我們利用歐拉角推導出了一根橫切面為圓形，但是無自發扭曲的均勻彈性杆之形狀方程式。我們證明了兩端閉合的細杆通常存在著平面形曲線形的解。我們探討了在外力與外力矩下，形成螺旋杆的邊界條件暨實驗條件。我們發現要形成螺旋形，Euler角 必須為常數並由自然曲率所決定。我們研究了螺旋杆在外力和外力矩下的彈性性質。我們嚴格證明了當力矩為零時，在改變外力時螺旋杆的伸長不會有突然跳躍的現象。這行為與一根有自發扭曲的均勻彈性杆之行為相當不同，並解釋了為何在巨觀力學實驗中難以觀察到這種跳躍。然而在非零值的固定外力矩下，改變外力則螺旋的伸長可能會有一次突然的轉變。

We derive the shape equations in terms of Euler angles for a uniform elastic filament with circular cross section but free of spontaneous torsion. We show that in general there are planar curve solutions for a closed rod. We study the boundary conditions (i.e., experimental conditions in a force experiment) to form a helical filament under external force and twisting. We find that to form a helix, the Euler angle must be a constant determined by the spontaneous curvatures. We study the elasticity of a helical filament under different conditions. We find that the extension of a helix under fixed and finite torque may subject to a one-step sharp transition with increasing stretching force. However, we show exactly that there is not jump of extension for a helical filament free of external torque. This behavior is quite different from a uniform elastic rod with circular cross section and spontaneous torsion, and provides another very important reason why one cannot observe the sharp jump of extension for most macroscopic helical springs.

第一章    緒論                                            1
   § 1-1  歷史簡介 ....................................   1
   § 1-2  動機與目的 ..................................   2
   § 1-3  主要內容 ....................................   5
第二章    模型                                             9
   § 2-1  一般模型 ....................................   9
   § 2-2  本工作考慮的模型 ............................  12
第三章    相關長杆的解析解                               14
   § 3-1  兩端封閉之長杆的形狀方程及幾個簡單的解析解     14
   § 3-2  兩端開放之長杆的形狀方程、邊界條件             15
   § 3-3  兩端開放之長杆的形狀方程之螺旋解               17
第四章    固定外力矩時螺旋杆的彈性性質                   24
   § 4-1   固定外力矩時外力與相對伸長的關係 ...........  24
   § 4-1-1 當外力矩為零時，外力與相對伸長的關係 .......  24
   § 4-1-2 當外力矩與自然曲率為零時，外力與伸長的關係 .  28
   § 4-1-3 當外力矩不為零時，外力與相對伸長的關係 .....  28
   § 4-1-4 當外力矩大於零時，彎曲(扭曲)剛度對伸長的影    31
第五章    固定超螺旋度時螺旋杆的彈性性質                 35
   § 5-1   固定超螺旋度時外力與相對伸長的關係 .......... 35
第六章    固定外力時螺旋杆的彈性性質                     39
   § 6-1   固定外力時外力矩與相對伸長的關係 ............ 39
   § 6-2   固定外力時超螺旋度與相對伸長的關係 .......... 43
第七章    結論                                           46
參考文獻 ............................................... 49

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