Relaxation Mechanisms in the Unfolding of Thin Sheets

B. Thiria and M. Adda-Bedia

Phys. Rev. Lett. 107, 025506 – Published 8 July 2011

Abstract

When a thin sheet is crumpled, creases form in which plastic deformations are localized. Here we study experimentally the relaxation process of a single fold in a thin sheet subjected to an external strain. The unfolding process is described by a quick opening at first and then a progressive slow relaxation of the crease. In the latter regime, the necessary force needed to open the folded sheet at a given displacement is found to decrease logarithmically in time, allowing its description through an Arrhenius activation process. We accurately determine the parameters of this law and show its general character by performing experiments on both Mylar and paper sheets.

Received 10 March 2011

DOI:https://doi.org/10.1103/PhysRevLett.107.025506

Authors & Affiliations

B. Thiria¹ and M. Adda-Bedia²

¹Laboratoire de Physique et Mécanique des Milieux Hétérogènes, CNRS-ESPCI-UPMC Paris 6, Université Paris-Diderot, 10 rue Vauquelin, 75005 Paris, France
²Laboratoire de Physique Statistique, Ecole Normale Supérieure, UPMC Paris 6, Université Paris-Diderot, CNRS, 24 rue Lhomond, 75005 Paris, France

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Vol. 107, Iss. 2 — 8 July 2011

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Images

Figure 1
Uncontrolled experiment: time evolution of the angle of a crease made in a rectangular sheet of Mylar of thickness 0.35 mm (see the inset). The dashed line is a logarithmic fit $θ (t) = a \log (t) + b$ with $a = 0.055$ and $b = 1.392$ . Inset: Superimposed images of the sheet during the relaxation process.Reuse & Permissions
Figure 2
(a) Schematics of the controlled experiment. A folded sheet is clamped on both sides and maintained at a fixed strain $2 y$ . The force and angle are measured simultaneously. (b) Illustration of the protocol for the fold preparation. The crease is obtained by applying a constant load on a folded sheet into two equal parts during a given time (typically between 10 and 20 min). (c) Superimposed images of the initial stage of the sheet during the unfolding and relaxation processes.Reuse & Permissions
Figure 3
Evolution of the force $F$ as the fold relaxes in a Mylar sheet of thickness $h = 0.35 mm$ and for a spacing $2 y = 6.5 cm$ between the plates. At short times ( $t ≲ 150 s$ ), the force decreases following an exponential law $γ \exp (- t / t_{0})$ (thin dashed curve), and then a slow relaxation regime described by a logarithmic law $a \log (t) + b$ (thick dashed curve) takes place. Inset: The corresponding evolution of the angle $θ (t)$ of the crease showing the same features as $F (t)$ .Reuse & Permissions
Figure 4
Evolution of the dimensionless force $\tilde{F} (θ)$ measured for two different runs (○). Experiments were performed in Mylar sheets of thickness $h = 0.35 mm$ at the same imposed initial spacing $y / L = 0.7625$ . The predictions of the elastica for the experimentally imposed spacing (dashed line) and for $y / L = 0.7357$ (continuous line) are also shown.Reuse & Permissions
Figure 5
Experimental runs showing the two protocols followed for the characterization of the force relaxation. (a) The fold spacing is increased periodically by an increment $Δ y$ starting from an initial spacing $y_{0}$ . (b) The fold spacing is alternately increased and decreased by an increment $Δ y$ . The time interval $Δ t$ between two successive events and $Δ y$ are the control parameters of the experiment.Reuse & Permissions
Figure 6
Logarithmic relaxation of the force. (a) Results of protocol I experiments. We started from various initial spacings $y_{0}$ and applied an increment $Δ y = 0.5 cm$ during waiting times $Δ t = 15$ , 20, and 25 min. Inset: Protocol II experiments with $Δ t = 20 \min$ and $Δ y = 5.5$ , 11, and 16.5 cm. The constants are determined from fittings by $a \log (t - t_{0}) + b$ . (b) Constants determined from fittings by $a_{⋆} \log (t) + b_{⋆}$ of the same data as in (a). Experiments were performed by using 0.35 mm Mylar sheets (open symbols) and printing paper of thickness 0.1 (filled squares) and 0.2 mm (filled triangles).Reuse & Permissions

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