JElast https://doi.org/10.1007/s10659-018-9675-4 Whirl Mappings on Generalised Annuli and the Incompressible Symmetric Equilibria of the Dirichlet Energy 1 1 Charles Morris · Ali Taheri Received: 27 March 2017 © Springer Science+Business Media B.V., part of Springer Nature 2018 Abstract In this paper we show a striking contrast in the symmetries of equilibria and ex- tremisers 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 ,...,u ): 1 N u = div(P(x) cof ∇u) in X, det ∇u =1in X, EL[u, X]= u ≡ ϕ 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 sym- metric 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 coun- terparts of these results in higher dimensions by way
Journal of Elasticity – Springer Journals
Published: May 29, 2018
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