We study whether and how the energy scalings based on the single-ridgeapproximation are revised in an actual crumpled sheet; namely, in the presenceof ridge-ridge interactions. Molecular Dynamics Simulation is employed for thispurpose. In order to improve the data quality, modifications are introduced tothe common protocol. As crumpling proceeds, we find that the average storingenergy changes from being proportional to one-third of the ridge length to alinear relation, while the ratio of bending and stretching energies decreasesfrom 5 to 2. The discrepancy between previous simulations and experiments onthe material-dependence for the power-law exponent is resolved. We furtherdetermine the averaged ridge length to scale linearly with the crumpled ballsize $R$, the ridge number as $1/R^2$, and the average storing energy per unitridge length as $1/R^{2.364\sim 2.487}$. These results are consistent with themean-field predictions. Finally, we extend the existent simulations to thehigh-pressure region for completeness, and verify the existence of a newscaling relation that is more general than the familiar power law at coveringthe whole density range.
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机译：我们研究在实际的皱褶纸中是否以及如何修改基于单脊线近似的能量缩放；也就是说，在存在脊-脊相互作用的情况下。为此，采用了分子动力学模拟。为了提高数据质量，对通用协议进行了修改。随着折皱的进行，我们发现平均储能从正比于脊长的三分之一变为线性关系，而弯曲和拉伸能的比值从5降低到2。以前的模拟和实验在材料依赖性上的差异幂律指数得到解决。我们进一步确定平均脊长以与皱缩的球形尺寸$ R $，脊数为$ 1 / R ^ 2 $以及每单位脊长的平均存储能量为$ 1 / R ^ {2.364 \ sim 2.487} $线性缩放。这些结果与主题领域的预测是一致的。最后，为了完整起见，我们将现有的模拟扩展到高压区域，并在覆盖整个密度范围的情况下，验证了存在比熟悉的幂定律更笼统的新比例关系。
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