Saint-Venant torsion of anisotropic shafts

Theoretical frameworks, extremal bounds and affine transformations

Tungyang Chen, Chia Jung Wei

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

We study the Saint-Venant torsion of anisotropic shafts. Theoretical frameworks for torsion of anisotropic composite shafts are derived in terms of warping function, conjugate function as well as stress potential, parallel to the existing frameworks for torsion of isotropic shafts. We prove an extremal property for the torsional rigidity of anisotropic composite shafts. For homogeneous shafts, an affine coordinate transformation is introduced in the formulation, which demonstrates how the cross-sectional shape of the shaft is deformed (stretching and rotation) under the mapping, and how the warping field and the torsional rigidity of an anisotropic shaft are correlated to those of an isotropic one. We find that a certain class of anisotropic elliptical shafts, simply- or multiply-connected, will not warp under an applied torque. Of all homogeneous shafts with a given cross-sectional area and the same shear rigidity matrix, the torsional rigidity, associated with zero warping displacement, can be proven as extremal upper bounds. Finally, families of anisotropic shafts that are equivalent to isotropic ones, including elliptical and hollow elliptical shafts, and cylindrical shafts with specific cross-sections of parallelogram and triangle shape, are characterized.

Original languageEnglish
Pages (from-to)269-287
Number of pages19
JournalQuarterly Journal of Mechanics and Applied Mathematics
Volume58
Issue number2
DOIs
Publication statusPublished - 2005 May 1

Fingerprint

Rigidity
Torsional stress
Affine transformation
torsion
Torsion
Warping
rigidity
Composite materials
Composite
Stretching
Conjugate functions
Parallelogram
Torque
Coordinate Transformation
Framework
Triangle
Multiplication
Cross section
parallelograms
Upper bound

All Science Journal Classification (ASJC) codes

  • Condensed Matter Physics
  • Mechanics of Materials
  • Mechanical Engineering
  • Applied Mathematics

Cite this

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abstract = "We study the Saint-Venant torsion of anisotropic shafts. Theoretical frameworks for torsion of anisotropic composite shafts are derived in terms of warping function, conjugate function as well as stress potential, parallel to the existing frameworks for torsion of isotropic shafts. We prove an extremal property for the torsional rigidity of anisotropic composite shafts. For homogeneous shafts, an affine coordinate transformation is introduced in the formulation, which demonstrates how the cross-sectional shape of the shaft is deformed (stretching and rotation) under the mapping, and how the warping field and the torsional rigidity of an anisotropic shaft are correlated to those of an isotropic one. We find that a certain class of anisotropic elliptical shafts, simply- or multiply-connected, will not warp under an applied torque. Of all homogeneous shafts with a given cross-sectional area and the same shear rigidity matrix, the torsional rigidity, associated with zero warping displacement, can be proven as extremal upper bounds. Finally, families of anisotropic shafts that are equivalent to isotropic ones, including elliptical and hollow elliptical shafts, and cylindrical shafts with specific cross-sections of parallelogram and triangle shape, are characterized.",
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