Engineering Dept., University of Cambridge, Trumpington St., Cambridge, CB2 1PZ, UK.
Soft Matter. 2020 Dec 28;16(48):10935-10945. doi: 10.1039/d0sm01192d. Epub 2020 Nov 3.
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 of 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.
编码有收缩/拉伸模式的平面薄片会变成曲面。如果这些表面具有高斯曲率，那么产生的驱动效果可以是强大的。我们利用高斯-博内定理来推断出在具有空间变化方向的均匀幅度收缩/拉伸模式中编码的高斯曲率，这在图案化液晶弹性体中是常见的。这种方法揭示了两种截然不同的贡献：一种是结构曲率，它取决于图案的精确形式，另一种是由收缩方向缺陷产生的拓扑曲率。这些曲率以不同的收缩/拉伸幅度函数形式增长，解释了以前对简单+1 缺陷的计算与平滑无缺陷图案之间的明显矛盾。我们通过对编码有简单高阶收缩缺陷的薄片进行数值壳计算来验证这些结构和拓扑贡献，以揭示它们的激活形态。最后，我们计算了具有空间变化幅度和方向的图案产生的高斯曲率，这导致结构项中出现额外的幅度梯度贡献。我们预计这种形式在幅度和方向是自然变量的情况下将非常有用，包括描述肌肉沿着其图案化纤维方向收缩的情况，或者组织通过延长其细胞来生长的情况。
