Stretching Hookean ribbons part I: relative edge extension underlies transverse compression and buckling instability
Stretching Hookean ribbons part I: relative edge extension underlies transverse compression and buckling instability
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DOI：
10.1140/epje/s10189-021-00092-z
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发表时间：
2020-12
期刊：
影响因子：
--
通讯作者：
M. Xin;B. Davidovitch
中科院分区：
文献类型：
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作者：
M. Xin;B. Davidovitch
摘要拉伸矩形薄板时产生的褶皱图案引起了"极限力学"界的极大兴趣。然而，这一显著现象的关键方面仍然难以捉摸。具体地说，平面状态不稳定性背后的压应力的来源是什么？随之而来的分歧的性质是什么？形状如何从临界的、接近阈值的状态演变成渗透大部分片材的平行褶皱的充分发展的图案？在本文中，我们解决了一些这些问题，通过数值模拟和分析研究的平面状态在胡克表。我们表明，横向压缩是一种边界效应，这源于相对于横向收缩，压缩自由的大部分片材的夹紧边缘的相对延伸，并绘制这种边缘诱导的压缩和莫法特涡在粘性，空腔驱动流之间的类比。接下来，我们解决的平面状态的不稳定性，并表明它会引起一个屈曲模式，本地化的夹紧边缘附近，它的演变-在增加拉伸载荷-皱纹侵入未压缩部分的片材。最重要的是，我们证明了这个过程的关键方面--从横向压缩区的形成，到随之而来的平面状态的不稳定性和褶皱图案的出现--可以在胡克框架内理解，其中非线性响应的唯一起源是几何的，而不是非胡克应力应变关系。图形摘要
AbstractThe wrinkle pattern exhibited upon stretching a rectangular sheet has attracted considerable interest in the “extreme mechanics” community. Nevertheless, key aspects of this notable phenomenon remain elusive. Specifically—what is the origin of the compressive stress underlying the instability of the planar state? what is the nature of the ensuing bifurcation? how does the shape evolve from a critical, near-threshold regime to a fully developed pattern of parallel wrinkles that permeate most of the sheet? In this paper we address some of these questions through numerical simulations and analytic study of the planar state in Hookean sheets. We show that transverse compression is a boundary effect, which originates from the relative extension of the clamped edges with respect to the transversely contracted, compression-free bulk of the sheet, and draw analogy between this edge-induced compression and Moffatt vortices in viscous, cavity-driven flow. Next, we address the instability of the planar state and show that it gives rise to a buckling pattern, localized near the clamped edges, which evolves—upon increasing the tensile load—to wrinkles that invade the uncompressed portion of the sheet. Crucially, we show that the key aspects of the process—from the formation of transversely compressed zones, to the consequent instability of the planar state and the emergence of a wrinkle pattern—can be understood within a Hookean framework, where the only origin of nonlinear response is geometric, rather than a non-Hookean stress–strain relation.Graphic abstract