Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem

D. Y. Gao, R. W. Ogden

Research output: Contribution to journalArticle

35 Citations (Scopus)

Abstract

The pure azimuthal shear problem for a circular cylindrical tube of nonlinearly elastic material, both isotropic and anisotropic, is examined on the basis of a complementary energy principle. For particular choices of strain-energy function, one convex and one non-convex, closed-form solutions are obtained for this mixed boundary-value problem, for which the governing differential equation can be converted into an algebraic equation. The results for the non-convex strain energy function provide an illustration of a situation in which smooth analytic solutions of a nonlinear boundary-value problem are not global minimizers of the energy in the variational statement of the problem. Both the global minimizer and the local extrema are identified and the results are illustrated for particular values of the material parameters.
Original languageEnglish
Pages (from-to)498-517
Number of pages21
JournalZeitschrift für Angewandte Mathematik und Physik
Volume59
Issue number3
Early online date6 Aug 2007
DOIs
Publication statusPublished - May 2008

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Global Minimizer
Elasticity Problem
Strain Energy
Large Deformation
Strain energy
Energy Function
Closed-form Solution
Boundary value problems
Elasticity
elastic properties
Mixed Boundary Value Problem
Elastic Material
Nonlinear Boundary Value Problems
Extremum
Energy
Analytic Solution
boundary value problems
Algebraic Equation
Governing equation
Tube

Keywords

  • 74B20
  • 74P99
  • finite elasticity
  • large deformations
  • nonlinear elasticity
  • azimuthal shear
  • complementary variational principle
  • triality theory

Cite this

Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem. / Gao, D. Y.; Ogden, R. W.

In: Zeitschrift für Angewandte Mathematik und Physik, Vol. 59, No. 3, 05.2008, p. 498-517.

Research output: Contribution to journalArticle

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