry and topology, the study of distance and shape. When civil engineering PhD student Aparna Deshmukh approached the math department looking for help in her project to create stronger concrete structures through 3D printing, Healy volunteered to collaborate. A pattern to follow It’s expensive to 3D print concrete, so instead of printing shapes and then testing what structural properties the 4 • IN FOCUS • January, 2021
finished product has, Healy is helping Deshmukh take a “math-up” approach by using modeling to understand what properties a finished product might have once it actually exists. To do that, he’s examining existing research and determining what physical properties certain designs have once they’re created and printed. These properties include parameters like a Poisson’s ratio – a measure of the deformation of a material perpendicular to where pressure is applied – as an anisotropic An example of a strain, and if a material deforms more repeating pattern in one direction than another. “The idea is to look at all of the systems that have been used so far … and say, these systems all have x, y, or z properties and this is how they work,” Healy explained. “If you want a certain property, look at the collection of systems that had that property. What, mathematically speaking, do those systems have in common that you can try to emulate?” The designs that Healy is researching are made up of repeating patterns – think of the design of lines and rhombuses on sheet metal flooring, for example – and those patterns have different properties that affect the integrity of the concrete. “What I noticed when I was being shown all of these (patterns) is that mathematically speaking, we understand all of them,” Healy said. “It turns out there are exactly 17 different kinds of patterns that these things can have, and these are called the wallpaper groups.” Wallpaper groups are a way to classify repeating patterns, and mathematicians like Healy have long studied their characteristics. “The question is, to what extent are values (like anisotropic strain and Poisson’s ratio) influenced by the wallpaper group associated to the pattern that it was modeled on? That’s what we’re trying to figure out,” he added.
