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.     

Extension of the Vook–Witt and inverse Vook–Witt elastic grain-interaction models to general loading states

The recently developed Vook–Witt and inverse Vook–Witt elastic grain-interaction models have been employed for the calculation of mechanical elastic constants and diffraction (X-ray) stress factors of, in particular, thin films. However, their applicability is limited to a planar, rotationally symmetric state of macroscopic, mechanical stress. For such a loading state (and an, at least, transversely, elastically isotropic specimen), only two mechanical elastic constants are necessary to describe mechanical elastic behaviour and only the sum of two diffraction (X-ray) stress factors is needed to relate lattice strains to the one independent component of the mechanical stress tensor. The restriction to a planar, rotationally symmetric state of mechanical stress will be removed in this work. Calculation of the full stiffness tensor and all six diffraction (X-ray) stress factors then becomes possible. It was found previously that the Vook–Witt and inverse Vook–Witt models become (but only approximately) equivalent to the Eshelby–Kröner model for certain ideal grain-shape textures. For this reason, results of numerical calculations of mechanical elastic constants and diffraction (X-ray) stress factors, based on the Vook–Witt and inverse Vook–Witt models, will be presented and compared to corresponding results obtained from the Eshelby--Kröner grain-interaction model considering ideal grain-shape (‘morphological’) textures.     

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.     

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.     

The determination of stresses in thin films; modelling elastic grain interaction

X-ray diffraction is frequently employed for the analysis of mechanical stresses in polycrystalline specimens. To this end, suitable so-called diffraction elastic constants are needed for determining the components of the mechanical stress tensor from measured lattice strains. These diffraction elastic constants depend on the single-crystal elastic constants of the material considered and the so-called grain interaction, describing the distribution of stresses and strains over the crystallographically differently oriented crystallites composing the specimen. Well-known grain interaction models, as due to Voigt, to Reuss, to Neerfeld and Hill and to Eshelby and Kröner, may be applied to bulk specimens, but they are generally not suitable for thin films. In this paper, an average 'effective' grain interaction model is proposed that consists of a linear combination of basic extreme models including new models specially suited to thin films. Experimental verification has been achieved by X-ray diffraction strain measurements performed on a sputter-deposited copper film. This is the first time that anisotropic grain interaction has been analysed quantitatively.     

Deformation textures produced in diamond anvil experiments, analysed in radial diffraction geometry

Diamond anvil cells may not only impose pressure upon a sample but also a compressive stress that produces elastic and plastic deformation of polycrystalline samples. The plastic deformation may result in texture development if the material deforms by slip or mechanical twinning, or if grains have a non-equiaxed shape. In radial diffraction geometry, texture is revealed by variation of intensity along Debye rings relative to the compression direction. Diffraction images (obtained by CCD or image plate) can be used to extract quantitative texture information. Currently the most elegant and powerful method is a modified Rietveld technique as implemented in the software package MAUD. From texture data one can evaluate the homogeneity of strain in a diamond anvil cell, the strain magnitude and deformation mechanisms, the latter by comparing observed texture patterns with results from polycrystal plasticity simulations. Some examples such as olivine, magnesiowuestite, MgSiO(3) perovskite and ε-iron are discussed.     

Effect of elastic strain energy on grain growth and texture in AZ31 magnesium alloy by phase-field simulation

A phase-field model is modified to investigate the grain growth and texture evolution in AZ31 magnesium alloy during stressing at elevated temperatures. The order parameters are defined to represent a physical variable of grain orientation in terms of three angles in spatial coordinates so that the grain volume of different order parameters can be used to indicate the texture of the alloy. The stiffness tensors for different grains are different because of elastic anisotropy of the magnesium lattice. The tensor is defined by transforming the standard stiffness tensor according to the angle between the (0001) plane of a grain and the direction of applied stress. Therefore, different grains contribute to different amounts of work under applied stress. The simulation results are well-explained by using the limited experimental data available, and the texture results are in good agreement with the experimental observations. The simulation results reveal that the applied stress strongly influences AZ31 alloy grain growth and that the grain-growth rate increases with the applied stress increasing, particularly when the stress is less than 400 MPa. A parameter (△d) is introduced to characterize the degree of grain-size variation due to abnormal grain growth; the △d increases with applied stress increasing and becomes considerably large only when the stress is greater than 800 MPa. Moreover, the applied stress also results in an intensive texture of the 〈0001〉 axis parallel to the direction of compressive stress in AZ31 alloy after growing at elevated temperatures, only when the applied stress is greater than 500 MPa.     

Evolution of structural and magnetic properties of sputtered nanocrystalline Co thin films with thermal annealing

Ultrafine grain films of cobalt prepared using ion-beam sputtering have been studied using X-ray diffraction (XRD), X-ray reflectivity (XRR), atomic force microscopy (AFM) and magneto-optical Kerr effect (MOKE) measurements. As-prepared films have very smooth surface owing to the ultrafine nature of the grains. Evolution of the structure and morphology of the film with thermal annealing has been studied and the same is correlated with the magnetic properties. Above an annealing temperature of 300 °C, the film gradually transforms from HCP to FCC phase that remains stable at room temperature. A significant contribution of the surface energy, due to small grain size, results in stabilisation of the FCC phase at room temperature. It is found that other processes like stress relaxation, grain texturing and growth also exhibit an enhanced rate above 300 °C, and may be associated with an enhanced mobility of the atoms above this temperature. Films possess a uniaxial anisotropy, which exhibits a non-monotonous behaviour with thermal annealing. The observed variation in the anisotropy and coercivity with annealing can be understood in terms of variations in the internal stresses, surface roughness, and grain structure.     

Statistical theory of the microdeformation of polycrystals. II

Starting from an isotropic random distribution of the elastic moduli tensor components of a polycrystal caused by grain disorientation, an attempt is made to find the density distribution function of grain yield points in the direction of expansion at the first and second phases of microdeformation. This permits the solution of the problem of the change in the relative number of plastically deformed grains with increase in the external stress. The nature of the involvement of the grains in plastic deformation is analyzed in materials with various grains sizes.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii Fizika, No. 10, pp. 73–78, October, 1971.     

Microstresses in textured polycrystals studied by the multireflection diffraction method and self-consistent model

A new version of the X-ray diffraction method for determining macrostresses and microstresses in textured polycrystalline material is presented. In this method the lattice strains for various orientation of the scattering vector as well as for various crystallographic planes {hkl} are measured. The interpretation of the experimental data is based on the least-squares fitting procedure, in which the diffraction elastic constants and theoretical values of microstresses are used. The diffraction elastic constants and the microstresses are calculated by the self-consistent model. The new method was successfully applied for stress determination in one- and two-phase steels subjected to elastoplastic deformation, and the significant anisotropy of the incompatibility stresses was observed in textured samples.     

Ion-beam induced changes in magnetic and microstructural properties of thin iron films

Changes in magnetic and structural properties of 60–82 nm iron films induced by heavy-ion implantation were studied using the magneto-optical Kerr effect, M?ssbauer spectroscopy, Rutherford backscattering spectroscopy, X-ray diffraction, and X-ray absorption fine structure. The influence of ion-beam parameters (ion mass, fluence) and of sample parameters (external magnetic field and stress during implantation) were investigated. The Fe films, some of them containing a thin ⁵⁷Fe marker layer for M?ssbauer spectroscopy, were deposited on Si(100) substrates, by electron-beam and effusion-cell evaporation. The films were irradiated with ²⁰Ne, ⁵⁶Fe, ⁸⁶Kr and ¹³²Xe ions at energies chosen so that the implantation profiles peaked near the middle of the Fe films. The as-deposited films were magnetically isotropic and had a high coercivity. After ion implantation, the coercivity decreased and magnetic anisotropy developed. Both changes correlated with a decrease in the internal film stress. External mechanical stress applied during the irradiation had hardly any influence on the magnetic texture, opposite to an external magnetic field applied during or before ion implantation. The results are compared with those obtained for ion-irradiated polycrystalline Ni films and epitaxial Fe films and discussed with respect to the role of radiation-induced extended defects as pinning centers.     

X-ray line profiles analysis of plastically deformed metals

The theoretical basis of X-ray line profile analysis and its application to microstructural characterization of plastically deformed metallic alloys is presented. The microstructure is described in terms of coherent domain size, planar fault density, dislocation density and a dislocation arrangement parameter. Two evaluation methods are introduced: the momentum method and the extended Convolutional Multiple Whole Profile fit procedure. Their use is exemplified on plastically deformed single crystals, single grains residing in the bulk of a polycrystal and family of grains making up texture components. The selected examples show the potential of X-ray line profile analysis applied to diffraction patterns recorded with laboratory or synchrotron sources.     

Residual stress delaying phase transformation in Y-TZP bio-restorations

Engineering favorable residual stress for the complex geometry of bi-layer porcelain-zirconia crowns potentially prevents crack initiation and improves the mechanical performance and lifetime of the dental restoration. In addition to external load, the stress field depends on initial residual stress before loading. Residual stress is the result of factors such as the thermal expansion mismatch of layers and compliance anisotropy of zirconia grains in the process of sintering and cooling. Stress induced phase transformation in zirconia extensively relaxes the residual stress and changes the stress state. The objective of this study is to investigate the coupling between tetragonal to monoclinic phase transformations and residual stress. Residual stress, on the surface of the sectioned single load to failure crown, at 23 points starting from the pure tetragonal and ending at a fully monoclinic region were measured using the micro X-ray diffraction sin² ψ method. An important observation is the significant range in measured residual stress from a compressive stress of ?400?MPa up to tensile stress of 400?MPa and up to 100% tetragonal to monoclinic phase transformation.     

The viscoplastic behaviour of ice in polar ice sheets: experimental results and modelling

The slow motion of polar ice sheets is governed by the viscous deformation of anisotropic ices. Physical mechanisms controlling the deformation of ice crystal and polycrystal are reviewed. For the low stress conditions prevailing in ice sheets, the stress exponent of the flow law is lower than 2 and the deformation is dominated by the glide of dislocations on the basal plane. The mismatch of slip at grain boundaries induces large strain inhomogeneities partially relieved in ice sheets by grain growth and recrystallisation. The hard X-ray diffraction technique can be used to describe the orientation gradients within grains. The structure of ice along deep ice cores in Antarctica and Greenland exhibits significant changes in the shape, size and orientation of grains. A large variation of ice viscosity with depth is therefore expected. Polycrystal deformation models accounting for the changing rheological properties of polar ice are discussed. These models must predict and take into account the intracrystalline field heterogeneity. To cite this article: M. Montagnat, P. Duval, C. R. Physique 5 (2004).     

纳米复合Fe-R-O(R=Hf Nd Dy)薄膜面内铁磁共振研究

用反应溅射方法制备了FeRO(R=Hf, Nd, Dy) 薄膜，并在400℃时对样品进行退火处理，x射线衍射和电子衍射结果显示纳米量级的Fe晶粒镶嵌在非晶氧化物基质中.用面内铁磁共振技术仔细测量了样品的共振吸收谱，并分析了局域磁化强度Ms与晶粒尺寸的关系.制备态样品呈现出显著的面内单轴各向异性，退火后单轴各向异性显著减弱，取而代之的是较弱的磁晶各向异性.利用公式(ω/γ)2=(Hres+HK)(Hres+HK+4πMs)求出局域磁化强度Ms，它随晶粒尺寸减小而减小，在晶粒尺寸为5nm时仅约为Fe体材料饱和磁化强度的30%.局域磁化强度与根据Fe的体积百分比算出的体磁化强度相比偏小，并与晶粒尺寸的倒数呈线性关系，说明在晶粒表面存在较强的磁矩钉扎效应. 关键词： 铁磁共振 局域磁化强度 单轴各向异性 磁晶各向异性     

Anisotropic rheology during grain boundary diffusion creep and its relation to grain rotation,grain boundary sliding and superplasticity

The response of periodic microstructures to deformation can be analysed rigorously and this provides guidance in understanding more complex microstructures. When deforming by diffusion creep accompanied by sliding, irregular hexagons are shown to be anisotropic in their rheology. Analytic solutions are derived in which grain rotation is a key aspect of the deformation. If grain boundaries cannot support shear stress, the polycrystal viscosity is extremely anisotropic. There are two orthogonal directions of zero strength: sliding and rotation cooperate to allow strain parallel to these directions to be accomplished without any dissolution or plating. When a linear velocity/shear stress relationship is introduced for grain boundaries, the anisotropy is less extreme, but two weak directions still exist along which polycrystal strength is controlled only by the grain boundary “viscosity”. Irregular hexagons are characterised by four parameters. A particular subset of hexagons defined by two parameters, which includes regular hexagons as well as some elongate shapes, shows singular behaviour. Grain shapes that are close to that of the subset may exhibit large grain rotation rates and have no well-defined rheology unless there is a finite grain boundary viscosity. This new analysis explains why microstructures based on irregular but near equiaxed grains show high rotation rates during diffusion creep and it provides a framework for understanding strength anisotropy during diffusion creep.     

Calculation of the stability and mechanical and phonon properties of NbRuB,TaRuB, and NbOsB compounds

The structural, mechanical, and phonon properties of NbRuB, TaRuB, and NbOsB compounds with orthorhombic space groups Pmma (No. 51) and Pbam (No. 55) are investigated by using first-principles calculations. The elastic constants and moduli, Debye temperature, Poisson’s ration, Pugh’s ratio, elastic anisotropy factors, and minimum thermal conductivities have been predicted to understand the mechanical behavior of these ternary compounds. The mechanical anisotropy is discussed via several anisotropy indices and two-dimensional (2D) surface constructions. We observe that all compounds ought to be classified as ductile materials and exhibit elastically anisotropy. Their mechanical and dynamical stability is confirmed via the calculated elastic constants and phonon dispersion curves, respectively. This work should provide a useful guide for designing ternary boride materials that have excellent thermoelectric performance.     

Calculation of anisotropic properties of dental enamel from synchrotron data

Obtaining information about the intrinsic structure of polycrystalline materials is of prime importance owing to the anisotropic behaviour of individual crystallites. Grain orientation and its statistical distribution, i.e. the texture, have an important influence on the material properties. Crystallographic orientations play an important role in all kinds of polycrystalline materials such as metallic, geological and biological. Using synchrotron diffraction techniques the texture can be measured with high local and angular resolving power. Here methods are presented which allow the spatial orientation of the crystallites to be determined and information about the anisotropy of mechanical properties, such as elastic modulus or thermal expansion, to obtained. The methods are adapted to all crystal and several sample symmetries as well as to different phases, for example with overlapping diffraction peaks. To demonstrate the abilities of the methods, human dental enamel has been chosen, showing even overlapping diffraction peaks. Likewise it is of special interest to learn more about the orientation and anisotropic properties of dental enamel, since only basic information is available up to now. The texture of enamel has been found to be a tilted fibre texture of high strength (up to 12.5×). The calculated elastic modulus is up to 155 GPa and the thermal expansion up to 22.3 × 10^?6 °C^?1.     

Lattice constants and anisotropic microstrain at low temperature in 242Pu–Ga alloys

We attempted to characterize by neutron powder diffraction the monoclinic α′ phase that is known to form at low temperatures in dilute Pu–Ga alloys. This attempt was unsuccessful, as we did not detect any transformation to the α′ phase, but instead observed a line-broadening effect in the fcc?δ?phase. This effect is large enough to be visible in the raw diffraction data and is highly anisotropic in crystal space. The onset temperature of the line broadening (150?K) coincides with previous observations of the δ–α′ transformation. Bulk α′ was not observed. We believe that the development of α′ nuclei creates a spatially inhomogeneous stress distribution in the?δ?matrix, which in turn exhibits an anisotropic response, governed by its elastic anisotropy. We have analysed this observation of anisotropic microstrains in terms of the fictive microstresses required to produce them by elastic deformation. During the course of this work, we found a pseudo-isotope effect in the room temperature lattice constants of Pu–Ga alloys. The alloys made from nominal ²⁴²Pu isotope show systematically higher lattice constants than the corresponding ²³⁹Pu alloys, and the size of the effect is proportional to the Ga concentration. We believe that this effect is associated with the higher levels of radiation damage from isotopic impurities in the ²⁴²Pu alloys.