Triclinic Metamaterials by Tristable Origami with Reprogrammable Frustration

Modular chiral origami metamaterials

Metamaterials with multimodal deformation mechanisms resemble machines1,2, especially when endowed with autonomous functionality. A representative architected assembly, with tunable chirality, converts linear motion into rotation3. These chiral metamaterials with a machine-like dual modality have potential use in areas such as wave manipulation4, optical activity related to circular polarization5 and chiral active fluids6. However, the dual motions are essentially coupled and cannot be independently controlled. Moreover, they are restricted to small deformation, that is, strain ≤2%, which limits their applications. Here we establish modular chiral metamaterials, consisting of auxetic planar tessellations and origami-inspired columnar arrays, with decoupled actuation. Under single-degree-of-freedom actuation, the assembly twists between 0° and 90°, contracts in-plane up to 25% and shrinks out-of-plane more than 50%. Using experiments and simulations, we show that the deformation of the assembly involves in-plane twist and contraction dominated by the rotating-square tessellations and out-of-plane shrinkage dominated by the tubular Kresling origami arrays. Moreover, we demonstrate two distinct actuation conditions: twist with free translation and linear displacement with free rotation. Our metamaterial is built on a highly modular assembly, which enables reprogrammable instability, local chirality control, tunable loading capacity and scalability. Our concept provides routes towards multimodal, multistable and reprogrammable machines, with applications in robotic transformers, thermoregulation, mechanical memories in hysteresis loops, non-commutative state transition and plug-and-play functional assemblies for energy absorption and information encryption.

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.

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'.

Programming the mechanical properties of double-corrugated metamaterials by varying mountain-valley assignments

Origami metamaterials have gained significant attention in recent years, with extensive analysis conducted on their mechanical properties. Previous studies have primarily focused on the effects of design angles, panel side lengths, folding angles or other geometric and material parameters. However, mountain-valley crease assignments of origami patterns, which significantly effect both the geometric and mechanical properties, have yet to be studied in depth. In this article, we create a series of double-corrugated metamaterials with diverse mountain-valley assignments and analyse their Poisson's ratios and mechanical properties under compression loading. The findings of our study demonstrate that varying the mountain-valley assignments allows for the construction of metamaterials with consistent or distinct Poisson's ratios. These assignments have the capability to program the magnitude and to vary the rate of the folding angles. Furthermore, the mechanical properties of the corresponding metamaterials, in particular the specific energy absorption (SEA) and normalized stiffness, exhibit positive correlations with the respective folding angles. Our study highlights the significance of varying mountain-valley assignments as a promising approach for designing origami metamaterials and programming their mechanical properties.This article is part of the theme issue 'Origami/Kirigami-inspired structures: from fundamentals to applications'.

Group-theoretic analysis of symmetry-preserving deployable structures and metamaterials

Many deployable structures in nature, as well as human-made mechanisms, preserve symmetry as their configurations evolve. Examples in nature include blooming flowers, dilation of the iris within the human eye, viral capsid maturation and molecular and bacterial motors. Engineered examples include opening umbrellas, elongating scissor jacks, variable apertures in cameras, expanding Hoberman spheres and some kinds of morphing origami structures. In these cases, the structures either preserve a discrete symmetry group or are described as an evolution from one discrete symmetry group to another of the same type as the structure deploys. Likewise, elastic metamaterials built from lattice structures can also preserve symmetry type while passively deforming and changing lattice parameters. A mathematical formulation of such transitions/deployments is articulated here. It is shown that if [Formula: see text] is Euclidean space, [Formula: see text] is a continuous group of motions of Euclidean space and [Formula: see text] is the type of the discrete subgroup of [Formula: see text] describing the symmetries of the deploying structure, then the symmetry of the evolving structure can be described by time-dependent subgroups of [Formula: see text] of the form [Formula: see text], where [Formula: see text] is a time-dependent affine transformation. Then, instead of considering the whole structure in [Formula: see text], a 'sector' of it that lives in the orbit space [Formula: see text] can be considered at each instant in time, and instead of considering all motions in [Formula: see text], only representatives from right cosets in the space [Formula: see text] need to be considered. This article is part of the theme issue 'Current developments in elastic and acoustic metamaterials science (Part 1)'.

Emergent disorder and mechanical memory in periodic metamaterials

Ordered mechanical systems typically have one or only a few stable rest configurations, and hence are not considered useful for encoding memory. Multistable and history-dependent responses usually emerge from quenched disorder, for example in amorphous solids or crumpled sheets. In contrast, due to geometric frustration, periodic magnetic systems can create their own disorder and espouse an extensive manifold of quasi-degenerate configurations. Inspired by the topological structure of frustrated artificial spin ices, we introduce an approach to design ordered, periodic mechanical metamaterials that exhibit an extensive set of spatially disordered states. While our design exploits the correspondence between frustration in magnetism and incompatibility in meta-mechanics, our mechanical systems encompass continuous degrees of freedom, and thus generalize their magnetic counterparts. We show how such systems exhibit non-Abelian and history-dependent responses, as their state can depend on the order in which external manipulations were applied. We demonstrate how this richness of the dynamics enables to recognize, from a static measurement of the final state, the sequence of operations that an extended system underwent. Thus, multistability and potential to perform computation emerge from geometric frustration in ordered mechanical lattices that create their own disorder.

Zero modes activation to reconcile floppiness, rigidity, and multistability into an all-in-one class of reprogrammable metamaterials

Existing mechanical metamaterials are typically designed to either withstand loads as a stiff structure, shape morph as a floppy mechanism, or trap energy as a multistable matter, distinct behaviours that correspond to three primary classes of macroscopic solids. Their stiffness and stability are sealed permanently into their architecture, mostly remaining immutable post-fabrication due to the invariance of zero modes. Here, we introduce an all-in-one reprogrammable class of Kagome metamaterials that enable the in-situ reprogramming of zero modes to access the apparently conflicting properties of all classes. Through the selective activation of metahinges via self-contact, their architecture can be switched to acquire on-demand rigidity, floppiness, or global multistability, bridging the seemingly uncrossable gap between structures, mechanisms, and multistable matters. We showcase the versatile generalizations of the metahinge and remarkable reprogrammability of zero modes for a range of properties including stiffness, mechanical signal guiding, buckling modes, phonon spectra, and auxeticity, opening a plethora of opportunities for all-in-one materials and devices.

Unravelling Size-Dependent and Coupled Properties in Mechanical Metamaterials: A Couple-Stress Theory Perspective

The lack of material characteristic length scale prevents classical continuum theory (CCT) from recognizing size effect. Additionally, the even-order material property tensors associated with CCT only characterize the materials' centrosymmetric behavior and overlook their intrinsic chirality and polarity. Moreover, CCT is not reducible to 2D and 1D space without adding couples and higher-order deformation gradients. Despite several generalized continuum theories proposed over the past century to overcome the limitations of CCT, the broad application of these theories in the field of mechanical metamaterials has encountered significant challenges. These obstacles primarily arise from a limited understanding of the material coefficients associated with these theories, impeding their widespread adoption. Implementing a bottom-up approach based on augmented asymptotic homogenization, a consistent and self-sufficient effective couple-stress theory for materials with microstructures in 3D, 2D, and 1D spaces is presented. Utilizing the developed models, material properties associated with axial-twist, shear-bending, bending-twist, and double curvature bending couplings in mechanical metamaterials are characterized. The accuracy of these homogenized models is investigated by comparing them with the detailed finite element models and experiments performed on 3D-printed samples. The proposed models provide a benchmark for the rational design, classification, and manufacturing of mechanical metamaterials with programmable coupled deformation modes.

Anisotropic Morphing in Bistable Kirigami through Symmetry Breaking and Geometric Frustration

Shape morphing in bistable kirigami enables remarkable functionalities appealing to a diverse range of applications across the spectrum of length scale. At the core of their shape shifting lies the architecture of their repeating unit, where highly deformable slits and quasi-rigid rotating units often exhibit multiple symmetries that confer isotropic deployment obeying uniform scaling transformation. In this work, symmetry breaking in bistable kirigami is investigated to access geometric frustration and anisotropic morphing, enabling arbitrarily scaled deployment in planar and spatial bistable domains. With an analysis on their symmetry properties complemented by a systematic investigation integrating semi-analytical derivations, numerical simulations, and experiments on elastic kirigami sheets, this work unveils the fundamental relations between slit symmetry, geometric frustration, and anisotropic bistable deployment. Furthermore, asymmetric kirigami units are leveraged in planar and flat-to-3D demonstrations to showcase the pivotal role of shear deformation in achieving target shapes and functions so far unattainable with uniformly stretchable kirigami. The insights provided in this work unveil the role of slit symmetry breaking in controlling the anisotropic bistable deployment of soft kirigami metamaterials, enriching the range of achievable functionalities for applications spanning deployable space structures, wearable technologies, and soft machines.

Origami Metamaterials Enable Low-Stress-Driven Giant Elastocaloric Effect

Elastocaloric materials, capable of achieving reversible thermal changes in response to a uniaxial stress, have attracted considerable attention for applications in advanced thermal management technologies, owing to their environmental friendliness and economic benefits. However, most elastocaloric materials operating on the basis of first/second-order phase transition often exhibit a limited caloric response, field hysteresis, and restricted working temperature ranges. This study resorts to origami engineering for realizing multifunctional metamaterials with exceptional elastocaloric effects at both nano (exemplified by computational simulations for graphene) and meso (demonstrated by experimentation on thermoplastic polyurethane elastomers) scales. The findings uncover that the graphene origami exhibits low-stress-driven reversible and giant elastocaloric effects without a hysteresis loss and with a high elastocaloric strength. These effects are achieved across a wide working temperature range (100-600 K) and are tailorable by fine-tuning the topological parameters and configurational status of the origami metamaterials. We demonstrate the scalability of the origami design strategy for magnifying the elastocaloric effect by the 3D printing of a mesoscale origami-inspired thermoplastic polyurethane metastructure that showcases enhanced elastocaloric performance at room temperature. This study presents the potential for the realization of architected elastocaloric materials through surface functionalization and origami engineering. The findings impart exciting prospects of elastocaloric origami metamaterials as the next generation of multiscale and sustainable thermal management technologies.

Algorithmic encoding of adaptive responses in temperature-sensing multimaterial architectures

We envision programmable matters that can alter their physical properties in desirable manners based on user input or autonomous sensing. This vision motivates the pursuit of mechanical metamaterials that interact with the environment in a programmable fashion. However, this has not been systematically achieved for soft metamaterials because of the highly nonlinear deformation and underdevelopment of rational design strategies. Here, we use computational morphogenesis and multimaterial polymer 3D printing to systematically create soft metamaterials with arbitrarily programmable temperature-switchable nonlinear mechanical responses under large deformations. This is made possible by harnessing the distinct glass transition temperatures of different polymers, which, when optimally synthesized, produce local and giant stiffness changes in a controllable manner. Featuring complex geometries, the generated structures and metamaterials exhibit fundamentally different yet programmable nonlinear force-displacement relations and deformation patterns as temperature varies. The rational design and fabrication establish an objective-oriented synthesis of metamaterials with freely tunable thermally adaptive behaviors. This imbues structures and materials with environment-aware intelligence.

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.

In Situ Activation of Snap-Through Instability in Multi-Response Metamaterials through Multistable Topological Transformation

Snap-through instability has been widely leveraged in metamaterials to attain non-monotonic responses for a specific subset of applications where conventional monotonic materials fail to perform. In the remaining more plentiful set of ordinary applications, snap-through instability is harmful, and current snapping metamaterials become inadequate because their capacity to snap cannot be suppressed post-fabrication. Here, a class of topology-transformable metamaterials is introduced to enable in situ activation and deactivation of the snapping capacity, providing a remarkable level of versatility in switching between responses from monotonic to monostable and bistable snap-through. Theoretical analysis, numerical simulations, and experiments are combined to unveil the role played by contact in the topological transformation capable of increasing the geometry incompatibility and confinement stiffness of selected architectural members. The strategy here presented for post-fabrication reprogrammability of matter and on-the-fly response switching paves the way to multifunctionality for application in multiple sectors from mechanical logic gates, and adjustable energy dissipators, to in situ adaptable sport equipment.

Oribron: An Origami-Inspired Deformable Rigid Bronchoscope for Radial Support

The structure of a traditional rigid bronchoscope includes proximal, distal, and body, representing an important means to treat hypoxic diseases. However, the body structure is too simple, resulting in the utilization rate of oxygen being usually low. In this work, we reported a deformable rigid bronchoscope (named Oribron) by adding a Waterbomb origami structure to the body. The Waterbomb's backbone is made of films, and the pneumatic actuators are placed inside it to achieve rapid deformation at low pressure. Experiments showed that Waterbomb has a unique deformation mechanism, which can transform from a small-diameter configuration (#1) to a large-diameter configuration (#2), showing excellent radial support capability. When Oribron entered or left the trachea, the Waterbomb remained in #1. When Oribron is working, the Waterbomb transforms from #1 to #2. Since #2 reduces the gap between the bronchoscope and the tracheal wall, it effectively slows down the rate of oxygen loss, thus promoting the absorption of oxygen by the patient. Therefore, we believe that this work will provide a new strategy for the integrated development of origami and medical devices.