Search for: All records

A periodic surface is one that is invariant by a two-dimensional lattice of translations. Deformation modes that stretch the lattice without stretching the surface are effective membrane modes. Deformation modes that bend the lattice without stretching the surface are effective bending modes. For periodic piecewise smooth simply connected surfaces, it is shown that the effective membrane modes are, in a sense, orthogonal to effective bending modes. This means that if a surface gains a membrane mode, it loses a bending mode, and conversely, in such a way that the total number of modes, membrane and bending combined, can never exceed 3. Various examples, inspired from curved-crease origami tessellations, illustrate the results. This article is part of the theme issue ‘Origami/Kirigami-inspired structures: from fundamentals to applications’. 

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. 

Shell mechanisms are patterned surface-like structures with compliant deformation modes that allow them to change shape drastically. Examples include many origami and kirigami tessellations as well as other periodic truss mechanisms. The deployment paths of a shell mechanism are greatly constrained by the inextensibility of the constitutive material locally and by the compatibility requirements of surface geometry globally. With notable exceptions (e.g., Miura-ori), the deployment of a shell mechanism often couples in-plane stretching and out-of-plane bending. Here, we investigate the repercussions of this kinematic coupling in the presence of geometric confinement, specifically in tubular states. We demonstrate that the confinement in the hoop direction leads to a frustration that propagates axially, as if by buckling. We fully characterize this phenomenon in terms of amplitude, wavelength, and mode shape in the asymptotic regime, where the size of the unit cell of the mechanismris small compared to the typical radius of curvatureρ. In particular, we conclude that the amplitude and wavelength of the frustration are of order $\sqrt{r/\rho }$and that the mode shape is an elastica solution. Derivations are carried out for a particular pyramidal truss mechanism. The findings are supported by numerical solutions of the exact kinematics.