Experimental Pole Figures Research Articles

Evolution of texture and intergranular stresses of αZr and minority phases in Zr-2.5Nb pressure tube through synchrotron X-ray diffraction

This study investigates the impact of different processing steps on the crystallographic texture evolution of αZr and minority phases βZr and ωZr, and intergranular stresses as a function of grain orientation in a Zr2.5Nb pressure tube. This was done by analysis of diffraction data recorded in the 1-ID line of the Advanced Photon Source for high energy X-Rays (∼80 Kev) in transmission geometry on three samples with different processing conditions: extruded, cold rolled and after autoclaving. The crystallographic texture of both αZr and βZr phases changes slightly after cold rolling, with the Burgers orientation relationship between the two phases remaining intact. The texture of βZr and ωZr shows that βZr decomposition during thermal treatment is sensitive to grain orientation. A new method was proposed to obtain the orientation dependence of the intergranular stresses from the experimental pole figures. Intergranular stresses in αZr grains were low after extrusion, but rolling significantly increased them, with values with magnitude as higher as 350 MPa and a strong dependence on grain orientation. The autoclaving treatment leads to important stress relaxation, but the dependence on grain orientation remains. The βZr phase grains exhibit significant intergranular strains in both extruded and cold-rolled conditions; the hydrostatic part was associated to the presence of Nb in solid solution, with content of ∼20 % and an orientation dependence of approximately ±1 at%. The deviatoric strain was attributed to intergranular stresses, with stresses below 80 MPa for the extruded sample and ranging from −600 to 400 MPa after cold rolling.

EFFECT OF COLD ROLLING PROCESS ON TEXTURE EVOLUTION IN AUSTENITIC STEEL SHEETS

The investigation presents the evolution of texture and microstructure of cold-rolled austenitic AISI 301 stainless steel sheets. The austenite steel was deformed by cold-rolling to 30%, 50% and 80% of rolling reduction. It was found, that during cold plastic deformation, phase transformation of austenite into martensite takes place. The transformation progress was confirmed by the microscopic investigation, as well as by X-ray diffraction methods. Texture analysis was performed on the basis of the orientation distribution functions (ODFs) calculated from the experimental pole figures. The texture of the deformed austenite is dominated by fiber orientations α =110║KN, mainly Goss orientation {110}001 and alloy type orientation {110}112. On the other hand, plastic deformation-induced martensitic transformation of the type (γ') significantly affects the development of martensite texture ' in the cold rolling process. After rolling with a 30% of deformation degree in the texture of  martensite, the strongest orientation is {001}110, while after maximum deformation (about 80%), the {111}112 and {112}110 type orientations dominate. It was found that in AISI 301 steel structure after deformation range from 30% to 80% proportion of α’ martensite phase changes from about 26.7% to about 67.8%.

Restoration of Orientation Distribution Function Using Texture Components with Radial Normal Distributions

The methods for restoration of the orientation distribution function (ODF) from experimental pole figures have been compared for materials possessing a low symmetry of specimen (by example of the median section of the hot-pressed band from Mg–4.5% Nd magnesium alloy), namely, the method of texture components using radial normal distributions on SO(3) with different spreading and the method of superposition of a large number of normal distributions with equivalent small spreading. Both approaches have demonstrated similar ODFs. In this case, the former method, which is less sensitive to measurement errors of pole figures, is based on nonlinear optimization with a complex choice of initial approximations of the parameters of the model. The latter approach is more sensitive and easier to use.

On accounting for preferred crystallite orientations in determination of average elastic strain by diffraction

Standard diffraction-based measurements of elastic strains in polycrystalline materials rely on shifts of Bragg peaks. Measurement results are usually given in the form of a single tensor assumed to represent the average stress in the material, but the question about the true relationship between the tensor and the average stress generally goes without notice. This paper describes a novel procedure for analysis of data obtained from such measurements. It is applicable in cases when spatial correlations in the material are ignored and statistical information about the polycrystalline specimen is limited to texture-related intensity pole figures and strain pole figures. A tensor closest to auxiliary strain tensors linked to the results of measurements in particular specimen directions is considered to represent the strain state. This tensor is shown to be a good approximation of the average strain tensor. A closed-form expression allowing for its direct computation from experimental pole figures is given. The performance of the procedure is illustrated using simulated data.

Quantitative Analysis of the Crystallographic Orientation Relationship Between the Martensite and Austenite in Quenching–Partitioning–Tempering Steels

The orientation relationships (ORs) between the martensite and the retained austenite in low- and medium-carbon steels after quenching–partitioning–tempering process were studied in this work. The ORs in the studied steels are identified by selected-area electron diffraction (SAED) as either K–S or N–W ORs. Meanwhile, the ORs were also studied based on numerical fitting of electron backscatter diffraction data method suggested by Miyamoto. The simulated K–S and N–W ORs in the low-index directions generally do not well coincide with the experimental pole figure, which may be attributed to both the orientation spread from the ideal variant orientations and high symmetry of the low-index directions. However, the simulated results coincide well with experimental pole figures in the high-index directions {123}bcc. A modified method with simplicity based on Miyamoto’s work was proposed. The results indicate that the ORs determined by modified method are similar to those determined by Miyamoto’ method, that is, the OR is near K–S OR for the low-carbon Q–P–T steel, and with the increase of carbon content, the OR is closer to N–W OR in medium-carbon Q–P–T steel.

ECCI, EBSD and EPSC characterization of rhombohedral twinning in polycrystalline α-alumina deformed in a D-DIA apparatus

Rhombohedral twinning in alumina (aluminium oxide, α-Al2O3) is an important mechanism for plastic deformation under high-temperature–pressure conditions. Rhombohedral twins in a polycrystalline alumina sample deformed in a D-DIA apparatus at 965 K and 4.48 GPa have been characterized. Three classes of grains were imaged, containing single, double and mosaic twins, using electron channeling contrast imaging (ECCI) in a field emission scanning electron microscope. These twinned grains were analyzed using electron backscatter diffraction (EBSD). The methodology for twin identification presented here is based on comparison of theoretical pole figures for a rhombohedral twin with experimental pole figures obtained with EBSD crystal orientation mapping. An 85°〈02{\overline 2}1〉 angle–axis pair of misorientation was identified for rhombohedral twin boundaries in alumina, which can be readily used in EBSD post-processing software to identify the twin boundaries in EBSD maps and distinguish the rhombohedral twins from basal twins. Elastic plastic self-consistent (EPSC) modeling was then used to model the synchrotron X-ray diffraction data from the D-DIA experiments utilizing the rhombohedral twinning law. From these EPSC models, a critical resolved shear stress of 0.25 GPa was obtained for rhombohedral twinning under the above experimental conditions, which is internally consistent with the value estimated from the applied load and Schmid factors determined by EBSD analysis.

A methodology to determine the elastic moduli of crystals by matching experimental and simulated lattice strain pole figures using discrete harmonics

Determining reliable single crystal material parameters for complex polycrystalline materials is a significant challenge for the materials community. In this work, a novel methodology for determining those parameters is outlined and successfully applied to the titanium alloy, Ti-6Al-4V. Utilizing the results from a lattice strain pole figure experiment conducted at the Cornell High Energy Synchrotron Source, an iterative approach is used to optimize the single crystal elastic moduli by comparing experimental and simulated lattice strain pole figures at discrete load steps during a uniaxial tensile test. Due to the large number of unique measurements taken during the experiments, comparisons were made by using the discrete spherical harmonic modes of both the experimental and simulated lattice strain pole figures, allowing the complete pole figures to be used to determine the single crystal elastic moduli.

Comparison of two methods for the ODF modeling for magnesium alloy with low symmetry

In this work we compare two methods for the reconstruction of orientation distribution function (ODF) from experimental pole figures. The magnesium alloy sample Mg-4.5%Nd is used for the investigation. The component fit method with the central normal distributions on SO(3) and the orientation grid method demonstrate similar reconstructed ODF. The component fit method is more robust to experimental errors, however it relies on nonlinear optimization with a set of initial approximations provided by a complicated selection problem. The orientation grid method is more sensitive, but easier to implement.

Uncertainty quantification of microstructural properties due to variability in measured pole figures

Experimental pole figures are an important input for microstructure homogenization models. In this paper, we derive an exact analytical formulation to quantify the uncertainties in homogenized properties due to uncertainty in the experimentally measured pole figures. The pole figures are acquired from a set of Ti-7Al alloy samples. These samples were obtained from the same process: by compressing a beta forged ingot at room temperature followed by annealing. The samples were taken from different regions of the original ingot, and this created variability in the resulting pole figures. The joint multivariate probability distributions of the computed orientation distribution function (ODF) is then found using the method of characteristic functions based on a Gaussian model of the pole figures. Engineering properties such as elastic modulus can be obtained by volume averaging over the ODF. We also show that uncertainty in elastic properties can be analytically obtained using direct transformation of random variables in the homogenization approach.

Optimization of the procedure for determining integral texture parameters of products from zirconium-based alloys using the orientation distribution function

Integral texture Kearns parameters, or f-parameters, are the one of the characteristics that is controlled during the tube production of Zr-based alloys now. These parameters define the anisotropy of physical and mechanical properties of the products. f-parameters are the sum of effective fractions of basal normals aligned in a selected sample direction. However, so far there is no standardized procedure for their calculation. The widespread availability of X-ray diffractometers, the high statistical significance of the results and the software development for texture analysis causes the optimality of the f-parameters determination by complete pole figures (0001) calculated through orientation distribution function (ODF). Accuracy analysis of the ODF extraction by several incomplete pole figures using LaboTex software was carried out in this paper. The comparison of the calculated data was performed with experimental complete pole figures (0001) obtained by «sewing» method. Various texture types, different number and combinations of incomplete pole figures were considered. For this purpose, 6 incomplete pole figures - - were registered for each of three mutually perpendicular surfaces of the CANDU-tube sample. By comparison of experimental and calculated by ODF complete pole figures as well as f-parameters, procedures of their determination were optimized for products from Zr-based alloys.

Orientation relationship between austenite and non-modulated martensite in Ni–Mn–Ga single crystals

In-situ electron back scatter diffraction experiments were used to accurately determine the orientation relationships between austenite and martensite in non-modulated Ni–Mn–Ga single crystals. Using a heating stage device direct orientation measurements of austenite and martensite in single-, two- and self-accommodated state were performed. The initial single crystals show a typical self-accommodated microstructure composed of a mixture of variants allowing the distribution of martensitic variants with minimal macroscopic shear. The martensitic variants are observed on different length scales: starting from a few nanometers up to a few micrometers. The corresponding experimental pole figures reveal a strongly asymmetric distribution of martensitic variants with respect to the parent cubic orientation. Combining the microstructural and crystallographic orientation information it is obvious that a hierarchy in twin formation exists. At first the main martensitic variants arise by Bain strain and rigid body rotation. Subsequently, the remaining variants follow a specific crystallographic orientation relationship.Using training procedure a single- or two-variant state in martensite can be obtained. In-situ heating experiments reveal that such configurations show a different orientation relationship between the martensitic variants and austenite compared to that in a self-accommodated state. It strongly suggests that the orientation relationship between these two phases is not strictly fixed, as given by the main models describing this relation, but also depends on the number and width of martensitic variants. The results are discussed with respect to lattice parameters, number of variants and existing models in the literature.

Quantifying texture evolution during hot rolling of AZ31 Twin Roll Cast strip

Multi-pass rolling experiments with an AZ31 Twin Roll Cast (TRC) alloy were performed on an industrial scaled four-high rolling mill. Within the rolling with an intermediate annealing the evolution of texture was investigated. To quantify the extent of preferred crystallographic orientation experimental X-ray pole figures were measured after different process steps and analyzed using the free and open Matlab® toolbox MTEX for texture analysis. The development of the fiber texture was observed and analyzed in dependence on rolling conditions. In the initial state the specimen exhibits a texture composed of a weak basal texture and a cast texture with {0001}-planes oriented across the rolling direction. During the following rolling process a fiber texture was developed. The expected strength increment of the fiber texture was quantitatively confirmed in terms of volume portions of the orientation density function around the fiber and in terms of the canonical parameters of fitted pseudo Bingham distributions. On the results of this work a model for prediction of the texture evolution during the strip rolling of magnesium in the examined parameter range was developed.

Structure, texture, and mechanical properties of an MA2-1hp magnesium alloy after two-stage equal-channel angular pressing and intermediate annealing

The effect of two-stage equal-channel angular pressing (ECAP) on the microstructure, the texture, and the mechanical properties of an MA2-1hp magnesium alloy is analyzed. ECAP leads to the formation of a submicrocrystalline structure with an average grain size of 640 nm, which includes Mg17Al12 phase particles with an average grain size of 240 nm and a volume fracture of 5.5%. A scattered tilted basal texture forms after ECAP, and its experimental pole figures are used for calculating orientation distribution functions and determining the volume fractions of the main orientations and the Schmid factors for different deformation systems. An increased activation of basal slip is found after both the first and the second stages of ECAP. As a result of two-stage ECAP, the strength properties of the alloy that correspond to the minimum acceptable values achieved by direct compression are obtained. Ductility is 44 and 18% after the first stage of ECAP plus subsequent annealing and after the second stage, respectively, which is almost four and two times higher than the initial value. The resulting strength mechanical properties of the alloy after the first and the second ECAP stages are analyzed using the Hall-Petch relation.

Quantifying texture evolution during hot rolling of magnesium Twin Roll Cast strip

Multi-pass rolling experiments with an AZ31 Twin Roll Cast (TRC) alloy were performed on an industrial scaled four-high rolling mill. During rolling with an intermediate annealing the evolution of texture and microstructure was investigated. To quantify the extent of preferred crystallographic orientation experimental X-ray pole figures were measured after different process steps and analyzed using the free and open Matlab® toolbox mtex for texture analysis. The evolution of both the fiber texture and the microstructure was observed and analyzed in dependence on rolling conditions. In the initial state the specimen exhibits a texture composed of a weak basal texture and a cast texture with {0001}-planes oriented across the rolling direction. During the following rolling process a fiber texture evolved. The expected strength increment of the fiber texture was quantitatively confirmed in terms of volume portions of the orientation density function around the fiber and in terms of the canonical parameters of fitted pseudo Bingham distributions. Based on these results a model to predict the texture evolution during the strip rolling of magnesium in the examined parameter range was developed.

Effects of thermal modification on wood ultrastructure analyzed with crystallographic texture

Abstract The changes in ultrastructure of thermally modified beech and poplar wood have been studied by means of the crystallographic texture analysis. The experimental pole figures have been registered based on the X-ray diffraction (XRD). The 3D texture function, i.e., the orientation distribution function (ODF) was calculated with the use of the procedure implemented for materials with the monoclinic lattice symmetry. The inverse pole figures were recalculated from the obtained ODF. Areas with specific crystallographic organization in wood were identified and gradual changes in intensities of moderately oriented areas were observed. Simultaneously, additional reorganization of crystalline cellulose was observed as a result of heat treatment. The integrated skeleton lines confirmed the gradual decomposition of hemicelluloses as well as the temporary increase of cellulose self-organization.

Analysis of Deformation Texture in AISI 304 Steel Sheets

The textures of cold-rolled AISI 304 austenitic steel were the object of the investigations. The austenite steel was deformed by cold-rolling to 20, 40 and 70% reduction. A significant amount of martensite, formed due to the strain induced (γ®a’) transformation, was detected in the deformed structure by applying magnetic and X-ray diffraction methods. Texture analysis was performed on the basis of the orientation distribution functions (ODFs) calculated from the experimental pole figures. The texture measurements of both phases were conducted from the center layers of the cold-rolled strip. In the case of metastable austenite AISI 304 steel the texture development was very complex because three processes were proceded simultaneously during the cold-rolling, namely: plastic deformation of the austenitic g-phase, strain induced phase transformation γ®a’ and deformation of the formed a’-martensite. These processes resulted in the presence of two phases in the structure of the steel with a definite crystallographic relationship and orientation changes of both phases with increasing of the deformation. Thus, the resultant deformation texture of the investigated steels is described by the austenite and martensite texture components. The rolling texture of γ-phase describes mainly orientations from the fiber α =<110>║ND and the major components of the martensite deformation texture are orientations from the fibers α1=<110>║RD and γ ={111}║ND.

A two-scale methodology for determining the residual stresses in polycrystalline solids using high energy X-ray diffraction data

This paper presents a new method for ascertaining residual stress fields in engineering components. Diffraction data are employed with a finite element discretization to determine the macroscopic (continuum) residual stress field over the workpiece simultaneously with the crystal-scale distribution of elastic strains at each diffraction measurement point. Stress equilibrium and traction free boundary conditions constrain the solution at the continuum scale. The thousands of lattice strain measurements made at each diffraction volume ensure that the stress solution is consistent with crystal-scale elastic distortions. Finally, integrated over all orientations within each aggregate, the crystal-scale stresses must match the continuum stress. The method was demonstrated using a shrink fit specimen with a nickel-base superalloy disk fit over a stainless steel shaft. High energy synchrotron X-ray diffraction experiments conducted at the Advanced Photon Source, Argonne National Laboratory, provided nearly 1 million lattice strain measurements. Using these data, the new methodology produced a stress field that satisfied the macroscopic constraints and matched the crystal-scale distortions at each diffraction volume as manifest by spherical harmonic expansions of the lattice strain tensor over orientation space. The residual stress distribution exhibits the general features of an axisymmetric analytical approximation, but also contains details that likely arise from the pre-existing material state and fabrication of the shrink fit assembly. Projections of the spherical harmonic expansions of the crystal-scale lattice strain tensor matched the experimental lattice strain pole figures. Issues related to diffraction volume spacing (spatial resolution) and specimen shadowing are also discussed.

The effect of deformation mechanisms on the high temperature plasticity of Zircaloy-4

In situ compression and tension tests were performed over a range of temperatures on annealed Zircaloy-4 plate in the rolling and transverse directions (RD and TD) at strain rates ranging from 10−4s−1 to 10−1s−1. Post deformation microstructures were indicative of twinning at higher strain rates (10−1s−1 and 10−2s−1) with little or no twinning occurring at slower rates. Analysis of post deformation pole figures showed no texture evolution during tensile tests and basal poles remained orientated ±30° between the normal direction (ND) and the transverse direction (TD), as in the as received material. In compression, no evolution was seen in specimens tested in the RD. Those tested in the TD showed a variation in post deformation pole figures as strain rate increased. At slow strain rates a slight rotation of basal poles around the ND was noted, indicative of slip. Increasing the strain rate decreased this effect, and at strain rates of 10−1s−1 a total reorientation of the basal poles to lie solely in the RD was seen, which is indicative of twinning. Evolution of Kearns’ factors with strain agree with these results. An elasto-plastic self-consistant model was used to analyse the experimental results. The model results indicated that increasing strain rate caused an increase in twinning density, as suggested by the experimental microstructures and pole figures.

Quantitative neutron diffraction texture measurement applied to α-phase alumina and Ti3AlC2

Orientation distribution functions, essential for making a quantitative connection between single-crystal and polycrystal properties, have been determined for extruded α-phase alumina, hot-pressed Ti3AlC2and cold isostatically pressed Ti3AlC2using experimental pole figures recorded on the fixed-wavelength neutron diffractometer KOWARI. Some practical improvements to the calculation of the pole-figure density from the raw area-detector data, and for constructing pole figures on ann×n° hemispherical grid, are presented. The textures give some insight into particle flow during manufacture. Directly measured material textures were compared with one-dimensional pole density functions, such as the March and Rietveld functions commonly used for the correction of preferred orientation in Rietveld refinements, as a means of assessing the utility of the latter for the computation of diffraction elastic constants and other polycrystal properties from a given set of single-crystal properties.

The mechanisms of the fcc–bcc martensitic transformation revealed by pole figures

Face-centered cubic (fcc) to body-centered cubic (bcc) martensitic transformations occur in many materials, such as steels, FeNi meteorites or brass. The phenomenological theory has been the accepted theory for these transformations for more than half a century. However, it cannot explain the continuous singular features in the experimental electron backscatter diffraction or X-ray diffraction pole figures. Here we show that such patterns can be simulated by one discrete orientation relationship and two continuous rotations that correspond to a trace of the transformation mechanisms. A new theory of martensite transformation that is in full agreement with the experimental pole figures is proposed. In this theory, the fcc–bcc transformation results from a fcc–hexagonal close-packed (hcp) step followed by an hcp–bcc step. The advantages of this two-step theory over the phenomenological theory are discussed.