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Box F, Erlich A, Guan JH, Thorogood C. Gigantic floating leaves occupy a large surface area at an economical material cost. SCIENCE ADVANCES 2022; 8:eabg3790. [PMID: 35138898 PMCID: PMC8827653 DOI: 10.1126/sciadv.abg3790] [Citation(s) in RCA: 1] [Impact Index Per Article: 0.5] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Indexed: 05/09/2023]
Abstract
The giant Amazonian waterlily (genus Victoria) produces the largest floating leaves in the plant kingdom. The leaves' notable vasculature has inspired artists, engineers, and architects for centuries. Despite the aesthetic appeal and scale of this botanical enigma, little is known about the mechanics of these extraordinary leaves. For example, how do these leaves achieve gigantic proportions? We show that the geometric form of the leaf is structurally more efficient than those of other smaller species of waterlily. In particular, the spatially varying thickness and regular branching of the primary veins ensures the structural integrity necessary for extensive coverage of the water surface, enabling optimal light capture despite a relatively low leaf biomass. Leaf gigantism in waterlilies may have been driven by selection pressures favoring a large surface area at an economical material cost, for outcompeting other plants in fast-drying ephemeral pools.
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Affiliation(s)
- Finn Box
- Department of Physics and Astronomy, University of Manchester, Manchester M13 9PL, UK
- Gulliver UMR CNRS 7083, ESPCI Paris and PSL University, 75005 Paris, France
| | - Alexander Erlich
- Institut de Recherche sur les Phénomènes Hors Equilibre (IRPHE), Aix-Marseille Université, 49 rue Frédéric Joliot-Curie, 13384 Marseille, France
- Institut de Biologie du Développement de Marseille (IBDM), Aix-Marseille Université, 163 av de Luminy, 13009 Marseille, France
| | - Jian H. Guan
- Department of Mathematics, University of North Carolina at Chapel Hill, NC 27599, USA
| | - Chris Thorogood
- Department of Plant Sciences, University of Oxford, Oxford OX1 3RB, UK
- University of Oxford Botanic Garden and Arboretum, Oxford OX1 4AZ, UK
- Corresponding author.
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Timounay Y, Hartwell AR, He M, King DE, Murphy LK, Démery V, Paulsen JD. Sculpting Liquids with Ultrathin Shells. PHYSICAL REVIEW LETTERS 2021; 127:108002. [PMID: 34533328 DOI: 10.1103/physrevlett.127.108002] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 05/19/2021] [Accepted: 08/12/2021] [Indexed: 06/13/2023]
Abstract
Thin elastic films can spontaneously attach to liquid interfaces, offering a platform for tailoring their physical, chemical, and optical properties. Current understanding of the elastocapillarity of thin films is based primarily on studies of planar sheets. We show that curved shells can be used to manipulate interfaces in qualitatively different ways. We elucidate a regime where an ultrathin shell with vanishing bending rigidity imposes its own rest shape on a liquid surface, using experiment and theory. Conceptually, the pressure across the interface "inflates" the shell into its original shape. The setup is amenable to optical applications as the shell is transparent, free of wrinkles, and may be manufactured over a range of curvatures.
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Affiliation(s)
- Yousra Timounay
- Department of Physics, Syracuse University, Syracuse, New York 13244, USA
- BioInspired Syracuse: Institute for Material and Living Systems, Syracuse University, Syracuse, New York 13244, USA
| | | | - Mengfei He
- Department of Physics, Syracuse University, Syracuse, New York 13244, USA
- BioInspired Syracuse: Institute for Material and Living Systems, Syracuse University, Syracuse, New York 13244, USA
| | - D Eric King
- Department of Physics, Syracuse University, Syracuse, New York 13244, USA
| | - Lindsay K Murphy
- Department of Physics, Syracuse University, Syracuse, New York 13244, USA
| | - Vincent Démery
- Gulliver UMR CNRS 7083, ESPCI Paris, Université PSL, (10 rue Vauquelin), 75005 Paris, France
- Université Lyon, ENS de Lyon, Université Claude Bernard Lyon 1, CNRS, Laboratoire de Physique, F-69342 Lyon, France
| | - Joseph D Paulsen
- Department of Physics, Syracuse University, Syracuse, New York 13244, USA
- BioInspired Syracuse: Institute for Material and Living Systems, Syracuse University, Syracuse, New York 13244, USA
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Xin M, Davidovitch B. Stretching Hookean ribbons part II: from buckling instability to far-from-threshold wrinkle pattern. THE EUROPEAN PHYSICAL JOURNAL. E, SOFT MATTER 2021; 44:94. [PMID: 34241720 DOI: 10.1140/epje/s10189-021-00088-9] [Citation(s) in RCA: 0] [Impact Index Per Article: 0] [Reference Citation Analysis] [Abstract] [Track Full Text] [Subscribe] [Scholar Register] [Received: 12/24/2020] [Accepted: 06/02/2021] [Indexed: 06/13/2023]
Abstract
We address the fully developed wrinkle pattern formed upon stretching a Hookean, rectangular-shaped sheet, when the longitudinal tensile load induces transverse compression that far exceeds the stability threshold of a purely planar deformation. At this "far-from-threshold" parameter regime, which has been the subject of the celebrated Cerda-Mahadevan model (Cerda and Mahadevan in Phys Rev Lett 90:074302, 2003), the wrinkle pattern expands throughout the length of the sheet and the characteristic wavelength of undulations is much smaller than its width. Employing Surface Evolver simulations over a range of sheet thicknesses and tensile loads, we elucidate the theoretical underpinnings of the far-from-threshold framework in this setup. We show that the evolution of wrinkles comes in tandem with collapse of transverse compressive stress, rather than vanishing transverse strain (which was hypothesized by Cerda and Mahadevan in Phys Rev Lett 90:074302, 2003), such that the stress field approaches asymptotically a compression-free limit, describable by tension field theory. We compute the compression-free stress field by simulating a Hookean sheet that has finite stretching modulus but no bending rigidity, and show that this singular limit encapsulates the geometrical nonlinearity underlying the amplitude-wavelength ratio of wrinkle patterns in physical, highly bendable sheets, even though the actual strains may be so small that the local mechanics is perfectly Hookean. Finally, we revisit the balance of bending and stretching energies that gives rise to a favorable wrinkle wavelength, and study the consequent dependence of the wavelength on the tensile load as well as the thickness and length of the sheet.
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Affiliation(s)
- Meng Xin
- Physics Department, University of Massachusetts, Amherst, MA, 01003, USA
| | - Benny Davidovitch
- Physics Department, University of Massachusetts, Amherst, MA, 01003, USA.
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Davidovitch B, Guinea F. Indentation of solid membranes on rigid substrates with van der Waals attraction. Phys Rev E 2021; 103:043002. [PMID: 34005936 DOI: 10.1103/physreve.103.043002] [Citation(s) in RCA: 4] [Impact Index Per Article: 1.3] [Reference Citation Analysis] [Abstract] [Track Full Text] [Journal Information] [Subscribe] [Scholar Register] [Received: 08/06/2020] [Accepted: 03/11/2021] [Indexed: 11/07/2022]
Abstract
We revisit the indentation of a thin solid sheet of size R_{sheet} suspended on a circular hole of radius R≪R_{sheet} in a smooth rigid substrate, addressing the effects of boundary conditions at the hole's edge. Introducing a basic theoretical model for the van der Waals (vdW) sheet-substrate attraction, we demonstrate the dramatic effect of replacing the clamping condition (Schwerin model) with a sliding condition, whereby the supported part of the sheet is allowed to slide towards the indenter and relax the induced hoop compression through angstrom-scale deflections from the thermodynamic equilibrium (determined by the vdW potential). We highlight the possibility that the indentation force F may not exhibit the commonly anticipated cubic dependence on the indentation depth (F∝δ^{3}), in which the proportionality constant is governed by the sheet's stretching modulus and the hole's radius R, but rather a pseduolinear response F∝δ, whereby the proportionality constant is governed by the bending modulus, the vdW attraction, and the sheet's size R_{sheet}≫R.
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Affiliation(s)
- Benny Davidovitch
- Department of Physics, University of Massachusetts Amherst, Amherst, Massachusetts 01003, USA
| | - Francisco Guinea
- IMDEA Nanoscience, C/Faraday 9, 28049 Madrid, Spain.,Donostia International Physics Center, Paseo Manuel de Lardizábal 4, 20018 San Sebastián, Spain
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