Department of Chemical & Biological Engineering, University of Wisconsin-Madison, 1415 Engineering Drive, Madison, WI 53706, USA.
Soft Matter. 2017 Apr 5;13(14):2620-2633. doi: 10.1039/c6sm02113a.
We study the dynamics of piecewise rigid sheets containing predefined crease lines in shear flow. The crease lines act like hinge joints along which the sheet may fold rigidly, i.e. without bending any other crease line. We choose the crease lines such that they tessellate the sheet into a two-dimensional array of parallelograms. Specifically, we focus on a particular arrangement of crease lines known as a Miura-pattern in the origami community. When all the hinges are fully open the sheet is planar, whereas when all are closed the sheet folds over itself to form a compact flat structure. Due to rigidity constraints, the folded state of a Miura-sheet can be described using a single fold angle. The hinged sheet is modeled using the framework of constrained multibody systems in the absence of inertia. The hydrodynamic drag on each of the rigid panels is calculated based on an inscribed elliptic disk, but intra-panel hydrodynamic interactions are neglected. We find that when the motion of a sheet remains symmetric with respect to the flow-gradient plane, after a sufficiently long time, the sheet either exhibits asymptotically periodic tumbling and breathing, indicating approach to a limit cycle; or it reaches a steady state by completely unfolding, which we show to be a half-stable node in the phase space. In the case of asymmetric motion of the sheet with respect to the flow-gradient plane, we find that the terminal state of motion is one of - (i) steady state with a fully unfolded or fully folded configuration, (ii) asymptotically periodic tumbling, indicating approach to a limit cycle, (iii) cyclic tumbling without repetition, indicating a quasiperiodic orbit, or (iv) cyclic tumbling with repetition after several cycles, indicating a resonant quasiperiodic orbit. No chaotic behavior was found.
我们研究了含有预定义折线的分段刚片在剪切流中的动力学。这些折线充当了铰链，使得薄片可以沿着这些铰链刚性地折叠，即在不弯曲任何其他折线的情况下折叠。我们选择的折线使薄片能够被分割成二维平行四边形阵列。具体来说，我们关注的是折纸社区中一种特殊的折线排列，即三浦折叠。当所有的铰链完全打开时，薄片是平面的，而当所有的铰链都关闭时，薄片会折叠起来形成一个紧凑的平面结构。由于刚性约束，三浦薄片的折叠状态可以用单个折叠角来描述。在没有惯性的情况下，使用约束多体系统的框架来模拟带铰链的薄片。基于内接椭圆盘，计算每个刚性面板的水动力阻力，但忽略面板内的水动力相互作用。我们发现，当薄片的运动相对于流-梯度平面保持对称时，经过足够长的时间后，薄片要么表现出渐近周期性的翻滚和呼吸，表明接近极限环；要么通过完全展开达到稳定状态，我们证明这是相空间中的半稳定节点。对于薄片相对于流-梯度平面的不对称运动，我们发现运动的最终状态是：（i）完全展开或完全折叠的稳定状态，（ii）渐近周期性的翻滚，表明接近极限环，（iii）无重复的周期性翻滚，表明准周期轨道，或（iv）经过几个周期后重复的周期性翻滚，表明共振准周期轨道。没有发现混沌行为。
