## Analysis & PDE

• Anal. PDE
• Volume 12, Number 1 (2019), 245-258.

### On a boundary value problem for conically deformed thin elastic sheets

Heiner Olbermann

#### Abstract

We consider a thin elastic sheet in the shape of a disk that is clamped at its boundary such that the displacement and the deformation gradient coincide with a conical deformation with no stretching there. These are the boundary conditions of a so-called “d-cone”. We define the free elastic energy as a variation of the von Kármán energy, which penalizes bending energy in $Lp$ with $p∈(2,83)$ (instead of, as usual, $p=2$). We prove ansatz-free upper and lower bounds for the elastic energy that scale like $hp∕(p−1)$, where $h$ is the thickness of the sheet.

#### Article information

Source
Anal. PDE, Volume 12, Number 1 (2019), 245-258.

Dates
Revised: 5 March 2018
Accepted: 10 April 2018
First available in Project Euclid: 16 August 2018

https://projecteuclid.org/euclid.apde/1534384918

Digital Object Identifier
doi:10.2140/apde.2019.12.245

Mathematical Reviews number (MathSciNet)
MR3842912

Zentralblatt MATH identifier
06930187

Keywords
thin elastic sheets d-cone

#### Citation

Olbermann, Heiner. On a boundary value problem for conically deformed thin elastic sheets. Anal. PDE 12 (2019), no. 1, 245--258. doi:10.2140/apde.2019.12.245. https://projecteuclid.org/euclid.apde/1534384918

#### References

• P. Bella and R. V. Kohn, “Wrinkles as the result of compressive stresses in an annular thin film”, Comm. Pure Appl. Math. 67:5 (2014), 693–747.
• H. Ben Belgacem, S. Conti, A. DeSimone, and S. Müller, “Energy scaling of compressed elastic films: three-dimensional elasticity and reduced theories”, Arch. Ration. Mech. Anal. 164:1 (2002), 1–37.
• D. P. Bourne, S. Conti, and S. Müller, “Energy bounds for a compressed elastic film on a substrate”, J. Nonlinear Sci. 27:2 (2017), 453–494.
• J. Brandman, R. V. Kohn, and H. Nguyen, “Energy scaling laws for conically constrained thin elastic sheets”, J. Elasticity 113:2 (2013), 251–264.
• E. Cerda, S. Chaieb, F. Melo, and L. Mahadevan, “Conical dislocations in crumpling”, Nature 401 (1999), 46–49.
• P. G. Ciarlet, “A justification of the von Kármán equations”, Arch. Rational Mech. Anal. 73:4 (1980), 349–389.
• P. G. Ciarlet, Mathematical elasticity, II: Theory of plates, Studies in Mathematics and its Applications 27, North-Holland, Amsterdam, 1997.
• S. Conti and F. Maggi, “Confining thin elastic sheets and folding paper”, Arch. Ration. Mech. Anal. 187:1 (2008), 1–48.
• I. Fonseca and W. Gangbo, Degree theory in analysis and applications, Oxford Lecture Series in Mathematics and its Applications 2, Clarendon, Oxford, 1995.
• G. Friesecke, R. D. James, and S. Müller, “A theorem on geometric rigidity and the derivation of nonlinear plate theory from three-dimensional elasticity”, Comm. Pure Appl. Math. 55:11 (2002), 1461–1506.
• G. Friesecke, R. D. James, and S. Müller, “A hierarchy of plate models derived from nonlinear elasticity by gamma-convergence”, Arch. Ration. Mech. Anal. 180:2 (2006), 183–236.
• R. V. Kohn and H.-M. Nguyen, “Analysis of a compressed thin film bonded to a compliant substrate: the energy scaling law”, J. Nonlinear Sci. 23:3 (2013), 343–362.
• N. H. Kuiper, “On $C^1$-isometric imbeddings, I”, Nederl. Akad. Wetensch. Proc. Ser. A. 17 (1955), 545–556.
• N. H. Kuiper, “On $C^1$-isometric imbeddings, II”, Nederl. Akad. Wetensch. Proc. Ser. A. 17 (1955), 683–689.
• H. Le Dret and A. Raoult, “The nonlinear membrane model as variational limit of nonlinear three-dimensional elasticity”, J. Math. Pures Appl. $(9)$ 74:6 (1995), 549–578.
• M. Lewicka and M. R. Pakzad, “Convex integration for the Monge–Ampère equation in two dimensions”, Anal. PDE 10:3 (2017), 695–727.
• A. Lobkovsky and T. A. Witten, “Properties of ridges in elastic membranes”, Phys. Rev. E 55:2 (1997), 1577–1589.
• A. Lobkovsky, S. Gentges, H. Li, D. Morse, and T. Witten, “Scaling properties of stretching ridges in a crumpled elastic sheet”, Science 270:5241 (1995), 1482–1485.
• S. Müller and H. Olbermann, “Conical singularities in thin elastic sheets”, Calc. Var. Partial Differential Equations 49:3-4 (2014), 1177–1186.
• J. Nash, “$C^1$ isometric imbeddings”, Ann. of Math. $(2)$ 60 (1954), 383–396.
• L. Nirenberg, “On elliptic partial differential equations”, Ann. Scuola Norm. Sup. Pisa $(3)$ 13 (1959), 115–162.
• H. Olbermann, “Energy scaling law for the regular cone”, J. Nonlinear Sci. 26:2 (2016), 287–314.
• H. Olbermann, “Energy scaling law for a single disclination in a thin elastic sheet”, Arch. Ration. Mech. Anal. 224:3 (2017), 985–1019.
• H. Olbermann, “The shape of low energy configurations of a thin elastic sheet with a single disclination”, Anal. PDE 11:5 (2018), 1285–1302.
• H. Triebel, Interpolation theory, function spaces, differential operators, North-Holland Mathematical Library 18, North-Holland, Amsterdam, 1978.
• H. Triebel, Theory of function spaces, Monographs in Mathematics 78, Birkhäuser, Basel, 1983.
• S. C. Venkataramani, “Lower bounds for the energy in a crumpled elastic sheet: a minimal ridge”, Nonlinearity 17:1 (2004), 301–312.
• T. A. Witten, “Stress focusing in elastic sheets”, Rev. Modern Phys. 79:2 (2007), 643–675.