Equivalent Unit Cell Research Articles

Rate-dependent constitutive behavior and mechanism of CMDB under tension loading

The composite modified double-base propellants (CMDB) were experimentally investigated for mechanical behavior under wide tension strain rate loading from 0.0001 s−1 to 2500 s−1, based on Instron mechanical machine and modified Hopkinson tension bar technique. Stress-strain curves were obtained for its strain rate effect and integrality evaluation. The results indicated that CMDB presents high rate-dependence with flow stress distinctly increase as loading strain rate increasing. A succinct constitutive formulation is established with only five parameters, to characterize well the rate-dependence and strain hardening behavior. The fracture morphologies were investigated by scanning electron microscopy, and it is indicated that they are also rate-dependent: the cavitation and matrix damage induced from matrix deformation work less but more RDX particles fractures with strain rate increasing. Equivalent unit cell model with brittle cracking was established to simulate the mechanical behavior and failure characteristics of CMDB. The results reveal that with increasing loading strain rates, CMDB presents a tough-brittle transition, with less cavitation and matrix damage induced by matrix deformation, while more RDX particles fracture. Series of simulated results confirm qualitatively the experimental observations, and the obtained stress contours facilitate to validate the observed characteristics and propose reasonable mechanisms.

Reduced relaxed micromorphic modeling of harmonically loaded metamaterial plates: investigating boundary effects in finite-size structures

In this paper, we propose an approach for describing wave propagation in finite-size microstructured metamaterials using a reduced relaxed micromorphic model. This method introduces an additional kinematic field with respect to the classical Cauchy continua, allowing to capture the effects of the underlying microstructure with a homogeneous model. We show that the reduced relaxed micromorphic model is not only effective for studying infinite-size metamaterials, but also efficient for numerical simulations and analysis on specimens of finite size. This makes it an essential tool for designing and optimizing metamaterials structures with specific wave propagation properties. The proposed model’s efficiency is assessed through numerical simulations for finite-size benchmark problems, and shows a good agreement for a wide range of frequencies. The possibility of producing the same macroscopic metamaterial with different but equivalent unit cell “cuts” is also analyzed, showing that, even close to the boundary, the reduced relaxed micromorphic model is capable of giving accurate responses for the considered loading and boundary conditions.

Synthesis, characterization and anticancer activities of cationic η6-p-cymene ruthenium(II) complexes containing phosphine and nitrogenous ligands

Ruthenium-based anticancer agents have created a center of attention in the field of inorganic medicinal chemistry. The first fully characterized cationic ruthenium(II)-arene complexes [Ru(η6-p-cymene) (PAr3)LNCl]+ with highly lipophilic PAr3 ligands where Ar = 3,5-((CH3)3C)2C6H3– (L1), 3,5-(CH3)2C6H3– (L2), 4-CH3O-3,5-(CH3)2C6H2– (L3) and 4-CH3O-C6H4– (L4) with N = 3-methylpyridine (1–4, respectively), or L4 and 4-methylpyridine (5), or L4 and CH3CN (6) were obtained (yields 67–91%) as solids stable to light and air. Electrical conductance indicates that all the complexes are 1:1 electrolytes in solution. Their composition and purity have been unambiguously established by single-crystal X-ray diffraction, NMR spectroscopy and elemental analysis. The coordination geometries are uniform for all six complexes and each structure consist of a unipositive complex cation bearing the phosphine ligands L1-L4 and LN = 3-methylpyridine, 4-methylpyridine or CH3CN attached to the organometallic fragment. The equivalent unit cell volumes per formula unit decrease with 1 > 3 > 2 > 4 > 5 > 6, accurately reflecting the decreasing sizes of the phosphines L1-L4, and a greater occupied volume for 3-methyl- vs. 4-methylpyridine, and the smallest volume contribution from CH3CN. Electrochemical studies showed mixed electrochemical mechanisms (EC/ECE) from partial substitution of p-cymene by CH3CN ligands from the solvent. A large electrochemical stability window (>2.2 V) for Ru(II) was observed extending beyond the physiological E° range. The complexes were cytotoxic against human cancer cell lines in vitro, and some complexes altered cell morphology.

Design of a free space metamaterial lens based on LC parameters at S-band

A metamaterial lens was designed based on a two-dimensional host transmission line network loaded with inductors and capacitors. The planar negative refractive index (NRI) periodic structure, consisting of a unit cell array with a concave side, works as a curved lens matched to free space operating at S-Band. An equivalent unit cell dispersion relation was developed to determine the transmission characteristics at the operating frequency range. The challenge in the metamaterial lens design is to increase the focal distance. In the simulated system, a high energy level throughout a focal region was observed in the whole frequency range. An equivalent bulky material is used as a resource to reduce computational effort.

Two-node method for the effective elastic properties of periodic cellular truss materials and experiment verification via stereolithography

A two-node method is proposed in this paper to predict the effective elastic properties of periodic cellular truss materials based on the traditional representative volume element (RVE) method. Through the two-node method, the original unit cell with multiple boundary nodes is transformed into a new equivalent unit cell with only two boundary nodes, and the effective elastic properties of the original unit cell can be indirectly obtained by the new equivalent unit cell. The accuracy of the present method is theoretically and numerically validated by using the asymptotic homogenization (AH) method and experimentally validated by the quasi-static uniaxial compression tests of the 3D printed cellular structures. Both theoretical and numerical results show that the present two-node method is more accurate and easier to implement than the traditional RVE method and the AH method when the size of the unit cell is negligible relative to the size of the macrostructure. For periodic cellular truss materials with rod elements, the present method can provide results with the same accuracy as the AH method but with more efficiency. For periodic cellular materials with beam elements, the present method can provide more accurate lower bound of the effective elastic properties than the traditional RVE method. The uniaxial compression tests results show that the elastic modulus of the equivalent structure obtained by the two-node method is almost the same as its original structures and is in good agreement with the calculation results obtained by the AH method.

Analytical and numerical modeling on resonant response of particles in polymer matrix under blast wave

A novel methodology is proposed in this paper to design polymer matrix with particles for protecting against the blast wave. Through the adoption of vibration analysis, two kinds of basic equivalent unit cell models introducing the massive springs are developed and the corresponding relations for structural parameters considering the Poisson effects are derived. The resonant frequencies of particles are obtained by theoretical prediction and compared with the results of finite element analysis, in which a good agreement is achieved. Based on the theoretical analysis, two types of composite materials with different geometry configurations are proposed and the 3D periodic models are established to investigate the dynamic performance under blast wave. The energy history of system is calculated to study the energy transfer effect among components. Additionally, the numerical parametric studies are conducted to investigate the effects of coating properties and particle volume fraction on the wave propagating in these two unit cell models. The acquired results show that the attenuation of blast wave is caused by the interaction between the elastic wave motion and the resonant particles. The coating properties and particle volume fraction are significant factors to affect the wave propagation and attenuation. The dominant frequencies of spectrum obtained by the simulations are well coincident with the theoretical solutions.

Cladding mode coupling in long-period gratings in index-guided microstructured optical fibers

To inscribe the long-period gratings (LPGs) in an index-guiding (or the solid-core) microstructured optical fiber (MOF), the opto-geometrical parameters of the fiber have to be determined, at first. Using our earlier developed analytical field model, we have evaluated the effective index for the fundamental core mode of the triangular lattice-based MOF, and the effective index of the fundamental cladding mode is obtained by approximating hexagonal unit cell by a circular unit cell. We demonstrate that the grating period of the LPGs obtained using the radius of the equivalent circular unit cell of Λ/2 and $$\varLambda \left( {{{\sqrt 3 } \mathord{\left/ {\vphantom {{\sqrt 3 } {2\pi }}} \right. \kern-0pt} {2\pi }}} \right)^{1/2}$$ , which are the two widely accepted radii for evaluating the effective index of the fundamental cladding mode, do not match well with the available experimental results. Therefore, to achieve better agreement in the results, we have proposed the linear combination of these two radii. Comparisons with available experimental results have also been included.

Analytical analysis of slow and fast pressure waves in a two-dimensional cellular solid with fluid-filled cells.

Wave propagation in cellular and porous media is widely studied due to its abundance in nature and industrial applications. Biot's theory for open-cell media predicts the existence of two simultaneous pressure waves, distinguished by its velocity. A fast wave travels through the solid matrix, whereas a much slower wave is carried by fluid channels. In closed-cell materials, the slow wave disappears due to a lack of a continuous fluid path. However, recent finite element (FE) simulations done by the authors of this paper also predict the presence of slow pressure waves in saturated closed-cell materials. The nature of the slow wave is not clear. In this paper, an equivalent unit cell of a medium with square cells is proposed to permit an analytical description of the dynamics of such a material. A simplified FE model suggests that the fluid-structure interaction can be fully captured using a wavenumber-dependent spring support of the vibrating cell walls. Using this approach, the pressure wave behavior can be calculated with high accuracy, but with less numerical effort. Finally, Rayleigh's energy method is used to investigate the coexistence of two waves with different velocities.

Thermo-mechanical analyses of heterogeneous materials with a strongly anisotropic phase: the case of cast iron

This work presents a systematic study of thermo-mechanical behaviour of macroscopically isotropic heterogeneous materials with anisotropic constituents based on microstructural modelling. As an example, lamellar cast iron is considered, whose microstructure is composed of spatially interconnected anisotropic graphite particles embedded in ferrite/pearlite matrix. The complex three-dimensional microstructure of lamellar cast iron is represented here by an idealised unit cell model which captures in a simplified manner the main morphological features of the material. The thermal, mechanical and thermo-mechanical response of the unit cell incorporating the highly anisotropic phase is analysed by comparing the results for the equivalent unit cell with the isotropic constituents and considering both fully fixed and loose interface conditions. The major conclusion drawn from these analyses is that the anisotropy of microstructural phases plays crucial role in determining both the effective as well as local response of the material. The simplifying isotropy assumption leads to significantly different predictions at both scales.

Perovskites and their potential use in solar energy applications.

1. Introduction A material may be described as having a perovskite structure1 if it has same type of crystal structure as perovskite--calcium titanium oxide (CaTi[O.sub.3])--does (Figure 1). Perovskite was first discovered in 1839 by Gustav Rose, in the Ural mountains in Russia, and is named after the Russian mineralogist L. A. Perovski (1792-1856), who first characterised the material. The general formula for perovskites is [ABX.sub.3], with 'A' and 'B' being two cations of significantly different sizes ('A'>'B'), while X is an that binds with both cations. The perovskite structure is adopted by many oxides that have an elemental composition: AB[O.sub.3]. The ideal cubic-structure has the 'B' cation in a 6-fold coordination, surrounded by an octahedron of anions, with the 'A' cation in a 12-fold cuboctahedral coordination. Cations 'A' occupy the cube corner positions (0, 0, 0), while cations 'B' occupy the body centred positions (1/2, 1/2, 1/2) with oxygen anions 'O' being located at face centred positions (1/2, 1/2, 0). Figure 1 shows edges for an equivalent unit cell with 'A' in body centre, 'B' at the comers, and 'O' in mid-edge. The requirements of relative ionic radii are quite exacting to maintain a stable cubic structure, meaning that even relatively minor degrees of buckling and distortion can result in a number of alternative versions with lower symmetry, in which the coordination numbers of either the 'A' cations, 'B' cations, or both, are reduced. Tilting of the B06 octahedron reduces the coordination of a too-small 'A' cation from 12 down to as low as 8. Conversely, when a small 'B' cation is brought off-centre, within its octahedral coordination, a stable bonding arrangement can be obtained. Such distortions can create an electric dipole and it is for this reason that perovskites such as BaTi03, which distort in this manner, exhibit the property of ferroelectricity. The most usual non-cubic forms of perovskites are the orthorhombic and tetragonal variants. There are also some more complex perovskite structures which contain two different 'B'-site cations, with the result that ordered and disordered variants are possible. [FIGURE 1 OMITTED] Under the high pressure conditions of the Earth's lower mantle, the pyroxene enstatite, MgSi[O.sub.3], is converted to a more dense perovskite-type polymorph, and indeed it is speculated that this particular phase of the material might be the most common mineral in the Earth (2). It has a perovskite structure, with an orthorhombic distortion, and is stable at pressures from ~24 GPa to ~110 GPa. [For comparison, the pressure at the centre of the Earth is ca 300 GPa.] However, it is stable only at depths of several hundred kilometres and could not be transported back to the Earth's surface without reforming into less dense materials. At yet greater pressures, MgSi[O.sub.3] perovskite undergoes a transformation to form post-perovskite. Although the most common perovskite compounds contain oxygen, perovskites containing fluoride anions are known, e.g. NaMg[F.sub.3]. Metallic perovskite compounds also exist (1), with the general formula [RT.sub.3]M, where R represents a rare-earth or other relatively large cation, T is a transition metal ion and M represents light metalloids (anions) which occupy the octahedrally coordinated 'B' sites, e.g. [RPd.sub.3]B, [RRh.sub.3]B and Ce[Ru.sub.3]C. MgC[Ni.sub.3] is a metallic perovskite compound, and is of particular interest on account of its superconducting properties. A further category has mixed oxide-aurides of Cs and Rb, such as [Cs.sub.3]AuO, which contain large alkali metal cations in the traditional anion sites, bonded to [O.sup.2-] and [Au.sup.-] anions. 2. Properties of perovskites As noted, the perovskite structure is imparted with an appreciable element of structural pliancy, and the ideal cubic structure (Figure 1) can be distorted in many different ways. Thus, the octahedra may become tilted, the cations can be displaced from the centres of their coordination polyhedra, and the octahedra might be distorted at the behest of electronic factors (e. …

Dynamical Diffraction Simulations in FePt—I

A series of multislice simulations to quantify the effect of various degrees of order, composition, and thickness on the electron diffracted intensities were performed using the L1₀ FePt system as the case study. The dynamical diffraction studies were done in both a convergent electron beam diffraction and selected area electron diffraction condition. The L1₀ symmetry demonstrated some peculiar challenges in the simulation, in particular between the {111} plane normal and the <111> direction, which are not equivalent because of tetragonality. A hybrid weighting function atom of Fe-Pt was constructed to account for S < 1 or nonequiatomic compositions. This statistical approach reduced the complexity of constructing a crystal with the probability that a particular atom was at a particular lattice site for a given order parameter and composition. Considerations of accelerating voltage, convergent angle, and thermal effects are discussed. The simulations revealed significant differences in intensity ratios between films of various compositions but equivalent unit cell numbers and degree of order.

Noncovalent Interactions under Extreme Conditions: High-Pressure and Low-Temperature Diffraction Studies of the Isostructural Metal−Organic Networks (4-Chloropyridinium)2[CoX4] (X = Cl, Br)

The crystal structures of the two isostructural metal-organic salts, (4-chloropyridinium) 2[CoX 4], X = Cl ( 1), Br ( 2), have each been determined at nine temperatures from 30 to 300 K and at nine pressures from atmospheric pressure to 4.2 GPa. A 5% reduction in unit cell volume is observed upon temperature reduction, whereas an 18-19% reduction is observed upon increasing the pressure over the ranges studied. The structures adopt a tape arrangement propagated by bifurcated N-H...X 2Co hydrogen bonds and by Co-X...Cl-C halogen bonds. Intertape interactions include type I Co-X...Cl-C and C-Cl...Cl-C halogen-halogen interactions as well as offset pi-stacking between the aromatic rings of the cations. Although little anisotropy in compression is seen upon temperature reduction, marked anisotropy in compression is observed upon pressure increase. Compression between tapes far exceeds compression along the tape, consistent with the strong attractive nature of the intratape N-H...X 2Co and Co-X...Cl-C interactions and the weaker dispersion dominated Co-X...Cl-C and C-Cl...Cl-C halogen-halogen interactions and pi-stacking interactions. Increased distortion of the [CoX 4] (2-) anions from idealized tetrahedral geometry arises upon pressure increase, consistent with local changes in the electric field that result from compression of the pairs of pi-stacked cations. The study of the isostructural pair of compounds permits a rare opportunity for quantitative evaluation of the "internal" or "chemical" pressure exerted by changing the [CoBr 4] (2-) anion for the smaller [CoCl 4] (2-) anion. Thus, crystal structures of 1 and 2 with equivalent unit cell volumes require an additional pressure of ca. 1 GPa exerted upon the structure containing the larger [CoBr 4] (2-) anion ( 2). This internal pressure increases to ca. 1.9 GPa at the highest pressures used in this study. Most significant is that examination of the isovolumetric pairs of structures shows that the structures containing the [CoCl 4] (2-) anion are contracted along the <101> vector, the direction of tape propagation, by ca. 1.2% and correspondingly expanded in other directions, relative to those containing the [CoBr 4] (2-) anion. Since the effect of difference in anion size has been removed by application of pressure, this anisotropy in dimensions clearly indicates that the N-H...X 2Co hydrogen bonds and Co-X...Cl-C halogen bonds are more strongly attractive for X = Cl rather than X = Br. Use of internal pressure thereby provides unique insight into the relative strength of intermolecular interactions.

Analysis of the fundamental space-filling mode of photonic crystal fibres: a symmetry point of view

We analyse the fundamental space-filling mode of photonic crystal fibres using known results from group theory. In photonic crystal fibres based on both hexagonal lattices and square lattices, this mode can be analytically computed most accurately from an equivalent circular unit cell having the same area as the original unit cell and the boundary conditions are perfect electric and perfect magnetic conductors for the electromagnetic field. We also show how the boundary conditions for the elementary piece used in numerical analysis are associated with the symmetry-induced results and the degeneracy of the fundamental space-filling mode of a rectangular photonic crystal fibre is lifted.

Improved fully vectorial effective index method for photonic crystal fibers: evaluation and enhancement

In the effective index method, which models photonic crystal fibers by means of a step-index waveguide analogy, two critical parameters are the effective index of the cladding and the effective core radius. It has been shown that the use of an effective core radius as a function of the relative air hole diameter, or also of the relative wavelength, can improve the accuracy of this method. We show, by comparison with a rigorous finite-difference frequency-domain method, that the reported improved fully vectorial effective index methods have commonly adopted a radius of the equivalent circular unit cell, which does not give the best accurate effective cladding index as compared with the use of an equivalent circular unit cell having the same area as the hexagonal unit cell. Furthermore, by defining both the radius of the equivalent circular unit cell and the effective core radius as a function of the relative air hole diameter, and the relative wavelength, we believe that the fully vectorial effective index method can be further enhanced in terms of accuracy of both the effective cladding index and the modal index.

The Samson phase, β-Mg2Al3, revisited

Co-Authors: Michael Feuerbacher, Carsten Thomas, Julien P. A. Makongo, Stefan Hoffmann, Wilder Carrillo-Cabrera, Raul Cardoso, Yuri Grin, Guido Kreiner, Jean-Marc Joubert, Thomas Schenk, Joseph Gastaldi, Henri Nguyen-Thi, Nathalie Mangelinck-Noël, Bernard Billia, Patricia Donnadieu, Aleksandra Czyrska-Filemonowicz, Anna Zielinska-Lipiec, Beata Dubiel, Thomas Weber, Philippe Schaub, Günter Krauss, Volker Gramlich, Jeppe Christensen, Sven Lidin, Daniel Fredrickson, Marek Mihalkovic, Wieslawa Sikora, Janusz Malinowski, Stefan Brühne, Thomas Proffen, Wolf Assmus, Marc de Boissieu, Francoise Bley, Jean-Luis Chemin, Jürgen Schreuer Abstract. The Al−Mg phase diagram has been reinvestigated in the vicinity of the stability range of the Samson phase, β-Mg2Al3 (cF1168). For the composition Mg 38.5 Al 61.5, this cubic phase, space group Fd-3m (no 227), a = 28.242(1) Å, V = 22526(2) Å3, undergoes at 214 °C a first-order phase transition to rhombohedral β′-Mg2Al3(hR293), a = 19.968(1) Å, c = 48.9114(8) Å, V = 16889(2) Å3, (i.e. 22519 Å3 for the equivalent cubic unit cell) space group R3m (no 160), a subgroup of index four of Fd-3m. The structure of the β-phase has been redetermined at ambient temperature as well as in situ at 400 °C. It essentially agrees with Samson's model, even in most of the many partially occupied and split positions. The structure of β′-Mg2Al3is closely related to that of the β-phase. Its atomic sites can be derived from those of the β-phase by group-theoretical considerations. The main difference between the two structures is that all atomic sites are fully occupied in case of the β′-phase. The reciprocal space, Bragg as well as diffuse scattering, has been explored as function of temperature and the β- to β′-phase transition was studied in detail. The microstructures of both phases have been analyzed by electron microscopy and X-ray topography showing them highly defective. Finally, the thermal expansion coefficients and elastic parameters have been determined. Their values are somewhere in between those of Al and Mg.

From random microstructures to representative volume elements

A unified treatment of random microstructures proposed in this contribution opens the way to efficient solutions of large-scale real world problems. The paper introduces a notion of statistically equivalent periodic unit cell (SEPUC) that replaces in a computational step the actual complex geometries on an arbitrary scale. A SEPUC is constructed such that its morphology conforms with images of real microstructures. Here, the appreciated two-point probability function and the lineal path function are employed to classify, from the statistical point of view, the geometrical arrangement of various material systems. Examples of statistically equivalent unit cells constructed for a unidirectional fibre tow, a plain weave textile composite and an irregular-coursed masonry wall are given. A specific result promoting the applicability of the SEPUC as a tool for the derivation of homogenized effective properties that are subsequently used in an independent macroscopic analysis is also presented.

Solution of the fundamental space-filling mode of photonic crystal fibers: numerical method versus analytical approaches

The accuracy of the solution of the fundamental space-filling mode of photonic crystal fibers by scalar and vectorial analytical approaches and its effect on the effective index models are investigated. Using a plane wave method as a benchmark, we show that the optimal choice of the radius of the equivalent circular unit cell used in the approximations is different for the two approaches and this value has a great effect on the accuracy of the solution of the fundamental space-filling mode. We also show that the vectorial approach with a properly defined value of the radius is highly accurate over a wide parameter range, whereas the scalar approach causes the main error in the scalar effective index model. We also confirm that a fully vectorial effective index model is accurate and efficient in the case of photonic crystal fibers with large air filling fractions.

Analytical solution of the fundamental space filling mode of photonic crystal fibers

Analytical solution of the fundamental space filling mode of photonic crystal fibers is revisited based on previous results by Midrio et al. [J Lightwave Technol 2000;18(7):1031–7]. The fundamental space filling mode is designated HE 11 mode following the conventional mode classification. A comparison is made between the vectorial method and the scalar method when identical parameters are used in the analytical solution. A more accurate radius of the equivalent circular unit cell is determined.

Designing a 161-element Ku-band microstrip reflectarray of variable size patches using an equivalent unit cell waveguide approach

We present an equivalent unit cell waveguide approach (WGA) to designing a multilayer microstrip reflectarray of variable size patches. A normal incidence of a plane wave on an infinite periodic array of radiating elements is considered to obtain reflection coefficient phase curves for the reflectarray's elements. It is shown that this problem is equivalent to the problem of reflection of the dominant TEM mode in a waveguide with patches interleaved by layers of dielectric. This waveguide problem is solved using a field matching technique and the method of moments (MoM). Based on this solution, a fast computer algorithm is developed to generate reflection coefficient phase curves for a multilayer microstrip patch reflectarray. The validity of the developed algorithm is tested against alternative approaches and Agilent high frequency structure simulator (HFSS). Having confirmed the validity of the WGA approach, a small offset feed two-layer microstrip patch array is designed and developed. This reflectarray is tested experimentally and shows good performance.

A unit cell waveguide model of a reflectarray formed by microstrip patches and slots

AbstractAn equivalent unit cell waveguide approach (WGA) is described to obtain reflection coefficient phase curves for designing a microstrip patch reflectarray supported by a ground plane with periodic apertures or slots. Based on the presented theory, a computer algorithm for determining the reflection coefficient of a plane wave normally incident on a multi‐layer structure of patches and apertures is developed. The validity of the developed algorithm is verified by comparing the obtained results with those published in the literature and the ones generated by Agilent High Frequency Structure Simulator (HFSS). A good agreement in all the presented examples is obtained, proving that the developed theory and computer algorithm can be an effective tool for designing multi‐layer microstrip reflectarrays with a periodically perforated ground plane. © 2003 Wiley Periodicals, Inc. Microwave Opt Technol Lett 36: 206–210, 2003; Published online in Wiley InterScience (www.interscience.wiley.com). DOI 10.1002/mop.10721