Stress determination through diffraction: establishing the link between Kröner and Voigt/Reuss limits

2015 ◽  
Vol 30 (2) ◽  
pp. 99-103 ◽  
Author(s):  
Conal E. Murray ◽  
Jean L. Jordan-Sweet ◽  
Stephen W. Bedell ◽  
E. Todd Ryan
Keyword(s):  
Elastic Constants ◽  
Mechanical Response ◽  
Elastic Anisotropy ◽  
Polycrystalline Materials ◽  
X Rays ◽  
Least Squares Regression ◽  
Multiple Reflections ◽  
Cubic Symmetry ◽  
Grain Interaction ◽  
Eshelby Inclusion

The quantification of stress in polycrystalline materials by diffraction-based methods relies on the proper choice of grain interaction model that links the observed strain to the elastic stress state in the aggregate. X-ray elastic constants (XEC) relate the strain as measured using X-rays to the state of stress in a quasi-isotropic ensemble of grains. However, the corresponding interaction models (e.g., Voigt and Reuss limits) often possess unlikely assumptions as to mechanical response of the individual grains. The Kröner limit, which employs a self-consistent scheme based on the Eshelby inclusion method, is based on a more physical representation of isotropic grain interaction. For polycrystalline aggregates composed of crystals with cubic symmetry, Kröner limit XEC are equal to those calculated from a linear combination of Reuss and Voigt XEC, where the weighting fraction, xKr, is solely a function of the single-crystal elastic constants and scales with the material's elastic anisotropy. This weighting fraction can also be experimentally determined using a linear, least-squares regression of diffraction data from multiple reflections. Data on metallic thin films reveals that this optimal experimental weighting fraction, x*, can vary significantly from xKr, as well as that of the Neerfeld limit (x = 0.5).

Different Grain Interaction Models Used for Interpretation of Lattice Strain Data Collected Using Grazing Incidence X-Ray Diffraction

2013 ◽  
Vol 768-769 ◽  
pp. 26-30
Author(s):  
Marianna Marciszko ◽  
Andrzej Baczmański ◽  
Mirosław Wróbel ◽  
Wilfrid Seiler ◽  
Chedly Braham ◽  
...  
Keyword(s):  
Elastic Constants ◽  
Elastic Anisotropy ◽  
Grazing Incidence ◽  
Polycrystalline Materials ◽  
X Ray Diffraction ◽  
Surface Layers ◽  
X Ray ◽  
Grain Interaction ◽  
Ni Alloy ◽  
Strain Data

Multireflection grazing incidence X-ray diffraction (MGIXD) was applied to measure residual stresses in thin surface layers and the problem of X-ray elastic constants (XEC) used for the interpretation of results was studied. To show the influence of the X-ray elastic constants on the interpretation of MGIDX results, polycrystalline materials having low (Ti alloy) and high elastic anisotropy of crystallites (Ni alloy) were investigated.

The dislocations contrast factors of cubic crystals in the Zener constant range between zero and unity

2002 ◽  
Vol 17 (2) ◽  
pp. 104-111 ◽  
Author(s):  
I. C. Dragomir ◽  
T. Ungár
Keyword(s):  
Elastic Constants ◽  
Elastic Anisotropy ◽  
Profile Analysis ◽  
Anisotropy Constant ◽  
Diffraction Peak ◽  
Polycrystalline Materials ◽  
Ionic Crystals ◽  
Strain Anisotropy ◽  
Cubic Crystals ◽  
Lattice Distortions

Diffraction peak profiles broaden due to the smallness of crystallites and the presence of lattice defects. Strain broadening of powders of polycrystalline materials is often anisotropic in terms of the hkl indices. This kind of strain anisotropy has been shown to be well interpreted assuming dislocations as one of the major sources of lattice distortions. The knowledge of the dislocation contrast factors are inevitable for this interoperation. In a previous work the theoretical contrast factors were evaluated for cubic crystals for elastic constants in the Zener constant range 0.5≤Az≤8. A large number of ionic crystals and many refractory metals have elastic anisotropy, Az, well below 0.5. In the present work the contrast factors for this lower anisotropy-constant range are investigated. The calculations and the corresponding peak profile analysis are tested on ball milled PbS and Nb and nanocrystalline CeO2.

Implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for any crystal symmetry

2019 ◽  
Vol 34 (2) ◽  
pp. 103-109
Author(s):  
Arnold C. Vermeulen ◽  
Christopher M. Kube ◽  
Nicholas Norberg
Keyword(s):  
Elastic Constants ◽  
Ultrasonic Waves ◽  
The Self ◽  
Crystal Symmetry ◽  
Intermediate Step ◽  
X Rays ◽  
X Ray ◽  
Eshelby Inclusion ◽  
Eshelby Model ◽  
Self Consistent

In this paper, we will report about the implementation of the self-consistent Kröner–Eshelby model for the calculation of X-ray elastic constants for general, triclinic crystal symmetry. With applying appropriate symmetry relations, the point groups of higher crystal symmetries are covered as well. This simplifies the implementation effort to cover the calculations for any crystal symmetry. In the literature, several models can be found to estimate the polycrystalline elastic properties from single crystal elastic constants. In general, this is an intermediate step toward the calculation of the polycrystalline response to different techniques using X-rays, neutrons, or ultrasonic waves. In the case of X-ray residual stress analysis, the final goal is the calculation of X-ray Elastic constants. Contrary to the models of Reuss, Voigt, and Hill, the Kröner–Eshelby model has the benefit that, because of the implementation of the Eshelby inclusion model, it can be expanded to cover more complicated systems that exhibit multiple phases, inclusions or pores and that these can be optionally combined with a polycrystalline matrix that is anisotropic, i.e., contains texture. We will discuss a recent theoretical development where the approaches of calculating bounds of Reuss and Voigt, the tighter bounds of Hashin–Shtrikman and Dederichs–Zeller are brought together in one unifying model that converges to the self-consistent solution of Kröner–Eshelby. For the implementation of the Kröner–Eshelby model the well-known Voigt notation is adopted. The 4-rank tensor operations have been rewritten into 2-rank matrix operations. The practical difficulties of the Voigt notation, as usually concealed in the scientific literature, will be discussed. Last, we will show a practical X-ray example in which the various models are applied and compared.

Standard Deviations in X-Ray Stress and Elastic Constants Due to Counting Statistics

1988 ◽  
Vol 32 ◽  
pp. 377-388 ◽  
Author(s):  
Masanori Kurita
Keyword(s):  
Elastic Constants ◽  
Stress Measurement ◽  
Steel Specimen ◽  
Polycrystalline Materials ◽  
X Rays ◽  
Confidence Limits ◽  
X Ray Diffraction ◽  
X Ray ◽  
Standard Deviations ◽  
Counting Statistics

AbstractX-ray diffraction can be used to nondestructively measure residual stress of polycrystalline materials. In x-ray stress measurement, it is important to determine a stress constant experimentally in order to measure the stress accurately. However, every value measured by x-ray diffraction has statistical errors arising from counting statistics. The equations for calculating the standard deviations of the stress constant and elastic constants measured by x-rays are derived analytically in order to ascertain the reproducibility of the measured values. These standard deviations represent the size of the variability caused by counting statistics, and can be calculated from a single set of measurements by using these equations. These equations can apply Lu any meuhud for x-ray stress ifiesuremenL. The variances of the x-ray stress and elastic constants are expressed in terms of the linear combinations of the variances of the peak position. The confidence limits of these constants of a quenched and tempered steel specimen were determined by the Gaussian curve method. The 95% confidence limits of the stress constant were -314 ± 25 MFa/deg.

scholarly journals Determining elastic anisotropy of textured polycrystals using resonant ultrasound spectroscopy

2021 ◽  
Vol 56 (16) ◽  
pp. 10053-10073
Author(s):  
Jordan A. Evans ◽  
Blake T. Sturtevant ◽  
Bjørn Clausen ◽  
Sven C. Vogel ◽  
Fedor F. Balakirev ◽  
...  
Keyword(s):  
Elastic Constants ◽  
Elastic Anisotropy ◽  
Transversely Isotropic ◽  
Ultrasonic Pulse ◽  
Polycrystalline Materials ◽  
Induced Anisotropy ◽  
Resonant Ultrasound Spectroscopy ◽  
Aluminum Single Crystal ◽  
Resonant Ultrasound ◽  
Pulse Echo

AbstractPolycrystalline materials can have complex anisotropic properties depending on their crystallographic texture and crystal structure. In this study, we use resonant ultrasound spectroscopy (RUS) to nondestructively quantify the elastic anisotropy in extruded aluminum alloy 1100-O, an inherently low-anisotropy material. Further, we show that RUS can be used to indirectly provide a description of the material’s texture, which in the present case is found to be transversely isotropic. By determining the entire elastic tensor, we can identify the level and orientation of the anisotropy originated during extrusion. The relative anisotropy of the compressive (c11/c33) and shear (c44/c66) elastic constants is 1.5% ± 0.5% and 5.7% ± 0.5%, respectively, where the elastic constants (five independent elastic constants for transversely isotropic) are those associated with the extrusion axis that defines the symmetry of the texture. These results indicate that the texture is expected to have transversely isotropic symmetry. This finding is confirmed by two additional approaches. First, we confirm elastic constants and the degree of elastic anisotropy by direct sound velocity measurements using ultrasonic pulse echo. Second, neutron diffraction (ND) data confirm the symmetry of the bulk texture consistent with extrusion-induced anisotropy, and polycrystal elasticity simulations using the elastic self-consistent model with input from ND textures and aluminum single-crystal elastic constants render similar levels of polycrystal elastic anisotropy to those measured by RUS. We demonstrate the ability of RUS to detect texture-induced anisotropy in inherently low-anisotropy materials. Therefore, as many other common materials have intrinsically higher elastic anisotropy, this technique should be applicable for similar levels of texture, providing an efficient general diagnostic and characterization tool.

Structural, Electronic, and Mechanical Properties of CoN and NiN: An Ab Initio Study

2017 ◽  
Vol 72 (4) ◽  
pp. 321-330 ◽  
Author(s):  
A. Amudhavalli ◽  
M. Manikandan ◽  
A. Jemmy Cinthia ◽  
R. Rajeswarapalanichamy ◽  
K. Iyakutti
Keyword(s):  
Elastic Constants ◽  
Electronic Structures ◽  
Elastic Anisotropy ◽  
Structural Phase ◽  
Lattice Parameter ◽  
Stable Phase ◽  
Ab Initio Study ◽  
Zinc Blende ◽  
Experimental Values ◽  
Parameter Values

AbstractThe structural stabilities of cobalt mononitride (CoN) and nickel mono-nitride (NiN) were investigated among the crystal structures, namely, NaCl (B1), CsCl (B2), and zinc blende (B3). It was found that the zinc blende (B3) phase was the most stable phase for both nitrides. A pressure-induced structural phase transition from B3 to B1 phase was predicted in these nitrides. The computed lattice parameter values were in agreement with the experimental values and other theoretical values. The electronic structures reveal that these nitrides are metallic at zero pressure. The computed elastic constants indicate that CoN and NiN are mechanically stable in the B1 and B3 phases. The variations of the elastic constants, bulk modulus, shear modulus, Poisson’s ratio, and elastic anisotropy factor with pressure were investigated. The Debye temperature θD values are reported for both the nitrides in their B1 and B3 phases. The high-pressure NaCl phase of both CoN and NiN were found to be ferromagnetic.

scholarly journals Estimation of 8th, 10th and 12th Order ODF Coefficients From Elastic Properties in Cold Rolled Steel Sheets by Adjustment of Single Crystal Elastic Constants

1990 ◽  
Vol 12 (1-3) ◽  
pp. 175-185 ◽  
Author(s):  
Kei Sakata ◽  
Dominique Daniel ◽  
John J. Jonas
Keyword(s):  
Single Crystal ◽  
Elastic Constants ◽  
Elastic Energy ◽  
X Rays ◽  
Rolled Steel ◽  
Steel Sheets ◽  
Fitting Procedure ◽  
Pole Figures ◽  
Cold Rolled ◽  
Fiber Components

In an earlier paper (Sakata et al., 1989), it was shown that the 4th and 6th order ODF coefficients could be successfully derived from Young's modulus measurements using the elastic energy method. However, the values of some of the coefficients fell beyond the expected error ranges. In this study, more appropriate single crystal elastic constants are selected by means of a fitting procedure. Then the ODF coefficients are again estimated in the manner described previously. As a result, the values of the C411, C611, C612 and C614 coeffioents, which were somewhat inaccurate in the previous calculation, are improved considerably. The volume fractions of the principal preferred orientations are then employed to predict the 8th order coefficients and the fiber components of the l = 10 and l = 12 (C1011, C1211 and C1221) coefficients. With the aid of the coefficients obtained in this way, both pole and inverse pole figures are drawn, which are in better agreement with those based on X-rays than when only the 4th order coefficients are employed.

A software for diffraction stress factor calculations for textured materials

2012 ◽  
Vol 27 (2) ◽  
pp. 114-116 ◽  
Author(s):  
Thomas Gnäupel-Herold
Keyword(s):  
Distribution Function ◽  
Pole Figure ◽  
Elastic Constants ◽  
Orientation Distribution Function ◽  
Lattice Strain ◽  
Orientation Distribution ◽  
Crystallite Orientation ◽  
Interaction Models ◽  
Grain Interaction ◽  
Textured Materials

A software for the calculation of diffraction elastic constants (DEC) for materials both with and without preferred orientation was developed. All grain-interaction models that can use the crystallite orientation distribution function (ODF) are incorporated, including Kröner, Hill, inverse Kröner, and Reuss. The functions of the software include: reading the ODF in common textual formats, pole figure calculation, calculation of DEC for different (hkl,φ,ψ), calculation of anisotropic bulk constants from the ODF, calculation of macro-stress from lattice strain and vice versa, as well as mixture ratios of (hkl) of overlapped reflections in textured materials.

scholarly journals The Elastic Constants of the Single Crystal of the Mg-Zn-Zr-REM Alloy from the Data of the Elastic Anisotropy and the Texture of the Polycrystalline Sheet

2014 ◽  
Vol 2014 ◽  
pp. 1-6
Author(s):  
S. V. San’kova ◽  
N. M. Shkatulyak ◽  
V. V. Usov ◽  
N. A. Volchok
Keyword(s):  
Single Crystals ◽  
Young’S Modulus ◽  
Elastic Constants ◽  
Elastic Anisotropy ◽  
Young's Modulus ◽  
Rare Earth Metals ◽  
Alloy Sheet ◽  
Pole Figures ◽  
Integral Characteristics ◽  
Experimental Values

The measuring of the constants of single-crystals requires the availability of crystals of relatively big size. In this paper the elastic constants of the single crystals of magnesium alloy with zinc, zirconium, and rare earth metals (REM) were determined by means of the experimental anisotropy of Young’s modulus and integral characteristics of texture (ICT), which were found from pole figures. Using these constants the anisotropy of Young’s modulus of alloy sheet ZE10 was calculated. Deviation of calculated values from experimental values did not exceed 2%.

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