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.
Origami describes rules for creating folded structures from patterns on a flat sheet, but does not prescribe how patterns can be designed to fit target shapes. Here, starting from the simplest periodic origami pattern that yields one degree-of-freedom collapsible structures, we show that scale-independent elementary geometric constructions and constrained optimization algorithms can be used to determine spatially modulated patterns that yield approximations to given surfaces of constant or varying curvature. Paper models confirm the feasibility of our calculations. We also assess the difficult…
Origami-based mechanical metamaterials have recently received significant scientific interest due to their versatile and reconfigurable architectures. However, it is often challenging to account for all possible geometrical configurations of the origami assembly when each origami cell can take multiple phases. Here, we investigate the reconfigurability of a tessellation of origami-based cellular structures composed of bellows-like unit cells, specifically Tachi-Miura Polyhedron (TMP). One of the unique features of the TMP is that a single cell can take four different phases in a rigid foldable…
Mechanical metamaterials with complex microstructures have superior physical properties such as graded stiffness, negative Poisson's ratio, and advantage in energy absorption. In recent years, origami provide many inspirations in the geometry structure of the metamaterials. Here we present a modular origami inspired reconfigurable metamaterial. Our approach exploits a transformable module consisted of the Sarrus linkage and planar four-bar linkage, and the modules are connected following a tessellation pattern to form a periodic structure. Different arrangements, module shapes and grid pattern…
The principles of origami design have proven useful in a number of technological applications. Origami tessellations in particular constitute a class of morphing metamaterials with unusual geometric and elastic properties. Although inextensible in principle, fine creases allow origami metamaterials to effectively deform non-isometrically. Determining the strains that are compatible with coarse-grained origami kinematics as well as the corresponding elasticity functionals is paramount to understanding and controlling the morphing paths of origami metamaterials. Here, within a unified theory, we…
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 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…
We study the three-dimensional equilibrium shape of a shell formed by a deployed accordion-like origami, made from an elastic sheet decorated by a series of parallel creases crossed by a central longitudinal crease. Surprisingly, while the imprinted crease network does not exhibit a geodesic curvature, the emergent structure is characterized by an effective curvature produced by the deformed central fold. Moreover, both finite element analysis and manually made mylar origamis show a robust empirical relation between the imprinted crease network's dimensions and the apparent curvature. A detail…
Rotational erection system (RES) represents an origami-based design method for generating a three-dimensional (3D) structure from a planar sheet without compression. Its rotational and translational kinematics is fully encoded in a form of prescribed cuts and folds that has only zero degrees of freedom in the geometric limit. Here we characterize mechanical and geometric properties of a threefold symmetric RES by combining finite element numerical simulation and physical experiment. We demonstrate that a plate bending in RES creates a physical route connecting the two energetically separated c…
The article studies the elastic and locomotive properties of Miura-ori-type paper origami. The mechanics of a single paper crease is studied experimentally, and its non-elastic properties turn out to be crucial. The entire origami construction is then described as a collection of individual creases, its capability to launch small objects is evaluated, and the equation of motion is found. Thus, the height of the launched ball is studied theoretically and experimentally as a function of governing parameters.
Non-rigid origami patterns could provide more versatile performance than their rigid counterparts in the design of mechanical metamaterials owing to the simultaneous deformation of facets and creases, but their complex deformation modes make quantitative characterization and programmability of mechanical properties a challenging task. Here, we investigated the tensile behavior of a non-rigid square-twist origami structure with rotational symmetry by combining biaxial tension experiments and finite element modeling. A three-stage deformation process, including tightening, unlocking, and flatten…
State Key Laboratory for Turbulence and Complex Systems, Department of Mechanics and Engineering Science, BIC-ESAT, College of Engineering, Peking University（北京大学工学院湍流与复杂系统国家重点实验室）；CAPT, HEDPS and IFSA Collaborative Innovation Center of MoE, Peking University（北京大学CAPT、HEDPS与IFSA协同创新中心）
Flexible robotics are capable of achieving various functionalities by shape morphing, benefiting from their compliant bodies and reconfigurable structures. Here we construct and study a class of origami springs generalized from the known interleaved origami spring, as promising candidates for shape morphing in flexible robotics. These springs are found to exhibit nonlinear stretch-twist coupling and linear/nonlinear mechanical response in the compression/tension region, analyzed by the demonstrated continuum mechanics models, experiments, and finite element simulations. To improve the mechanic…
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 m…
While memory effects emerge from systems of wildly varying length- and time-scales, the reduction of a complex system with many interacting elements into one simple enough to be understood without also losing the complex behavior continues to be a challenge. Here, we investigate how bistable cylindrical origamis provide such a reduction via tunably-interactive memory behaviors. We base our investigation on folded sheets of Kresling patterns that function as two-state memory units. By linking several units, each with a selected activation energy, we construct a one-dimensional material that exh…
Folding nanopatterned flat sheets into complex 3D structures enables the fabrication of meta-biomaterials that combine a rationally designed 3D architecture (e.g., to tune mechanical and mass transport properties) with nanoscale surface features (e.g., to guide the differentiation of stem cells). Self-folding is an attractive approach for realizing such materials. However, self-folded lattices are generally too compliant as there is an inherent competition between the ease-of-folding requirements and the final load-bearing characteristics. Inspired by sheet metal forming, we propose an alterna…
Nonlinear Mechanics and Predictable Bifurcation of Multi-Cell Kresling Origami Chains
Institute for Infrastructure and Environment, School of Engineering, The University of Edinburgh（爱丁堡大学工程学院基础设施与环境研究所）；Civil, Environmental and Geomatic Engineering, Faculty of Engineering Sciences, University College London（伦敦大学学院工程科学学院土木、环境与地理信息工程系）
We numerically investigate the mechanical and geometrical properties of dense wet granular particles with irreversible attractive interaction. The shear modulus exhibits two inflection points as the packing fraction increases, and the bulk modulus shows a non-monotonic behavior. The coordination number also exhibits two inflection points. The peak position in the pair correlation function shifts to a lower value due to attractive interaction. The Voronoi tessellation of the particle configuration reveals that the probability density function for the volume of the Voronoi cell broadens as the p…
Molecular dynamics simulations are often used to provide feedback in the design workflow of DNA nanostructures. However, even with coarse-grained models, convergence of distributions from unbiased simulation is slow, limiting applications to equilibrium structural properties. Given the increasing interest in dynamic, reconfigurable, and deformable devices, methods that enable efficient quantification of large ranges of motion, conformational transitions, and mechanical deformation are critically needed. Metadynamics is an automated biasing technique that enables the rapid acquisition of molecu…
Kirigami tessellations, regular planar patterns formed by cutting flat, thin sheets, have attracted recent scientific interest for their rich geometries, surprising material properties and promise for technologies. Here we pose and solve the inverse problem of designing the number, size, and orientation of cuts that allows us to convert a closed, compact regular kirigami tessellation of the plane into a deployment that conforms approximately to any prescribed target shape in two and three dimensions. We do this by first identifying the constraints on the lengths and angles of generalized kirig…
To study the elastic properties of rod-like DNA nanostructures, we perform long simulations of these structure using the oxDNA coarse-grained model. By analysing the fluctuations in these trajectories we obtain estimates of the bend and twist persistence lengths, and the underlying bend and twist elastic moduli and couplings between them. Only on length scales beyond those associated with the spacings between the interhelix crossovers do the bending fluctuations behave like those of a worm-like chain. The obtained bending persistence lengths are much larger than that for double-stranded DNA an…
From the complex motions of robots to the oxygen binding of hemoglobin, the function of many mechanical systems depends on large, coordinated movements of their components. Such movements arise from a network of physical interactions in the form of links that transmit forces between constituent elements. However, the principled design of specific movements is made difficult by the number and nonlinearity of interactions. Here, we model mechanical systems as linkages of rigid bonds (edges) connected by joints (nodes), and formulate a simple but powerful framework for designing full nonlinear co…
Creases are purposely introduced to thin structures for designing deployable origami, artistic geometries, and functional structures with tunable nonlinear mechanics. Modeling the mechanics of creased structures is challenging because creases introduce geometric discontinuity and often have complex mechanical responses due to the local material damage. In this work, we propose a continuous description of the sharp geometry of creases and apply it to the study of \emph{creased annuli}, made by introducing radial creases to annular strips with the creases annealed to behave elastically. We find …
Mechanical metamaterials owe their extraordinary properties and functionalities to their micro-/nanoscale design of which shape, including both geometry and topology, is perhaps the most important aspect. 4D printing enables programmed, predictable, and precise change in the shape of mechanical metamaterials to achieve multi-functionality, adaptive properties, and the other types of desired behaviors that cannot be achieved using simple 3D printing. This paper presents an overview of 4D printing as applied to mechanical metamaterials. It starts by presenting a systematic definition of what 4D …
Mechanical and phononic metamaterials exhibiting negative elastic moduli, gapped vibrational spectra, or topologically protected modes enable precise control of structural and acoustic functionalities. While much progress has been made in their experimental and theoretical characterization, the inverse design of mechanical metamaterials with arbitrarily programmable spectral properties and mode localization remains an unsolved problem. Here, we present a flexible computational inverse-design framework that allows the efficient tuning of one or more gaps at nearly arbitrary positions in the spe…
Department of Mechanical Engineering, Massachusetts Institute of Technology（麻省理工学院机械工程系）；Department of Mechanical Engineering, University of California Berkeley（加州大学伯克利分校机械工程系）
Atomically thin sheets, such as graphene, are widely used in nanotechnology. Recently they have also been used in applications including kirigami and self-folding origami, where it becomes important to understand how they respond to external loads. Motivated by this, we investigate how isotropic sheets respond to uniaxial tension by employing the self-consistent screening analysis method and molecular dynamics simulations. Previously, it was shown that for freely suspended sheets thermal fluctuations effectively renormalize elastic constants, which become scale-dependent beyond a characteristi…
The mechanical properties of thermally excited two-dimensional crystalline membranes can depend dramatically on their geometry and topology. A particularly relevant example is the effect on the crumpling transition of holes in the membrane. Here we use molecular dynamics simulations to study the case of elastic frames (sheets with a single large hole in the center) and find that the system approaches the crumpled phase through a sequence of origami-like folds at decreasing length scales when temperature is increased. We use normal-normal correlation functions to quantify the temperature-depend…
