Lattice mechanics of origami tessellations

Coarse-grained fundamental forms for characterizing isometries of trapezoid-based origami metamaterials

Investigations of origami tessellations as effective media reveal the ability to program the components of their elasticity tensor, and thus control the mechanical behavior of thin sheets. However, existing efforts focus on crease patterns that are composed of parallelogram faces where the parallel lines constrain the quasi-static elastic response. In this work, crease patterns composed of more general trapezoid faces are considered and their low-energy linear response is explored. Deformations of such origami tessellations are modeled as linear isometries that do not stretch individual panels at the small scale yet map to non-isometric changes of coarse-grained fundamental forms that quantify how the effective medium strains and curves at the large scale. Two distinct mode shapes, a rigid breathing mode and a nonrigid shearing mode, are identified in the continuum model. A specific example, which we refer to as Arc-Morph origami, is presented with analytical expressions for its deformations in both the discrete and continuous models. A developable specimen is fabricated and tested to validate the analytical predictions. This work advances the continuum modeling of origami tessellations as effective media with the incorporation of more generic faces and ground states, thereby enabling the investigation of novel designs and applications.

Improving mass lumping and stiffness parameters of bar and hinge model for accurate modal dynamics of origami structures

The bar and hinge framework uses truss elements and rotational springs to efficiently model the structural behaviour of origami. The framework is especially useful to investigate origami metamaterials as they have repeating geometry, which makes conventional finite element simulations very expensive due to a large number of degrees of freedom. This work proposes improvements to the parameters of bar and hinge model within the context of structural dynamics, specifically modal analysis under small deformations, which has not been carried out previously in the literature. A range of low-frequency modes involving origami folding and panel bending deformations that can be accurately captured by the bar and hinge framework are identified. Within this range, bar and hinge parameters like the lumped masses and the rotational spring stiffness values are derived using conservation laws and finite element tests. The best among the proposed schemes is found to predict natural frequencies of the considered origami structures to within 10% maximum error, improving the accuracy by more than three times from existing schemes. In most cases, the errors in natural frequencies are less than 5%. This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Discrete symmetries control geometric mechanics in parallelogram-based origami

Geometric compatibility constraints dictate the mechanical response of soft systems that can be utilized for the design of mechanical metamaterials such as the negative Poisson's ratio Miura-ori origami crease pattern. Here, we develop a formalism for linear compatibility that enables explicit investigation of the interplay between geometric symmetries and functionality in origami crease patterns. We apply this formalism to a particular class of periodic crease patterns with unit cells composed of four arbitrary parallelogram faces and establish that their mechanical response is characterized by an anticommuting symmetry. In particular, we show that the modes are eigenstates of this symmetry operator and that these modes are simultaneously diagonalizable with the symmetric strain operator and the antisymmetric curvature operator. This feature reveals that the anticommuting symmetry defines an equivalence class of crease pattern geometries that possess equal and opposite in-plane and out-of-plane Poisson's ratios. Finally, we show that such Poisson's ratios generically change sign as the crease pattern rigidly folds between degenerate ground states and we determine subfamilies that possess strictly negative in-plane or out-of-plane Poisson's ratios throughout all configurations.

Hidden symmetries generate rigid folding mechanisms in periodic origami

We consider the zero-energy deformations of periodic origami sheets with generic crease patterns. Using a mapping from the linear folding motions of such sheets to force-bearing modes in conjunction with the Maxwell-Calladine index theorem we derive a relation between the number of linear folding motions and the number of rigid body modes that depends only on the average coordination number of the origami's vertices. This supports the recent result by Tachi [T. Tachi, Origami 6, 97-108 (2015)] which shows periodic origami sheets with triangular faces exhibit two-dimensional spaces of rigidly foldable cylindrical configurations. We also find, through analytical calculation and numerical simulation, branching of this configuration space from the flat state due to geometric compatibility constraints that prohibit finite Gaussian curvature. The same counting argument leads to pairing of spatially varying modes at opposite wavenumber in triangulated origami, preventing topological polarization but permitting a family of zero-energy deformations in the bulk that may be used to reconfigure the origami sheet.

Non-Euclidean origami

Traditional origami starts from flat surfaces, leading to crease patterns consisting of Euclidean vertices. However, Euclidean vertices are limited in their folding motions, are degenerate, and suffer from misfolding. Here we show how non-Euclidean 4-vertices overcome these limitations by lifting this degeneracy, and that when the elasticity of the hinges is taken into account, non-Euclidean 4-vertices permit higher order multistability. We harness these advantages to design an origami inverter that does not suffer from misfolding and to physically realize a tristable vertex.

Mechanics of generically creased disks

Folded structures are often idealized as a series of rigid faces connected by creases acting as revolute hinges. However, real folded structures can deform between creases. An example of particular interest is a disk decorated by multiple radial creases. Such disks are bistable, snapping between a "natural" and "inverted" shape. We investigate the mechanical behavior of these creased disks and propose a new analytical approach to describe their mechanics. Detailed experiments are performed which show that, when indented at the center, a localized dimple forms, precluding the conical shape assumed in previous studies. As the indentation depth increases this dimple expands radially until reaching the disk edge when it snaps to the inverted shape, which has a conical form. We develop an analytical model which approximates each face as a series of rigid facets connected by hinges that can both rotate and stretch. Energy expressions are derived relating hinge rotation and stretching to compatible shell deformations of the facets and equilibrium enforced by minimizing the total strain energy. By increasing the number of facets, the mechanics of the continuum shell is approached asymptotically. The analysis shows that membrane stretching of the faces is required when a conical form of deformation is enforced. However, in the limit of zero thickness, the forming and propagation of a localized dimple is inextensional. This new approach relates the kinematic analysis of rigid origami to the mechanics of thin shells, offering an efficient method to predict the behavior of folded structures.

Theory and practice of origami in science

No longer just the purview of artists and enthusiasts, origami engineering has emerged as a potentially powerful tool to create three dimensional structures on disparate scales. Whether origami (and the closely related kirigami) engineering can emerge as a useful technology will depend crucially on both fundamental theoretical advances as well as the development of further fabrication tools.

Beyond power amplification: latch-mediated spring actuation is an emerging framework for the study of diverse elastic systems

Rapid biological movements, such as the extraordinary strikes of mantis shrimp and accelerations of jumping insects, have captivated generations of scientists and engineers. These organisms store energy in elastic structures (e.g. springs) and then rapidly release it using latches, such that movement is driven by the rapid conversion of stored elastic to kinetic energy using springs, with the dynamics of this conversion mediated by latches. Initially drawn to these systems by an interest in the muscle power limits of small jumping insects, biologists established the idea of power amplification, which refers both to a measurement technique and to a conceptual framework defined by the mechanical power output of a system exceeding muscle limits. However, the field of fast elastically driven movements has expanded to encompass diverse biological and synthetic systems that do not have muscles - such as the surface tension catapults of fungal spores and launches of plant seeds. Furthermore, while latches have been recognized as an essential part of many elastic systems, their role in mediating the storage and release of elastic energy from the spring is only now being elucidated. Here, we critically examine the metrics and concepts of power amplification and encourage a framework centered on latch-mediated spring actuation (LaMSA). We emphasize approaches and metrics of LaMSA systems that will forge a pathway toward a principled, interdisciplinary field.

Strain Reversal in Actuated Origami Structures

Origami in engineering is gaining interest for its potential as deployable or shape-adaptive structures. Practical systems could employ a network of actuators distributed across the structure to induce these deformations. Selecting the actuator locations requires an understanding of how the effect of a single actuator propagates spatially in an origami structure. We combine experimental results, finite element analysis, and reduced-order bar-and-hinge models to show how a localized static actuation decays elastically in Miura-ori tubes and sheets. We observe a strain reversal, before the origami structure springs back to the initial configuration further away from the point of actuation. The strain reversal is the result of bending of the facets, while the spring back requires in-plane facet deformations.

The compressive strength of crumpled matter

Crumpling a sheet creates a unique, stiff and lightweight structure. Use of crumples in engineering design is limited because there are not simple, physically motivated structure-property relations available for crumpled materials; one cannot trust a crumple. On the contrary, we demonstrate that an empirical model reliably predicts the reaction of a crumpled sheet to a compressive force. Experiments show that the prediction is quantitative over 50 orders of magnitude in force, for purely elastic and highly plastic polymer films. Our data does not match recent theoretical predictions based on the dominance of building-block structures (bends, folds, d-cones, and ridges). However, by directly measuring substructures, we show clearly that the bending in the stretching ridge is responsible for the strength of both elastic and plastic crumples. Our simple, predictive model may open the door to the engineering use of a vast range of materials in this state of crumpled matter.

Crumpled matter hasn’t been widely used to solve real world engineering problems largely due to the lack of quantitative models. Croll et al. show that it is the bending in ridges making both elastic and plastic sheets resistant to compression and describe the mechanical response using an empirical model.

Architected Origami Materials: How Folding Creates Sophisticated Mechanical Properties

Origami, the ancient Japanese art of paper folding, is not only an inspiring technique to create sophisticated shapes, but also a surprisingly powerful method to induce nonlinear mechanical properties. Over the last decade, advances in crease design, mechanics modeling, and scalable fabrication have fostered the rapid emergence of architected origami materials. These materials typically consist of folded origami sheets or modules with intricate 3D geometries, and feature many unique and desirable material properties like auxetics, tunable nonlinear stiffness, multistability, and impact absorption. Rich designs in origami offer great freedom to design the performance of such origami materials, and folding offers a unique opportunity to efficiently fabricate these materials at vastly different sizes. Here, recent studies on the different aspects of origami materials-geometric design, mechanics analysis, achieved properties, and fabrication techniques-are highlighted and the challenges ahead discussed. The synergies between these different aspects will continue to mature and flourish this promising field.

Nonlinear mechanics of non-rigid origami: an efficient computational approach

Origami-inspired designs possess attractive applications to science and engineering (e.g. deployable, self-assembling, adaptable systems). The special geometric arrangement of panels and creases gives rise to unique mechanical properties of origami, such as reconfigurability, making origami designs well suited for tunable structures. Although often being ignored, origami structures exhibit additional soft modes beyond rigid folding due to the flexibility of thin sheets that further influence their behaviour. Actual behaviour of origami structures usually involves significant geometric nonlinearity, which amplifies the influence of additional soft modes. To investigate the nonlinear mechanics of origami structures with deformable panels, we present a structural engineering approach for simulating the nonlinear response of non-rigid origami structures. In this paper, we propose a fully nonlinear, displacement-based implicit formulation for performing static/quasi-static analyses of non-rigid origami structures based on 'bar-and-hinge' models. The formulation itself leads to an efficient and robust numerical implementation. Agreement between real models and numerical simulations demonstrates the ability of the proposed approach to capture key features of origami behaviour.

Self-folding origami at any energy scale

Programmable stiff sheets with a single low-energy folding motion have been sought in fields ranging from the ancient art of origami to modern meta-materials research. Despite such attention, only two extreme classes of crease patterns are usually studied; special Miura-Ori-based zero-energy patterns, in which crease folding requires no sheet bending, and random patterns with high-energy folding, in which the sheet bends as much as creases fold. We present a physical approach that allows systematic exploration of the entire space of crease patterns as a function of the folding energy. Consequently, we uncover statistical results in origami, finding the entropy of crease patterns of given folding energy. Notably, we identify three classes of Mountain-Valley choices that have widely varying ‘typical' folding energies. Our work opens up a wealth of experimentally relevant self-folding origami designs not reliant on Miura-Ori, the Kawasaki condition or any special symmetry in space.

Origami is widely practiced in the design of foldable structures for smart applications and usually consists of stiff sheets that only deform along prescribed creases. Pinson et al. take a statistical physics approach to design and characterize arbitrary patterns as a function of folding energy.

Dynamics of a bistable Miura-origami structure

Origami-inspired structures and materials have shown extraordinary properties and performances originating from the intricate geometries of folding. However, current state of the art studies have mostly focused on static and quasistatic characteristics. This research performs a comprehensive experimental and analytical study on the dynamics of origami folding through investigating a stacked Miura-Ori (SMO) structure with intrinsic bistability. We fabricate and experimentally investigated a bistable SMO prototype with rigid facets and flexible crease lines. Under harmonic base excitation, the SMO exhibits both intrawell and interwell oscillations. Spectrum analyses reveal that the dominant nonlinearities of SMO are quadratic and cubic, which generate rich dynamics including subharmonic and chaotic oscillations. The identified nonlinearities indicate that a third-order polynomial can be employed to approximate the measured force-displacement relationship. Such an approximation is validated via numerical study by qualitatively reproducing the phenomena observed in the experiments. The dynamic characteristics of the bistable SMO resemble those of a Helmholtz-Duffing oscillator (HDO); this suggests the possibility of applying the established tools and insights of HDO to predict origami dynamics. We also show that the bistability of SMO can be programmed within a large design space via tailoring the crease stiffness and initial stress-free configurations. The results of this research offer a wealth of fundamental insights into the dynamics of origami folding, and provide a solid foundation for developing foldable and deployable structures and materials with embedded dynamic functionalities.

Dynamics of Miura-patterned foldable sheets in shear flow

We study the dynamics of piecewise rigid sheets containing predefined crease lines in shear flow. The crease lines act like hinge joints along which the sheet may fold rigidly, i.e. without bending any other crease line. We choose the crease lines such that they tessellate the sheet into a two-dimensional array of parallelograms. Specifically, we focus on a particular arrangement of crease lines known as a Miura-pattern in the origami community. When all the hinges are fully open the sheet is planar, whereas when all are closed the sheet folds over itself to form a compact flat structure. Due to rigidity constraints, the folded state of a Miura-sheet can be described using a single fold angle. The hinged sheet is modeled using the framework of constrained multibody systems in the absence of inertia. The hydrodynamic drag on each of the rigid panels is calculated based on an inscribed elliptic disk, but intra-panel hydrodynamic interactions are neglected. We find that when the motion of a sheet remains symmetric with respect to the flow-gradient plane, after a sufficiently long time, the sheet either exhibits asymptotically periodic tumbling and breathing, indicating approach to a limit cycle; or it reaches a steady state by completely unfolding, which we show to be a half-stable node in the phase space. In the case of asymmetric motion of the sheet with respect to the flow-gradient plane, we find that the terminal state of motion is one of - (i) steady state with a fully unfolded or fully folded configuration, (ii) asymptotically periodic tumbling, indicating approach to a limit cycle, (iii) cyclic tumbling without repetition, indicating a quasiperiodic orbit, or (iv) cyclic tumbling with repetition after several cycles, indicating a resonant quasiperiodic orbit. No chaotic behavior was found.

Decoupling local mechanics from large-scale structure in modular metamaterials

A defining feature of mechanical metamaterials is that their properties are determined by the organization of internal structure instead of the raw fabrication materials. This shift of attention to engineering internal degrees of freedom has coaxed relatively simple materials into exhibiting a wide range of remarkable mechanical properties. For practical applications to be realized, however, this nascent understanding of metamaterial design must be translated into a capacity for engineering large-scale structures with prescribed mechanical functionality. Thus, the challenge is to systematically map desired functionality of large-scale structures backward into a design scheme while using finite parameter domains. Such "inverse design" is often complicated by the deep coupling between large-scale structure and local mechanical function, which limits the available design space. Here, we introduce a design strategy for constructing 1D, 2D, and 3D mechanical metamaterials inspired by modular origami and kirigami. Our approach is to assemble a number of modules into a voxelized large-scale structure, where the module's design has a greater number of mechanical design parameters than the number of constraints imposed by bulk assembly. This inequality allows each voxel in the bulk structure to be uniquely assigned mechanical properties independent from its ability to connect and deform with its neighbors. In studying specific examples of large-scale metamaterial structures we show that a decoupling of global structure from local mechanical function allows for a variety of mechanically and topologically complex designs.

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 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.

Elastic theory of origami-based metamaterials

Origami offers the possibility for new metamaterials whose overall mechanical properties can be programed by acting locally on each crease. Starting from a thin plate and having knowledge about the properties of the material and the folding procedure, one would like to determine the shape taken by the structure at rest and its mechanical response. In this article, we introduce a vector deformation field acting on the imprinted network of creases that allows us to express the geometrical constraints of rigid origami structures in a simple and systematic way. This formalism is then used to write a general covariant expression of the elastic energy of n-creases meeting at a single vertex. Computations of the equilibrium states are then carried out explicitly in two special cases: the generalized waterbomb base and the Miura-Ori. For the waterbomb, we show a generic bistability for any number of creases. For the Miura folding, however, we uncover a phase transition from monostable to bistable states that explains the efficient deployability of this structure for a given range of geometrical and mechanical parameters. Moreover, the analysis shows that geometric frustration induces residual stresses in origami structures that should be taken into account in determining their mechanical response. This formalism can be extended to a general crease network, ordered or otherwise, and so opens new perspectives for the mechanics and the physics of origami-based metamaterials.

Origami building blocks: Generic and special four-vertices

Four rigid panels connected by hinges that meet at a point form a four-vertex, the fundamental building block of origami metamaterials. Most materials designed so far are based on the same four-vertex geometry, and little is known regarding how different geometries affect folding behavior. Here we systematically categorize and analyze the geometries and resulting folding motions of Euclidean four-vertices. Comparing the relative sizes of sector angles, we identify three types of generic vertices and two accompanying subtypes. We determine which folds can fully close and the possible mountain-valley assignments. Next, we consider what occurs when sector angles or sums thereof are set equal, which results in 16 special vertex types. One of these, flat-foldable vertices, has been studied extensively, but we show that a wide variety of qualitatively different folding motions exist for the other 15 special and 3 generic types. Our work establishes a straightforward set of rules for understanding the folding motion of both generic and special four-vertices and serves as a roadmap for designing origami metamaterials.