Abstract
Problems for a round shaft of variable diameter subjected to torsion are studied. By transforming the governing equation into oblate spheroidal coordinates, a general solution is obtained in terms of the associated Legendre functions of order 2. Two illustrative problems, one being a shaft with a hyperbolic notch, the other a cylindrical shaft containing a small oblate spheroidal cavity located on its central axis, are solved. On the basis of the ensuing stress concentration, the important connection between the deep notch and crack is exploited. Several previously known solutions can be recovered. Results are extended to the cases of composites and transversely isotropic materials.
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
Love AEH (1927) A treatise on the mathematical theory of elasticity, 4th edn. Cambridge University Press, Cambridge
Sokolnikoff IS (1956) Mathematical theory of elasticity. McGraw-Hill, New York
Timoshenko SP, Goodier JN (1970) Theory of elasticity, 3rd edn. McGraw-Hill, New York
Neuber H (1946) Theory of notches. Edwards, Michigan
Pilkey WD (1997) Peterson’s stress concentration handbook. Wiley, New York
Hobson EW (1931) The theory of spherical and ellipsoidal harmonics. Cambridge University Press, Cambridge
Abbassi MM (1955) Torsion of circular shafts of variable diameter. J Appl Mech 22: 530–532
Davis EA, Tuba IS (1966) Elastic–plastic solutions for notched shafts in torsion. J Appl Mech 33: 79–84
Creager M, Paris PC (1967) Elastic field equations for blunt cracks with reference to stress corrosion cracking. Int J Fract Mech 3: 247–252
Smith E (2006) The mode III elastic stress distribution near the root of (a) a intrusion-type notch and (b) a key-hole notch. Int J Eng Sci 44: 340–344
Chiang CR (2009) Some elasticity problems of a general anisotropic solid subjected to anti-plane loadings. Acta Mech 203: 49–61
Kassir MK, Sih GC (1975) Three-dimensional crack problems. Noordhoff, Leyden
Tada H, Paris PC, Irwin GR (1985) The stress analysis of cracks handbook. Paris Production, St. Louis
Chowdhury KL (1983) Solution of the problem of a concentrated torque on a semi-space by similarity transforms. J Elast 13: 87–90
Das SC (1954) Stress concentrations around a small spherical or spheroidal inclusion on the axis of a circular cylinder in torsion. J Appl Mech 21: 83–87
Chiang CR (1989) Mode III interface crack propagation. Eng Fract Mech 32: 545–550
Broek D (1986) Elementary engineering fracture mechanics. Martinus Nijhoff, Dordrecht
Lekhnitskii SG (1963) Theory of elasticity of an anisotropic elastic body. Holden-Day, San Francisco
Bose SC (1965) On the torsion of a transversely isotropic circular cylinder having a small spherical elastic inclusion of an isotropic material. Z Angew Math Mech 45: 133–135
Chen WT (1966) On a spheroidal elastic inclusion in a transversely isotropic material under an axisymmetric torsion field. J Appl Mech 33: 944–945
Chiang CR (2003) Stress concentration around a strongly oblate spheroidal cavity in a transversely isotropic medium. Int J Fract 119: L91–L97
Chiang CR (2004) Some crack problems in transversely isotropic solids. Acta Mech 170: 1–9
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Chiang, CR. Torsion of a round shaft of variable diameter. J Eng Math 77, 119–130 (2012). https://doi.org/10.1007/s10665-012-9549-x
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s10665-012-9549-x