本文主要探討圓形桿件內含有圓形內含物之扭轉問題，藉由複數諧和函數建立解析問題的架構，而求得翹曲函數與扭轉剛度，並以扭轉剛度之數值結果與文獻比較，映證扭轉剛度正確性；另外，依據扭轉剛度界限的觀點，評估扭轉剛度數值結果的正確性，並加以探討桿件之幾何參數與材料對扭轉剛度界限的影響。接著，考慮圓形桿件內含有週期排列內含物。利用旋轉對稱的特性，簡化界面連續條件，減少待定係數與線性方程式數目；另外，探討週期排列內含物之零翹曲曲線之特性，與級數項次對應扭轉剛度收斂的關係。最後，根據中性內含物之定義，由本文的架構推導圓形桿件內有中性複合圓柱之限制條件，並以數值結果呈現翹曲函數圖與扭轉剛度。

We consider the Saint-Venant's torsion of circular shafts containing a number of circular fibers. The complex variable method together with a conformal mapping technique is employed to derive the warping field and torsional rigidity of composite shafts. Numerical results of warping function and torsional rigidity are presented for a few simple configurations and are compared with existing solutions in the literature. In addition, the results are verified with rigorous bounds on the torsional rigidity based on variational principles. Next, we consider circular shafts containing circular inclusions with a periodic arrangement. We find that the zero warping lines can be identified. Due to the periodicity, we can also simplify the interface continuity conditions so that the number of unknown coefficients to be fulfilled can be much reduced. Finally, the definition of neutral inclusions allows us to construct the constraint conditions of circular shaft containing neutral coated fibers.

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