Abstract
We consider a thin elastic sheet in the shape of a disk whose reference metric is that of a singular cone. That is, the reference metric is flat away from the center and has a defect there. We define a geometrically fully nonlinear free elastic energy and investigate the scaling behavior of this energy as the thickness h tends to 0. We work with two simplifying assumptions: Firstly, we think of the deformed sheet as an immersed 2-dimensional Riemannian manifold in Euclidean 3-space and assume that the exponential map at the origin (the center of the sheet) supplies a coordinate chart for the whole manifold. Secondly, the energy functional penalizes the difference between the induced metric and the reference metric in \(L^\infty \) (instead of, as is usual, in \(L^2\)). Under these assumptions, we show that the elastic energy per unit thickness of the regular cone in the leading order of h is given by \(C^*h^2|\log h|\), where the value of \(C^*\) is given explicitly.
Similar content being viewed by others
References
Ambrosio, L.: Some fine properties of sets of finite perimeter in Ahlfors regular metric measure spaces. Adv. Math. 159(1), 51–67 (2001)
Ambrosio, L., Fusco, N., Pallara, D.: Functions of Bounded Variation and Free Discontinuity Problems. Oxford Mathematical Monographs. The Clarendon Press, Oxford University Press, New York (2000)
Ben Amar, M., Pomeau, Y.: Crumpled paper. Proc. R. Soc. Lond. Ser. A 453(1959), 729–755 (1997)
Bernstein, F.: Über die isoperimetrische Eigenschaft des Kreises auf der Kugeloberfläche und in der Ebene. Math. Ann. 60(1), 117–136 (1905)
Borisov, J.F.: On the connection between the spatial form of smooth surfaces and their intrinsic geometry. Vestnik Leningrad. Univ. 14(13), 20–26 (1959)
Borisov, J.F.: On the question of parallel displacement on a smooth surface and the connection of space forms of smooth surfaces with their intrinsic geometries. Vestnik Leningrad. Univ. 15(19), 127–129 (1960)
Borisov, J.F.: \(C^{1,\,\alpha }\)-isometric immersions of Riemannian spaces. Dokl. Akad. Nauk SSSR 163, 11–13 (1965)
Borisov, J.F.: Irregular surfaces of the class \(C^{1,\beta }\) with an analytic metric. Sibirsk. Mat. Zh. 45(1), 25–61 (2004)
Brandman, J., Kohn, R.V., Nguyen, H.-M.: Energy scaling laws for conically constrained thin elastic sheets. J. Elast. 113(2), 251–264 (2013)
Cerda, E., Chaieb, S., Melo, F., Mahadevan, L.: Conical dislocations in crumpling. Nature 401, 46–49 (1999)
Cerda, E., Mahadevan, L.: Conical surfaces and crescent singularities in crumpled sheets. Phys. Rev. Lett. 80, 2358–2361 (1998)
Conti, S., Maggi, F.: Confining thin elastic sheets and folding paper. Arch. Ration. Mech. Anal. 187(1), 1–48 (2008)
Didonna, B.A., Witten, T.A.: Anomalous strength of membranes with elastic ridges. Phys. Rev. Lett. 87(20), 206105 (2001)
do Carmo, M.P.: Differential geometry of curves and surfaces. Prentice-Hall Inc., Englewood Cliffs (1976). Translated from the Portuguese
Fonseca, Irene, Gangbo, Wilfrid: Degree theory in analysis and applications, volume 2 of Oxford Lecture Series in Mathematics and its Applications. The Clarendon Press, Oxford University Press, New York (1995). Oxford Science Publications
Friesecke, G., James, R.D., Müller, S.: A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity. Comm. Pure Appl. Math. 55(11), 1461–1506 (2002)
Gilbarg, D., Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Classics in Mathematics. Springer, Berlin (2001)
Gromov, M.: Partial Differential Relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9. Springer, Berlin (1986)
Jin, W., Kohn, R.V.: Singular perturbation and the energy of folds. J. Nonlinear Sci. 10(3), 355–390 (2000)
Kohn, Robert V., Müller, Stefan: Relaxation and regularization of nonconvex variational problems. In: Proceedings of the Second International Conference on Partial Differential Equations (Italian) (Milan, 1992), volume 62, pp. 89–113 (1994), 1992
Kohn, R.V., Müller, S.: Surface energy and microstructure in coherent phase transitions. Comm. Pure Appl. Math. 47(4), 405–435 (1994)
Kramer, E.M., Witten, T.A.: Stress condensation in crushed elastic manifolds. Phys. Rev. Lett. 78, 1303–1306 (1997)
Kramer, E.M.: The von Karman equations, the stress function, and elastic ridges in high dimensions. J. Math. Phys. 38(2), 830–846 (1997)
Kuiper, N.H.: On \(C^1\)-isometric imbeddings. I, II. Nederl. Akad. Wetensch. Proc. Ser. A. 58 =. Indag. Math. 17(545–556), 683–689 (1955)
Liang, T., Witten, T.A.: Spontaneous curvature cancellation in forced thin sheets. Phys. Rev. E 73(4), 046604 (2006)
Lidmar, J., Mirny, L., Nelson, D.R.: Virus shapes and buckling transitions in spherical shells. Phys. Rev. E 68(5), 051910 (2003)
Lobkovsky, A., Gentges, S., Li, H., Morse, D., Witten, T.A.: Scaling properties of stretching ridges in a crumpled elastic sheet. Science 270(5241), 1482–1485 (1995)
Lobkovsky, A.E., Witten, T.A.: Properties of ridges in elastic membranes. Phys. Rev. E 55, 1577–1589 (1997)
Lobkovsky, A.E.: Boundary layer analysis of the ridge singularity in a thin plate. Phys. Rev. E (3) 53(4, part B), 3750–3759 (1996)
Miranda, M.: Functions of bounded variation on “good” metric spaces. J. de mathématiques pures et appliquées 82(8), 975–1004 (2003)
Müller, S., Olbermann, H.: Almost conical deformations of thin sheets with rotational symmetry. SIAM J. Math. Anal. 46(1), 25–44 (2014)
Müller, S., Olbermann, H.: Conical singularities in thin elastic sheets. Calc. Var. Partial Differ. Equ. 49(3–4), 1177–1186 (2014)
Nash, J.: \(C^1\) isometric imbeddings. Ann. Math. 2(60), 383–396 (1954)
Pogorelov, A.V.: Extrinsic geometry of convex surfaces. American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations, Translations of Mathematical Monographs, Vol. 35
Seung, H.S., Nelson, D.R.: Defects in flexible membranes with crystalline order. Phys. Rev. A 38, 1005–1018 (1988)
Spivak, M.: A Comprehensive Introduction to Differential Geometry, vol. II, 2nd edn. Publish or Perish Inc, Wilmington (1979)
Venkataramani, S.C.: Lower bounds for the energy in a crumpled elastic sheet–a minimal ridge. Nonlinearity 17(1), 301–312 (2004)
Witten, T.A.: Stress focusing in elastic sheets. Rev. Mod. Phys. 79, 643–675 (2007)
Witten, T.A., Li, H.: Asymptotic shape of a fullerene ball. EPL (Europhysics Letters) 23(1), 51 (1993)
Acknowledgments
The author would like to thank Stefan Müller for helpful discussions.
Author information
Authors and Affiliations
Corresponding author
Ethics declarations
Conflict of Interest
The author declares that he has no conflict of interest.
Human participants / animals
The research carried out for this article did not involve human participants or animals.
Additional information
Communicated by Robert V. Kohn.
Rights and permissions
About this article
Cite this article
Olbermann, H. Energy Scaling Law for the Regular Cone. J Nonlinear Sci 26, 287–314 (2016). https://doi.org/10.1007/s00332-015-9275-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00332-015-9275-4