Bella Peter, Kohn Robert V
Institute of Mathematics, Leipzig University, Augustusplatz 10, 04109 Leipzig, Germany
Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY 10012, USA.
Philos Trans A Math Phys Eng Sci. 2017 May 13;375(2093). doi: 10.1098/rsta.2016.0157.
We consider a disc-shaped thin elastic sheet bonded to a compliant sphere. (Our sheet can slip along the sphere; the bonding controls only its normal displacement.) If the bonding is stiff (but not too stiff), the geometry of the sphere makes the sheet wrinkle to avoid azimuthal compression. The total energy of this system is the elastic energy of the sheet plus a (Winkler-type) substrate energy. Treating the thickness of the sheet as a small parameter, we determine the leading-order behaviour of the energy as tends to zero, and we give (almost matching) upper and lower bounds for the next-order correction. Our analysis of the leading-order behaviour determines the macroscopic deformation of the sheet; in particular, it determines the extent of the wrinkled region, and predicts the (non-trivial) radial strain of the sheet. The leading-order behaviour also provides insight about the length scale of the wrinkling, showing that it must be approximately independent of the distance from the centre of the sheet (so that the number of wrinkles must increase with ). Our results on the next-order correction provide insight about how the wrinkling pattern should vary with Roughly speaking, they suggest that the length scale of wrinkling should be exactly constant-rather, it should vary slightly, so that the number of wrinkles at radius can be approximately piecewise constant in its dependence on , taking values that are integer multiples of with [Formula: see text]This article is part of the themed issue 'Patterning through instabilities in complex media: theory and applications'.
我们考虑一个与柔性球体相连的圆盘形薄弹性片。（我们的薄片可以沿着球体滑动；连接仅控制其法向位移。）如果连接是刚性的（但不太硬），球体的几何形状会使薄片起皱以避免方位向压缩。该系统的总能量是薄片的弹性能量加上一个（温克勒型）基底能量。将薄片的厚度视为一个小参数，我们确定当趋于零时能量的主导阶行为，并给出下一阶修正的（几乎匹配的）上下界。我们对主导阶行为的分析确定了薄片的宏观变形；特别是，它确定了起皱区域的范围，并预测了薄片的（非平凡的）径向应变。主导阶行为还提供了关于起皱长度尺度的见解，表明它必须大致与距薄片中心的距离无关（这样皱纹的数量必须随增加）。我们关于下一阶修正的结果提供了关于起皱模式应如何随变化的见解。大致来说，它们表明起皱的长度尺度不应完全恒定——相反，它应该略有变化，使得半径处的皱纹数量在其对的依赖关系中可以大致分段恒定，取值为的整数倍，其中[公式：见正文]本文是主题为“通过复杂介质中的不稳定性进行图案化：理论与应用”的特刊的一部分。
