### 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 language | English |
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Pages (from-to) | 498-517 |

Number of pages | 21 |

Journal | Zeitschrift für Angewandte Mathematik und Physik |

Volume | 59 |

Issue number | 3 |

Early online date | 6 Aug 2007 |

DOIs | |

Publication status | Published - May 2008 |

### Keywords

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

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## Cite this

Gao, D. Y., & Ogden, R. W. (2008). Closed-form solutions, extremality and nonsmoothness criteria in a large deformation elasticity problem.

*Zeitschrift für Angewandte Mathematik und Physik*,*59*(3), 498-517. https://doi.org/10.1007/s00033-007-7047-1