Crystallite Distribution Function Research Articles

Physical Asymmetry of the Crystallographic Texture of an Interstitial-Free Sheet Steel

The texture of a mill-produced, Nb-bearing, interstitial-free, hot-rolled sheet steel was evaluated at a number of sheet thickness positions. The texture was also studied following laboratory cold rolling and following recrystallization. A “nonsymmetric case” crystallite orientation distribution function analysis applicable to the analysis on nonsymmetric pole figures was used. While sheet steels are generally assumed to have orthotropic physical symmetry, it was found that an appreciable degree of physical asymmetry existed. The asymmetry was with respect to the rolling direction (RD) and normal direction (ND) mirror planes, while transverse direction (TD) mirror-plane symmetry existed.

Coordinate Free Tensorial Representation of the OrientationDistribution Function With Harmonic Polynomials

The crystallite orientation distribution function (CODF) is reviewed in terms of classical spherical function representation and more recent coordinate free tensorial representation (CFTR). A CFTR is a Fourier expansion wherein the coefficients are tensors in the three‐dimensional space. The equivalence between homogeneous harmonic polynomials of degree k and symmetric and traceless tensors of rank k allows a realization of these tensors by the method of harmonic polynomials. Such a method provides for the rapid assembly of a tensorial representation from microstructural orientation measurement data. The coefficients are determined to twenty‐first order and expanded in the form of a crystallite orientation distribution function, and compared with previous calculations.

Group theory and representation of microstructure and mechanical behavior of polycrystals

We Review the group theoretical basis of the result that the observed mechanical behavior of a material can be represented by constitutive (differential) equations which govern the evolution of state variables and that these variables are even-rank irreducible tensors. On the other hand, microscopic observations of the internal structure of a polycrystal produce functions that are defined on “curved” objects such as the unit sphere of directions or the set of distinct orientations of a cube, etc. We show, in terms of an example (the crystallite orientation distribution function for a macroscopically homogeneous polycrystal composed of grains of a cubic crystalline solid), that representations of such functions give rise to Fourier coefficients that are also irreducible tensors. The tensorial state variables will be related to these tensorial Fourier coefficients. A major problem of the mechanics of materials is to develop methods that enable one, for a given material and for a given purpose, to extract tensorial state variables and the laws for their evolution from the knowledge obtained from the studies of the microstructure and behavior of the material.

Deformation mechanism of nylon 6 gel and melt films estimated in terms of orientation distribution function of crystallites

Abstract Nylon 6 films were prepared by gelation/crystallization from solutions. The solvents used were a formic acid-chloroform mixture, trifluoroethanol and benzyl alcohol. Maximum values for Young's modulus and the orientational degree of the b-axes (the crystal fibre axes), were obtained for the drawn films prepared from formic acid-chloroform solution. These values, however, were lower than those of melt films. To study the reason for this, the deformation mechanism of gel films prepared from the formic acid-chloroform cosolvent was investigated and compared with the deformation mechanism of melt film using the orientation distribution function of crystallites. The distribution functions of both films exhibited similar profiles, indicating the preferential orientation of the b-axis with respect to the stretching direction by the rotation of crystallites around the c-axis, leading to taut tie molecules. This mechanism was found to be quite different from the molecular orientation of ultradrawn polyethylene gel films associated with significant crystal transformation from a folded to a fibrous type.

Nondestructive characterization of texture and plastic strain ratio of metal sheets with electromagnetic acoustic transducers : Journal of the acoustical society of America, vol. 87, no. 2, pp. 681–690 (Feb. 1990)

All the lowest‐order independent CODF (crystallite orientation distribution function) coefficients W400, W420, and W440 of rolled steel sheets, with thickness less than 1 mm, have been obtained nondestructively by using the known single‐crystal elastic constants and the relative ultrasonic velocities measured with EMATs (electromagnetic acoustic transducers). The ultrasonic velocities of all the waves, longitudinal waves, shear waves propagating in the through thickness direction, and SH0 (shear horizontal) plate waves, were measured by the EMATs. No acoustic coupling medium was necessary, making possible quick, accurate, and reproducible measurements. It is shown that the CODF coefficients can be obtained as the solutions of three linear equations in which the measured relative ultrasonic velocities are included. There was an overall correspondence between the ultrasonic pole figures drawn using the obtained CODF coefficients and x‐ray pole figures. All nine elastic constants of a steel sheet were also c...

Tensorial Representation of the Orientation Distribution Functionin Cubic Polycrystals

A precise definition for the crystallite orientation distribution function (codf) of cubic polycrystals is given in terms of the set of distinct orientations of a cube. Elements of the classical Fourier representation of the codf, in terms of (symmetrized) generalized spherical harmonics, are reviewed. An alternative Fourier representation is defined in which the coefficients of the series expansion are irreducible tensors. Since tensors can be defined without the benefit of a coordinate frame, the tensorial representation is coordinate free. A geometrical association between irreducible tensors and a bouquet of lines passing through a common origin is discussed. Algorithms are given for computing the irreducible tensors and basis functions for cubic polycrystals.

Texture and Mechanical Anısotropy in Cold Rolled and Annealed Pure Ti Sheets

Hot rolled bands of pure Ti obtained by thermomechanical treatments were old rolled by varying rolling reductions between 50 and 90%. By annealing these specimens at 750degC for 1h, the effect of cold rolling reductions on the development of the annealing texture was systematically investigated with the crystallite orientation distribution function analysis. Also yield stresses and r values of these specimens were determined and their anisotropies were correlated with their textures. It was found that the main orientation of the annealing texture was always (anti 1013)(1anti 210). It increased significantly with increasing rolling reduction up to 80%. At higher rolling reductions, it decreased remarkably. Both yield stresses and r values of these specimens showed pronounced anisotropies, giving minimum values at the rolling direction, and increasing linearly with increasing angle to the rolling direction. By using Sachs' model of polycrytal deformation and by assuming only prismatic slip as the deformation mode, these anisotropies were predicted from the orientation distribution functions of these specimens. Anisotropies of the yield stress calculated with this method agree qualitatively well with those observed experimentally. However, disagreements between calculated and observed anisotropies of r values were considerably large. It was calculated that, since r values were measured after tensile straining of about 10%, contributions of twinning cannot be neglected in the calculation. (orig.).

Transformation Textures in Ti–15V–3Cr–3Sn–3Al Alloy Sheets

Cold rolling and recrystallization textures of the metastable β phase (bcc) and transformation textures of the α phase (hcp) precipitated during aging from the β matrices having these textures in a Ti-15V-3Cr-3Sn-3Al alloy were investigated in conjunction with their microstructures. Furthermore, the measured transformation textures were compared with the a phase textures predicted theoretically from the parent β phase textures using the crystallite orientation distribution function to discuss the mechanism of texture formation caused by the β to α transformation

Comparison of Incomplete Pole-Figure Methods for Surfaces Perpendicular to Rolling, Transverse and Normal Directions

Coefficients for a generalized-spherical-harmonic expansion of the crystallite orientation distribution function (ODF) through L=16 were obtained by an incomplete pole-figure method from a deep-drawing aluminum-killed sheet steel sample with surface perpendicular to the sheet-normal direction (ND). These coefficients were subsequently transformed from the RD, TD, ND reference frame to –ND, TD, RD and ND, RD, TD reference frames. Spherical-surface-harmonic expansions of incomplete {110}, {100}, and {112} pole-figures were calculated for each reference frame and used as input data to calculate ODF coefficients for each frame. The thus-calculated coefficients were transformed to the RD, TD, ND frame in each case. Series expansions of pole-figures and ODF for each frame are compared with the initial data.

Effects of transformation on texture and iodine stress corrosion cracking resistance of zircaloy sheet

A two-step heat treatment was developed as a means of maximizing the basal texture of Zircaloy-4 sheet. The heat treatment, consisting of a brief beta heat treatment and subsequent controlled cooling, resulted in a Kearns number (fn) of 0.83 with a microstructure of uniformly fine grains and high hardness. The texture component, as determined by the crystallite orientation distribution function (CODF), was (0002)(10−10). The result was discussed in terms of phase transformation, either diffusional or martensitic, incurred by controlled cooling. The rate of stress corrosion cracking (SCC) of the alloy was evaluated with constant deflection tests in an iodine environment. The exponentn of the relationshipda/dt = C · (Kln was found to correlate with fn for KI, ranging from 4 to 17 MPa√m.

Development of Cold-Rolling Textures in Pure Ti

Hot bands of pure Ti, obtained with thermomechanical processing, were cold-rolled up to 90% reduction in thickness, and the development of the cold-rolling texture was investigated in detail with the crystallite orientation distribution function analysis. Through the thermomechanical processing, which consisted of 97.5% hot rolling at 700deg C, followed by annealing for 1 h at the same temperature, a strong {l brace}1013{r brace} recrystallization texture wad developed in these hot bands. This texture was quite unstable during cold-rolling, so that it was considerably weakened after 30% cold-rolling. This texture change was caused mainly by extensive twinning which was followed by slip rotations. At rolling reductions between 30 and 50%, a basal texture having its center at the {l brace}0001{r brace} orientation was developed remarkably. To this texture change, slip rotations contributed much more than twinning. At rolling reductions above 50%, deformation by slip induced this {l brace}0001{r brace} orientation to rotate about its axis lying parallel to the rolling direction toward the {l brace}2115{r brace} orientation. This rotation was terminated at the {l brace}2115{r brace} orientation, and at rolling reductions above 80%, the {l brace}2115{r brace} orientation become the stable end orientation of the cold-rolling texture. (orig.).

Earing in cup drawing face-centered cubic single crystals and polycrystals

A simple model of flange behavior during cup drawing was used to predict the earing profile of deep-drawn cups. The relationship between yield surface shape and earing tendency was established, with plane stress yielding corresponding to no hold-down pressure on the flange and plane strain corresponding to no thickening. Using the Schmid law, the earing model was applied to the case of a single crystal in cube position and compared to Tucker’s well-known results.[6] For the plane strain case, good agreement was obtained with the experiment; but for plane stress, the predicted profile did not agree with the experimental one. Using the Taylor/ Bishop and Hill (TBH) theory[8,9] and measured crystallite orientation distribution functions (CODF), the model was applied to the case of high-purity aluminum sheet with various cold-rolling reductions (35, 60, 80, and 90 pct). The major experimental trends were again correctly predicted by the plane strain case.

Biaxial creep testing of textured Ti-3AI-2.5V tubing

Creep anisotropy of annealed Ti-3A1-2.5V tubing has been studied under biaxial stress conditions at 673 K using internal pressurization combined with axial loading. Biaxial strains were measuredin situ during creep using laser and linear variable differential transformer (LVDT) extensometers. Creep data were obtained for different stress ratios (α = σθ/gsz@#@), and the steady-state creep rates were found to obey power law with a stress exponent of 4.5 ±0.2 essentially independent of the stress state. Trie experimentally determined creep locus constructed at a constant value of the dissipative work function(W) deviated significantly from isotropy, indicating anisotropy of the material caused by crystallographic texture. The anisotropy parameters(R andP) in the modified Hill’s equation were obtained from the locus fitted to the experimental data to be 5.9 and 1.0, respectively. The crystallographic texture of the material was characterized through inverse and direct pole figures using X-ray diffraction techniques. The crystallite orientation distribution function (CODF) was derived from the pole figure data and represented graphically in the form of Euler plots. This CODF, along with the lower-bound plasticity model, was employed for model predictions of the anisotropy parameters and the creep loci assuming the dominance of basal, prismatic, and pyramidal slip systems. The texture-based predictions differ from the experimental results, and probable reasons for the discrepancy are discussed.

Nondestructive characterization of texture and plastic strain ratio of metal sheets with electromagnetic acoustic transducers

All the lowest-order independent CODF (crystallite orientation distribution function) coefficients W400, W420, and W440 of rolled steel sheets, with thickness less than 1 mm, have been obtained nondestructively by using the known single-crystal elastic constants and the relative ultrasonic velocities measured with EMATs (electromagnetic acoustic transducers). The ultrasonic velocities of all the waves, longitudinal waves, shear waves propagating in the through thickness direction, and SH0 (shear horizontal) plate waves, were measured by the EMATs. No acoustic coupling medium was necessary, making possible quick, accurate, and reproducible measurements. It is shown that the CODF coefficients can be obtained as the solutions of three linear equations in which the measured relative ultrasonic velocities are included. There was an overall correspondence between the ultrasonic pole figures drawn using the obtained CODF coefficients and x-ray pole figures. All nine elastic constants of a steel sheet were also calculated. Average Young’s modulus Ē and planar variation in Young’s modulus dE calculated using the nine elastic constants were consistent with average plastic strain ratio r̄ and planar variation in plastic strain ratio dr, respectively, showing that the ultrasonic method here is well adapted for practical use.

Easy-axis distribution function of permanent magnets

A new method is presented for the measurement of the easy-axis angular distribution function of crystallites in permanent magnets. The method is based on the singular point detection technique and it applies to the case of unaxial materials and to specimens having rotational symmetry around the orientation axis.

Localized plastic flow, anisotropic mechanical properties and crystallographic texture in Zircaloy sheet

Mechanical properties of Zircaloys, like many other hexagonal close-packed (hcp) metals, are sensitive to the textures developed during fabrication processes. These alloys exhibit highly anisotropic mechanical properties, a thorough knowledge of which is needed for selecting materials with good formability characteristics as well as for the predictability of inservice behavior. This paper describes texture analysis performed on recrystallized Zircaloy-4 sheet using x-ray diffraction techniques, crystallite orientation distribution function (CODF) analysis using pole figure intensity data, predictions on mechanical anisotropy and formability using CODF and appropriate plasticity models, and experimental characterization of mechanical anisotropy using grid technique. In addition, the localized plastic flow analysis, which gives the upper limits on the strains that the material can withstand without failure, is also considered. The texture based model predictions of the anisotropy parametersR andP and the formability parameterB show an excellent agreement with the experimental measurements. Analysis of the true stress-strain behavior indicates that necking in this material starts much earlier than the maximum load, and the localized plastic flow analysis gives the maximum strains along the rolling (RD) and transverse (TD) directions of the sheet that the material can be subjected to without the initiation of diffuse or local necking. The material parameters obtained in these analyses control the formability during thermomechanical processing, and the inservice behavior of the material.

Inference of stress and texture from angular dependence of ultrasonic plate mode velocities: Thompson, R.B.; Smith, J.F.; Lee, S.S. Analytical ultrasonics in materials research and testing. Proceedings of a conference, NASA Lewis Research Center, Ohio (United States), 13–14 Nov. 1984. pp. 169–180. NASA conference publication 2383 (1986)

The theory for the angular dependence of the ultrasonic wave velocity in a symmetry plane of an orthorhombic, stressed material is presented. The two waves having polarizations in this plane are shown to have velocities which can be estimated from measurements of the SH sub 0 and S sub 0 guided modes of a thin plate: the relationship being exact for the SH sub 0 mode and requiring a 10% correction for the S sub 0 mode at long wavelength. It is then shown how stress and texture can be independently inferred from various features of the angular dependence of these two velocities. From the SH sub 0 data, the ability to determine the directions and differences in magnitudes of principal stresses is described and supported by experimental data on several materials. From a combination of the SH sub 0 and S sub 0 data, a procedure is proposed for determining the coefficients W sub 400, W sub 420 and W sub 440 of an expansion of the crystallite orientation distribution function in terms of generalized Legendre functions. Possible applications in process control are indicated.

Contractile strain ratios of textured cp-titanium (Grade II) sheet

The mechanical anisotropy and strain localization were investigated on cp-titanium sheet using grid analysis method which yielded the transverse contractile strain ratios (R,P), formability parameter (B), and critical strains to necking and fracture. Crystallographic texture pole figures were determined using X-ray diffraction (XRD) techniques from which crystallite orientation distribution functions (CODFs) were derived. The anisotropy and formability parameters were evaluated using CODF in conjunction with crystal plasticity models, and are correlated with experimental results.

Seismic anisotropy of the inner core

Recent studies of seismic travel times and free oscillations have been interpreted in terms of an anisotropic inner core with cylindrical symmetry aligned with the earth's axis of rotation. The inner core is believed to consist of hexagonal close‐packed (hcp) iron. In this letter the velocity anisotropy is used to determine the coefficients Wlmn of a series expansion of the crystallite orientation distribution function in generalised spherical harmonics. This function gives the probability of a crystallite having a given orientation with respect to a set of axes fixed in the core. The solution to this problem enables construction of crystallographic pole figures, which suggest a strong alignment of hcp Fe c‐axes along the earth's axis of rotation.

Correlation of biaxial yield behavior in copper with crystallographic texture for strains in the range 10 −4–10 −2

Tension-torsion yield surfaces in 101 alloy copper tubing measured at effective strains ranging from 10 −4 to 10 −2 are compared with the predictions from two polycrystalline models: the Bishop-Hill model and a uniform stress lower-bound model. Both models use the crystallite orientation distribution function (CODF) to predict anisotropy in the yield surface. Neither model predicted the yield surface well at an effective strain of 10 −4. At strains above 5×10 −4 both models predict the observed anisotropy between the axial and shear yield stress well. Overall, the lower-bound model prediction was found to exhibit better agreement with the measured yield surface for strains up to 10 −2. Prediction of bulk elastic constants using single crystal compliance values is also discussed. Using the CODF as a weighting factor bulk compliances were predicted to within 4% of the experimental measurements.