Quantifying the Boundary Conditions of Molting Behavior in Snakes and Caterpillars

By Lina Sorg

Origami is well known as the traditional Japanese art of paper folding. The resulting three-dimensional creations are thin, lightweight, versatile, and can manifest in a variety of intricate shapes. As such, origami-like structures are ubiquitous in a variety of real-world engineering applications, including spacecraft design, architecture, medicine, and robotics.

During the 10th International Congress on Industrial and Applied Mathematics, which is currently taking place in Tokyo, Japan, Taiju Yoneda of Kyushu University expanded the interpretation of origami beyond the context of folding and used mechanical experiments to investigate the geometry of molting patterns in snakes and caterpillars. “We would like to quantify the shape and dynamics of thin structures, including bending and stretching,” Yoneda said. “If the origami structure can be used effectively beyond the context of folding, it would make movement even more convenient.”

Rather than create a completely new structure, Yoneda sought to imitate existing real-life origami designs that exhibit good movement in the bodies of living creatures. He introduced several examples of flora and fauna that successfully utilize thin structures, such as the elastic expansion of insect wings, the noncommittal development of blooming morning glories, and the highly efficient packaging of mitochondria. Ultimately, however, Yoneda concentrated on molting: the process by which invertebrates shed their outer exoskeletons. “Molting is one of the most common movements of the thin structure of living things,” he said. 

Yoneda outlined four different types of molting—“space suit,” “inverted socks,” “slouchy socks,” and “doughnut socks”—each of which corresponds to a separate animal(s) (see Figure 1). For example, during the “space suit” molting of crabs, cicadas, and other hard-skinned insects, the shedded skin retains its original shape. And when lizards molt, their skin rolls down the body in the shape of a doughnut. But for the purposes of this study, Yoneda focused on the “inverted socks” and “slouchy socks” styles of molting, which are respectively characteristic of snakes and caterpillars. Despite the similarities in their cylindrical bodies, these creatures molt in distinct ways. Snakes molt from their heads, and their discarded skin turns inside out and takes the shape of a removed, inverted sock. In contrast, caterpillars molt by wrinkling and buckling their skin towards their tails so that is resembles an oversized, slouchy sock. 

Yoneda employed a simplified model to identify the factors that cause this distinction. “We may be able to strip the packaging, streamline thin film production, and understand the optimization of the evolution of living things,” he said. In his simplified experiment, the boundary conditions determine the style of molting and resistance depends on both friction and gap size (between the body and skin). The geometry/material parameters of the model include the radius, thickness, coefficient of friction, and Young’s modulus. Because real-life organisms have different shapes, external force boundary conditions, and molting progressions, Yoneda aimed to reveal the boundary conditions that are energetically advantageous to molting.

To play with these conditions, Yoneda formed skin models from a synthetic rubber called silicone. “There is little expansion and contraction during creation, and Hooke’s law of elasticity is valid within the scope of experiments,” he said. Next, he molded a cylinder and flowed the silicone over the surface from both ends to offest a thickness gradient and ultimately generate a thin cylindrical skin film of uniform thickness. After attaching rigid rings to the ends of the cylinder and constraining the ends to be parallel to the axis, Yoneda used a compression tester to obtain a displacement and measure the load.

Once everything was appropriately positioned, he changed the boundary conditions. “The difference between the snake and caterpillar mode depends largely on how the boundaries are moved,” Yoneda said, adding that the angle always remains constant because it holds the upper boundary handle. “In snake mode, the direction of the angle vector is up and the boundary handles are fixed outside of the skin. In caterpillar mode, the direction of the angle vector is down.”

Yoneda then addressed the impact of friction as well as the gap between the body and skin. “We controlled the gap by using the body model as a roll-up film and reducing the radius of the body by peeling off the film because it is not possible to form skins of various sizes,” he said. He powdered the machine to measure the moving force with both high and low levels of friction. 

Finally, Yoneda compared the snake and caterpillar modes in their current parameter range with different-sized gaps and changing levels of friction. “Under all conditions, the snake mode required less force to move,” he said. “The tapered shape that is characteristic of the caterpillar and its short body may also be an important factor. If the bending stiffness were lower, the force of the caterpillar mode might be smaller.”

Looking ahead, Yoneda hopes to expand the study by incorporating a wider range of parameters that cannot be accounted for in physical simulations. “In particular, we have not been able to conduct experiments under ‘hard’ conditions that would prevent the skin from stretching,” he said. “If the skin is hard, it may not be possible to molt in snake mode because it is difficult to invert the skin first.” He is also considering possible paper-based experiments in order to return to the origins of origami.

Lina Sorg is the managing editor of SIAM News.