Principal Anisotropy Research Articles

Paraxial Fields of a Wedge with Anisotropic Impedance and Perfect Electric Conductor Faces Excited by a Dipole

A three-dimensional dyadic Green's function is derived for electromagnetic scattering due to a point electric current source radiating in the vicinity of a wedge with anisotropic impedance and perfect electric conductor faces. The anisotropic impedance face is characterized by surface impedances along directions parallel and orthogonal to the edge axis (principal anisotropy axis). Arbitrary surface impedance is assumed in one direction only (either parallel or orthogonal to principal anisotropy axis), and vanishing impedance is assumed on the other. Assuming reactive surface impedance, the derived forms involve a summation over an angular wavenumber and a longitudinal spectral integral, which may be evaluated asymptotically to describe the fields in paraxial region, where the source and observation points are in close proximity to the apex but widely separated. The final asymptotic result reveals three different scattering mechanisms: edge-guided waves, surface waves, and guided waves in the classical sense. Numerical simulations are performed for capacitive and inductive surfaces using the asymptotic results. Simulation results are also compared to those of a commercially available electromagnetic field solver, which assumes uniform surface impedance.

On the Equivalent In-Plane Permeability

Simulation of RTM processes is usually performed in 2D. Thus, only a mesh of the middle plane is needed with the associated degrees of freedom savings with respect to fully 3D modeling. However such modeling needs the definition of an equivalent in-plane permeability representing the ignored dimension (the thickness). The definition of such permeability is not a trivial task because each ply in the thickness direction can be anisotropic, being the principal anisotropy direction different from one ply to the neighbor plies. In this work we propose a novel fully 3D modeling whose computational cost is equivalent to a 2D solution. It allows addressing properly the equivalent in-plane permeability issue.

High‐frequency asymptotic solutions benchmarking skew incidence diffraction by anisotropic impedance half and full planes

The diffraction of plane waves obliquely incident on the edge of anisotropic impedance half and full planes is investigated. Homogeneous anisotropic impedance boundary conditions are defined on both faces of the canonical structures under study, the principal anisotropy axes being parallel and perpendicular to the edge. Rigorous integral representations for the longitudinal field components are derived by applying the Sommerfeld‐Maliuzhinets method for a set of specific electrical configurations. Explicit uniform asymptotic expressions for the fields are given in the format of the Uniform Geometrical Theory of Diffraction (UTD). Although obtained for a specific class of geometrical and electrical configurations, these high‐frequency solutions provide a contribution for investigating the effects of material anisotropy on edge diffraction, as they represent a new set of reference cases for either developing perturbative solutions or testing numerical and approximate analytical methods of more general validity. Furthermore, these solutions extend the applicability of the Sommerfeld‐Maliuzhinets technique and represent a step forward to the solution of more general wedge canonical problems.

Pore fabric shape anisotropy in porous sandstones and its relation to elastic wave velocity and permeability anisotropy under hydrostatic pressure

To understand the relationship between pore space anisotropy and petrophysical properties, we developed a novel apparatus capable of simultaneously measuring permeability, porosity and ultrasonic velocities at hydrostatic pressures up to 100 MPa. First, we use magnetic susceptibilities and acoustic wave velocities to identify the principal anisotropy axes under ambient laboratory conditions. This directional anisotropy data is then used to guide experiments on two sandstones (Bentheim and Crab Orchard) under hydrostatic pressure from 5 to 90 MPa. We find the structural anisotropy formed by the void space is well described by velocity anisotropy in both cases. Under hydrostatic pressure, the acoustic anisotropy of Crab Orchard sandstone (COS) decreases from 3% and 7% at 5 MPa (P-wave and S-wave) to 1.5% and 1%, respectively, at effective pressures over 40 MPa; for Bentheim sandstone the decrease is considerably less. Permeability of COS is 125×10 −18 m 2, decreasing rapidly as effective pressure increases, with permeability parallel to bedding approximately twice that normal to bedding. In contrast, permeability of Bentheim sandstone is 0.86×10 −12 m 2, and varies little with effective pressure or coring direction. We relate many of our measurements made under hydrostatic pressure to the contrasting pore fabric between the two rock types, and infer that a critical pressure is required for the initiation of crack closure.

Half-ordered state in the anisotropic Haldane-gap antiferromagnetNi(C5D14N2)2N3(PF6)

Neutron diffraction experiments performed on the Haldane gap material NDMAP in high magnetic fields applied at an angle to the principal anisotropy axes reveal two consecutive field-induced phase transitions. The low-field phase is the gapped Haldane state, while at high fields the system exhibits 3-dimensional long-range Neel order. In a peculiar phase found at intermediate fields only half of all the spin chains participate in the long-range ordering, while the other half remains disordered and gapped.

Electromagnetic Diffraction of a very Obliquely Incident Plane Wave Field by a Wedge with Anisotropic Impedance Faces

An asymptotic solution in the format of the uniform geometrical theory of diffraction (UTD) for the field diffracted by an anisotropic impedance wedge has been presented, when the wedge is illuminated by an arbitrarily polarized plane wave impinging at very skew incidence on its edge. The surface impedance tensor principal anisotropy axes are along directions parallel and perpendicular to the edge. The solution is derived by resorting to the perturbation technique with respect to the oblique incident angle based on an exactly solved special case: grazing to the edge incidence. The numerical examples provide a uniform and continuous behaviour of the field with respect to the observation angle.

Diffraction by two classes of anisotropic impedance half-planes with nonaxial anisotropy tensors

The three-dimensional plane wave scattering from anisotropic impedance half-planes having their principal anisotropy axes arbitrarily oriented with respect to the diffracting edge is analyzed. Two classes of exactly solvable half-plane configurations are identified and the corresponding rigorous analytical expressions for the total scattered field are derived in terms of a generalization of the Maliuzhinets special function. Then, explicit uniform asymptotic field expressions are provided along with samples of numerical results proving their validity.

Skew Incidence Diffraction by an Anisotropic Impedance Half Plane With a PEC Face and Arbitrarily Oriented Anisotropy Axes

High-frequency expressions for the field scattered by a half-plane with a perfectly conducting and an anisotropic impedance face are provided in the format of the uniform geometrical theory of diffraction (UTD), when the half-plane is illuminated by an arbitrarily polarized plane wave obliquely incident on its edge. The loaded face is characterized by a tensor surface impedance with principal anisotropy axes arbitrarily oriented with respect to the edge; a vanishing surface impedance is exhibited in one of the principal directions. This kind of tensor surface impedance can be suitably applied for analyzing the effects on the scattered field of corrugated surfaces or grounded dielectric slabs periodically loaded by metallic strips. This solution extends previous high-frequency formulations valid in those cases in which the direction of corrugations or strips is either parallel or perpendicular to the edge. The analysis is performed by resorting to the Sommerfeld-Maliuzhinets method. To determine the spectral solution, a special function is needed that differs from the standard Maliuzhinets one and was originally introduced to study the electromagnetic scattering by a wedge embedded in a gyroelectric medium.

AN APPROXIMATE SOLUTION FOR SKEW INCIDENCE DIFFRACTION BY AN INTERIOR RIGHT-ANGLED ANISOTROPIC IMPEDANCE WEDGE

The scattering by an anisotropic impedance interior rightangled wedge is analyzed when the principal anisotropy directions on the two faces are parallel and perpendicular to the edge. The problem is first approached by directly applying geometrical optics (GO); this allows us to identify the conditions under which the edge diffracted contribution vanishes. For those configurations not satisfying the above conditions, a perturbative technique, based on the Sommerfeld-Maliuzhinets method, is developed to determine an approximate edge diffracted field solution, valid when the normalized surface impedances on the anisotropic faces assume small values. The perturbative corrections to the field are asymptotically evaluated in the context of the Uniform Geometrical Theory of Diffraction (UTD).

Influence of the Direction of the Principal Anisotropy Axes in a Rectilinearly Orthotropic Material on the Stress State of a Compound Solid of Revolution Subject to Nonaxisymmetric Heating

A technique for analysis of the temperature fields and the stress state of isotropic and orthotropic laminated bodies of revolution under nonaxisymmetric loading is described. The influence of the direction of the principal anisotropy axes in a rectilinearly orthotropic material on the stress state of a three-layer body of revolution under nonaxisymmetric loading is studied

Estimation of fracture parameters from reflection seismic data—Part III: Fractured models with monoclinic symmetry

Geophysical and geological data acquired over naturally fractured reservoirs often reveal the presence of multiple vertical fracture sets. Here, we discuss modeling and inversion of the effective anisotropic parameters of two types of fractured media with monoclinic symmetry. The first model is formed by two different nonorthogonal sets of rotationally invariant vertical fractures in an isotropic host rock; the other contains a single set of fractures with microcorrugated faces. In monoclinic media with two fracture sets, the shear‐wave polarizations at vertical incidence and the orientation of the NMO ellipses of pure modes in a horizontal layer are controlled by the fracture azimuths as well as by their compliances. While theS-wave polarization directions depend only on the tangential compliances, the axes of the P-wave NMO ellipse are also influenced by the normal compliances and therefore have a different orientation. This yields an apparent discrepancy between the principal anisotropy directions obtained using P and S data that does not exist in orthorhombic media. By first using the weak‐anisotropy approximation for the effective anisotropic parameters and then inverting the exact equations, we devise a complete fracture characterization procedure based on the vertical velocities of theP- and two split S-waves (or converted PS-waves) and their NMO ellipses from a horizontal reflector. Our algorithm yields the azimuths and compliances of both fracture systems as well as the P- and S-wave velocities in the isotropic background medium. In the model with a single set of microcorrugated fractures, monoclinic symmetry stems from the coupling between the normal and tangential (to the fracture faces) slips, or jumps in displacement. We demonstrate that for this model the shear‐wave splitting coefficient at vertical incidence varies with the fluid content of the fractures. Although conventional fracture models that ignore microcorrugation predict no such dependence, our conclusions are supported by experimental observations showing that shear‐wave splitting for dry cracks may be substantially greater than that for fluid‐filled ones.

High-frequency EM scattering by edges in artificially hard and soft surfaces illuminated at oblique incidence

Uniform high-frequency expressions describing the field scattered by edges in anisotropic impedance surfaces illuminated at oblique incidence are provided. The specific anisotropic impedance boundary condition considered here exhibits a vanishing surface impedance along a principal anisotropy axis and an arbitrary one in the orthogonal direction. In certain circumstances, this tensor surface impedance may represent an accurate model for describing the scattering properties of artificially hard and soft surfaces. In order to simplify the analysis but without losing pertinence with real problems, in all canonical configurations we consider a face of the wedge to be perfectly conducting. The anisotropic impedance face is characterized by a tensor surface impedance with the principal anisotropy axes parallel and perpendicular to the edge.

Research on dependence of resistivity changing anisotropy on microcracks extending in rock with experiment

Multiple electrodes are arranged on the surfaces of cubic granite samples of different sizes according to pre-designed patterns. Samples are fully saturated with water in vacuum. Waterproof insulation glue is coated on the measuring surface and a small parts (including two arrises) of the adjacent surfaces of the saturated sample to ensure that the electric current flows only within the sample through the connection between the electrodes. The multiple electrodes are combined form arrays of different direction and spacing with symmetrical four-electrode method according to need of measuring of resistivity changing anisotropy, electric profiling and electric sounding. The samples are placed into container filled with water. The samples are uniaxially compressed along the direction parallel to the longest dimension of the cubic, and the variation of resistivity during the whole loading process is observed. In the experiments, some samples are loaded to rupture with macro-fractures, some are only loaded to the stage, which shows obvious precursors in variation of resistivity associated with the indication of forthcoming rupture. Finally a quantitative comparison between the dominant orientation of pre-existing cracks in photo-micrography of unruptured samples and those macro-fractures in ruptured sample is made, together with their respective resistivity changing anisotropy behaviors. The experimental results are the following: (1) For measuring points in areas that are passed by cracks or rupture bands, the directions of principal anisotropy axes deduced from four kinds of combined equation sets are essentially identical with each other, and accord with the orientation of cracks or main rupture bands approximately. For measuring points in area without crack or rupture band passing through, either the directions of calculated principal anisotropy axes by different combinatorial arrays are inconsistent with each other, or the principal anisotropy axis cannot be determined, especially in the cases where the crack plane is parallel to the measuring surface. (2) The dominant orientation of microfractures or rupture bands shown from micrographs is close to the direction of principal anisotropy axis along which the variation in resistivity is the greatest. (3) The results of electric profiling can be used for detecting the localization of cracks.

Electromagnetic diffraction of an obliquely incident plane wave by a right-angled anisotropic impedance wedge with a perfectly conducting face

The diffraction of an arbitrarily polarized electromagnetic plane wave obliquely incident on the edge of a right-angled anisotropic impedance wedge with a perfectly conducting face is analyzed. The impedance tensor on the loaded face has its principal anisotropy axes along directions parallel and perpendicular to the edge, exhibiting arbitrary surface impedance values in these directions. The proposed solution procedure applies both to the exterior and the interior right-angled wedges. The rigorous spectral solution for the field components parallel to the edge is determined through the application of the Sommerfeld-Maliuzhinets technique. A uniform asymptotic solution is provided in the framework of the uniform geometrical theory of diffraction (UTD). The diffracted field is expressed in a simple closed form involving ratios of trigonometric functions and the UTD transition function. Samples of numerical results are presented to demonstrate the effectiveness of the asymptotic expressions proposed and to show that they contain as limit cases all previous three-dimensional (3-D) solutions for the right-angled impedance wedge with a perfectly conducting face.

EM scattering from anisotropic impedance half-plane

A rigorous spectral solution for three-dimensional (3D) electromagnetic scattering by the edge of an anisotropic impedance half-plane with a perfect electric conducting (PEC) face is presented. The surface impedance tensor of the loaded face is characterised by: (i) principal anisotropy axes arbitrarily oriented with respect to the diffracting edge; (ii) a vanishing surface impedance along a principal anisotropy axis and an arbitrary impedance in the orthogonal direction.

EM scattering by metallic half plane loaded by anisotropic face with arbitrarily oriented anisotropy axes

Electromagnetic scattering by the edge of an anisotropic impedance half plane with a perfect electric conducting (PEC) face is analysed. Although only valid at normal incidence, the solution has some interesting features with respect to previous analyses presented for anisotropic impedance canonical configurations. Indeed, it allows the investigation of the case in which the principal anisotropy axes on the loaded face are arbitrarily oriented with respect to the diffracting edge. The specific tensor surface impedance considered exhibits a vanishing surface impedance along a principal anisotropy axis and an arbitrary one in the orthogonal direction.

Studies of the magnetic behaviour of ErAl 3(BO 3) 4 and the effects of the crystal field

Single crystals of ErAl 3(BO 3) 4 were produced by flux growing techniques. The principal magnetic susceptibility, χ ⊥, perpendicular to the symmetry axis of the crystal was measured in the range 300-14.2 K, and the principal anisotropy ( χ ∥ − χ ⊥) was determined from 300 to 77 K, which turned out to be negative throughout. A crystal field conforming to D 3 site symmetry (obtained from structural data) accounts very well for the observed thermal variation of the magnetic susceptibilities; this yields g factors, g ∥ and g ⊥, of 2.32 and 8.95, respectively. The electronic heat capacity is computed, and shows a Schottky anomaly at around 45 K. Calculated quadrupole splitting of 167Er due to crystal field effects shows a smooth decrease with decreasing temperature. The hyperfine heat capacity, contributed mainly by magnetic hyperfine splitting, displays a rounded maximum at about 8 mK.

EM Scattering from Anisotropic Impedance Wedges at Oblique Incidence: Application to Artificially Hard and Soft Surfaces

ABSTRACT Three-dimensional electromagnetic scattering of plane waves from a class of anisotropic impedance wedges is analyzed. All the configurations considered are characterized by anisotropic impedance faces with their principal anisotropy axes parallel and perpendicular to the edge, with one of the two elements characterizing the impedance tensors vanishing or diverging. Rigorous integral representations for the field are determined by resorting to the Maliuzhinets’ method; they are defined along the Sommerfeld's integration path, so that are suitable for their asymptotic evaluation. Although valid only for specific configurations, we note that the new rigorous solutions can be used for testing the accuracy of numerical or approximate analytical methods, applicable to more general cases. The particular category of impedance tensors considered for the wedge faces can be usefully applied for analyzing the specific case of the artificially hard and soft surfaces.

Magnetic behavior of Ho 3+ in HoAl 3(BO 3) 4

Single crystals of HoAl 3(BO 3) 4 were flux grown and their principal magnetic susceptibilities, parallel and perpendicular to the symmetry axis of the crystal, χ ‖ and χ ⊥, measured in the range 300-14.5 K. The principal anisotropy χ ‖- χ ⊥ turns out to be positive throughout. A theoretical analysis of the experimental results invoking the crystal field interaction of the Ho 3+ ion with D 3 site symmetry, obtained from structural data, provide an excellent explanation of the observed behavior of the ion over the entire temperature range in our study; this rules out exchange or dipolar effects of any significance down to 14.5 K. The energy level structure of this potential laser crystal for all the low lying multiplets of the ground term is given and the anomalies in the electronic heat capacity located. The quadrupole splitting of 165Ho due to crystal field effects has been computed, which gives rise to a peak in the hyperfine heat capacity at about 5.5 mK.

Magnetocrystalline Anisotropy of Pure and Doped Hematite

The magnetocrystaline anisotropy of synthetic single crystals of hematite ($\ensuremath{\alpha}\ensuremath{-}{\mathrm{Fe}}_{2}{\mathrm{O}}_{3}$), both pure and doped with ${\mathrm{Ga}}^{3+}$, ${\mathrm{Al}}^{3+}$, and ${\mathrm{Ti}}^{4+}$, has been measured. The principal anisotropy constant $K_{1}^{}{}_{}{}^{\ensuremath{'}}$ has been determined below the low-temperature (Morin) transition by spin-flop measurements. An analysis of these data that considers the origins of the anisotropy has permitted the prediction of the transition temperature. Although the transition temperature is a sensitive function of doping, the predicted values are in excellent agreement with those observed experimentally. The magnetocrystalline anisotropy in the basal plane above the transition temperature has been studied with ferromagnetic-resonance techniques. The in-plane anisotropy is observed to be remarkably sensitive to stress but only slightly dependent on doping.