Article

Smooth Triaxial Weaving with Naturally Curved Ribbons

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Abstract

Triaxial weaving is a handicraft technique that has long been used to create curved structures using initially straight and flat ribbons. Weavers typically introduce discrete topological defects to produce nonzero Gaussian curvature, albeit with faceted surfaces. We demonstrate that, by tuning the in-plane curvature of the ribbons, the integrated Gaussian curvature of the weave can be varied continuously, which is not feasible using traditional techniques. Further, we reveal that the shape of the physical unit cells is dictated solely by the in-plane geometry of the ribbons, not elasticity. Finally, we leverage the geometry-driven nature of triaxial weaving to design a set of ribbon profiles to weave smooth spherical, ellipsoidal, and toroidal structures.

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... It has been realized that determining how to form curved 3D structures with continuous geometry and small topological defects using planar structures such as ribbons or rods has been a huge challenge. To address this problem, Baek et al. [8] proposed a new three-dimensional curved surface structure forming method with the in-plane geometry of ribbons. Additionally, by tuning the in-plane curvature of the ribbons, the integrated Gaussian curvature of the weave can be varied continuously. ...
... Additionally, by tuning the in-plane curvature of the ribbons, the integrated Gaussian curvature of the weave can be varied continuously. In addition, Baek et al. [8] used two kinds of curved ribbons to fabricate the different ellipsoidal weaves of aspect ratios. This ellipsoid woven structure had better continuity and was closer to a perfect ellipsoid. ...
... However, Ref. [8] only studied the geometric problems of an ellipsoidal weave, and did not deal with the buckling behavior of an ellipsoidal weave. Existing studies on shell structures found that elastic spherical shell structures such as table tennis under vertical loads have showed that CFRP cylinders with end stiffeners exhibited high compressive properties. ...
... It has been realized that determining how to form curved 3D structures with continuous geometry and small topological defects using planar structures such as ribbons or rods has been a huge challenge. To address this problem, Baek et al. [8] proposed a new three-dimensional curved surface structure forming method with the in-plane geometry of ribbons. Additionally, by tuning the in-plane curvature of the ribbons, the integrated Gaussian curvature of the weave can be varied continuously. ...
... Additionally, by tuning the in-plane curvature of the ribbons, the integrated Gaussian curvature of the weave can be varied continuously. In addition, Baek et al. [8] used two kinds of curved ribbons to fabricate the different ellipsoidal weaves of aspect ratios. This ellipsoid woven structure had better continuity and was closer to a perfect ellipsoid. ...
... However, Ref. [8] only studied the geometric problems of an ellipsoidal weave, and did not deal with the buckling behavior of an ellipsoidal weave. Existing studies on shell structures found that elastic spherical shell structures such as table tennis under vertical loads have showed that CFRP cylinders with end stiffeners exhibited high compressive properties. ...
Article
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The woven structure made of naturally curved (in-plane) ribbons has smooth geometry and fewer geometric imperfections, but there is no study of its buckling mechanical properties under vertical loads. The aim of this paper is to investigate buckling mechanical properties of spherical woven structures. Three spherical woven structures with different ribbon types and six new spherical woven structures with different ribbon widths and thicknesses were designed and the quasi-static vertical compression tests were carried out. The buckling load of spherical woven structures were studied by nonlinear finite element and ring buckling theory. Results indicate that the failure mode of the spherical weave structure under vertical loading can be divided into two stages, where a flat contact region forms between the spherical weave structure and the rigid plate and inward dimple of ribbons. Spherical weave structures using naturally curved (in-plane) ribbon weaving have better buckling stability than those woven with straight ribbon. Based on theoretical and finite element analysis, we propose a buckling load equation and buckling correction factor equation for the new spherical weave structure under vertical compression load. The formula is validated and has good agreement with the test results, which could help to design the stability of spherical weave structures with in-plane ribbons.
... However, most 3D structures woven from straight ribbons have topological defects. Baek et al. [ 15]proposed a method to weave smoother continuous 3D surface structures using naturally curved (in-plane) ribbons, obtained a new surface structure with relatively continuous variation of Gaussian curvature, and analyzed its geometric properties. We believe that this new 3D surface structure with smooth geometric properties must correspond to new mechanical properties. ...
... How to form some curved 3D structures with continuous geometry using planar structures such as ribbons or rods has been a huge challenge. To solve this problem, Baek et al. [15] proposed a 3D surface forming method based on traditional weaving technology, containing ribbons with in-plane curvatures to weave, and using rivets FIG. 1: 3D structure fabricated by planar structure: (a) 3D structure based on paper-cutting substrate driven by buckling [13]; (b) Inflatable folding structure [10]. ...
... The 3D structure woven by ribbons with in-plane curvatures has fewer geometric defects, and the rivets greatly increase the flexibility of the structure. Baek et al. [15] also deduced the formula of Gaussian curvature, laying a theoretical foundation for the formation of new woven structures. The formula for calculating the Gaussian curvature of a woven structure is shown below: ...
Preprint
Weaving is an ancient and effective structural forming technique characterized by the ability to convert two-dimensional ribbons to three-dimensional structures. However, most 3D structures woven from straight ribbons have topological defects. Baek et al. proposed a method to weave smoother continuous 3D surface structures using naturally curved (in-plane) ribbons, obtained a new surface structure with relatively continuous variation of Gaussian curvature, and analyzed its geometric properties. We believe that this new 3D surface structure with smooth geometric properties must correspond to new mechanical properties. To this end, we investigated a 3D surface structure using naturally curved (in-plane) ribbon weaving, and the results of calculations and experiments show that such structures have better buckling stability than those woven with straight ribbons. It is observed that the number of ribbons influences the buckling behavior of different types of woven structures.
... It has been realized that determining how to form curved 3D structures with continuous geometry and small topological defects using planar structures such as ribbons or rods has been a huge challenge. To address this problem, Baek et al. [8] proposed a new three-dimensional curved surface structure forming method with the in-plane geometry of ribbons. Additionally, by tuning the in-plane curvature of the ribbons, the integrated Gaussian curvature of the weave can be varied continuously. ...
... Additionally, by tuning the in-plane curvature of the ribbons, the integrated Gaussian curvature of the weave can be varied continuously. In addition, Baek et al. [8] used two kinds of curved ribbons to fabricate the different ellipsoidal weaves of aspect ratios. This ellipsoid woven structure had better continuity and was closer to a perfect ellipsoid. ...
... However, Ref. [8] only studied the geometric problems of an ellipsoidal weave, and did not deal with the buckling behavior of an ellipsoidal weave. Existing studies on shell structures found that elastic spherical shell structures such as table tennis under vertical loads have showed that CFRP cylinders with end stiffeners exhibited high compressive properties. ...
Preprint
Weaving technology can convert two-dimensional structures such as ribbons into three-dimensional structures by specific connections. However, most of the 3D structures fabricated by conventional weaving methods using straight ribbons have some topological defects. In order to obtain smoother continuous 3D surface structures, Baek et al. proposed a novel weaving method using naturally curved (in-plane) ribbons to fabricated three-dimensional curved structures and using this method to weave new spherical weave structures that are closer to perfect spheres. We believe that this new spherical weave structure with smooth geometric properties must correspond to new mechanical properties. To this end, we investigated the buckling characteristics of different types of spherical weave structures by the combination of test and finite element method. The results of calculations and experiments show that the failure mode of the spherical weave structure under vertical loading can be divided into two stages: a flat contact region forms between the spherical weave structure and the rigid plate and inward dimple of ribbons. The spherical weave structures using naturally curved (in-plane) ribbon weaving have better buckling stability than those woven with straight ribbons. The vertical buckling load of spherical weave structures using naturally curved ribbon increases with the width and thickness of the ribbon. In addition, this paper combines test, theoretical and finite element analysis to propose the buckling load equation and buckling correction factor equation for the new spherical weave structure under vertical compression load.
... In recent years, there have been a surging interest in the study of the mechanics of slender structures due to their potential applications in designing deployable and morphing structures [1][2][3][4][5][6][7], complex 3D shapes [8][9][10][11], flexible electronics and actuators [12][13][14], flexbile medical device [15,16], and soft robots [17][18][19]. These application scenarios are often achieved through large deformations, bistability/multistability, and snapping behaviors of slender structures such as strips and rods. ...
... Slender structures such as thin strips and rods can easily undergo large deformations, exhibit buckling behaviors, and have multiple equilibria, resulting in rich and complex nonlinear mechanical behaviors. While a single strip or rod can exhibit a variety of bifurcations, multistability, and complicated buckling patterns [22,[27][28][29][30][31][32][33][34][35], coupling multiple strips or rods with joints through tailored geometric designs can create deployable structures [1,24,25], gridshells [36], and complex surfaces [8,37]. Large deformations of thin strips and rods are generally dominated by bending and twisting, with negligible stretching of their centerlines. ...
Preprint
Full-text available
Serpentine structures, composed of straight and circular strips, have garnered significant attention as potential designs for flexible electronics due to their remarkable stretchability. When subjected to stretching, these serpentine strips buckle out of plane, and previous studies have identified two distinct buckling modes whose order of appearance may interchange in serpentine structures with a single cell. In this study, we employ anisotropic rod theory to model serpentine strips as a multi-segment boundary value problem (BVP), with continuity conditions enforced at the interface between the straight and curved strips. We solve the BVP using methods of continuation, and our results reveal that: 1) the exchange of the two buckling modes in a single-cell serpentine strip is induced by a double-eigenvalue and associated secondary bifurcations, which also alter the stability of the two buckling modes; 2) a variety of stable states with reversible symmetry can be manually obtained in tabletop models and are found to be disconnected from the planar branch in numerical continuation. Furthermore, we demonstrate that modulating the strip thickness across different cells leads to the initiation of buckling in the thinnest section, thereby allowing for the tuning of buckling modes in serpentine strips. In structures with two cells, the sequence of the two buckling modes can also be controlled by designing serpentine strips with nonuniform height. This work could enhance the mechanical design of serpentine-interconnect-based flexible structures and could have applications in multistable actuators and mechanical memory devices.
... It is known that the mechanics of hollow structures would be different with both 1D beams/rods and 2D plates/shells [17,18]. For the simulation of elastic gridshells, Baek et al. first introduced the concept of stiff spring into DER framework to ensure the non-deviation condition between two rods at joint area [19] graphics community mainly focusing on visual effects while the mechanical responses of the structures experiencing thermal load are seldom addressed, which is crucial for the real engineering applications. ...
... Here, we develop a numerical method to simulate the mechanical response of a planar gridshell under thermal load. An elastic gridshell is first separated into multiple independent rods with each one characterized by the DER approach [19]. A specific mapping algorithm is later introduced to explore the mechanics of a rigid joint between two intersected rods, leading to a fully implicit numerical framework. ...
Article
We introduce a discrete model to predict the buckling instability and the vibration performance of an elastic gridshell which experiences a thermal load. With both thermal stress and temperature-dependent material property considered, a discrete framework for the simulation of hollow grid is developed based on a well-established Discrete Elastic Rods (DER) method and a specific mapping technique. It is found that the critical temperatures which induce a structural instability are identical for single rod system and multiple rods network, regardless of rod density. Meanwhile, its natural frequency linearly enhances with the growth of rod number. Moreover, the increase of environmental temperature will result in that the resonant peak of displacement response moves toward a lower frequency. Interestingly, we found that the natural frequency of gridshell is closer to the beam model, while its displacement response distribution shows similarity to plate theory. The free vibration frequencies of one-dimensional (1D) beam and two-dimensional (2D) plate with a simply-supported boundary condition under thermal effects are also derived analytically for cross-validation. We hope our findings could provide a fundamental insight in structural dynamics and further facilitate the designs in mechanical and aerospace engineering, e.g., deployable structures and smart metamaterials.
... However, owing to several fundamental physical and geometric constraints, it is generally impossible to transform a flat surface into a smooth one with arbitrary curvature. To overcome this difficulty, several novel ideas have recently been developed, such as the use of non-Euclidean deformation due to nonuniform growth [1], origami or kirigami [2][3][4][5][6][7][8][9][10][11], swelling pressure [12,13], and other computer-assisted shape-programming methods [14,15]. These emerging technologies are pushing the limits of the shape-morphing abilities of functional surfaces and expanding the range of their applications. ...
Article
Full-text available
Generating a three-dimensional curved surface from flat sheets using simple mechanical actuation is a crucial step in developing functional shape-shifting materials. Among the methodologies relying on emergent concepts, the traditional cellular solid-based plate, with its highly porous structures, provides a versatile tool for creating doubly curved surfaces through planar bending actuation. Leveraging a combination of digital fabrication, physical experiments, finite element simulation (FES), and linear elasticity theory, we demonstrate how such a cellular metaplate can exhibit doubly curved shapes. By extending Lamb's classical theory of the anticlastic effect in thin elastic plates to our cellular metaplate, we develop a scaling law for the crossover length below which the doubly curved surface appears as b c ∼ b Lamb / ρ , where b Lamb ∼ R w , with R being the externally imposed radius of curvature, w being the plate thickness, and ρ being the relative density of a given cellular geometry. The prediction is validated by our experiment and FES. The proposed framework is versatile and emphasizes the fundamental physical principles governing the mechanics of shape-morphing surfaces. Published by the American Physical Society 2024
... Additionally, we anticipate that the friction effect can be incorporated into our framework, as it has already been realized in the one-dimensional (1D) discrete simulation. 55,58 ...
... Designing the proper geometry of a thin structure is crucial for realizing its functionality, and has become a central topic in the field of soft matter mechanics [1,2]. For example, by choosing suitable geometrical and structural arrangements for the assembly of thin strips and harnessing their elastic forces, various curved (pseudo)surfaces and their transformations have been achieved [3][4][5]. Typically, the fundamental building blocks for such advanced materials are intrinsically straight and/or uniformly curved rods, ribbons, and plates. The intrinsic curvature of a slender object introduces novel types of elastic instabilities and morphologies, which may find various applications, such as mechanical metamaterials [6], jumping soft robotics [7,8], and bioinspired actuators [9]. ...
Article
In this study, we investigate the morphology and mechanics of a naturally curved elastic arch loaded at its center and frictionally supported at both ends on a flat, rigid substrate. Through systematic numerical simulations, we classify the observed behaviors of the arch into three configurations in terms of the arch geometry and the coefficient of static friction with the substrate. A linear theory is developed based on a planar elastica model combined with Amontons–Coulomb's frictional law, which quantitatively explains the numerically constructed phase diagram. The snapping transition of a loaded arch in a sufficiently large indentation regime, which involves a discontinuous force jump, is numerically observed. The proposed model problem enables a fully analytical investigation and demonstrates a rich variety of mechanical behaviors owing to the interplay among elasticity, geometry, and friction. This study provides a basis for understanding more common but complex systems, such as a cylindrical shell subjected to a concentrated load and simultaneously supported by frictional contact with surrounding objects.
... In addition, our ambition extends to the generalization of our findings into the realm of surfaces, akin to the approach outlined in [15], where the Gauss-Bonnet theorem, as expounded in [14], will play a pivotal role. This theorem provides a means of classifying topological surfaces based on their total Gaussian curvature and geodesic curvature, affording us a broader perspective on the mathematical representation of music within a geometric context. ...
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Full-text available
This manuscript endeavors to establish a framework for the mapping of music onto a three-dimensional structure. Our objective is to transform the guitar choruses of Beatles songs into curves, with each chorus corresponding to its respective curve. We aim to investigate and characterize the intricacy of each song by employing mathematical techniques derived from differential geometry, specifically focusing on the total curvature of the chorus curve. Given that a single song may possess varying chord progressions in different verses, the performer can determine the geometric representation they aim to convey through the number of loops and the direction of the curve. The overarching objective of our study is to enable viewers to identify specific songs or motives by visually examining an object and exploring its geometric properties. Furthermore, we posit that these ideas can provide composers with a fresh perspective on their own musical compositions while also granting non-professional audiences a glimpse into the intricacies involved in the process of composing.
... A low-profile actuator array can be created using a surface of bending actuators. Morphing into a target structure can be achieved by adjusting the curvature of each ribbon/beam (36,37). Since the pixels of bending actuators are coupled to each other, the height of each pixel is influenced by the surrounding pixels. ...
Article
Reconfigurable morphing surfaces provide new opportunities for advanced human-machine interfaces and bio-inspired robotics. Morphing into arbitrary surfaces on demand requires a device with a sufficiently large number of actuators and an inverse control strategy. Developing compact, efficient control interfaces and algorithms is vital for broader adoption. In this work, we describe a passively addressed robotic morphing surface (PARMS) composed of matrix-arranged ionic actuators. To reduce the complexity of the physical control interface, we introduce passive matrix addressing. Matrix addressing allows the control of N2 independent actuators using only 2N control inputs, which is substantially lower than traditional direct addressing (N2 control inputs). Using machine learning with finite element simulations for training, our control algorithm enables real-time, high-precision forward and inverse control, allowing PARMS to dynamically morph into arbitrary achievable predefined surfaces on demand. These innovations may enable the future implementation of PARMS in wearables, haptics, and augmented reality/virtual reality.
... A low-profile actuator array can be created using a surface of bending actuators. Morphing into a target structure can be achieved by adjusting the curvature of each ribbon/beam (36,37). Since the pixels of bending actuators are coupled to each other, the height of each pixel is influenced by the surrounding pixels. ...
Preprint
Reconfigurable morphing surfaces provide new opportunities for advanced human-machine interfaces and bio-inspired robotics. Morphing into arbitrary surfaces on demand requires a device with a sufficiently large number of actuators and an inverse control strategy that can calculate the actuator stimulation necessary to achieve a target surface. The programmability of a morphing surface can be improved by increasing the number of independent actuators, but this increases the complexity of the control system. Thus, developing compact and efficient control interfaces and control algorithms is a crucial knowledge gap for the adoption of morphing surfaces in broad applications. In this work, we describe a passively addressed robotic morphing surface (PARMS) composed of matrix-arranged ionic actuators. To reduce the complexity of the physical control interface, we introduce passive matrix addressing. Matrix addressing allows the control of independent actuators using only 2N control inputs, which is significantly lower than control inputs required for traditional direct addressing. Our control algorithm is based on machine learning using finite element simulations as the training data. This machine learning approach allows both forward and inverse control with high precision in real time. Inverse control demonstrations show that the PARMS can dynamically morph into arbitrary pre-defined surfaces on demand. These innovations in actuator matrix control may enable future implementation of PARMS in wearables, haptics, and augmented reality/virtual reality (AR/VR).
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Straight flat strips of inextensible material can be bent into curved strips aligned with arbitrary space curves. The large shape variety of these so-called rectifying strips makes them candidates for shape modeling, especially in applications such as architecture where simple elements are preferred for the fabrication of complex shapes. In this paper, we provide computational tools for the design of shapes from rectifying strips. They can form various patterns and fulfill constraints which are required for specific applications such as gridshells or shading systems. The methodology is based on discrete models of rectifying strips, a discrete level-set formulation and optimization-based constrained mesh design and editing. We also analyse the geometry at nodes and present remarkable quadrilateral arrangements of rectifying strips with torsion-free nodes.
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Magneto‐elastic materials facilitate features such as shape programmability, adaptive stiffness, and tunable strength, which are critical for advances in structural and robotic materials. Magneto‐elastic networks are commonly fabricated by employing hard magnets embedded in soft matrices to constitute a monolithic body. These architected network materials have excellent mechanical properties but damage incurred in extreme loading scenarios are permanent. To overcome this limitation, we present a novel design for elastic bars with permanent fixed dipole magnets at their ends and demonstrate their ability to self‐assemble into magneto‐elastic networks under random vibrations. The magneto‐elastic unit configuration, most notably the orientation of end dipoles, is shown to dictate the self‐assembled network topology, which can range from quasi‐ordered triangular lattices to stacks or strings of particles. Network mechanics are probed with uniaxial tensile tests and design criteria for forming stable lightweight 2D networks are established. It is shown that these magneto‐elastic networks rearrange and break gracefully at their magnetic nodes under large excitations and yet recover their original structure at moderate random excitations. This work paves the way for structural materials that can be self‐assembled and repaired on‐the‐fly with random vibrations, and broadens the applications of magneto‐elastic soft materials.
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Shape-morphing structures are at the core of future applications in aeronautics1, minimally invasive surgery2, tissue engineering3 and smart materials4. However, current engineering technologies, based on inhomogeneous actuation across the thickness of slender structures, are intrinsically limited to one-directional bending5. Here, we describe a strategy where mesostructured elastomer plates undergo fast, controllable and complex shape transformations under applied pressure. Similar to pioneering techniques based on soft hydrogel swelling6–10, these pneumatic shape-morphing elastomers, termed here as ‘baromorphs’, are inspired by the morphogenesis of biological structures11–15. Geometric restrictions are overcome by controlling precisely the local growth rate and direction through a specific network of airways embedded inside the rubber plate. We show how arbitrary three-dimensional shapes can be programmed using an analytic theoretical model, propose a direct geometric solution to the inverse problem, and illustrate the versatility of the technique with a collection of configurations. Elastomer sheets with programmable air channel organization swiftly shape into complex three-dimensional structures upon the application of pressure.
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This paper explores how computational methods of representation can support and extend kagome handcraft towards the fabrication of interlaced lattice structures in an expanded set of domains, beyond basket making. Through reference to the literature and state of the art, we argue that the instrumentalisation of kagome principles into computational design methods is both timely and relevant; it addresses a growing interest in such structures across design and engineering communities; it also fills a current gap in tools that facilitate design and fabrication investigation across a spectrum of expertise, from the novice to the expert. The paper describes the underlying topological and geometrical principles of kagome weave, and demonstrates the direct compatibility of these principles to properties of computational triangular meshes and their duals. We employ the known Medial Construction method to generate the weave pattern, edge 'walking' methods to consolidate geometry into individual strips, physics based relaxation to achieve a materially informed final geometry and projection to generate fabrication information. Our principle contribution is the combination of these methods to produce a principled workflow that supports design investigation of kagome weave patterns with the constraint of being made using straight strips of material. We evaluate the computational workflow through comparison to physical artefacts constructed ex-ante and ex-post.
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Deployable structures are physical mechanisms that can easily transition between two or more geometric configurations; such structures enable industrial, scientific, and consumer applications at a wide variety of scales. This paper develops novel deployable structures that can approximate a large class of doubly-curved surfaces and are easily actuated from a flat initial state via inflation or gravitational loading. The structures are based on two-dimensional rigid mechanical linkages that implicitly encode the curvature of the target shape via a user-programmable pattern that permits locally isotropic scaling under load. We explicitly characterize the shapes that can be realized by such structures---in particular, we show that they can approximate target surfaces of positive mean curvature and bounded scale distortion relative to a given reference domain. Based on this observation, we develop efficient computational design algorithms for approximating a given input geometry. The resulting designs can be rapidly manufactured via digital fabrication technologies such as laser cutting, CNC milling, or 3D printing. We validate our approach through a series of physical prototypes and present several application case studies, ranging from surgical implants to large-scale deployable architecture.
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Fabrication from developable parts is the basis for arts such as papercraft and needlework, as well as modern architecture and CAD in general, and it has inspired much research. We observe that the assembly of complex 3D shapes created by existing methods often requires first fabricating many small parts and then carefully following instructions to assemble them together. Despite its significance, this error prone and tedious process is generally neglected in the discussion. We present the concept of zippables - single, two dimensional, branching, ribbon-like pieces of fabric that can be quickly zipped up without any instructions to form 3D objects. Our inspiration comes from the so-called zipit bags [zipit 2017], which are made of a single, long ribbon with a zipper around its boundary. In order to "assemble" the bag, one simply needs to zip up the ribbon. Our method operates in the same fashion, but it can be used to approximate a wide variety of shapes. Given a 3D model, our algorithm produces plans for a single 2D shape that can be laser cut in few parts from fabric or paper. A zipper can then be attached along the boundary by sewing, or by gluing using a custom-built fastening rig. We show physical and virtual results that demonstrate the capabilities of our method and the ease with which shapes can be assembled.
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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 difficulty of realizing these geometric structures by quantifying the energetic barrier that separates the metastable flat and folded states. Moreover, we characterize the trade-off between the accuracy to which the pattern conforms to the target surface, and the effort associated with creating finer folds. Our approach enables the tailoring of origami patterns to drape complex surfaces independent of absolute scale, as well as the quantification of the energetic and material cost of doing so.
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Significance Curved crystals cannot comprise hexagons alone; additional defects are required by both topology and energetics that depend on the system size. These constraints are present in systems as diverse as virus capsules, soccer balls, and geodesic domes. In this paper, we study the structure of defects of the crystalline dimpled patterns that self-organize through curved wrinkling on a thin elastic shell bound to a compliant substrate. The dimples are treated as point-like packing units, even if the shell is a continuum. Our results provide quantitative evidence that our macroscopic wrinkling system can be mapped into and described within the framework of curved crystallography, albeit with some important differences attributed to the far-from-equilibrium nature of our patterns.
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Geodesic curves in surfaces are not only minimizers of distance, but they are also the curves of zero geodesic (sideways) curvature. It turns out that this property makes patterns of geodesics the basic geometric entity when dealing with the cladding of a freeform surface with wooden panels which do not bend sideways. Likewise a geodesic is the favored shape of timber support elements in freeform architecture, for reasons of manufacturing and statics. Both problem areas are fundamental in freeform architecture, but so far only experimental solutions have been available. This paper provides a systematic treatment and shows how to design geodesic patterns in different ways: The evolution of geodesic curves is good for local studies and simple patterns; the level set formulation can deal with the global layout of multiple patterns of geodesics; finally geodesic vector fields allow us to interactively model geodesic patterns and perform surface segmentation into panelizable parts.
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In this paper, we show how to create plain-weaving over an arbitrary surface. To create a plain-weaving on a surface, we need to create cycles that cross other cycles (or themselves) by alternatingly going over and under. We use the fact that it is possible to create such cycles, starting from any given manifold-mesh surface by simply twisting every edge of the manifold mesh. We have developed a new method that converts plain-weaving cycles to 3D thread structures. Using this method, it is possible to cover a surface without large gaps between threads by controlling the sizes of the gaps. We have developed a system that converts any manifold mesh to a plain-woven object, by interactively controlling the shapes of the threads with a set of parameters. We have demonstrated that by using this system, we can create a wide variety of plain-weaving patterns, some of which may not have been seen before.
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We present a discrete treatment of adapted framed curves, parallel transport, and holonomy, thus establishing the language for a discrete geometric model of thin flexible rods with arbitrary cross section and undeformed configuration. Our approach differs from existing simulation techniques in the graphics and mechanics literature both in the kinematic description---we represent the material frame by its angular deviation from the natural Bishop frame---as well as in the dynamical treatment---we treat the centerline as dynamic and the material frame as quasistatic. Additionally, we describe a manifold projection method for coupling rods to rigid-bodies and simultaneously enforcing rod inextensibility. The use of quasistatics and constraints provides an efficient treatment for stiff twisting and stretching modes; at the same time, we retain the dynamic bending of the centerline and accurately reproduce the coupling between bending and twisting modes. We validate the discrete rod model via quantitative buckling, stability, and coupled-mode experiments, and via qualitative knot-tying comparisons.
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Basket weaving is a traditional craft for creating curved surfaces as an interwoven array of thin, flexible, and initially straight ribbons. The three-dimensional shape of a woven structure emerges through a complex interplay of the elastic bending behavior of the ribbons and the contact forces at their crossings. Curvature can be injected by carefully placing topological singularities in the otherwise regular weaving pattern. However, shape control through topology is highly non-trivial and inherently discrete, which severely limits the range of attainable woven geometries. Here, we demonstrate how to construct arbitrary smooth free-form surface geometries by weaving carefully optimized curved ribbons. We present an optimization-based approach to solving the inverse design problem for such woven structures. Our algorithm computes the ribbons' planar geometry such that their interwoven assembly closely approximates a given target design surface in equilibrium. We systematically validate our approach through a series of physical prototypes to show a broad range of new woven geometries that is not achievable by existing methods. We anticipate our computational approach to significantly enhance the capabilities for the design of new woven structures. Facilitated by modern digital fabrication technology, we see potential applications in material science, bio- and mechanical engineering, art, design, and architecture.
Conference Paper
Digital fabrication devices are powerful tools for creating tangible reproductions of 3D digital models. Most available printing technologies aim at producing an accurate copy of a tridimensional shape. However, fabrication technologies can also be used to create a stylistic representation of a digital shape. We refer to this class of methods as stylized fabrication methods. These methods abstract geometric and physical features of a given shape to create an unconventional representation, to produce an optical illusion, or to devise a particular interaction with the fabricated model. In this course, we classify and overview this broad and emerging class of approaches and also propose possible directions for future research.
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We propose a method for computing global Chebyshev nets on triangular meshes. We formulate the corresponding global parameterization problem in terms of commuting PolyVector fields, and design an efficient optimization method to solve it. We compute, for the first time, Chebyshev nets with automatically-placed singularities, and demonstrate the realizability of our approach using real material.
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Traditional computer-aided design systems are mainly intended for expert users, but research involving systems incorporating computer graphics and interactive techniques that are easy to use by novices is also active. In this paper, we describe a design support system, BandWeavy, that can be used by a novice to easily design a craft band artwork using his or her desired pattern. We propose an algorithm that can automatically calculate cuboid and cylinder shapes according to the sizes desired by the users.
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We present X-shells, a new class of deployable structures formed by an ensemble of elastically deforming beams coupled through rotational joints. An X-shell can be assembled conveniently in a flat configuration from standard elastic beam elements and then deployed through force actuation into the desired 3D target state. During deployment, the coupling imposed by the joints will force the beams to twist and buckle out of plane to maintain a state of static equilibrium. This complex interaction of discrete joints and continuously deforming beams allows interesting 3D forms to emerge. Simulating X-shells is challenging, however, due to unstable equilibria at the onset of beam buckling. We propose an optimization-based simulation framework building on a discrete rod model that robustly handles such difficult scenarios by analyzing and appropriately modifying the elastic energy Hessian. This real-time simulation method forms the basis of a computational design tool for X-shells that enables interactive design space exploration by varying and optimizing design parameters to achieve a specific design intent. We jointly optimize the assembly state and the deployed configuration to ensure the geometric and structural integrity of the deployable X-shell. Once a design is finalized, we also optimize for a sparse distribution of actuation forces to efficiently deploy it from its flat assembly state to its 3D target state. We demonstrate the effectiveness of our design approach with a number of design studies that highlight the richness of the X-shell design space, enabling new forms not possible with existing approaches. We validate our computational model with several physical prototypes that show excellent agreement with the optimized digital models.
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We propose FlexMaps, a novel framework for fabricating smooth shapes out of flat, flexible panels with tailored mechanical properties. We start by mapping the 3D surface onto a 2D domain as in traditional UV mapping to design a set of deformable flat panels called FlexMaps. For these panels, we design and obtain specific mechanical properties such that, once they are assembled, the static equilibrium configuration matches the desired 3D shape. FlexMaps can be fabricated from an almost rigid material, such as wood or plastic, and are made flexible in a controlled way by using computationally designed spiraling microstructures.
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Packed atlases, consisting of 2D parameterized charts, are ubiquitously used to store surface signals such as texture or normals. Tight packing is similarly used to arrange and cut-out 2D panels for fabrication from sheet materials. Packing efficiency, or the ratio between the areas of the packed atlas and its bounding box, significantly impacts downstream applications. We propose Box Cutter, a new method for optimizing packing efficiency suitable for both settings. Our algorithm improves packing efficiency without changing distortion by strategically cutting and repacking the atlas charts or panels. It preserves the local mapping between the 3D surface and the atlas charts and retains global mapping continuity across the newly formed cuts. We balance packing efficiency improvement against increase in chart boundary length and enable users to directly control the acceptable amount of boundary elongation. While the problem we address is NP-hard, we provide an effective practical solution by iteratively detecting large rectangular empty spaces, or void boxes, in the current atlas packing and eliminating them by first refining the atlas using strategically placed axis-aligned cuts and then repacking the refined charts. We repeat this process until no further improvement is possible, or until the desired balance between packing improvement and boundary elongation is achieved. Packed chart atlases are only useful for the applications we address if their charts are overlap-free; yet many popular parameterization methods, used as-is, produce atlases with global overlaps. Our pre-processing step eliminates all input overlaps while explicitly minimizing the boundary length of the resulting overlap-free charts. We demonstrate our combined strategy on a large range of input atlases produced by diverse parameterization methods, as well as on multiple sets of 2D fabrication panels. Our framework dramatically improves the output packing efficiency on all inputs; for instance with boundary length increase capped at 50% we improve packing efficiency by 68% on average.
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The geometry of simple knots and catenanes is described using the concept of linear line segments (sticks) joined at corners. This is extended to include woven linear threads as members of the extended family of knots. The concept of transitivity that can be used as a measure of regularity is explained. Then a review is given of the simplest, most ‘regular’ 2- and 3-periodic patterns of polycatenanes and weavings. Occurrences in crystal structures are noted but most structures are believed to be new and ripe targets for designed synthesis.
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Curved folded surfaces, given their ability to produce elegant freeform shapes by folding flat sheets etched with curved creases, hold a special place in computational Origami. Artists and designers have proposed a wide variety of different fold patterns to create a range of interesting surfaces. The creative process, design, as well as fabrication is usually only concerned with the static surface that emerges once folding has completed. Folding such patterns, however, is difficult as multiple creases have to be folded simultaneously to obtain a properly folded target shape. We introduce string actuated curved folded surfaces that can be shaped by pulling a network of strings, thus, vastly simplifying the process of creating such surfaces and making the folding motion an integral part of the design. Technically, we solve the problem of which surface points to string together and how to actuate them by locally expressing a desired folding path in the space of isometric shape deformations in terms of novel string actuation modes. We demonstrate the validity of our approach by computing string actuation networks for a range of well-known crease patterns and testing their effectiveness on physical prototypes. All the examples in this article can be downloaded for personal use from http://geometry.cs.ucl.ac.uk/projects/2017/string-actuated/.
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Despite recent advances in the synthesis of increasingly complex topologies at the molecular level, nano- and microscopic weaves have remained difficult to achieve. Only a few diaxial molecular weaves exist—these were achieved by templation with metals. Here, we present an extended triaxial supramolecular weave that consists of self-assembled organic threads. Each thread is formed by the self-assembly of a building block comprising a rigid oligoproline segment with two perylene-monoimide chromophores spaced at 18 Å. Upon π stacking of the chromophores, threads form that feature alternating up- and down-facing voids at regular distances. These voids accommodate incoming building blocks and establish crossing points through CH–π interactions on further assembly of the threads into a triaxial woven superstructure. The resulting micrometre-scale supramolecular weave proved to be more robust than non-woven self-assemblies of the same building block. The uniform hexagonal pores of the interwoven network were able to host iridium nanoparticles, which may be of interest for practical applications.
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We propose a computational tool for designing Kirchhoff-Plateau Surfaces---planar rod networks embedded in pre-stretched fabric that deploy into complex, three-dimensional shapes. While Kirchhoff-Plateau Surfaces offer an intriguing and expressive design space, navigating this space is made difficult by the highly nonlinear nature of the underlying mechanical problem. In order to tackle this challenge, we propose a user-guided but computer-assisted approach that combines an efficient forward simulation model with a dedicated optimization algorithm in order to implement a powerful set of design tools. We demonstrate our method by designing a diverse set of complex-shaped Kirchhoff-Plateau Surfaces, each validated through physically-fabricated prototypes.
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We present a computational approach for designing CurveUps, curvy shells that form from an initially flat state. They consist of small rigid tiles that are tightly held together by two pre-stretched elastic sheets attached to them. Our method allows the realization of smooth, doubly curved surfaces that can be fabricated as a flat piece. Once released, the restoring forces of the pre-stretched sheets support the object to take shape in 3D. CurveUps are structurally stable in their target configuration. The design process starts with a target surface. Our method generates a tile layout in 2D and optimizes the distribution, shape, and attachment areas of the tiles to obtain a configuration that is fabricable and in which the curved up state closely matches the target. Our approach is based on an efficient approximate model and a local optimization strategy for an otherwise intractable nonlinear optimization problem. We demonstrate the effectiveness of our approach for a wide range of shapes, all realized as physical prototypes.
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Computational manufacturing technologies such as 3D printing hold the potential for creating objects with previously undreamed-of combinations of functionality and physical properties. Human designers, however, typically cannot exploit the full geometric (and often material) complexity of which these devices are capable. This STAR examines recent systems developed by the computer graphics community in which designers specify higher-level goals ranging from structural integrity and deformation to appearance and aesthetics, with the final detailed shape and manufacturing instructions emerging as the result of computation. It summarizes frameworks for interaction, simulation, and optimization, as well as documents the range of general objectives and domain-specific goals that have been considered. An important unifying thread in this analysis is that different underlying geometric and physical representations are necessary for different tasks: we document over a dozen classes of representations that have been used for fabrication-aware design in the literature. We analyze how these classes possess obvious advantages for some needs, but have also been used in creative manners to facilitate unexpected problem solutions. © 2017 The Author(s) Computer Graphics Forum © 2017 The Eurographics Association and John Wiley & Sons Ltd. Published by John Wiley & Sons Ltd.
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We present a computational method for interactive 3D design and rationalization of surfaces via auxetic materials, i.e., flat flexible material that can stretch uniformly up to a certain extent. A key motivation for studying such material is that one can approximate doubly-curved surfaces (such as the sphere) using only flat pieces, making it attractive for fabrication. We physically realize surfaces by introducing cuts into approximately inextensible material such as sheet metal, plastic, or leather. The cutting pattern is modeled as a regular triangular linkage that yields hexagonal openings of spatially-varying radius when stretched. In the same way that isometry is fundamental to modeling developable surfaces, we leverage conformal geometry to understand auxetic design. In particular, we compute a global conformal map with bounded scale factor to initialize an otherwise intractable non-linear optimization. We demonstrate that this global approach can handle non-trivial topology and non-local dependencies inherent in auxetic material. Design studies and physical prototypes are used to illustrate a wide range of possible applications.
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We present a computational tool for designing ornamental curve networks---structurally-sound physical surfaces with user-controlled aesthetics. In contrast to approaches that leverage texture synthesis for creating decorative surface patterns, our method relies on user-defined spline curves as central design primitives. More specifically, we build on the physically-inspired metaphor of an embedded elastic curve that can move on a smooth surface, deform, and connect with other curves. We formalize this idea as a globally coupled energy-minimization problem, discretized with piece-wise linear curves that are optimized in the parametric space of a smooth surface. Building on this technical core, we propose a set of interactive design and editing tools that we demonstrate on manually-created layouts and semi-automated deformable packings. In order to prevent excessive compliance, we furthermore propose a structural analysis tool that uses eigenanalysis to identify potentially large deformations between geodesically-close curves and guide the user in strengthening the corresponding regions. We used our approach to create a variety of designs in simulation, validated with a set of 3D-printed physical prototypes.
Article
We present a computational method for designing wire sculptures consisting of interlocking wires. Our method allows the computation of aesthetically pleasing structures that are structurally stable, efficiently fabricatable with a 2D wire bending machine, and assemblable without the need of additional connectors. Starting from a set of planar contours provided by the user, our method automatically tests for the feasibility of a design, determines a discrete ordering of wires at intersection points, and optimizes for the rest shape of the individual wires to maximize structural stability under frictional contact. In addition to their application to art, wire sculptures present an extremely efficient and fast alternative for low-fidelity rapid prototyping because manufacturing time and required material linearly scales with the physical size of objects. We demonstrate the effectiveness of our approach on a varied set of examples, all of which we fabricated.
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We investigate the influence of curvature and topology on crystalline dimpled patterns on the surface of generic elastic bilayers. Our numerical analysis predicts that the total number of defects created by adiabatic compression exhibits universal quadratic scaling for spherical, ellipsoidal, and toroidal surfaces over a wide range of system sizes. However, both the localization of individual defects and the orientation of defect chains depend strongly on the local Gaussian curvature and its gradients across a surface. Our results imply that curvature and topology can be utilized to pattern defects in elastic materials, thus promising improved control over hierarchical bending, buckling, or folding processes. Generally, this study suggests that bilayer systems provide an inexpensive yet valuable experimental test bed for exploring the effects of geometrically induced forces on assemblies of topological charges.
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Shape-morphing systems can be found in many areas, including smart textiles, autonomous robotics, biomedical devices, drug delivery and tissue engineering. The natural analogues of such systems are exemplified by nastic plant motions, where a variety of organs such as tendrils, bracts, leaves and flowers respond to environmental stimuli (such as humidity, light or touch) by varying internal turgor, which leads to dynamic conformations governed by the tissue composition and microstructural anisotropy of cell walls. Inspired by these botanical systems, we printed composite hydrogel architectures that are encoded with localized, anisotropic swelling behaviour controlled by the alignment of cellulose fibrils along prescribed four-dimensional printing pathways. When combined with a minimal theoretical framework that allows us to solve the inverse problem of designing the alignment patterns for prescribed target shapes, we can programmably fabricate plant-inspired architectures that change shape on immersion in water, yielding complex three-dimensional morphologies.
Article
We introduce Dual Strip Weaving, a novel concept for the interactive design of quad layouts, i.e. partitionings of freeform surfaces into quadrilateral patch networks. In contrast to established tools for the design of quad layouts or subdivision base meshes, which are often based on creating individual vertices, edges, and quads, our method takes a more global perspective, operating on a higher level of abstraction: the atomic operation of our method is the creation of an entire cyclic strip, delineating a large number of quad patches at once. The global consistency-preserving nature of this approach reduces demands on the user's expertise by requiring less advance planning. Efficiency is achieved using a novel method at the heart of our system, which automatically proposes geometrically and topologically suitable strips to the user. Based on this we provide interaction tools to influence the design process to any desired degree and visual guides to support the user in this task.
Article
We present a computational tool for fabrication-oriented design of flexible rod meshes. Given a deformable surface and a set of deformed poses as input, our method automatically computes a printable rod mesh that, once manufactured, closely matches the input poses under the same boundary conditions. The core of our method is formed by an optimization scheme that adjusts the cross-sectional profiles of the rods and their rest centerline in order to best approximate the target deformations. This approach allows us to locally control the bending and stretching resistance of the surface with a single material, yielding high design flexibility and low fabrication cost.
Article
We present a novel approach to remesh a surface into an isotropic triangular or quad-dominant mesh using a unified local smoothing operator that optimizes both the edge orientations and vertex positions in the output mesh. Our algorithm produces meshes with high isotropy while naturally aligning and snapping edges to sharp features. The method is simple to implement and parallelize, and it can process a variety of input surface representations, such as point clouds, range scans and triangle meshes. Our full pipeline executes instantly (less than a second) on meshes with hundreds of thousands of faces, enabling new types of interactive workflows. Since our algorithm avoids any global optimization, and its key steps scale linearly with input size, we are able to process extremely large meshes and point clouds, with sizes exceeding several hundred million elements. To demonstrate the robustness and effectiveness of our method, we apply it to hundreds of models of varying complexity and provide our cross-platform reference implementation in the supplemental material.
Article
During experiments aimed at understanding the mechanisms by which long-chain carbon molecules are formed in interstellar space and circumstellar shells1, graphite has been vaporized by laser irradiation, producing a remarkably stable cluster consisting of 60 carbon atoms. Concerning the question of what kind of 60-carbon atom structure might give rise to a superstable species, we suggest a truncated icosahedron, a polygon with 60 vertices and 32 faces, 12 of which are pentagonal and 20 hexagonal. This object is commonly encountered as the football shown in Fig. 1. The C60 molecule which results when a carbon atom is placed at each vertex of this structure has all valences satisfied by two single bonds and one double bond, has many resonance structures, and appears to be aromatic.
Article
The equations for the equilibrium of a thin elastic ribbon are derived by adapting the classical theory of thin elastic rods. Previously established ribbon models are extended to handle geodesic curvature, natural out-of-plane curvature, and a variable width. Both the case of a finite width (Wunderlich's model) and the limit of small width (Sadowksky's model) are recovered. The ribbon is assumed to remain developable as it deforms, and the direction of the generatrices is used as an internal variable. Internal constraints expressing inextensibility are identified. The equilibrium of the ribbon is found to be governed by an equation of equilibrium for the internal variable involving its second-gradient, by the classical Kirchhoff equations for thin rods, and by specific, thin-rod-like constitutive laws; this extends the results of Starostin and van der Heijden (2007) to a general ribbon model. Our equations are applicable in particular to ribbons having geodesic curvature, such as an annulus cut out in a piece of paper. Other examples of application are discussed. By making use of a material frame rather than the Fr\'enet-Serret's frame, the present work unifies the description of thin ribbons and thin rods.
Article
The phenomenon of elastic boundary layers under quasistatic loading is investigated using the Floquet-Bloch formalism for two-dimensional, isotropic, periodic lattices. The elastic boundary layer is a region of localised elastic deformation, confined to the free-edge of a lattice. Boundary layer phenomena in three isotropic lattice topologies are investigated: the semi-regular Kagome lattice, the regular hexagonal lattice and the regular fully-triangulated lattice. The boundary layer depth is on the order of the strut length for the hexagonal and the fully-triangulated lattices. For the Kagome lattice, the depth of boundary layer scales inversely with the relative density. Thus, the boundary layer in a Kagome lattice of low relative density spans many cells.
Article
Imposing curvature on crystalline sheets, such as 2D packings of colloids or proteins, or covalently bonded graphene leads to distinct types of structural instabilities. The first type involves the proliferation of localized defects that disrupt the crystalline order without affecting the imposed shape, whereas the second type consists of elastic modes, such as wrinkles and crumples, which deform the shape and also are common in amorphous polymer sheets. Here, we propose a profound link between these types of patterns, encapsulated in a universal, compression-free stress field, which is determined solely by the macroscale confining conditions. This "stress universality" principle and a few of its immediate consequences are borne out by studying a circular crystalline patch bound to a deformable spherical substrate, in which the two distinct patterns become, respectively, radial chains of dislocations (called "scars") and radial wrinkles. The simplicity of this set-up allows us to characterize the morphologies and evaluate the energies of both patterns, from which we construct a phase diagram that predicts a wrinkle-scar transition in confined crystalline sheets at a critical value of the substrate stiffness. The construction of a unified theoretical framework that bridges inelastic crystalline defects and elastic deformations opens unique research directions. Beyond the potential use of this concept for finding energy-optimizing packings in curved topographies, the possibility of transforming defects into shape deformations that retain the crystalline structure may be valuable for a broad range of material applications, such as manipulations of graphene's electronic structure.
Article
Studies were carried out to determine the feasibility of weaving triaxial fabrics (Doweave) in tight, low-porosity con figurations, and to investigate the stability and isotropy of triaxially woven fabrics. Detailed analyses of yarn motions required in triaxial weaving cycles disclosed no fundamental problems to prevent the achievement of tight fabrics woven of fine yarns. A breadboard loom, assembled to produce samples, confirmed to a degree the analytic predictions of loom configuration precision required to yield quality triaxial fabrics. Guidelines were generated for extending weaving capa bilities for full-scale production.
Article
A defect description of liquids and metallic glasses is developed. In two dimensions, surfaces of constant negative curvature contain an irreducible density of point disclinations in a hexatic order parameter. Analogous defect lines in an icosahedral order parameter appear in three-dimensional flat space. Frustration in tetrahedral particle packings forces disclination lines into the medium in a way reminiscent of Abrikosov flux lines in a type-II superconductor and of uniformly frustrated spin-glasses. The defect density is determined by an isotropic curvature mismatch, and the resulting singular lines run in all directions. The Frank-Kasper phases of transition-metal alloys are ordered networks of these lines, which, when disordered, provide an appealing model for structure in metallic glasses.
Article
Hexagons can easily tile a flat surface, but not a curved one. Introducing heptagons and pentagons (defects with topological charge) makes it easier to tile curved surfaces; for example, soccer balls based on the geodesic domes of Buckminster Fuller have exactly 12 pentagons (positive charges). Interacting particles that invariably form hexagonal crystals on a plane exhibit fascinating scarred defect patterns on a sphere. Here we show that, for more general curved surfaces, curvature may be relaxed by pleats: uncharged lines of dislocations (topological dipoles) that vanish on the surface and play the same role as fabric pleats. We experimentally investigate crystal order on surfaces with spatially varying positive and negative curvature. On cylindrical capillary bridges, stretched to produce negative curvature, we observe a sequence of transitions-consistent with our energetic calculations-from no defects to isolated dislocations, which subsequently proliferate and organize into pleats; finally, scars and isolated heptagons (previously unseen) appear. This fine control of crystal order with curvature will enable explorations of general theories of defects in curved spaces. From a practical viewpoint, it may be possible to engineer structures with curvature (such as waisted nanotubes and vaulted architecture) and to develop novel methods for soft lithography and directed self-assembly.
Article
We study isolated dislocations and disclinations in flexible membranes with internal crystalline order, using continuum elasticity theory and zero-temperature numerical simulation. These defects are relevant, for instance, to lipid bilayers in vesicles or in the Lbeta phase of lyotropic smectic liquid crystals. We first simulate defects in flat membranes, obtaining numerical results in good agreement with plane elasticity theory. Disclinations and dislocations eventually exhibit a buckling transition with increasing membrane radius. We generalize the continuum theory to include such buckled defects, and solve the disclination equations in the inextensional limit. The critical radius at which buckling starts to screen out internal elastic stresses is determined numerically. Computer simulation of buckled defects confirms predictions of the disclination energies and gives evidence for a finite dislocation energy.