SIAM Journal on Applied Mathematics, Volume 80, Issue 3, Page 1083-1100, January 2020.
Transverse wrinkles are known to appear in thin rectangular elastic sheets when stretched in the long direction. Numerically computed bifurcation diagrams for extremely thin, highly stretched films indicate entire orbits of wrinkling solutions; cf. Healey, Li, and Cheng [J. Nonlinear Sci., 23 (2013), pp. 777--805]. These correspond to arbitrary phase shifts of the wrinkled pattern in the transverse direction. While such behavior is normally associated with problems in the presence of a continuous symmetry group, an unloaded rectangular sheet possesses only a finite symmetry group. In order to understand this phenomenon, we consider a simpler problem more amenable to analysis---a finite-length beam on a nonlinear softening foundation under axial compression. We first obtain asymptotic results via amplitude equations that are valid as a certain nondimensional beam length becomes sufficiently large. We deduce that any two phase shifts of a solution differ from one another by exponentially small terms in that length. We validate this observation with numerical computations, indicating the presence of solution orbits for sufficiently long beams. We refer to this as “hidden asymptotic symmetry.”

中文翻译：

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长弹性结构中的隐藏渐近对称性

SIAM应用数学杂志，第80卷，第3期，第1083-1100页，2020年1月。
当在长方向上拉伸时，已知在薄的矩形弹性片中会出现横向皱纹。极薄，高拉伸薄膜的数值分叉图表明了皱纹溶液的整个轨道。cf. Healey，Li和Cheng [J. 非线性科学，23（2013），第777--805页。这些对应于皱纹图案在横向方向上的任意相移。尽管这种行为通常与存在连续对称组的问题有关，但未加载的矩形板仅具有有限对称组。为了理解这种现象，我们考虑一个更易于分析的简单问题-轴向压缩下在非线性软化基础上的有限长梁。我们首先通过振幅方程获得渐近结果，该幅方程在一定的无量纲光束长度变得足够大时有效。我们推断出，解决方案的任何两个相移在该长度上相差很小。我们用数值计算验证了这一观察结果，表明存在足够长的光束的解轨道。我们称此为“隐藏渐近对称”。