Rigid Squares Research Articles

Flexible kirigami with local cylindrical shell design for stretchable microstrip antenna

This paper proposes a flexible kirigami structure. It has rotating rigid squares and cylindrical shells to improve the flexibility of the structure and produce snap-through instability. First, a mechanical spring model based on a quarter of unit cell was established to predict the mechanical characteristics of its deployment. The elastic deformation energy stored in the mechanical system was calculated, and the applied force was obtained after deriving the displacement. By assuming that the rotational stiffness of the spring is related to the length, elastic modulus, and thickness of the cylindrical shell, the coefficients used to quantify the stiffness of the rotational spring were obtained through a finite element calculation and the mechanical characteristics of the structure were obtained. The accuracy was verified through experiments and the finite element method. Then, the deployment mechanism and parameterization of the structure were analyzed. The snap-through characteristics of the cylindrical shell were analyzed, and the curvature of the second steady state was obtained using the minimum potential energy. The influence of the geometric parameters of the unit cell, such as its length, thickness, and the center angle of the cylindrical shell, on the snap-through instability was analyzed. By stretching or compressing it at different positions, different mechanical characteristics of the deployment were obtained. Finally, based on the designed structure, a stretchable microstrip antenna was designed to achieve high gain while the center frequency remained fixed. An antenna was fabricated and measured, and the test results are in good agreement with the simulated results. This work may provide theoretical guidance for the design and application of deformable structures.

Investigation of Modified Auxetic Structures from Rigid Rotating Squares.

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.

Wave propagation of 2D elastic metamaterial with rotating squares and hinges

Auxetic metamaterials have attracted widespread attention because of their untraditional physical properties and practical application values. In this paper, we focus on an elastic auxetic metamaterial composed of hinged rotating rigid squares. Theoretical analysis based on a mass-spring system is adopted to investigate the wave propagation behavior and vibration characteristic of the elastic metamaterial (EM). The motion equations are derived based on a discrete rigid-squares hinged model. The dispersion curves and eigenmodes under two configurations are studied based on Bloch's theorem with linear assumption. Moreover, the evolution process of the bandgap with the change of configuration parameter is revealed systematically and the transmission of the finite periodic hinged squares are explored under harmonic excitation. The results show that with proper parameter configuration, the considered EM structure can exhibit a complete bandgap. The formation mechanism of the bandgap is translational and rotational mode resonance. Moreover, the expressions of bandgap boundary frequencies with parameters are analytically derived, which provides explicit guidance for the adjustment of the bandgap characteristics and the parameter design of the structure. Furthermore, the harmonic transmission analysis validates the wave attenuation predicted by the bandgap. The structure can convert part of the energy into rotational energy, thereby obtaining a better vibration control capability. This work provides significant guidance for the design of elastic metamaterials with potential applications of vibration control.

Computational method for highly constrained molecular dynamics of rigid bodies: Coarse-grained simulation of auxetic two-dimensional protein crystals.

The increasing number of protein-based metamaterials demands reliable and efficient theoretical and computational methods to study the physicochemical properties they may display. In this regard, we develop a simulation strategy based on Molecular Dynamics (MD) that addresses the geometric degrees of freedom of an auxetic two-dimensional protein crystal. This model consists of a network of impenetrable rigid squares linked through massless rigid rods. Our MD methodology extends the well-known protocols SHAKE and RATTLE to include highly non-linear holonomic and non-holonomic constraints, with an emphasis on collision detection and response between anisotropic rigid bodies. The presented method enables the simulation of long-time dynamics with reasonably large time steps. The data extracted from the simulations allow the characterization of the dynamical correlations featured by the protein subunits, which show a persistent motional interdependence across the array. On the other hand, non-holonomic constraints (collisions between subunits) increase the number of inhomogeneous deformations of the network, thus driving it away from an isotropic response. Our work provides the first long-timescale simulation of the dynamics of protein crystals and offers insights into promising mechanical properties afforded by these materials.

Spatial pseudo-rigid body model for the analysis of a tubular mechanical metamaterial

In this paper, a pseudo-rigid body model is proposed for the analysis of a spatial mechanical metamaterial and its application is demonstrated. Using this model, the post-buckling behavior of the mechanical metamaterial can be determined without the need to consider the whole elastic structure, e.g., using finite-element procedures. This is done by analyzing a porous cylindrical mechanical metamaterial using a rigid body mechanism, consisting of rigid squares that are connected at their corners. Stiffness in this model comes from torsion springs placed at the connections between rigid parts. The theory of the model is presented and the results of two versions of this model are compared through experiments. One version describes the metamaterial in the free state, while the other, more extended, version includes clamped boundaries, matching the conditions of the experimental set-up. It is shown that the mechanical behavior of the spatial metamaterial is captured by the models and that the shape of the metamaterial in the deformed state can be obtained from the more extended model.

Influence of Negative Poisson′s Ratio on Stent Applications

Stents are usually defined as small tubular structures that are inserted into the diseased region which provide mechanical support of damaged artery or some other hollow organ to restore lumen and blood flow conditions in arteries. Recently developed stents may accumulate functions; for example, they can be used for drug delivery or for treatment of bleeding esophageal varices. Further clinical failures such as crushing, removing, restenosis are some of the most serious reasons for continuing stent improvements. Therefore, our research study has the design and manufacture of an auxetic structure with enhanced mechanical properties through the tailored negative Poisson’s ratio, and its configuration as an auxetic stent to be used for example in the palliative treatment of esophageal cancer and for the prevention of dysphagia as the main objective. Background: The use of stents for esophageal diseases has evolved greatly over the past 30 years. Esophageal cancer is the growth of cancer cells in the esophagus tube which usually originates in the inner layers of the lining of the esophagus and grows outward. In time, the tumor can obstruct the passage of food and liquid, making swallowing painful and difficult. Since most patients are not diagnosed until the late stages of the disease, esophageal cancer is associated with poor quality of life and low survival rates. Method: This research study is organized in three parts devoted to fabrication of auxetic structure and stent samples, experimental testing and discussion of results. For this purpose a model of an auxetic rotating-square geometry with a circular hole structure is developed by laser cutting apolyurethane sheet. The auxeticoesophageal stent was made by folding and gluing the auxetic sheet (with rigid squares and circular holes) into a cylindrical shape. Combined videoextensometry with compressive and tensile testing are employed to determine the deformation response of the auxetic sheet and stent to obtain stress-strain curves and estimation of Poisson’s ratio.Conclusions: The stenting outcomes are improved through mechanical design with tailored negative Poisson’s ratio through cell geometry of the esophagus stent. Theauxetic stent geometry of rotating squares with circular holes will be lighter in weight; reduces obstruction and food impaction because of its unique deformation mechanism; and can reduces migration of the implant stent by embedding inside the tissue.

Reading Steven Crane’s “War Is Kind” to My Husband, and: Surface Warfare, and: Sea-Change, and: Ithaca, and: Penelope, Stateside, and: The Rooted Bed

Reading Steven Crane’s “War Is Kind” to My Husband, and: Surface Warfare, and: Sea-Change, and: Ithaca, and: Penelope, Stateside, and: The Rooted Bed Jehanne Dubrow (bio) Reading Steven Crane’s “War Is Kind” to My Husband I packed your seabag today: six pairs of pants, shirts folded in their rigid squares, your socks balled up like tan grenades. I put my photo in as well, laid it there between the Kevlar vest and heap of clothes. Don’t weep, the poet warns, don’t weep. On 60 Minutes, a soldier turns his face toward us, shows the camera his burns, small metal slivers still embedded in the skin, his mouth a scrap of ragged tin. The young man’s face was beautiful before, smooth, unblemished as my own. For war [End Page 54] is kind, I read. Great isthe battle-god and great the auguries, the firing squad, the neon green of night vision that cuts the darkness open at its seams, gutted and spilling on the sand. Great is the Glock, the Aegis Weapons System, the Blackhawk circling. Great are the Ka-Bar fighting knives, the shells that sing through air, as though alive. Surface Warfare Our arguments move across the surfaces of things, smooth flat areas where silence floats for weeks. The rule: whoever speaks [End Page 55] first loses. If he patrols the living room, then I control our bed, an Atlantic filled with my insomnia, the quilts too thick to wade through. Some nights I think drowning would be easier and drink mouthfuls of salt. No shallows here, only the fathoms of marriage, and we anchored side by side, the darkness wide, percussive as a mine. Sea-Change Imagine this: saltwater scrubbing sand into my husband’s skin, his fingers pale anemones, his hands turned coral reef, and in his eyes the nacreous pearls of Ariel. This could be my husband, drowning in the swell. A sea-change means a shift, a change of heart, and how the oceans turn [End Page 56] glass shards into a jewel, rip apart familiar things. Waves churn. The surf is a liquid body that peels a carrier from bow to stern, the keel bent back, steel bands pliable as kelp. And long before I wake, the sailors drown. No point in calling help. Each night, my husband shakes me out of sleep. I cannot reach for him or drag him to the surface so he’ll swim. Ithaca There’s war beyond the shores. But here there’s Dairy Queen and Taco Bell, the Westfield Shopping Mall, the cell phone superstore, Home Depot, Sears. And home remains a metaphor for something else: a wife who tries to guard her chastity, ties it like a yellow ribbon to her door, sticks it to the bumper of her car, so that the neighbors know she sleeps alone, almost a widow to the Trojan war, her love preserved in plastic wrap like some dessert too beautiful to taste. [End Page 57] At PTA meetings, she’s chased by divorcés and other glum suitors. Nobody seems to care that she still wears a wedding ring. Odysseus is gone—same thing as being dead. And so men stare at her when she buys groceries or takes the dog out for a pee. She’s Ithaca, trapped in her own body, an island circled by the seas. Penelope, Stateside On an island called America, start fantasizing of the sex you had with him. Go shop for bras and lacey thongs at the PX, black garters, bustiers, a cream that leaves your body woven silk, a self-help book for self-esteem, a bag of M&Ms, skim milk to keep you thin, and Lean Cuisine (you hate to cook-for-one). Or buy a pair of True Religion jeans, the denim pressing on each thigh so that there’s no sensation but blue fabric like a second skin, [End Page 58] no lover’s touch more intimate than the zipper pressing in. But don’t forget. He may come home so torn that purchases won’t mean a thing, not the Posturepedic foam pillowtop mattress, or the sateen duvet. He won...

Auxetic behaviour from rotating semi‐rigid units

AbstractAuxetics (i.e. systems with negative Poisson's ratios) exhibit the unexpected feature of becoming fatter when stretched and narrower when compressed. This property is highly desirable as it imparts many beneficial effects on the material's macroscopic properties. Recent research suggests that in an idealised scenario, systems composed of connected ‘rigid squares’ can exhibit auxetic behaviour (Poisson's ratio = –1) due to a mechanism involving relative rotation of the squares. This paper shows through force‐field based molecular modelling simulations that although ‘rotating squares’ are responsible for negative Poisson's ratios in various zeolite frameworks, in these real materials, the squares are not rigid and the auxeticity is not as pronounced as in the ‘idealised’ model. In view of this, a new model system made from connected ‘semi‐rigid’ squares is proposed and analytical equations for the mechanical properties of this new model system are derived and discussed. It will be shown that the Poisson's ratios in this new model are highly dependent on the extent of rigidity of the squares and the direction of loading. It will also be shown that this new model provides a better description for the behaviour of auxetic zeolite frameworks than the original ‘rotating rigid squares’ model. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Self-assembly of a novel mixed-valence tetranuclear molecular square [FeII3FeIIIL2(HL)2]3+and [M(HL)]44+squares (M = Ni, Zn and Cd) {H2L = bis(2-acetylpyridine) thiocarbazone}

A novel mixed-valence rigid molecular square [FeII3FeIIIL2(HL)2]3+ was constructed via self-assembly using bis(2-acetylpyridine) thiocarbazone (H2L) as both linker and reducing agent. Single crystal structure analysis shows that each cation is located at an inversion center with four iron atoms at the corners of a square with edge length Fe⋯Fe ca. 4.2 Å. Magnetic measurement exhibits that there is only one FeIII center in each of the tetranuclear squares. There is no significant difference of the coordinated bonds among the four iron centers, indicating that it is impossible to recognize the FeIII in the four iron centers of a square. Electrochemical measurement and UV–VIS–NIR spectral studies demonstrate the Robin–Day class II behavior of the mixed-valence compound with appreciable iron–iron interaction. New rigid metallocyclic squares [Ni(HL)]44+, [Zn(HL)]44+ and [Cd(HL)]44+ were also assembled and structurally characterized for comparison. ESI-MS spectra reveal that the fragments of tetranuclear molecular squares are stable even in solution.

Auxetic behavior from rotating squares

Auxetic materials exhibit the very unusual properties of becoming wider when stretched and narrower when squashed [1], that is they have negative Poisson’s ratios. Apart from the pure scientific interest of having materials showing such an unconventional property, a negative Poisson’s ratio gives a material several other beneficial effects such as an increased shear stiffness, an increased plane strain fracture toughness and an increased indentation resistance. These properties make auxetics superior to conventional materials for many practical applications [1, 2]. In recent years several auxetics have been fabricated by modifying the microstructure of existing materials, including foams [2] and microporous polymers [3]. A number of molecular auxetics have also been proposed [4–9] one example being α-cristobalite [7]. The auxetic behavior in these materials can be explained in terms of their geometry and deformation mechanism. Thus, the hunt for new auxetic materials is frequently approached through searching for geometric features which may give such behavior [9, 10]. In this letter we present a new mechanism to achieve a negative Poisson’s ratio. This is based on an arrangement involving rigid squares connected together at their vertices by hinges as illustrated in Fig. 1. This may be viewed as a two dimensional arrangement of squares or as a projection of a particular plane of a three dimensional structure. This latter type of geometry is commonly found in inorganic crystalline materials [8, 9, 11]. Referring to Fig. 1, for squares of side length “l” at an angle θ to each other, the dimensions of the unit cell in the Oxi directions are given by:

Elastic properties of a disordered continuum of regular fractal structure: Self-consistent approach and finite-element method simulations

Macromolecular structures, as well as aggregation of filler in polymer-based composites, often may be described properly as fractals. Scaling behavior of the elastic moduli of a modeled fractal, the Sierpinski carpet, was the subject of this study. Sheng and Tao [1] and Patlazhan [2] found that, in the case of voids in on elastic host, axial and shear moduli exhibit distinct scaling dependencies on the size of the system. Nevertheless, it is widely accepted that moduli of random isotropic fractals (percolation clusters) scale with the same exponents. Explanation of the discrepancy is one of the main targets of the paper. The self-consistent approach and position space renormalization group technique (PSRG) have been applied for this goal. The mapping, corresponding to PSRG, was constructed numerically using the finite-element method (FEM) in the cases of voids and rigid inclusions. The self-consistent approach gives scaling behavior with exponents of values of about 0.11, independent of the modulus and type of inclusion, at developed stages of the fractal. It has been shown that mappings of PSRG on the plane, for two ratios of three independent moduli, have stable fixed points. This means that different elastic moduli exhibit scaling behavior with the same exponents (0.29 for voids and 0.17 for rigid squares) for developed fractal structure. The discrepancy in the exponent values obtained in the previous simulations is caused by the analysis of the initial stages of the structure. We believe that analogous results are valid for the wide class of self-similar fractals, and the dimension is the main parameter that governs the exponents and fixed point values.

Photoresponsive and Redox-Active Supramolecular Systems with Rigid Sp Carbon Chain Spacers

The construction and investigation of photoresponsive and redox-active supramolecular systems with acetylenyl (polyyne) and cumulenic sp hybridized carbon chains (Cn) are reported here. Novel types of one-dimensional rigid molecular rods and macrocyclic molecular squares with controllable and yet fixed chain lengths or cavity sizes are explored. Such systems will have the desired features such as synthetic accessibility, coordination versatility, rigidity, directionality, photostability, electronic conductivity, controllable distances between donors and acceptors, and controllable cavity sizes in macrocyclic molecular squares. These features are extremely important to the future applications of these new rigid supramolecular systems in molecular photonic and electronic devices, light-energy conversion, molecular recognition, and host-guest chemistry. In addition to the exploration of synthetic features, the solution state redox characteristics and molecular photophysics of the metal-containing supramolecular systems with sp carbon chains are also under investigation. This includes the modulation of electronic communication between multiple redox centers spanned by rigid spacers of various chain lengths and the study of long-range, directional photoinduced energy/electron transfer across such spacers. In addition, covalent surface confinement of systems containing sp carbon chains is also investigated here. Specifically, UV/vis and X-ray photoelectron spectroscopy have been used as spectroscopic tools to monitor the growth of mono-and multilayer thin films.

Rigid 3-Dimensional Compacta Whose Squares are Manifolds

Rigid 3-Dimensional Compacta Whose Squares are Manifolds

Rigid Finite Dimensional Compacta Whose Squares are Manifolds

Rigid Finite Dimensional Compacta Whose Squares are Manifolds

Mercerization of cellulose: 1. Determination of the structure of Mercerized cotton

The structure of Mercerized cellulose II has been determined using the intensity data from Mercerized cotton fibres and rigid body least squares refinement techniques. The crystal structure consists of an array of antiparallel chain molecules essentially identical to that found for regenerated cellulose II. The unit cell is monoclinic with dimensions a = 8.02 A ̊ , b = 8.99 A ̊ , c = 10.36 A ̊ (fibre axis), and γ = 116.6°; the space group is P2 1 and the cell contains sections of two cellulose chains which pass through the centre and corner of the ab projection. The final crystallographic R value was 0.263, based on intensity data for 30 observed and 11 unobserved non-meridional reflections. The CH 2OH groups of the corner chains are oriented near to the gt position while those of the centre chain are near to the tg position. The possibility of alternative side group orientations, most likely on the exterior of the cellulose II crystallites, has been demonstrated for both the Mercerized and regenerated structures. The percentage of these other orientations is small in the case of Fortisan (rayon) but substantial in the case of Mercerized cotton.

The exact determination of the orientation of organic molecules in crystal structures by least squares refinement of the Eulerian angles

AbstractA method for the exact determination of the orientation of the molecules in a crystalline organic structure by X‐ray analysis is described. The Eulerian angles are refined by a special rigid body least squares procedure. The theoretical background is given.

Can the `Lattice' Model of a Gas describe both Liquefaction and Solidification?

A preliminary study is made of a two-dimensional model of an imperfect gas of `attracting rigid squares' which is a limiting case of the two-dimensional Ising model with second-nearest neighbour interactions studied by Domb and Potts. The qualitative behaviour of the high order Mayer cluster-sums is determined by a method used by Temperley. It is found that at high temperatures these sums alternate in sign, which is what would be expected if the approaching transition were the formation of an ordered structure. As we lower the temperature, the analytic behaviour of the cluster-sums progressively changes to what was found by Yang and Lee when the approaching transition involves the formation of large domains, analogous to liquefaction. It is tenatively concluded that this version of the Ising model may show two transitions below a certain critical temperature, just like a real imperfect gas.

Virial Coefficients of ``Parallel Square'' and ``Parallel Cube'' Gases

The second, third, fourth, and fifth virial coefficients are evaluated exactly for gases whose molecules interact in two dimensions as rigid elastic squares with their sides parallel or in three dimensions as rigid elastic cubes with their edges parallel. Upper and lower bounds on the virial coefficients of gases composed of rigid circles or rigid spheres are obtained in terms of the virial coefficients for the parallel squares and parallel cubes.

Diffraction by Rigid Disks and Rigid Squares

Diffraction by Rigid Disks and Rigid Squares