Thin sheets have long been known to experience an increase in stiffness when they are bent, buckled, or assembled into smaller interlocking structures. We introduce a unique orientation for coupling rigidly foldable origami tubes in a “zipper” fashion that substantially increases the system stiffness and permits only one flexible deformation mode through which the structure can deploy. The flexible deployment of the tubular structures is permitted by localized bending of the origami along prescribed fold lines. All other deformation modes, such as global bending and twisting of the structural system, are substantially stiffer because the tubular assemblages are overconstrained and the thin sheets become engaged in tension and compression. The zipper-coupled tubes yield an unusually large eigenvalue bandgap that represents the unique difference in stiffness between deformation modes. Furthermore, we couple compatible origami tubes into a variety of cellular assemblages that can enhance mechanical characteristics and geometric versatility, leading to a potential design paradigm for structures and metamaterials that can be deployed, stiffened, and tuned. The enhanced mechanical properties, versatility, and adaptivity of these thin sheet systems can provide practical solutions of varying geometric scales in science and engineering.

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We explore the recently-proposed Virtual Element Method (VEM) for numerical solution of boundary value problems on arbitrary polyhedral meshes. More specifically, we focus on the elasticity equations in three-dimensions and elaborate upon the key concepts underlying the first-order VEM. While the point of departure is a conforming Galerkin framework, the distinguishing feature of VEM is that it does not require an explicit computation of the trial and test spaces, thereby circumventing a barrier to standard finite element discretizations on arbitrary grids. At the heart of the method is a particular kinematic decomposition of element deformation states which, in turn, leads to a corresponding decomposition of strain energy. By capturing the energy of linear deformations exactly, one can guarantee satisfaction of the engineering patch test and optimal convergence of numerical solutions. The decomposition itself is enabled by local projection maps that appropriately extract the rigid body motion and constant strain components of the deformation. As we show, computing these projection maps and subsequently the local stiffness matrices, in practice, reduces to the computation of purely geometric quantities. In addition to discussing aspects of implementation of the method, we present several numerical studies in order to verify convergence of the VEM and evaluate its performance for various types of meshes.

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