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DRAPING WOVEN SHEETS

Part of: Hyperbolic equations and systems Equilibrium (steady-state) problems

Published online by Cambridge University Press:  07 January 2021

P. D. HOWELL Open the ORCID record for P. D. HOWELL [Opens in a new window] ,
H. OCKENDON Open the ORCID record for H. OCKENDON [Opens in a new window]  and
J. R. OCKENDON Open the ORCID record for J. R. OCKENDON [Opens in a new window]
P. D. HOWELL*
Affiliation:
OCIAM, Mathematical Institute, Andrew Wiles Building, Oxford OX2 6GG, UK; e-mail: ockendon@maths.ox.ac.uk, ock@maths.ox.ac.uk.
H. OCKENDON
Affiliation:
OCIAM, Mathematical Institute, Andrew Wiles Building, Oxford OX2 6GG, UK; e-mail: ockendon@maths.ox.ac.uk, ock@maths.ox.ac.uk.
J. R. OCKENDON
Affiliation:
OCIAM, Mathematical Institute, Andrew Wiles Building, Oxford OX2 6GG, UK; e-mail: ockendon@maths.ox.ac.uk, ock@maths.ox.ac.uk.
*
e-mail: howell@maths.ox.ac.uk
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Abstract

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Motivated by the manufacture of carbon fibre components, this paper considers the smooth draping of loosely woven fabric over rigid obstacles, both smooth and nonsmooth. The draped fabric is modelled as the continuum limit of a Chebyshev net of two families of short rigid rods that are freely pivoted at their joints. This approach results in a system of nonlinear hyperbolic partial differential equations whose characteristics are the fibres in the fabric. The analysis of this system gives useful information about the drapability of obstacles of many shapes and also poses interesting theoretical questions concerning well-posedness, smoothness and computability of the solutions.

Keywords

drapingwoven fabricshyperbolic systemssingularity formation

MSC classification

Primary: 35L10: Second-order hyperbolic equations
Secondary: 74G99: None of the above, but in this section

Information

Type
Research Article
Information
The ANZIAM Journal , Volume 62 , Issue 4 , October 2020 , pp. 355 - 385
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Copyright
© Australian Mathematical Society 2020

References

Baek, C., Sageman-Furnas, A. D., Javed, M. K. and Reis, P. M., “Form finding in elastic gridshells”, Proc. Nat. Acad. Sci. USA 115 (2018) 75–80; doi:10.1073/pnas.1713841115.CrossRefGoogle ScholarPubMed
Barton, D., Ockendon, H., Piette, B. and Whittaker, R., “New techniques for composite wing manufacture”, Study Group Report, ESGI 138, Bath, UK, 2018, available at http://miis.maths.ox.ac.uk/miis/754/.Google Scholar
Bianchi, L., Lezioni di geometria differenziale, 3rd edn, Volume 1, (E. Spoerri, Pisa, 1922) 153–162, available at https://books.google.co.uk/books?id=chk0jxO2S2MC.Google Scholar
Cerda, E., Mahadevan, L. and Pasini, J. M., “The elements of draping”, Proc. Nat. Acad. Sci. USA 101 (2004) 1806–1810; doi:10.1073/pnas.0307160101.CrossRefGoogle ScholarPubMed
Demaine, E. D. and O’Rourke, J., Geometric folding algorithms: linkages, origami, polyhedra (Cambridge University Press, Cambridge, 2007), available at https://books.google.co.uk/books?id=ycYLAQAAQBAJ.CrossRefGoogle Scholar
Ghys, É., “Sur la coupe des vêtements: Variation autour d’un thème de Tchebychev”, Enseign. Math. 57 (2011) 165–208; doi:10.4171/LEM/57-1-8.CrossRefGoogle Scholar
Hazzidakis, J. N., “Ueber einige Eigenschaften der Flächen mit constantem Krümmungsmaass”, J. Reine Angew. Math. 88 (1879) 68–73; doi:10.1515/crll.1880.88.68.Google Scholar
Jiang, C., Gast, T. and Teran, J., “Anisotropic elastoplasticity for cloth, knit and hair frictional contact”, ACM Trans. Graph. 36 (2017) 152:1–152:14; doi:10.1145/3072959.3073623.CrossRefGoogle Scholar
Koenderink, J. and van Doorn, A., “Shape from Chebyshev nets”, in: European Conference on Computer Vision, Volume 1407 of Lect. Notes Comput. Sci. (Springer, Berlin, 1998) 215–225; doi:10.1007/BFb0054743.Google Scholar
Masson, Y. and Monasse, L., “Existence of global Chebyshev nets on surfaces of absolute Gaussian curvature less than $2\pi$”, J. Geom. 108 (2017) 25–32; doi:10.1007/s00022-016-0319-1.CrossRefGoogle Scholar
Mitani, J. and Igarishi, T., “Interactive design of planar curved folding by reflection”, in: Pacific Graphics short papers (The Eurographics Association, 2011); doi:10.2312/PE/PG/PG2011short/077-081.CrossRefGoogle Scholar
Ockendon, J., Howison, S., Lacey, A. and Movchan, A., Applied partial differential equations (Oxford University Press, Oxford, 2003), available at https://books.google.co.uk/books?id=CdA6jcJWCToC.Google Scholar
Papadopoulos, A., “Euler and Chebyshev: from the sphere to the plane and backwards”, Preprint, 2016, arXiv:1608.02724.Google Scholar
Pipkin, A. C., “Equilibrium of Tchebychev nets”, in: The breadth and depth of continuum mechanics (Springer, Berlin, 1986); doi:10.1007/978-3-642-61634-1_12.Google Scholar
Poincloux, S., Adda-Bedin, M. and Lechenault, F., “Geometry and elasticity of a knitted fabric”, Phys. Rev. X 8 (2018) 021075; doi:10.1103/PhysRevX.8.021075.Google Scholar
Postle, J. R. and Postle, R., “The dynamics of fabric drape”, Text. Res. J. 69 (1999) 623–629; doi:10.1177/004051759906900901.CrossRefGoogle Scholar
Quinn, G. and Gengnagel, C., “A review of elastic grid shells, their erection methods and the potential use of pneumatic formwork”, in: Mobile and rapidly assembled structures IV (eds De Temmerman, N. and Brebbia, C. A.), (WIT Press, Southampton, 2014) 129–145; doi:10.2495/MAR140111.CrossRefGoogle Scholar
Reinhardt, W. P. and Walker, P. L., “Jacobian elliptic functions”, in: Digital Library of Mathematical Functions, Version 1.0.28 (NIST, Gaithersburg, MD, US Department of Commerce, Release 15-9-2020), Chapter 22, available at http://dlmf.nist.gov/22.Google Scholar
Robertson, R. E., Chu, T.-J., Gerard, R. J., Kim, J.-H., Park, M., Kim, H.-G. and Peterson, R. C., “Three-dimensional fiber reinforcement shapes obtainable from flat, bidirectional fabrics without wrinkling or cutting. Part 1. A single four-sided pyramid”, Compos. Part A Appl. Sci. Manuf. 31 (2000) 703–715; doi:10.1016/S1359-835X(00)00013-0.CrossRefGoogle Scholar
Robertson, R. E., Chu, T.-J., Gerard, R. J., Kim, J.-H., Park, M., Kim, H.-G. and Peterson, R. C., “Three-dimensional fiber reinforcement shapes obtainable from flat, bidirectional fabrics without wrinkling or cutting. Part 2: a single n-sided pyramid, cone, or round box”, Compos. Part A Appl. Sci. Manuf. 31 (2000) 1149–1165; doi:10.1016/S1359-835X(00)00120-2.CrossRefGoogle Scholar
Samelson, S. L. and Dayawansa, W. P., “On the existence of global Tchebychev nets”, Trans. Amer. Math. Soc. 347 (1995) 651–660; doi:10.1090/S0002-9947-1995-1233983-3.CrossRefGoogle Scholar
Servant, M., “Sur l’habillage des surfaces”, C. R. Acad. Sci. 135 (1902) 575–577, available at https://gallica.bnf.fr/ark:/12148/bpt6k64435428/f581.item#.Google Scholar
Servant, M., “Sur l’habillage des surfaces”, C. R. Acad. Sci. 137 (1903) 112–115, available at https://gallica.bnf.fr/ark:/12148/bpt6k6262248g/f12.item#.Google Scholar
Steigmann, D. J., “Continuum theory for elastic sheets formed by inextensible crossed elasticae”, Int. J. Non Linear Mech. 106 (2018) 324–329; doi:10.1016/j.ijnonlinmec.2018.03.012.CrossRefGoogle Scholar
Wang, W. B. and Pipkin, A. C., “Inextensible networks with bending stiffness”, Q. J. Mech. Appl. Math. 39 (1985) 343–359; doi:10.1093/qjmam/39.3.343.CrossRefGoogle Scholar
Weatherburn, C. E., Differential geometry of three dimensions (Cambridge University Press, Cambridge, 1930), available at https://books.google.co.uk/books?id=fGWcCwAAQBAJ.Google Scholar
Willmore, T. J., An introduction to differential geometry (Oxford University Press, Oxford, 1959), available at https://books.google.co.uk/books?id=dbIAAQAAQBAJ.Google Scholar