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1. How periodic surfaces bend without stretching

   Nassar, Hussein; Weber, Andrew (August 2024, International Association for Shell and Spatial Structures)

   Many compliant shell mechanisms are periodically corrugated or creased. Being thin, their preferred deformation modes are inextensional, i.e., isometric. Here, we report on a recent characterization of the isometric deformations of periodic surfaces. In a way reminiscent of Gauss theorem, the result builds a constraint that relates the ways in which the periodic surface stretches, effectively but isometrically, to the ways in which it bends and twists. Several examples and use cases are presented. 

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2. Hidden symmetries generate rigid folding mechanisms in periodic origami

   https://doi.org/10.1073/pnas.2005089117  

   McInerney, James; Chen, Bryan Gin-ge; Theran, Louis; Santangelo, Christian D.; Rocklin, D. Zeb (November 2020, Proceedings of the National Academy of Sciences)

   We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell–Calladine index theorem we derive a relation between the number of linear folding motions and the number of rigid body modes that depends only on the average coordination number of the origami’s vertices. This supports the recent result by Tachi [T. Tachi,Origami6, 97–108 (2015)] which shows periodic origami sheets with triangular faces exhibit two-dimensional spaces of rigidly foldable cylindrical configurations. We also find, through analytical calculation and numerical simulation, branching of this configuration space from the flat state due to geometric compatibility constraints that prohibit finite Gaussian curvature. The same counting argument leads to pairing of spatially varying modes at opposite wavenumber in triangulated origami, preventing topological polarization but permitting a family of zero-energy deformations in the bulk that may be used to reconfigure the origami sheet. 

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3. Real-time Wing Deformation Simulations for Flying Insects

   https://doi.org/10.1145/3641519.3657434  

   Chen, Qiang; Deng, Zhigang; Li, Feng; Fang, Yuming; Lu, Tingsong; Tong, Yang; Zuo, Yifan (July 2024, ACM)

   Realistic simulation of the intricate wing deformations seen in flying insects not only deepens our comprehension of insect fight mechanics but also opens up numerous applications in fields such as computer animation and virtual reality. Despite its importance, this research area has been relatively under-explored due to the complex and diverse wing structures and the intricate patterns of deformation. This paper presents an efficient skeleton-driven model specifically designed to real-time simulate realistic wing deformations across a wide range of flying insects. Our approach begins with the construction of a virtual skeleton that accurately reflects the distinct morphological characteristics of individual insect species. This skeleton serves as the foundation for the simulation of the intricate deformation wave propagation often observed in wing deformations. To faithfully reproduce the bending effect seen in these deformations, we introduce both internal and external forces that act on the wing joints, drawing on periodic wing-beat motion and a simplified aerodynamics model. Additionally, we utilize mass- spring algorithms to simulate the inherent elasticity of the wings, helping to prevent excessive twisting. Through various simulation experiments, comparisons, and user studies, we demonstrate the effectiveness, robustness, and adaptability of our model. 

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4. Periodic tilings and auxetic deployments

   https://doi.org/10.1177/1081286520948116  

   Borcea, Ciprian_S; Streinu, Ileana (August 2020, Mathematics and Mechanics of Solids)

   We investigate geometric characteristics of a specific planar periodic framework with three degrees of freedom. While several avatars of this structural design have been considered in materials science under the name of chiral or missing rib models, all previous studies have addressed only local properties and limited deployment scenarios. We describe the global configuration space of the framework and emphasize the geometric underpinnings of auxetic deformations. Analogous structures may be considered in arbitrary dimension. 

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5. Self-organized dynamics of a viscous drop with interfacial nematic activity

   https://doi.org/10.1103/PhysRevResearch.7.L012054  

   Firouznia, Mohammadhossein; Saintillan, David (March 2025, Physical Review Research)

   We study emergent dynamics in a viscous drop subject to interfacial nematic activity. Using hydrodynamic simulations, we show how the interplay of nematodynamics, activity-driven flows in the fluid bulk, and surface deformations gives rise to a sequence of self-organized behaviors of increasing complexity, from periodic braiding motions of topological defects to chaotic defect dynamics and active turbulence, along with spontaneous shape changes and translation. Our findings recapitulate qualitative features of experiments and shed light on the mechanisms underpinning morphological dynamics in active interfaces. 

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