Topological Elasticity of Flexible Structures

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.

Popular Summary

The theory of elasticity succeeds in capturing how solid objects resist attempts to change their shapes, which ultimately depends on atomic structure, via just a few easily measured elastic constants. However, systems at the edge of elasticity—such as loosely packed granular materials, networks of flexible fibers, and cutting-edge flexible mechanical metamaterials—defy this picture by undergoing large deformations that vary abruptly from one component to its neighbor. We introduce a new micromorphic elastic theory that accounts for these microscopic changes in shape.

Our theory shows that the microstructure generates energy costs that lie on the surface of the system, in addition to the conventional bulk elastic moduli. These terms cause the surface of such a system to respond quite differently from that of a conventional solid, with deformations extending much more deeply into the bulk, at a new elastic length scale set by both the type of deformation and the size of the microscopic constituents.

These modes of surface deformation are topological in nature, meaning that they persist under changes to the system and are determined by the bulk structure. The bulk structure determines a topological invariant, which in turn sets the relative numbers of zero- or low-energy deformations on opposite surfaces. Although such topological correspondences are typically found in lattice theories, our work and other recent results have extended them from the atomic scale up to the macroscopic scale.

Figure 1

A periodic spring network has a periodic microstructure consisting of bonds connecting sites, as in the generalized kagome lattice shown here. A solid bond is located at $\mathbf{p}$ relative to the center $\mathbf{r}$ of the cell in which it lies. The bond connects two sites with relative position $\mathbf{b}$, undergoing displacements ${\mathbf{u}}_1(\mathbf{r}),{\mathbf{u}}_2(\mathbf{r})$ that cause extension or compression of the bond. For continuum fields, the displacements may differ greatly for each site within the repeating cell but vary smoothly across the cells.

Figure 2

(a) On a periodic system, we apply a particular strain ${\epsilon }_{ij}(\mathbf{r})=({\epsilon }_{xx},{\epsilon }_{yy})e^{i(\mathbf{q}·\mathbf{r})}$. This strain causes bonds to stretch (green) or compress (red). (b) The system then relaxes by projecting onto the space of self-stresses that mostly capture particles displacing over short distances but reducing the energy cost overall.

Figure 3

The contour used to establish the relationship between the bulk structure and the locations of the zero modes. Via complex analysis, the full contour counts the difference in numbers of long-wavelength zero modes on the left and right edges while remaining insensitive to the short-wavelength modes. As described in the text, for this particular contour, the dashed portions may be neglected, such that the locations of the zero modes are determined solely by the bulk physics captured in the solid lines.

Figure 4

(a) Topological transition as we deform the kagome lattice. The geometry of the system is parametrized as $\mathbf{g}(x)=x{\mathbf{g}}_1+(1-x){\mathbf{g}}_2$, where ${\mathbf{g}}_1$, ${\mathbf{g}}_2$ are geometric configurations of two kagome systems with respective topological polarizations 0 and 2 along the ${\widehat{r}}_n$ direction. Note that ${\kappa }_{\pm }$ are the signed inverse decay lengths, such that, when one vanishes, the associated mode lies in the bulk and, when both have the same sign, the system is polarized. (b) The system in the ${\mathbf{g}}_1$ configuration with trivial polarization: There is a zero mode on the left side of the considered orientation ${\widehat{r}}_n$ and one on the opposite side. (c) The system in nontrivial polarization ($\mathrm{\Delta }N=2$): This time, the two modes are on the left side of the direction. In all three figures, the purple dashed lines represent the shape of the system at the topological transition (when $x=0.7$).

Figure 5

(a) For a fixed system, we numerically compute the signed inverse decay lengths of each mode [${\kappa }_{-}({\theta }_n),{\kappa }_{+}({\theta }_n)$] and the topological polarization $\mathrm{\Delta }N({\theta }_n)$ as a function of the normal direction ${\widehat{r}}_n({\theta }_n)$. (b) The polarization changes as the considered direction crosses either soft direction (orange lines). One can then find the regions of the lattice where the $+$ or $-$ edge modes are located (blue or red arcs).

Figure 6

Shape of the edge modes in a system with polarization $\mathrm{\Delta }N=2$. We impose a mode with wavelength $\lambda =15.6|{\mathbf{l}}_1|$ on the boundary ${\widehat{r}}_t$ (where ${\mathbf{l}}_1$ is the first lattice primitive vector). This mode then propagates through the bulk and decays along the direction ${\widehat{r}}_n$. Each mode (red or blue) varies sinusoidally along its corresponding soft direction; therefore, it also varies along the boundary with a longer wavelength. The modes decay into the bulk over length scales much longer than this wavelength.