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Finally, because the principles and methods we describe are purely geometric, they open the door for developing design paradigms independent of length scale and material system.
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Three-dimensional models of different geometries were designed in a CAD software. The non-Euclidean geometries helicoid and hemisphere were fabricated using a commercial 3D printer Stratys Inc. Before filling the mold, the 1. The hemispherical and helicoid shells studied were 1 mm thick, and the crease had a rectangular cross-section 0.
Only samples without structural defects were included for testing. Owing to their Euclidean nature, cylinders could be fabricated using a conventional 2D technique. Helicoids with different creases were clamped on one edge, and deformed along the crease using a rigid indenter by hand. Composite images using frames at equal time intervals from these movies were created by using alpha blending.
For the sample with a snap-through, frames were chosen to be ms apart. For the sample with a planar crease, frames are 1 and 6 s apart for deformation on either side of the torsional hinge. Lastly, for the sample with helical crease, frames were 1. A custom-built force displacement device, combining a linear translation stage Zaber Technologies Inc.
A crease placed on a surface is parametrized by a space curve c s , with s an arc length variable that runs along the curve.
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These vectors are characterized by the following relationship:. The surface of the shell is composed of two regions that are divided by the crease, each parametrized by a local orthonormal frame. These vectors are related in a similar fashion to the Frenet frame:. The relationship between the surface vectors and the crease vectors is given in terms of the linear combination.
This corresponds to the folded and unfolded states shown in Fig. To determine the stability of the folded state we first write the energy for deformation of a thin shell:. While this limit is singular, it provides a simple geometric interpretation of nearly free deformations and yields insight into the stability and foldability of general shells. Written using our nomenclature, this indicates that.
Lecture 8.7: Basic Analysis
If the crease has zero normal curvature, however, the component of the second fundamental form h v v is unconstrained by the crease. The folding angle and torsion, however, are constrained by. The geodesic torsion physically corresponds to the rate of rotation of the normal to the surface along the curve. Together, these results can be used to infer a number of things.
First, finite normal curvature implies that there is an energy barrier, which implies that a subcritical bifurcation may occur. Furthermore, zero normal curvature explicitly means that one of the components of the curvature tensor vanishes identically; specifically, the curvature of the surface in the direction of the crease is always zero. For a general surface with nonpositive Gaussian curvature, at every point there exists a pair of asymptotic curves such that the normal curvature along these curves is zero.
The ability to fold the helicoid about any of these construction lines follows trivially from rigid-body rotations, and any energetic cost is associated with the fold, not the bending of the surface. These arguments are all local, and there may be global constraints that lead to an energetic barrier, but this depends specifically on the type of surface and shape of the crease. The authors thank Jesse L.
Silverberg, Thomas C. Hull, Douglas P. Holmes, and Dominic Vella for illuminating discussions. We are grateful to Michael J. Imburgia, Alfred J. Crosby, Mindy Dai, and Sam R. Nugen for assistance with both the 3D printer and laser cutter, and to Pedro Reis for discussions regarding the fundamentals of shell mechanics and insight on elastomer shells. The authors declare no conflict of interest. This article contains supporting information online at www.
Marin , c Itai Cohen , d Ryan C. Hayward , a, 2 and Christian D. Santangelo b, 2.
Arthur A. Luis A. Ryan C.
Christian D. Email: ude. Edited by Tom C. Copyright notice. This article has been cited by other articles in PMC. Significance Shape-programmable structures have recently used origami to reconfigure using a smooth folding motion, but are hampered by slow speeds and complicated material assembly. Keywords: buckling instability, origami inspired, snap-through, creased shell, programmable matter.here
Bendings of Surfaces and Stability of Shells - Alekse_ Vasil_evich Pogorelov - Google книги
Abstract Curvature and mechanics are intimately connected for thin materials, and this coupling between geometry and physical properties is readily seen in folded structures from intestinal villi and pollen grains to wrinkled membranes and programmable metamaterials. Geometrical Mechanics of Folding a Shell Inspired by these ideas from origami, we consider the folding of curved surfaces with creases. Open in a separate window. Negative Gaussian Curvature: Helicoid To explore snapping behavior and the lack thereof in negative Gaussian curvature surfaces, we specifically choose the helicoid.
Positive Gaussian Curvature: Sphere Given their axisymmetry, spherical shells are well suited to quantitative analytical, computational, and experimental analysis. Conclusion The ability to introduce tunable bistability into a curved shell via structural inhomogeneity represents a major step in generating programmable materials with rapid actuation capabilities.
Materials and Methods Shell Fabrication.