我們對三維單張紙的揉皺了解很多，我們接著兩張紙的揉皺很感興趣。尤其是兩種彎曲係數差很多的材料一起揉。從模擬上我們發現混合揉皺的半徑和外力也存在冪次關係，而且其平均皺折的長度和單張揉的相差甚遠，因此，我們提出了一個「棋盤」模型。它不僅能解釋其冪次關係，還能給出對的平均皺折隨變曲係數不同所改變的趨勢。除此之外，由這簡單的模型還能得到平均皺折和體密度倒數的正比關係，此結果與模擬上所得到的結果完全吻合。
除了三維揉皺，我們也做了一維揉皺的模擬，並發現其高度和外力關係可分為三個區間。這三個區間分別對應到內部結構的三種狀態。三角晶格面的平均角度是很好的有序參數，用來幫助我們區別前兩個區間。在第二個區間，存在冪次關係，其指數約為1/3，稍大於在三維發現的值。藉由對三個區間的觀察，我們發現造成一維的冪次關係和三維的機制是完全不同的。並提出一個「橢球」模型來解釋冪次關係，並得到完全一致的次方值。

Our research group has been interested at the ubiquitous, yet still poorly understood
phenomenon of crumpling. In the past we have focused on designing high-pressure chambers
for the purpose of crumpling thin sheets under dierent magnitudes of ambient pressure. In
this thesis I shall try to attack the problem from a dierent front; that is, analytically and
by Molecular Dynamics (MD) simulations. I also want to ask (1) how these mechanical and
statistical properties derived from experiments come about? and (2) how they are aected
when two sheets of dierent composites are crumpled at the same time. Analytically, I
can show that, within a simple \chessboard" model, how the average ridge lengths for each
sheets evolve with the compaction, for which the average ridge length is positively related to
bending modulus and inversely proportional to the density under the knowledge background
from simulation.
These results are consistent with the simulation results. One big advantage of studying
MD is that we can get a clear physical picture for how the potential energy is allotted. We
nd that, during cocumpling, the sheet with larger average facet takes the major blunt of
external force and thus absorbs more energy in our simulations. In contrast to crumpling
under an ambient pressure, I also check if the scaling relation persists when I crush the sheet
unidirectionally. Unlike its 3-D counterpart, the mechanical properties of a 1-D crumpled
ball depend more sensitively on the initial condition. More specically, the compaction of
the precrumpled ball via a 3-D pressure aects its later resistance in 1-D.
We found that the force versus height relation consisted of three regions, similar to
the 3-D case. Although the ball exhibits the scaling law in 3-D, the law does not exist
when we change the nature of force to 1-D. We call this stage the rst region. When the
height decreases, the scaling law appears and the power is between 0.25 to 0.35, somewhat
larger than the exponent in 3-D for the same parameters of sheet. We reported possible
connection between these regions and the change of inner structure. With the knowledge of
these features, we considered an ellipsoid model that can predict both the power law and the
right magnitude of exponent. Furthermore, the model revealed that the density is a better
parameter than the height in the second region for the mechanical behavior. In the end, we
investigated the density distribution of layers and found two kinds of distribution, which are
dissimilar to that in 3-D crumpling.
The nal part is a summary of conclusions and discussions on my previous eort to
investigate the quantized conductance of threadlike mercury. This was initially designed to
combine my experience with the classic drop-breakup problem with the expertise of my labmate
on quantized conductance in quantum point contacts. We hope to utilize the micro-size
neck of mercury to investigate the correctness of report in the literature that the quantized
conductance can persist in room temperature and pressure for liquid metal.

[1] E. Sultan and A. Boudaoud, Phys. Rev. Lett. 96, 136103 (2006).
[2] G. A. Vliegenthart and G. Gompper, Nat Mater 5, 216 (2006).
[3] Y. C. Lin, Y. L. Wang, Y. Liu, and T. M. Hong, Phys. Rev. Lett. 101,
125504 (2008).
[4] T. Tallinen, Numerical Studies on Membrane Crumpling, PhD thesis,
University of Jyvaskyla, Finland, 2009.
[5] W. Bai, Y.-C. Lin, T.-K. Hou, and T.-M. Hong, Phys. Rev. E 82, 066112
(2010).
[6] T. A. Witten, Rev. Mod. Phys. 79, 643 (2007).
[7] A. Wood, Physica A 313, 83 (2002).
[8] C.-C. Lo, Membrane crumpling with molecular dynamical simulation,
Master's thesis, National Tsing Hua University, 2012.
[9] L. D. Landau and E. M. Lifshitz, Theory of Elasticity, 2nd (Pergamon,
Oxford, 1970).
[10] J. L. Costa-Kramer et al., Phys. Rev. B 55, 5416 (1997).
53