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Whirl mappings on generalised annuli and the incompressible symmetric equilibria of the dirichlet energy

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posted on 2023-06-09, 13:11 authored by Charles Morris, Ali TaheriAli Taheri
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=(u1,…,uN) : EL[u,X]=??????u=div(P(x)cof?u)det?u=1u=finX,inX,on?X, where X is a finite, open, symmetric N -annulus (with N=2 ), P=P(x) is an unknown hydrostatic pressure field and f is the identity mapping, we prove that, despite the inherent rotational symmetry in the system, when N=3 , the problem possesses no non-trivial symmetric equilibria whereas in sharp contrast, when 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 and discuss a number of closely related issues.

History

Publication status

  • Published

File Version

  • Accepted version

Journal

Journal of Elasticity

ISSN

0374-3535

Publisher

Springer

Issue

2

Volume

133

Page range

201-222

Department affiliated with

  • Mathematics Publications

Research groups affiliated with

  • Analysis and Partial Differential Equations Research Group Publications

Full text available

  • Yes

Peer reviewed?

  • Yes

Legacy Posted Date

2018-05-09

First Open Access (FOA) Date

2019-05-29

First Compliant Deposit (FCD) Date

2018-05-09

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