Elastic theory of origami-based metamaterials

Lagrangian approach to origami vertex analysis: kinematics

The use of origami in engineering has significantly expanded in recent years, spanning deployable structures across scales, folding robotics and mechanical metamaterials. However, finding foldable paths can be a formidable task as the kinematics are determined by a nonlinear system of equations, often with several degrees of freedom. In this article, we leverage a Lagrangian approach to derive reduced-order compatibility conditions for rigid-facet origami vertices with reflection and rotational symmetries. Then, using the reduced-order conditions, we derive exact, multi-degree of freedom solutions for degree 6 and degree 8 vertices with prescribed symmetries. The exact kinematic solutions allow us to efficiently investigate the topology of allowable kinematics, including the consideration of a self-contact constraint, and then visually interpret the role of geometric design parameters on these admissible fold paths by monitoring the change in the kinematic topology. We then introduce a procedure to construct lower-symmetry kinematic solutions by breaking symmetry of higher-order kinematic solutions in a systematic way that preserves compatibility. The multi-degree of freedom solutions discovered here should assist with building intuition of the kinematic feasibility of higher-degree origami vertices and also facilitate the development of new algorithmic procedures for origami-engineering design.This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Origami Morphing Surfaces with Arrayed Quasi-Rigid-Foldable Polyhedrons

Artificial morphing surfaces, inspired by the high adaptability of biological tissues, have emerged as a significant area of research in recent years. However, the practical applications of these surfaces, constructed from soft materials, are considerably limited due to their low shear stiffness. Rigid-foldable cylinders are anisotropic structures that exhibit high adaptability and shear stiffness. Thus, they have the potential to address this issue. However, changes in shape and area at both ends during folding can lead to collisions or gaps on the morphing surface. Here, a quasi-rigid-foldable (QRF) rate is first introduced to quantify the rigid-foldability of a foldable structure and validate it through experiments. More importantly, a QRF polyhedron is then proposed, which is not only notably anisotropic, similar to a rigid-foldable cylinder, but also exhibits a zero-Poisson's ratio property, making it suitable for arraying as morphing surfaces without any collisions or gaps. Such surfaces have a myriad of applications, including modulating electromagnetic waves, gripping fragile objects, and serving as soles for climbing robots.

4D printed origami-inspired accordion, Kresling and Yoshimura tubes

Applying tessellated origami patterns to the design of mechanical materials can enhance properties such as strength-to-weight ratio and impact absorption ability. Another advantage is the predictability of the deformation mechanics since origami materials typically deform through the folding and unfolding of their creases. This work focuses on creating 4D printed flexible tubular origami based on three different origami patterns: the accordion, the Kresling and the Yoshimura origami patterns, fabricated with a flexible polylactic acid (PLA) filament with heat-activated shape memory effect. The shape memory characteristics of the self-unfolding structures were then harnessed at 60°C, 75°C and 90°C. Due to differences in the folding patterns of each origami design, significant differences in behaviour were observed during shape programming and actuation. Among the three patterns, the accordion proved to be the most effective for actuation as the overall structure can be compressed following the folding crease lines. In comparison, the Kresling pattern exhibited cracking at crease locations during deformation, while the Yoshimura pattern buckled and did not fold as expected at the crease lines. To demonstrate a potential application, an accordion-patterned origami 4D printed tube for use in hand rehabilitation devices was designed and tested as a proof-of-concept prototype incorporating self-unfolding origami.

Proportions of conical motifs: Optimal packing via the spherical image

We present a scheme for calculating the shape of two well-known conical motifs: the d-Cone and the e-Cone. Each begins as a thin, flat disk, before buckling during loading into a deformed shape with distinctive, asymmetrical conical features and a localised apex. Various deformed equilibrium models rightly assume a developable shape, with a particular focus on determining how much of the disk detaches from how it is supported during buckling; they are, nevertheless, extensively curated analytically, and must confront (some, ingeniously) the question of singular, viz., infinite properties at the conical apex. In this study, we find an approximate description of shape that reveals the extent of detachment, from an analogous mobile vertex that packages optimally according to its constraints. To this end, we further develop the usage of Gauss's Mapping and the associated spherical image, which has been used previously, but only to confirm known properties of deformed shape. Despite the simplicity of our approach, remarkably good predictions are availed, perhaps because such problems of extreme deformation are geometrically (rather than equilibrium) dominated.

Connecting the branches of multistable non-Euclidean origami by crease stretching

Non-Euclidean origami is a promising technique for designing multistable deployable structures folded from nonplanar developable surfaces. The impossibility of flat foldability inherent to non-Euclidean origami results in two disconnected solution branches each with the same angular deficiency but opposite handedness. We show that these regions can be connected via "crease stretching," wherein the creases exhibit extensibility in addition to torsional stiffness. We further reveal that crease stretching acts as an energy storage method capable of passive deployment and control. Specifically, we show that in a Miura-Ori system with a single stretchable crease, this is achieved via two unique, easy to realize, equilibrium folding pathways for a certain wide set of parameters. In particular, we demonstrate that this connection mostly preserves the stable states of the non-Euclidean system, while resulting in a third stable state enabled only by the interaction of crease torsion and stretching. Finally, we show that this simplified model can be used as an efficient and robust tool for inverse design of multistable origami based on closed-form predictions that yield the system parameters required to attain multiple, desired stable shapes. This facilitates the implementation of multistable origami for applications in architecture materials, robotics, and deployable structures.

Multistability, symmetry and geometric conservation in eightfold waterbomb origami

The nonlinearities inherent in the mechanics of origami make it a rich design space for multistable structures and mechanical metamaterials. Here, we investigate the multistability of a classic origami base: the symmetric eightfold waterbomb. We prove that the waterbomb is bistable for certain crease properties, and derive bounds on, and closed-form approximations of, its stable states. We introduce a simplified form of the waterbomb kinematics and present a design procedure for tuning the depth and the symmetry/asymmetry of its energy wells. By incorporating the concept of pretensioned torsional springs, we also demonstrate the existence of tristable cases for the waterbomb fold pattern. We then apply the analysis of a single waterbomb to study quasi-one-dimensional arrays of waterbombs, where we discover a conserved geometric-kinematic quantity in which the number of popped-up and popped-down vertices is determined uniquely through analysis of the origami structure’s boundaries. This culminates with a discussion of how the quasi-one-dimensional array may be designed to achieve stable states with various degeneracies, kinematics and gaps between energy levels. Collectively, this work presents an alternative approach for characterizing origami multistability properties and reveals an origami design motif that has potential applications in physically reconfigurable structures, mechanical energy absorption and metamaterials.

Shape reconfiguration through origami folding sets an upper limit on drag

Mechanisms of drag reduction through shape reconfiguration have been extensively studied on model geometries of plates and beams that deform primarily in bending. Adding an origami crease pattern to such plates produces a distinct class of deformation modes, with large shape changes along selected degrees of freedom. Here, we investigate the impact of those creases on reconfiguration processes and on drag, focusing on the waterbomb base as a generic case. When placed in a uniform airflow, this origami unit folds into a compact structure, whose frontal area collapses with increasing flow velocity. It enhances drag reduction to the point that fluid loading eventually ceases to increase with flow speed, reaching an upper limit. We further show that this limit is adjustable through the origami structural parameters: the stiffness and rest angle of the folds, and their pattern. Experimental results, corroborated by a fluid–elastic theoretical model, point to a scenario consistent with the previous literature: reconfiguration is governed by a dimensionless Cauchy number that measures the competition between fluid loading and elastic resistance to deformation, here embodied in creases. This foldable system yet stands out through the rare passive drag-capping lever it provides, a valuable asset for self-protection in strong wind.

New Use of BIM-Origami-Based Techniques for Energy Optimisation of Buildings

Outstanding properties and advanced functionalities of thermal–regulatory by origami-based architecture materials have been shown at various scales. However, in order to model and manage its programmable mechanical properties by Building Information Modelling (BIM) for use in a covering structure is not a simple task. The aim of this study was to model an element that forms a dynamic shell that prevents or allows the perpendicular incidence of the sun into the infrastructure. Parametric modelling of such complex structures was performed by Grasshopper and Rhinoceros 3D and were rendered by using the V-ray’s plugin. The elements followed the principles of origami to readjust its geometry considering the sun position, changing the shadow in real time depending on the momentary interest. The results of the project show that quadrangular was the most suitable Origami shape for façade elements. In addition, a BIM-based automated system capable of modifying façade elements considering the sun position was performed. The significance of this research relies on the first implementation and design of an Origami constructive element using BIM methodology, showing its viability and opening outstanding future research lines in terms of sustainability and energy efficiency.

Large-displacement symmetrical foldable cones

A foldable cone, or f-cone, is a discrete vertex folded from flat with curved faces, and is an exemplar of nonrigid origami. We determine the exact equilibrium shape of a well-folded symmetrical f-cone by geometrical considerations alone, by treating as cones intersecting along original fold lines expressing equal fold angles. When moderate displacements prevail, the shape is found alternatively from the spherical image of the vertex, from deploying Gauss's mapping. The analytical working is much less compared to direct linearization of the exact solution framework; expressions for shape parameters are availed in closed form, and some of their variation is accurately retained compared to the exact case.

Plasticity and aging of folded elastic sheets

We investigate the dissipative mechanisms exhibited by creased material sheets when subjected to mechanical loading, which comes in the form of plasticity and relaxation phenomena within the creases. After demonstrating that plasticity mostly affects the rest angle of the creases, we devise a mapping between this quantity and the macroscopic state of the system that allows us to track its reference configuration along an arbitrary loading path, resulting in a powerful monitoring and design tool for crease-based metamaterials. Furthermore, we show that complex relaxation phenomena, in particular memory effects, can give rise to a nonmonotonic response at the crease level, possibly relating to the similar behavior reported for crumpled sheets. We describe our observations through a classical double-logarithmic time evolution and obtain a constitutive behavior compatible with that of the underlying material. Thus the lever effect provided by the crease allows magnified access to the material's rheology.

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.

Folding of Tubular Waterbomb

Origami has recently emerged as a promising building block of mechanical metamaterials because it offers a purely geometric design approach independent of scale and constituent material. The folding mechanics of origami-inspired metamaterials, i.e., whether the deformation involves only rotation of crease lines (rigid origami) or both crease rotation and facet distortion (nonrigid origami), is critical for fine-tuning their mechanical properties yet very difficult to determine for origami patterns with complex behaviors. Here, we characterize the folding of tubular waterbomb using a combined kinematic and structural analysis. We for the first time uncover that a waterbomb tube can undergo a mixed mode involving both rigid origami motion and nonrigid structural deformation, and the transition between them can lead to a substantial change in the stiffness. Furthermore, we derive theoretically the range of geometric parameters for the transition to occur, which paves the road to program the mechanical properties of the waterbomb pattern. We expect that such analysis and design approach will be applicable to more general origami patterns to create innovative programmable metamaterials, serving for a wide range of applications including aerospace systems, soft robotics, morphing structures, and medical devices.

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.

Exploring multistability in prismatic metamaterials through local actuation

Metamaterials are artificial materials that derive their unusual properties from their periodic architecture. Some metamaterials can deform their internal structure to switch between different properties. However, the precise control of these deformations remains a challenge, as these structures often exhibit non-linear mechanical behavior. We introduce a computational and experimental strategy to explore the folding behavior of a range of 3D prismatic building blocks that exhibit controllable multifunctionality. By applying local actuation patterns, we are able to explore and visualize their complex mechanical behavior. We find a vast and discrete set of mechanically stable configurations, that arise from local minima in their elastic energy. Additionally these building blocks can be assembled into metamaterials that exhibit similar behavior. The mechanical principles on which the multistable behavior is based are scale-independent, making our designs candidates for e.g., reconfigurable acoustic wave guides, microelectronic mechanical systems and energy storage systems.

So far, mostly 2D building blocks have been used to make origami-type multistable metamaterials that can keep their shape when deformed. Here, the authors introduce a local actuation strategy to explore the energy landscape of multistable metamaterials fabricated from 3D prismatic building blocks.

Foldable cones as a framework for nonrigid origami

The study of origami-based mechanical metamaterials usually focuses on the kinematics of deployable structures made of an assembly of rigid flat plates connected by hinges. When the elastic response of each panel is taken into account, novel behaviors take place, as in the case of foldable cones (f-cones): circular sheets decorated by radial creases around which they can fold. These structures exhibit bistability, in the sense that they can snap through from one metastable configuration to another. In this work, we study the elastic behavior of isometric f-cones for any deflection and crease mechanics, which introduce nonlinear corrections to a linear model studied previously. Furthermore, we test the inextensibility hypothesis by means of a continuous numerical model that includes both the extended nature of the creases, stretching and bending deformations of the panels. The results show that this phase-field-like model could become an efficient numerical tool for the study of realistic origami structures.

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.

Local mechanical description of an elastic fold

To go beyond the simple model for the fold as two flexible surfaces or faces linked by a crease that behaves as an elastic hinge, we carefully shape and anneal a crease within a polymer sheet and study its mechanical response. First, we carry out an experimental study that involves recording both the shape of the fold in various loading configurations and the associated force needed to deform it. Then, an elastic model of the fold is built upon a continuous description of both the faces and the crease as a thin sheet with a non-flat reference configuration. The comparison between the model and experiments yields the local fold properties and explains the significant differences we observe between tensile and compression regimes. Furthermore, an asymptotic study of the fold deformation enables us to determine the local shape of the crease and identify the origin of its mechanical behaviour.

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.

Bioinspired spring origami

Origami enables folding of objects into a variety of shapes in arts, engineering, and biological systems. In contrast to well-known paper-folded objects, the wing of the earwig has an exquisite natural folding system that cannot be sufficiently described by current origami models. Such an unusual biological system displays incompatible folding patterns, remains open by a bistable locking mechanism during flight, and self-folds rapidly without muscular actuation. We show that these notable functionalities arise from the protein-rich joints of the earwig wing, which work as extensional and rotational springs between facets. Inspired by this biological wing, we establish a spring origami model that broadens the folding design space of traditional origami and allows for the fabrication of precisely tunable, four-dimensional-printed objects with programmable bioinspired morphing functionalities.

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.

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.

Geometry and design of origami bellows with tunable response

Origami folded cylinders (origami bellows) have found increasingly sophisticated applications in space flight and medicine. In spite of this interest, a general understanding of the mechanics of an origami folded cylinder has been elusive. With a newly developed set of geometrical tools, we have found an analytic solution for all possible cylindrical rigid-face states of both Miura-ori and triangular tessellations. Although an idealized bellows in both of these families may have two allowed rigid-face configurations over a well-defined region, the corresponding physical device, limited by nonzero material thickness and forced to balance hinge and plate-bending energy, often cannot stably maintain a stowed configuration. We have identified the parameters that control this emergent bistability, and we have demonstrated the ability to design and fabricate bellows with tunable deployability.

k-cones and kirigami metamaterials

We are inspired by the tensile buckling of a thin sheet with a slit to create a foldable planar metamaterial. The buckled shape comprises two pairs of identical e-cones connected to the slit, which we refer to as a k-cone. We approximate this shape as discrete vertices that can be folded out of plane as the slit is pulled apart. We determine their kinematics and we calculate generic shape properties using a simple elastic model of the folded shape. We then show how the folded sheet may be tessellated as a unit cell within a larger sheet, which may be constructed a priori by cutting and folding the latter in a regular way, in order to form a planar kirigami structure with a single degree of freedom.

Fundamental conical defects: The d-cone, its e-cone, and its p-cone

We consider well-known surface disclinations by cutting, joining, and folding pieces of paper card. The resulting shapes have a discrete, folded vertex whose geometry is described easily by Gauss's mapping, in particular, we can relate the degree of angular excess, or deficit, to the size of fold line rotations by the area enclosed by the vector diagram of these rotations. This is well known for the case of a so-called "d-cone" of zero angular deficit, and we formulate the same for a general disclination. This method allows us to observe kinematic properties in a meaningful way without needing to consider equilibrium. Importantly, the simple vector nature of our analysis shows that some disclinations are primitive; and that other types, such as d-cones, are amalgamations of them.