Liquid crystal elastomers (LCEs) can undergo large reversible contractions along their nematic director upon heating or illumination. A spatially patterned director within a flat LCE sheet, thus, encodes a pattern of contraction on heating, which can morph the sheet into a curved shell, akin to how a pattern of growth sculpts a developing organism. Here, we consider theoretically, numerically, and experimentally patterns constructed from regions of radial and circular director, which, in isolation, would form cones and anticones. The resultant surfaces contain curved ridges with sharp $V$-shaped cross sections, associated with the boundaries between regions in the patterns. Such ridges may be created in positively and negatively curved variants and, since they bear Gauss curvature (quantified here via the Gauss–Bonnet theorem), they cannot be flattened without energetically prohibitive stretch. Our experiments and numerics highlight that, although such ridges cannot be flattened isometrically, they can deform isometrically by trading the (singular) curvature of the $V$ angle against the (finite) curvature of the ridge line. Furthermore, in finite thickness sheets, the sharp ridges are inevitably non-isometrically blunted to relieve bend, resulting in a modest smearing out of the encoded singular Gauss curvature. We close by discussing the use of such features as actuating linear features, such as probes, tongues, and grippers. We speculate on similarities between these patterns of shape change and those found during the morphogenesis of several biological systems.

中文翻译：

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集中高斯曲率的形状规划线

液晶弹性体 (LCE) 可以在加热或光照下沿其向列指向矢发生大的可逆收缩。因此，平面 LCE 薄片内的空间图案导向器编码了加热收缩模式，这可以将薄片变形为弯曲的外壳，类似于生长模式如何塑造发育中的有机体。在这里，我们从理论上、数值上和实验上考虑由径向和圆形导向器区域构建的图案，这些区域孤立地将形成锥体和反锥体。生成的表面包含带有锐利的弯曲脊$伏$形横截面，与图案中区域之间的边界相关联。此类脊可能以正弯曲和负弯曲变体产生，并且由于它们具有高斯曲率（此处通过 Gauss-Bonnet 定理量化），因此如果没有能量上的禁止拉伸，它们就无法变平。我们的实验和数值强调，虽然这样的脊不能等轴展平，但它们可以通过交换（单一）曲率来等轴变形。$伏$与脊线的（有限）曲率的角度。此外，在有限厚度的片材中，尖锐的脊不可避免地非等距钝化以减轻弯曲，导致编码的奇异高斯曲率的适度涂抹。我们通过讨论诸如致动线性特征（例如探针、舌片和夹具）之类的特征的使用来结束。我们推测这些形状变化模式与在几个生物系统的形态发生过程中发现的那些模式之间的相似性。