Euclidean Frustrated Ribbons

Open Access

Euclidean Frustrated Ribbons

Emmanuel Siéfert, Ido Levin, and Eran Sharon

Phys. Rev. X 11, 011062 – Published 29 March 2021

Abstract

Geometrical frustration in thin sheets is ubiquitous across scales in biology and becomes increasingly relevant in technology. Previous research identified the origin of the frustration as the violation of Gauss’s Theorema Egregium. Such “Gauss frustration” exhibits rich phenomenology; it may lead to mechanical instabilities, anomalous mechanics, and shape-morphing abilities that can be harnessed in engineering systems. Here we report a new type of geometrical frustration, one that is as general as Gauss frustration. We show that its origin is the violation of Mainardi-Codazzi-Peterson compatibility equations and that it appears in Euclidean sheets. Combining experiments, simulations, and theory, we study the specific case of a Euclidean ribbon with radial and geodesic curvatures. Experiments, conducted using different materials and techniques, reveal shape transitions, symmetry breaking, and spontaneous stress focusing. These observations are quantitatively rationalized using analytic solutions and geometrical arguments. We expect this frustration to play a significant role in natural and engineering systems, specifically in slender 3D printed sheets.

Received 20 October 2020
Revised 21 December 2020
Accepted 9 February 2021

DOI:https://doi.org/10.1103/PhysRevX.11.011062

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI.

Published by the American Physical Society

Physics Subject Headings (PhySH)

Buckling Elastic deformation Elastic forces Phase diagrams Surface instabilities

Composite materials Membrane structures

Finite-element method Scaling methods

Polymers & Soft Matter

Authors & Affiliations

Emmanuel Siéfert ^*,†, Ido Levin ^*, and Eran Sharon

Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel

^*These authors contributed equally to this work.
^†Corresponding author. emmanuel.siefert@mail.huji.ac.il

Popular Summary

Self-shaping of thin sheets is abundant in biological systems, from plant leaves to organ walls. Inspired by natural systems, many responsive materials have been developed in recent years to program shape-shifting structures, with applications in soft robotics, mini-invasive surgery, and smart textiles. In many cases, the responsiveness induces spontaneous curvature, which controls the tendency of the sheet to bend. However, despite its ubiquity in manufacturing techniques, the impact of spontaneous curvature on the shape selection and on the stability of the structures is poorly understood and has only been studied in simple cases. Through experiments, simulation, and theory, we present a new mechanism to engineer residually stressed thin sheets by exploiting such spontaneous curvature.

In a suite of experiments ranging from prestretched bilayers to heat-actuated 3D-printed polymers, we demonstrate that flat ribbons with spatially varying spontaneous curvature exhibit rich morphologies, instabilities, and transitions typical of prestressed sheets. We identify the geometrical origin of the residual stress and characterize through scaling laws the transitions observed. We also analytically derive quantitative predictions for the shape selection using simple geometrical arguments.

Our work shows that shape selection in thin sheets is richer than currently believed: Spontaneous curvature aligned along a single direction alone may cause internal stresses and should be taken into account to accurately program shapes. Our results require a fundamental overhaul of existing frameworks describing prestrained sheets, while our experimental results provide concrete directions to build such a theory.

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Vol. 11, Iss. 1 — January - March 2021

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Figure 1
Configurations of MCP-frustrated ribbons. (a) Curved ribbon with constant radial reference curvature $κ$ . (b) Curved ribbon with constant azimuthal reference curvature $κ$ . (c) Straight ribbon with linearly varying transverse reference curvature along the longitudinal direction (ranging from 0 to $π / w$ ). (d) Illustration of MCP frustration via the nonconservation of the normal for the example (c). The normal evolution along the ribbon’s boundary in coordinate space results in no rotation along the long direction, since there is no reference curvature in this direction. At one end of the ribbon, it rotates by $π$ along the width, whereas it does not rotate at the other end. Therefore, after integrating the normal evolution over this closed path, we obtain a nonzero rotation of the normal, contradicting the MCP equations, thus leading to an incompatible reference geometry.
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Figure 2
Morphologies of radially curved ribbons. (a) Phase diagram of the three different types of observed shapes: toroidal (blue), intermediate (orange), and faceted (green). Full markers correspond to different experimental techniques: latex bilayers (circles), latex or polypropylene bilayers (diamonds), and 3D-printed PLA (triangles). Open markers correspond to finite-element (FE) simulation (see Sec. 4 for more details on the experimental techniques and the simulation). (b) Examples of the observed morphologies: toroidal (top), intermediate (center), and faceted (bottom) for latex bilayers (left), 3D-printed PLA (center), and FE simulations (right). Movie 1 in Supplemental Material shows the deformation of 3D-printed ribbons when immersed in hot water [43].
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Figure 3
Overcurvature and buckling of bending-dominated ribbons. (a) In this regime, ribbons adopt a toroidal shape, inducing hoop tension and compression in the inner and outer part, respectively. (b) When closed, the ribbon outline buckles out of plane, with an amplitude depending on the total radial rotation angle $κ w$ ; experiments (left and center) and simulation (right) of buckled tori. (c) Open ribbons remain planar and overcurve by a factor $λ = θ / θ_{0}$ . (d) Left: azimuthal strain profile across the width of the ribbon for various overcurvature factors $λ$ (with $κ w = π$ , $R / w = 10$ ). Coiling induces a reduction of the strains in the ribbon, until an optimal configuration (black line) is reached. Right: corresponding stretching energy as a function of $λ$ . (e) Overcurvature factor $λ$ as a function of the total radial rotation angle $κ w$ . Latex bilayers (circles), 3D-printed PLA (triangles), and numerical simulations (open squares) fall on the analytic curve given by Eq. (10).
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Figure 4
Faceted stretching-dominated ribbons. (a) Sketch of the locally tubular configuration observed in experiments. (b) Closed ribbons (made here of polypropylene or latex bilayers) exhibit polygonal shapes, the number of sides corresponding to the number of locally tubular portions. When the width is gradually reduced, the number of sides increases. (c) Finite-element simulation of a portion of a curved stretching-dominated ribbon. The color codes for the bending energy density: it increases with the angle $θ$ [as predicted by Eq. (14)], except along the inner boundary. (d) Simulation of full structures, with clear bending energy focusing at the junction between tubular sections. (e) Geometrical rule to predict the typical angle extension $θ_{c}$ of the sides and hence the number of sides. (f) Number of sides $n$ as a function of the slenderness ratio $R / w$ of the curved ribbons. Simulations and experimental data (pictures of some experiments in inset) are quantitatively predicted by the simple geometric construction [Eq. (15)].
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Figure 5
Stretched latex bilayers. (a) A latex sheet is stretched radially by a predetermined factor. The sheet is then glued to an unstretched sheet made of latex or polypropylene. The radial cuts release the hoop stress resulting from the Poisson effect, inducing a quasiradial stretch. (b) When cut into a curved ribbon and thus released, a radial reference curvature is imposed to the structure. This curvature is, however, not exactly constant. Depending on the size of the radial cuts, and more specifically on the ratio $r_{\min} / r_{\max}$ (c), the radial strain, and thus the target curvature, vary as a function of the radial coordinate $r$ (d). This effect is, however, neglected in our study.
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Figure 6
3D-printed thermoplastic ribbons. (a) Radial lines (second layer) are printed on top of a layer printed along a Hilbert curve (first layer). (b) Long polymeric chains straighten in the nozzle due to the extensional flow. Cooling down, this elongated configuration results in residual stresses along the printing direction. When heated above the glass transition temperature, the printed thermoplastic material shrinks in the printing direction to release the residual stresses. (c) While the top layer shrinks radially, the bottom layer is mostly passive due to its thickness and the isotropic pattern, inducing thus a radial reference curvature (see Movie 1 in Supplemental Material [43]).
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