TY - JOUR

T1 - On the relative energies of the Rivlin and cavitation instabilities for compressible materials

AU - Lloyd, J. G.

AU - Sivaloganathan, J.

PY - 2012/6/1

Y1 - 2012/6/1

N2 - In 1948, Rivlin showed that if a cube of incompressible neo-Hookean material is subjected to a sufficiently large uniform normal dead-load on its boundary, then an asymmetric deformation is the minimizer of the energy in the class of homogeneous deformations. Ball showed in 1982 that, for classes of compressible elastic materials, if a ball of the material is subjected to a sufficiently large uniform radial dead-load, then a deformation forming a cavity is the minimizer of the energy in the class of radial deformations. In this paper we consider compressible hyperelastic materials and show that under such dead-loading, if a local minimizer of the radial energy forms a cavity, then there necessarily exists an asymmetric homogeneous deformation with less energy. Our approach extends and generalizes previous results of Abeyaratne and Hou for the incompressible case.

AB - In 1948, Rivlin showed that if a cube of incompressible neo-Hookean material is subjected to a sufficiently large uniform normal dead-load on its boundary, then an asymmetric deformation is the minimizer of the energy in the class of homogeneous deformations. Ball showed in 1982 that, for classes of compressible elastic materials, if a ball of the material is subjected to a sufficiently large uniform radial dead-load, then a deformation forming a cavity is the minimizer of the energy in the class of radial deformations. In this paper we consider compressible hyperelastic materials and show that under such dead-loading, if a local minimizer of the radial energy forms a cavity, then there necessarily exists an asymmetric homogeneous deformation with less energy. Our approach extends and generalizes previous results of Abeyaratne and Hou for the incompressible case.

UR - http://www.scopus.com/inward/record.url?scp=84861952465&partnerID=8YFLogxK

UR - http://dx.doi.org/10.1177/1081286511418275

U2 - 10.1177/1081286511418275

DO - 10.1177/1081286511418275

M3 - Article

SN - 1081-2865

VL - 17

SP - 338

EP - 350

JO - Mathematics and Mechanics of Solids

JF - Mathematics and Mechanics of Solids

IS - 4

ER -