Hayase Yumino, Aonuma Hitoshi, Takahara Satoshi, Sakaue Takahiro, Kaneko Shun'ichi, Nakanishi Hiizu
Department of Physics, Kyushu University, Fukuoka 819-0395, Japan.
Research Institute for Electronic Science, Hokkaido University, Sapporo 060-0812, Japan.
Phys Rev E. 2021 Aug;104(2-2):025005. doi: 10.1103/PhysRevE.104.025005.
Hand-crumpled paper balls involve intricate structure with a network of creases and vertices, yet show simple scaling properties, which suggests self-similarity of the structure. We investigate the internal structure of crumpled papers by the microcomputed tomography (micro-CT) without destroying or unfolding them. From the reconstructed three-dimensional (3D) data, we examine several power laws for the crumpled square sheets of paper of the sizes L=50-300 mm and obtain the mass fractal dimension D_{M}=2.7±0.1 by the relation between the mass and the radius of gyration of the balls and the fractal dimension 2.5≲d_{f}≲2.8 for the internal structure of each crumpled paper ball by the box counting method in the real space and the structure factors in the Fourier space. The data for the paper sheets are consistent with D_{M}=d_{f}, suggesting that the self-similarity in the structure of each crumpled ball gives rise to the similarity among the balls with different sizes. We also examine the cellophane sheets and the aluminium foils of the size L=200 mm and obtain 2.6≲d_{f}≲2.8 for both of them. The micro-CT also allows us to reconstruct 3D structure of a line drawn on the crumpled sheets of paper. The Hurst exponent for the root-mean-square displacement along the line is estimated as H≈0.9 for the length scale shorter than the scale of the radius of gyration, beyond which the line structure becomes more random with H∼0.5.
手工揉皱的纸球具有复杂的结构，带有褶皱和顶点网络，但却呈现出简单的缩放特性，这表明结构具有自相似性。我们通过微型计算机断层扫描（micro-CT）在不破坏或展开纸球的情况下研究揉皱纸张的内部结构。从重建的三维（3D）数据中，我们研究了尺寸为L = 50 - 300毫米的揉皱方形纸张的几个幂律，并通过球的质量与回转半径之间的关系获得质量分形维数(D_{M}=2.7±0.1)，以及通过实空间中的盒计数法和傅里叶空间中的结构因子，得到每个揉皱纸球内部结构的分形维数(2.5≲d_{f}≲2.8)。纸张的数据与(D_{M}=d_{f})一致，这表明每个揉皱球结构中的自相似性导致了不同尺寸球之间的相似性。我们还研究了尺寸为L = 200毫米的玻璃纸和铝箔，两者的(d_{f})都在(2.6≲d_{f}≲2.8)范围内。微型计算机断层扫描还使我们能够重建绘制在揉皱纸张上的线的三维结构。对于长度尺度小于回转半径尺度的情况，沿该线的均方根位移的赫斯特指数估计为(H≈0.9)，超过该尺度后，线结构变得更加随机，(H∼0.5)。
