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## Orientation Distribution Function (orientation + distribution_function)
## Selected Abstracts## Thermodynamic consistent modelling of defects and microstructures in ferroelectrics GAMM - MITTEILUNGEN, Issue 2 2008Ralf MüllerAbstract The paper describes the main phenomena associated with fatigue in ferroelectricmaterials due to defects and microstructural effects. An analysis the modelling on different length scales is presented. Starting from a thermodynamic analysis of the macroscopic material behavior other microscopic aspects are addressed. The introduction of an orientation distribution function allows for a computationally efficient extension of a single crystal model to realistic 3D structures. Additionally, the thermodynamic treatment of defects and domain wall motion is discussed to provide a better understanding of various micro-mechanisms. It is explained by the concept of configurational/driving forces, how defects influence each other and how the mobility of domain walls is reduced in the presence of defects. (© 2008 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] ## Seismic anisotropy of shales GEOPHYSICAL PROSPECTING, Issue 5 2005C.M. SayersABSTRACT Shales are a major component of sedimentary basins, and they play a decisive role in fluid flow and seismic-wave propagation because of their low permeability and anisotropic microstructure. Shale anisotropy needs to be quantified to obtain reliable information on reservoir fluid, lithology and pore pressure from seismic data, and to understand time-to-depth conversion errors and non-hyperbolic moveout. A single anisotropy parameter, Thomsen's , parameter, is sufficient to explain the difference between the small-offset normal-moveout velocity and vertical velocity, and to interpret the small-offset AVO response. The sign of this parameter is poorly understood, with both positive and negative values having been reported in the literature. , is sensitive to the compliance of the contact regions between clay particles and to the degree of disorder in the orientation of clay particles. If the ratio of the normal to shear compliance of the contact regions exceeds a critical value, the presence of these regions acts to increase ,, and a change in the sign of ,, from the negative values characteristic of clay minerals to the positive values commonly reported for shales, may occur. Misalignment of the clay particles can also lead to a positive value of ,. For transverse isotropy, the elastic anisotropy parameters can be written in terms of the coefficients W200 and W400 in an expansion of the clay-particle orientation distribution function in generalized Legendre functions. For a given value of W200, decreasing W400 leads to an increase in ,, while for fixed W400, , increases with increasing W200. Perfect alignment of clay particles with normals along the symmetry axis corresponds to the maximum values of W200 and W400, given by and . A comparison of the predictions of the theory with laboratory measurements shows that most shales lie in a region of the (W200, W400)-plane defined by W400/W200,Wmax400/Wmax200. [source] ## An anisotropic strength criterion for jointed rock masses and its application in wellbore stability analyses INTERNATIONAL JOURNAL FOR NUMERICAL AND ANALYTICAL METHODS IN GEOMECHANICS, Issue 6 2008X. ChenAbstract In this paper, an anisotropic strength criterion is established for jointed rock masses. An orientation distribution function (ODF) of joint connectivity, is introduced to characterize the anisotropic strength of jointed rock masses related to directional distributed joint sets. Coulomb failure condition is formulated for each plane of jointed rock masses by joint connectivity, where the friction coefficient and cohesion of the jointed rock mass are related to those of the intact rock and joint and become orientation dependent. When approximating joint connectivity by its second-order fabric tensor, an anisotropic strength criterion is derived through an approximate analytical solution to the critical plane problem. To demonstrate the effects of joint distribution on the anisotropic strength of jointed rock masses, the failure envelopes are worked out for different relative orientations of material anisotropy and principal stress axes. The anisotropic strength criterion is also applied to wellbore stability analyses. It is shown that a borehole drilled in the direction of the maximum principal in situ stress is not always the safest due to the anisotropic strength of the jointed rock mass. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## Synchrotron texture analysis with area detectors JOURNAL OF APPLIED CRYSTALLOGRAPHY, Issue 4 2003H.-R. WenkThe wide availability of X-ray area detectors provides an opportunity for using synchrotron radiation based X-ray diffraction for the determination of preferred crystallite orientation in polycrystalline materials. These measurements are very fast compared to other techniques. Texture is immediately recognized as intensity variations along Debye rings in diffraction images, yet in many cases this information is not used because the quantitative treatment of texture information has not yet been developed into a standard technique. In special cases it is possible to interpret the texture information contained in these intensity variations intuitively. However, diffraction studies focused on the effects of texture on materials properties often require the full orientation distribution function (ODF) which can be obtained from spherical tomography analysis. In cases of high crystal symmetry (cubic and hexagonal) an approximation to the full ODF can be reconstructed from single diffraction images, as is demonstrated for textures in rolled copper and titanium sheets. Combined with area detectors, the reconstruction methods make the measurements fast enough to study orientation changes during phase transformations, recrystallization and deformation in situ, and even in real time, at a wide range of temperature and pressure conditions. The present work focuses on practical aspects of texture measurement and data processing procedures to make the latter available for the growing community of synchrotron users. It reviews previous applications and highlights some opportunities for synchrotron texture analysis based on case studies on different materials. [source] ## Regularized, fast, and robust analytical Q-ball imaging MAGNETIC RESONANCE IN MEDICINE, Issue 3 2007Maxime DescoteauxAbstract We propose a regularized, fast, and robust analytical solution for the Q-ball imaging (QBI) reconstruction of the orientation distribution function (ODF) together with its detailed validation and a discussion on its benefits over the state-of-the-art. Our analytical solution is achieved by modeling the raw high angular resolution diffusion imaging signal with a spherical harmonic basis that incorporates a regularization term based on the Laplace,Beltrami operator defined on the unit sphere. This leads to an elegant mathematical simplification of the Funk,Radon transform which approximates the ODF. We prove a new corollary of the Funk,Hecke theorem to obtain this simplification. Then, we show that the Laplace,Beltrami regularization is theoretically and practically better than Tikhonov regularization. At the cost of slightly reducing angular resolution, the Laplace,Beltrami regularization reduces ODF estimation errors and improves fiber detection while reducing angular error in the ODF maxima detected. Finally, a careful quantitative validation is performed against ground truth from synthetic data and against real data from a biological phantom and a human brain dataset. We show that our technique is also able to recover known fiber crossings in the human brain and provides the practical advantage of being up to 15 times faster than original numerical QBI method. Magn Reson Med 58:497,510, 2007. © 2007 Wiley-Liss, Inc. [source] ## The relationship of the hyperspherical harmonics to SO(3), SO(4) and orientation distribution functions ACTA CRYSTALLOGRAPHICA SECTION A, Issue 4 2009J. K. MasonThe expansion of an orientation distribution function as a linear combination of the hyperspherical harmonics suggests that the analysis of crystallographic orientation information may be performed entirely in the axis,angle parameterization. Practical implementation of this requires an understanding of the properties of the hyperspherical harmonics. An addition theorem for the hyperspherical harmonics and an explicit formula for the relevant irreducible representatives of SO(4) are provided. The addition theorem is useful for performing convolutions of orientation distribution functions, while the irreducible representatives enable the construction of symmetric hyperspherical harmonics consistent with the crystal and sample symmetries. [source] ## Elevating tensor rank increases anisotropy in brain areas associated with intra-voxel orientational heterogeneity (IVOH): a generalised DTI (GDTI) study NMR IN BIOMEDICINE, Issue 1 2008L. MinatiAbstract Rank-2 tensors are unable to represent multi-modal diffusion associated with intra-voxel orientational heterogeneity (IVOH), which occurs where axons are incoherently oriented, such as where bundles intersect or diverge. Under this condition, they are oblate or spheroidally shaped, resulting in artefactually low anisotropy, potentially masking reduced axonal density, myelinisation and integrity. Higher rank tensors can represent multi-modal diffusion, and suitable metrics such as generalised anisotropy (GA) and scaled entropy (SE) have been introduced. The effect of tensor rank was studied through simulations, and analysing high angular resolution diffusion imaging (HARDI) data from two volunteers, fit with rank-2, rank-4 and rank-6 tensors. The variation of GA and SE as a function of rank was investigated through difference maps and region of interest (ROI)-based comparisons. Results were correlated with orientation distribution functions (ODF) reconstructed with q-ball, and with colour-maps of the principal and second eigenvectors. Simulations revealed that rank-4 tensors are able to represent multi-modal diffusion, and that increasing rank further has a minor effect on measurements. IVOH was detected in subcortical regions of the corona radiata, along the superior longitudinal fasciculus, in the radiations of the genu of the corpus callosum, in peritrigonal white matter and along the inferior fronto-occipital and longitudinal fascicula. In these regions, elevating tensor rank increased anisotropy. This was also true for the corpus callosum, cingulum and anterior limb of the internal capsule, where increasing tensor rank resulted in patterns that, although mono-modal, were more anisotropic. In these regions the second eigenvector was coherently oriented. As rank-4 tensors have only 15 distinct elements, they can be determined without acquiring a large number of directions. By removing artefactual underestimation of anisotropy, their use may increase the sensitivity to pathological change. Copyright © 2007 John Wiley & Sons, Ltd. [source] ## The relationship of the hyperspherical harmonics to SO(3), SO(4) and orientation distribution functions ACTA CRYSTALLOGRAPHICA SECTION A, Issue 4 2009J. K. MasonThe expansion of an orientation distribution function as a linear combination of the hyperspherical harmonics suggests that the analysis of crystallographic orientation information may be performed entirely in the axis,angle parameterization. Practical implementation of this requires an understanding of the properties of the hyperspherical harmonics. An addition theorem for the hyperspherical harmonics and an explicit formula for the relevant irreducible representatives of SO(4) are provided. The addition theorem is useful for performing convolutions of orientation distribution functions, while the irreducible representatives enable the construction of symmetric hyperspherical harmonics consistent with the crystal and sample symmetries. [source] ## Simulation of polycrystalline ferroelectrics based on discrete orientation distribution functions PROCEEDINGS IN APPLIED MATHEMATICS & MECHANICS, Issue 1 2005Ingo KurzhöferFerroelectric materials exhibit a spontaneous polarization, which can be reversed by an applied electric field of sufficient magnitude. The resulting nonlinearities are expressed by characteristic dielectric and butterfly hysteresis loops. These effects are correlated to the structure of the crystal and especially to the axis of spontaneous polarization in case of single crystals. We start with a representative meso scale, where the domains consist of unit cells with equal spontaneous polarization. Each domain is modeled within a coordinate invariant formulation for an assumed transversely isotropic material as presented in [10], in this context see also [8]. In this investigation we obtain the macroscopic polycrystalline quantities via a simple homogenization procedure, where discrete orientation distribution functions are used to approximate the different domains. (© 2005 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim) [source] |