Orientation Density Function Research Articles

Microstructure and residual stress evolution during cyclic elastoplastic deformation of AISI316L fabricated via laser powder bed fusion

In metal additive manufacturing (MAM), microstructural properties such as texture, residual stresses, and dislocation density have emerged as key factors ruling the resulting mechanical performances. In this study, cylindrical AISI 316L specimens, fabricated with laser powder bed fusion (LPBF), were tested under cyclic elastoplastic (EP) deformation using a constant strain amplitude to highlight the evolution of residual stresses (RS), dislocation density and texture with increasing number of EP cycles, N, across the hardening-softening (H–S) transition stage, in the attempt to find correlations between relevant microstructural parameters and macroscopic properties. The structural and microstructural analysis is carried out through whole powder pattern modeling (WPPM) of neutron diffraction (ND) data and Electron Back-Scattering Diffraction (EBSD) analysis. The H–S transition is found to occur within 7–9 cycles, with RS fading out already after 5 cycles. Across the H–S transition, the trend of the maximum tensile stress correlates closely with the trend of WPPM-calculated total dislocation density, suggesting a major role of dislocations’ characteristics in the evolution of macroscopic mechanical properties. EBSD analysis reveals the rearrangement of geometrically necessary dislocations (GND) into cellular structures, and moderate grain refinement, which are deemed to be responsible for the quick fading of RS in the very early stage of EP loading. ND-based texture analysis reveals a (220) preferential orientation retained throughout the EP tests but with orientation density functions (ODFs) changing non-monotonically with N, suggesting preliminary partial randomization of grains around the deformation axis followed by the recovery of crystallographic anisotropy and more localized ODFs.

Efficient numerical solution of the Fokker–Planck equation using physics-conforming finite element methods

Abstract We consider the Fokker–Planck equation (FPE) for the orientation probability density of fiber suspensions. Using the continuous Galerkin method, we express the numerical solution in terms of Lagrange basis functions that are associated with N nodes of a computational mesh for a domain in the 3D physical space and M nodes of a mesh for the surface of a unit sphere representing the configuration space. The NM time-dependent unknowns of our finite element approximations are probabilities corresponding to discrete space locations and orientation angles. The framework of alternating-direction methods enables us to update the numerical solution in parallel by solving N evolution equations on the sphere and M three-dimensional advection equations in each (pseudo-)time step. To ensure positivity preservation as well as the normalization property of the probability density, we perform algebraic flux correction for each equation and synchronize the correction factors corresponding to different orientation angles. The velocity field for the spatial advection step is obtained using a Schur complement method to solve a generalized system of the incompressible Navier–Stokes equations (NSE). Fiber-induced subgrid-scale effects are taken into account using an effective stress tensor that depends on the second- and fourth-order moments of the orientation density function. Numerical studies are performed for individual subproblems and for the coupled FPE-NSE system.

Effect of annealing twins, strain-recrystallization processing and δ-phase fraction on microtexture and evaluation of mechanical properties of nickel-based superalloy 718

The Nickel-based superalloy 718 is a precipitation-hardened material, especially by the ordered and coherent phases γ” (Ni3Nb) and γ″ (Ni3 (Al, Ti)). In addition to these phases, precipitation of carbides and δ-phase is possible. This material has high strength, creep, and corrosion resistance at temperatures up to 650 °C. However, despite the alloy’s outstanding performance under harsh environments, research effort has been put into optimizing its properties and increasing the operation temperature. Manipulation of microstructural parameters such as grain boundary character distribution (GBCD), microtexture, second phase fraction, etc. is frequently betaken in order ot achieve better properties. In this context, this work aimed to assess the influence of Thermomechanical Processing on the recrystallization texture of the material and its influence on the alloy’s mechanical behavior. The material was submitted to four iterative processing routes, composed of deformation-heat treatment cycles. The alloy’s microtexture data were obtained through the Electron Backscattered Diffraction technique. Data processing was performed using the MTEX toolbox, where Orientation Density Functions were obtained for microtexture analysis. It was observed that the deformation imposed during thermomechanical processing significantly affects the recrystallization texture of the material so that smaller deformations induce the formation of typical recrystallization texture components; on the other hand, high degrees of deformation lead to a retained deformation texture after annealing. Mechanical properties were evaluated by depicting the influence of different strengthening mechanisms and δ-phase fraction on yield strength and ductility, respectively. It was observed that the main strengthening parameters for the alloy were grain boundaries and intrinsic resistance (Pierls-Nabarro and solid solution), meanwhile ductility was greatly influenced by δ-phase and ∑3n boundaries fractions.

Optimized rectification of fiber orientation density function with background threshold

PurposeTo describe an optimized fiber orientation density function (fODF) rectification procedure that removes negative values and absorbs all features below a specified threshold into a constant background. Theory and methodsThe fODF for a white matter imaging voxel describes the angular density of axons. Because of signal noise and Gibbs ringing, fODFs estimated with diffusion MRI may take on unphysical negative values in some directions and contain spurious peaks. In order to suppress such artifacts, an fODF rectification procedure is proposed that both eliminates all negative values and incorporates all features below a specified threshold, η, into a constant background while at the same time minimizing the mean square deviation from the original, unrectified fODF. Calculating this fODF is straightforward, and the directions and shapes of peaks not absorbed into the background are preserved. The rectification method is illustrated for an analytic fODF model and for experimental diffusion MRI data obtained in healthy human brain, with the original fODFs being obtained from fiber ball imaging. ResultsExamples of optimal rectified fODFs are given for three choices of the background threshold referred to as minimal rectification (η = 0), average-level rectification (η ≈ 0.08), and fractional-anisotropy-axonal-based rectification (η ≈ 0.1). As η is increased, artifacts and other small features are more strongly suppressed, but the major fODF peaks are largely unaffected for the range of η values illustrated by these three alternatives. ConclusionArtifactual features of fODFs estimated with diffusion MRI can be effectively suppressed by applying the proposed optimized rectification procedure. Since it minimizes fODF distortion in the mean square sense, it may be useful in the study of how fODF fine structure is affected by aging and disease.

Impact of intra-axonal kurtosis on fiber orientation densityfunctions estimated with fiber ball imaging.

To determine the impact of an intra-axonal kurtosis on estimates of the fiber orientation density function (fODF) obtained with fiber ball imaging (FBI). Standard FBI assumes Gaussian diffusion within individual axons and estimates the fODF by applying an inverse generalized Funk transform to diffusion MRI data for b-values of 4000 s/mm2 or higher. However, recent work based on numeric simulations shows that diffusion inside axons is non-Gaussian with an intra-axonal kurtosis of ∼ 0.4. Here, the theory underlying FBI is extended to incorporate an intra-axonal kurtosis. This is done to first order in the intra-axonal kurtosis without making assumptions about the details of the diffusion dynamics and to all orders for a specific model based on a gamma distribution of diffusivities. The first order approximation is used to assess the effect of an intra-axonal kurtosis on FBI estimates for the fODF and axonal water fraction. The gamma distribution model is used to test the validity of the approximation. The first order approximation indicates the estimated fODF is altered by a few percent for an intra-axonal kurtosis of 0.4 in comparison to predictions of standard FBI. If one neglects the intra-axonal kurtosis, the angular resolution of the point spread function for the fODF is changed by <1°, whereas the axonal water fraction is overestimated by ∼ 5%. The gamma distribution model shows that the first order approximation is accurate to within a few percent. The intra-axonal kurtosis has a small impact on fODFs estimated with FBI.

Viscoelastic-viscoplastic homogenization of short glass-fiber reinforced polyamide composites (PA66/GF) with progressive interphase and matrix damage: New developments and experimental validation

In this paper, an original probabilistic micromechanics damage framework involving multi-deformation mechanisms, based on the modified Mori-Tanaka and Transformation Field Analysis (MT-TFA) techniques, is developed to predict monotonic and oligocyclic stress-strain responses in short fiber-reinforced polyamide composites. The proposed model allows simulating actual injection-induced fiber arrangement, which is characterized by arbitrary fractions of randomly oriented fibers distributed in the laminate plane. Furthermore, the modified MT-TFA approach employs a phenomenological model consisting of 4 Kelvin-Voigt branches and a viscoplastic branch, formulated under the thermodynamics framework, to describe the rate-dependent viscoelastic-viscoplastic deformation and the ductile damage of the polymer matrix phase. In addition, the Weibull probabilistic density function is utilized to simulate initiation and coalescence of the void-type discrete damage in the vicinity of the fiber/matrix interphase, induced by the fiber/matrix debonding as observed experimentally. The parameters of the developed model are calibrated against the experimental response of glass/polyamide (PA66/GF35) composites via uniaxial loading/unloading tests, by taking into account the actual fiber orientation density function (ODF). The reliability and efficiency of the modified Mori-Tanaka and TFA schemes are assessed vis-à-vis the separate and hold-out experimental data subjected to uniaxial and oligocyclic loading at various loading rates. Progressive matrix and interphase damage are compared in support of the modified MT-TFA technique's capabilities to capture the experimentally observed damage mechanisms. To accurately capture the experimental response, the progressive degradation of the load transfer between the fiber and matrix phases is introduced through a reduction of the active fiber length. The latter is introduced by considering the effect of the interphase void-damage content. The new mean-field formulation provides accurate predictions of the overall response under complex loading paths. It can be combined with other techniques in our future work, such as cycle-jump, towards simulating high-cycle fatigue damage in short-fiber composite structures.

High fidelity fiber orientation density functions from fiber ball imaging.

The fiber orientation density function (fODF) in white matter is a primary physical quantity that can be estimated with diffusion MRI. It has often been employed for fiber tracking and microstructural modeling. Requirements for the construction of high fidelity fODFs, in the sense of having good angular resolution, adequate data to avoid sampling errors, and minimal noise artifacts, are described for fODFs calculated with fiber ball imaging. A criterion is formulated for the number of diffusion encoding directions needed to achieve a given angular resolution. The advantages of using large b-values (≥6000 s/mm2 ) are also discussed. For the direct comparison of different fODFs, a method is developed for defining a local frame of reference tied to each voxel's individual axonal structure. The Matusita anisotropy axonal is proposed as a scalar fODF measure for quantifying angular variability. Experimental results, obtained at 3 T from human volunteers, are used as illustrations.

Orientation Analysis

Simple figures illustrate the basic concepts: orientation, Euler angles, Euler space, orientation density function, pole density function. The iteration that decisively influenced the development of orientation analysis follows directly from the relationship between the two density functions. The minimum principle defines the initial function and the structure of the iteration. Using model orientation density function, we prove that this kind of orientation analysis is extremely effective.

Constrained spherical deconvolution of nonspherically sampled diffusion MRI data.

Constrained spherical deconvolution (CSD) of diffusion‐weighted MRI (DW‐MRI) is a popular analysis method that extracts the full white matter (WM) fiber orientation density function (fODF) in the living human brain, noninvasively. It assumes that the DW‐MRI signal on the sphere can be represented as the spherical convolution of a single‐fiber response function (RF) and the fODF, and recovers the fODF through the inverse operation. CSD approaches typically require that the DW‐MRI data is sampled shell‐wise, and estimate the RF in a purely spherical manner using spherical basis functions, such as spherical harmonics (SH), disregarding any radial dependencies. This precludes analysis of data acquired with nonspherical sampling schemes, for example, Cartesian sampling. Additionally, nonspherical sampling can also arise due to technical issues, for example, gradient nonlinearities, resulting in a spatially dependent bias of the apparent tissue densities and connectivity information. Here, we adopt a compact model for the RFs that also describes their radial dependency. We demonstrate that the proposed model can accurately predict the tissue response for a wide range of b‐values. On shell‐wise data, our approach provides fODFs and tissue densities indistinguishable from those estimated using SH. On Cartesian data, fODF estimates and apparent tissue densities are on par with those obtained from shell‐wise data, significantly broadening the range of data sets that can be analyzed using CSD. In addition, gradient nonlinearities can be accounted for using the proposed model, resulting in much more accurate apparent tissue densities and connectivity metrics.

Optimized rectification of fiber orientation density function.

To demonstrate an optimized rectification strategy for fiber orientation density functions (fODFs). In white matter, fODFs can be estimated with diffusion MRI. However, because of signal noise, imaging artifacts and other factors, experimentally determined fODFs may take on unphysical negative values in some directions. Here, we show how to rectify such fODFs to eliminate all negative values while minimizing the mean square difference between the original and rectified fODFs. The method is demonstrated for a mathematical model and for fODFs estimated from experimental human data using both constrained spherical deconvolution and fiber ball imaging. Comparison with an alternative nonoptimized rectification approach is also provided. For the mathematical model, it is found that the optimized rectification procedure removes negative fODF values while at the same time reducing the mean square error. Relative to the alternative rectification approach, the optimized fODFs are substantially more accurate. For the experimental data, the optimized fODFs have a lower average fractional anisotropy axonal and often fewer small peaks than the original, unrectified fODFs. The calculation of optimized fODFs is straightforward where the main step is the finding of the root to an equation in one variable, as may be efficiently accomplished with the bisection method. Unphysical negative fODF values can be easily eliminated in a manner that minimizes the mean square difference between the original and rectified fODFs. Optimized fODF rectification may be useful in applications for which negative values are problematic.

A 3D phenomenological yield function with both in and out-of-plane mechanical anisotropy using full-field crystal plasticity spectral method for modelling sheet metal forming of strong textured aluminum alloy

In this paper, a multi-scale model for AA3104-H19 and AA2024-T3 aluminium alloy with strong cold rolled crystallographic texture is proposed. This model consists of two components: (i) a full-field crystal plasticity spectral method using fast Fourier transformation (CPFFT) that can be used to predict anisotropic mechanical properties in both the in and out-of-plane uniaxial and multi-axial stress states and (ii) a 3D new phenomenological anisotropic yield function with enough flexibility to describe the in and out-of-plane mechanical anisotropies of strongly textured aluminium alloy sheet. This multi-scale model was implemented into a commercial finite element (FE) software Abaqus via a user-defined material subroutine UMAT to predict the anisotropic plastic deformation in a multi-step forming process (deep drawing). Results showed that this new modelling approach is able to accurately predict earing profiles with multiple ears in deep drawn cups based on the crystallographic texture information data obtained from orientation density functions (ODFs).

Optimized reconstruction of the crystallographic orientation density function based on a reduced set of orientations.

Crystallographic textures, as they develop for example during cold forming, can have a significant influence on the mechanical properties of metals, such as plastic anisotropy. Textures are typically characterized by a non-uniform distribution of crystallographic orientations that can be measured by diffraction experiments like electron backscatter diffraction (EBSD). Such experimental data usually contain a large number of data points, which must be significantly reduced to be used for numerical modeling. However, the challenge in such data reduction is to preserve the important characteristics of the experimental data, while reducing the volume and preserving the computational efficiency of the numerical model. For example, in micromechanical modeling, representative volume elements (RVEs) of the real microstructure are generated and the mechanical properties of these RVEs are studied by the crystal plasticity finite element method. In this work, a new method is developed for extracting a reduced set of orientations from EBSD data containing a large number of orientations. This approach is based on the established integer approximation method and it minimizes its shortcomings. Furthermore, the L 1 norm is applied as an error function; this is commonly used in texture analysis for quantitative assessment of the degree of approximation and can be used to control the convergence behavior. The method is tested on four experimental data sets to demonstrate its capabilities. This new method for the purposeful reduction of a set of orientations into equally weighted orientations is not only suitable for numerical simulation but also shows improvement in results in comparison with other available methods.

On the Orientation Average Based on Central Orientation Density Functions for Polycrystalline Materials

The present work continues the investigation first started by Lobos et al. (J. Elast. 128(1):17–60, 2017) concerning the orientation average of tensorial quantities connected to single-crystal physical quantities distributed in polycrystals. In Lobos et al. (J. Elast. 128(1):17–60, 2017), central orientation density functions were considered in the orientation average for fourth-order tensors with certain index symmetries belonging to single-crystal quantities. The present work generalizes the results of Lobos et al. (J. Elast. 128(1):17–60, 2017) for the orientation average of tensors of arbitrary order by presenting a clear connection to the Fourier expansion of central orientation density functions and of the general orientation density function in terms of tensorial texture coefficients. The closed form of the orientation average based on a central orientation density function is represented in terms of the Fourier coefficients (referred to as texture eigenvalues) and the central orientation of the central orientation density function. The given representation requires the computation of specific isotropic tensors. A pragmatic algorithm for the automated generation of a basis of isotropic tensors is given. Applications and examples are presented to show that the representation of the orientation average offers a low-dimensional parametrization with major benefits for optimization problems in materials science. A simple implementation in Python 3 for the reproduction of all examples is offered through the GitHub repository https://github.com/mauricio-fernandez-l/centralODF-average.

Optimization of data acquisition and analysis for fiber ball imaging

The inverse Funk transform of high angular resolution diffusion imaging (HARDI) data provides an estimate for the fiber orientation density function (fODF) in white matter (WM). Since the inverse Funk transform is a straightforward linear transformation, this technique, referred to as fiber ball imaging (FBI), offers a practical means of calculating the fODF that avoids the need for a response function or nonlinear numerical fitting. Nevertheless, the accuracy of FBI depends on both the choice of b-value and the number of diffusion-encoding directions used to acquire the HARDI data. To inform the design of optimal scan protocols for its implementation, FBI predictions are investigated here with in vivo data from healthy adult volunteers acquired at 3 T for b-values spanning 1000 to 10,000 s/mm2, for diffusion-encoding directions varying in number from 30 to 256 and for TE ranging from 90 to 120 ms. Our results suggest b-values above 4000 s/mm2 with at least 64 diffusion-encoding directions are adequate to achieve reasonable accuracy with FBI for calculating axon-specific diffusion measures and for performing WM fiber tractography (WMFT).

Numerical investigation of evolution of earing, anisotropic yield and plastic potentials in cold rolled FCC aluminium alloy based on the crystallographic texture measurements

The mechanical anisotropy of FCC aluminium alloy is influenced by the polycrystalline orientations transformed by cold rolling. The relationship between mechanical anisotropy and evolution of crystallographic textures was established using the crystal plasticity finite element method (CPFEM) based on measured polycrystalline orientation density functions (ODFs) and used to determine the parameters of a phenomenological convex anisotropic yield function which are solved with an implicit finite element (FE) stress integration code. A multi-scale modelling system is generated by combining CPFEM and the phenomenological yield function enabling accurate simulations of forming operations without the need for conventional physical mechanical testing of the material.

A framework for multi-component analysis of diffusion MRI data over the neonatal period

We describe a framework for creating a time-resolved group average template of the developing brain using advanced multi-shell high angular resolution diffusion imaging data, for use in group voxel or fixel-wise analysis, atlas-building, and related applications. This relies on the recently proposed multi-shell multi-tissue constrained spherical deconvolution (MSMT-CSD) technique. We decompose the signal into one isotropic component and two anisotropic components, with response functions estimated from cerebrospinal fluid and white matter in the youngest and oldest participant groups, respectively. We build an orientationally-resolved template of those tissue components from data acquired from 113 babies between 33 and 44 weeks postmenstrual age, imaged as part of the Developing Human Connectome Project. These data were split into weekly groups, and registered to the corresponding group average templates using a previously-proposed non-linear diffeomorphic registration framework, designed to align orientation density functions (ODF). This framework was extended to allow the use of the multiple contrasts provided by the multi-tissue decomposition, and shown to provide superior alignment. Finally, the weekly templates were registered to the same common template to facilitate investigations into the evolution of the different components as a function of age. The resulting multi-tissue atlas provides insights into brain development and accompanying changes in microstructure, and forms the basis for future longitudinal investigations into healthy and pathological white matter maturation.

Mapping fine-scale anatomy of gray matter, white matter, and trigeminal-root region applying spherical deconvolution to high-resolution 7-T diffusion MRI

ObjectivesWe assessed the use of high-resolution ultra-high-field diffusion magnetic resonance imaging (dMRI) to determine neuronal fiber orientation density functions (fODFs) throughout the human brain, including gray matter (GM), white matter (WM), and small intertwined structures in the cerebellopontine region.Materials and methodsWe acquired 7-T whole-brain dMRI data of 23 volunteers with 1.4-mm isotropic resolution; fODFs were estimated using constrained spherical deconvolution.ResultsHigh-resolution fODFs enabled a detailed view of the intravoxel distributions of fiber populations in the whole brain. In the brainstem region, the fODF of the extra- and intrapontine parts of the trigeminus could be resolved. Intrapontine trigeminal fiber populations were crossed in a network-like fashion by fiber populations of the surrounding cerebellopontine tracts. In cortical GM, additional evidence was found that in parts of primary somatosensory cortex, fODFs seem to be oriented less perpendicular to the cortical surface than in GM of motor, premotor, and secondary somatosensory cortices.ConclusionWith 7-T MRI being introduced into clinical routine, high-resolution dMRI and derived measures such as fODFs can serve to characterize fine-scale anatomic structures as a prerequisite to detecting pathologies in GM and small or intertwined WM tracts.

Functional Informed Fiber Tracking Using Combination of Diffusion and Functional MRI.

Fiber tractography using diffusion weighted MRI (DWI) is a primary tool for mapping structural connectivity in the human brain in vivo. However, this method suffers from a number of inherent limitations that have a significant impact on its capability in faithfully constructing fiber bundles for specific function. In this paper, a novel tractography algorithm combining DWI and functional MRI (fMRI) was proposed. Specifically, a spatio-temporal correlation tensor that characterizes the anisotropy of fMRI signals in white matter was introduced to complement the estimation of fiber orientation density function from DWI. The proposed method has been demonstrated to identify functional pathways implicated in fMRI task. It can effectively follow tracts in the genu of the corpus callosum that connects to the frontal lobe cortex, obtain connections between the thalamus and the anterior insula under sensory simulation, and reconstruct optic radiations in the visual circuit under visual stimulation. Taken together, the method we proposed in this work may benefit our understanding of structure-function relations in the human brain.

Building area extraction from the high spatial resolution remote sensing imagery

An approach to building area extraction from high-resolution remote sensing imagery is proposed based on the local gradient orientation density function (LGODF) and the bimodal density function (BDF). Firstly, the LGODF is calculated by moving the window of 35 × 35 based on the gradient magnitude and the gradient direction of each pixel in the image. Then, the BDF is obtained by multiplying the movable bimodal Gaussian mixture function and LGODF in each window. Finally, peaks with difference of 90° are searched for in the BDF, and the central point of the corresponding windows are determined as building pixels. For the validity of the proposed method, seven representative sub-images from PLEIADES images covering Shenzhen China are selected. Experimental results reveal that the precision achieve 94.08% and the recall up to 96.70%.

Combined texture and microstructure analysis of deformed crystals by high-energy X-ray diffraction

It is shown that high-energy X-ray diffraction allows a fast and accurate texture and microstructure analysis of crystals, which can help to set up optimal industrial procedures for materials manufacturing. This paper presents the experimental and theoretical aspects of quantitative texture analysis using high-energy synchrotron beams. Intensity corrections are less important in this approach than in classical laboratory methods; however, the most important correction, related to the Lorentz factor, can introduce relative fraction changes of up to about 40% compared to the uncorrected case. The resolution of the orientation density function also influences the results. For example, the usual 5° resolution leads to relative deviations of up to 30% in the fraction of some components. The method allowed detection of small changes taking place during the recovery and continuous recrystallization of a cold-rolled Al–TiB2 nanocomposite. Texture information was combined with the results of line profile analysis, evidencing the evolution of the average dislocation density and coherent domain size of the selected grain families. It was found that recovery, as described in terms of dislocation annihilation and coherent domain coarsening, takes place at similar rates in all components.