Origami and kirigami have emerged as potential tools for the design of mechanical metamaterials whose properties such as curvature, Poisson’s 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-one dimensional 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.
B. G-g. Chen, B. Liu, A. A. Evans, J. Paulose, I. Cohen, V. Vitelli, C. D. Santangelo (submitted) preprint
Curvature and mechanics are intimately connected for thin materials, and this coupling between geometry and physical properties is readily seen in folded structures from intestinal villi and pollen grains, to wrinkled membranes and programmable metamaterials. While the well-known rules and mechanisms behind folding a flat surface have been used to create deployable structures and shape transformable materials, folding of curved shells is still not fundamentally understood. Shells naturally deform by simultaneously bending and stretching, and while this coupling gives them great stability for engi- neering applications, it makes folding a surface of arbitrary curvature a non-trivial task. Here we discuss the geometry of folding a creased shell, and demonstrate theoretically the conditions under which it may fold smoothly. When these conditions are violated we show, using experiments and simulations, that shells undergo rapid snapping motion to fold from one stable configuration to another. Although material asymmetry is a proven mechanism for creating this bifurcation of stability, for the case of a creased shell, the inherent geometry itself serves as a barrier to folding. We discuss here how two fundamental geometric concepts, creases and curvature, combine to allow rapid transitions from one stable state to another. Independent of material system and length scale, the design rule that we introduce here explains how to generate snapping transitions in arbitrary surfaces, thus facilitating the creation of programmable multi-stable materials with fast actuation capabilities.
Origami-based design holds promise for developing materials whose mechanical properties are tuned by crease patterns introduced to thin sheets. Although there has been heuristic developments in constructing patterns with desirable qualities, the bridge between origami and physics has yet to be fully developed. To truly consider origami structures as a class of materials, methods akin to solid mechanics need to be developed to understand their long-wavelength behavior. We introduce here a lattice theory for examining the mechanics of origami tessellations in terms of the topology of their crease pattern and the relationship between the folds at each vertex. This formulation provides a general method for associating mechanical properties with periodic folded structures, and allows for a concrete connection between more conventional materials and the mechanical metamaterials constructed using origami-based design.
A. A. Evans, J. L. Silverberg, and C. D. Santangelo, Phys. Rev. E, 92, 013205 (2015) preprint
Origami is used beyond purely aesthetic pursuits to design responsive and customizable mechanical metamaterials. However, a generalized physical understanding of origami remains elusive, owing to the challenge of determining whether local kinematic constraints are globally compatible and to an incomplete understanding of how the folded sheet’s material properties contribute to the overall mechanical response. Here, we show that the traditional square twist, whose crease pattern has zero degrees of freedom (DOF) and therefore should not be foldable, can nevertheless be folded by accessing bending deformations that are not explicit in the crease pattern. These hidden bending DOF are separated from the crease DOF by an energy gap that gives rise to a geometrically driven critical bifurcation between mono- and bistability. Noting its potential utility for fabricating mechanical switches, we use a temperature-responsive polymer-gel version of the square twist to demonstrate hysteretic folding dynamics at the sub-millimetre scale.
J. L. Silverberg, J.-H. Na, A. A. Evans, B. Liu, T. C. Hull, C. D. Santangelo, R. J. Lang, R. C. Hayward, I. Cohen, “Origami structures with a critical transition to bistability arising from hidden degrees of freedom”, Nat. Mater. DOI: 10.1038/nmat4232 (2015) pdf SI
Patterning deformation within the plane of thin elastic sheets represents a powerful tool for the definition of complex and stimuli-responsive 3D buckled shapes. Previous experimental methods, however, have focused on sheets that access a limited number of shapes pre-programmed into the sheet, restricting the degree of dynamic control. Here, we demonstrate on-demand reconfigurable buckling of poly(N-isopropylacrylamide-co-acrylic acid) (PNIPAM) hydrogel network films containing gold nanoparticles (AuNPs) by patterned photothermal deswelling. Predictable, easily controllable, and reversible transformations from a single flat gel sheet to numerous different three-dimensional forms are shown. Importantly, the response time is limited by poroelastic mass transport, rather than photochemical switching kinetics, enabling reconfiguration of shape on timescales of several seconds, with further increases in speed possible by reducing film thickness.
A. W. Hauser, A. A. Evans, J.-H. Na, R. C. Hayward, Angew. Chem. Int. Ed., DOI: 10.1002/anie.201412160 (2015) pdf SI
Developments in origami mathematics over the past few dec-ades have enabled the systematic design of folded structures with arbitrary complexity, extending the capabilities of the form well beyond the diversity of shapes achieved with traditional paper art, and highlighting its power as a tool for the fabrication of 3D objects from 2D sheets. More recently, an area of considerable interest has been the development of self- folding structures that undergo autonomous transformations between programmed shapes in response to external triggers or changes in their environment. The ability to use origami design principles to controllably fold, unfold, and refold thin sheets prepared by planar fabrication techniques would offer great promise for applications in biomimetic systems, soft robotics, and mechanical metamaterials, especially for fabrication of structures on small length scales, where traditional manufacturing processes fail.
J.-H. Na, A. A. Evans, J. Bae, M. C. Chiappelli, C. D. Santangelo, R. J. Lang, R. C. Hayward, Adv. Mater., 27 79-85 (2015) pdf SI
Although broadly admired for its aesthetic qualities, the art of origami is now being recognized also as a framework for mechanical metamaterial design. Working with the Miura-ori tessellation, we find that each unit cell of this crease pattern is mechanically bistable, and by switching between states, the compressive modulus of the overall structure can be rationally and reversibly tuned. By virtue of their interactions, these mechanically stable lattice defects also lead to emergent crystallographic structures such as vacancies, dislocations, and grain boundaries. Each of these structures comes from an arrangement of reversible folds, highlighting a connection between mechanical metamaterials and programmable matter. Given origami’s scale-free geometric character, this framework for metamaterial design can be directly transferred to milli-, micro-, and nanometer-size systems.
J. L. Silverberg, A. A. Evans, L. McLeod, R. C. Hayward, T. Hull, C. D. Santangelo, and I. Cohen, Science, 345, 647 (2014) pdf SI
When a flexible filament is confined to a fluid interface, the balance between capillary attraction, bending resistance, and tension from an external source can lead to a self-buckling instability. We perform an analysis of this instability and provide analytical formulae that compare favorably with the results of detailed numerical computations. The stability and long-time dynamics of the filament are governed by a single dimensionless elastocapillary number quantifying the ratio between capillary to bending stresses. Complex, folded filament configurations such as loops, needles, and racquet shapes may be reached at longer times, and long filaments can undergo a cascade of self-folding events.
A. A. Evans, S. E. Spagnolie, D. Bartolo, and E. Lauga, Soft Matter, 9 1711-1720 (2013) pdf
We develop a continuum elastic approach to examining the bending mechanics of semiflexible filaments with a local internal degree of freedom that couples to the bending modulus. We apply this model to study the nonlinear mechanics of a double-stranded DNA oligomer (shorter than its thermal persistence length) whose free ends are linked by a single-stranded DNA chain. This construct, studied by H. Qu and G. Zocchi [Europhys. Lett. 94, 18003 (2011)], displays nonlinear strain softening associated with the local melting of the double-stranded DNA under applied torque and serves as a model system with which to study the nonlinear elasticity of DNA under large energy deformations. We show that one can account quantitatively for the observed bending mechanics using an augmented wormlike chain model, the helix-coil wormlike chain. We also predict that the highly bent and partially molten dsDNA should exhibit particularly large end-to-end fluctuations associated with the fluctuation of the length of the molten region, and propose appropriate experimental tests. We suggest that the augmented wormlike chain model discussed here is a useful analytic approach to the nonlinear mechanics of DNA or other biopolymer systems.
A. A. Evans and A. J. Levine, High energy deformation of filaments with internal structure and localized torque-induced melting of DNA, Phys. Rev. E, 85 051915 (2012) pdf
Motivated by recent advances in vesicle engineering, we consider theoretically the locomotion of shape- changing bilayer vesicles at low Reynolds number. By modulating their volume and membrane composition, the vesicles can be made to change shape quasi-statically in thermal equilibrium. When the control parameters are tuned appropriately to yield periodic shape changes, which are not time- reversible, the result is a net swimming motion over one cycle of shape deformation. For two classical vesicle models (spontaneous curvature and bilayer coupling), we numerically determine the sequence of vesicle shapes through an enthalpy minimization, as well as the fluid-body interactions by solving a boundary integral formulation of the Stokes equations. For both models, net locomotion can be obtained either by continuously modulating fore-aft asymmetric vesicle shapes or by crossing a continuous shape-transition region and alternating between fore-aft asymmetric and fore-aft symmetric shapes. The obtained hydrodynamic efficiencies are similar to those of other low Reynolds number biological swimmers and suggest that shape-changing vesicles might provide an alternative to flagella-based synthetic microswimmers.
A. A. Evans, S. E. Spagnolie, and E. Lauga, Stokesian jellyfish: Viscous locomotion of bilayer vesicles, Soft Matter, 6 1737-1747 (2010) pdf