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

Morris, Charles and Taheri, Ali (2018) Whirl mappings on generalised annuli and the incompressible symmetric equilibria of the dirichlet energy. Journal of Elasticity, 133 (2). pp. 201-222. ISSN 0374-3535

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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=(u1,…,uN) :

EL[u,X]=⎧⎩⎨⎪⎪Δu=div(P(x)cof∇u)det∇u=1u≡φinX,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 φ 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.

Item Type: Article
Schools and Departments: School of Mathematical and Physical Sciences > Mathematics
Research Centres and Groups: Analysis and Partial Differential Equations Research Group
Depositing User: Ali Taheri
Date Deposited: 09 May 2018 09:20
Last Modified: 01 Jul 2019 12:01
URI: http://sro.sussex.ac.uk/id/eprint/75682

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