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Let $${A=\{{\bf x} \in \mathbb{R}^n : a  0}$$ , be the region occupied by an incompressible, nonlinearly hyperelastic material in its reference configuration. Deformations $${{\bf u} : {\it A} \rightarrow \mathbb{R}^n}$$ are therefore isochoric maps and so satisfy the incompressibility constraint$${\rm det}\nabla{\bf u} =1.$$    The displacement and the mixed displacement/zero-traction boundary-value problems are considered. Here a displacement boundary condition of the form$${\bf u} ({\bf x} ) =\sigma {\bf x}$$    is prescribed on one of the boundary components (where σ > 0 is a given constant) and the displacement on the remaining boundary component is either prescribed in the same form (in the case of the pure displacement boundary-value problem) or left unspecified (in the case of the mixed boundary-value problem). In this paper isoperimetric arguments are used to prove that the radially symmetric solutions to these problems are global energy minimisers in various classes of (possibly non-symmetric) isochoric deformations of the region A.

On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity I: Incompressible Materials