Determination of the elastic constants by coherent neutron scattering in polycrystals

It is shown that elastic constants which usually can be determined in single crystals only may be measured in polycrystals by inelastic coherent neutron scattering. Measurements are reported for polycrystalline samples of aluminium, copper and stainless steel. The method is best suited for the determination of the elastic constants connected with transverse phonons.     

On the scatter ranges for the elastic moduli of random aggregates of general anisotropic crystals

A randomly inhomogeneous material may have macroscopic properties (elasticity, conductivity) scattered over some uncertainty intervals, despite the idealistic uniqueness assumption of homogenization theory. Based on minimum energy principles and certain statistical isotropy-symmetry hypotheses, our partly third-order bounds on the effective properties of random polycrystals are expected to estimate those scatter ranges. Explicit expressions are given and calculated for the elastic moduli of the random aggregates of some known monoclinic and triclinic crystals, which yield results in agreement with those calculated for higher-symmetry crystals: the moduli are determinable within an accuracy of two or three significant digits in most cases. It is shown, however, that with some real-world exotic crystals the bounds may fall far apart, and further theoretical and experimental studies on them deserve attention.     

New estimates for macroscopic elastic moduli of random polycrystalline aggregates

Elastic behaviour of random polycrystals, the shape and crystalline orientations of the constituent grains of which are uncorrelated, is considered. A geometrical restriction in earlier derived bounds on the aggregates’ elastic moduli is released leading to new estimates for the general random aggregates. The bounds are expected to predict the scatter ranges for the observed macroscopic moduli. Application to a number of cubic crystals’ aggregates indicates that their moduli are determinable within the accuracy of up to 2 or 3 significant digits in most cases.     

Mechanical elastic constants and diffraction stress factors of macroscopically elastically anisotropic polycrystals: the effect of grain-shape (morphological) texture

The effect of the grain-shape (‘morphological’) texture of a polycrystal on the mechanical elastic constants and diffraction (X-ray) stress factors is investigated. To this end, the Eshelby–Kröner grain interaction model originally devised for polycrystals consisting of spherical grains is extended to ellipsoidal grain morphology. Results obtained for the mechanical elastic constants show that a polycrystal consisting of ellipsoidal grains with their principal axes aligned along common directions (i.e. when an ideal grain-shape texture occurs) is macroscopically elastically anisotropic. Also the diffraction (X-ray) stress factors are affected by the grain-shape texture; they reflect the macroscopic elastic anisotropy by resulting in nonlinear so-called sin²?ψ plots. In general, a grain-shape texture can have a moderate effect on the mechanical elastic constants and a pronounced effect on the diffraction elastic constants, depending on the crystal symmetry and single-crystal elastic anisotropy.     

Elastic constants of bismuth polycrystal containing tin,lead, antimony and tellurium impurities

Investigations of ultrasonic velocity and internal friction are carried out in bismuth polycrystals containing Sn, Pb, Sb and Te impurities in the concentration range of 0–1 atomic percent using the composite oscillator technique, and elastic constants are estimated from velocity and density data. The variations of elastic constants are interpreted in terms of changes of lattice parameter, valency of the impurity, the electron-atom ratio, atomic size and Fermi energy. It is observed that the elastic constants are less affected in Bi-Te alloys compared with Bi-Sb, Bi-Pb and Bi-Sn alloys in the above concentration range.     

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.     

High-pressure structure and elastic properties of tantalum single crystal:First principles investigation

Since knowledge of the structure and elastic properties of Ta at high pressures is critical for addressing the recent controversies regarding the high-pressure stable phase and elastic properties, we perform a systematical study on the highpressure structure and elastic properties of the cubic Ta by using the first-principles method. Results show that the initial body-centered cubic phase of Ta remains stable even up to 500 GPa and the high-pressure elastic properties are excellently consistent with the available experimental results. Besides, the high-pressure sound velocities of the single- and polycrystals Ta are also calculated based on the elastic constants, and the predications exhibit good agreement with the existing experimental data.     

具有六方晶系结构的多晶体材料弹性常数——Y弹性常数

利用最近提出的新的物理参量——Ｙ弹性常数,将其应用于具有六方晶系结构的多晶体材料.推导了六方晶系结构的多晶体材料之Ｙ弹性常数,通过算例与具有六方晶系结构的多晶体材料之X射线弹性常数进行了比较.运用这个Ｙ弹性常数进一步推导出的多晶体材料整体之机械弹性常数的表达式与Kneer的研究结果中的表达式虽然形式不同,但针对具体材料所计算的结果却完全符合. 关键词： Y弹性常数 六方晶系 多晶体材料     

Constrained grain-boundary sliding and inelasticity of polycrystals

A model of inelastic behavior of polycrystals that is based on the idea of constrained grain-boundary sliding is developed. This model assumes that grain-boundary sliding is accommodated only through grain elastic deformation, which is valid under low stresses. By way of examples, the dynamic inelastic behavior and a decrease in the elastic moduli of polycrystals subjected to a small-amplitude ultrasonic field are investigated. It is shown that some of the available experimental data obtained on ultra-fine-grained materials can be explained in terms of the model. The theory predicts that the decrease in the elastic moduli is a size effect that is bound to be observed when the grain size is small. These predictions are supported experimentally.     

Elastic parameters of single-crystal and polycrystalline wurtzite-like oxides BeO and ZnO: Ab initio calculations

The structural and elastic parameters (elastic constants; bulk, shear, and Young’s moduli; Poisson’s ratios; and Lamé coefficients) for ideal wurtzite-like beryllium and zinc monoxides are calculated by the full-potential linearized augmented-plane-wave (FLAPW) method with the WIEN2k code. These parameters are approximated to those for polycrystalline oxides BeO and ZnO in the framework of the Voigt-Reuss-Hill model. The results of calculations are compared with the available experimental data and used to estimate numerically the velocities of sound and the Debye temperatures for BeO and ZnO polycrystals.     

Elastic property of multiphase composites with random microstructures

We propose a computational method with no ad hoc empirical parameters to determine the elastic properties of multiphase composites of complex geometries by numerically solving the stress–strain relationships in heterogeneous materials. First the random microstructure of the multiphase composites is reproduced in our model by the random generation-growth method. Then a high-efficiency lattice Boltzmann method is employed to solve the governing equation on the multiphase microstructures. After validated against a few standard solutions for simple geometries, the present method is used to predict the effective elastic properties of real multiphase composites. The comparisons between the predictions and the existing experimental data have shown that the effects of pores/voids in composites are not negligible despite their seemingly tiny amounts. Ignorance of such effects will lead to over-predictions of the effective elastic properties compared with the experimental measurements. When the pores are taken into account and treated as a separate phase, the predicted Young’s modulus, shear modulus and Poisson’s ratio agree well with the available experimental data. The present method provides an alternative tool for analysis, design and optimization of multiphase composite materials.     

Resonant ultrasound spectroscopy and homogeneity in polycrystals

Resonant ultrasound spectroscopy (RUS) is capable of determining the bulk elastic properties of a solid from its characteristic vibration frequencies, given the dimensions, density and shape of the sample. The model used for extracting values of the elastic constants assumes perfect homogeneity, which can be approximated by average-isotropic polycrystals. This approximation is excellent in the small grain regime assumed for most averaging procedures, but for real samples with indeterminate grain size distributions, it is not clear where the approximation breaks down. RUS measurements were made on pure copper samples where the grain size distribution was changed by progressive heat treatments in order to find a quantitative limit for the loss of homogeneity. It is found that when a measure of the largest grains is 15% of the sample’s smallest dimension, the deviation in RUS fits indicates elastic inhomogeneity.     

具有立方晶系结构的多晶体材料的弹性常数——Y弹性常数

提出了一个新的物理参量“Ｙ弹性常数”,并阐述了其物理含义.并将其应用于具有立方晶系结构的多晶体材料,推导了立方晶系结构的多晶体材料的Ｙ弹性常数,通过算例与具有立方晶系结构的多晶体材料的X射线弹性常数进行了比较.运用这个Ｙ弹性常数进一步推导出的多晶体材料整体的机械弹性常数的表达式与Krner的研究结果完全符合. 关键词： Y弹性常数 立方晶系 多晶体材料     

Effective thermal conduction in composite materials

The problem of determining the bounds and/or estimating the effective thermal conductivity (λ _eff) of a composite (multiphase) system given the volume fractions and the conductivities of the components has been investigated. A comparison between the measured data and the results predicted by theoretical models has been made for seven heterogeneous samples. The tested models include those of the effective medium theory (EMT), Hashin and Shtrikman (HS) bounds, and Wiener bounds. These models can be used to characterize macroscopic homogeneous and isotropic multiphase composite materials either by determining the bounds for the effective thermal conductivity and/or by estimating the overall conductivity of the random mixture. It turns out that the most suitable one of these models to estimate λ _eff is the EMT model. This model is a mathematical model based on the homogeneity condition which satisfies the existence of a statistically homogeneous medium that encloses inclusions of different phases. Numerical values of thermal conductivity for the samples that satisfy the homogeneity condition imposed by the effective medium theory are in best agreement with the experimentally measured ones.     

Elastic mechanical grain interactions in polycrystalline materials; analysis by diffraction-line broadening

Abstract

Experimental investigations have revealed that the Neerfeld–Hill and Eshelby–Kröner models, for grain interactions in massive, bulk (in particular, macroscopically isotropic) polycrystals, and a recently proposed effective grain-interaction model for macroscopically anisotropic polycrystals, as thin films, provide good estimates for the macroscopic (mechanical and) X-ray elastic constants and stress factors of such polycrystalline aggregates. These models can also be used to calculate the strain variation among the diffracting crystallites, i.e. the diffraction-line broadening induced by elastic grain interactions can thus be predicted. This work provides an assessment of diffraction-line broadening induced by elastic loading of polycrystalline specimens according to the various grain-interaction models. It is shown that the variety of environment, and thus the heterogeneity of the stress–strain states experienced by each of the individual grains exhibiting the same crystallographic orientation in a real polycrystal, cannot be accounted for by traditional grain-interaction models, where all grains of the same crystallographic orientation in the specimen frame of reference are considered to experience the same stress–strain state. A significant degree of broadening which is induced by the heterogeneity of the environments of the individual crystallites is calculated on the basis of a finite element algorithm. The obtained results have vast implication for diffraction-line broadening analysis and modelling of the elastic behaviour of massive polycrystals.     

Elastic moduli approximation of higher symmetry for the acoustical properties of an anisotropic material

The issue of how to define and determine an optimal acoustical fit to a set of anisotropic elastic constants is addressed. The optimal moduli are defined as those which minimize the mean-squared difference in the acoustical tensors between the given moduli and all possible moduli of a chosen higher material symmetry. The solution is shown to be identical to minimizing a Euclidean distance function, or equivalently, projecting the tensor of elastic stiffness onto the appropriate symmetry. This has implications for how to best select anisotropic constants to acoustically model complex materials.     

Elastic properties of nanostructured materials with layered grain boundary structure

Atomistic calculations of the elastic constants for a bulk nanostructured material that consists of a layered structure where alternating layers meet along high angle grain boundaries and where atoms interact via a Lennard-Jones potential are presented. The calculations of the elastic constants were performed in the frame of homogeneous deformations for a wide range of layer widths ranging from 2.24 up to 74.62 nm. The results showed that the relaxation of the atomic structure affects the elastic constants for the cases where more than 5% of atoms are located in the GB region. Also it was found that the way that external stresses are applied on the system affects the values of the obtained elastic properties, with the elastic constants related to the characteristic directions of the grain boundary being the most affected ones. The findings of this work are of interest for the fabrication methods of nanostructured materials, the measurement methods of their elastic properties as well as multiscale modeling schemes of nanostructured materials.     

Elastic dipole interaction of point defects in a sphere of elastically anisotropic cubic material

The elastic field of centres of dilatation at arbitrary positions in a spherical specimen with cubic elastic constants is studied. The influence of the (free) boundary, which is of central importance due to the long-range nature of the elastic interaction, is included. For materials with small elastic anisotropy expansions in terms of vector spherical harmonics are obtained for the displacements due to one defect, as well as for the interaction energy of two defects. Infinite material expressions and surface contributions are given separately. The general treatment is exemplified by presenting explicit results for the case of hydrogen atoms dissolved in vanadium.     

Investigating the temperature dependence of elastic constants by thermal fluctuation formula

The differences between the calculated values of elastic constants of materials and the experimental data are consistently restricting the application of thermal fluctuation formula to the mechanical properties of materials. In this work, the temperature dependence of elastic constants of many-body potentials is studied by thermal fluctuation formula. The differences between the calculated values and the experimental data are investigated in detail. Our studies show that the differences come from the thermal expansion of the materials: the calculated zero-stress states are bigger than the experimental zero-stress states of the materials, and this deviation makes Born terms of the thermal fluctuation formula decrease sharply as the temperature increases, while the fluctuation terms and the kinetic terms change little. As a result, the elastic constants, which are the sum of these three terms, decrease faster than the experimental data as the temperature increases. Our studies show that when the experimental zero-stress states are used as the reference states in constant volume and constant energy (NVE) simulations, the elastic constants calculated by thermal fluctuation formula are in good agreement with the experimental values.