Trigonal Materials Research Articles

Laplace domain BEM for anisotropic transient elastodynamics

In this paper, we describe Laplace domain boundary element method (BEM) for transient dynamic problems of three-dimensional finite homogeneous anisotropic linearly elastic solids. The employed boundary integral equations for displacements are regularized using the static traction fundamental solution. Modified integral expressions for the dynamic parts of anisotropic fundamental solutions and their first derivatives are obtained. Boundary elements with mixed approximation of geometry and field variables with the standard nodal collocation procedure are used for spatial discretization. In order to obtain time-domain solutions, the classic Durbin’s method is applied for numerical inversion of Laplace transform. Problem of alleviating Gibbs oscillations is addressed. Dynamic boundary element analysis of the model problem involving trigonal material is performed to test presented formulation. Obtained results are compared with finite element solutions.

Green’s functions for a trigonal piezoelectric half-plane belonging to 3m crystal class

Piezoelectric materials have a wide range of industrial applications in different branches of engineering due to their electromechanical coupling. So, investigating their responses to either mechanical or electric loadings helps engineers for efficient design of smart systems. However, most of the studies have assessed the well-known 6 mm piezoelectric materials or piezoceramics and few papers have studied other piezoelectric crystals despite of their application in industry. In this paper, fundamental solutions of a trigonal piezoelectric half-plane belonging to 3m crystal class is obtained. The governing differential equations are derived and solved analytically using potential method. It is shown that the solution for the 3m material can be degenerated to 6 mm solution as a special case. The contour lines were depicted for two practical piezoelectric materials belonging to 3m and 6 mm crystal classes including lithium niobate and PZT-4 and they were compared to each other. The numerical results showed that the response of the trigonal material is asymmetric due to anisotropy and the effect of anisotropy on some responses is considerable causing totally different behavior from 6 mm piezoelectric material.

Magnetic properties of ferrimagnetic Mn3Si2Se6

The physical properties of Mn3Si2Se6 have been investigated using single crystals grown by iodine-assisted vapor transport. The material possesses a ferrimagnetic ground state and the properties are similar to those in the isostructural compound Mn3Si2Te6. In these trigonal materials, the dominant magnetic exchanges are antiferromagnetic and intrinsically frustrated, leading to a variety of competing ground states. In Mn3Si2Se6, the Curie temperature is TC = 67(1)K, which is slightly lower than that of Mn3Si2Te6 where TC = 78K. The anisotropy field is also smaller in the selenide relative to the telluride. In both materials, short-range correlations likely exist well-above TC. Neutron single crystal diffraction data on Mn3Si2Se6 suggest a collinear ferrimagnetic structure with the moments tilting out of the ab-plane in zero-field. Finally, the magnetization of vapor grown Mn3Si2Te6 is shown for comparison; these vapor grown crystals do not show a sharp onset of magnetic anisotropy above 300K as was previously observed for melt grown Mn3Si2Te6 crystals.

Symmetry-enforced band crossings in trigonal materials: Accordion states and Weyl nodal lines

Nonsymmoprhic symmetries, such as screw rotations or glide reflections, can enforce band crossings within high-symmetry lines or planes of the Brillouin zone. When these band degeneracies are close to the Fermi energy, they can give rise to a number of unusual phenomena: e.g., anomalous magnetoelectric responses, transverse Hall currents, and exotic surface states. In this paper, we present a comprehensive classification of such nonsymmorphic band crossings in trigonal materials with strong spin-orbit coupling. We find that in trigonal systems there are two different types of nonsymmorphic band degeneracies: (i) Weyl points protected by screw rotations with an accordion-like dispersion, and (ii) Weyl nodal lines protected by glide reflections. We report a number of existing materials, where these band crossings are realized near the Fermi energy. This includes Cu2SrSnS4 and elemental tellurium (Te), which exhibit accordion Weyl points; and the tellurium-silicon clathrate Te16Si38, which shows Weyl nodal lines. The ab-initio band structures and surface states of these materials are studied in detail, and implications for experiments are briefly discussed.

Characterization of Te nanoparticles synthesized by plasma processing

The local structure of Tellurium nanoparticles synthesized by arc plasma were studied by X-ray absorption spectroscopy. Nanoparticles were characterized by using different techniques to reveal particles size, morphology and composition. Te K-edge EXAFS spectra reveal a structure similar to trigonal Te, where both the inter-chain and intra-chain bonds, typical of Te crystal structure, are present. Evolution of the Debye-Waller factor with temperature reveals an increased static disorder in the nanoparticles respect to the trigonal material.

Effects of triaxial magnetic field on the anisotropic nanoplates

In this study, the influences of triaxial magnetic field on the wave propagation behavior of anisotropic nanoplates are studied. In order to include small scale effects, nonlocal strain gradient theory has been implemented. To study the nanoplate as a continuum model, the three-dimensional elasticity theory is adopted in Cartesian coordinate. In our study, all the elastic constants are considered and assumed to be the functions of (x, y, z), so all kind of anisotropic structures such as hexagonal and trigonal materials can be modeled, too. Moreover, all types of functionally graded structures can be investigated. eigenvalue method is employed and analytical solutions for the wave propagation are obtained. To justify our methodology, our results for the wave propagation of isotropic nanoplates are compared with the results available in the literature and great agreement is achieved. Five different types of anisotropic structures are investigated in present paper and then the influences of wave number, material properties, nonlocal and gradient parameter and uniaxial, biaxial and triaxial magnetic field on the wave propagation analysis of anisotropic nanoplates are presented. From the best knowledge of authors, it is the first time that three-dimensional elasticity theory and nonlocal strain gradient theory are used together with no approximation to derive the governing equations. Moreover, up to now, the effects of triaxial magnetic field have not been studied with considering size effects in nanoplates. According to the lack of any common approximations in the displacement field or in elastic constant, present theory has the potential to be used as a bench mark for future works.

Continuum-based free vibration of circular trigonal and isotropic plates

The free vibration behavior of completely unrestrained elastic circular plates with trigonal and isotropic material symmetry is studied with an approach involving approximate continuum solutions to the three-dimensional theory of linear elasticity. Of primary interest are (1) the influence of trigonal material symmetry on the modes of free vibration and (2) the accuracy of thin plate theory relative to the more exact three-dimensional theory. Resonant frequencies are calculated from the weak form of the equations of motion for the plate through the use of approximation functions and the Ritz method formulated in cylindrical coordinates. This approach enables the resulting eigenvalue problem to be split through group-theoretical symmetry analysis. Representative examples are given and quantitative limits are discussed.

The Cowin–Mehrabadi theorem for an axis of symmetry

The well-known Cowin–Mehrabadi Theorem deals with necessary and sufficient conditions for a normal n to a symmetry plane. Necessary conditions require that n be a common eigenvector of C ijkk , C ikjk and C ijkl n j n l . It is shown that a vector parallel to an axis of symmetry must also satisfy these conditions. An axis of rotational symmetry is also a normal to a plane of symmetry except in the case of a trigonal material. Being a common eigenvector of C ijkk and C ikjk belonging to a nondegenerate eigenvalue guarantees it to be an axis of symmetry.

First-Principles Investigations of the Physical Properties of Magnesium Nitridoboride

In this article, we investigate the electronic, dielectric, dynamic, and elastic properties of new magnesium nitridoboride (MgNB 9 ) using a first-principles approach based on density functional theory. No available experimental or theoretical investigations of these physical properties have been previously reported in the literature. Our work shows that MgNB 9 is a semiconducting positive uniaxial trigonal material with an indirect band gap (Z-L) of 1.76 eV and mixed ionic-covalent character. In addition, its electronic dielectric tensor is nearly isotropic, and the magnitude of its components is similar to those reported for ferroelectric materials. By contrast, its static dielectric tensor is strongly anisotropic in the plane orthogonal to its optical axis. This anisotropy is mainly governed by a highly polar low-frequency mode assigned to localized Mg motions. Furthermore, this material is mechanically stable, and its bulk and shear moduli are larger than those reported on III-V semiconductors. These results suggest that MgNB 9 could be a promising potential material for applications in optoelectronics.

Transverse waves in anisotropic elastic materials

With few exceptions, a transverse wave can propagate in an anisotropic elastic material along certain directions only. Equations that allow us to compute the direction n, the polarization vector a and the wave speed c are presented for a general anisotropic elastic material. In contrast to longitudinal waves, the directions along which a transverse wave can propagate are not finite isolated directions. There is a one-parameter family of n along which a transverse wave can propagate. On the surface of the unit sphere n 1 2 + n 2 2 + n 3 3 = 1 , the n along which a transverse wave can propagate traces out at least three curves. The intersection of two or more curves is the direction along which a longitudinal wave can propagate. We present solutions for n, a and c for orthotropic, tetragonal and trigonal materials. The corresponding solutions for hexagonal and cubic materials have been obtained earlier.

Traction-free vibrations of finite trigonal elastic cylinders.

The unrestrained, traction-free vibrations of finite elastic cylinders with trigonal material symmetry are studied using two approaches, based on the Ritz method, which formulate the weak form of the equations of motion in cylindrical and rectangular coordinates. Elements of group theory are used to divide approximation functions into orthogonal subsets, thus reducing the size of the computational problem and classifying the general symmetries of the vibrational modes. Results for the special case of an isotropic cylinder are presented and compared with values published by other researchers. For the isotropic case, the relative accuracy of the formulations in cylindrical and rectangular coordinates can be evaluated, because exact analytical solutions are known for the torsional modes. The calculation in cylindrical coordinates is found to be more accurate for a given number of terms in the series approximation functions. For a representative trigonal material, langatate, calculations of the resonant frequencies and the sensitivity of the frequencies on each of the elastic constants are presented. The dependence on geometry (ratio of length to diameter) is briefly explored. The special case of a transversely isotropic cylinder (with the elastic stiffness C14 equal to zero) is also considered.

Chirality in the Torsion of Cylinders with Trigonal Symmetry

Solutions are obtained for a class of torsion problems for cylinders of a material with trigonal material symmetry. In particular, the solutions for elliptical, circular and equilateral triangular cross-sections are presented. These solutions show that the stress distributions are non-chiral and the same as they would be if the material were isotropic; however the in-plane displacements are chiral and different from the isotropic case. The results show that there will be transverse, in-plane stress interactions between the layers of a composite cylinder composed of concentric cylinders of different trigonal materials in torsional loading. Such composite cylinders are structural designs used by nature and by man.

Acoustic wave propagation along cladded fibers of trigonal crystal symmetry

An approximate method is proposed to analyze the acoustic wave propagation characteristics in nonpiezoelectric cladded LiNbO/sub 3/ fibers of trigonal crystal symmetry. The fiber axis coincides with the crystalline Z axis and the cladding thickness is infinite. This method utilizes coupled mode equations and exact solutions for corresponding cladded fibers of hexagonal crystal symmetry are used as the approximation base. The analysis concentrates on the trigonal materials of 32, 3 m, and ~3 m groups. Dispersion curves of flexural F/sub 11/, F/sub 21/ and F/sub 12/ modes in such a fiber are presented for the first time with a focus on the fundamental F/sub 11/ mode. Acoustic modal birefringence phenomenon (or polarization dispersion) has been observed, which means that the dispersion relation of a mode depends on the relative transverse polarizations of the corresponding coupled modes. It was also found that in trigonal cladded fibers there are generally no pure torsional and radial-axial modes, which agrees with results previously obtained for trigonal rods.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Simulation of material microstructure using a 3D voronoi tesselation: Calculation of effective thermal expansion coefficient of polycrystalline materials

The three-dimensional Poisson-Voronoi model is used to simulate a topologically equivalent material microstructure. A finite element package has been written to use this model to calculate the effective thermal expansion coefficient of the polycrystalline aggregate using the single crystal thermoelastic properties. The results have been found to be close to the experimental values and the values calculated from the Hashin's exact equation for hexagonal, tetragonal and trigonal materials.

Simple reflection in anisotropic elastic media and its relation to exceptional waves and supersonic surface waves (II): examples

Illustrative examples of space of simple reflection and supersonic surface waves in some cubic, hexagonal and trigonal materials are presented. The examples are discussed in terms of the general analysis in the preceding paper on the interrelation between the space of simple reflection, exceptional transonic states and subsonic surface waves and the existence rules for supersonic surface waves derived there. Leaky wave branches to which the undamped supersonic surface waves belong, have also been calculated.

Near-coincidence orientations in hexagonal materials: from a unified twin approach to a quasiperiodic description

In materials belonging to the hexagonal crystal family (hexagonal or trigonal crystal systems), for which the irrationality arises primarily from the lattice parameters, the concept of near-coincidence orientation has to be introduced in order to characterize experimental grain boundaries. The practical use of this concept can be simplified if a twin approach is introduced: high-Σ specific coincidence orientations are described as a deviation from very low-Σ twin orientations defined among a unique set of limiting Σ. Consequently, for real hexagonal or trigonal materials, each orientation relationship defined by a quaternion (m, u, v, w), all relatively prime integers, can be described, for any c/a, uniquely by a quasiperiodic arrangement of elementary `twin' coincidences. Experimental cases of interfaces in hexagonal and rhombohedral crystals (h.c.p. metals, tungsten carbide, alumina) are analysed.

Study of Order-Disorder in Rock-Salt-Related Structures by Infrared Spectroscopy

Infrared spectra were determined for a series of oxides related by ordering and distortion to the rock-salt structure. The expected spectrum of the tetragonal 1:1 ordering (space group D4h19) contains 5 ir-active and 8 Raman-active bands. For the trigonally distorted structure (space group D3d5) factor group calculations yield 4 ir and 2 Raman modes. The expected additional structure appears in the spectra of the tetragonal materials but is broadened and washed out by the long-range polarization field. The spectra of the trigonal materials contain twice the predicted number of modes. The broadened bands of the 6-coordinated structures are compared with the sharp spectra of 4-coordinated γ-LiAlO2.