Theoretical examination of ultrasonic pole figures via comparison with the results analyzed by finite element polycrystal model

The ultrasonic wave velocities in a polycrystalline aggregate are sensitively influenced by texture development due to plastic deformation. According to Sayer's model, it is possible to construct ultrasonic pole figures via the crystallite orientation distribution function (CODF), which can be calculated by using ultrasonic wave velocities. In the previous papers, the theoretical modeling to simulate ultrasonic wave velocities propagating in solid materials under plastic deformation has been proposed by the authors and proved to be a good agreement with experimental results. Generally, wave velocities are dependent upon the propagating wave frequency; hence to evaluate texture development via ultrasonic pole figures it is necessary to examine an influence of frequency dependence on the ultrasonic wave velocities. In the present paper, the proposed theoretical modeling is applied to the texture characterization in polycrystalline aggregates of FCC metals under various plastic strain histories via ultrasonic pole figures, and also the frequency dependence is examined by using Granato-Lücke's dislocation strings model. Then the simulated ultrasonic pole figures are compared with the pole figures analyzed by the finite element polycrystal model (FEPM). The good qualitative agreement between both results suggests the sufficient accuracy of our proposed theoretical modeling.     

Relationship between elastic anisotropy and texture in metal-matrix composites

The relationship between elastic anisotropy and texture in two-phase metal-matrix composites has been developed under certain conditions. Using measurements of the six independent ultrasonic velocities V_ij in samples of the aluminium alloys 8091 and 7064 containing up to 20% SiC particles and the formulation given by Bunge, the fourt-order expansion coefficients of the orientation distribution function are determined. The Young's moduli in different directions are also obtained from ultrasonic velocity measurements. Linear correlations between anisotropy described by Young's moduli and texture determined by orientation distribution function expansion coefficients are obtained, and confirm developed relationships for two-phase metal-matrix composites. This result shows that ultrasonic measurements provide a technique for the characterization of texture and elastic anisotropy in these materials.     

Determination of orientation distribution function of anisotropic paramagnetic species by analysis of ESR spectra angular dependence

A method has been developed to determine orientation distribution function (ODF) of anisotropic paramagnetic species by analysis of the angular dependence of the ESR spectra. The method is based on computational spectra simulation. The ODF is represented as an expansion in terms of orthonormal functions. The expansion coefficients are determined through minimization of discrepancies between simulated spectra and experimental ones. By means of the suggested method we have determined the orientation distribution functions for radical probe 2,2,6,6-tetramethyl-4-ol-piperidinooxyl in 4-n-amyl-4'-cyanobiphenyl aligned by magnetic field and 2-septadecyl-2,3,4,5,5-pentamethylimidazolidine in polyethylene stretched films. In each case, thermal evolution of the ODF has been investigated.     

Effect of texture and grain shape on ultrasonic backscattering in polycrystals

An ultrasonic backscattering model is developed for textured polycrystalline materials with orthotropic or trigonal grains of ellipsoidal shape. The model allows us to simulate realistic microstructures and orthotropic macroscopic material textures resulting from thermomechanical processing for a broad variety of material symmetries. The 3-D texture is described by a modified Gaussian orientation distribution function (ODF) of the crystallographic orientation of the grains along the macroscopic texture direction. The preferred texture directions are arbitrary relative to the axes of the ellipsoidal grains. The averaged elastic covariance and the directional anisotropy of the backscattering coefficient are obtained for a wave propagation direction arbitrary relative to the texture and grain elongation directions. One particular application of this analysis is the backscattering solution for cubic crystallites with common textures such as Cube, Goss, Brass and Copper. In our analysis, in the texture-defined coordinates the matrix of elastic constants for cubic crystallites takes the form of orthotropic or trigonal symmetry. Numerical results are presented, discussed and compared to the experimental data available in the literature illustrating the dependence of the backscattering coefficient on texture and grain shape.     

On the third- and fourth-order constants of incompressible isotropic elasticity

Consider the constitutive law for an isotropic elastic solid with the strain-energy function expanded up to the fourth order in the strain and the stress up to the third order in the strain. The stress-strain relation can then be inverted to give the strain in terms of the stress with a view to considering the incompressible limit. For this purpose, use of the logarithmic strain tensor is of particular value. It enables the limiting values of all nine fourth-order elastic constants in the incompressible limit to be evaluated precisely and rigorously. In particular, it is explained why the three constants of fourth-order incompressible elasticity μ, ā, and D are of the same order of magnitude. Several examples of application of the results follow, including determination of the acoustoelastic coefficients in incompressible solids and the limiting values of the coefficients of nonlinearity for elastic wave propagation.     

The texture problem in Mössbauer spectroscopy

The theory of preferred orientation determination (crystallographic- or spintexture) by Mössbauer effect is presented for the case of unpolarized and polarized source radiation. The maximum information one can get out from Mössbauer measurements for the case of dipole radiation (M1) are nine expansion coefficients of the texture distribution expanded in a series of spherical harmonics. These coefficients can be obtained by a fitting procedure of the experimentally determined line intensity ratios in an absorption spectrum with rotated source and absorber. The general theory is applied for M1 radiation (Fe⁵⁷) and the cases of pure Zeeman or quadrupole splitting of source and absorber. Extension to higher multipole radiation can be easily done. Some graphs of possible textures are shown.     

Coefficients of Clebsch-Gordan for the holomorphic discrete series

We prove a theorem for the tensor product of representations of the holomorphic discrete series analogous to the classical theorem of Clebsch-Gordan and give an asymptotic formula for corresponding coefficients of Clebsch-Gordan. For small groups we compute the coefficients explicitly.Partially supported by NSF Grant MCS 78-01826.     

Direction-dependent elastic grain-interaction models – a comparative study

Mechanical and diffraction (X-ray) elastic constants (diffraction (X-ray) stress factors for macroscopically elastically anisotropic specimens) can be calculated for polycrystalline specimens from single-crystal elastic data by employing elastic grain-interaction models. Traditionally, only so-called isotropic grain-interaction models are considered: all directions in the polycrystal are taken equivalent with respect to the grain interaction. Only recently, so-called direction-dependent, i.e. anisotropic grain-interaction models, have been proposed. These models can express the effects of the reduced dimensionality of thin films, of the surface anisotropy of bulk polycrystals and of a grain-shape (morphological) texture on the elastic properties of polycrystals. In this work, the available, recently proposed direction-dependent grain-interaction models will be compared, in particular on the basis of numerical calculations of diffraction and mechanical elastic constants, of variances of certain orientation-dependent stress and strain tensor components and of the distributions of strains in the Euler (orientation) space. It will be demonstrated that the so-called Vook–Witt and inverse Vook–Witt models become (but only approximate) equivalent to the Eshelby–Kröner model for certain grain-shape textures.     

Attenuation of ultrasonic waves in rolled metals

Scattering of ultrasonic waves in polycrystals with texture is studied in this article. The attenuations of the three wave modes are determined as a function of dimensionless frequency and propagation direction, respectively, for given orientation distribution coefficients (ODCs). The calculation is done in the case of a statistically orthorhombic sample made up of cubic crystallites. The wave propagation and scattering model is formulated by the Dyson equation using an anisotropic Green's function approach. Within the limits of the first-order smoothing approximation, the Dyson equation is solved in the spatial Fourier transform domain. The results presented are shown to be directional dependent, frequency dependent, and especially dependent on the texture coefficients (ODCs) for the quasilongitudinal and two quasishear waves. The theoretical results presented may be used to improve the understanding of the microstructure during recrystallization processes.     

用于YBCO高温超导涂层的立方织构铜基带制备

Cu是制备双轴织构YBCO高温超导涂层的优良基带材料之一,立方织构的铜带可以提供良好的外延生长环境。文中采用轧制辅助双轴织构基带技术(RABiTS),研究了立方织构铜基带的制备工艺。将铜锭在初轧变形量达到88.4%后,进行400℃退火30min,再经二次轧制总变形量达到99.07%后,在氩气环境125—850℃之间不同温度下退火30min。用March-Dollase函数计算该系列样品的r值,其中850℃退火的样品具有最强的200择优取向,r值为0.25,用不完整极图对该样品的织构情况做进一步分析,用ODF函数定量计算了该样品立方织构的体积百分含量。该样品形成了较强的{001}<100>立方织构,体积百分含量为70.8%。     

Elastic deformation of polycrystals

We propose a framework to model elastic properties of polycrystals by coupling crystal orientational degrees of freedom with elastic strains. Our model encodes crystal symmetries and takes into account explicitly the strain compatibility induced long-range interaction between grains. The coupling of crystal orientation and elastic interactions allows for the rotation of individual grains by an external load. We apply the model to simulate uniaxial tensile loading of a 2D polycrystal within linear elasticity and a system with elastic anharmonicities that describe structural phase transformations. We investigate the constitutive response of the polycrystal and compare it to that of single crystals with crystallographic orientations that form the polycrystal.     

The group with Grassmann structureUOSP(1.2)

The finite-dimensional representations of the Lie superalgebraosp(1.2) and the group with Grassmann structureOSP(1.2) have been studied. The explicit expression of the projection operator of the superalgebraosp(1.2) has been found. The operator permits an arbitrary finite-dimensional representation to be expanded in the components multiple to the irreducible ones. The Clebsch-Gordan coefficients for the tensor product of two arbitrary irreducible representations have been obtained. The matrix elements of the irreducible representations of the groupUOSP(1.2) [the analoque of the compact form of the groupOSP(1.2)] are studied. The explicit form of these matrix elements, the differential equations satisfied by them, and the integral of their product have been found.     

Pressure Dependences of Elastic Constants of AMg6 Aluminum–Magnesium Alloy and n-AMg6/С60 Nanocomposite Alloy

The ultrasonic study results for dependence of the elastic wave velocities and second-order elasticity coefficients of the polycrystalline aluminum alloy AMg6 and its nanocomposite n-AMg6/C₆₀ on hydrostatic pressure up to 1.6 GPa have been described. The ultrasonic research has been carried out using a highpressure ultrasonic piezometer based on the piston-cylinder device. The pressure derivatives of the secondorder elastic constants of these materials established in the present study have been compared with the results of the third-order elastic constants measurements of the test alloys using the Thurston–Brugger method. Involving available literature data, we determined the relationships between the pressure derivatives of the second-order elastic constants of the AMg6 alloy and the Mg-content and nanostructuring.

     

Modelling magnetic polarisation J50 by different methods

Two different methods for modelling the angular behaviour of magnetic polarisation at 5000 A/m (J₅₀) of electrical steels were evaluated and compared. Both methods are based upon crystallographic texture data. The texture of non-oriented electrical steels with silicon content ranging from 0.11 to 3%Si was determined by X-ray diffraction. In the first method, J₅₀ was correlated to the calculated value of the average anisotropy energy in each direction, using texture data. In the second method, the first three coefficients of the spherical harmonic series of the ODF and two experimental points were used to estimate the angular variation of J₅₀. The first method allows the estimation of J₅₀ for samples with different textures and Si contents using only the texture data, with no need of magnetic measurement, and this is advantageous, because texture data can be acquired with less than 2 g of material. The second method may give better adjust in some situations but besides the texture data, it requests magnetic measurements in at least two directions, for example, rolling and transverse directions.     

Punktdefekte im kubischen elastischen Kontinuum

The determination of the displacement and strain fields of a point defect in a cubic crystal requires even in the framework of continuum elasticity theory numerical calculation. These fields of elastic dipoles are expanded in suitable vector and tensor fields. The coefficients of this expansion are calculated up to polynomials of 5th and 4th order in the direction cosines using the ratios of elastic constants as parameters. With this expansion the interaction of elastic dipoles in a cubic medium can be calculated. The results have been applied to the interaction of F-centres and of O₂ ^?-centres in alkali halides.     

On Sakharov's theory of gravitation

The Sakharov theory of gravitation is examined from the viewpoint of the analogy between gravitation and elasticity. It is found that, by using the Cattaneo-Zel'manov projection technique, the deformation tensor connected with the gravitational field can be considered the deformation tensor of a suitable elastic medium. By supposing that transversal waves propagate in this medium with velocityc, one can find an explicit expression for the time dependence of the gravitational constant. Some applications of cosmological interest are briefly discussed.     

Crystal algebra

We define the crystal algebra, an algebra which has a base of elements of crystal bases of a quantum group. The multiplication is defined by the tensor product rule of crystal bases. A universal n-colored crystal algebra is defined. We study the relation between those algebras and the tensor algebras of the crystal algebra of U _q (sl(2)) and give a presentation by generators and relations for the case of U _q (sl(n)).     

Stress functions and internal stresses in linear three-dimensional anisotropic elasticity

A stress function method is presented in order to give a general solution of the incompatibility problem of three-dimensional linear anisotropic elasticity theory. A relation between the internal stress tensor in terms of derivatives of a sixth-order stress function tensor will be derived. Using this formulation the so far open problem of the representation of the second-order stress function tensor by the fourth-order stress function tensor in anisotropic elasticity theory is solved in general.     

Die Abweichungen von den Cauchy-Relationen

Qualitative rules for the deviations from the Cauchy relations are derived from experimental data. These departures are the components of a second rank tensor. The coefficients of “lateral interaction”c _iijj dominate in most cases over the correspondent shear resistancesc _ijij. Extreme effects of this type are generated by asymmetric lattice particles. Covalent bonds and other strong bonds with preferential orientation as well as strong overlap cause opposite effects. The departures from the Cauchy relations and the atomistic binding properties are very closely correlated.