How a Curved Elastic Strip Opens
Résumé
An elastic strip is transversely clamped in a curved frame. The induced curvature decreases as the strip opens and connects to its flat natural shape. Various ribbon profiles are measured and the scaling law for the opening length validates a description where the in-plane stretching gradually relaxes the bending stress. An analytical model of the strip profile is proposed and a quantitative agreement is found with both experiments and simulations of the plates equations. This result provides a unique illustration of smooth nondevelopable solutions in thin sheets. Geometry-induced rigidity is a fundamental feature of thin structures [1], which has long been used in engineering and architecture to design stiff fuselages, hulls, roofs, and deployable structures [2–4]. It is also widely encountered in living structures, such as in plant leaves where curvature can prevent the collapse of the leaves under their own weight [5]. A simple illustration of this rigidity induced by curvature is given by a strip of paper held at one end. When the strip is flat, it is unable to sustain its own weight and bends downward under gravity. However, if the end of the paper is slightly curved transversely, the strip straightens up and becomes much stiffer. Rigidity in these systems arises because bending in one direction is coupled to the transverse curvature and cannot occur without stretching the sheet—a costly mode of deformation in thin plates in terms of elastic energy [6]. Knowing the distance over which an induced curvature spreads is thus an important issue for predicting the rigidity of thin plates and shells. In this Letter, we address the question of the persistence length of curvature in thin sheets on a minimal system: a flat elastic ribbon of thickness t, width W, and length L W, which is clamped at one end over a cylinder of radius R [Fig. 1(a)]. After what distance from the clamp does the ribbon unfold and recover its flat natural shape? This deceptively simple problem is actually not straightforward as, to unfold, the ribbon has to stretch—a forbidden mode of deformation in the inextensible limit. In thin sheets, this constraint is usually resolved by focusing the stretch in elastic defects or singularities, such as the ridges and peaks of a crumpled paper, the rest of the surface being fully developable (i.e. free of stretching) [7–14]. However, another way to obtain the stretching of a thin sheet is to consider that the curvature variation on large distances is associated with a regular stretching, i.e., without defects. Surprisingly, the first insights of this approach are found in the studies of defects such as ridges [15] and pinches [16], where both the focused-stress and the diffuse-stress are present [17]. In each of these situations, regular developable solutions exist away from the defect but they are not observed as the bending energy can be progressively released by a small in-plane stretching (see also [18,19]). The stretching over large distances is also involved in the shape of drapes [20] and curtains [21], the tearing of sheets [22], or the dynamics of curved ribbons [23,24]. Our prototypal system provides a reduced model to probe these situations and the transition between smooth and singular solutions in strained sheets. A first observation of the opening is displayed in Fig. 1(b), using an acetate elastic sheet (t ¼ 110 m, W ¼ 4.5 cm) clamped in a circular frame of radius R ¼ 2.5 cm. The strip is positioned vertically to limit the out-of-plane deflection caused by its weight. Away from the clamping, the strip opens and its curvature decreases and connects to the flat stress-free region over a finite length L p. The opening also results in a small deflection of the strip corresponding to a tilt angle θ of the centerline. The persistence length of curvature L p can be estimated from a balance between the stretching and bending elastic energies using scaling arguments similar to [15]. On one hand, the bending energy of the ribbon scales as E b ∼ EtWL p ðt=RÞ 2 , where E is the Young modulus of the medium. On the other hand, the opening of the ribbon requires the stretching of the edges of the ribbon over a length L p. This is associated with a stretching energy E s ∼ EtWL p 2 , where ∼ Z 2 =2L 2 p is the typical in-plane strain and Z ∼ W 2 =8R is the out-of-plane deflection of the ribbon at the clamp [see Fig. 1(b) inset]. The trade off between the two energies E s and E b gives L p ∼ W 2 = ffiffiffiffiffi tR p or equivalently [15] L p ∼ W ffiffiffiffiffiffiffiffi Z=t p. The persistence length of the curved region is independent of the Young modulus and increases when the thickness is reduced.
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