Session K14: Extreme Mechanics: Topology in Mechanics

Topological mechanics of gyroscopic meta-materials

Topological mechanical meta-materials are artificial structures whose unusual properties are protected very much like their electronic and optical counterparts. I will present an experimental and theoretical study\footnote{Topological mechanics of gyroscopic meta-materials, Lisa M. Nash, Dustin Kleckner, Alismari Read, Vincenzo Vitelli, Ari M. Turner and William T.M. Irvine, PNAS (In press).} of a new kind of active meta-material – comprised of coupled gyroscopes on a lattice – that breaks time-reversal symmetry. The vibrational spectrum displays a sonic gap populated by topologically protected edge modes which propagate in only one direction and are unaffected by disorder. We observe these edge modes in experiment and verify their robustness to disorder and the insertion of obstacles. Controlled distortions of the underlying lattice can induce a topological phase transition that switches the edge mode chirality. This effect allows the direction of the edge current to be determined on demand.   

Kirigami!

We explore and develop a simple set of rules that apply to cutting, pasting, and folding honeycomb lattices. We consider origami-like structures that are extrinsically flat away from zero dimensional sources of Gaussian curvature and one-dimensional sources of mean curvature, and our cutting and pasting rules maintain the intrinsic bond lengths on both the lattice and its dual lattice. We find that a small set of rules is allowed providing a framework for exploring and building kirigami - folding, cutting, and pasting the edges of paper.   

Mechanisms and nonlinear waves from topological modes

Topological protection can arise in mechanical structures such as linkages, frames, or rigid origami. The key ingredients are a balance of degrees of freedom and constraints away from the boundaries. In this setting certain zero energy modes of the system can be made robust against a broad class of perturbations and noise. However, since there are no restoring forces to these modes to linear order, they result in flexes and mechanisms which must be treated as nonlinear waves. I will discuss several simple and concrete examples which illustrate these ideas.   

Topological Toughening of graphene and other 2D materials

It has been claimed that graphene, with the elastic modulus of 1TPa and theoretical strength as high as 130 GPa, is the strongest material. However, from an engineering point of view, it is the fracture toughness that determines the actual strength of materials, as crack-like flaws (i.e., cracks, holes, notches, corners, etc.) are inevitable in the design, fabrication, and operation of practical devices and systems. Recently, it has been demonstrated that graphene has very low fracture toughness, in fact close to that of ideally brittle solids. These findings have raised sharp questions and are calling for efforts to explore effective methods to toughen graphene. Recently, we have been exploring the potential use of topological effects to enhance the fracture toughness of graphene. For example, it has been shown that a sinusoidal graphene containing periodically distributed disclination quadrupoles can achieve a mode I fracture toughness nearly twice that of pristine graphene. Here we report working progresses on further studies of topological toughening of graphene and other 2D materials. A phase field crystal method is adopted to generate the atomic coordinates of material with specific topological patterns. We then perform molecular dynamics simulations of fracture in the designed samples, and observe a variety of toughening mechanisms, including crack tip blunting, crack trapping, ligament bridging, crack deflection and daughter crack initiation and coalescence.   

Control of defect localization in crystalline wrinkling by curvature and topology

We investigate the influence of curvature and topology on crystalline wrinkling patterns in generic elastic bilayers. Our numerical analysis predicts that the total number of defects created by adiabatic compression exhibits universal quadratic scaling for spherical, ellipsoidal and toroidal surfaces over a wide range of system sizes. However, both the localization of individual defects and the orientation of defect chains depend strongly on the local Gaussian curvature and its gradients across a surface. Our results imply that curvature and topology can be utilized to pattern defects in elastic materials, thus promising improved control over hierarchical bending, buckling or folding processes. Generally, this study suggests that bilayer systems provide an inexpensive yet valuable experimental test-bed for exploring the effects of geometrically induced forces on assemblies of topological charges.