Mechanical Properties of Long Leaves: Experiment and Theory
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Mechanical Properties of Long Leaves: Experiment and Theory A. Jakubska‑Busse1 · M. W. Janowicz2 · L. Ochnio3 · B. Jackowska‑Zduniak4 · J. M. A. Ashbourn5 Received: 15 July 2020 / Accepted: 15 October 2020 © The Author(s) 2020
Abstract The static properties of leaves with parallel venation from terrestrial orchids of the genus Epipactis were modelled as coupled elastic rods using the geometrically exact Cosserat theory and the resulting boundary-value problem was solved numerically using a method from Shampine, Muir and Xu. The response of the leaf structure to the applied force was obtained from preliminary measurements. These measurements allowed the Young’s modulus of the Epipactis leaves to be determined. The appearance of wrinkles and undulation characteristics for some leaves has been attributed to the small torsional stiffness of the leaf edges. Keywords Long leaves · Elasticity · Cosserat rods · Epipactis · Orchids
* A. Jakubska‑Busse anna.jakubska‑[email protected] * M. W. Janowicz [email protected] 1
Department of Botany, Institute of Environmental Biology, University of Wrocław, Kanonia 6/8, 50‑328 Wrocław, Poland
2
Faculty of Applications of Informatics and Mathematics, Department of Applied Mathematics, Warsaw University of Life Sciences-SGGW, ul. Ciszewskiego 8, 02‑786 Warsaw, Poland
3
Faculty of Applications of Informatics and Mathematics, Department of Econometrics and Statistics, Warsaw University of Life Sciences-SGGW, ul. Nowoursynowska 159, 02‑776 Warsaw, Poland
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Faculty of Applications of Informatics and Mathematics, Department of Computer Science, Warsaw University of Life Sciences - SGGW, ul. Nowoursynowska 159, 02‑776 Warsaw, Poland
5
Department of Engineering Science, University of Oxford, Parks Road, Oxford OX1 3PJ, UK
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A. Jakubska‑Busse et al.
1 Introduction The modelling of plant organs remains an open problem due to the complexity of plant architecture, regardless of the particular organ or any particular taxonomical group of plants being considered. This study focusses on leaves with parallel venation, namely, leaves with a relatively simple structure of veins which can nonetheless exhibit interesting features such as wrinkles and undulation. Leaf undulation is an interesting and relatively well-studied phenomenon observed in the monocotyledons including orchids. It has been found that the waves in the leaf blades in monocots usually appear perpendicular to the leaf length, which demonstrates that as the leaf surface grows it changes correspondingly lengthwise to the leaf blade. It is worth noting that although undulation normally occurs perpendicular to the leaf length, in the initial stages it occurs alongside its length. Displacements in the wrinkled leaf also occur across the leaf blade and whichever way these appear, they demonstrate that the pace of their growth is irregular since wrinkles in a leaf can be of different lengths. Hejnowicz (1992) found that spatial and temporal fluctuations in the pH of the epidermal cell
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