Plane Deformations in Incompressible Nonlinear Elasticity |
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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, U.S.A;(2) Division of Mathematics and Computer Science, Maryville College, Maryville, TN, 37804, U.S.A |
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Abstract: | The substantially general class of plane deformation fields, whose only restriction requires that the angular deformation
not vary radially, is considered in the context of isotropic incompressible nonlinear elasticity. Analysis to determine the
types of deformations possible, that is, solutions of the governing systems of nonlinear partial differential equations and
constraint of incompressibility, is developed in general. The Mooney-Rivlin material model is then considered as an example
and all possible solutions to the equations of equilibrium are determined. One of these is interpreted in the context of nonradially
symmetric cavitation, i.e., deformation of an intact cylinder to one with a double-cylindrical cavity. Results for general
incompressible hyperelastic materials are then discussed. The novel approach taken here requires the derivation and use of
a material formulation of the governing equations; the traditional approach employing a spatial formulation in which the governing
equations hold on an unknown region of space is not conducive to the study of deformation fields containing more than one
independent variable. The derivation of the cylindrical polar coordinate form of the equilibrium equations for the nominal
stress tensor (material formulation) for a general hyperelastic solid and a fully arbitrary cylindrical deformation field
is also given.
This revised version was published online in August 2006 with corrections to the Cover Date. |
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Keywords: | nonsymmetric inhomogeneous equilibrium plane hyperelasticity incompressible nominal stress material formulation of cylindrical problems cavitation |
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