Kinematics of stretched sheets

In a new study now published as a report and also illustrated as the online cover-page of Science Advances, Julien Chopin, Arshad Kudrolli, and a research team in Physics in the U.S. and Brazil showed how twisted hyper-elastic ...

Shape transformation of sheets Traditionally, twisting sheets can form functional yarns that rely on millennia of human practice to form catgut bow strings , surgical sutures and fabric wearables; however, the practice still lacks overarching principles that guide the intricacy of such architectures. Scrolled yarns with nested structures can be used to harness energy on batteries and in embedding amorphous materials . Tensional twist folding can transform flat sheets into layered structures via remote boundary regulation. Twist-folding and scrolling can be used to reconfigure and repurpose flat sheets as seen with the multifunctional Rajasthan turban.

To understand shape transformation of sheets and the interplay between topology and large shape transformations, Chopin et al used three-dimensional X-ray scanning to detail the spontaneous formation of twisted, multilayered yarns with ordered internal architectures. It is, however, still challenging to model the large shape transformations and configurations. Recent studies have incorporated elastic plate models including the Föppl–von Kármán (FvK) equation to solve the initial growth above the onset of primary instability , but such methods remain to explain the transformation of a flat sheet into scrolled yarns . In this work, Chopin et al developed a new framework to combine the kinematics of structured sheets, and used origami to explain these observations. The team showed how the folded sheets showed regular polygonal shapes as described by Schläfli symbols and how origami kinematics captured the main features of the structure to provide a framework that served as a guide to develop hyper-elastic materials with broad applications.

Experiments reveal a highly ordered transformation to yarns when sheets held under tension are twisted beyond the onset of primary instabilities. Examples of twisted, folded, and scrolled structures are the following: (A) wrapped candy, (B) multifunctional Rajashtani Turban (photo credit: Lauren Cohen), and (C) scrolled yarn from a polyethylene sheet (see section S4). (D to G) Shadowgraphs of a transparent PDMS sheet twisted through angle θ as shown in the inset (L/W = 1; t/W = 0.0028; ΔL/L = 0.1; θp = 60 ± 5°). Inset: Schematic and lab coordinate system. (D) Wrinkles observed just above the onset of primary instability. (E) Accordion folded sheet with self-contact. (F) A nested helicoid with folded layers that develop as the sheet is twisted further. (G) Secondary buckling instability occurs with further twisting, resulting in a yarn-like structure. The scale bar is the same in (D) to (G). (H) The measured torque shows a repeated increasing and decreasing sawtooth variation with twist. The amplitude of variation increases as L/W decreases. (I) A map delineating regions where the primary instability, self-contact, and secondary instability occur as a function of aspect ratio and twist. Lines are guides to the eye, except the primary instability for L/W > 3. Credit: Science Advances (2022). DOI: 10.1126/sciadv.abi8818 Read more

Online cover: A thin polydimethylsiloxane (PDMS) sheet is twisted into multilayered scrolled yarn. For millennia, humans have twisted stretchable sheets to form functional yarns to create clothing items, string instruments, and upcycle plastic. Chopin and Kudrolli , develop an elasto-geometric framework to understand the physical mechanisms involved in twisting stretchable sheets into self-assembled architectures for advanced manufacturing strategies. Credit: Science Advances (2022). DOI: 10.1126/sciadv.abi8818 Read more

An overview of the observation transformations with twist and the tensional twist-folding framework. The observed main transformations as a planar sheet experiences tensional twist-folding and scrolling with applied twist. The elastogeometric framework is shown, including the perturbative FvK formalism, the elastogeometric torque model that incorporates geometric nonlinearities to explain the stress-strain relation with twist, the Schläfli origami kinematic model, and the geometric yarn model. Credit: Science Advances (2022). DOI: 10.1126/sciadv.abi8818 Read more

Accordion folding through curvature localization. (A) The deformation of a polyvinyl siloxane (PVS) sheet twisted by θ = 120° obtained with x-ray tomography and rendered with mean curvature H given by the color bar on the right (L/W = 3; t/W = 0.009; θp = 75 ° ± 5°). The central 80% of the sheet away from the clamps is shown. (B) The spatial distribution H mapped to a rectangular domain shows symmetry breaking and localization of the sheet curvature with twist. (C) Bending content wb shows the localization of energy with creasing across the cross section indicated by the solid white line in (A). (D) The measured number of folds n compared with the relation given by the wavelength of the primary instability n = 2W/λp. The aspect ratios (t/W, L/W) are as follows: PVS a (0.009,2), PVS b (0.006,3), PDMS (0.003,1), and latex (0.003,2). The three materials are hyperelastic with Young’s modulus E = 1.2 MPa (PVS), 6.2 MPa (PDMS), and 3.6 MPa (latex). Credit: Science Advances (2022). DOI: 10.1126/sciadv.abi8818 Read more

Partial Schläfli origami explains layered architectures at half-twist. (A) Geometrical forms obtained by increasing the Schläfli symbols and number of facets. (B) Comparison of the experimental radiogram and Schläfli fold origami. Good correspondence is observed in all four cases. (C) The angle Ψi of the ith fold as a function of the calculated angle i α using the geometric model is in excellent agreement. (D) Comparison of the apex angle α as a function calculated α using various sheets and loading. (E) The apex angle as a function of triangle number is essentially constant. Credit: Science Advances (2022). DOI: 10.1126/sciadv.abi8818 Read more