Department of Physics, University of Massachusetts, Amherst, MA 01003, USA.
Proc Natl Acad Sci U S A. 2011 Nov 8;108(45):18227-32. doi: 10.1073/pnas.1108553108. Epub 2011 Oct 31.
The buckling and wrinkling of thin films has recently seen a surge of interest among physicists, biologists, mathematicians, and engineers. This activity has been triggered by the growing interest in developing technologies at ever-decreasing scales and the resulting necessity to control the mechanics of tiny structures, as well as by the realization that morphogenetic processes, such as the tissue-shaping instabilities occurring in animal epithelia or plant leaves, often emerge from mechanical instabilities of cell sheets. Although the most basic buckling instability of uniaxially compressed plates was understood by Euler more than two centuries ago, recent experiments on nanometrically thin (ultrathin) films have shown significant deviations from predictions of standard buckling theory. Motivated by this puzzle, we introduce here a theoretical model that allows for a systematic analysis of wrinkling in sheets far from their instability threshold. We focus on the simplest extension of Euler buckling that exhibits wrinkles of finite length--a sheet under axisymmetric tensile loads. The first study of this geometry, which is attributed to Lamé, allows us to construct a phase diagram that demonstrates the dramatic variation of wrinkling patterns from near-threshold to far-from-threshold conditions. Theoretical arguments and comparison to experiments show that the thinner the sheet is, the smaller is the compressive load above which the far-from-threshold regime emerges. This observation emphasizes the relevance of our analysis for nanomechanics applications.
近年来，物理学家、生物学家、数学家和工程师对薄膜的屈曲和起皱现象产生了浓厚的兴趣。这种兴趣的增长源于在不断减小的尺度上开发技术的需求，以及控制微小结构力学的必要性，同时也意识到形态发生过程（如动物上皮或植物叶片中出现的组织变形不稳定性）通常源于细胞片的力学不稳定性。尽管 Euler 在两个多世纪前就已经理解了单轴压缩板的最基本屈曲不稳定性，但最近对纳米级（超薄）薄膜的实验表明，标准屈曲理论的预测存在显著偏差。受此难题的启发，我们在这里引入了一个理论模型，该模型允许对远离不稳定性阈值的薄片进行系统的褶皱分析。我们专注于 Euler 屈曲的最简单扩展，该扩展表现出有限长度的褶皱——在轴对称拉伸载荷下的薄片。这项研究最早归因于 Lame，我们可以通过它来构建一个相图，该相图展示了从接近阈值到远离阈值条件下褶皱模式的剧烈变化。理论论证和与实验的比较表明，薄片越薄，在远离阈值的情况下出现的压缩载荷就越小。这一观察结果强调了我们的分析对于纳米力学应用的重要性。
