How much energy does it take to stamp a thin elastic shell flat? Motivated by recent experiments on the wrinkling patterns formed by thin shells floating on a water bath, we develop a rigorous method via $\Gamma$-convergence for answering this question to leading order in the shell's thickness and other small parameters. The experimentally observed patterns involve regions of well-defined wrinkling alongside other "disordered" regions where the local features of the patterns can, for thin enough shells, depend on the history of the experiment. Our goal is to explain the appearance and lack thereof of such "wrinkling domains". Rescaling by the energy of a typical wrinkling pattern, we derive a limiting area problem that asks (in an appropriately linearized way) to cover up as much area as possible in the plane with a length-shortening map. Convex analysis yields a boundary value problem characterizing optimal patterns via their effective strains or, what is equivalent, their "defect measures". Optimal defect measures are in general non-unique. Nevertheless, in some cases their restrictions to certain regions are uniquely determined and solution formulas exist. In this way, we can deduce from the principle of minimum energy the formation of the observed wrinkling domains.

中文翻译：

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薄弹性壳的曲率驱动起皱

将薄的弹性壳压平需要多少能量？受最近关于漂浮在水浴上的薄壳形成的皱纹图案的实验的启发，我们通过 $\Gamma$-convergence 开发了一种严格的方法来回答这个问题，以在壳的厚度和其他小参数中进行引导。实验观察到的图案涉及与其他“无序”区域一起定义明确的皱纹区域，其中图案的局部特征对于足够薄的壳，取决于实验的历史。我们的目标是解释这种“起皱域”的出现和缺乏。通过典型皱纹模式的能量重新缩放，我们推导出一个限制区域问题，该问题要求（以适当的线性化方式）用长度缩短图覆盖平面中尽可能多的区域。凸分析产生了一个边界值问题，通过它们的有效应变或等效的“缺陷度量”来表征最佳模式。最佳缺陷测量通常是非唯一的。然而，在某些情况下，它们对某些区域的限制是唯一确定的，并且存在解决方案。这样，我们可以从最小能量原理推导出观察到的起皱域的形成。在某些情况下，它们对某些区域的限制是唯一确定的，并且存在解决方案。这样，我们可以从最小能量原理推导出观察到的起皱域的形成。在某些情况下，它们对某些区域的限制是唯一确定的，并且存在解决方案。这样，我们可以从最小能量原理推导出观察到的起皱域的形成。