### Abstract

The authors consider the two-point boundary-value problem resulting from the equations of nonlinear elastostatics for azimuthal shear of a Blatz-Ko tube. Previous work on this problem by Simmonds and Warne includes a numerical study of these equations and indicates that smooth radial deformation solutions (no kinks) should exist regardless of the aspect ratio of the tube, provided that the dimensionless applied torque τ is small enough (τ <≈ 0.72). The numerics of Simmonds and Warne also indicated that the existence of smooth solutions for τ >≈ 0.72 depends on the geometry of the tube, and that for τ = √3, no smooth solution exists. Motivated by this numerical work, the authors prove via a topological shooting argument the existence and uniqueness of smooth solutions to this problem for τ ≤ τ_{cr} = √3/4^{4/3} ≈ 0.69, and the nonexistence of smooth solutions for τ = √3.

Original language | English (US) |
---|---|

Pages (from-to) | 53-69 |

Number of pages | 17 |

Journal | Mathematics and Mechanics of Solids |

Volume | 3 |

Issue number | 1 |

DOIs | |

State | Published - Jan 1 1998 |

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### All Science Journal Classification (ASJC) codes

- Mathematics(all)
- Materials Science(all)
- Mechanics of Materials

### Cite this

*Mathematics and Mechanics of Solids*,

*3*(1), 53-69. https://doi.org/10.1177/108128659800300104

}

*Mathematics and Mechanics of Solids*, vol. 3, no. 1, pp. 53-69. https://doi.org/10.1177/108128659800300104

**Existence and uniqueness of azimuthal shear solutions in compressible isotropic nonlinear elasticity.** / Paullet, Joseph E.; Warne, Debra Polignone; Warne, Paul G.

Research output: Contribution to journal › Article

TY - JOUR

T1 - Existence and uniqueness of azimuthal shear solutions in compressible isotropic nonlinear elasticity

AU - Paullet, Joseph E.

AU - Warne, Debra Polignone

AU - Warne, Paul G.

PY - 1998/1/1

Y1 - 1998/1/1

N2 - The authors consider the two-point boundary-value problem resulting from the equations of nonlinear elastostatics for azimuthal shear of a Blatz-Ko tube. Previous work on this problem by Simmonds and Warne includes a numerical study of these equations and indicates that smooth radial deformation solutions (no kinks) should exist regardless of the aspect ratio of the tube, provided that the dimensionless applied torque τ is small enough (τ <≈ 0.72). The numerics of Simmonds and Warne also indicated that the existence of smooth solutions for τ >≈ 0.72 depends on the geometry of the tube, and that for τ = √3, no smooth solution exists. Motivated by this numerical work, the authors prove via a topological shooting argument the existence and uniqueness of smooth solutions to this problem for τ ≤ τcr = √3/44/3 ≈ 0.69, and the nonexistence of smooth solutions for τ = √3.

AB - The authors consider the two-point boundary-value problem resulting from the equations of nonlinear elastostatics for azimuthal shear of a Blatz-Ko tube. Previous work on this problem by Simmonds and Warne includes a numerical study of these equations and indicates that smooth radial deformation solutions (no kinks) should exist regardless of the aspect ratio of the tube, provided that the dimensionless applied torque τ is small enough (τ <≈ 0.72). The numerics of Simmonds and Warne also indicated that the existence of smooth solutions for τ >≈ 0.72 depends on the geometry of the tube, and that for τ = √3, no smooth solution exists. Motivated by this numerical work, the authors prove via a topological shooting argument the existence and uniqueness of smooth solutions to this problem for τ ≤ τcr = √3/44/3 ≈ 0.69, and the nonexistence of smooth solutions for τ = √3.

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U2 - 10.1177/108128659800300104

DO - 10.1177/108128659800300104

M3 - Article

AN - SCOPUS:0032025545

VL - 3

SP - 53

EP - 69

JO - Mathematics and Mechanics of Solids

JF - Mathematics and Mechanics of Solids

SN - 1081-2865

IS - 1

ER -