以往探究薄膜揉皺的力學性質（紙團大小和壓力的關係）、摺痕長度的分佈以及所儲存的彎曲／拉伸位能比例、摺痕數目、揉皺過程所發出的噪音統計等，都只針對平面型的薄膜，並未深究薄膜具有內在曲率是否會影響這些結果。本研究以球殼為對象，嚴格檢驗非歐幾里德、封閉薄膜的揉皺性質。
　　方便和平面揉皺比較，本研究採用分子動力學模擬(MD)工具，專注在過往研究過的性質：(1)摺痕的《彎曲與拉伸位能比值》與體積密度的關係。根據針對平面揉皺的風箏模型，如果摺痕彼此不相干，這個比值可以被嚴格證明為5，而球殼的實驗結果給出的比值為1；但兩者都會在摺痕、摺點耦合時（即多體的影響），比值明顯的降低。(2)外力與紙團密度的關係，平面揉皺的情況可以透過平面面積與摺痕數目估算而得，但是球殼特有的「隕石坑」摺痕極可能使上述方式不再有效。最明顯的是平面揉皺的power law無法在球殼揉皺中得到。(3)摺痕所儲存的總位能與其長度的關係，對照風箏模型所給出的從早期的1/3冪次，到晚期的正比理論預測，與球殼揉皺早期所得的1次方關係極為不同。
　　從所得的模擬結果，可以結論平面揉皺所使用的風箏模型無法套用在球殼上。我們認為主要理由是球殼所形成的隕石坑與平面所形成的線形摺痕的差異。

　　Crumpled membranes exhibit many interesting mechanical and statistical properties. However, researchers, including us, have long focused on flat sheets and did not second-guess whether an object that carries extrinsic curvature (and/or closed, i.e., without boundaries) may behave differently when crushed. We use Molecular Dynamics (MD) simulation to study the crumple process of a thin sphere in three dimensions.
　　The first property we measure is how the ratio of bending energy and stretching energy changes with volume density of the crumpled ball. In sheet case, it can be proved to 5 but the result of sphere is 1. The same part of sheet and sphere is that the ratio of bending energy and stretching energy will decay obviously in many-body interaction stage. Second is the relation between external force and volume density. The most different part is power law can not be seen for sphere case. Third is how the average total energy stored in each crease scales with the ridge length. The power of sheet is 1/3 and sphere is 1.
　　Based on MD result, Kite model is no longer suit for sphere. The reason is the difference of deformation of sheet is ridge, but the deformation of sphere is pit.

摘要 2
Abstract 3
目錄 4
第一章 簡介 6
1.1 研究動機 6
1.2 相關文獻 7
1.2.1 Witten’s lecture 7
1.2.2 Crumpling a Thin Sheet 10
1.2.3 Crumpling under an Ambient Pressure 12
1.2.4 Scaling relation for a compact crumpled thin sheet 15
1.2.5 Forced crumpling of self-avoiding elastic sheets 18
1.2.6 The effect of plasticity in crumpling of thin sheets 19
1.2.7 Effect of Ridge-Ridge Interactions in Crumpled Thin Sheets 20
第二章 實驗原理 23
2.1 模擬實驗環境原理 23
2.1.1 Lennard-Jones potential 23
2.1.2 Langevin equation 23
2.2 模擬球殼原理 25
2.2.1 粗粒化模型 25
2.2.2 歐拉示性數(正多面體) 25
2.2.3 球殼形狀與缺陷 25
2.2.4 粒子間的能量形式 28
2.2.5 球殼的能量形式 29
第三章 實驗結果與討論 30
3.1 模擬實驗與真實球殼的對比 30
3.2 能量比值與體積密度之關係 30
3.3 球殼受力與體積密度之關係 35
3.4 摺痕長度與儲存的能量 39
3.5 不同厚度所產生之影響 43
3.6 凹坑與擠壓半徑之關係 45
第四章 結論與討論 47
參考資料 49

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