Elasticity and Geometry of Compliant Periodic Shells

Abstract: Compliant shell mechanisms are thin-walled structures adorned with folds and corrugations that enable shape change for locomotion, deployment, or adaptation in changing environments. Insect wings, umbrellas, and paper bags are familiar examples; other applications include deployable sunshields for telescopes, smart morphing airfoils, and several medical devices. In all cases, thin shells favor bending over stretching. Ideally, their compliant deformation modes are isometric, i.e., stretch-free. Identifying isometric deformations – or proving their nonexistence – has long been a difficult geometry problem that gained significance in mechanics with the advent of shell theory. Interest has recently resurged in the contexts of space missions, soft robotics, origami science, and computer graphics.
Here, we report a recent breakthrough in characterizing isometric deformations for thin periodic shells, be them creased or smoothly corrugated. The main result establishes a duality between in-plane and out-of-plane compliant deformation modes. In particular, it shows that if a periodic shell is compliant in twisting, then it resists shearing, and vice versa. The findings align with behaviors observed in physical samples (3D prints and origami tessellations) and further explain why some samples bend into saddles while others bend into domes. Shell finite element simulations are employed to verify the theoretical predictions and capture finer elastic effects, such as small but nonzero stretching. The work leverages ideas and techniques from differential geometry and the theory of composites; it uncovers novel design principles for compliant shell mechanisms and, hopefully, builds new connections between communities interested in mechanics, geometry, architecture, and the visual arts.  

Bio: Hussein Nassar is an Assistant Professor of Mechanical and Aerospace Engineering at the University of Missouri – Columbia (MU). He holds a double degree in Mathematics and Engineering from Sorbonne Université and École des Mines – PSL as well as a PhD in Mechanics from Université Gustave Eiffel. He joined the faculty at MU in 2018 and was a Senior Visiting Scholar at École Polytechnique in the summer of 2024. His research investigates theoretical models of continuum mechanics applicable to architected solids and shells with emphasis on interactions between geometry and elasticity, both in static and dynamic regimes. His research has been supported by the NSF and the Army Research Office; he is a recipient of the NSF CAREER Award.

‼️ Le séminaire peut être suivi en ligne suivant le lien : https://univ-amu-fr.zoom.us/j/97579713636?pwd=XQuReTAbOoGMofJhCWzDA9SfVf4cad.1

Le mardi 18 mars 2025 à 15h00 / Amphithéâtre François Canac, LMA

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