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Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy

Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet... In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation u = ( u 1 , … , u N ) $u=(u_{1}, \ldots, u_{N})$ : E L [ u , X ] = { Δ u = div ( P ( x ) cof ∇ u ) in  X , det ∇ u = 1 in  X , u ≡ φ on  ∂ X , $$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$ where X ${\mathbf {X}}$ is a finite, open, symmetric N $N$ -annulus (with N ≥ 2 $N \ge2$ ), P = P ( x ) $\mathscr{P}=\mathscr{P}(x)$ is an unknown hydrostatic pressure field and φ $\varphi$ is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when N = 3 $N=3$ , the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when N = 2 $N=2$ , the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions N ≥ 4 $N \ge4$ and discuss a number of closely related issues. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Journal of Elasticity Springer Journals

Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy

Morris, Charles; Taheri, Ali
Journal of Elasticity , Volume 133 (2) – May 29, 2018
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Publisher
Springer Journals
Copyright
Copyright © 2018 by Springer Science+Business Media B.V., part of Springer Nature
Subject
Physics; Classical Mechanics; Automotive Engineering
ISSN
0374-3535
eISSN
1573-2681
DOI
10.1007/s10659-018-9675-4
Publisher site
See Article on Publisher Site

Abstract

In this paper we show a striking contrast in the symmetries of equilibria and extremisers of the total elastic energy of a hyperelastic incompressible annulus subject to pure displacement boundary conditions. Indeed upon considering the equilibrium equations, here, the nonlinear second order elliptic system formulated for the deformation u = ( u 1 , … , u N ) $u=(u_{1}, \ldots, u_{N})$ : E L [ u , X ] = { Δ u = div ( P ( x ) cof ∇ u ) in  X , det ∇ u = 1 in  X , u ≡ φ on  ∂ X , $$ {\mathbb{E}} {\mathbb{L}}[u, {\mathbf {X}}] = \left \{ \textstyle\begin{array}{l@{\quad}l} \Delta u = \operatorname{div}(\mathscr{P} (x) \operatorname{cof} \nabla u) & \textrm{in }{\mathbf {X}},\\ \det\nabla u = 1 & \textrm{in }{\mathbf {X}},\\ u \equiv\varphi& \textrm{on }\partial{\mathbf {X}}, \end{array}\displaystyle \right . $$ where X ${\mathbf {X}}$ is a finite, open, symmetric N $N$ -annulus (with N ≥ 2 $N \ge2$ ), P = P ( x ) $\mathscr{P}=\mathscr{P}(x)$ is an unknown hydrostatic pressure field and φ $\varphi$ is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when N = 3 $N=3$ , the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when N = 2 $N=2$ , the problem possesses an infinite family of symmetric and topologically distinct equilibria. We extend and prove the counterparts of these results in higher dimensions by way of showing that a similar dichotomy persists between all odd vs. even dimensions N ≥ 4 $N \ge4$ and discuss a number of closely related issues.

Journal

Journal of Elasticity – Springer Journals

Published: May 29, 2018

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