Calculation Of Orientation Distribution Function Research Articles

Restoration of the orientation distribution function for materials with low lattice and sample symmetry using the harmonic method

A lot of the properties polycrystalline materials depend on their crystallographic texture. The most complete information about the texture can be obtained from the orientation distribution function (ODF). We present the results of recovering ODF using series expansion technique for materials with low crystal and sample symmetry. The technique of ODF restoration is based on its Fourier series expansion with symmetrical spherical harmonic functions. Real spherical harmonics which are linear combinations of general spherical harmonics were used. The model single-component texture as well as the real texture of magnesium alloy sample subjected to equal-channel angular pressing have been studied. Textures are characterized by hexagonal crystal symmetry and triclinic sample symmetry. In both cases RP-factors and ODF calculation errors that were used as reliability criteria of ODF reconstruction showed good agreement between the calculated and experimental data. It was also revealed that the ODF of a magnesium alloy sample subjected to equal-channel angular pressing contains two texture components (1216)[1211] and (1216)[1211] with maximum intensity values of 13.81 and 2.23, respectively. The results obtained can be used for texture studies of ceramics, rocks and other non-metallic materials characterized by a lower symmetry.

X-ray Diffraction for the Determination of Residual Stress of Crystalline Material: An Overview

Though there are a variety of experimental techniques available for residual stress measurements, diffraction-based measurements have the unique advantage of estimating the individual components of the residual strain matrix in a crystalline material. This is then converted to residual stresses with appropriate continuum elasticity model(s) and X-ray elastic constants. In particular, measurements based on electron or neutron diffractions have their complexities or availability issues. The laboratory X-ray diffraction, on the other hand, may provide an easy resource and an effective tool. Such measurements range from two tilt methods to more extended d-sin2ψ measurements and multiple {hkil} grazing incident X-ray diffraction. Measurements can even be conducted on single crystals with micro-Laue diffraction and extended to stress ODF (orientation distribution function) calculations. These techniques are unquestionably extremely specialized, where measurement uncertainty plays an important role in the effectiveness plus reproducibility of the data. Unfortunately, standard textbooks or review articles typically describe some, but not all, of the techniques. In this overview, different techniques of X-ray diffraction for the determination of residual stresses in crystalline material have been summarized. It is hoped that potential users may benefit from the deliberations.

Kriging‐based orthotropic closure for flow‐induced fiber orientation and the part stiffness predictions with experimental investigation

This study employs a new Kriging‐based orthotropic closure (KRI) and several current closures to solve the second‐order orientation tensor equation of motion, and the differences between closures and orientation distribution function calculation (DFC) are firstly compared by a new developed error metric. The mechanical properties of an injection molded short fiber reinforced composite directly depend on the orientation of fibers, the Halpin–Tsai and Tandon–Weng mechanical models are considered to evaluate the accuracy of the closures on stiffness predictions, quantitative comparisons between the numerical simulations and the experimental result are carried out for 30% weight fractions of short carbon fiber/poly ether ether ketone (PEEK) matrix composite. It is observed that the result of sequential KRI closure is in good agreement with the experimental data. The KRI closure is shown to be far more accurate than Hybrid closure and also performs better than orthotropic and natural closures. POLYM. COMPOS., 40:3844–3856, 2019. © 2019 Society of Plastics Engineers

The evolution of texture in an equal channel pressed aluminum AA1050

The aim of this work was to analyze the evolution of texture of an aluminium AA1050 processed by ECAP and further recrystallization in search of optimum processing conditions to improve formability. The initial material was processed by two different routes: the first set of samples came from a hot rolled coil with a mixture of cube, copper and brass textures and the second from roll casting, exhibiting a strong brass texture. The ECAP deformation followed routes A (no sample rotation between passes), Ay (no sample rotation between passes but keeping the normal to rolling direction parallel to the transverse extrusion axis) and Bc (with a 90o sample rotation around the extrusion axis between passes). Texture evaluation was performed by x-ray analysis and results were analyzed by regular texture and orientation distribution function calculations. The normal and planar anisotropy were measured for the deformed and annealed samples. ECAP orientations were generated mostly by rotation around the transverse axis. 〈111〉//ND; orientations can be generated from S, Goss or copper textures but not from brass or cube textures. Therefore route A and starting with the hot rolled or rolled textures should be considered if 〈111〉//ND orientations are desired. Using route Bc, rotating the existing orientations into a favorable position can generate the desired fiber. The strength of a texture after ECAP depends on the orientations of the initial texture, however the correlation starts vanishing after various ECAP steps and the texture finally smoothes out. When 〈111〉//ND was more intense there was an improvement in the planar anisotropy.

The Evolution of Texture and Deformation Anisotropy at an Equal Channel Extruded Aluminum 1050 Alloy

The aim of this work was to analyze the evolution of texture and the deformation anisotropy of a AA1050 after ECAP and further recrystallization. The initial material was processed by two different routes: the first set of samples came from a hot rolled band with a mixture of cube, copper and brass textures and the second from roll casting and presented a strong brass texture. The ECAP deformation followed routes A, Ay (no rotation between passes but keeping the normal to rolling direction parallel to the transverse extrusion axis) and Bc. Texture evaluation was performed by x-ray analysis. Results were analyzed by regular texture and orientation distribution function calculations. The deformation anisotropy was measured by tension tests with specimens oriented 0, 45 and 90o to the extrusion direction.

Measurement of uncertainty in orientation distribution function calculations

A technique to quantify the uncertainty in an estimated orientation distribution function (ODF) by propagating the uncertainty from measured pole figures is demonstrated in this paper using Monte Carlo techniques and the Matlab toolbox for quantitative texture analysis. Unlike the uncertainty in the measured pole figures, where small values of intensity correlate to high uncertainty, uncertainty as a function of orientation in an ODF is not directly related to intensity.

Analysis of Preferred Orientations in Linear Friction Welded (LFW) Aluminium Alloy Specimens using “One-shot” Multi-element Energy Dispersive Synchrotron X-ray Diffraction

Linear Friction Welding (LFW) has tremendous potential for joining components from similar and dissimilar materials, avoiding material melting and introducing minimal distortion and only moderate levels of residual stress. However, the significant amount of attendant shear introduces preferred crystal orientations that have not yet been well studied. The “one-shot” approach to the interpretation of multi-element energy-dispersive X-ray powder diffraction data allows preferred orientation analysis without any sample preparation (cutting or polishing) or sample rotation. The key step for texture analysis by X-ray powder diffraction is the derivation of the orientation distribution function (ODF) from experimental data. Matlab toolbox “MTEX” provides a powerful function “calcODF” based on the harmonics method for this purpose. In the study reported in this paper, energy dispersive X-ray diffraction patterns were collected using the “horseshoe” multi-element energy-dispersive Ge detector installed on the JEEP beamline at Diamond Light Source. A single exposure was used for each gauge volume of interest, and a line was scanned across an Aluminum 2024 alloy LFW sample. The patterns were converted into raw pole figures through single peak fitting and equal area projection. The ODF calculation was performed based on these pole figures using Matlab toolbox “MTEX”. As a result, full pole figures obtained after ODF calculation were obtained. These are presented and discussed. The results show that the thermal-mechanical processes that occur during the LFW process lead to significant modification of the orientation distribution, but cause only moderate changes in the texture index.

Neutron diffraction study of the crystallographic preferential orientation of metagabro mylonite

Neutron diffraction (ND) was used to investigate the crystallographic preferential orientation (CPO, texture) and structure parameters of four samples of metagabro mylonite collected from the eastern part of the metagabbro sheet at the Stare Mesto belt, Bohemia Massif, Czech Republic. The samples were selected to form a deformation-ordered series with the microstructure varying from non-deformed metagabbro protolith to strongly sheared ultramylonite and amphibole-rich ulramylonite. The specimens used in ND measurements were polished in form of an exact sphere with diameter 50 ± 0.1 mm. The obtained ND patterns were corrected for non-linear background and then evaluated using the Rietveld method implemented in the software package GSAS. Data recorded from powder preparded by milling material from the sampled rocks were used to refine the structure parameters of plagioclase (labradorite structure, triclinic space group C-1) and amphibole (monoclinic space group C2/m). The experiments were performed on the KSN-2 neutron diffractometer situated at the research reactor LVR-15 of the Nuclear Research Institute, plc. Rez, Czech Republic. The data sets of ND patterns measured on each of the four spherical specimens consisted of 90 diagrams collected for different diffraction vectors covering uniformly one orientation hemisphere of the specimen. Based on the collected data, the orientation distribution function (ODF) of crystalline grains was determined by Rietveld harmonic method (coefficients C(l,m,n) of the spherical harmonics expansion determined up to the order L = 8) for the two principal mineral phases - amphibole and plagioclase. The ODF was used to reconstruct (001), (020), (021), (110), (111) (plagioclase) and (001), (11-1), (020), (110), (200) (amphibole) pole figures (PFs). Direct method of ODF calculation implemented in ResMat software package was then used to calculate inverted pole figures (IPFs) of the plagioclase phase. CPO of the amphibole and plagioclase is then discussed in terms of the obtained texture indices and calculated PFs and IPFs and compared with data measured by other methods or available in literature.

On-Line Determination of Texture in Deep Drawing Steel Sheet by Two-Dimensional X-Ray Diffraction

This paper described the application of a diffraction system based on X-ray area detector on pole figure measurement as well as corresponding computation of orientation distribution functions and the principle of rapid measurement texture. The impact of calculates the orientation distribution function on the conditions of the two-dimensional X-ray diffraction was analyzed; this was illustrated by an example of deep drawing steel sheets texture measurement. ̙̈́˰͇̱̓˰̶̴̿̾ͅ˰̸̱̈́̈́˰̷̱̹͂̿̈́̈́̾˰̸̵̈́˰͇̈́̿˽̴̵̹̹̱̼̽̾̓̿̾˰̴̵̵̳̈́̈́̿͂˰̈́̿˰̸̵̈́˰̵̱̹̱̀̀͂̿̀͂̈́˰̂θ position then fix it, reduce the sample rotation; the texture determination time can be significantly reduced. Reduce the Measuring range of angle χ˰̴̱̾ φ˰̴̵̿̓˰̾̿̈́˰̶̶̵̱̳̈́˰̸̵̈́ calculation of orientation distribution function, it also can significantly reduce the measurement of diffraction data. Several technical problems appeared on the on-line determination of texture based on an X-ray two-dimensional detector system and the possibility to improve the measurement speed and accuracy in the industrial production applications were then discussed.

The Evolution of Texture in AA 1050 Alloy Deformed by Equal-Channel Angular Pressing

The aim of the present work is to analyze the evolution of texture after ECAP and post deformation heat treatment of an AA1050 alloy produced by roll casting. The initial plate exhibited a weak cube texture at the surface, decreasing in its intensity toward the sheet center, where the deformation texture consisted of Brass and Copper components. ECAP-deformation employed one to four passes at room temperature, following route A in a = 120o die. Texture evaluation was performed by x-ray analysis and by EBSD, using transverse cross section scans. Only the central part of the plate was studied. Results were analyzed by regular texture and orientation distribution function calculations. After 1, 2 and 4 passes the texture changed to a strong {111}// TD and after recrystallization at 350oC for 1 h this main orientation was maintained.

The modified component method for calculation of orientation distribution function from pole figures

The main problem of the quantitative texture analysis is discussed. The sharp texture of nanocrystalline magnesium is under investigation. The well-known harmonic (Roe–Bunge) method of texture determination does not give good results for the materials with low sample symmetry. The robust component fit method for the determination of the orientation distribution function from the set of incomplete X-ray pole figures is used in this work. A unique and reliable solution is achieved.

Preferred crystallographic texture of α-chitin as a microscopic and macroscopic design principle of the exoskeleton of the lobster Homarus americanus

The crystallographic texture of the crystalline α-chitin matrix in the biological composite material forming the exoskeleton of the lobster Homarus americanus has been determined using synchrotron X-ray pole figure measurements and the calculation of orientation distribution functions. The study has two objectives. The first one is to elucidate crystallographic building principles via the preferred synthesis of certain orientations in crystalline organic tissue. The second one is to study whether a general global design principle exists for the exoskeleton which uses preferred textures relative to the local coordinate system throughout the lobster cuticle. The first point, hence, pursues the question of the extent to which and why α-chitin reveals preferred textures in the lobster cuticle. The second point addresses the question of why and whether such preferred textures (and the resulting anisotropy) exist everywhere in the exoskeleton. Concerning the first aspect, a strong preference of a fiber texture of the orthorhombic α-chitin is observed which is characterized by a 〈0 2 0〉 crystal axis normal to the exoskeleton surface for the chitin matrix. The second question is tackled by studying samples from different parts of the carapace. While the first aspect takes a microscopic perspective at the basic structure of the biological composite, the second point aims at building a bridge between an understanding of the microstructure and the macroscopic nature of a larger biological construction. We observe that the texture is everywhere in the carapace optimized in such a way that the same crystallographic axis of the chitin matrix is parallel to the normal to the local tangent plane of the carapace. Notable differences in the texture are observed between hard mineralized parts on the one hand and soft membranous parts on the other. The study shows that the complex hierarchical microstructure of the arthropod cuticle can be well described by surprisingly simple crystallographic textures.

ECAE of Al-4%Cu Alloys: Experimental Study Assisted by Polycrystalline-FEM Simulations

The present work reports on the results obtained on equal channel angular extrusion experiments (ECAE) done on a laboratory-cast Al-4%Cu alloy, in the T4 condition, and the use of Polycrystalline-FEM simulations to assist in the interpretation of the experiments. The experimental setup consists on a die of approximately 15 x 15 mm2 sections intersecting at 120o. Deformation at room temperature consisted of up to 5 passes with no rotation between passes. After each extrusion pass, the samples were cut from the deformed billet along planes parallel to the extrusion direction and the preferential orientations were measured on surface and middle layers. Three pole figures, (111), (200) and (220) were measured by conventional x-ray diffraction techniques and used for Orientation Distribution Function calculation and analysis. In addition tensile tests and optical microscopy have been performed in each sample to provide a good estimation of the parameters that enter in the modeling process. A finite element code specially developed to model large deformation processes (Forge3Ò) was used with tetrahedral elements and an elastic-viscoplastic material model to investigate the influence of the different strain paths sustained by different areas of the samples. The calculated distribution of deformations agrees well with the theoretical result. The simulation was used to assist in the selection of sample-cutting procedures for texture measurements and to provide the strain paths needed for self-consistent polycrystal modeling of texture development.

Some Practical Results for Calculating Elastic Properties of Textured Cubic Polycrystals

The aim of this work is to compare elastic properties of cubic materials with different texture types calculated with orientation distribution function (ODF) obtained by different methods: iterative series expansion method, Williams-Imhof-Matthies-Vinel (WIMV) and texture component method. We have used the approach of mean values for the calculation of elastic properties with different approximations (Reuss, Voigt, Hill). The calculations were carried out on a rolled Al-6% Mg alloy sheet and on an extruded Cu rod. The experimental pole figures were obtained by means of neutron diffraction and used for ODF calculation. The conformity between the different results is discussed.

Quantitative analysis of preferential orientation components of ferroelectric thin films

A quantitative study of the preferred orientation of lanthanum and calcium modified lead titanate thin films, including the measurement of pole figures and calculation of orientation distribution function (OD), is presented. The main preferential orientation components are identified as <001> and <100> perpendicular to the film surface. Variations of the texture strength and the contribution of these components to the final texture are observed with changes in the film thickness or the substrate used. These variations are correlated to the ferroelectric properties of these films. This information can be used to optimise the preparation process to obtain improved thin films for pyroelectric and piezoelectric applications.

Optimal Calculation of ODF With the Canonical Normal Distribution on the Rotation Group

Macroscopic physical properties of most polycrystalline materials are controlled by orientation distribution of their grains. The orientation distribution function (ODF) of a polycrystal is seldom if ever determined directly from an experiment. Usually experimental data are represented by a set of pole figures (PFs), these latter are some integral projections of the ODF. The main problem of quantitative texture analysis is to recover ODF from its corresponding PFs. With any set of PFs the solution of this problem is non-unique. That is why some assumptions about ODF structure are necessary. We consider ODF as superposition of the canonical normal distribution (CND) on the rotation group SO(3).

Texture Transformations in Thermomechanically Treated Steels Having Increased Nitrogen Content and Alloys

In this study the results of mathematical simulation and experimental research of texture transition in stress assisted shear phase transformations are discussed. The orientation distribution function (ODF) calculation model to the any type crystalline lattice symmetry materials has been suggested. By crystalline lattice symmetry analysis this computer program determines necessary, amount of the initial experimental data. The experimental verification supports this model well. The ODF calculation model after shear phase transformation with equal use of all possible variants of orientation relationship or with various degrees of variant selection worked out. In this study the experimental and calculating data of the direct and reversal shear transformations or α ↔ γ in Fe-alloys and B2 ↔ B19' in Ti-Ni alloys are discussed. The stress assisted shear phase transformation model and computer program to calculate ODF after transformation have been worked out.

Estimation of the Minimum Grain Number for the Orientation Distribution Function Calculation from Individual Orientation Measurements on Fe–3%Si and Ti–4Al–6V Alloys

The calculation of characteristics describing texture as well as relations between orientations and morphological features of microstructure are based on single orientation measurements. For such experimental data, it is essential to estimate the number of necessary measurements of single orientations for a statistically significant representation of the investigated quantity, which, in the present paper, is the orientation distribution function (ODF). In a previous article [Pospiech, Jura &amp; Gottstein (1993) Mater Sci. Forum, 157–162, 407–412], this number has been estimated by a criterion that is used here for a cubic and a hexagonal material. This approach is very useful since it allows one to estimate the minimum orientation number with or without referring to an ODF calculated from pole figures measured by X-ray or neutron diffraction.

Practical Considerations in the Calculation of Orientation Distribution Functions from Electron Back-Scattered Diffraction Patterns

Practical Considerations in the Calculation of Orientation Distribution Functions from Electron Back-Scattered Diffraction Patterns

Structural changes during polystyrene orientation: A study of optical birefringence and wide angle X‐ray scattering

AbstractResults of coupled optical birefringence and wide‐angle X‐ray scattering (WAXS) measurements on uniaxially oriented, atactic, glassy polystyrene (PS) samples are reported. Orientations were achieved in tensile test experiments performed slightly above the glass transition temperature. The elongated samples were quenched after orientation. The applied force was kept constant during cooling. The second order moments of the chain orientation distribution function (ODF) were calculated from the azimuthal profile of the scattered X‐ray intensity from measurements made on five PS samples with different degrees of orientation. For ODF calculations the first, equatorial halo was used. The influence of the second, meridional halo was removed by resolving the WAXS patterns into three peaks. The intrinsic azimuthal linewidth of the WAXS profile was estimated and taken into account in the ODF calculations. The magnitude of the intrinsic birefringence Δnint was determined using the linear relationship between the birefringence and the second order moment of the ODF. A value for Δnint of −0.05 was obtained. This result was used to calculate the angle between the normal plane of the benzene ring and the chain axis. A comparison with previously published results indicates that the average relative orientation of the benzene ring with respect to the chain axis was changing in our samples during stretching. This structural difference is attributed to strong deformation of the network and retarded relaxation during sample preparation.