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

Let ${A=\{{\bf x} \in \mathbb{R}^n : a < |{\bf x}| < b\}, n \geqq 2, a > 0}Consider a homogeneous, isotropic, hyperelastic body occupying the region A = {x ? \mathbbRⁿ :  a <  |x |  < b }{A = \{{\bf x}\in\mathbb{R}^{n}\, : \,a <\,|{\bf x} |\,< b \}} in its reference state and subject to radially symmetric displacement, or mixed displacement/traction, boundary conditions. In Part I (Sivaloganathan and Spector in Arch Ration Mech Anal, 2009, in press) the authors restricted their attention to incompressible materials. For a large-class of polyconvex constitutive relations that grow sufficiently rapidly at infinity it was shown that to each nonradial isochoric deformation of A there corresponds a radial isochoric deformation that has strictly less elastic energy than the given deformation. In this paper that analysis is further developed and extended to the compressible case. The key ingredient is a new radial-symmetrisation procedure that is appropriate for problems where the symmetrised mapping must be one-to-one in order to prevent interpenetration of matter. For the pure displacement boundary-value problem, the radial symmetrisation of an orientation-preserving diffeomorphism u : A → A* between spherical shells A and A* is the deformation

urad(x)=\fracr(R)Rx,     R=|x|,                        (0.1){\bf u}^{\rm rad}({\bf x})=\frac{r(R)}{R}{\bf x}, \quad R=|{\bf x}|,\qquad\qquad\qquad\qquad(0.1)
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