Stretching Hookean ribbons part II: from buckling instability to far-from-threshold wrinkle pattern
Stretching Hookean ribbons part II: from buckling instability to far-from-threshold wrinkle pattern
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DOI：
10.1140/epje/s10189-021-00088-9
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发表时间：
2020-12
期刊：
影响因子：
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通讯作者：
M. Xin;B. Davidovitch
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文献类型：
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作者：
M. Xin;B. Davidovitch
当纵向拉伸载荷引起的横向压缩远远超过纯平面变形的稳定性阈值时，我们讨论了拉伸Hookean矩形板时形成的充分发展的褶皱图案。在该“远离阈值”的参数状态下，其是著名的Cerda-Mahadevan模型（塞尔达和Mahadevan，Phys Rev Lett 90：074302，2003）的主题，褶皱图案在整个片材的长度上扩展，并且起伏的特征波长远小于其宽度。采用表面进化模拟在一定范围内的片材厚度和拉伸载荷，我们阐明了在这种设置的理论基础的远离阈值的框架。我们表明，皱纹的演变来串联与横向压缩应力的崩溃，而不是消失的横向应变（这是假设由塞尔达和Mahadevan在物理评论快报90：074302，2003年），使应力场渐近接近无压缩极限，可描述的张力场理论。我们计算无压缩应力场通过模拟胡克片，具有有限的拉伸模量，但没有弯曲刚度，并表明这种奇异的限制封装的几何非线性的振幅-波长比的皱纹图案在物理，高度可弯曲的片材，即使实际的应变可能是如此之小，局部力学是完美的胡克。最后，我们重新审视了产生有利褶皱波长的弯曲和拉伸能量的平衡，并研究了波长对拉伸载荷以及板材厚度和长度的依赖性。图形摘要
AbstractWe address the fully developed wrinkle pattern formed upon stretching a Hookean, rectangular-shaped sheet, when the longitudinal tensile load induces transverse compression that far exceeds the stability threshold of a purely planar deformation. At this “far-from-threshold” parameter regime, which has been the subject of the celebrated Cerda–Mahadevan model (Cerda and Mahadevan in Phys Rev Lett 90:074302, 2003), the wrinkle pattern expands throughout the length of the sheet and the characteristic wavelength of undulations is much smaller than its width. Employing Surface Evolver simulations over a range of sheet thicknesses and tensile loads, we elucidate the theoretical underpinnings of the far-from-threshold framework in this setup. We show that the evolution of wrinkles comes in tandem with collapse of transverse compressive stress, rather than vanishing transverse strain (which was hypothesized by Cerda and Mahadevan in Phys Rev Lett 90:074302, 2003), such that the stress field approaches asymptotically a compression-free limit, describable by tension field theory. We compute the compression-free stress field by simulating a Hookean sheet that has finite stretching modulus but no bending rigidity, and show that this singular limit encapsulates the geometrical nonlinearity underlying the amplitude–wavelength ratio of wrinkle patterns in physical, highly bendable sheets, even though the actual strains may be so small that the local mechanics is perfectly Hookean. Finally, we revisit the balance of bending and stretching energies that gives rise to a favorable wrinkle wavelength, and study the consequent dependence of the wavelength on the tensile load as well as the thickness and length of the sheet.Graphic abstract