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Research Papers

Surface Texturing Through Cylinder Buckling

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

Department of Engineering,
University of Cambridge,
Trumpington Street,
Cambridge CB2 1PZ, UK
e-mail: kas14@cam.ac.uk

S. V. Stott

Department of Engineering,
University of Cambridge,
Trumpington Street,
Cambridge CB2 1PZ, UK

1Corresponding author.

Manuscript received August 13, 2013; final manuscript received December 20, 2013; accepted manuscript posted December 25, 2013; published online January 30, 2014. Assoc. Editor: Taher Saif.

J. Appl. Mech 81(6), 061001 (Jan 30, 2014) (7 pages) Paper No: JAM-13-1340; doi: 10.1115/1.4026331 History: Received August 13, 2013; Revised December 20, 2013; Accepted December 25, 2013

We consider the axial buckling of a thin-walled cylinder fitted onto a mandrel core with a prescribed annular gap. The buckling pattern develops fully and uniformly to yield a surface texture of regular diamond-shaped buckles, which we propose for novel morphing structures. We describe experiments that operate well into the postbuckling regime, where a classical analysis does not apply; we show that the size of buckles depends on the cylinder radius and the gap width, but not on its thickness, and we formulate simple relationships from kinematics alone for estimating the buckle proportions during loading.

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Figures

Grahic Jump Location
Fig. 1

Progressive formation of a surface texture during axial buckling of a thin-walled cylinder. (a) Initially, the cylinder is mounted as a close-fitting sleeve on a rigid cylindrical mandrel (not in view), and the wooden end pieces, or platens, are pushed together; a small axial gap allows the sleeve to compress without loading the mandrel. (b) Diamond-like buckles manifest locally at first, and their tendency to displace inwards is arrested by the mandrel, which allows their number to increase over the entire surface. (c) Further compression produces a high level of contact between the cylinder and mandrel. (d) The cylinder has length 150 mm and diameter 60 mm, and its radius differs from that of the mandrel by 1%. The material is Mylar with a thickness of 0.044 mm.

Grahic Jump Location
Fig. 2

(a) Yoshimura's facetted cylinder, created by folding a flat sheet along prescribed hinge lines, before joining the edges, courtesy of [6]. (b) Formal testing of our cylinders produces a distinctive Yoshimura pattern. (c)–(e) Close-up view of diamond buckle details for different cylinders. (c) The smallest mismatch gives ridges that curve and touch at adjacent corners. (d) A larger mismatch produces a more distinctive polygonal outline for each diamond, with significant adherence over the inner mandrel. (e) The largest mismatch produces adjacent ridges that tend to be separated. The features in (c)–(e) help to inform upon the simple circumferential mode-shapes presented in Fig. 8.

Grahic Jump Location
Fig. 3

Surface textures due to axial buckling of thin-walled tubes fitted onto three differently shaped mandrels of (a) circular, (b) elliptical, and (c) airfoil sections. Each shows that the buckle size is affected by the local radius of mandrel curvature. In (a), all are the same, whereas in (b) and (c), they become larger when the section becomes flatter, and vice versa. The wooden end-plates enable axial compression to be applied.

Grahic Jump Location
Fig. 4

(a) Manufacture of a cylinder by wrapping a flat Mylar sheet around a cylindrical mandrel. In order to create a precise annular gap, a rod is inserted as a shim to create extra circumferential length. A seam is created from a small overlapping joint. The shim is then removed to leave a slightly larger outer cylinder, where the mismatch in radii can be characterized by defining ξ=(R2/R1)-1. For example, in one case tested later, the radii of the mandrel and the shim rod are 50 mm and 4.76 mm, respectively, giving ξ=2.4%. (b) Formal compression scheme using an Instron tensometer. A sliding collar is fitted to the top platen so that only the cylinder is compressed while the bulk of its surface is in contact with the mandrel. In all of these experiments, the mandrel has a radius of 50 mm, and is approximately 300 mm tall.

Grahic Jump Location
Fig. 5

Force-displacement response of five cylinders tested as per the schematic in Fig. 4(b). Before recording data, each is nominally compressed so that buckles form everywhere, and some manual adjustment is required to ensure an evenly distributed pattern. The axial load is then decreased to zero while measuring the end displacement, so the data proceed from left to right: at zero load, the displacement is set to be zero, and all values referred to this datum. The cylinders have initial radii, R2, of 51.2, 51.8, 52.5, 53.3, and 55.0 mm, length l of 322, 326, 330, 335, and 345 mm, and all have the same thickness, t=0.044mm, giving a radius-to-thickness range of 1164–1250. The inner mandrel has a radius R1 of 50 mm, so that initial mismatch strains, ξ, are equal to 2.4, 3.7, 5.1, 6.6, and 9.9%, respectively, as stated in the legend. The axial strain, ɛ, is the end-wise displacement divided by the initial height, and the axial stress, σ¯, is found by dividing the axial force by the area of contact, 2πR2t, and then again by the value of the classical axial buckling stress for a cylinder, Et/(R23[1-v2]), where E is the Young's modulus (4905 MPa) and ν is the Poisson's ratio (0.34) of Mylar.

Grahic Jump Location
Fig. 6

Initial mismatch between the cylinder and mandrel, ξ=R2/R1-1, versus the number of circumferential buckles, n, in the postbuckling regime. The circles are obtained from the five Instron-tested cases in Fig. 5, where n is calculated by averaging their number on several axial levels. The squares apply to those manually compressed by hand on different mandrels (of radii 29.6, 31.6, and 63.5 mm), where n is averaged on two circumferences. Solid lines are predictions by Eq. (1) and Fig. 8(b), and dashed lines are predictions by Eq. (3) and Fig. 8(c).

Grahic Jump Location
Fig. 7

Average buckle height, h, (Fig. 8) versus axial compressive strain, ε, for the five Instron-tested cases from Fig. 5 The extra lines are predictions by Eq. (4), which uses n from Fig. 6 All have the same gradient of 1/ε, their mismatch strains are indicated alongside, and their colors match the legend symbols. Note that, generally, when the mismatch strain is smallest, all buckles deform in unison, and h is approximately governed by 1/ε. As the mismatch increases, buckles tend to form progressively rather than together, so that some of the buckles have roughly constant height.

Grahic Jump Location
Fig. 8

Schematic details for estimating the buckled geometry. (a) The number of facets, n, on a given circumference is given by the number of diamond features touching on adjacent corners. (b) Simplified polygonal description of the circumferential path in (a) with n straight sides interconnected by a small regions of constant radius of curvature, r. When r is zero, the maximum radial distance from the center is p equal to R1/cos(π/n). (c) The path is now described by n curved buckles of radius, R, subtending angle, 2β, but meeting discontinuously at points of infinite curvature. (d) Scheme for estimating the axial height, h, of buckles in which the edge is treated as a rod buckling between the inner mandrel and a fictitious outer constraint defined by the largest radial displacement, p, from (b). The axial displacement for a given facet is δ and the inner mandrel has radius, R1.

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