The Shape of Helically Creased Cylinders

2013 ◽  
Vol 80 (5) ◽  
Author(s):  
K. A. Seffen ◽  
N. Borner
Keyword(s):  
Large Deformation ◽  
Gaussian Curvature ◽  
Pitch Angle ◽  
Thin Shells ◽  
Flat Sheet ◽  
Cylindrical Form ◽  
Geometrical Concept ◽  
Mohr’S Circles ◽  
Doubly Curved ◽  
Mohr’S Circle

Creasing in thin shells admits large deformation by concentrating curvatures while relieving stretching strains over the bulk of the shell: after unloading, the creases remain as narrow ridges and the rest of the shell is flat or simply curved. We present a helically creased unloaded shell that is doubly curved everywhere, which is formed by cylindrically wrapping a flat sheet with embedded fold-lines not axially aligned. The finished shell is in a state of uniform self-stress and this is responsible for maintaining the Gaussian curvature outside of the creases in a controllable and persistent manner. We describe the overall shape of the shell using the familiar geometrical concept of a Mohr's circle applied to each of its constituent features—the creases, the regions between the creases, and the overall cylindrical form. These Mohr's circles can be combined in view of geometrical compatibility, which enables the observed shape to be accurately and completely described in terms of the helical pitch angle alone.

scholarly journals Universally bistable shells with nonzero Gaussian curvature for two-way transition waves

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Nikolaos Vasios ◽  
Bolei Deng ◽  
Benjamin Gorissen ◽  
Katia Bertoldi
Keyword(s):  
Gaussian Curvature ◽  
Energy Landscape ◽  
Propagation Distance ◽  
Energy Landscapes ◽  
Bending Energy ◽  
Nonlinear Dynamic Response ◽  
One Dimensional ◽  
Doubly Curved Shells ◽  
Doubly Curved ◽  
The Waves

AbstractMulti-welled energy landscapes arising in shells with nonzero Gaussian curvature typically fade away as their thickness becomes larger because of the increased bending energy required for inversion. Motivated by this limitation, we propose a strategy to realize doubly curved shells that are bistable for any thickness. We then study the nonlinear dynamic response of one-dimensional (1D) arrays of our universally bistable shells when coupled by compressible fluid cavities. We find that the system supports the propagation of bidirectional transition waves whose characteristics can be tuned by varying both geometric parameters as well as the amount of energy supplied to initiate the waves. However, since our bistable shells have equal energy minima, the distance traveled by such waves is limited by dissipation. To overcome this limitation, we identify a strategy to realize thick bistable shells with tunable energy landscape and show that their strategic placement within the 1D array can extend the propagation distance of the supported bidirectional transition waves.

scholarly journals Additive lattice kirigami

2016 ◽  
Vol 2 (9) ◽  
pp. e1601258 ◽  
Author(s):  
Toen Castle ◽  
Daniel M. Sussman ◽  
Michael Tanis ◽  
Randall D. Kamien
Keyword(s):  
Gaussian Curvature ◽  
Three Dimensional ◽  
Complex Structures ◽  
3D Structures ◽  
Flat Sheet ◽  
Simple Rules ◽  
New Material ◽  
3D Shapes ◽  
Huge Variety

Kirigami uses bending, folding, cutting, and pasting to create complex three-dimensional (3D) structures from a flat sheet. In the case of lattice kirigami, this cutting and rejoining introduces defects into an underlying 2D lattice in the form of points of nonzero Gaussian curvature. A set of simple rules was previously used to generate a wide variety of stepped structures; we now pare back these rules to their minimum. This allows us to describe a set of techniques that unify a wide variety of cut-and-paste actions under the rubric of lattice kirigami, including adding new material and rejoining material across arbitrary cuts in the sheet. We also explore the use of more complex lattices and the different structures that consequently arise. Regardless of the choice of lattice, creating complex structures may require multiple overlapping kirigami cuts, where subsequent cuts are not performed on a locally flat lattice. Our additive kirigami method describes such cuts, providing a simple methodology and a set of techniques to build a huge variety of complex 3D shapes.

Buckling of doubly curved shallow thin shells.

AIAA Journal ◽  
1968 ◽  
Vol 6 (5) ◽  
pp. 982-984 ◽  
Author(s):  
H. M. HAYDL
Keyword(s):  
Thin Shells ◽  
Doubly Curved

Lofting and Fabrication of Compound-Curved Plates

1993 ◽  
Vol 37 (02) ◽  
pp. 166-175
Author(s):  
John S. Letcher
Keyword(s):  
Plastic Strain ◽  
Numerical Solution ◽  
Gaussian Curvature ◽  
Strain Distribution ◽  
Sheet Material ◽  
Quantitative Description ◽  
Geometric Theory ◽  
Flat Sheet ◽  
Plane Plastic ◽  
The Relationship

The fabrication of plates with compound curvature (nonzero Gaussian curvature) from initially flat sheet material necessarily involves some degree of in-plane plastic strain. A basically geometric theory is presented for modeling and controlling this process, leading to a quantitative description of both the strain distribution required to achieve the compounding, and the relationship between points and curves on the curved surface and the flat material. The theory thus provides accurate methods for lofting (expansion or plane development) of curved plates, as well as quantitative control of the compounding process. Several methods of numerical solution are presented, with illustrative examples.

Free Vibration of Doubly Curved Thin Shells

2017 ◽  
Vol 140 (3) ◽  
Author(s):  
April Bryan
Keyword(s):  
Differential Equation ◽  
Vibration Analysis ◽  
Equation Of Motion ◽  
Thin Shells ◽  
Numerical Examples ◽  
Normal Coordinates ◽  
Spatial Equation ◽  
Doubly Curved ◽  
Analytical Approaches ◽  
Numerical Approaches

While several numerical approaches exist for the vibration analysis of thin shells, there is a lack of analytical approaches to address this problem. This is due to complications that arise from coupling between the midsurface and normal coordinates in the transverse differential equation of motion (TDEM) of the shell. In this research, an Uncoupling Theorem for solving the TDEM of doubly curved, thin shells with equivalent radii is introduced. The use of the uncoupling theorem leads to the development of an uncoupled transverse differential of motion for the shells under consideration. Solution of the uncoupled spatial equation results in a general expression for the eigenfrequencies of these shells. The theorem is applied to four shell geometries, and numerical examples are used to demonstrate the influence of material and geometric parameters on the eigenfrequencies of these shells.

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