本文主要推導任意形狀之桿件中含有塗層纖維內含物，於各種不完美界面下，其扭轉剛度界限；並探討其解析解之值及其存在條件。其中考慮不完美界面中之二大類，一者為低剪力模數之薄層，稱之為LS-type；一者為高剪力模數之薄層，即為HS-type，並以界面剪力模數及厚度所定義之界面參數來表示其界面特性。此外，我們推論當桿件中內含物為多層複合材料時，其扭轉剛度界限值與解析解之存在條件。最後，探討桿件外形對扭轉剛度上下限重合條件之影響，並發現當桿件外形較為複雜時，若假設界面參數為常數，則無法滿足扭轉剛度上下限重合條件；故而進一步假設界面參數沿界面而改變，以調整扭轉剛度界限，使上下限重合，得出解析解。

Under different kinds of imperfect interfaces, we derive upper and lower bounds for the torsional rigidity of cylindrically shafts with arbitrary cross-section containing a number of coated fibers with circular cross-section by variational principles. At the interfaces between the different materials two kinds of imperfect interfaces are considered for the Saint-Venant torsion problem of composite shafts: 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. We find that when an additional constraint in the outer interface is fulfilled, the upper and lower bounds will coincide. Furthermore, we deduce the bounds and exact solution for the torsional rigidity of composite shafts containing a number of multi-coated fibers based on the previous derivation.
In addition, the interface parameter varying along the interface will be considered to fulfill the conditions of the existence of the exact torsional rigidity for the different shapes of the shaft.

Benveniste, Y., A general interface model for a three-dimensional curved thin anisotropic interphase between two anisotropic media, J. Mech. Phys. Solids 54, pp.708-734, 2006.

Benveniste, Y. & Chen, T., On the Saint-Venant torsion of composite bars with imperfect interfaces, Proc. R. Soc. Lond. A 457, pp.231-255, 2001.

Bovik, P., On the modeling of thin interface layers in elastic and acoustic scattering problems, Quart. J. Mech. Appl. Math. 47, pp.17-40, 1994.

Chen, T., Thermal conduction of a circular inclusion with variable interface parameter, Int. J. Solids Struct. 38, pp.3081-3097, 2001.

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. Lond. A 458, pp.1719-1759, 2002.

Chen, T., An exactly solvable microgeometry in torsion: assemblage of multicoated cylinders, Proc. R. Soc. Lond. A 460, pp.1981-1993, 2004.

Chen, T. & Wei, C.J., Saint-Venant torsion of anisotropic shafts: theoretical frameworks, extremal bounds and affine transformations, Q. JI Mech. Appl. Math. 58 (2), pp.267-287, 2005.

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, pp.3291-3309, 2007.

Chen, T. & Chan, I-Tung, Rigorous bounds on the torsional rigidity of composite shafts with imperfect interfaces, J. Elasticity. 92, pp.91-108, 2008.

Lipton, R. & Chen, T., Bounds and extremal configurations for the torsional rigidity of coated fiber reinforced shafts, SIAM J. Appl. Math. 65, no.1, pp.299-315, 2004.

Lipton, R., Optimal configurations for maximum torsional rigidity, Arch. Ration. Mech. Analysis 144, pp.79-106, 1998.

Muskhelishvili, N. I., Some Basic Problems of the Mathematical Theory of Elasticity, Noordhoff, Gronongen, 1953.

Payne, L.E. & Weinberger, H.F., Some isoperimetric inequalities for membrane frequencies and torsional rigidity, J. Math. Anal. Appl. 2, 210-216, 1961.

Polya, G., Torsional rigidity, principal frequency, electrostatic capacity and symmetrization, Quart. Appl. Math. 6, pp.267-277, 1948.

Polya, G. & Weinstein, A., On the torsional rigidity of multiply connected cross sections, Ann. Math. 52, pp.155-163, 1950.

Sadd, M.H., Elasticity: Theory, Applications, and Numerics, Elsevier Academic Press, 2005.

Sokolnikoff, I.S., Mathematical Theory of Elasticity, McGraw-Hill, New York, 1956.

Timoshenko, S.P. & Gooder, J.N., Theory of Elasticity, McGraw-Hill, New York, 1970.

Tolstov, G.P., Fourier Series, Dover, New York, 1976; R.A. Silverman, Trans., original work in Russian.

詹益東, 不完美界面下複合桿件的扭轉剛度上下限, 國立成功大學土木工程所碩士論文, 2007.