论文标题

拉伸胡胡丝带第二部分:从屈曲不稳定到远距离阈值皱纹图案

Stretching Hookean ribbons Part II: from buckling instability to far-from-threshold wrinkle pattern

论文作者

Xin, Meng, Davidovitch, Benny

论文摘要

当纵向拉伸载荷引起横向压缩时,我们解决了拉伸矩形纸板时形成的完全发达的皱纹图案,远远超过了纯平面变形的稳定性阈值。在这个“远离阈值”参数状态下,这是著名的Cerda-Mahadevan(CM)模型的主题,皱纹模式在整个板的长度上扩展,并且散布的特征波长远小于其宽度。在此设置中,采用表面Evolver模拟在一系列薄板厚度和拉伸载荷上采用了远距离阈值框架的理论基础。我们表明,皱纹的演变与横向压缩应力的崩溃而不是消失的横向应变,因此应力场渐近地接近无压缩极限,可以通过张力场理论来描述。我们通过模拟具有有限拉伸模量但没有弯曲刚度的胡克恩纸来计算无压缩应力场,并表明这种奇异极限封装了在物理,高度弯曲的床上振幅波长比率的几何非线性,即使实际的应变是如此小,即使是如此之小的钩子,也可能是如此小。最后,我们重新审视弯曲和拉伸能量的平衡,从而产生有利的皱纹波长,并研究波长对拉伸负荷以及纸张的厚度和长度的依赖性。

We 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 (CM) model, 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 set-up. We show that the evolution of wrinkles comes in tandem with collapse of transverse compressive stress, rather than vanishing transverse strain, 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.

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