Abstract

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 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

中文翻译：

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拉伸胡克丝带第二部分：从屈曲不稳定性到远离阈值的皱纹图案

摘要

我们解决了在拉伸 Hookean 矩形片材时形成的完全发达的皱纹图案，当纵向拉伸载荷引起横向压缩远远超过纯平面变形的稳定性阈值时。在这个“远离阈值”的参数机制中，它一直是著名的 Cerda-Mahadevan 模型（Cerda 和 Mahadevan in Phys Rev Lett 90:074302, 2003）的主题，皱纹图案扩展到整个纸张的长度和起伏的特征波长远小于其宽度。在一定范围的板材厚度和拉伸载荷下使用 Surface Evolver 模拟，我们阐明了此设置中远离阈值框架的理论基础。我们表明皱纹的演变与横向压应力的坍塌同时发生，而不是消失的横向应变（这是由 Cerda 和 Mahadevan 在 Phys Rev Lett 90:074302, 2003 中假设的），使得应力场渐近接近无压缩极限，可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 Hookean 薄板来计算无压缩应力场，并表明该奇异极限包含了物理、高度可弯曲薄板中皱纹图案振幅-波长比的几何非线性，即使尽管实际应变可能很小，以至于局部力学完全是胡克式的。最后，我们重新审视产生有利皱纹波长的弯曲和拉伸能量的平衡，

图形摘要