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Technical Briefs

The Shape of Helically Creased Cylinders

[+] Author and Article Information
K. A. Seffen

e-mail: kas14@cam.ac.uk

N. Borner

Department of Engineering,
University of Cambridge,
Trumpington Street,
Cambridge, CB2 1PZUK

1Corresponding author.

Manuscript received March 6, 2012; final manuscript received January 31, 2013; accepted manuscript posted February 11, 2013; published online July 12, 2013. Assoc. Editor: George Kardomateas.

J. Appl. Mech 80(5), 054501 (Jul 12, 2013) (4 pages) Paper No: JAM-12-1094; doi: 10.1115/1.4023624 History: Received March 06, 2012; Revised January 31, 2013; Accepted February 11, 2013

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.

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Figures

Grahic Jump Location
Fig. 5

Predictions (circles) from Eq. (2) compared to measurements for a range of pitch angles of crease. Angle β refers to the direction along which there is zero surface curvature in each of the interlineal strips. Each cylinder is constructed and folded from A4 paper-card, as described in Fig. 1.

Grahic Jump Location
Fig. 4

Description of the surface shape of a creased cylinder using Mohr's circles of twisting curvature (or “twist’’) c¯ versus the curvature c. (a) Performance of a smooth right-circular cylinder of radius 1/κ. The principal diameter with end-points (0,0) and (κ,0) is rotated by 2α in the same sense as the fold-lines in Fig. 1, to reveal the local curvatures κxxG and κyyG, respectively, along and normal to the creased lines and the associated twist κxyG. Simple geometry gives: κxxG = κsin2α, κyyG = κsin2α, and κxyG = κsinαcosα. (b) Mohr's circle for the strip (“S”) regions outside of the creases. The curvature along κxxS and the twist κxyS are the same as in (a) but the curvature across κyyS is zero because the strips are flat in this direction. After rotating the diameter by 2β, a second direction of zero surface curvature is found, which corresponds to the measurements highlighted in Fig. 2(c). Finally, the equivalent Mohr's circle for a crease is obtained by subtracting (b) from (a) under the rules applied in Fig. 3. The resulting end-points always lie at the end of a principal diameter: there is no curvature along, so κxxC = 0 and there is no twist κxyC = 0; the curvature across is κyyC = κsin2α.

Grahic Jump Location
Fig. 2

Views relating to certain properties of the shape, which are amplified by the drawn lines. (a) In a direction widthwise and normal to the creased lines, the strip has no surface curvature as given by the straight pencil line: this is repeated over several strips. (b) A second direction of zero surface curvature, defined by β in Fig. 1, is found by rotating a flat narrow edge on the surface of each strip region until the edge makes contact everywhere on the strip. This is repeated over all strips in sequence and highlighted to show the contiguous line. (c) An axial line on the surface shows some gentle undulation and, hence, some (negative) Gaussian curvature. The rotation across each crease is clearly visible.

Grahic Jump Location
Fig. 1

Manufacture of a creased cylinder. (a) Parallel fold-lines are introduced first into a flat rectangular sheet at an angle, α to one of the edges. The sheet then wraps into a helicoidal form when simply bent about each fold-line. (b) The opposite short edges of the sheet are connected to form a right-circular cylinder overall with the top and bottom edges each being planar. The angle of pitch of the original fold-lines to the axis of cylinder is α and the coordinates x and y denote the directions, respectively along and across the interlineal region of strip between the fold-lines, which are now designated as creases. The angle β defines the direction of a line relative to the strip width direction whose properties are discussed in Fig. 2(c). A practical cylinder made of A4 paper-card with fold-lines 15 mm apart and inclined at α = 30°. A description of the drawn lines is also given in Fig. 2.

Grahic Jump Location
Fig. 3

Addition of Mohr's circles for an element in plane-stress equilibrium. Direct stresses are denoted by σ and shear stresses by τ. Top (I): a general state of stress associated with a coordinate system defined by the (a,b) axes. The Mohr's circle, which is plotted in (τ,σ) space, shows the diameter formed by the indicated end point coordinates, which follows the usual plotting convention for Mohr's circles familiar to many undergraduates. Also indicated are two vectors to reach the lower end point from the origin by way of the center of the circle. Middle (II): a second state of stress associated with a new set of axes (c,d), which is rotated from (a,b) by an arbitrary angle θ. The corresponding diameter (not labeled) is then rotated by 2θ in the opposite direction, to yield the new, and extra, stresses associated with the original direction. Bottom (III): The stress-states from (I) and (II) are combined into a third Mohr's circle by adding the values of stress associated with the same direction: σaaIII = σaaI + σaaII, σbbIII = σbbI + σbbII, and τabIII = τabI + τabII. The pair of vectors from (I) and (II) also combine faithfully to yield the designated end point.

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