Abstract
We consider a single disclination in a thin elastic sheet of thickness h. We prove ansatz-free lower bounds for the free elastic energy in three different settings: first, for a geometrically fully non-linear plate model; second, for three-dimensional nonlinear elasticity; and third, for the Föppl-von Kármán plate theory. The lower bounds in the first and third result are optimal in the sense that we find upper bounds that are identical to the respective lower bounds in the leading order of h.
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Audoly, B., Pomeau, Y.: Elasticity and Geometry: From Hair Curls to the Non-linear Response of Shells. Oxford University Press, Oxford, 2010
Ben Amar M., Pomeau Y.: Crumpled paper. Proc. R. Soc. Lond. Ser. A 453(1959), 729–755 (1997)
Brandman J., Kohn R.V., Nguyen H.-M.: Energy scaling laws for conically constrained thin elastic sheets. J. Elast. 113(2), 251–264 (2013)
Caspar, D.L.D., Klug, A.: Physical principles in the construction of regular viruses. Cold Spring Harb. Symp. Quant. Biol. 27, 1–24, 1962
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)
Cerda E., Mahadevan L.: Confined developable elastic surfaces: cylinders, cones and the elastica. Proc. R. Soc. Lond. Ser. A 461(2055), 671–700 (2005)
Conti S., Maggi F.: Confining thin elastic sheets and folding paper. Arch. Ration. Mech. Anal. 187(1), 1–48 (2008)
Dervaux J., Ciarletta P., Ben Amar M.: Morphogenesis of thin hyperelastic plates: a constitutive theory of biological growth in the Föppl-von Kármán limit. J. Mech. Phys. Solids 57(3), 458–471 (2009)
DiDonna B.A., Witten T.A.: Anomalous strength of membranes with elastic ridges. Phys. Rev. Lett. 87, 206105 (2001)
Efrati E., Sharon E., Kupferman R.: Buckling transition and boundary layer in non-Euclidean plates. Phys. Rev. E 80(1), 016602 (2009)
Efrati E., Sharon E., Kupferman R.: Elastic theory of unconstrained non-Euclidean plates. J. Mech. Phys. Solids 57(4), 762–775 (2009)
Fonseca, I., Gangbo, W.: Degree theory in analysis and applications, vol. 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. Commun. Pure Appl. Math. 55(11), 1461–1506 (2002)
Friesecke G., James R.D., Müller S.: A hierarchy of plate models derived from nonlinear elasticity by Gamma-convergence. Arch. Ration. Mech. Anal. 180(2), 183–236 (2006)
Gemmer J.A., Venkataramani S.C.: Shape selection in non-Euclidean plates. Phys. D Nonlinear Phenom. 240(19), 1536–1552 (2011)
Gemmer J.A.: Venkataramani S.C., Defects and boundary layers in non-Euclidean plates. Nonlinearity 25(12), 3553 (2012)
Gilbarg, D., Trudinger, N.S.: Elliptic partial differential equations of second order. Classics in Mathematics. Springer, Heidelberg, 2001
Hornung P.: Approximation of flat \({W^{2, 2}}\) isometric immersions by smooth ones. Arch. Ration. Mech. Anal. 199(3), 1015–1067 (2011)
Kim J., Hanna J.A., Byun M., Santangelo C.D., Hayward R.C.: Designing responsive buckled surfaces by halftone gel lithography. Science 335(6073), 1201–1205 (2012)
Klein Y., Efrati E., Sharon E.: Shaping of elastic sheets by prescription of non-Euclidean metrics. Science 315(5815), 1116–1120 (2007)
Kramer E.M., Witten T.A.: Stress condensation in crushed elastic manifolds. Phys. Rev. Lett. 78, 1303–1306 (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
Kupferman R., Moshe M., Solomon J.P.: Metric description of singular defects in isotropic materials. Arch. Ration. Mech. Anal. 216(3), 1009–1047 (2015)
Lecumberry M., Müller S.: Stability of slender bodies under compression and validity of the von Kármán theory. Arch. Ration. Mech. Anal. 193(2), 255–310 (2009)
Lewicka M., Pakzad M.R.: Scaling laws for non-Euclidean plates and the \({W^{2,2}}\) isometric immersions of Riemannian metrics. ESAIM Control Optim. Calc. Var. 17(4), 1158–1173 (2011)
Lidmar J., Mirny L., Nelson D.R.: Virus shapes and buckling transitions in spherical shells. Phys. Rev. E 68(5), 051910 (2003)
Lobkovsky A.: Boundary layer analysis of the ridge singularity in a thin plate. Phys. Rev. E 53, 3750–3759 (1996)
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., Witten T.A.: Properties of ridges in elastic membranes. Phys. Rev. E 55, 1577–1589 (1997)
Maz’ya, V.: Lectures on isoperimetric and isocapacitary inequalities in the theory of Sobolev spaces. In: Heat Kernels and Analysis on Manifolds, Graphs, and Metric Spaces (Paris, 2002), vol. 338 of Contemporary Mathematics, pp. 307–340. Amer. Math. Soc., Providence, 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)
Olbermann H.: Energy scaling law for the regular cone. J. Nonlinear Sci. 26, 287–314 (2016)
Pakzad M.R.: On the Sobolev space of isometric immersions. J. Differ. Geom. 66(1), 47–69 (2004)
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
Romanov, A.E.: Mechanics and physics of disclinations in solids. Eur. J. Mech. A Solids 22(5), 727–741, 2003. 5th Euromech Solid Mechanics Conference (Thessaloniki, 2003)
Seung H.S., Nelson D.R.: Defects in flexible membranes with crystalline order. Phys. Rev. A 38, 1005–1018 (1988)
Simon, T.: \({\Gamma}\)-Equivalence for Nonlinear Plate Theories. Master’s thesis, Universität Bonn, 2014
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. Europhys. Lett. 23(1), 51 (1993)
Yavari A., Goriely A.: Riemann-Cartan geometry of nonlinear disclination mechanics. Math. Mech. Solids 18(1), 91–102 (2013)
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Olbermann, H. Energy Scaling Law for a Single Disclination in a Thin Elastic Sheet. Arch Rational Mech Anal 224, 985–1019 (2017). https://doi.org/10.1007/s00205-017-1093-4
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DOI: https://doi.org/10.1007/s00205-017-1093-4