On Non-Symmetric Deformations of an Incompressible Nonlinearly Elastic Isotropic Sphere |
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Authors: | Debra Polignone Warne Paul G Warne |
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Institution: | (1) Department of Mathematics, University of Tennessee, Knoxville, TN 37996, USA;(2) Division of Mathematics and Computer Science, Maryville College, Maryville, TN 37804, USA |
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Abstract: | A class of non-symmetric deformations of a neo-Hookean incompressible nonlinearly elastic sphere are investigated. It is found
via the semi-inverse method that, to satisfy the governing three-dimensional equations of equilibrium and the incompressibility
constraint, only three special cases of the class of deformation fields are possible. One of these is the trivial solution,
one the solution describing radially symmetric deformation, and the other a (non-symmetric, non-homogeneous) deformation describing
inflation and stretching. The implications of these results for cavitation phenomena are also discussed. In the course of
this work, we also present explicitly the spherical polar coordinate form of the equilibrium equations for the nominal stress
tensor for a general hyperelastic solid. These are more complicated than their counterparts for Cauchy stresses due to the
mixed bases (both reference and deformed) associated with the nominal (or Piola-Kirchhoff) stress tensor, but more useful
in considering general deformation fields.
This revised version was published online in August 2006 with corrections to the Cover Date. |
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Keywords: | incompressible nonlinear elasticity non-symmetric deformations governing equations material formulation cavitation |
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