On the Symmetry of Energy-Minimising Deformations in Nonlinear Elasticity II: Compressible Materials

• Published in: Archive for rational mechanics and analysis Vol. 196; no. 2; pp. 395 - 431
• Main Authors: Sivaloganathan, Jeyabal, Spector, Scott J.
• Format: Journal Article
• Language: English
• Published: Berlin/Heidelberg Springer-Verlag 01.05.2010; Springer Nature B.V
• Subjects: Classical Mechanics; Complex Systems; Fluid- and Aerodynamics; Mathematical and Computational Physics; Physics; Physics and Astronomy; Theoretical; Nonlinear Elasticity; Distributional Derivative; Absolute Minimiser; Radial Deformation; Spherical Shell
• ISSN: 0003-9527; 1432-0673
• DOI: 10.1007/s00205-009-0262-5

Consider a homogeneous, isotropic, hyperelastic body occupying the region 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 that maps each sphere , of radius R  > 0, centred at the origin into another such sphere that encloses the same volume as u ( S R ). Since the volumes enclosed by the surfaces u ( S R ) and u rad ( S R ) are equal, the classical isoperimetric inequality implies that . The equality of the enclosed volumes together with this reduction in surface area is then shown to give rise to a reduction in total energy for many of the constitutive relations used in nonlinear elasticity. These results are also extended to classes of Sobolev deformations and applied to prove that the radially symmetric solutions to these boundary-value problems are local or global energy minimisers in various classes of (possibly nonsymmetric) deformations of a thick spherical shell.