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Bending-Neutral Deformations of Minimal Surfaces

Published 25 May 2024 in math.DG, math-ph, and math.MP | (2405.16169v2)

Abstract: Minimal surfaces are ubiquitous in nature. Here they are considered as geometric objects that bear a deformation content. By refining the resolution of the surface deformation gradient afforded by the polar decomposition theorem, we identify a bending content and a class of deformations that leave it unchanged. These are the bending-neutral deformations, fully characterized by an integrability condition; they preserve normals. We prove that (1) every minimal surface is transformed into a minimal surface by a bending-neutral deformation, (2) given two minimal surfaces with the same system of normals, there is a bending-neutral deformation that maps one into the other, and (3) all minimal surfaces have indeed a universal bending content.

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References (37)
  1. S. Hildebrandt and A. Tromba, The Parsimonious Universe. Shape and Form in the Natural World (Springer-Verlag, New York, 1996).
  2. S. S. Antman, Nonlinear Problems of Elasticity, Applied Mathematical Sciences, Vol. 107 (Springer, New York, 1995).
  3. E. Vitral and J. A. Hanna, Dilation-invariant bending of elastic plates, and broken symmetry in shells, J. Elast. 153, 571 (2023a).
  4. E. Vitral and J. A. Hanna, Energies for elastic plates and shells from quadratic-stretch elasticity, J. Elast. 153, 581 (2023b).
  5. E. G. Virga, Pure measures of bending for soft plates, Soft Matter 20, 144 (2024).
  6. A. Acharya, Mid-surface scaling invariance of some bending strain measures, J. Elast. , 5517 (2024).
  7. E. Vitral and J. A. Hanna, Assorted remarks on bending measures and energies for plates and shells, and their invariance properties (2024), arXiv:2405.06638 [cond-mat.soft] .
  8. C. E. Weatherburn, Differential Geometry of Three Dimensions, Vol. I (Cambridge University Press, Cambridge, 2016).
  9. M. Šilhavý, A new approach to curvature measures in linear shell theories, Math. Mech. Solids 26, 1241 (2021).
  10. S. Kralj, R. Rosso, and E. G. Virga, Curvature control of valence on nematic shells, Soft Matter 7, 670 (2011).
  11. R. Rosso, E. G. Virga, and S. Kralj, Parallel transport and defects on nematic shells, Continuum Mech. Thermodyn. 24, 643 (2012).
  12. C. E. Weatherburn, Differential Geometry of Three Dimensions, Vol. II (Cambridge University Press, Cambridge, 2016).
  13. E. Beltrami, Opere Matematiche, Vol. 1 (Hoepli, Milan, 1902) available from https://gallica.bnf.fr/ark:/12148/bpt6k99432q/f1.item.
  14. M. E. Gurtin, E. Fried, and L. Anand, The Mechanics and Thermodynamics of Continua (Cambridge University Press, Cambridge, 2010).
  15. C.-S. Man and H. Cohen, A coordinate-free approach to the kinematics of membranes, J. Elast. 16, 97 (1986).
  16. M. E. Gurtin and A. I. Murdoch, A continuum theory of elastic material surfaces, Arch. Rational Mech. Anal. 57, 291 (1975a), see also [42].
  17. W. Pietraszkiewicz, M. Szwabowicz, and C. Vallée, Determination of the midsurface of a deformed shell from prescribed surface strains and bendings via the polar decomposition, Int. J. Non-Linear Mech. 43, 579 (2008).
  18. B. Palais, R. Palais, and S. Rodi, A disorienting look at Euler’s theorem on the axis of a rotation, Am. Math. Mon. 116, 892 (2009).
  19. T. J. Hughes and F. Brezzi, On drilling degrees of freedom, Comput. Methods Appl. Mech. Engrg. 72, 105 (1989).
  20. D. Fox and J. Simo, A drill rotation formulation for geometrically exact shells, Comput. Methods Appl. Mech. Engrg. 98, 329 (1992).
  21. M. Mohammadi Saem, P. Lewintan, and P. Neff, On in-plane drill rotations for cosserat surfaces, Proc. R. Soc. Lond. A 477, 20210158 (2021).
  22. O. Rodrigues, Des lois géométriques qui régissent les déplacements d’un système solide dans l’espace, et de la variation des coordonnées provenant de ces déplacements considérés indépendamment des causes qui peuvent les produire, J. Math. Pures Appl. 5, 380 (1840), available from http://www.numdam.org/item/JMPA_1840_1_5__380_0/.
  23. C. D. Mladenova and I. M. Mladenov, Vector decomposition of finite rotations, Rep. Math. Phys. 68, 107 (2011).
  24. L. C. Martins and P. Podio-Guidugli, A variational approach to the polar decomposition theorem, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. 66, 487 (1979), available from http://www.bdim.eu/item?id=RLINA_1979_8_66_6_487_0.
  25. L. C. Martins and P. Podio-Guidugli, An elementary proof of the polar decomposition theorem, Amer. Math. Month. 87, 288 (1980).
  26. G. Grioli, Una proprietà di minimo nella cinematica delle deformazioni finite, Boll. Un. Math. Ital. 2, 252 (1940), available from https://www.uni-due.de//imperia/md/content/mathematik/ag_neff/grioli_deformatione.pdf.
  27. P. Neff, J. Lankeit, and A. Madeo, On Grioli’s minimum property and its relation to Cauchy’s polar decomposition, Int. J. Eng. Sci. 80, 209 (2014).
  28. O. Ozenda, A. M. Sonnet, and E. G. Virga, A blend of stretching and bending in nematic polymer networks, Soft Matter 16, 8877 (2020).
  29. O. Ozenda and E. G. Virga, On the Kirchhoff-Love hypothesis (revised and vindicated), J. Elast. 143, 359 (2021).
  30. S.-S. Chern, An elementary proof of the existence of isothermal parameters on a surface, Proc. Amer. Math. Soc. 6, 771 (1955).
  31. R. Osserman, A Survey of Minimal Surfaces (Dover, Mineola, NY, 1986).
  32. M. Spivak, A Comprehensive Introduction to Differential Geometry, 3rd ed., Vol. 4 (Publish or Perish, Houston,Texas, 1999).
  33. K. Weierstrass, Untersuchungen über die Flächen, deren mittlere Krümmung überall gleich Null ist, in Mathematische Werke: Herausgegeben unter Mitwirkung einer von der königlich preussischen Akademie der Wissenschaften eingesetzten Commission, edited by J. Knoblauch (Mayer & Müller, Berlin, 1903) pp. 39–52, available from https://archive.org/details/mathematischewer03weieuoft/page/38/mode/2up.
  34. K. Weierstrass, Untersuchungen über die Flächen, deren mittlere Krümmung überall gleich Null ist, in Mathematische Werke: Herausgegeben unter Mitwirkung einer von der königlich preussischen Akademie der Wissenschaften eingesetzten Commission, Cambridge Library Collection – Mathematics, edited by J. Knoblauch (Cambridge University Press, Cambridge, 2013) pp. 39–52.
  35. E. Bour, Théorie de la déformation des surfaces, J. Éc. Impériale Polytech. 39, 1 (1862).
  36. A. Cogliati and R. Rivis, The origins of the fundamental theorem of surface theory, Hist. Math. 61, 45 (2022).
  37. M. E. Gurtin and A. I. Murdoch, Addenda to our paper A continuum theory of elastic material surfaces, Arch. Rational Mech. Anal. 59, 389 (1975b).

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