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We develop a framework to understand the mechanics of metamaterial sheets on curved surfaces. Here we have constructed a continuum elastic theory of mechanical metamaterials by introducing an auxiliary, scalar gauge-like field that absorbs the strain along the soft mode and projects out the stiff ones. We propose a general form of the elastic energy of a mechanism based metamaterial sheet and specialize to the cases of dilational metamaterials and shear metamaterials conforming to positively and negatively curved substrates in the Föppl–Von Kármán limit of small strains. We perform numerical simulations of these systems and obtain good agreement with our analytical predictions. This work provides a framework that can be easily extended to explore non-linear soft modes in metamaterial elasticity in future. 

Bacterial growth is remarkably robust to environmental fluctuations, yet the mechanisms of growth-rate homeostasis are poorly understood. Here, we combine theory and experiment to infer mechanisms by which Escherichia coli adapts its growth rate in response to changes in osmolarity, a fundamental physicochemical property of the environment. The central tenet of our theoretical model is that cell-envelope expansion is only sensitive to local information, such as enzyme concentrations, cell-envelope curvature, and mechanical strain in the envelope. We constrained this model with quantitative measurements of the dynamics of E. coli elongation rate and cell width after hyperosmotic shock. Our analysis demonstrated that adaptive cell-envelope softening is a key process underlying growth-rate homeostasis. Furthermore, our model correctly predicted that softening does not occur above a critical hyperosmotic shock magnitude and precisely recapitulated the elongation-rate dynamics in response to shocks with magnitude larger than this threshold. Finally, we found that, to coordinately achieve growth-rate and cell-width homeostasis, cells employ direct feedback between cell-envelope curvature and envelope expansion. In sum, our analysis points to cellular mechanisms of bacterial growth-rate homeostasis and provides a practical theoretical framework for understanding this process. 

Self-folding origami, structures that are engineered flat to fold into targeted, three-dimensional shapes, have many potential engineering applications. Though significant effort in recent years has been devoted to designing fold patterns that can achieve a variety of target shapes, recent work has also made clear that many origami structures exhibit multiple folding pathways, with a proliferation of geometric folding pathways as the origami structure becomes complex. The competition between these pathways can lead to structures that are programmed for one shape, yet fold incorrectly. To disentangle the features that lead to misfolding, we introduce a model of self-folding origami that accounts for the finite stretching rigidity of the origami faces and allows the computation of energy landscapes that lead to misfolding. We find that, in addition to the geometrical features of the origami, the finite elasticity of the nearly-flat origami configurations regulates the proliferation of potential misfolded states through a series of saddle-node bifurcations. We apply our model to one of the most common origami motifs, the symmetric “bird's foot,” a single vertex with four folds. We show that though even a small error in programmed fold angles induces metastability in rigid origami, elasticity allows one to tune resilience to misfolding. In a more complex design, the “Randlett flapping bird,” which has thousands of potential competing states, we further show that the number of actual observed minima is strongly determined by the structure's elasticity. In general, we show that elastic origami with both stiffer folds and less bendable faces self-folds better. 

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.

 

No longer just the purview of artists and enthusiasts, origami engineering has emerged as a potentially powerful tool to create three dimensional structures on disparate scales. Whether origami (and the closely related kirigami) engineering can emerge as a useful technology will depend crucially on both fundamental theoretical advances as well as the development of further fabrication tools.