我們利用歐拉角（Euler angle），推導出一根具有自發曲率和自發扭曲率之等向性彎曲剛度的均勻彈性桿，在受到單軸力下，其形狀所滿足的方程式，並且研究長細螺旋桿在外力拉伸下的彈性性質與穩定性。我們發現，當使用超螺旋度作為獨立變數時，在拉伸螺旋桿的過程中沒有發生伸長跳躍的現象。這結果意味著用超螺旋度作為獨立變數時，螺旋桿的彈性性質完全不同於用外力矩作為獨立變數時的彈性性質。我們也發現大的負超螺旋度值會使螺旋桿不穩定。

We derive the shape equations in terms of Euler angles for an uniform elastic rod with isotropic bending rigidity and spontaneous curvature and spontaneous twist rate. We study the elasticity and stability of a helical filament under uniaxial force. We find that there is no sharp transition for the extention of a helix when using supercoiling degree as an independent variable. This result suggests that the elastic response of a helical filament using the supercoiling degree as independent variable must be quite different from using the torque as independent variable. We also find that a large negative supercoiling degree tends to destabilize the helical filament.

目錄
第一章	 簡介....................................1
§ 1-1  引言.......................................1
§ 1-2  動機與目的.................................2
§ 1-3  主要內容...................................3
第二章	 模型....................................5
§ 2-1  Frenet-Serret方程組........................5
§ 2-2  廣義的Frenet-Serret方程組..................9
§ 2-3  本工作使用的模型...........................12
第三章	 螺旋解..................................16
§ 3-1  GWLRC模型下的螺旋解........................16
§ 3-2  邊界條件...................................18
§ 3-3  穩定性條件.................................20
第四章	 長細螺旋桿的彈性性質與穩定性............22 
§ 4-1  固定超螺旋度時外力與螺旋桿伸長的關係.......22
§ 4-2  固定外力時超螺旋度與螺旋桿伸長的關係.......30
第五章    結論....................................35
參考文獻..........................................38

圖表目錄
圖(2-1) 長度為L的平面曲線...............................7
圖(2-2) 空間曲線.........................................7
圖(2-3) 曲率與扭曲的意義.................................9
圖(4-1)  時的情況....................................23
圖(4-2)  增加時的情況...................................24
圖(4-3)  而且 是虛數時的情況........................25
圖(4-4)  而且 是實數時的情況........................27
圖(4-5)  增加時的情況...................................28
圖(4-6)  和 的值是小的情況..........................29
圖(4-7)  的情況......................................31
圖(4-8)  ，而且 的值是小時，以及 的情況..........33
圖(4-9)  ，而且 的值是小時，以及 的情況..........34

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