Nonlinear conduction via solitons in a topological mechanical insulator

Abstract

Networks of rigid bars connected by joints, termed linkages, provide a minimal framework to design robotic arms and mechanical metamaterials built of folding components. Here, we investigate a chain-like linkage that, according to linear elasticity, behaves like a topological mechanical insulator whose zero-energy modes are localized at the edge. Simple experiments we performed using prototypes of the chain vividly illustrate how the soft motion, initially localized at the edge, can in fact propagate unobstructed all of the way to the opposite end. Using real prototypes, simulations, and analytical models, we demonstrate that the chain is a mechanical conductor, whose carriers are nonlinear solitary waves, not captured within linear elasticity. Indeed, the linkage prototype can be regarded as the simplest example of a topological metamaterial whose protected mechanical excitations are solitons, moving domain walls between distinct topological mechanical phases. More practically, we have built a topologically protected mechanism that can perform basic tasks such as transporting a mechanical state from one location to another. Our work paves the way toward adopting the principle of topological robustness in the design of robots assembled from activated linkages as well as in the fabrication of complex molecular nanostructures.topological matter | origami | isostaticity | jamming | active matter M echanical structures composed of folding components, such as bars or plates rotating around pivots or hinges, are ubiquitous in engineering, materials science, and biology (1). For example, complex origami-like structures can be created by folding a paper sheet along suitably chosen creases around which two nearby faces can freely rotate (2-4). Similarly, linkages can be viewed as 1D versions of origami where rigid bars (links) are joined at their ends by joints (vertices) that permit full rotation of the bars ( Fig. 1 A-C). Some of the joints can be pinned to the plane while the remaining ones rotate relative to each other under the constraints imposed by the network structure of the linkage (5). Familiar examples include the windshield wiper, robotic arms, biological linkages in the jaw and knee, and toys like the Jacob's ladder (6) and the Hoberman sphere. Moreover, linkages and origami can be used in the design of microscopic and structural metamaterials whose peculiar properties are controlled by the geometry of the unit cell (7,8).Many of these examples are instances of what mechanical engineers call mechanisms: structures where the degrees of freedom are nearly balanced by carefully chosen constraints so that the allowed free motions encode a desired mechanical function. However, as the number of components increases, more can go wrong: lack of precision machining or undesired perturbations. Robustness in this sense is a concern relevant to the design of complex mechanical structures from the microscopic to the architectural scale, typically addressed at the cost of higher manufacturing tolerances or active feedbac...

Epigenetic Transitions and Knotted Solitons in Stretched Chromatin

“…Similarly to solitons recently observed in meta-materials [64] and minimal surfaces [65], it is the non-trivial topology of the localised knot which keeps the structure together. The epigenetic state of the knotted soliton can "switch" from red to blue, or vice versa -switching events occur, in Figure 4, at about 2 10 5 τ Br and 3.5 10 5 τ Br ).…”

Geometric phase and topology of elastic oscillations and vibrations in model systems: Harmonic oscillator and superlattice

“…Phononic structures have also been shown recently to possess non-conventional topology as well as topologically constrained propagative properties. These properties have been achieved by breaking time-reversal symmetry through internal resonance or symmetry breaking structural features (e.g., chirality) [9][10][11][12][13][14][15][16][17][18][19] and without addition of energy from the outside. Energy can also be added to extrinsic topological elastic systems to break time reversal symmetry.…”

Topological Edge Floppy Modes in Disordered Fiber Networks

“…Introduction -Recent theoretical advances in applying concepts of topological states of matter to mechanical systems has led to the burgeoning new field of "topological mechanics", where nontrivial topologies of the phonon bands give rise to exotic mechanical and acoustic properties [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16][17][18][19][20].…”

“…Among many different types of topological mechanical systems, a particularly interesting class consists of "Maxwell lattices", which are central-force lattices with average coordination number z = 2d where d is the spatial dimension, and are thus at the verge of mechanical instability [2,3,[16][17][18][19][20]. Maxwell lattices host topologically protected phonon edge modes at zero frequency (floppy modes).…”

“…Nevertheless, floppy modes still correspond to null space ofC(q 1 , q 2 ) in momentum space. In order to have floppy modes, the condition detC = 0 (20) has to be met (where the determinant is taken in the q 1 , q 2 space). In particular, from Eq.…”