Research Papers

Curling of a Heated Annulus

[+] 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

Contributed by the Applied Mechanics Division of ASME for publication in the JOURNAL OF APPLIED MECHANICS. Manuscript received August 21, 2015; final manuscript received October 22, 2015; published online November 13, 2015. Assoc. Editor: Kyung-Suk Kim.

J. Appl. Mech 83(2), 021005 (Nov 13, 2015) (7 pages) Paper No: JAM-15-1443; doi: 10.1115/1.4031894 History: Received August 21, 2015; Revised October 22, 2015

We study the large-displacement behavior of a thin annulus subjected to thermal gradients. Following axisymmetrical deflections, the annulus buckles into a periodic mode shape, which eventually forms a number of a radial curls under sustained heating. We predict the number of curls using a simple plate-buckling analogy and an approximate energy method.

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Mansfield, E. H. , 1965, “ Bending, Buckling and Curling of a Heated Elliptical Plate,” Proc. R. Soc. London, A, 288(1414), pp. 396–417. [CrossRef]
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Seffen, K. A. , and McMahon, R. A. , 2007, “ Heating of a Uniform Wafer Disk,” Int. J. Mech. Sci., 49(2), pp. 230–238. [CrossRef]
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Grahic Jump Location
Fig. 4

Cross-over point data from Fig. 3 plotted on normal axes and logarithmic axes. Solid dots are data points (in each case being the lowest width for a given mode number) and lines (of the same color) are best-fit lines, see Table 2. The gray line is the simple buckling prediction for a straight plate, Eq. (4), the black dashed line is the inextensible model, Eq.(2), and the black circles are derived from the analysis associated with Fig. 8. The legend applies to both subfigures.

Grahic Jump Location
Fig. 1

Large subfigure: finite element analysis of annular plate subject to thermal gradient through its thickness. Transverse (out-of-plane) displacements are W, and the thermal gradient is embodied by κT, the stress-free curvature. The inner edge is restrained against rotation but all other displacements are unconstrained. Axisymmetrical behavior is relayed by the curve labelled “perfect.” Two sets of asymmetrical buckling curves pertain to different amplitudes of initial out-of-plane imperfection, equal to 0.1% and 1%, where solid lines are for edge points with maximum displacement, and dotted lines are for minimum, as indicated in the legend. The disk has the material and planform properties from Table 1, with thickness, t, equal to 0.001 m, and the ratio of inner-to-outer radii equal to 0.5. The bottom subfigure shows the transverse displacement of all points on the edge at increasing levels of heating for the 0.1% case.

Grahic Jump Location
Fig. 3

The mode number, n, from finite element data versus dimensionless annular width, 1−b/a, for three different initial thicknesses, t. The larger (colored) dots are for when the inner edge of annulus is completely restrained; the smaller dots, when the inner edge can displace radially (but not rotate). The thin lines track the widths where the mode number jumps in value. The dashed line is the prediction from the inextensible model of Fig. 5.

Grahic Jump Location
Fig. 2

Large-displacement buckled shapes for inner-to-outer radii ratios of 0.3, 0.5, 0.7, and 0.84 in (a)–(d), respectively. The mode number, n, is the number of curled “petals,” equal to 3, 4, 6, and 9. The coloring helps to convey the levels of displacement.

Grahic Jump Location
Fig. 5

Inextensible model for calculating mode number, n. (a) Initial planform geometry where b and a are inner and outer radii, respectively. The inscribed polygon (shown here as a triangle) has n sides. Inextensibility implies that the displacement field can only be cylindrical or flat: inside the polygon, the disk remains flat, and outside (in a typical hatched region), the disk is cylindrically deformed. (b) The inscribed polygon has side-length 2l1, and just touches the inner radius; when the inner radius is slightly larger (dashed) then the order of polygon must increase to avoid encroachment, to satisfy displacement compatibility. (c) The limiting inner radius when the polygon, of side-length 2l2, is just touching.

Grahic Jump Location
Fig. 6

Schematic deformation of annulus during (a) prebuckling (axisymmetric) and (b) postbuckling (periodic). (a) Conservation of radial lengths in the initial planform suggests that the outer edge of the deformed plate must contract, leading to circumferential compression; the variation overall must be self-equilibrating, so there is in tension on the inside and a nonuniform variation in between, as confirmed by finite element analysis. (b) Subsequent buckling into a periodic mode shape of wavelength, λ: thus, n × λ = 2πa.

Grahic Jump Location
Fig. 7

Variation of circumferential stress, σθ, during prebuckling, for three cases of b/a, obtained from finite element analysis. The thickness of disk in all cases is t = 0.001 m. In each subfigure, we show three curves, which highlight how σθ varies and increases with more heating; importantly we see compression on the outer edge and tension on the inner edge at all times and, as b/a increases, the profile becomes increasingly linear as the thermal gradient increases. The material properties are given in Table 1.

Grahic Jump Location
Fig. 8

Energy method prediction of the buckling factor, ρ, from Eq. (7), which relates to the level of thermal gradient required to initiate periodic buckling of the annulus. Each of the curves, from dark to light, is for the circumferential mode number, n, equal to 2, 3, 4, 5, and 6, respectively, and m = 1 in Eq. (7). The annulus thickness is 0.0001 m, with other properties in Table 1. The circles highlight the cross-over intersection points between successive n-curves. Note that the abscissae values are plotted as logarithmic but the ordinate values are not.



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