Principal Axes Of Anisotropy Research Articles

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.

Instability of an anisotropic power-law fluid in a basic state of plane flow

Initiation of cylindrical structures by buckling or necking in an anisotropic power-law fluid is treated for general plane flow. The principal axis of anisotropy, x′, in the stiffest direction in shortening or extension may be viewed as the trace of a foliation or lamination. Plane-flow constitutive relations between components of rate of deformation, D ′ x x and D ′ x y , and of deviatoric stress, s ′ x x and s ′ x y , for the fluid are D ′ x x = B ( Y ′ 2 ) [ ( n − 1 ) / 2 ] s ′ x x and D ′ x y = a 2 B ( Y ′ 2 ) [ ( n − 1 ) / 2 ] s ′ x y , where Y ′ 2 = ( s ′ x x ) 2 + a 2 ( s ′ x y ) 2 is an anisotropic invariant, a 2 is the anisotropy parameter, and n is the stress exponent. We determine the rate of amplification of wavelength components in the deflection of the foliation, θ, from a mean orientation parallel to x. Linearly independent, or non-interacting normal modes have a periodic, band-like form θ ( x , y ) ≅ ∂ ζ / ∂ x = − ( λ A ) sin [ λ ( x − v y ) ] , where ζ is the height of a foliation trace above its mean plane, v=tan β, where β is the angle between the normal to mean foliation and the axial surface, positive clockwise, and L=2π/ λ is the foliation-parallel wavelength. Evolution of a component may be followed through a finite bulk deformation provided θ remains ≪1. The growth rate of slope, λA, is independent of L. Components with axial plane normal to the foliation ( β=0) are strongly amplified in foliation-parallel shortening. If n>>1, internal necking (boudinage) occurs in foliation-parallel extension for components with axial plane inclined at a large angle to the foliation normal. In combined shortening and shear, the most rapidly growing component has an axial plane that dips steeply in the direction of shear. For n>1, maximum instability occurs for combined foliation-parallel shear and shortening rather than pure shortening. Weak instability is present in foliation-parallel shear. This anisotropic nonlinear fluid approximates the behavior of an isotropic power-law medium containing preferentially oriented but anastomosing slip surfaces, or that of a rock in which a stiffer component of lenticular form is embedded in a softer matrix.

Unit cell debonding analyses for arbitrary orientation of plastic anisotropy

Debonding of rigid inclusions embedded in the elastic–plastic aluminum alloy Al 2090-T3 is analyzed numerically using a unit cell model taking full account of finite strains. The cell is subjected to overall biaxial plane strain tension and periodical boundary conditions are applied to represent arbitrary orientations of plastic anisotropy. Plastic anisotropy is considered using two phenomenological anisotropic yield criteria, namely Hill [Proceedings of the Royal Society of London A 193 (1948) 281] and Barlat et al. [International Journal of Plasticity 7 (1991) 693]. For this material plastic anisotropy delays debonding compared to plastic isotropy except for the case of Hill’s yield function when the tensile directions coincided with the principal axes of anisotropy. For some inclinations of the principal axes of anisotropy relative to the tensile directions, the stress strain responses are identical but the deformation modes are mirror images of each other.

Anisotropic viscosity of a dispersion of aligned elliptical cylindrical clasts in viscous matrix

Self-consistent averaging, using an auxiliary solution for an elliptical anisotropic viscous inclusion in an anisotropic viscous host, provides estimates of the principal bulk viscosities of a dispersion of aligned elliptical viscous clasts in an isotropic or anisotropic viscous matrix. Analysis and results are for a two-dimensional analog of a composite rock with clasts that are cylinders with axes normal to the plane of flow. The ratio of principal viscosities, η n in clast parallel extension or shortening, η s in clast-parallel shear, m= η n/ η s, is smaller than that obtained using an auxiliary solution in which the host is isotropic. Results for the limiting case of rigid clasts indicates that the latter procedure overestimates the stress concentration in axis-parallel extension or shortening at intermediate clast volume fraction, f. If the matrix is anisotropic, bulk anisotropy derives from both the shape anisotropy and the intrinsic anisotropy of the matrix, and unsymmetrical relations for the principal viscosities and m( f) result. The results suggest that rheological anisotropy in rocks with a planar fabric is greatly reduced if the components are lenticular in form rather than continuous layers. A general solution is given for an elliptical inclusion for the case that the principal axes of anisotropy in both the host and the inclusion are oblique to the axes of the elliptical section and the host is subjected to homogeneous stress far from the inclusion.

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.

Wavefield forward modeling with the pseudo-spectral method in viscoelastic and azimuthally anisotropic media

When a seismic signal propagates through a realistic medium in the Earth, in add ition to anisotropy, the layers may show intrinsic viscoelastic behavior. Thus, an accurate description of wave propagation requires a rheology that accounts for the anisotropic and viscoelastic behavior of rocks. On this basis, the pseudo spectral recursive formulae of viscoelastic and azimuthally anisotropic media are derived. Wavefield forward modeling with the pseudo-spectral method was car ried out in viscoelastic and azimuthally anisotropic media. The numerical results from the pseudo-spectral method clearly show that seismic wavefield is closely related with the viscoelasticity and the azimuth of the principal axes of anis otropy.

Time Domain Version of the Uapo Solution for the Field Diffracted by an Anisotropic Impedance Half-Plane

This work deals with the time domain version of the Uniform Asymptotic Physical Optics solution for the field diffracted by the edge of a nonpenetrable half-plane characterized by an anisotropic impedance boundary condition. A transient plane wave is assumed obliquely incident on the scatterer in the more general case of principal axes of anisotropy not coincident with the edge. The time domain solution is obtained from the knowledge of the corresponding frequency domain expression in the hypothesis that the impedance tensor components along the principal axes of anisotropy do not depend on the frequency. An inverse Laplace's transform is used to this end. The response to an incident unit step function plane wave is explicitly considered.

Contact Problem for an Elastic Orthotropic Half-Strip

The paper discusses the problem on a rigid flat-based die forced into a half-strip. The material of the half-strip, whose semiinfinite sides are fixed, is orthotropic; the principal axes of anisotropy coincide with the Cartesian coordinate axes. Friction between the die and the half-strip is taken into account. The problem is solved by an asymptotic method developed by the authors. Some numerical results are presented

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

Macrodispersivity for transport in arbitrary nonuniform flow fields: Asymptotic and preasymptotic results

We use homogenization theory to investigate the asymptotic macrodispersion in arbitrary nonuniform velocity fields, which show small‐scale fluctuations. In the first part of the paper, a multiple‐scale expansion analysis is performed to study transport phenomena in the asymptotic limit ε ≪ 1, where ε represents the ratio between typical lengths of the small and large scale. In this limit the effects of small‐scale velocity fluctuations on the transport behavior are described by a macrodispersive term, and our analysis provides an additional local equation that allows calculating the macrodispersive tensor. For Darcian flow fields we show that the macrodispersivity is a fourth‐rank tensor. If dispersion/diffusion can be neglected, it depends only on the direction of the mean flow with respect to the principal axes of anisotropy of the medium. Hence the macrodispersivity represents a medium property. In the second part of the paper, we heuristically extend the theory to finite ε effects. Our results differ from those obtained in the common probabilistic approach employing ensemble averages. This demonstrates that standard ensemble averaging does not consistently account for finite scale effects: it tends to overestimate the dispersion coefficient in the single realization.

Revolution of Principal Axes of Plastic Anisotropy Developed during Deformation Process

The anisotropies of sheet metals originate from the texture developed by a cold rolling process. In the texture, the crystal grains assume a partial direction, and so the strength of the texture is different from that of the surrounding material. When the sheet metals are deformed with changes of the strain path, the difference in strength produces a new texture and defines a new anisotropic principal axis. Therefore, it is supposed that the principal axis of anisotropies does not behave like the material fiber printed on the sheet metals. To confirm the above assumption, a measuring method of revolution angles of principal axes is proposed by using a laser speckle surface strain meter. The revolution angle is measured by investigating the distribution of r-values with respect to the tensile directions. Finally, a dynamical revolution model of the principal axes is derived and its parameters are identified.

A direct method for the identification of the permeability field of an anisotropic porous medium

In a recent publication [Zhan and Yortsos, 2000] we proposed a direct method for the inversion of the permeability field of an isotropic porous medium based on the analysis of the displacement of a passive tracer. By monitoring the displacement front at successive time intervals (e.g., using a tomographic method), the permeability can be directly obtained from the solution of a nonlinear boundary value problem. In this paper we extend this approach to the case when the porous medium is anisotropic. When the principal axes of anisotropy are known and fixed, a procedure is proposed in which the tracer is injected two (or three) consecutive times along the two (or three) principal directions (for the case of a two‐dimensional (2‐D) (or three‐dimensional) problem). It is shown that the diagonal components can be obtained from the solution of two (or three) coupled boundary value problems involving the experimentally obtained fields of arrival times. Numerical examples show that the method works well when the permeability variation is not very sharp (e.g., for correlated distributions). When the permeability tensor is full and the principal axes vary in space, a procedure involving the injection in three different directions (for the case of a 2‐D problem) is also described.

SH wave in a transition layer of anisotropy

Shear wave, Horizontally polarized wave propagation in an anisotropic elastic medium with non-uniform orientation of the principal axis of anisotropy, which is a good mathematical model of an injection-molded short-fiber reinforced plastic (FRP), is investigated by means of the high frequency approximation. Variations of the phase, ray and power flow density of a plane SH wave are obtained in simple mathematical forms and are graphically illustrated. Wave front patterns are also constructed as an envelope of phase curves. As a result, major wave phenomena in the transition layer with the non-uniform orientation of the principal axis of anisotropy are clarified. The information obtained is found to contribute to further development of the ultrasonic inspection technique for detecting the fiber orientation in the FRP.

Identification of the permeability field of a porous medium from the injection of a passive tracer

We propose a method for the direct inversion of the permeability field of a porous medium from the analysis of the displacement of a passive tracer. By monitoring the displacement front at successive time intervals (for example, using a tomographic method), the permeability can be directly obtained from the solution of a nonlinear boundary-value problem. Well posedness requires knowledge of the pressure profile or the permeability at no-flow boundaries. The method is tested using synthetic data in two dimensions (2D) (and some 3D) geometries for a variety of heterogeneous fields and found to work well when the permeability contrast is not too large. However, it is sensitive to sharp variations in permeability. In the latter case, a modified approach based on the successive injection in both directions and the use of an optimization technique leads to improved estimates. The sensitivity to measurement errors is analyzed. An important feature of the direct method is that it also applies to anisotropic porous media. When the principal axes of anisotropy are known, a suitable procedure is proposed and demonstrated using synthetic data.

PIEZOTHERMOELASTIC ANALYSIS OF A CROSS-PLY LAMINATE CONSIDERING THE EFFECTS OF TRANSVERSE SHEAR AND COUPLING

In this study, we treat the behavior of a piezothermoelastic composite plate taking into account the effects of transverse shear and coupling among mechanical, thermal, and electrostatic fields. As an analytical model, we consider a rectangular laminate composed of several fiber-reinforced laminae and a piezoelectric layer. All the laminae are assumed to exhibit 2-mm symmetry and the principal axes of anisotropy to coincide with coordinate axes, that is, a cross-ply laminate is considered. We assume that the laminate is subjected to mechanical, thermal, and electrical loads with all edges simply supported and treat the problem as a coupled piezothermoelastic one. Numerical calculation is carried out to examine the effects of transverse shear and coupling on the behavior of the laminate. Moreover, the control of the deflection by applied voltage is examined.

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.

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.

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 byanisotropic 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.