過去幾十年來，薄膜揉皺的力學與統計性質已被廣泛研究，包括紙團大小與外力的關係、摺痕所儲存的彎曲和拉伸位能的分配、及摺痕形變的相關統計（例如數目、平均長度與儲存位能如何隨紙團大小改變）。以前大部分的研究針對二維平面薄膜，認為三維立體薄膜的性質應該差不多，因此沒有多加著墨，然而本實驗室在數年前研究球形薄膜時，赫然發現結果相當不同，然而歸根究底，到底是球面（或廣義上，兩個垂直方向的曲率皆不為零的所謂「非高斯曲面」）造成的？還是純粹「立體」這個事實就足以引進新的物理？如果是後者，那「立體」該如何界定？揉皺過的平面薄膜算不算立體？為了回答這些問題，本研究以不是「非高斯曲面」的紙盒為對象，先檢驗立體與平面薄膜的差異，再探討幾何特徵如何影響立體揉皺的力學性質。
　　我們先採用分子動力學模擬（Molecular Dynamics simulation），證實了有些平面揉皺的性質不適用於紙盒揉皺，例如：(1)彎曲和拉伸位能的比值與體積密度的關係，根據Witten的風箏模型（kite model）與實驗證實，該比值在平面揉皺早期會等於5；但是在紙盒或球殼卻單調遞增，不是常數。(2)受力與體積密度的關係，平面與球形薄膜的受力曲線在揉皺早期存在冪次關係（power law），但是其他立體揉皺的受力曲線卻不。(3)形變演化的相關統計，平面與球殼揉皺的形變數目皆隨揉皺過程增加，且形變尺寸也持續縮小；而紙盒揉皺的形變演化卻相當複雜。
　　為了探討紙盒與球殼如何在揉皺早期造成力學性質的差異，我們聚焦在單一尖角的形變，並設計一維壓縮的儀器去擠壓不同邊數與立體角的真實薄膜角錐。除了證實初期的受力曲線不存在冪次關係以外，發現擠壓位移和外力的曲線出現一個山峰（peak），我們認為它類似一維壓縮細長棍子，當壓縮力氣或形變到達臨界值時，會突然從壓縮變成彎曲形變。那平面薄膜在揉皺後的紙團不是也有許多尖錐嗎，為何不會給出這裏的山峰呢？我們預期它和前者的底部可移動有關。

Over the last few decades, crumpled membranes have been widely studied for their interesting mechanical and statistical properties, such as the power-law relation between the size of crumpled ball and pressure, the ratio of bending to stretching potential energies stored in the deformation, and statistics for the number and length of deformation. However, previous researches mainly focused on 2D flat membranes rather than more abundant 3D stereoscopic objects like pots, boxes, and cars. In this thesis, we employ Molecular Dynamics (MD) simulations to study the crumpling of a box in three dimensions and investigate whether and how the morphology of objects affects the mechanical properties. Simulation results indicated that most properties of 2D flat membranes are no longer applicable to 3D stereoscopic objects, including (1) the constant ratio of bending to stretching energies, (2) power-law relation between the volume density and pressure, and (3) time evolution of deformation.
To investigate and identify which part of the morphology is mainly responsible for the discrepancies, we focused our attention on and arranged experiments to study the linear crumpling of a single corner, as taken from the box. It turned out that the missing of power-law relation in mechanical responses for stereoscopic objects has to do with the standard compression-to-bending transition for a long rod taught in mechanics of materials 101. We were able to generalize the calculations to a hollow pyramid and predict how the transition is affected by the characterization of pyramid, such as the number of sides and tilt angle. Note that there also exists many vertices or sharp corners on a crumpled flat membrane. An interesting question to ask is why they do not give rise to such a peak in their strain-stress relation? We believe it has something to do with the base of these latter corners not being stationary.

致謝-----------------------------------------------------------------3
摘要-----------------------------------------------------------------5
Abstract-------------------------------------------------------------6
目錄-----------------------------------------------------------------7
第一章 簡介-----------------------------------------------------------8
1.1 研究動機---------------------------------------------------------8
1.2相關文獻---------------------------------------------------------10
1.2.1 Witten’s lecture----------------------------------------------10
1.2.2 Crumpling a thin sheet----------------------------------------11
1.2.3 Forced crumpling of self-avoiding elastic sheets--------------13
1.2.4 Crumpling under an ambient pressure---------------------------14
1.2.5 Scaling relation for a compact crumpled thin sheet------------16
1.2.6 Effect of ridge-ridge interactions in crumpled thin sheets----18
1.2.7 Crumpling an elastoplastic thin sphere------------------------21
第二章 實驗原理、器材與操作流程---------------------------------------23
2.1 分子動力學模擬原理-----------------------------------------------23
2.1.1 粗粒化模型與系統能量形式----------------------------------------23
2.1.2 Nosé–Hoover thermostat----------------------------------------25
2.2 實驗器材--------------------------------------------------------26
2.3 實驗操作流程-----------------------------------------------------27
第三章 實驗結果與討論------------------------------------------------28
3.1 紙盒揉皺的物理性質-----------------------------------------------28
3.1.1紙盒能量比值與擠壓程度之關係-------------------------------------28
3.1.2紙盒受力與擠壓程度之關係-----------------------------------------29
3.1.3紙盒形變演化與邊界缺陷-------------------------------------------30
3.2稜角形變---------------------------------------------------------33
3.2.1紙盒稜角形變----------------------------------------------------33
3.2.2 角錐揉皺------------------------------------------------------34
第四章 結論與討論----------------------------------------------------38
參考資料------------------------------------------------------------39

[1] Wen Jiang, Juan Chang, Joanita Jakana, Peter Weigele, Jonathan King, and Wah Chiu, Structure of epsilon15 bacteriophage reveals genome organization and DNA packaging/ injection apparatus, Nature 439, 612 (2006).
[2] Eytan Katzav, Mokhtar Adda-Bedia, and Arezki Boudaoud, A statistical approach to close packing of elastic rods and to DNA packaging in viral capsids, Proc. Natl. Acad. Sci. U.S.A. 103, 18900 (2006).
[3] David Nelson, Tsvi Piran, and Steven Weinberg, Statistical Mechanics of Membranes and Surfaces, (World Scientific Publishing Co. Pte. Ltd, USA, 2004).
[4] A. Jamie Wood, Witten’s lectures on crumpling, Physica A 313, 83-109 (2002).
[5] Y. C. Lin, Y. L. Wang, Y. Liu, and T. M. Hong, Crumpling under an Ambient Pressure, Phys. Rev. Lett. 101, 125504 (2008).
[6] Wubin Bai, Yen-Chih Lin, Tzon-Kun Hou, and Tzay-Ming Hong, Scaling relation for a compact crumpled thin sheet, Phys. Rev. E. 82, 066112 (2010).
[7] Shiuan-Fan Liou, Chun-Chao Lo, Ming-Han Chou, Pai-Yi Hsiao, and Tzay-Ming Hong, Effect of ridge-ridge interactions in crumpled thin sheets, Phys. Rev. E. 89, 022404 (2014).
[8] Hung-Chieh Fan Chiang, Li-Jie Chiu, Hsin-Huei Li, Pai-Yi Hsiao, and Tzay-Ming Hong, Crumpling an elastoplastic thin sphere, Phys. Rev. E. 103, 012209 (2021).
[9] Kittiwit Matan, Rachel B. Williams, Thomas A. Witten, and Sidney R. Nagel, Crumpling a Thin Sheet, Phys. Rev. Lett. 88, 076101 (2002).
[10] Gerrit A. Vliegenthart, and Gerhard Gompper, Force crumpling of self-avoiding elastic sheets, Nature Mater. 5, 216-221 (2006).
[11] William G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A. 31, 1695 (1985).