Mathematical Institute, Radcliffe Observatory Quarter, Woodstock Road, Oxford, OX2 6GG, UK.
Department of Physics, University of Massachusetts Amherst, Amherst, MA 01003, USA.
Soft Matter. 2017 Mar 15;13(11):2264-2278. doi: 10.1039/c6sm02451c.
We study the indentation of ultrathin elastic sheets clamped to the edge of a circular hole. This classical setup has received considerable attention lately, being used by various experimental groups as a probe to measure the surface properties and stretching modulus of thin solid films. Despite the apparent simplicity of this method, the geometric nonlinearity inherent in the mechanical response of thin solid objects renders the analysis of the resulting data a nontrivial task. Importantly, the essence of this difficulty is in the geometric coupling between in-plane stress and out-of-plane deformations, and hence is present in the behaviour of Hookean solids even when the slope of the deformed membrane remains small. Here we take a systematic approach to address this problem, using the membrane limit of the Föppl-von-Kármán equations. This approach highlights some of the dangers in the use of approximate formulae in the metrology of solid films, which can introduce large errors; we suggest how such errors may be avoided in performing experiments and analyzing the resulting data.
我们研究了夹在圆形孔边缘的超薄弹性片的压痕。这个经典的设置最近受到了相当多的关注，被各种实验组用作探针来测量薄膜的表面性质和拉伸模量。尽管这种方法表面上很简单，但薄固体物体的机械响应中的几何非线性使得分析由此产生的数据成为一项复杂的任务。重要的是，这种困难的本质在于面内应力和面外变形之间的几何耦合，因此即使在变形膜的斜率仍然很小时，胡克固体的行为也存在这种耦合。在这里，我们采用系统的方法来解决这个问题，使用 Föppl-von-Kármán 方程的膜极限。这种方法强调了在固体薄膜计量中使用近似公式的一些危险，这些公式可能会引入大的误差；我们建议在进行实验和分析所得到的数据时如何避免这种误差。
