Flexible mechanical metamaterials possess repeating structural motifs that imbue them with novel, exciting properties including programmability, anomalous elastic moduli, and nonlinear and robust response. We address such structures via micromorphic continuum elasticity, which allows highly nonuniform deformations (missed in conventional elasticity) within unit cells that nevertheless vary smoothly between cells. We show that the bulk microstructure gives rise to boundary elastic terms. Discrete lattice theories have shown that critically coordinated structures possess a topological invariant that determines the placement of low-energy modes on edges of such a system. We show that in continuum systems, a new topological invariant emerges, which relates the difference in the number of such modes between two opposing edges. Guided by the continuum limit of the lattice structures, we identify macroscopic experimental observables for these topological properties that may be observed independently on a new length scale above that of the microstructure.

中文翻译：

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柔性结构的拓扑弹性

柔性机械超材料具有重复的结构图案，这些图案赋予它们新颖，令人兴奋的特性，包括可编程性，异常弹性模量以及非线性和鲁棒的响应。我们通过微形连续弹性来解决此类结构，该结构允许在单元格内发生高度不均匀的变形（传统弹性中缺少），但在单元格之间会平滑变化。我们表明，整体微观结构引起边界弹性项。离散晶格理论表明，临界协调的结构具有拓扑不变性，该不变性决定了低能模式在此类系统边缘的位置。我们表明，在连续系统中，出现了一个新的拓扑不变量，它与两个相对边缘之间的这种模式的数量有关。