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Research Papers

Prestressed Morphing Bistable and Neutrally Stable Shells

[+] Author and Article Information
Keith A. Seffen1

Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UKkas14@cam.ac.uk

Simon D. Guest

Department of Engineering, University of Cambridge, Trumpington Street, Cambridge CB2 1PZ, UKsdg13@cam.ac.uk

1

Corresponding author.

J. Appl. Mech. 78(1), 011002 (Oct 08, 2010) (6 pages) doi:10.1115/1.4002117 History: Received August 20, 2009; Revised July 02, 2010; Posted July 07, 2010; Published October 08, 2010; Online October 08, 2010

This study deals with prestressed shells, which are capable of “morphing” under large deflexions between very different load-free configurations. Prestressing involves plastically curving a flat, thin shell in orthogonal directions either in the opposite or same sense, resulting in two unique types of behavior for isotropic shells. Opposite-sense prestressing produces a bistable, cylindrically curved shell provided the prestress levels are large enough and similar in size: This effect forms the basis of a child’s “flick” bracelet and is well known. On the other hand, same-sense prestressing results in a novel, neutrally stable shell provided the levels are also sufficiently large but identical: The shell has to be made precisely, otherwise, it is monostable and is demonstrated here by means of a thin, helically curved strip. The equilibrium states associated with both effects are quantified theoretically and new expressions are determined for the requisite prestress levels. Furthermore, each stability response is revealed in closed form where it is shown that the neutrally stable case occurs only for isotropic materials, otherwise, bistability follows for orthotropic materials, specifically, those, which have a shear modulus different from the isotropic value. Finally, prestressing and initial shape are considered together and, promisingly, it is predicted that some shells can be neutrally stable and bistable simultaneously.

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Figures

Grahic Jump Location
Figure 1

A thin, prestressed bistable smooth shell of diameter 50 mm and thickness 0.1 mm and made from thin age-hardened copper beryllium sheet. Left: the original flat metallic disk—note the rectangular sticker tab fashioned to the upper surface. Middle: the curved shell after plastic coiling around a pencil stem. Right: a second curved shape of shell, formed by plastic coiling in the opposite direction, about an axis at right angles to the middle shape. The shell is subsequently bistable between the final two curved shapes.

Grahic Jump Location
Figure 2

Three views of a neutrally stable shell-strip made by plastically coiling the original flat strip along and across its length (1). In each view, the strip rests such that it is coiled about an axis inclined to the strip length at: left, 90 deg to become fully coiled; middle, approximately 45 deg to form a spiral; right, 0 deg to yield an open strip. The transition between shapes is effected by twisting the strip between its ends and there is no stiffness. The strip is made from age-hardened copper beryllium sheet and has a length of 180 mm; the radius of coiling is approximately 11 mm.

Grahic Jump Location
Figure 3

Stability landscape for originally flat shells with prestress levels k¯3 and k¯4 in orthogonal directions according to Eq. 16. Each subplot refers to a specific value of the relative shear modulus, α: left, 4×(1−ν)/2; middle, 1.1×(1−ν)/2; right, 1.01×(1−ν)/2 (the isotropic value being (1−ν)/2). All of the darkest regions conform to bistable shells and the lightest are monostable. The first and third quadrants in each subfigure are mirror symmetrical in both axes, as are the second and fourth quadrants. The white points denote the closed form solutions asserting the onset of bistable behavior when the levels of prestress are identical, given by Eq. 20 (first and third) and Eq. 19 (second and fourth). The value of ϕ is given by Eq. 3 where a circular shell is taken with ρ=1 and b4/t2R2 is set to a nominal value of unity. The Poisson’s ratio is set to 0.3.

Grahic Jump Location
Figure 4

Regions of stable equilibria for shells with combinations of same-sense prestressing k¯5 and initial spherical curvature κ¯0. The darkest regions conform to bistable configurations. The lightest region exhibits neutral stability whilst the second darkest regions in the top-right corner of the subfigure are configurations that possess a neutrally stable mode and a singly stable inverted shape. The second lightest regions are monostable configurations only. All shells are circular, ρ=1, b4/t2R2=1, and ν=0.3.

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