Rigorous Bounds on the Torsional Rigidity of Composite Shafts with Imperfect Interfaces
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.
KeywordsTorsional rigidity Bounds Imperfect interfaces
Mathematics Subject Classifications (2000)73B27 73K05
Unable to display preview. Download preview PDF.
- 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
- 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.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