本論文包含兩個主題：
第一部分：圓柱薄膜形變的力學與統計性質
　　受擠壓的圓柱在工程上有廣泛的應用，如用於支撐的實心樑柱，而堆疊的寶特瓶和回收時壓扁的空罐則是中空的例子，它們造成的形變截然不同。綜合這兩者的日常現象為捲袖子或行走時的褲管，此時空心圓柱內部多了一個有間隙的實心核，研究的人相對比較少。經驗中有許多相關的性質值得探討，例如相較於反摺，捲起袖子所產生的結構十分鬆散，自然界是否有相似的情況？這兩種結構有相同的長度但其中的能量差別又是多少？這些問題都尚未有人討論。
　　為了方便釐清各個參數（例如外殼薄膜的厚度和楊氏係數、實心核的截面積、間隙大小、甚至塑性多寡）的作用，我們使用分子動力學來模擬圓柱薄膜的受力擠壓，在得到初步的應力-應變關係、形變種類和相圖後，再配合實驗來確認。我們發現這個系統的形變非常豐富，根據形貌上的結構，大致分成五類；藉由控制薄膜的厚度和楊氏係數，及它與內核的間隙寬度，可以繪製出不同形變結構的相圖。在眾多有趣的性質中，我們特別指出全彈性的材料在受到擠壓後，會自發性反摺來釋放能量，原先的皺褶結構也因此恢復平滑。此一現象在加入塑性後消失，且力學反應也從遞增變為在定值附近震盪。

第二部分：軟包裝釋放顆粒流之物理現象
　　顆粒的流動和釋放並不只限於沙漏或土石流，在過去被用來改善穀倉的設計與工業包裝，如綜合堅果、麥片等。過去討論的系統邊界大都模擬鐵板或塑膠罐的硬牆(solid wall)，得到的結論，諸如受震盪產生顆粒分層現象的（反）巴西堅果效應、穀倉中層層堆積的穀粒在一定高度後維持壓力定值的Janssen’s law等，是否依然適用如塑膠袋、紙袋等的軟牆壁，是我們想釐清的課題。
　　本研究利用分子動力學模擬軟牆壁對顆粒流釋放的影響，以硬牆壁做為對照，發現過去將顆粒流簡單分成漏斗流（funnel flow）與mass flow的情況已不再適用，且流場的範圍也會隨著流動而左右擺盪，這是過去沒有發現過的現象。而軟牆壁在裝了顆粒後的形變，隨著釋放顆粒後減少，對顆粒出力大小都不同，導致過去硬牆所得的結論僅能在定性上勉強相符。而顆粒高度與壓力的關係遠比流體複雜得多。

This thesis is composed of two parts which are connected by their being nonlinear phenomena that I studied out of curiosity and to sharpen my skills in Molecular Dynamics simulation. In the process, I teamed up with labmates who would perform experiments to compare with and confirm my predictions.
Part I: Compressed cylindrical shell with a rigid core and a gap
　　Examples of compressed cylinders are abound around us, e.g., columns in architecture structure and enginerring, squashed aluminum cans before recycling, etc. Depending on whether the cylinder is solid, hollow or in between, the deformation under compression can be very different. We shall study the less discussed case of a cylinder with a gapped core in this thesis. A ready example is the roll-up sleeves or trousers. There are many interesting and practical problems we can ask about these systems. For instance, why does a casual push or stack on our sleeves often result in a frabby structure, as opposed to the more stable one by folding? Are there analogies or applications in nature? And what are its applications?
　　By combining Molecular Dynamics simulations and experiments, we conclude that the deformations can roughly be classified into five categories, depending on the thickness and Young’s modulus of shell and the gap width. Besides plotting them in a mode diagram, we find that these results are sensitive to plasticity. For example, while an elastic material can spontaneously generate sagging during compression, the deformation quickly gets wiped out and the stored energy released upon being released. To counter this phenomenon, plasticity is crucial.

Part II: Discharging granular system with soft boundary
　　How a granular system discharges is the central theme and concern in the design of silo and for packing industries of nuts and cereal. Past discussions mainly focused on a solid wall as is appropriate for an iron board or plastic can. However, other systems may exhibit a soft boundary, e.g., a plastic or paper bag. It worths investigating whether the conclusions drawn from assuming solid walls still apply to a soft boundary. Notable properties include the segregation phenomena of common and inverse Brazil nuts effects that are triggered by vibrations, and the Janssen’s law that predicts a saturation for pressure as depth increases for grains in a silo.
　　As in Part I except requiring quite a few technical modifications, we study the discharging granular system via Molecular Dynamics simulations, focusing on the comparison of a solid and soft wall. The classification of discharging flow into funnel flow and mass flow is found to be no longer useful. In addition, the range of velocity field changes and oscillates during discharge. Furthermore, the soft boundary allows an intimate correlation between the varying shape and the force acting on the grains, which leads to qualitatively different properties from the hard-wall case. And the relation between depth and local pressure is also much more complicated than the fluid case.

摘要 2
Abstract 4
第一部分 圓柱薄膜形變的力學與統計性質 10
第一章 簡介 10
1.1 研究動機 10
1.2 相關文獻 11
1.2.1 Stress-Dependent Finite Growth in Soft Elastic Tissues 11
1.2.2 Pattern Selection in Growing Tubular Tissues 12
1.2.3 Pattern Transitions in a Soft Cylindrical Shell 14
第二章 實驗原理 15
2.1 實驗設置 15
2.1.1 實驗設置 15
2.2 模擬實驗原理 16
2.2.1 粗粒化模型 16
2.2.2 Lennard-Jones potential 16
2.2.3 Langevin equation 17
2.2.4 粒子間的能量形式 18
第三章 實驗結果與討論 19
3.1 模擬實驗與真實薄膜的對比 19
3.2 能量與長度比例之關係 21
3.3 薄膜厚度與形變之關係 26
3.4 薄膜受力與擠壓長度之關係 32
3.5 薄膜的形變種類與擠壓長度之關係 39
3.6 薄膜受擠壓後釋放的力學性質 43
第四章 結論與討論 48
第二部分 軟包裝釋放顆粒流之物理現象 49
第五章 簡介 49
5.1 研究動機 49
5.2 相關文獻 50
5.2.1 Effects of horizontal vibration on hopper flows of granular materials 50
5.2.2 Effects of vertical vibration on hopper flows of granular material 53
5.2.3 Evolution of pressure profiles during the discharge of a silo 55
5.2.4 Three-dimensional discrete element simulations in hoppers and silos 58
5.2.5 Pattern Formation in Flowing Sand 60
5.2.6 Beyond Stevin's law: the Janssen effect 62
5.2.7 Continuum modeling of granular segregation during hopper discharge 65
第六章 實驗原理 66
6.1 模擬實驗原理 66
6.1.1 Hertz theory 66
6.1.2 Voronoi Cell 67
6.2 模擬實驗設置 68
6.2.1 固體牆邊界設定 68
6.2.2 可形變牆邊界設定 68
第七章 模擬實驗結果與討論 70
7.1 模擬實驗設置的可信度 70
7.1.1 顆粒流 70
7.1.2 Janssen effect 72
7.2 可形變牆壁之釋放顆粒流 75
7.3垂直震盪中可形變容器的釋放顆粒流 86
7.4 Janssen Effect Revisited 93
7.4.1 靜止與填充的圓柱系統（Silo） 93
7.4.2 流動圓柱系統（Silo） 110
7.4.3 靜止與填充的斜面系統（Hopper） 117
7.4.4 流動的斜面系統（Hopper） 120
7.4.5 流動系統中的震盪波 129
第八章 結論與討論 140
參考資料 141

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