Journal of Elasticity

, Volume 92, Issue 1, pp 91–108 | Cite as

Rigorous Bounds on the Torsional Rigidity of Composite Shafts with Imperfect Interfaces

  • Tungyang Chen
  • I-Tung Chan


We derive upper and lower bounds for the torsional rigidity of cylindrical shafts with arbitrary cross-section containing a number of fibers with circular cross-section. Each fiber may have different constituent materials with different radius. At the interfaces between the fibers and the host matrix two kinds of imperfect interfaces are considered: one which models a thin interphase of low shear modulus and one which models a thin interphase of high shear modulus. Both types of interface will be characterized by an interface parameter which measures the stiffness of the interface. The exact expressions for the upper and lower bounds of the composite shaft depend on the constituent shear moduli, the absolute sizes and locations of the fibers, interface parameters, and the cross-sectional shape of the host shaft. Simplified expressions are also deduced for shafts with perfect bonding interfaces and for shafts with circular cross-section. The effects of the imperfect bonding are illustrated for a circular shaft containing a non-centered fiber. We find that when an additional constraint between the constituent properties of the phases is fulfilled for circular shafts, the upper and lower bounds will coincide. In the latter situation, the fibers are neutral inclusions under torsion and the bounds recover the previously known exact torsional rigidity.


Torsional rigidity Bounds Imperfect interfaces 

Mathematics Subject Classifications (2000)

73B27 73K05 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Love, A.E.H.: A Treatise on the Mathematical Theory of Elasticity. Dover, New York (1944)MATHGoogle Scholar
  2. 2.
    Polya, G.: Torsional rigidity, principal frequency, electrostatic capacity and symmetrization. Quart. Appl. Math. 6, 267–277 (1948)MATHMathSciNetGoogle Scholar
  3. 3.
    Polya, G., Weinstein, A.: On the torsional rigidity of multiply connected cross sections. Ann. Math. 52, 155–163 (1950)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Payne, L.E., Weinberger, H.F.: Some isoperimetric inequalities for membrane frequencies and torsional rigidity. J. Math. Anal. Appl. 2, 210–216 (1961)MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Payne, L.E.: Some isoperimetric inequalities in the torsion problem for multiply connected regions. In: Studies in Mathematical Analysis and Related Topics. Essay in honor of G. Polya, Stanford University Press, CA (1962)Google Scholar
  6. 6.
    Milton, G.W.: The Theory of Composites. Cambridge University Press, Cambridge (2002)MATHGoogle Scholar
  7. 7.
    Chen, T., Benveniste, Y., Chuang, P.C.: Exact solutions in torsion of composite bars: thickly coated neutral inhomogeneities and composite cylinder assemblages. Proc. R. Soc. A 458, 1719–1759 (2002)MATHCrossRefADSGoogle Scholar
  8. 8.
    Chen, T.: An exactly solvable microgeometry in torsion: assemblage of multicoated cylinders. Proc. R. Soc. A 460, 1981–1993 (2004)MATHCrossRefADSGoogle Scholar
  9. 9.
    Benveniste, Y., Miloh, T.: Soft neutral elastic inhomogeneities with membrane-type interface conditions. J. Elasticity 88, 87–111 (2007)MATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Lipton, R.: Optimal fiber configurations for maximum torsional rigidity. Arch. Ration Mech. Anal. 144, 79–106 (1998)MATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Lipton, R.: An Isoperimetric inequality for the torsional rigidity of imperfectly bonded fiber reinforced shafts. J. Elasticity 55, 1–10 (1999)MATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Lipton, R., Chen, T.: Bounds and extremal configurations for the torsional rigidity of coated fiber reinforced shafts. SIAM J. Appl. Math. 65, 299–315 (2004)MATHCrossRefMathSciNetGoogle Scholar
  13. 13.
    Chen, T., Lipton, R.: Bounds for the torsional rigidity of shafts with arbitrary cross-sections containing cylindrically orthotropic fibers or coated fibers. Proc. R. Soc. A. 463, 3291–3309 (2007)CrossRefADSMathSciNetMATHGoogle Scholar
  14. 14.
    Bovik, P.: On the modelling of thin interface layers in elastic and acoustic scattering problems. Quart. J. Mech. Appl. Math. 47, 17–40 (1994)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Benveniste, Y.: A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media. J. Mech. Phys. Solids 54, 708–734 (2006)MATHCrossRefADSMathSciNetGoogle Scholar
  16. 16.
    Torquato, S., Rintoul, M.D.: Effect of the interface on the properties of composite media. Phys. Rev. Lett. 75, 4067–4070 (1995)CrossRefADSGoogle Scholar
  17. 17.
    Niklasson, A.J., Datta, S.K., Dunn, M.L.: On approximate guided wave in plates with thin anisotropic coatings by means of effective boundary conditions. J. Acoust. Soc. Am. 108(Pt 1), 924–933 (2000)CrossRefADSGoogle Scholar
  18. 18.
    Ting, T.C.T.: Mechanics of a thin anisotropic elastic layer and a layer that is bonded to an anisotropic elastic body or bodies. Proc. R. Soc. A. 463, 2223–2239 (2007)MATHCrossRefADSMathSciNetGoogle Scholar
  19. 19.
    Lipton, R., Vernescu, B.: Variational methods, size effects, and extremal microgeometries for elastic composites with imperfect interface. Math. Meth. Mod Appl. Sci. 5, 1139–1173 (1995)MATHCrossRefMathSciNetGoogle Scholar
  20. 20.
    Lipton, R., Vernescu, B.: Composites with imperfect interface. Proc. R. Soc. A. 452, 329–358 (1996)MATHCrossRefADSMathSciNetGoogle Scholar
  21. 21.
    Benveniste, Y., Chen, T.: On the Saint-Venant torsion of composite bars with imperfect interfaces. Proc. R. Soc. A 457, 231–255 (2001)MATHCrossRefADSMathSciNetGoogle Scholar
  22. 22.
    Povstenko, Y.Z.: Theoretical investigation of phenomena caused by heterogeneous surface tension in solids. J. Mech. Phys. Solids 41, 1499–1514 (1993)MATHCrossRefADSMathSciNetGoogle Scholar
  23. 23.
    Shenoy, V.B.: Size-dependent rigidities of nanosized torsional elements. Int. J. Solids Struct. 39, 4039–4052 (2002)MATHCrossRefGoogle Scholar
  24. 24.
    Sokolnikoff, I.S.: Mathematical Theory of Elasticity. McGraw-Hill, New York (1956)MATHGoogle Scholar
  25. 25.
    Horgan, C.O., Knowles, J.K.: Recent developments concerning the Saint-Venant’s principle. In: Hutchinson, J.W. (ed.) Advances in Applied Mechanics, vol. 23, pp.170–269. Academic, New York (1983)Google Scholar
  26. 26.
    Chen, T., Chiu, M.S., Weng, C.N.: Derivation of the generalized Young-Laplace equation of curved interfaces in nano-scaled solids. J. Appl. Phys. 100, 074308(1–5) (2006)Google Scholar

Copyright information

© Springer Science+Business Media B.V. 2007

Authors and Affiliations

  1. 1.Department of Civil EngineeringNational Cheng Kung UniversityTainanTaiwan

Personalised recommendations