A traditional paper faucet is a piece of art. Each fold in origami leads to the transformation of a single square sheet of paper into a bird, dragon or flower. Origami discourages gluing, marking or cutting the paper, but in the art of kirigami, strategically placed cuts can transform the shape of the paper even further, creating complex structures from simple crevices. A well-known example of this is a pop-up book, where depending on how the flat paper is cut, a different set of shapes – a heart, a frog, a series of skyscrapers – will appear when the book is opened.
In production, kirigami changes the game of what is possible. As with paper, the repeated laser cutting of a sheet opens up the possibility for complex shape distortion, driven by the opening and closing of slits. Due to the freedom available in slot design, this creates a wide choice of geometries that have highly adaptable properties compared to traditional materials† In real-world applications, you might see such a material being used in robotics or space, for example a snakeskin-like material inspired by kirigami that allows a robot to crawl or a morphing airframe. But before these materials can be adapted for professional use, we need to better understand how kirigami materials change shape under typical engineering stresses and loads. While the rules for simple building blocks are well known, the rules for their collective shape-shifting interactions remain largely unclear.
In a recent article published in Physical Assessment Letters, a multidisciplinary team of researchers from USC, University of Illinois at Chicago and Stony Brook University has derived a new mathematical equation for categorizing the behavior of kirigami-inspired materials to better predict how they will move when pushed or pulled. The team includes USC assistant professor Paul Plucinsky and postdoctoral researcher Yue Zheng; Stony Brook University Assistant Professor Paolo Celli and Graduate Research Assistant Imtiar Niloy; and University of Illinois-Chicago assistant professor Ian Tobasco.
Plucinsky said, “The geometry of these materials is somewhat arbitrarily tuned. So we need rules about how you can choose the architectures that you’re going to fabricate. Once you have those rules, you also need to be able to model the system like that. you make a reasonable prediction of how it will deform if pushed or pulled.”
Plucinsky says previous models of material behavior don’t apply to kirigami materials because they aren’t sensitive to the complicated geometry of their designs. “If you want to be able to use these materials, you first have to understand why, when you introduce these patterns to loads, they produce a very uneven response.”
When a material is cut, it produces “cells” or contains spaces that repeat in a pattern, such as parallelograms, Plucinsky said. In the case of kirigami materials, these cells can be categorized to behave in two ways: wave-like or decaying along elliptical arcs, and this depends only on whether the pattern compresses or expands perpendicular to the pull direction. A mathematical equation governs the geometric behavior of things like water flow, Plucinsky said, but for solids like this it’s harder to deduce. A partial differential equation (PDE) is what Plucinsky and his team have been able to develop and set forth as the first piece of a larger puzzle needed to make kirigami materials practically applicable.
A modeling problem
Right now, Plucinsky says, there is a fundamental modeling problem that prevents its use, Plucinsky says. Plucinsky said not much is known about how kirigami materials function under base load conditions. “If you don’t have a good tool to model the systems in question, you would have a hard time exploring the design space and making comprehensive predictions about the individual patterns,” Plucinsky said.
In light of that, Plucinsky and his research team thought, ‘Is there a simple one? mathematical equation who could characterize these materials?’ “The equation,” he said, “would allow you to predict the behavior of the system in a numerically efficient way.”
The key to the equation was to realize that while kirigami cells themselves have complicated building blocks, they can be conceptualized as atoms in a lattice (a repeating 2D array of atoms), as in a conventional crystalline solid, where the units are identical. and be repetitive. From there it was easy to derive an equation that managed to represent the changes in the physical structure of such a material when manipulated. The equation provides insight into real-world scenarios, such as how a kirigami-based space object might react if a moon rock landed on it.
Puzzle pieces of design
Kirigami patterns, Plucinsky said, are beneficial for many reasons, including that they are material-independent in many ways. “These kind of parallels run nicely with additive manufacturing where they can now basically go in there and create carefully designed patterns at different scales. The point is that the pattern matters, so if you design the pattern correctly, the choice of materials does. you don’t use that’ it doesn’t have to be that important.”
See the success of the mathematical model in predicting kirigami-inspired materials open the doors to using such modeling for other materials, Plucinsky said. “We’re working towards the idea that if you have something with a repeating pattern, you can find an equation that models it accurately. From there, we can turn this on its head, so that if you want to design a particular property, you can say: ‘oh, it must contain an x-type pattern’ and reverse engineer it.”
Yue Zheng et al, Continuum field theory for the deformations of planar Kirigami, Physical Assessment Letters (2022). DOI: 10.1103/PhysRevLett.128.208003
University of Southern California
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