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Wrinkling of an elastic sheet floating on a liquid sphere

Peter Bella, Carlos Román

Abstract

A thin circular elastic sheet floating on a drop-like liquid substrate is deformed due to incompatibility between the curved substrate and the planar sheet. We adopt a variational viewpoint by minimizing the non-convex membrane energy together with a higher-order convex bending energy. Focusing on thin sheets, we expand the minimum of the energy in terms of a small thickness ratio $h$, and identify the first two terms of this expansion. The leading-order term arises from the minimization of a family of one-dimensional relaxed problems, while for the next-order term we establish lower and upper bounds. This generalizes the previous work [P. Bella and R.V. Kohn. Wrikling of a thin circular sheet bonded to a spherical substrate, Philos. Trans. Roy. Soc. A, 375(2017). arXiv:1611.01781] to the physically relevant case of a liquid substrate.

Wrinkling of an elastic sheet floating on a liquid sphere

Abstract

A thin circular elastic sheet floating on a drop-like liquid substrate is deformed due to incompatibility between the curved substrate and the planar sheet. We adopt a variational viewpoint by minimizing the non-convex membrane energy together with a higher-order convex bending energy. Focusing on thin sheets, we expand the minimum of the energy in terms of a small thickness ratio , and identify the first two terms of this expansion. The leading-order term arises from the minimization of a family of one-dimensional relaxed problems, while for the next-order term we establish lower and upper bounds. This generalizes the previous work [P. Bella and R.V. Kohn. Wrikling of a thin circular sheet bonded to a spherical substrate, Philos. Trans. Roy. Soc. A, 375(2017). arXiv:1611.01781] to the physically relevant case of a liquid substrate.
Paper Structure (17 sections, 5 theorems, 54 equations)

This paper contains 17 sections, 5 theorems, 54 equations.

Table of Contents

  1. Introduction
  2. The model
  3. Heuristic arguments
  4. Main results
  5. Preliminaries
  6. The energy
  7. Energetic cost of elastic deformations of a circle: analysis of Wr
  8. Effective functional
  9. Lower bound
  10. The functional Fh
  11. Energy estimates towards the lower bound
  12. Energetic cost of changing the wavenumber
  13. Proof of the lower bound part of the main results
  14. Upper bound
  15. Strategy for the upper bound
  16. ...and 2 more sections

Key Result

Theorem 1.1

Let $\beta\in(0,2]$ and $\alpha_s\in \left(0,2^{-8}r_0^4R^{-4}\right)$ if $\beta=2$. There exist positive constants $h_0$, $c_0$, $c_1$, $c_2$, $c_3$, $c_4$ depending on $\alpha_s$, $r_0$, and $R$ (with $h_0$ also depending on $\beta$ if $\beta\leq \frac{2}{3}$), such that for any $0<h<h_0$, we have

Theorems & Definitions (17)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Remark 1.3
  • Theorem 1.2
  • Remark 1.4
  • Remark 2.1
  • Proposition 3.1
  • proof
  • Proposition 3.2
  • ...and 7 more