Rodrigues Space Research Articles

Texture Componets Scattering Parameter Selection for Polycrystalline Materials when Clustering Rodrigues Space

The article considers the problem of maximum scattering cluster value selection at texture components formation within Rodrigues space for texture analysis representation. Texture analysis is carried out in terms of direct pole figure. Orientation cloud is being formed within Rodrigues space by calculating Rodrigues vectors for each found orientation. Texture components describing the texture are being formed by maximin algorithm clustering. The algorithm is considered finished when a predetermined value of maximum scattering in a cluster is reached. In this case, maximum scattering in clusters will be determined automatically. The algorithm enables texture presentation with required detail level corresponding to the posed problem. Implementation of Rodrigues space to form texture components enables texture analysis results visualization. The article gives principal expressions for Rodrigues-vector operations. The calculation of mean Rodrigues vector in a cluster is shown. The algorithm that allows comparison of texture analysis results obtained by two pole figures, taken for one sample. Comparison is carried out by calculation of distances between mean Rodrigues vectors of two pole figure clusters for different scattering parameter values. Texture representation is chosen as the best correspondence of texture components obtained by two pole figures. An example of aluminum tape texture analysis is presented. It is shown that formation of eight texture components with maximum cluster scattering 11.3° by pole figure of family {111} and of six texture components with maximum cluster scattering 10.15° by pole figure {200} their correspondence is 85 %.

Study of the texture of polycrystalline FCC materials using one incomplete direct pole figure

The article deals with the algorithm for texture analysis of polycrystalline materials using one direct pole figure (DPF). It is shown that the incomplete direct polar figure {111} for fcc materials contains the necessary information about the material texture. The algorithm provides identification of the preferred texture components in a multicomponent texture material and determination of their properties. The proposed algorithm is as follows. The upper hemisphere of the digital representation of the DPF is scanned by a polar complex of vectors that are normal to the reflection planes. Then the reliability parameters for each orientation are calculated and a set of the most reliable orientations is formed. The chosen orientations are recalculated to the Rodrigues space wherein the preferred texture components are formed by clustering. At the same time, an iterative algorithm with symmetry operators is used to avoid the umklapp effect. Each texture component is represented by the following parameters: Rodrigues mean vector, Miller indices, and Euler angles. The share and scattering of the texture component are also calculated. A method for selecting the optimal number of clusters providing presentation of the texture with the desired degree of detail is proposed. This is achieved by comparing two incomplete direct pole figures taken for {111} and {200} to select the maximum cluster scattering value on which the number of formed predominant texture components depend. The developed algorithm seems promising for rapid texture analysis, in analysis of sharp and weak textures and when there are less than three DPFs.

Investigation of the Relationship Between Microstructural Features and Strain Localization in Polycrystalline 316 L

In this paper, strains are monitored in-situ at the surface of a polycrystalline 316 L sample loaded quasi-statically in uniaxial tension. Initial orientation data collected over thousands of grains is compared with the strain data evolution in order to investigate, with statistical significance, the relationship between the microstructural features and the strain localization patterns emerging at the grain scale. It is shown quantitatively that high values of strain appear early and primarily around grain boundaries, before growing into a network spanning several grains and grain boundaries. Strains are also heterogeneous within a grain, with the heterogeneity tending to be more pronounced in larger grains. Statistical analyses demonstrate that the usual microstructural descriptors—such as Taylor factors, average intragranular misorientation angles or intergranular misorientation angles—do not display any correlation with the localization network. Furthermore, higher Schmid factors, albeit exhibiting some weak correlation with higher strains, fail to systematically identify the grains that experience the largest strains. It is thus proposed to analyze three-dimensional descriptors of orientation expressed in the Rodrigues space and they are shown to help readily identify orientations that exhibit high strains early in the loading. Additionally, the study of the full three-dimensional misorientation information, expressed in the reduced Rodrigues space, clearly highlights the fact that some misorientations oppose localization at certain boundaries in the microstructure at hand. Hence, considering the full three-dimensional orientation (and misorientation) data instead of just scalar descriptors is demonstrated to be of great interest when investigating the relationship between microstructural features and strain localization.

Computational modeling of crystallographic texture evolution over cubochoric space

The present work addresses representation of texture evolution in face-centered cubic (fcc) microstructures in cubochoric orientation space. The microstructure is quantified with the orientation distribution function (ODF), which models volume density in the fundamental region of crystallographic space. The ODF is discretized using a finite element scheme in the cubochoric fundamental region. This scheme shows superior features over the classical techniques in global spaces, such as spherical harmonics or Fourier space solutions, since it can represent a large variety of textures, including very sharp textures such as a single crystal. The texture evolution during a particular deformation process is associated with the evolution of the ODF in time, which is governed by the conservation equation and crystal constitutive relations. The transformation in between cubochoric space and other popular angle-axis representation such as Rodrigues space is performed with a two-step approach including the transformations from Rodrigues domain to homochoric domain, and homochoric domain to cubochoric domain through a numerical scheme. The ODF evolution in an fcc material during different deformation processes, such as tension, plane strain compression and shear, is compared across both Rodrigues and cubochoric spaces, and similar patterns are observed.

Orientation sampling for dictionary-based diffraction pattern indexing methods

A general framework for dictionary-based indexing of diffraction patterns is presented. A uniform sampling method of orientation space using the cubochoric representation is introduced and used to derive an empirical relation between the average disorientation between neighboring sampling points and the number of grid points sampled along the semi-edge of the cubochoric cube. A method to uniformly sample misorientation iso-surfaces is also presented. This method is used to show that the dot product serves as a proxy for misorientation. Furthermore, it is shown that misorientation iso-surfaces in Rodrigues space are quadractic surfaces. Finally, using the concept of Riesz energies, it is shown that the sampling method results in a near optimal covering of orientation space.

Определение структуры поликристаллических материалов с помощью алгоритма объектно-векторного представления плоскостей отражения и визуализация результатов в пространстве Родрига

Определение структуры поликристаллических материалов с помощью алгоритма объектно-векторного представления плоскостей отражения и визуализация результатов в пространстве Родрига

GrainSpotter: a fast and robust polycrystalline indexing algorithm

A new approach for indexing multigrain diffraction data is presented. It is based on the use of a monochromatic beam simultaneously illuminating all grains. By operating in sub-volumes of Rodrigues space, a powerful vertex-finding algorithm can be applied, with a running time that is compatible with online analysis. The resulting program,GrainSpotter, is sufficiently fast to enable online analysis during synchrotron sessions. The program applies outlier rejection schemes, leading to more robust and accurate data. By simulations it is shown that several thousand grains can be retrieved. A new method to derive partial symmetries, called pseudo-twins, is introduced. Uniquely,GrainSpotterincludes an analysis of pseudo-twins, which is shown to be critical to avoid erroneous grains resulting from the indexing.

A method for intragranular orientation and lattice strain distribution determination

A novel approach to quantifying intragranular distributions is developed and applied to the α → ∊ phase transition in iron. The approach captures both the distribution of lattice orientation within a grain and the orientation dependence of the lattice strain. Use of a finite element discretization over a ball in Rodrigues space allows for the efficient use of degrees of freedom in the numerical approach and provides a convenient framework for gradient-based regularization of the inverse problem. Application to the α → ∊ phase transition in iron demonstrates the utility of the method in that intragranular orientation and lattice strain distributions in the α phase are related to the observed ∊ orientations. Measurement of the lattice strain distribution enables quantitative analysis of the driving forces for ∊ variant selection. The measurement and analysis together indicate quantitatively that the Burgers mechanism is operative under the experimental conditions examined here.

Common Region between Euler Subspace and Rodrigues Fundamental Zone for a Cubic Crystal and Orthotropy Sample Symmetry

Due to crystal and sample symmetry, texture data can be presented uniquely in a smaller domain of Euler box or Rodrigues space. As these sub-domains are obtained by different criteria, a one-to-one relation does not always exist for conversion between Euler and Rodrigues coordinates. This work presents a methodology for finding the common region between the Euler sub-space and Rodrigues fundamental zone for the case of cubic crystal and orthotropy sample symmetry, and its size is found to be 0.0864068.

Microstructure model reduction and uncertainty quantification in multiscale deformation processes

The quantification and propagation of uncertainty in multiscale deformation processes is considered. A reduced-order model for representing the data-driven stochastic microstructure input is presented. The multiscale random field representing the random microstructure is decomposed into few modes in different scales (the Rodrigues space for representing texture on mesoscale and the continuum macroscale space). Realizations from a stochastic simulation are used to obtain a small number of modes approximating the stochastic field. An example of a multiscale closed-die forging problem is provided in which the effects of uncertain initial geometry and texture on the macroscale properties are studied.

The effect of multiple sources of uncertainty on the convex hull of material properties of polycrystals

Quantification and propagation of uncertainty in process conditions and initial microstructure on the final product properties in a deformation process are presented. The stochastic deformation problem is modeled using a sparse grid collocation approach that allows the utilization of a deterministic simulator to build interpolants of the main solution variables in the stochastic support space. The ability of the method in estimating the statistics of the macro-scale microstructure-sensitive properties and constructing the convex hull of these properties is shown through examples featuring randomness in initial texture and process parameters. A data-driven model reduction methodology together with a maximum entropy approach are used for representing randomness in initial texture in Rodrigues space. Comparisons are made with the results obtained from the Monte-Carlo method.

Determination of volume fraction of the texture components in the Rodrigues fundamental region

The fundamental region of Rodrigues space was discretized with finite elements as Euler space was discretized with 5° cell structures. Texture components were modeled with Gaussian distributions in the Rodrigues fundamental region. Spherical texture components are described by a sphere with a Gaussian distribution centered at its ideal position. Ideal fiber orientations are represented by a straight line or a skeleton line in Rodrigues space. The usual fiber textures are represented by a tube with a Gaussian distribution centered along a fiber line in Rodrigues space. The volume fractions of typical texture components can be collected by misorientation approach. The volume collections of spherical components of interest begin with the calculation of misorientation angles between the ideal components and their adjacent orientations. The radius of the orientation sphere is an angular distance. Adjacent components with smaller misorientation angles than a given criterion or a cut-off value locate inside the sphere and contribute to the volume fraction of the spherical component. For fibers, angular distances between the orientations along a skeleton line and adjacent orientations are compared with a cut-off. In addition, discretization effects of the Rodrigues fundamental region on the ODF and texture analysis were investigated. The volume fractions of texture components measured from a cold-rolled gold sheet and bonding wire were analyzed using the misorientation approach proposed.

Linear analysis of texture–property relationships using process-based representations of Rodrigues space

Techniques for exploring texture–property–process relationships in various deformation processes are derived from a reduced-order model of texture evolution. The orientation distribution function (ODF) in polycrystals is represented over the Rodrigues space in a discrete form using finite element interpolation techniques. Linear programming algorithms are developed for retrieving ODFs with extremal or optimal properties from the complete ODF space. The relationship of optimal textures with processing is addressed by representing texture evolution in a space of reduced basis coefficients called the process plane. This involves generation of orthogonal basis functions for representing spatial variations of the ODF in a given process using proper orthogonal decomposition. These basis functions are found to work in interpolatory and extrapolatory processing modes and allow representation of texturing for deviations in the process variables. Optimization problems are posed in the reduced space for retrieving textures with desired properties. A graphical technique is discussed that allows identification of optimal processing paths for reaching desired textures in association with process plane databases.

A novel optimization-based pole-figure inversion method: comparison with WIMV and maximum entropy methods

An optimization-based method for pole-figure (PF) inversion that utilizes the orientation distribution function (ODF) gradient for conditional control of the solution is presented. The novel PF inversion method, coined the hybrid {\cal H}^1-seminorm minimization (HHSM), is empirically shown to be versatile, general and robust in the presence of simulated experimental errors. Finite elements (FE) and Rodrigues space are used for the representation and parameterization of the orientation space throughout. The versatility of the FE representation is significantly enhanced from previous implementations by introducing a method for obtaining discrete approximations to spherical harmonic modes from the local FE basis functions. A comparative study with similar implementations of the basic WIMV algorithm and the maximum entropy method is undertaken using several model ODFs of varying sharpness and symmetry. Randomly distributed noise is added to the synthetic PFs to simulate experimental errors and assess the stability of each method. Solution consistency is assessed by inverting two sets of measured PFs, one complete, one incomplete, using several meshes on the orientation space with an increasing number of degrees of freedom. The HHSM method is shown to compare favorably in tests with both the WIMV method and the maximum entropy method.

Long-range current flow and percolation in RABiTS-type conductors and the relative importance of out-of-plane and in-plane misorientations in determining Jc

Calculations of long-range current flow using an advanced percolation model show that with the presently observed texture in RABiTS substrates, the dependence of Jc on length as a function of width is greatly reduced. Furthermore, this dependence becomes almost negligible in applied fields. These results suggest that sub-division of a wide conductor into narrow filaments should be possible without loss in Jc. The relative importance of the out-of-plane texture in affecting intergranular Jc was also explored by fabricating RABiTS substrates with different out-of-plane textures but approximately the same in-plane texture. This was accomplished by using TiN as a seed layer for which significant sharpening of the out-of-plane texture is observed. Similar Jc was found for samples with differing out-of-plane texture but almost the same in-plane texture. Finally, separation of the total misorientation in GB networks into in-plane and out-of-plane misorientations using manipulations in Rodrigues space shows that Jc correlates best with in-plane texture.

Analysis of local orientation gradients in deformed single crystals

Grain fragmentation and local orientation gradients in deformed single crystals are characterized using electron backscatter diffraction (EBSD) to obtain statistically reliable information. Interrogation of the dislocation substructure is accomplished by extracting information gleaned from small point-to-point misorientations as measured by EBSD. Along with an estimate of the geometrically necessary dislocation (GND) content, the point-to-point deviation from an average grain orientation is described by an orientation difference vector defined in Rodrigues space. Mapping of parameters such as GND, and divergence and gradient fields created from analysis of the difference vectors provide an alternative approach to obtain quantitative information and images from EBSD data.

Microtextural study of variant selection by EBSD in a bainitic steel containing retained austenite

The phenomenon of variant selection in a bainitic steel containing retained austenite was investigated. The steel was hot-rolled below the Tnr to true strains of epsilon = 0.2 and epsilon = 0.8 in the austenite temperature range. The orientation relationships between bainite and austenite are expressed in Rodrigues space. A strain of 0.2 is shown to be insufficient to induce variant selection. A hot deformation of 0.8 leads to a much smaller number of variants being created, which are shown to be related to the slip systems active during austenite deformation.

A proper orthogonal decomposition approach to microstructure model reduction in Rodrigues space with applications to optimal control of microstructure-sensitive properties

A novel reduced-order modeling approach to represent crystallographic texture in polycrystals is presented. This method involves generation of a reduced orthogonal basis for representing polycrystal texture in Rodrigues space. The reduced basis is generated from the textures obtained from different deformation modes (shear, uniaxial tension, etc.) using proper orthogonal decomposition and the method of snapshots. This model reduction can account for all essential information needed to describe the microstructure and leads to tremendous savings in computational time. Based on the above method a multiscale model for microstructure evolution during deformation is proposed driven by the macroscale velocity gradient. A continuum sensitivity-based gradient optimization framework is introduced in the above model for control of microstructure-sensitive material properties. Emphasis is given to relevant design problems that demonstrate the effectiveness of the above method for optimal control of texture-dependent material properties during deformation such as the magnetic hysteresis losses, the yield stress and the Lankford parameter R.

Pole Figure Inversion Using Finite Elements Over Rodrigues Space

Using finite elements over Rodrigues space, methods are developed for the formation and inversion of pole figures. The methods take advantage of the properties of Rodrigues space, particularly the fact that geodesics corresponding to pole figure projection paths are straight lines. Both discrete and continuous pole figure data may be inverted to obtain orientation distribution functions (ODFs) in Rodrigues space, and we include sample applications for both types of data.

Numerical simulation of deformation induced anisotropy of polycrystals

Aspects of the homogenization problem for polycrystalline materials using finite element based representative volume elements (RVE) are discussed. Special attention is paid to the boundary conditions and the representation of initially isotropic states with limited numbers of grains. The RVE is used for simulations of shear tests with aluminium specimens as reported by Williams [O.W. Williams, Shear textures in copper, brass, aluminium, iron and zirconium, Trans. Met. Soc. AIME 224 (1962) 129–139.]. On the grain level, slip system theory, combining anisotropic elasticity with a viscoplastic flow rule, is used. The resulting texture is represented in the Rodrigues space and by pole figures. Initial and subsequent yield surfaces are investigated using different yield criteria.