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 of Elasticity – Springer Journals
Published: May 29, 2018
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