Topological mechanics of origami and kirigami


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


Geometrically controlled snapping transitions in shells with curved creases

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.

arxiv  pdf

Lattice Mechanics of Origami Tessellations

miura_fig 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 structures with a critical transition to bistability arising from hidden degrees of freedom

sq_twistOrigami 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

Photothermally Reprogrammable Buckling of Nanocomposite Gel Sheets


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

Membrane Rheology

Surfactant monolayers and lipid bilayers are intrinsically two- dimensional structures with viscoelastic mechanical properties. Monolayers display a plethora of complex broken symmetry phases, each with its own rheological signature, while bilayers are of fundamental biological importance in forming the cell membrane and the principal internal partitions of the cell. Understanding the low energy excitations and mechanical response of these materials is thus an important probe of novel two-dimensional phases and essential to biomechanics at the cellular level, cell cell recognition, and trans- port across membranes; as such, a number of macroscopic and microscopic techniques have been developed to explore the rheological properties of mono- layers and membranes. In this chapter we review the fundamental physics and rheology of molecularly thin membranes, paying particular attention to the fact that these systems are necessarily bounded on one or both sides by an aqueous fluid. We develop the basic theory of both the in- and out-of-plane viscoelastic response of membranes and monolayers, and apply this theory to the study of particle transport in the surface. Such transport measurements form the basis of typical rheological experiments. We also report on more recent investigations regarding the role of nontrivial membrane geometry on particle transport, and examine a novel approach to monolayer and mem- brane microrheology using the thermal fluctuations of particles submerged beneath the membrane. We conclude with a discussion of open questions in the field and some speculations on future research directions.

A. A. Evans and A. J. Levine, Membrane Rheology, forthcoming in “Complex Fluids in Biological Systems” (Springer)

Programming Reversibly Self-Folding Origami with Micropatterned Photocrosslinkable Polymer Trilayers


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

Using origami design principles to fold reprogrammable mechanical metamaterials


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, Science345, 647 (2014) pdf SI

Probing interfacial dynamics and mechanics using submerged particle microrheology

Microrheology relies on tracking the thermal or driven motion of microscopic particles in a soft material. It is well suited to the study of materials that have no three-dimensional realization, which makes them difficult to study using a macroscopic rheometer. For this reason, microrheology is becoming an important rheological probe of Langmuir monolayers and membranes. Interfacial microrheology, however, has been difficult to reconcile quantitatively with more traditional macroscopic approaches. We suggest that uncertainties in accounting for the mechanical coupling of the tracer particle to the interface or membrane are responsible for these discrepancies. To resolve them, we propose a new non-contact approach to interfacial microrheology that uses particles submerged in the subphase a known distance below the interface. In this first of two papers, we present calculations of the response function (and thus the equilibrium fluctuation spectrum) of a spherical particle submerged below a viscoelastic surface that has a finite surface tension and/or bending modulus. In the second paper, we compare these results to submerged particle microrheology in a few example systems, showing quantitative agreement.

R. Shlomovitz, A. A. Evans, T. Boatwright, M. Dennin, and A. J. Levine, Phys. Fluids26 071903 (2014) pdf

T. Boatwright, M. Dennin, R. Shlomovitz, A. A. Evans, and A. J. Levine, Phys. Fluids26 071904 (2014) pdf

Reflection and Refraction of Flexural Waves at Geometric Boundaries


We present a theory of flexural wave propagation on elastic shells having nontrivial geometry and develop an analogy to geometric optics. The transport of momentum within the shell itself is anisotropic due to the curvature, and as such complex classical effects such as birefringence are generically found. We determine the equations of reflection and refraction of such waves at boundaries between different local geometries, showing that waves are totally internally reflected, especially at boundaries between regions of positive and negative Gaussian curvature. We verify these effects by using finite element simulations and discuss the ramifications of these effects for the statistical mechanics of thin curved materials.

A. A. Evans and A. J. Levine, Phys, Rev. Lett., 111 038101 (2013) pdf SI

Measurement of Monolayer Viscosity Using Non-Contact Microrheology


Microrheological studies of phospholipid monolayers, bilayers, and other Langmuir monolayer systems are traditionally performed by observing the thermal fluctuations of tracers attached to the membrane or interface. Measurements of this type obtain surface moduli that are orders of magnitude different from those obtained using macroscopic or active techniques. These large discrepancies can result from uncertainties in the tracer’s coupling to the monolayer or the local disruption of the monolayer by the tracer. To avoid such problems, we perform a microrheological experiment with the tracer particle placed at a known depth beneath the monolayer; this avoids the issues mentioned at the cost of generating a weaker, purely hydrodynamic coupling between the tracer and the monolayer. We calculate the appropriate response functions for this submerged particle microrheology and demonstrate the technique on three model monolayer systems.

R. Shlomovitz, A. A. Evans, T. Boatwright, M. Dennin, and A. J. Levine, Phys. Rev. Lett.110 137802 (2013) pdf SI 

Elastocapillary self-folding: buckling, wrinkling, and collapse of floating filaments


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 Matter9 1711-1720 (2013) pdf

High energy deformation of filaments with internal structure and localized torque-induced melting of DNA


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. E85 051915 (2012) pdf


Orientational order in concentrated suspensions of spherical microswimmers


We use numerical simulations to probe the dynamics of concentrated suspensions of spherical microswimmers interacting hydrodynamically. Previous work in the dilute limit predicted orientational instabilities of aligned suspensions for both pusher and puller swimmers, which we confirm computationally. Unlike previous work, we show that isotropic suspensions of spherical swimmers are also always unstable. Both types of initial conditions develop long-time polar order of a nature which depends on the hydrodynamic signature of the swimmer but very weakly on the volume fraction up to very high volume fractions.

A. A. Evans, T. Ishikawa, T. Yamaguchi, and E. Lauga, Phys. Fluids23 111702 (2011) pdf


Fluid transport by active elastic membranes

activemembraneA flexible membrane deforming its shape in time can self-propel in a viscous fluid. Alternatively, if the membrane is anchored, its deformation will lead to fluid transport. Past work in this area focused on situations where the deformation kinematics of the membrane were prescribed. Here we consider models where the deformation of the membrane is not prescribed, but instead the membrane is internally forced. Both the time-varying membrane shape and the resulting fluid motion result then from a balance between prescribed internal active stresses, internal passive resistance, and external viscous stresses. We introduce two specific models for such active internal forcing: one where a distribution of active bending moments is prescribed, and one where active inclusions exert normal stresses on the membrane by pumping fluid through it. In each case, we asymptotically calculate the membrane shape and the fluid transport velocities for small forcing amplitudes, and recover our results using scaling analysis.

A. A. Evans and E. Lauga, Fluid transport by active elastic membranes, Phys. Rev. E, 84 031924 (2011) pdf

Propulsion of passive filaments and active flagella near boundaries

propulsiveflagellaConfinement and wall effects are known to affect the kinematics and propulsive characteristics of swimming microorganisms. When a solid body is dragged through a viscous fluid at constant velocity, the presence of a wall increases fluid drag, and thus the net force required to maintain speed has to increase. In contrast, recent optical trapping experiments have revealed that the propulsive force generated by human spermatozoa is decreased by the presence of boundaries. Here, we use a series of simple models to analytically elucidate the propulsive effects of a solid boundary on passively actuated filaments and model flagella. For passive flexible filaments actuated periodically at one end, the presence of the wall is shown to increase the propulsive forces generated by the filaments in the case of displacement-driven actuation, while it decreases the force in the case of force-driven actuation. In the case of active filaments as models for eukaryotic flagella, we demonstrate that the manner in which a solid wall affects propulsion cannot be known a priori, but is instead a nontrivial function of the flagellum frequency, wavelength, its material characteristics, the manner in which the molecular motors self-organize to produce oscillations 􏰖prescribed activity model or self-organized axonemal beating model􏰗, and the boundary conditions applied experimentally to the tethered flagellum. In particular, we show that in some cases, the increase in fluid friction induced by the wall can lead to a change in the waveform expressed by the flagella, which results in a decrease in their propulsive force.

A. A. Evans and E. Lauga, Propulsion by passive filaments and active flagella near boundaries, Phys. Rev. E82 041915 (2010) pdf

Stokesian jellyfish: Viscous locomotion of bilayer vesicles

stokesian_jellyMotivated 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 Matter1737-1747 (2010) pdf

Adhesion transition in flexible sheets

adhesion Intermolecular forces are known to precipitate adhesion events between solid bodies. Inspired by a macro- scale experiment showing the hysteretic adhesion of a piece of flexible tape over a plastic substrate, we develop here a model of far-field dry adhesion between two flexible sheets interacting via a power-law potential. We show that phase transitions from unadhered to adhered states occur as dictated by a dimensionless bending parameter representing the ratio of interaction strength to bending stiffness. The order of the adhesion transitions, as well as their hysteretic nature, is shown to depend on the form of the interaction potential between the flexible sheets. When three or more sheets interact, additional geometrical considerations determine the hierarchical or sequential nature of the adhesion transitions.

A. A. Evans and E. Lauga, Adhesion transition of flexible sheets, Phys. Rev. E 79 066116 (2009) pdf