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In this paper we propose a general method for creating a new type of hierarchical structures at any level
in both 2D and 3D. A simple rule based on a rotate-and-mirror procedure is introduced to achieve multilevel
hierarchies. These new hierarchical structures have remarkably few degrees of freedom compared
to existing designs by other methods. More importantly, these structures exhibit synchronized motions
during opening or closure, resulting in uniform and easily-controllable deformations. Furthermore, a
simple analytical formula is found which can be used to avoid collision of units of the structure during
the closing process. The novel design concept is verified by mathematical analyses, computational
simulations and physical experiments.

To read the full-text of this research,

you can request a copy directly from the authors.

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... Several recent developments of Resch's interconnected assemblies considered 'hierarchical' generalisations of these structures 45,75,80,[82][83][84][85][86][87][88] . In hierarchical structures/materials, a distinct structural pattern repeats in different scales 89 , that is why they are also called 'multiscale' structures/materials 7 . ...

... bone, nacre, diatoms, and spider silkhave hierarchical structures, resulting in some beneficial mechanical properties such as increased toughness and resistance to crack propagation 90-94 . In general, by increasing the level of the hierarchy, the number of degrees of freedom (DoFs) of a pivotally-interconnected hierarchical structure will increase 82,85 . Seifi et al. 82 introduced the method of rotate-and-mirror (RAM) to limit the number of DoFs when the hierarchical level increases. ...

... In general, by increasing the level of the hierarchy, the number of degrees of freedom (DoFs) of a pivotally-interconnected hierarchical structure will increase 82,85 . Seifi et al. 82 introduced the method of rotate-and-mirror (RAM) to limit the number of DoFs when the hierarchical level increases. However, it was concluded that this method is only able to generate hierarchical assemblies with odd-numbered modules (e.g., the module in the upper part of Fig. 1a), whereas even-numbered modules (e.g., the module in the upper part of Fig. 1b) must be avoided. ...

Mechanical metamaterials are man-made structures capable of achieving different intended mechanical properties through their artificial, structural design. Specifically, metamaterials with negative Poisson’s ratio, known as auxetics, have been of widespread interest to scientists. It is well-known that some pivotally interconnected polygons exhibit auxetic behaviour. While some hierarchical variations of these structures have been proposed, generalising such structures presents various complexities depending on the initial configuration of their basic module. Here, we report the development of pivotally interconnected polygons based on even-numbered modules, which, in contrast to odd-numbered ones, are not straightforward to generalize. Particularly, we propose a design method for such assemblies based on the selective removal of rotational hinges, resulting in fully-deployable structures, not achievable with previously known methods. Analytical and numerical analyses are performed to evaluate Poisson’s ratio, verified by prototyping and experimentation. We anticipate this work to be a starting point for the further development of such metamaterials.

... This solution relies on previous work [28], which showed a set of simple cuts to obtain a wide range of desired shapes and patterns, including hinges consisting of partially cut units. Such solutions lead to stress concentration at the hinges because, as reported in [28,29], the hinge design represents a trade-off between hinge failure and hinge stiffness. ...

... This solu previous work [28], which showed a set of simple cuts to obtain a wide ran shapes and patterns, including hinges consisting of partially cut units. Such s to stress concentration at the hinges because, as reported in [28,29], the hing resents a trade-off between hinge failure and hinge stiffness. ...

Auxetic structures exhibit unusual changes in size, expanding laterally upon stretching instead of contracting. This paper presents this effect in a failsafe mode in structures made of rigid squares. We applied the concept of auxetic structures made of rigid rotating squares (from Grima and Evans) and offer a novel solution for connecting them. By introducing axes of rotation on the surface of the squares, a reliable working system is obtained, free from stress, in which the squares can come into contact with each other and completely cover the surface of the structure, or, in the open position, form regularly arranged pores. Herein, we present a new 2D auxetic metamaterial that is mathematically generated based on a theoretical relationship of the angle between the edges of a square and the position of the axis of rotation. Physical models were generated in the form of a planar structure and in the form of a circular closed structure. Such physical models confirmed our initial considerations and the geometrical relationships, offering new application possibilities. The novel structure that was designed and manufactured for the purpose of the paper can be considered as a new proposal in the market of auxetic materials.

... Such solutions lead to stress concentration at the hinges because, as reported in [28,29], the hinge design represents a trade-off between hinge failure and hinge stiffness. ...

The article deals with auxetic structures made on the basis of suitably connected rigid rotating squares (from Grima and Evans), with axes of rotation on the squares’ surface. The geometric model and the resulting relationships that allow for the determination of Poisson’s ratio are considered in detail. It is demonstrated that changing the rotation axis position does not affect the negative Poisson’s ratio, equal to -1. The models built confirm the initial considerations and can offer new application possibilities.

... This means that the transverse stiffness and strength can be increased by transferring materials from the middle part of the cell wall closer to the vertexes or by replacing the vertex with a smaller structure [35] . Based on this concept, some researchers have also explored the vertex-based triangle and quadrilateral hierarchical honeycomb structures [36][37][38][39]] . Another typical self-similar hierarchical design method is replacing the cell walls with other smaller structures [40][41][42] . ...

In order to enhance the energy absorption capacity of the honeycomb structure, a novel self-similar hierarchical honeycomb is proposed by adding smaller hexagons in the center and the vertex position of the hexagonal honeycomb (HH), and therefore named as center-vertex honeycomb (CVH). An analytical model is built to investigate the in-plane crushing responses of the newly proposed honeycomb, which is in good agreement with the simulation and experimental results. The in-plane quasi-static compression characteristics and energy absorption capabilities of CVH are investigated systematically by finite element method and are compared with that of HH and other hierarchical honeycombs. Two plateau stress regions in the stress–strain curves of CVH are found under the quasi-static compression, and the second plateau stress is over three times higher than the first one. In order to understand the strengthening mechanism clearly, stable unicellular deformation and global modes are revealed. Two typical plateau stress are deduced theoretically based on the collapse modes to achieve performance prediction, which are respectively decided by the hexagonal structures located on both sides and the six vertexes of the representative unit cell. The results show that CVH can absorb much more energy than HH and the vertex-based hierarchical structure with the same masses under quasi-static compression. Furthermore, a parametric study is performed to explore the influence of the impact velocity, the order number, and the wall thickness on the plateau stress, specific energy absorption, and collapse deformation modes of CVH. It can be concluded that CVH is a better choice for energy absorption. The present study provides significant suggestions and guidance for the design-oriented multi-stage energy absorbing reinforced honeycomb structure with special functions.

... Elastic properties of these fractal structures could be derived using the iterative averaging approach (Novikov et al., 2001). Recently, hierarchical design of the rotating rigid structures brings rich mechanical properties, such as, control the extent of auxeticity (Gatt et al., 2015), strength enhancement (Tang et al., 2015), increased toughness (Sun and Pugno, 2013), increased versatility and tunability (Dudek et al., 2017), and synchronized deformations (Seifi et al., 2017;Lu et al., 2018). Generally, auxetic materials composed of ligaments (Prall and Lakes, 1997) or ribs (Lakes and Wojciechowski, 2008) were very light. ...

A novel two-dimensional (2D) mechanical metamaterial with highly programmable mechanical response under compressive load is presented in this paper by introducing hierarchical rotating structures, in which flexible structures are used to replace the rigid part in conventional rotating rigid structures. Multi-step deformation pathways and negative Poisson’s ratio were observed in in situ compression experiments and finite element simulations, suggesting that the mechanical behavior of the hierarchical metastructure is highly ordered and can be programmed by constraining angles of the proposed metamaterial. This work offers new insights to create mechanical metamaterial using hierarchical rotating structures for flexible devices and crashworthiness applications.

... Auxetic metamaterials are typically classified into re-entrant [16][17][18][19][20][21][22][23], chiral [24][25][26][27][28], rotating [29][30][31][32][33][34], and hierarchical laminate structures [35][36][37] according to their deformation mechanisms. In general, the re-entrant, chiral, and rotating structures are porous and composed of a single component, whereas the hierarchical laminate structures are solid and consist of two or more components with different Poisson's ratios. ...

An auxetic metamaterial is a type of mechanical metamaterial that has a negative Poisson's ratio. Most auxetic metamaterials are truss-based or originate from Boolean operations of simple geometries. Herein, we introduce a new 3D auxetic metamaterial that is mathematically generated from an implicit expression. Further, this metamaterial is fabricated by 3D printing using a flexible material, which allows it to recover from large deformations. The buckling-induced auxetic behavior of the metamaterial was first evaluated via compression tests and finite element analyses. A nickel layer was then plated onto the surface to enhance its stiffness, strength, and conductivity without loss of auxeticity and resilience. The integration of 3D printing and electroless plating enabled accurate control over the mechanical and conduction properties of the auxetic metamaterial; these properties are presented as contour maps for guidance in functional applications.
We propose a novel 3D auxetic metamaterial derived from a mathematically defined triply periodic minimal surface. The stiffness, strength, and conductivity of the metamaterial are enhanced by nickel plating without loss of auxeticity and resilience. The effective mechanical and conduction properties were mapped against geometric parameters, including relative density and nickel layer thickness. These data maps provide insight for tuning its performance over a broad range.

... The second mechanism is to arrange cells on a regular lattice and connect them at their corners, such that the cells (e.g., rigid squares, triangles, etc.) rotate around each other to expand in both directions [40,61]. Such auxetic materials can also be arranged in a hierarchical manner [95,148], which allows for a better control over the degrees of freedom. These rotating cells can be produced by simply perforating sheets, 20 such as demonstrated by Shan et al. [150]. ...

Digital fabrication machines such as 3D printers excel at producing arbitrary shapes, such as for decorative objects. In recent years, researchers started to engineer not only the outer shape of objects, but also their internal microstructure. Such objects, typically based on 3D cell grids, are known as metamaterials. Metamaterials have been used to create materials that, e.g., change their volume, or have variable compliance. While metamaterials were initially understood as materials, we propose to think of them as devices. We argue that thinking of metamaterials as devices enables us to create internal structures that offer functionalities to implement an input-process-output model without electronics, but purely within the material’s internal structure. In this thesis, we investigate three aspects of such metamaterial devices that implement parts of the input-process-output model: (1) materials that process analog inputs by implementing mechanisms based on their microstructure, (2) that process digital signals by embedding mechanical computation into the object’s microstructure, and (3) interactive metamaterial objects that output to the user by changing their outside to interact with their environment. The input to our metamaterial devices is provided directly by the users interacting with the device by means of physically pushing the metamaterial, e.g., turning a handle, pushing a button, etc. The design of such intricate microstructures, which enable the functionality of metamaterial devices, is not obvious. The complexity of the design arises from the fact that not only a suitable cell geometry is necessary, but that additionally cells need to play together in a well-defined way. To support users in creating such microstructures, we research and implement interactive design tools. These tools allow experts to freely edit their materials, while supporting novice users by auto-generating cells assemblies from high-level input. Our tools implement easy-to-use interactions like brushing, interactively simulate the cell structures’ deformation directly in the editor, and export the geometry as a 3D-printable file. Our goal is to foster more research and innovation on metamaterial devices by allowing the broader public to contribute.

... On a microscopic level-if we think of the unit cells-they can be seen as many small compliant mechanisms that are interconnected on a grid. While regular tilings of such cells are well understood [14,15,27,38,39], researchers only recently started to vary the parameters across a metamaterial [24,26], yet maintain the same topology of cells. ...

In this paper, we establish the underlying foundations of mechanisms that are composed of cell structures---known as metamaterial mechanisms. Such metamaterial mechanisms were previously shown to implement complete mechanisms in the cell structure of a 3D printed material, without the need for assembly. However, their design is highly challenging. A mechanism consists of many cells that are interconnected and impose constraints on each other. This leads to unobvious and non-linear behavior of the mechanism, which impedes user design. In this work, we investigate the underlying topological constraints of such cell structures and their influence on the resulting mechanism. Based on these findings, we contribute a computational design tool that automatically creates a metamaterial mechanism from user-defined motion paths. This tool is only feasible because our novel abstract representation of the global constraints highly reduces the search space of possible cell arrangements.

... The rotate-and-mirror method of Seifi et al. [32] can design hierarchical structures with significantly reduced DOFs, within which different component groups represent different DOFs. These component groups scattered across the whole structure, resulting in a non-uniform deformation pattern. ...

In this paper, we propose a general mechanism to realize a uniform global motion of an n-level hierarchical structure constructed by base components of various shapes, which has only n degrees of freedom. The uniform global motion of the components at the same level of hierarchy is synchronized and independent of movements at other levels. The significantly reduced number of degrees of freedom is achieved by introducing a parallelogram linkage loop to the structure while the hierarchy is obtained from the similarity between the structure and its representative components at different levels. Theoretical analysis reveals the kinematic equations that govern the expansion and retraction of the deployable devices. Numerical simulation and physical prototyping verify the theoretical prediction. This study paves a way towards designing deployable and easily controllable devices and structures for many applications in aeronautics, electronics, optics, and MEMS.

Introducing the hierarchy into cellular materials has attracted increasing attention in the effort to pursue improved absorbed-energy abilities and impact resistance. In this paper, the dynamic crushing properties and energy absorption capacities of joint-based hierarchical honeycombs with different topologies were explored by means of explicit dynamic finite element (FE) analysis using ANSYS/LS-DYNA. Four types of joint-based hierarchical honeycombs with uniform cell-wall thickness were firstly constructed by substituting each vertex of regular honeycombs with a smaller self-similar cell (hexagon or square). The respective influences of hierarchical parameters and impact velocities on in-plane dynamic deformation modes, mechanical characteristic and energy absorption of joint-based hierarchical honeycombs were discussed. Research results showed that the hierarchy had a far greater influence on the in-plane deformation modes of honeycombs. Compared with regular honeycombs, the dynamic plateau stress and specific energy absorption of joint-based hierarchical honeycombs can be improved if the proper hierarchical parameters were chosen. Adding the joint-based hierarchy into regular honeycombs can enhance the crushing stress efficiency (CSE) of the specimens. In addition, by introducing a non-dimensional dynamic sensitivity index, the dynamic shock enhancement of hierarchical honeycombs was also investigated. These researches are useful for the multi-objective dynamic optimization design and controllable properties of cellular materials.

Kirigami—the Japanese art of cutting paper—has recently inspired the design of highly stretchable and morphable mechanical metamaterials that can be easily realized by embedding an array of cuts into a sheet. This study focuses on thin plastic sheets perforated with a hierarchical pattern of cuts arranged to form an array of hinged squares. It is shown that by tuning the geometric parameters of this hierarchy as well as thickness and material response of the sheets not only a variety of different buckling‐induced 3D deformation patterns can be triggered, but also the stress–strain response of the surface can be effectively programmed. Finally, it is shown that when multiple hierarchical patterns are brought together to create one combined heterogeneous surface, the mechanical response can be further tuned and information can be encrypted into and read out via the applied mechanical deformation. The mechanical response of kirigami thin sheets with a hierarchical pattern of cuts arranged to form an array of hinged squares is investigated. The combined experimental and numerical results indicate that by tuning the geometric parameters of this hierarchy not only a variety of different buckling‐induced 3D deformation patterns can be triggered, but also the stress–strain response of the surface can be effectively programmed.

Over the past decade, the area of stretchable inorganic electronics has evolved very rapidly, in part because the results have opened up a series of unprecedented applications with broad interest and potential for impact, especially in bio‐integrated systems. Low modulus mechanics and the ability to accommodate extreme mechanical deformations, especially high levels of stretching, represent key defining characteristics. Most existing studies exploit structural material designs to achieve these properties, through the integration of hard inorganic electronic components configured into strategic 2D/3D geometries onto patterned soft substrates. The diverse structural geometries developed for stretchable inorganic electronics are summarized, covering the designs of functional devices and soft substrates, with a focus on fundamental principles, design approaches, and system demonstrations. Strategies that allow spatial integration of 3D stretchable device layouts are also highlighted. Finally, perspectives on the remaining challenges and open opportunities are provided. Diverse material structures for stretchable inorganic electronics are summarized, covering both the functional devices and soft substrates, with a focus on the fundamental principles, design approaches, and system demonstrations. Strategies that allow spatial integration of 3D stretchable device configurations are also highlighted. Finally, perspectives on remaining challenges and open opportunities are provided.

Hard and soft structural composites found in biology provide inspiration for the design of advanced synthetic materials. Many examples of bio-inspired hard materials can be found in the literature; far less attention has been devoted to soft systems. Here we introduce deterministic routes to low-modulus thin film materials with stress/strain responses that can be tailored precisely to match the non-linear properties of biological tissues, with application opportunities that range from soft biomedical devices to constructs for tissue engineering. The approach combines a low-modulus matrix with an open, stretchable network as a structural reinforcement that can yield classes of composites with a wide range of desired mechanical responses, including anisotropic, spatially heterogeneous, hierarchical and self-similar designs. Demonstrative application examples in thin, skin-mounted electrophysiological sensors with mechanics precisely matched to the human epidermis and in soft, hydrogel-based vehicles for triggered drug release suggest their broad potential uses in biomedical devices.

Auxetic mechanical metamaterials are engineered systems that exhibit the unusual macroscopic property of a negative Poisson's ratio due to sub-unit structure rather than chemical composition. Although their unique behaviour makes them superior to conventional materials in many practical applications, they are limited in availability. Here, we propose a new class of hierarchical auxetics based on the rotating rigid units mechanism. These systems retain the enhanced properties from having a negative Poisson's ratio with the added benefits of being a hierarchical system. Using simulations on typical hierarchical multi-level rotating squares, we show that, through design, one can control the extent of auxeticity, degree of aperture and size of the different pores in the system. This makes the system more versatile than similar non-hierarchical ones, making them promising candidates for industrial and biomedical applications, such as stents and skin grafts. H ierarchical materials and structures are a class of systems which are composed of structural elements which themselves have structure 1. These naturally occurring or man-made systems benefit from significantly enhanced mechanical properties 1,2 such as lightweight high-strength characteristics and an increased resistance to crack propagation 2. These qualities are needed by biological structures like bones 3 , wood 4

In this paper, a new kind of hierarchical tube with a negative Poisson's ratio (NPR) is proposed. The first level tube is constructed by rolling up an auxetic hexagonal honeycomb. Then, the second level tube is produced by substituting the arm of the auxetic sheet with the first level tube and rolling it up. The Nth (N >= 1) level tube can be built recursively. Based on the Euler beam theory, the equivalent elastic parameters of the NPR hierarchical tubes under small deformations are derived. Under longitudinal axial tension, instead of shrinking, all levels of the NPR hierarchical tubes expand in the transverse direction. Using these kinds of auxetic tubes as reinforced fibers in composite materials would result in a higher resistance to fiber pullout. Thus, this paper provides a new strategy for the design of fiber reinforced hierarchical bio-inspired composites with a superior pull-out mechanism, strength and toughness. An application with super carbon nanotubes concludes the paper.

The transverse compression and shear collapse mechanisms of a second order hierarchical corrugated truss structure have been analyzed. The two competing collapse modes of a first order corrugated truss are elastic buckling or plastic yielding of the truss members. In second order trusses, elastic buckling and yielding of the larger and smaller struts, shear buckling of the larger struts, and wrinkling of the face sheets of the larger struts have been identified as the six competing modes of failure. Analytical expressions for the compressive and shear collapse strengths in each of these modes are derived and used to construct collapse mechanism maps for second order trusses. The maps are useful for selecting the geometries of second order trusses that maximize the collapse strength for a given mass. The optimization reveals that second order trusses made from structural alloys have significantly higher compressive and shear collapse strengths than their equivalent mass first order counterparts for relative densities less than about 5%. A simple sheet metal folding and dip brazing method of fabrication has been used to manufacture a prototype second order truss with a relative density of about 2%. The experimental investigation confirmed the analytical strength predictions of the second order truss, and demonstrate that its strength is about ten times greater than that of a first order truss of the same relative density.

Many biological tissues, such as wood and bone, are fiber composites with a hierarchical structure. Their exceptional mechanical properties are believed to be due to a functional adaptation of the structure at all levels of hierarchy. This article reviews the basic principles involved in designing hierarchical biological materials, such as cellular and composite architectures, adapative growth and as well as remodeling. Some examples that are found to utilize these strategies include wood, bone, tendon, and glass sponges – all of which are discussed.

Auxetic materials and structures exhibit the unexpected behaviour of getting wider when stretched and thinner when compressed. This behaviour requires the structures (the internal structure in the case of materials) to have geometric features, which must deform in a way that results in the structure expanding when stretched. This paper assesses the potential for auxetic behaviour of a novel class of two-dimensional periodic structures which can be described as “connected stars” as they contain star-shaped units of different rotational symmetry which are connected together to form two-dimensional periodic structures. These structures will be studied through a technique based on force-field based methods (the EMUDA technique) and it will be shown that some, but not all, of these structures can exhibit auxetic behaviour. An attempt is made to explain the reasons for the presence or absence of a negative Poisson's ratio in these systems.

Mineralized biological materials such as bone, sea sponges or diatoms provide load-bearing and armor functions and universally feature structural hierarchies from nano to macro. Here we report a systematic investigation of the effect of hierarchical structures on toughness and defect-tolerance based on a single and mechanically inferior brittle base material, silica, using a bottom-up approach rooted in atomistic modeling. Our analysis reveals drastic changes in the material crack-propagation resistance (R-curve) solely due to the introduction of hierarchical structures that also result in a vastly increased toughness and defect-tolerance, enabling stable crack propagation over an extensive range of crack sizes. Over a range of up to four hierarchy levels, we find an exponential increase in the defect-tolerance approaching hundred micrometers without introducing additional mechanisms or materials. This presents a significant departure from the defect-tolerance of the base material, silica, which is brittle and highly sensitive even to extremely small nanometer-scale defects.

The indentation resistance of foams, both of conventional structure and of a novel re-entrant structure giving rise to negative Poisson's ratio, was studied using holographic interferometry. In holographic indentation tests, re-entrant foams had higher yield strengths sigma(y) and lower stiffness E than conventional foams of the same original relative density. Damage in both kinds of foam occurred primarily directly under the indenter. Calculated energy absorption for dynamic impact is considerably higher for re-entrant foam than conventional foam.

The role of structural hierarchy in determining bulk material properties is examined. Dense hierarchical materials are discussed, including composites and polycrystals, polymers, and biological materials. Hierarchical cellular materials are considered, including cellular solids and the prediction of strength and stiffness in hierarchical cellular materials.

Structural materials in nature exhibit remarkable designs with building blocks, often hierarchically arranged from the nanometer to the macroscopic length scales. We report on the structural properties of biosilica observed in the hexactinellid sponge Euplectella sp. Consolidated, nanometer-scaled silica spheres are arranged in well-defined microscopic concentric rings glued together by organic matrix to form laminated spicules. The assembly of these spicules into bundles, effected by the laminated silica-based cement, results in the formation of a macroscopic cylindrical square-lattice cagelike structure reinforced by diagonal ridges. The ensuing design overcomes the brittleness of its constituent material, glass, and shows outstanding mechanical rigidity and stability. The mechanical benefits of each of seven identified hierarchical levels and their comparison with common mechanical engineering strategies are discussed.

Applying hierarchical cuts to thin sheets of elastomer generates super-stretchable and reconfigurable metamaterials, exhibiting highly nonlinear stress-strain behaviors and tunable phononic bandgaps. The cut concept fails on brittle thin sheets due to severe stress concentration in the rotating hinges. By engineering the local hinge shapes and global hierarchical structure, cut-based reconfigurable metamaterials with largely enhanced strength are realized.

Auxetic metamaterials are synthetic materials with microstructures engineered to achieve negative Poisson's ratios. Auxetic metamaterials are of great interest because of their unusual properties and various potential applications. However, most of the previous research has been focused on auxetic behaviour of elastomers under elastic deformation. Inspired by our recent finding of the loss of auxetic behaviour in metallic auxetic metamaterials, a systematic experimental and numerical investigation has been carried out to explore the mechanism behind this phenomenon. Using an improved methodology of generating buckling-induced auxetic metamaterials, several samples of metallic auxetic metamaterials have been fabricated using a 3D printing technique. The experiments on those samples have revealed the special features of auxetic behaviour for metallic auxetic metamaterials and proved the effectiveness of our structural modification. Parametric studies have been performed through experimentally validated finite element models to explore the auxetic performance of the designed metallic metamaterials. It is found that the auxetic performance can be tuned by the geometry of microstructures, and the strength and stiffness can be tuned by the plasticity of the base material while maintaining the auxetic performance.

Significance
Most materials can be stretched to a small degree, depending on their elastic limits and failure properties. For most materials the maximum elastic dilatation is very small, implying that the macroscopic shapes to which an elastic body can be deformed is severely limited. The present work addresses the simple modification of any material via hierarchical cut patterns to allow for extremely large strains and shape changes and a large range of macroscopic shapes. This is an important step in the development of shape-programmable materials. We provide the mathematical foundation, simulation results, and experimental demonstrations of the concept of fractal cut. This approach effectively broadens the design space for engineered materials for applications ranging from flexible/stretchable devices and photonic materials to bioscaffolds.

Elastic instability of soft cellular solids plays an increasingly important role in the creation of metamaterials with smart properties. Inspiration for much of this research comes from a planar metamaterial with negative Poisson's ratio behavior induced by elastic instability. Here we extend the concept of buckling induced pattern switch further to the design of a new series of three-dimensional metamaterials with negative Poisson's ratio over a large strain range. The highlight of this work is that our designs are based on very simple initial geometric shapes.Different deformation patterns of materials without and with auxetic behavior.

Hierarchical lattices are made of finer lattices in successive smaller scales. This paper analytically studies the effect of hierarchy on the stiffness and strength of self-similar and hybrid type lattices, made by combining two distinct variants of topologies, governed by the bending and stretching dominated architectures. Scaling argument and physical reasoning are used to explain the behaviour of these lattices. The results show that the in-plane stiffness and the elastic buckling strength of the bending-bending lattices progressively improve with hierarchy; in contrast, only the buckling strength improves substantially for the stretching-stretching lattices, while the stiffness decreases. Low density bending-stretching lattices are unique with a significant improvement in stiffness, buckling, plastic collapse or crushing strength with hierarchy, whereas the stretching-bending lattices exhibit flexibility with lower strength. Despite no gain in stiffness, substantial gain in out-of-plane compressive strength is obtained with hierarchy because of the enhanced elastic and plastic buckling strength. Thus the advantage of combining lattices at multiple length scales provides a wide spectrum of choices for tailoring the properties for target applications including high performance core material, energy absorption or packaging.

Hierarchical structures are predicted to have ultra-light weight and superior mechanical properties, including excellent anti-buckling ability and energy absorption capability. Based on the improvement of making technology, hierarchical composite honeycombs (HCHs) have been designed, made and tested. With woven textile sandwich walls, the HCH is ultra-light and renders relatively ideal complete stress–strain curve with a stable displacement plateau at a relative high stress level in compression. A plastic model was suggested based on tested failure maps to reveal the plastic deformation and energy absorbing mechanism of the HCH. The plastic model well fitted the tested curves and defines a lower limit of the plastic deformation curve.

We investigated the mechanical behavior of two-dimensional hierarchical honeycomb structures using analytical, numerical and experimental methods. Hierarchical honeycombs were constructed by replacing every three-edge vertex of a regular hexagonal lattice with a smaller hexagon. Repeating this process builds a fractal-appearing structure. The resulting isotropic in-plane elastic properties (effective elastic modulus and Poisson’s ratio) of this structure are controlled by the dimension ratios for different hierarchical orders. Hierarchical honeycombs of first and second order can be up to 2.0 and 3.5 times stiffer than regular honeycomb at the same mass (i.e., same overall average density). The Poisson’s ratio varies from nearly 1.0 (when planar ‘bulk’ modulus is considerably greater than Young’s modulus, so the structure acts ‘incompressible’ for most loadings) to 0.28, depending on the dimension ratios. The work provides insight into the role of structural organization and hierarchy in regulating the mechanical behavior of materials, and new opportunities for developing low-weight cellular structures with tailorable properties.

The mechanical properties (linear and nonlinear elastic and plastic) of two-dimensional cellular materials, or honeycombs, are analysed and compared with experiments. The properties are well described in terms of the bending, elastic buckling and plastic collapse of the beams that make up the cell walls.

Introducing hierarchy into structures has been credited with improving elastic properties and damage tolerance. Specifically, adding hierarchical sub-structures to honeycombs, which themselves have good-density specific elastic and energy-absorbing properties, has been proposed in the literature. An investigation of the elastic properties and structural hierarchy in honeycombs was undertaken, exploring the effects of adding hierarchy into a range of honeycombs, with hexagonal, triangular or square geometry super and sub-structure cells, via simulation using finite elements. Key parameters describing these geometries included the relative lengths of the sub- and super-structures, the fraction of mass shared between the sub- and super-structures, the co-ordination number of the honeycomb cells, the form and extent of functional grading, and the Poisson’s ratio of the sub-structure. The introduction of a hierarchical sub-structure into a honeycomb, in most cases, has a deleterious effect upon the in-plane density specific elastic modulus, typically a reduction of 40 to 50% vs a conventional non-hierarchical version. More complex sub-structures, e.g. graded density, can recover values of density specific elastic modulus. With careful design of functionally graded unit cells it is possible to exceed, by up to 75%, the density specific modulus of conventional versions. A negative Poisson’s ratio sub-structure also engenders substantial increases to the density modulus versus conventional honeycombs.

Buckling is exploited to design a new class of three dimensional metamaterials with negative Poisson's ratio. A library of auxetic building blocks is identified and procedures are defined to guide their selection and assembly. The auxetic properties of these materials are demonstrated both through experiments and finite element simulations and exhibit excellent qualitative and quantitative agreement.

The wave propagation in and the vibration of sandwich plates with cellular core are analyzed and controlled. Negative Poisson’s ratio (auxetic) core materials of different geometry placed periodically in the plate introduce the proper impedance mismatch necessary to obstruct the propagation of waves over specified frequency bands (stop bands) and in particular directions. The location and the extension of the stop bands and the directions of wave propagation can be modified by proper selection of the periodicity and of the geometrical and physical properties of the core.
A finite element model is developed to predict the dynamic response of three-layered sandwich panels with honeycomb core. The finite element model along with the theory of periodic structures evaluates the influence of core materials of different geometry placed periodically along the two dimensions of the structure. This combined analysis yields the phase constant surfaces for the considered sandwich plates, which define location and extension of the stop bands, as well as the directions of wave propagation at assigned frequency values. The analysis of the phase constant surfaces and the evaluation of the harmonic response at specified frequencies demonstrate the plates’ directional properties, whose spatial patterns strongly depend on the configuration of the periodic core and on the excitation frequency. Auxetic honeycombs are here considered as core materials in order to obtain maximum design flexibility. Their elastic and inertial characteristics in fact vary substantially with their internal geometry. For given configurations they outcast up to five times the corresponding properties of traditional hexagonal honeycombs.
The presented numerical results demonstrate the unique characteristics of this class of two-dimensional periodic structures, which behave as directional mechanical filters. The findings of the study suggest that optimal configurations for the periodic cellular core may be identified in order to design passive composite panels, which are stable and quiet over desired frequency bands and which fit desired transmissibility levels in particular directions. Such unique filtering capabilities are achieved without requiring additional passive or active control devices and therefore without compromising the size and the weight of the layered structure.

A 3D hierarchical computational model of deformation and stiffness of wood, which takes into account the structures of wood at several scale levels (cellularity, multilayered nature of cell walls, composite-like structures of the wall layers) is developed. At the mesoscale, the softwood cell is presented as a 3D hexagon-shape-tube with multilayered walls. The layers in the softwood cell are considered as considered as composite reinforced by microfibrils (celluloses). The elastic properties of the layers are determined with Halpin–Tsai equations, and introduced into mesoscale finite element cellular model. With the use of the developed hierarchical model, the influence of the microstructure, including microfibril angles (MFAs, which characterizes the orientation of the cellulose fibrils with respect to the cell axis), the thickness of the cell wall, the shape of the cell cross-section and the cell dimension (wood density), on the elastic properties of softwood was studied.

A three-dimensional cellular system that may be made to exhibit some very unusual but highly useful mechanical properties, including negative Poisson's ratio (auxetic), zero Poisson's ratio, negative linear and negative area compressibility, is proposed and discussed. It is shown that such behaviour is scale-independent and may be obtained from particular conformations of this highly versatile system. This model may be used to explain the auxetic behaviour in auxetic foams and in other related cellular systems; such materials are widely known for their superior performance in various practical applications. It may also be used as a blueprint for the design and manufacture of new man-made multifunctional systems, including auxetic and negative compressibility systems, which can be made to have tailor-made mechanical properties.

Natural selection and evolution develop a huge amount of biological materials in different environments (e.g. lotus in water and opuntia in desert). These biological materials possess many inspiring properties, which hint scientists and engineers to find some useful clues to create new materials or update the existing ones. In this review, we highlight some well-studied (e.g. nacre shell) and newly-studied (e.g. turtle shell) natural materials, and summarize their hierarchical structures and mechanisms behind their mechanical properties, from animals to plants. These fascinating mechanisms suggest to researchers to investigate natural materials deeply and broadly, and to design or fabricate new bio-inspired materials to serve our life.

Auxetics exhibit the unusual property of expanding when uniaxially stretched (negative Poisson's ratio), a property that is usually linked to particular geometric features and deformation mechanisms. One of the mechanisms which results in auxetic behaviour is the one involving rotating rigid units, for which, systems made from triangles, squares or rectangles have already been considered. In this work we extend this study by considering systems which can be constructed from either connected rhombi or connected parallelograms. We show that various types of such systems can exist and we discuss in detail the properties of one type of ‘rotating rhombi' and one type of ‘rotating parallelograms'. We also show that the Poisson's ratio of these systems, which can be positive or negative, is anisotropic and dependent on the shape of the parallelograms/rhombi and the degree of openness of the system. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

The force-field-based molecular simulations were used to predict negative Poisson's ratios in a number of idealized zeolite cage structures. The analysis was carried through a combination of the framework geometry and simple deformation mechanisms acting within the frameworks. The presence of cations and molecular-sized pores in the structure of zeolites helped to study their potential as molecular auxetics. The results depicted the relation between the effect of geometry and deformation mechanisms on framework nanostructures and the auxetic behavior of idealized zeolite structures.

Conceptual hardware architecture of skin-like circuits is described. An elastomeric skin carries rigid islands on which active subcircuits are made. The subcircuit islands are interconnected by stretchable metallization. We concentrate on recent advances in stretchable thin-film conductors, by covering their construction, evaluation, and laboratory and theoretical analysis. Reversibly stretchable conductors with electrically-critical strains ranging from 10% to 100% have been made. r 2004 Elsevier B.V. All rights reserved.

Auxetic materials exhibit the unexpected feature of becoming fatter when stretched and narrower when compressed, in other words, they exhibit a negative Poisson's ratio. This counter-intuitive behaviour imparts many beneficial effects on the material's macroscopic properties that make auxetics superior to conventional materials in many commercial applications. Recent research suggests that auxetic be-haviour generally results from a cooperative effect between the material's internal structure (geometry setup) and the deformation mechanism it undergoes when submitted to a stress. Auxetic behaviour is also known to be scale-independent, and thus, the same geometry/deformation mechanism may operate at the macro-, micro- and nano- (molecular) level. A considerable amount of research has been focused on the ‘re-entrant honeycomb structure’ which exhibits auxetic behaviour if deformed through hinging at the joints or flexure of the ribs, and it was proposed that this ‘re-entrant’ geometry plays an impor- tant role in generating auxetic behaviour in various forms of materials ranging from nanostructured polymers to foams. This paper discusses an alternative mode of deformation involving ‘rotating rigid units’ which also results in negative Poisson's ratios. In its most ideal form, this mechanism may be construc- ted in two dimensions using ‘rigid polygons’ connected together through hinges at their vertices. On application of uniaxial loads, these ‘rigid polygons’ rotate with respect to each other to form a more open structure hence giving rise to a negative Poisson's ratio. This paper also discusses the role that ‘rotating rigid units’ are thought to have in various classes of materials to give rise to negative Poisson's ratios. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)

A new mechanism to achieve a negative Poisson's ratio is presented. An arrangement is made which involves rigid squares connected together at their vertices by hinges. The off-axis mechanical properties obtained from the standard transformation equations show that the idealized system is isotropic. The Poisson's ratio has a value of -1 irrespective of the direction of loading. The geometry modelled is the projection of a plane in inorganic crystalline materials and involves octahedrally co-ordinated atoms.

The auxetic behavior of materials with a negative Poisson's ratio is discussed. This property gives a material several beneficial effects such as increased shear stiffness, increased plane strain fracture toughness, increased indentation resistance and improved acoustic damping properties. The Poisson's ratio is not constant and varies with strain and the initial geometry parameters. The deformations of the triangles inevitably occur in parallel with the rotations, which reduce the extent of the auxetic effect and allows the system to shear. The force-field based simulations suggest that auxetic behavior is retained, which illustrates that the rotating triangles mechanism can be a very effective way for introducing negative Poisson's ratios in real materials.

Materials with negative Poisson's ratios (auxetic) get fatter when stretched and thinner when compressed. This paper discusses a new explanation for achieving auxetic behaviour in foam cellular materials, namely a ‘rotation of rigid units’ mechanism. Such auxetic cellular materials can be produced from conventional open-cell cellular materials if the ribs of cell are slightly thicker in the proximity of the joints when compared to the centre of the ribs with the consequence that if the conventional cellular material is volumetrically compressed (and then ‘frozen’ in the compressed conformation), the cellular structure will deform in such a way which conserves the geometry at the joints (i.e. behave like ‘rigid units’) whilst the major deformations will occur along the length of the more flexible ribs which form ‘kinks’ at their centres as a result of the extensive buckling. It is proposed that uniaxial tensile loading of such cellular systems will result auxetic behaviour due to re-unfolding of these ‘kinks’ and re-rotation of the ‘rigid joints’.

Two-dimensional (2D) hierarchical cellular materials made up of sandwich walls are predicted to have superior mechanical properties compared with solid-wall cellular materials. Equations of the stiffness, the buckling strength, the plastic collapse strength, the brittle failure strength and the fracture toughness were deduced. The enhancement of the mechanical properties of 2nd order hierarchical honeycombs is substantial (even an order of magnitude). The hierarchical honeycomb is much more damage tolerant and insensitive to wavy imperfections of the cell wall. Sandwich struts also enhance the buckling strength of the stretching-dominated 2nd order lattice grid material. Made up of sandwich struts, the hierarchical honeycomb has comparable mechanical properties with the stretching-dominated lattice grid material.

Recent advances in mechanics and materials provide routes to integrated circuits that can offer the electrical properties
of conventional, rigid wafer-based technologies but with the ability to be stretched, compressed, twisted, bent, and deformed
into arbitrary shapes. Inorganic and organic electronic materials in microstructured and nanostructured forms, intimately
integrated with elastomeric substrates, offer particularly attractive characteristics, with realistic pathways to sophisticated
embodiments. Here, we review these strategies and describe applications of them in systems ranging from electronic eyeball
cameras to deformable light-emitting displays. We conclude with some perspectives on routes to commercialization, new device
opportunities, and remaining challenges for research.

Collagen is a protein material with superior mechanical properties. It consists of collagen fibrils composed of a staggered array of ultra-long tropocollagen (TC) molecules. Theoretical and molecular modeling suggests that this natural design of collagen fibrils maximizes the strength and provides large energy dissipation during deformation, thus creating a tough and robust material. We find that the mechanics of collagen fibrils can be understood quantitatively in terms of two critical molecular length scales chi(S) and chi(R) that characterize when (i) deformation changes from homogeneous intermolecular shear to propagation of slip pulses and when (ii) covalent bonds within TC molecules begin to fracture, leading to brittle-like failure. The ratio chi(S)/chi(R) indicates which mechanism dominates deformation. Our modeling rigorously links the chemical properties of individual TC molecules to the macroscopic mechanical response of fibrils. The results help to explain why collagen fibers found in nature consist of TC molecules with lengths in the proximity of 300 nm and advance the understanding how collagen diseases that change intermolecular adhesion properties influence mechanical properties.

Design of hierarchically cut hinges for highly stretchable and reconfigurable metamaterials with enhanced strength

- J A Rogers
- T Someya
- Y Huang

Rogers, J. A., Someya, T. & Huang, Y. Design of hierarchically cut hinges for highly stretchable and reconfigurable metamaterials
with enhanced strength. Science 327, 1603 (2010).

- R Gatt

Gatt, R. et al. Hierarchical auxetic mechanical metamaterials. Sci. Rep. 5, 8395 (2015).

- L Zhao
- Q Zheng
- H Fan
- F Jin

Zhao, L., Zheng, Q., Fan, H. & Jin, F. Hierarchical composite honeycombs. Mater. Design 40, 124-129 (2012).