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1. Deterministic and stochastic control of kirigami topology

   https://doi.org/10.1073/pnas.1909164117  

   Chen, Siheng; Choi, Gary P. T.; Mahadevan, L. (February 2020, Proceedings of the National Academy of Sciences)

   Kirigami, the creative art of paper cutting, is a promising paradigm for mechanical metamaterials. However, to make kirigami-inspired structures a reality requires controlling the topology of kirigami to achieve connectivity and rigidity. We address this question by deriving the maximum number of cuts (minimum number of links) that still allow us to preserve global rigidity and connectivity of the kirigami. A deterministic hierarchical construction method yields an efficient topological way to control both the number of connected pieces and the total degrees of freedom. A statistical approach to the control of rigidity and connectivity in kirigami with random cuts complements the deterministic pathway, and shows that both the number of connected pieces and the degrees of freedom show percolation transitions as a function of the density of cuts (links). Together, this provides a general framework for the control of rigidity and connectivity in planar kirigami. 

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2. Explosive rigidity percolation in kirigami

   https://doi.org/10.1098/rspa.2022.0798  

   Choi, Gary P.; Liu, Lucy; Mahadevan, L. (March 2023, Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences)

   Controlling the connectivity and rigidity of kirigami, i.e. the process of cutting paper to deploy it into an articulated system, is critical in the manifestations of kirigami in art, science and technology, as it provides the resulting metamaterial with a range of mechanical and geometric properties. Here, we combine deterministic and stochastic approaches for the control of rigidity in kirigami using the power of k choices, an approach borrowed from the statistical mechanics of explosive percolation transitions. We show that several methods for rigidifying a kirigami system by incrementally changing either the connectivity or the rigidity of individual components allow us to control the nature of the explosive transition by a choice of selection rules. Our results suggest simple lessons for the design of mechanical metamaterials. 

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3. Geometrical control of the magnetic anisotropy in six coordinate cobalt complexes

   https://doi.org/10.1039/D0CC03238G  

   Saber, Mohamed R.; Singh, Mukesh K.; Dunbar, Kim R. (July 2020, Chemical Communications)

   The geometry of cobalt( ii ) ions in the axially distorted octahedral cation in [Co(MeCN) 6 ](BF 4 ) 2 ( 1 ) was compared to the trigonal prismatic cation in [CoTp py ]PF 6 ( 2 ) which revealed significant differences in magnetic anisotropy. Combined experimental and ab initio CASSCF/NEVPT2 calculations support the observed zero field SMM behaviour for 2 , with easy axis anisotropy, attributed to the rigidity of the trigonal prismatic ligand. Strong transverse anisotropy for 1 leads to significant quantum tunnelling processes. 

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4. Point-Based Models: A Unified Approach for Geometric Constraint Analysis of Planar n -Bar Mechanisms With Rotary, Sliding, and Rolling Joints

   https://doi.org/10.1115/1.4066963  

   Lyu, Zhijie; Purwar, Anurag (April 2025, Journal of Mechanisms and Robotics)

   Abstract Kinematic simulation of planar n-bar mechanisms has been an intense topic of study for several decades now. However, a large majority of efforts have focused on position analysis of such mechanisms with limited links and joint types. This article presents a novel, unified approach to the analysis of geometric constraints of planar n-bar mechanisms with revolute joint (R-joint), prismatic joint (P-joint), and rolling joint. This work is motivated by a need to create and program a system of constraint equations that deal with different types of joints in a unified way. A key feature of this work is that the rolling joint constraints are represented by four-point models, which enables us to use the well-established undirected graph rigidity analysis algorithms. As a result, mechanisms with an arbitrary combination of revolute-, prismatic joints, and wheel/gear/wheel-belt chains without any limitations on their actuation scheme can be analyzed and simulated efficiently for potential implementation in interactive computer software and large-scale data generation. 

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5. Van der Waals interaction affects wrinkle formation in two-dimensional materials

   https://doi.org/10.1073/pnas.2025870118  

   Ares, Pablo; Wang, Yi Bo; Woods, Colin R.; Dougherty, James; Fumagalli, Laura; Guinea, Francisco; Davidovitch, Benny; Novoselov, Kostya S. (March 2021, Proceedings of the National Academy of Sciences)

   null (Ed.)

   Nonlinear mechanics of solids is an exciting field that encompasses both beautiful mathematics, such as the emergence of instabilities and the formation of complex patterns, as well as multiple applications. Two-dimensional crystals and van der Waals (vdW) heterostructures allow revisiting this field on the atomic level, allowing much finer control over the parameters and offering atomistic interpretation of experimental observations. In this work, we consider the formation of instabilities consisting of radially oriented wrinkles around mono- and few-layer “bubbles” in two-dimensional vdW heterostructures. Interestingly, the shape and wavelength of the wrinkles depend not only on the thickness of the two-dimensional crystal forming the bubble, but also on the atomistic structure of the interface between the bubble and the substrate, which can be controlled by their relative orientation. We argue that the periodic nature of these patterns emanates from an energetic balance between the resistance of the top membrane to bending, which favors large wavelength of wrinkles, and the membrane-substrate vdW attraction, which favors small wrinkle amplitude. Employing the classical “Winkler foundation” model of elasticity theory, we show that the number of radial wrinkles conveys a valuable relationship between the bending rigidity of the top membrane and the strength of the vdW interaction. Armed with this relationship, we use our data to demonstrate a nontrivial dependence of the bending rigidity on the number of layers in the top membrane, which shows two different regimes driven by slippage between the layers, and a high sensitivity of the vdW force to the alignment between the substrate and the membrane. 

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