论文标题
有缺陷的肿瘤发生：可编程形状响应板中具有拓扑缺陷的高斯曲率
Defective nematogenesis: Gauss curvature in programmable shape-responsive sheets with topological defects
论文作者
论文摘要
扁平板上编码的收缩/伸长模式变成弯曲的表面。如果表面带有高斯曲率，则最终的致动可能是强大而强大的。我们部署高斯骨定理来推断出以均匀刻度的收缩/伸长仪的模式与空间变化的方向编码的高斯曲率，就像在图案化的液晶弹性体中通常实现的那样。该方法揭示了两个根本不同的贡献：一种结构曲率，取决于模式的精确形式，以及在收缩方向上缺陷产生的拓扑曲率。这些曲率随着不同功能而增长的收缩/伸长幅度，解释了简单+1缺陷的先前计算与平滑缺陷模式之间的明显矛盾。我们通过对用简单的高阶收缩缺陷编码的纸张进行数值壳计算来验证这些结构和拓扑贡献，以揭示其活化的形态。最后，我们计算由具有空间变化的大小和方向的模式产生的高斯曲率，从而导致对结构项的额外梯度贡献。我们预计，每当大小和方向是自然变量时，包括描述肌肉沿其图案纤维方向的收缩或通过拉长其细胞生长的组织的收缩时，这种形式将是有用的。
Flat sheets encoded with patterns of contraction/elongation morph into curved surfaces. If the surfaces bear Gauss curvature, the resulting actuation can be strong and powerful. We deploy the Gauss-Bonnet theorem to deduce the Gauss curvature encoded in a pattern of uniform-magnitude contraction/elongation with spatially varying direction, as is commonly implemented in patterned liquid crystal elastomers. This approach reveals two fundamentally distinct contributions: a structural curvature which depends on the precise form of the pattern, and a topological curvature generated by defects in the contractile direction. These curvatures grow as different functions the contraction/elongation magnitude, explaining the apparent contradiction between previous calculations for simple +1 defects, and smooth defect-free patterns. We verify these structural and topological contributions by conducting numerical shell calculations on sheets encoded with simple higher-order contractile defects to reveal their activated morphology. Finally we calculate the Gauss curvature generated by patterns with spatially varying magnitude and direction, which leads to additional magnitude gradient contributions to the structural term. We anticipate this form will be useful whenever magnitude and direction are natural variables, including in describing the contraction of a muscle along its patterned fiber direction, or a tissue growing by elongating its cells.
