Topological Mechanics of Origami and Kirigami

Abstract

Origami and kirigami have emerged as potential tools for the design of mechanical metamaterials whose properties such as curvature, Poisson ratio, and existence of metastable states can be tuned using purely geometric criteria. A major obstacle to exploiting this property is the scarcity of tools to identify and program the flexibility of fold patterns. We exploit a recent connection between spring networks and quantum topological states to design origami with localized folding motions at boundaries and study them both experimentally and theoretically. These folding motions exist due to an underlying topological invariant rather than a local imbalance between constraints and degrees of freedom. We give a simple example of a quasi-1D folding pattern that realizes such topological states. We also demonstrate how to generalize these topological design principles to two dimensions. A striking consequence is that a domain wall between two topologically distinct, mechanically rigid structures is deformable even when constraints locally match the degrees of freedom. DOI: 10.1103/PhysRevLett.116.135501 Recent interest in origami mechanisms has been spurred by advances in fabrication and manufacturing [1][2][3], as well as a realization that folded structures can form the basis of mechanical metamaterials [4][5][6][7][8]. The ability to identify kinematic mechanisms-allowable folding motions of a crease pattern-is critical to the use of origami to design new deployable structures and mechanical metamaterials. For example, the mechanism in the celebrated Miura ori that allows it to furl and unfurl in a single motion [9,10] is also the primary determinant of the fold pattern's negative Poisson ratio [4,5]. Identifying these mechanisms becomes more challenging when the number of apparent constraints matches the number of degrees of freedom (DOF).When there is an exact balance between DOF and constraints in a periodic structure, the structure is marginally rigid [11,12]. In such a case, new mechanical properties such as nonlinear response to small perturbations emerge [13][14][15][16]. A recent realization is that the flexibility of such solids may be influenced by nontrivial topology in the phonon band structure [17,18]. Here, we show how to extend these topological ideas to origami and kirigami. We show that periodically folded sheets may exhibit distinct mechanical "phases" characterized by a topological invariant called the topological polarization, recently introduced by Kane and Lubensky [17] using a mapping of mechanically marginal structures to topological insulators [19]. The importance of this invariant has emerged in the study of the soft modes of spring networks [18], and the nonlinear mechanics of linkages [20] and buckling [21]. As in these examples, the phases in our origami and kirigami structures exhibit localized vibrational modes on certain boundaries, and transitions from between topological phases are characterized by the appearance of bulk modes that cost zero energy. These are the hallmarks of topol...

Mechanical Weyl Modes in Topological Maxwell Lattices

“…In quantum condensed matter systems, topological invariants guarantee the existence and robustness of electronic states at free surfaces and domain walls in polyacetylene [4,5], quantum Hall systems [6,7], and topological insulators [8-13] whose bulk electronic spectra are fully gapped (i.e., conduction and valence bands separated by a gap at all wave numbers). More recently, topological phononic and photonic states have been identified in suitably engineered classical materials as well [14][15][16][17][18][19][20][21][22][23][24][25][26][27][28][29][30][31][32][33], provided that the band structure of the corresponding wavelike excitations has a nontrivial topology.A special class of topological mechanical states occurs in Maxwell lattices, periodic structures in which the number of constraints equals the number of degrees of freedom in each unit cell [34]. In these mechanical frames, zero-energy modes and states of self-stress (SSSs) are the analogs of particles and holes in electronic topological materials [16].…”

Branches of Triangulated Origami Near the Unfolded State

“…To better understand the phenomena of multiple folding pathways and misfolding, it is useful to distinguish two notions of floppiness in an origami structure: (1) the number of degrees of freedom D, which is the dimensionality of the space of motions and scales with the number of boundary sides of a generic origami crease pattern [11,21,22], and (2) the number of distinct branches B, or folding pathways. As we discuss later, the flat unfolded configuration of an origami structure is a singularity in the space of origami configurations where B branches of dimension D intersect.…”

Sonic Landau Levels and Synthetic Gauge Fields in Mechanical Metamaterials

“…We show how this domain-wall-bound mode exhibits robustness against a type of disorder that may come in the manufacturing of acoustic metamaterialsdisorder in the stiffness of each component. Like other realizations of topological states [15,16] in mechanical [17][18][19][20][21][22][23][24][25][26][27], acoustic [28][29][30][31][32][33][34][35][36], and photonic [37] metamaterials, this characterization may help with the design of robust devices. We show that introducing dissipation on just one of the two sublattices enhances the domain-wall-bound sound mode.…”