Young’s modulus and Poisson’s ratio for seven-constant tetragonal crystals and nano/microtubes

In the paper, the elasticity theory was applied to consider the mechanical properties of rectilinearly anisotropic seven-constant tetragonal crystals and their cylindrically anisotropic nano/microtubes with and with no chiral angle, being the angle between the crystallographic symmetry axis and elongated tube axis. Pt is found that the number of crystals with negative Poisson’s ratio is the least for rectilinear anisotropy and is much larger for curvilinear anisotropy. With a nonzero chiral angle, all nano/microtubes can have negative Poisson’s ratio. The elastic problem on axial tension of cylindrical nano/microtubes is solved for radially inhomogeneous stresses: three normal stresses and one shear stress.     

Young’s modulus and the Poisson’s ratio of planar and nanotubular supracrystalline structures

A method has been proposed for calculating the two-dimensional Young??s modulus and the Poisson??s ratio for planar and nanotubular structures through the components of the two-dimensional elastic rigidity tensor obtained by numerical methods. The method has been tested for graphene and two-dimensional supracrystalline sheets.     

Negative Poisson’s ratio for cubic crystals and nano/microtubes

The paper systemizes numerous cubic crystals which can have both positive and negative Poisson’s ratios (the so-called partial auxetics) depending on the specimen orientation in tension. Several complete cubic auxetics whose Poisson’s ratio is always negative are indicated. The partial cubic auxetics are classified with the use of two dimensionless elastic parameters. For one of the parameters, a critical value is found at which the orientation behavior of the crystals changes qualitatively. The behavior of mesotubes obtained by rolling up plates of cubic crystals (crystals with rectilinear anisotropy) is considered in detail. Such mesotubes with curvilinear cubic anisotropy can have micron and nanometer lateral dimensions. It is shown that uniform tension of nano/microtubes of cubic crystals is possible only in the particular case of zero chirality angle (the angle between the crystallographic axis and the axis of a stretched tube). It is demonstrated by the semi-inverse Saint-Venant method that solution of the axial tension problem for cylindrically anisotropic nano/microtubes of cubic crystals with a non-zero chirality angle is possible with radially inhomogeneous fields of three normal stresses and one shear stress. In the examples considered, the cylindrically anisotropic nano/microtubes of cubic crystals are auxetics even if they are initially non-auxetics with rectilinear anisotropy.     

Correlation between Atomic Size Ratio and Poisson＇s Ratio in Metallic Glasses

We report the correlation between atomic size ratio and Poisson＇s ratio in various metallic glasses. It is found that atomic size ratio has an influence on the atomic packing density of metallic glasses, which would significantly impact the shear modulus rather than bulk modulus. The findings may be helpful for understanding the structural origin of Poisson＇s ratio in metallic glasses, and are instructive for designing tough metallic glasses with large Poisson ＇s ratio.     

Young＇s Modulus Anisotropy and Magnetomechanical Damping of [110] Oriented Tb0.3Dy0.7Fe1.95 Alloy

The △E effect and the magnetoelastic or magnetomechanical damping of vibrations in [110] oriented Tbo.aDyo.TFe1.95 alloy are experimentally studied by quasi-static stress-strain measurements under various constant magnetic fields between 0 and 3000 Oe, where stress ranges are taken from about 0 to -25, -50, -75 MPa. The linear stressstrain behaviour is obtained when the magnetic field is high enough （9000 Oe）. However, the modulus E varies with the angle θ between the compressive stress and magnetic field （9000 Oe）, which shows that the anisotropy of the modulus E is related to the magnetic domain distribution.     

Temperature dependence of ratio between dielectric anisotropy and order parameter in fluorinated nematic liquid crystals

Temperature dependence of ratio between dielectric anisotropy and order parameter of fluorinated nematic liquid crystal is investigated by using a semi-empirical molecular orbital package that can accurately calculate an angle between molecular dipole moment and long axis. We optimize the molecular conformations with three semi-empirical Hamiltonians AM1, PM3 and PM5, and then make a comparison between computational results and experimental measurements. It is shown that the results obtained from AM1 method are in good agreement with the measurements. The present study offers an applicable method to predict the dielectric properties of liquid crystal material.     

Young’s modulus and fracture during Knoop indentation of potassium, rubidium, cesium, and ammonium acid phthalate single crystals on the (010) plane

The correlation of the anisotropy of the Young??s modulus of organic single crystals of potassium, rubidium, cesium, and ammonium acid phthalates with strain and fracture patterns during Knoop indentation on the (010) cleavage plane in the [001] and [100] directions has been studied. The data on the maximum anisotropy of the strain and fracture patterns of the ammonium acid phthalate single crystal have been discussed in view of the published data on the structure, mechanical, elastic, and X-ray spectral properties of these crystals.     

Comment on Revision of Kaup-Newell＇s Works on IST for DNLS Equation

In this note, it is shown that the revision of the Kaup-Newell＇s works on 1ST for DNLS equation is only available in the ease of solving the bright one-soliton solution to the equation. An example is taken to illustrate our point of view.     

On Kaup and Newell＇s Method for Solving DNLS Equation

Kaup and Newell＇s revised inverse scattering transform for the derivative nonlinear Schrodinger （DNLS） equation is investigated. We compared it with a more reasonable approach proposed recently, which is rigorously proven by the Liouville theorem. It is conduded that Kanp and Newell＇s revision is only suitable for giving single-soliton solution and can not be generalized to multi-soliton case.     

Noether＇s and Poisson＇s methods for solving differential equation xs^（m） = Fs（t,xk^（m-2） , xk（m-1）

This paper studies integration of a higher-order differential equation which can be reduced to a second-order ordinary differential equation. The solution of the second-order equation can be obtained by the Noether method and the Poisson method. Then the solution of the higher-order equation can be obtained by integrating the solution of the second-order equation.     

Representation surfaces of Young''s modulus and Poisson''s ratio for BCC transition metals

 The general expressions for the compliance , Young's modulus E(h k l) and Poisson's ratio υ(h k l) along an arbitrary direction [h k l] are given for cubic crystals. The representation surfaces, for which the length of the radius vector in the [h k l] direction equals to E(h k l) or υ(h k l), are constructed for seven BCC transition metals Cr, Fe, Mo, Nb, Ta, V and W. Neglecting W, which is isotropic, both representation surfaces of Young's modulus and Poisson's ratio are spherical surfaces. The remaining BCC transition metals may be grouped into two classes. In the first group (Cr, Mo, Nb and V) with negative values of s_A, Young's modulus surface has eight depressed corners and six rounded protuberances at the centers of the faces. In the second group (Fe and Ta) with positive values of s_A, on the contrary, Young's modulus surface has eight rounded protuberance corners and six depressions at the centers of the faces. The contrary conclusions are obtained for Poisson's ratio.     

Young's modulus surface and Poisson's ratio curve for cubic metals

The general expressions for the compliance , Young's modulus E(h k l) and Poisson's ratio υ(h k l, θ) along arbitrary loading direction [h k l] are given for cubic crystals. The representation surface for which the length of the radius vector in the [h k l] direction equals to E(h k l) and representation curve for which the length of the radius vector with angle θ deviated from the reference directions , and equals to υ(100, θ), υ(110, θ) and υ(111, θ) for example, are constructed for six FCC metals Ag, Al, Au, Cu, Ni, Pb and seven BCC metals Cr, Fe, Mo, Nb, Ta, V, W.     

Elastic nonlinearity parameters of tetragonal crystals

The equations of motion for the propagation of finite amplitude elastic waves in crystals of tetragonal symmetry have been derived starting from the expression for the elastic strain energy. The equations have been solved for a finite amplitude sinusoidal wave propagating along the pure mode directions which are [100], [110] and [001] for the tetragonal group TI. The solutions corresponding to longitudinal wave propagation yield expressions for the amplitudes of the fundamental and generated second harmonic for these directions in terms of certain combinations of second and third order elastic constants of the medium. The results will aid the experimenter to determine these constants using ultrasonic harmonic generation technique.     

Rupture stress and modulus of elasticity for Nd:YAG crystals

A method for determining the rupture stress and modulus of elasticity of Nd:YAG crystals is described, and measurements are given for various kinds of surface finish.     

Ultrasonic measurement of the elastic properties of benzoyl glycine single crystals

Certain organic crystals are found to possess high non-linear optical coefficients, often one to two orders of magnitude higher than those of the well-known inorganic non-linear optical materials. Benzoyl glycine is one such crystal whose optical second-harmonic generation efficiency is much higher than that of potassium dihydrogen phosphate. Single crystals of benzoyl glycine are grown by solvent evaporation technique usingN, N-dimethyl formamide as the solvent. All the nine second-order elastic stiffness constants of this orthorhombic crystal are determined from ultrasonic wave velocity measurements employing the pulse echo overlap technique. The anisotropy of elastic wave propagation in this crystal is demonstrated by plotting the phase velocity, slowness, Young’s modulus and linear compressibility surfaces along symmetry planes. The volume compressibility, bulk modulus and relevant Poisson’s ratios are also determined. Variation of the diagonal elastic stiffness constants with temperature over a limited range are measured and reported.     

Heusler合金Pd2 CrAl四方变形、磁性及弹性常数的第一性原理计算

采用基于密度泛函理论的投影缀加波方法研究了Heusler合金Pd₂CrAl的四方变形、磁性和弹性常数. 四方变形中,Pd₂CrAl在c/a≈1.20处出现总能的局域最小值,对应一个稳定的马氏体. Pd₂CrAl的L2₁结构和四方结构的单胞总磁矩分别为3.825μ_B和3.512μ_B. 在这两种结构中Cr原子均是Pd₂CrAl总磁矩的主要贡献者,Pd和Cr原子间存在很强的杂化作用,Cr的3d电子的t_2g和e_g两个亚能带是Pd₂CrAl磁性的主要来源. 弹性常数的计算结果显示,Pd₂CrAl的L2₁结构和四方结构的弹性常数均满足相应结构的稳定性判据. 关键词： Heusler合金 四方变形 磁性 弹性常数     

Solitons in surface stabilized ferroelectric liquid crystals and the determination of the twist elastic constant

Excitation of solitary waves and their propagation in surface-stabilized ferroelectric liquid crystal cells under alternating external electric field is investigated both theoretically and experimentally. The effect of solitary waves on electro-optic response spectra is analyzed for different amplitudes of applied fields, temperatures, and sample thicknesses. It is shown that solitons can only be excited within narrow ranges of frequencies of the sufficiently strong electric fields. The minimal frequency, at which soliton waves appear in ferroelectric smectic liquid crystals, is found to be related to the material constants of these systems. It is proved that measuring this threshold frequency gives the possibility to determine one of the material parameters, if the others are known. In this way, the intra-smectic-layer elastic constant is found for systems with the chevron geometry.     

Lattice dynamical study of body-centred tetragonal indium

The phonon dispersion curves, phonon frequency distribution function as well as the lattice specific heat of body-centred tetragonal indium have been deduced using a lattice dynamical model which includes central, angular and volume forces. Six elastic constants, four zone boundary frequencies and an equilibrium condition were used in the evaluation of the force constants. It is shown that this model is elastically consistent and satisfies the symmetry requirements of the lattice, the phonon frequencies of indium deduced from it are in very good agreement with the experimental values of Reichardt and Smith and the theoretical values of Garrett and Swihart, and theθ _D values compare well with the experimental values over a wide temperature range. The apparent discrepancies in the phonon dispersion curves and theθ _D-T curves obtained from deficient models, importance of umklapp processes and the significance of angular forces in the lattice dynamical models are discussed.     

光子晶体介质性质及手性特征的一种判别方法

通过研究各向同性以及各向异性介质中电磁场的波矢量与波印廷矢量之间的角度关系,给出了利用电磁场的单位波矢量与单位群速度的点积值来判断光子晶体介质性质及其手性特征的一种方法.并指出,可以利用光子晶体手性特征临界点的特性来解决多光束的合束问题.最后,利用解析计算讨论了上述结论.上述研究结果为进一步丰富“电动力学”等相关课程的教学内容打下了基础.     

Structure,elasticity and phase transitions in liquid crystals with deformations

ABSTRACT

A molecular-statistical theory describing the nematic liquid crystals (LCs) with spherical inclusions (or point defects) is proposed. At given size of inclusions and nematic order parameters at the surfaces of inclusions (zero in the case of point defects) and far from inclusions (where the nematic LC is almost uniform), the distribution of nematic order parameters in the bulk of LC with inclusions was found to be fully determined by the elastic constants of LC. We have found and explained the two-step heat-driven transformation from the nematic phase into the isotropic phase, with the intermediate phase in between. The nematic order parameters and the elastic constants are evaluated in the framework of a unified approach based on the features of pair interaction potentials of the individual LC molecules. It is shown that, in the case of K₃₃ < K₁₁, the point defects should destroy the conventional nematic phase.