Azimuthal shear of a transversely isotropic elastic solid

Fotios Kassianidis, Raymond William Ogden, Jose Merodio, Thomas Pence

Research output: Contribution to journalArticle

29 Citations (Scopus)

Abstract

In this paper we study the problem of (plane strain) azimuthal shear of a circular cylindrical tube of incompressible transversely isotropic elastic material subject to finite deformation. The preferred direction associated with the transverse isotropy lies in the planes normal to the tube axis and is at an angle with the radial direction that depends only on the radius. For a general form of strain-energy function the considered deformation yields simple expressions for the azimuthal shear stress and the associated strong ellipticity condition in terms of the azimuthal shear strain. These apply for a sense of shear that is either “with” or “against” the preferred direction (anticlockwise and clockwise, respectively), so that material line elements locally in the preferred direction either extend or (at least initially) contract, respectively. For some specific strain-energy functions we then examine local loss of uniqueness of the shear stress—strain relationship and failure of ellipticity for the case of contraction and the dependence on the geometry of the preferred direction. In particular, for a reinforced neo-Hookean material, we obtain closed-form solutions that determine the domain of strong ellipticity in terms of the relationship between the shear strain and the angle (in general, a function of the radius) between the tangent to the preferred direction and the undeformed radial direction. It is shown, in particular, that as the magnitude of the applied shear stress increases then, after loss of ellipticity, there are two admissible values for the shear strain at certain radial locations. Absolutely stable deformations involve the lower magnitude value outside a certain radius and the higher magnitude value within this radius. The radius that separates the two values increases with increasing magnitude of the shear stress. The results are illustrated graphically for two specific forms of energy function.

Original languageEnglish
Pages (from-to)690-724
Number of pages35
JournalMathematics and Mechanics of Solids
Volume13
Issue number8
Early online date10 Sep 2007
DOIs
Publication statusPublished - Nov 2008

Fingerprint

Transversely Isotropic
Shear strain
Ellipticity
Shear stress
Radius
Strain energy
Energy Function
Shear Stress
Strain Energy
Tube
Transverse Isotropy
Anticlockwise
Angle
Clockwise
Finite Deformation
Plane Strain
Elastic Material
Geometry
Direction compound
Closed-form Solution

Keywords

  • large deformations
  • finite elasticity
  • transverse isotropy
  • azimuthal shear
  • loss of ellipticity

Cite this

Azimuthal shear of a transversely isotropic elastic solid. / Kassianidis, Fotios; Ogden, Raymond William; Merodio, Jose; Pence, Thomas.

In: Mathematics and Mechanics of Solids, Vol. 13, No. 8, 11.2008, p. 690-724.

Research output: Contribution to journalArticle

Kassianidis, F, Ogden, RW, Merodio, J & Pence, T 2008, 'Azimuthal shear of a transversely isotropic elastic solid', Mathematics and Mechanics of Solids, vol. 13, no. 8, pp. 690-724. https://doi.org/10.1177/1081286507079830
Kassianidis, Fotios ; Ogden, Raymond William ; Merodio, Jose ; Pence, Thomas. / Azimuthal shear of a transversely isotropic elastic solid. In: Mathematics and Mechanics of Solids. 2008 ; Vol. 13, No. 8. pp. 690-724.
@article{10f1a3ef56a548c595845941b0797e30,
title = "Azimuthal shear of a transversely isotropic elastic solid",
abstract = "In this paper we study the problem of (plane strain) azimuthal shear of a circular cylindrical tube of incompressible transversely isotropic elastic material subject to finite deformation. The preferred direction associated with the transverse isotropy lies in the planes normal to the tube axis and is at an angle with the radial direction that depends only on the radius. For a general form of strain-energy function the considered deformation yields simple expressions for the azimuthal shear stress and the associated strong ellipticity condition in terms of the azimuthal shear strain. These apply for a sense of shear that is either “with” or “against” the preferred direction (anticlockwise and clockwise, respectively), so that material line elements locally in the preferred direction either extend or (at least initially) contract, respectively. For some specific strain-energy functions we then examine local loss of uniqueness of the shear stress—strain relationship and failure of ellipticity for the case of contraction and the dependence on the geometry of the preferred direction. In particular, for a reinforced neo-Hookean material, we obtain closed-form solutions that determine the domain of strong ellipticity in terms of the relationship between the shear strain and the angle (in general, a function of the radius) between the tangent to the preferred direction and the undeformed radial direction. It is shown, in particular, that as the magnitude of the applied shear stress increases then, after loss of ellipticity, there are two admissible values for the shear strain at certain radial locations. Absolutely stable deformations involve the lower magnitude value outside a certain radius and the higher magnitude value within this radius. The radius that separates the two values increases with increasing magnitude of the shear stress. The results are illustrated graphically for two specific forms of energy function.",
keywords = "large deformations, finite elasticity, transverse isotropy, azimuthal shear, loss of ellipticity",
author = "Fotios Kassianidis and Ogden, {Raymond William} and Jose Merodio and Thomas Pence",
year = "2008",
month = "11",
doi = "10.1177/1081286507079830",
language = "English",
volume = "13",
pages = "690--724",
journal = "Mathematics and Mechanics of Solids",
issn = "1081-2865",
publisher = "SAGE Publications Inc.",
number = "8",

}

TY - JOUR

T1 - Azimuthal shear of a transversely isotropic elastic solid

AU - Kassianidis, Fotios

AU - Ogden, Raymond William

AU - Merodio, Jose

AU - Pence, Thomas

PY - 2008/11

Y1 - 2008/11

N2 - In this paper we study the problem of (plane strain) azimuthal shear of a circular cylindrical tube of incompressible transversely isotropic elastic material subject to finite deformation. The preferred direction associated with the transverse isotropy lies in the planes normal to the tube axis and is at an angle with the radial direction that depends only on the radius. For a general form of strain-energy function the considered deformation yields simple expressions for the azimuthal shear stress and the associated strong ellipticity condition in terms of the azimuthal shear strain. These apply for a sense of shear that is either “with” or “against” the preferred direction (anticlockwise and clockwise, respectively), so that material line elements locally in the preferred direction either extend or (at least initially) contract, respectively. For some specific strain-energy functions we then examine local loss of uniqueness of the shear stress—strain relationship and failure of ellipticity for the case of contraction and the dependence on the geometry of the preferred direction. In particular, for a reinforced neo-Hookean material, we obtain closed-form solutions that determine the domain of strong ellipticity in terms of the relationship between the shear strain and the angle (in general, a function of the radius) between the tangent to the preferred direction and the undeformed radial direction. It is shown, in particular, that as the magnitude of the applied shear stress increases then, after loss of ellipticity, there are two admissible values for the shear strain at certain radial locations. Absolutely stable deformations involve the lower magnitude value outside a certain radius and the higher magnitude value within this radius. The radius that separates the two values increases with increasing magnitude of the shear stress. The results are illustrated graphically for two specific forms of energy function.

AB - In this paper we study the problem of (plane strain) azimuthal shear of a circular cylindrical tube of incompressible transversely isotropic elastic material subject to finite deformation. The preferred direction associated with the transverse isotropy lies in the planes normal to the tube axis and is at an angle with the radial direction that depends only on the radius. For a general form of strain-energy function the considered deformation yields simple expressions for the azimuthal shear stress and the associated strong ellipticity condition in terms of the azimuthal shear strain. These apply for a sense of shear that is either “with” or “against” the preferred direction (anticlockwise and clockwise, respectively), so that material line elements locally in the preferred direction either extend or (at least initially) contract, respectively. For some specific strain-energy functions we then examine local loss of uniqueness of the shear stress—strain relationship and failure of ellipticity for the case of contraction and the dependence on the geometry of the preferred direction. In particular, for a reinforced neo-Hookean material, we obtain closed-form solutions that determine the domain of strong ellipticity in terms of the relationship between the shear strain and the angle (in general, a function of the radius) between the tangent to the preferred direction and the undeformed radial direction. It is shown, in particular, that as the magnitude of the applied shear stress increases then, after loss of ellipticity, there are two admissible values for the shear strain at certain radial locations. Absolutely stable deformations involve the lower magnitude value outside a certain radius and the higher magnitude value within this radius. The radius that separates the two values increases with increasing magnitude of the shear stress. The results are illustrated graphically for two specific forms of energy function.

KW - large deformations

KW - finite elasticity

KW - transverse isotropy

KW - azimuthal shear

KW - loss of ellipticity

U2 - 10.1177/1081286507079830

DO - 10.1177/1081286507079830

M3 - Article

VL - 13

SP - 690

EP - 724

JO - Mathematics and Mechanics of Solids

JF - Mathematics and Mechanics of Solids

SN - 1081-2865

IS - 8

ER -