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Multistable structures have widespread applications in the design of deployable aerospace systems, mechanical metamaterials, flexible electronics, and multimodal soft robotics due to their capability of shape reconfiguration between multiple stable states. Recently, the snap-folding of rings, often in the form of circles or polygons, has shown the capability of inducing diverse stable configurations. The natural curvature of the rod segment (curvature in its stress-free state) plays an important role in the elastic stability of these rings, determining the number and form of their stable configurations during folding. Here, we develop a general theoretical framework for the elastic stability analysis of segmented rings (e.g., polygons) based on an energy variational approach. Combining this framework with finite element simulations, we map out all planar stable configurations of various segmented rings and determine the natural curvature ranges of their multistable states. The theoretical and numerical results are validated through experiments, which demonstrate that a segmented ring with a rectangular cross-section can show up to six distinct planar stable states. The results also reveal that, by rationally designing the segment number and natural curvature of the segmented ring, its one- or multiloop configuration can store more strain energy than a circular ring of the same total length. We envision that the proposed strategy for achieving multistability in the current work will aid in the design of multifunctional, reconfigurable, and deployable structures. 

Abstract Ring origami has emerged as a robust strategy for designing foldable and deployable structures due to its impressive packing abilities achieved from the snap-folding mechanism. In general, polygonal rings with rationally designed geometric parameters can fold into compacted three-loop configurations with curved segments, which result from the internal bending moment in the folded state. Inspired by the internal bending moment-induced curvature in the folded state, we explore how this curvature can be tuned by introducing initial natural curvature to the segments of the polygonal rings in their deployed stress-free state, and study how this initial curvature affects their folded configurations. Taking a clue from straight-segmented polygonal rings that fold into overlapping curved loops, we find it is possible to reverse the process by introducing curvature into the ring segments in the stress-free initial state such that the rings fold into a straight-line looped pattern with “zero” area. This realizes extreme packing. In this work, by a combination of experimental observation, finite element analysis, and theoretical modeling, we systematically study the effect of segment curvature on folding behavior, folded configurations, and packing of curved ring origami with different geometries. It is anticipated that curved ring origami can open a new avenue for the design of foldable and deployable structures with simple folded configurations and high packing efficiency. 

Significant evolutionary shifts in locomotor behaviour often involve comparatively subtle anatomical transitions. For dinosaurian and avian evolution, medial overhang of the proximal femur has been central to discussions. However, there is an apparent conflict with regard to the evolutionary origin of the dinosaurian femoral head, with neontological and palaeontological data suggesting seemingly incongruent hypotheses. To reconcile this, we reconstructed the evolutionary history of morphogenesis of the proximal end of the femur from early archosaurs to crown birds. Embryological comparison of living archosaurs (crocodylians and birds) suggests the acquisition of the greater overhang of the femoral head in dinosaurs results from additional growth of the proximal end in the medial-ward direction. On the other hand, the fossil record suggests that this overhang was acquired by torsion of the proximal end, which projected in a more rostral direction ancestrally. We reconcile this apparent conflict by inferring that the medial overhang of the dinosaur femoral head was initially acquired by torsion, which was then superseded by mediad growth. Details of anatomical shifts in fossil forms support this hypothesis, and their biomechanical implications are congruent with the general consensus regarding broader morpho-functional evolution on the avian stem. 

The locomotion strategies of fossil invertebrates are typically interpreted on the basis of morphological descriptions. However, it has been shown that homologous structures with disparate morphologies in extant invertebrates do not necessarily correlate with differences in their locomotory capability. Here, we present a new methodology for analysing locomotion in fossil invertebrates with a rigid skeleton through an investigation of a cornute stylophoran, an extinct fossil echinoderm with enigmatic morphology that has made its mode of locomotion difficult to reconstruct. We determined the range of motion of a stylophoran arm based on digitized three-dimensional morphology of an early Ordovician form, Phyllocystis crassimarginata . Our analysis showed that efficient arm-forward epifaunal locomotion based on dorsoventral movements, as previously hypothesized for cornute stylophorans, was not possible for this taxon; locomotion driven primarily by lateral movement of the proximal aulacophore was more likely. Three-dimensional digital modelling provides an objective and rigorous methodology for illuminating the movement capabilities and locomotion strategies of fossil invertebrates.