Showing papers on "Isotropy published in 1984"

Local isotropy and the decay of turbulence in a stratified fluid

Abstract: The validity of the assumption of local isotropy is investigated using measurements of three orthogonal components of the turbulent velocity fields associated with initially high-Reynolds-number geophysical turbulence The turbulent fields, generated by various large-scale internal motions caused by tidal flows over an estuarine sill, decay under the influence of stable mean density gradients With measurements from sensors mounted on a submersible, we examine the evolution of spectral shapes and of ratios of cross-stream to streamwise components, as well as the degree of high-wavenumber universality, for the observational range of the parameter I≡ ks/kb = lb/ls This ratio is a measure of separation between the Kolmogoroff wavenumber ks≡ (e/ν3)¼ ≡ 2π/ls typical of scales by which turbulent kinetic energy has been dissipated (at rate e), and the buoyancy wavenumber kb ≡ (N3/e)½ ≡ 2π/lb typical of scales at which the ambient stratification parameter N ≡ (−gρz/ρ0)½ becomes important For values of I larger than ∼ 3000, inertial subranges are observed in all spectra, and the spectral ratio ϕ22/ϕ11 of cross-stream to streamwise spectral densities reaches the isotropic value of 4/3 for about a decade in wavenumber As ks/kb decreases, inertial subranges vanish, but spectra of the cross-stream and streamwise components continue to satisfy isotropic relationships at dissipation wavenumbers We provide a criterion for when e may safely be estimated from a single measured component of the dissipation tensor, and also explore questions of appropriate low-wavenumber normalization for buoyancy-modified turbulence

Theory of nonlocal elasticity and some applications

Abstract: : Constitutive equations of finite nonlocal elasticity are obtained. Thermodynamic restriction are studied. The linear theory is given for anisotropic and isotropic solids. The physical and mathematical properties of the nonlocal elastic moduli are explored through lattice dynamics and dispersive wave propagations. The theory is applied to the problems of surface waves, screw dislocation and a crack. Excellent agreements with the results known in atomic lattice dynamics and experiments display the power and potential of the theory.

Seismic waves in stratified anisotropic media

Abstract: Summary. The response of a structure composed of anisotropic strata can be built up from the reflection and transmission properties of individual interfaces using a slightly modified version of the recursion scheme of Kennett. This scheme is conveniently described in terms of scatterer operators and scatterer products. The effects of a free surface and the introduction of a simple point source at any depth can be accommodated in a manner directly analogous to the treatment for isotropic structures. As in the isotropic case the results so obtained are stable to arbitrary wavenumbers. For isotropic media, synthetic seismograms can be constructed by computing the structure response as a function of frequency and radial wavenumber, then performing the appropriate Fourier and Hankel transforms to obtain the wavefield in time-distance space. Such a scheme is convenient for any system with cylindrical symmetry (including transverse isotropy). Azimuthally anisotropic structures, however, do not display cylindrical symmetry; for these the transverse component of the wavenumber vector will, in general, be non-zero, with the result that phase, group, and energy velocities may all diverge. The problem is then much more conveniently addressed in Cartesian coordinates, with the frequency-wavenumber to time-distance transformation accomplished by 3-D Fourier transform.

Elasticity and Damping of Porous Materials and Impregnated Materials

Abstract: Earlier, a method was developed by the present author which predicts the elastic moduli of isotropic two-phase materials with arbitrary phase geometry. The primary scope of the present article is to demonstrate how this method works when used to predict elasticity (Young's modulus) and damping (loss tangent) of porous materials and impregnated materials made of elastic and/or viscoelastic Components. Simple equations are developed which are directly applicable to practical situations. Their qualities are tested successfully against a number of experimental data.

Spatial Smoothing on the Sphere

Abstract: The equivalence of taking an isotropic, moving, spatial average of a two-dimensional field on the sphere to multiplying the coefficients in its spherical harmonics representation with factors that depend only on the total wavenumber n is discussed. Equivalent spatial averaging operators for several such spectral filters are displayed.

Radiative transfer in a two-dimensional rectangular medium exposed to diffuse radiation

Abstract: The integral equation for radiative transfer in a two-dimensional rectangular scattering medium exposed to diffuse radiation is solved numerically by removing the singularity. This method yielded accurate results except at very large optical thicknesses. Graphical and tabular results for the source function, flux, and intensity are presented. The source function is also calculated using the first term of a Taylor series expansion. The Taylor series is fairly accurate for small optical thicknesses and columnar geometries. A method is presented for extending these results to the problem of a strongly anisotropic scattering phase function which is made up of a spike in the forward direction superimposed on an isotropic phase function.

THREE‐TERM TAYLOR SERIES FOR t2 ‐ x2‐CURVES OF P‐ AND S‐WAVES OVER LAYERED TRANSVERSELY ISOTROPIC GROUND*

Abstract: The arrival-time curve of a reflection from a horizontal interface, beneath a homogeneous isotropic layer, is a hyperbola in the x - t -domain If the subsurface is one-dimensionally inhomogeneous (horizontally layered), or if some or all of the layers are transversely isotropic with vertical axis of symmetry, the statement is no longer strictly true, though the arrival-time curves are still hyperbola-like In the case of transverse isotropy, however, classical interpretation of these curves fails Interval velocities calculated from t 2 - x 2 -curves do not always approximate vertical velocities and therefore cannot be used to calculate depths of reflectors To study the relationship between velocities calculated from t 2 - x 2 -curves and the true velocities of a transversely isotropic layer, we approximate t 2 - x 2 -curves over a vertically inhomogeneous transversely isotropic medium by a three-term Taylor series and calculate expressions for these terms as a function of the elastic parameters It is shown that both inhomogeneity and transverse isotropy affect slope and curvature of t 2 - x 2 -curves For P-waves the effect of transverse isotropy is that the t 2 - x 2 -curves are convex upwards; for SV-waves the curves are convex downwards For SH-waves transverse isotropy has no effect on curvature

Constitutive Model for (Geological) Materials

Abstract: A general procedure based on polynomial expansion of yield function in terms of invariants of the stress tensor is proposed in the context of associated plasticity for isotropic materials undergoing isotropic hardening. The procedure can be used to evolve one or more models for a material by using appropriate laboratory test results. One of the functions showing invariance at ultimate and a single function to describe continuous yield and ultimate yield behavior is investigated in detail. Based on comprehensive series of bests on cubical specimens for different (geological) materials, a hardening or growth function is defined in terms of the trajectory of plastic strain and the ratio of deviatoric to total plastic strain. The predictions of the proposed model are verified with respect to the observed results from tests with different stress paths. The model provides highly satisfactory predictions for both stress‐strain and volumetric strain responses from various stress paths. The proposed model shows po...

Anisotropy of X‐ray susceptibility and Bragg reflections in cubic crystals

Abstract: The previous theory of X-ray diffraction in crystals with anisotropic X-ray susceptibility [Dmitrienko (1983). Acta Cryst. A39, 29-35] is applied to cubic crystals. Such a theory is needed if the X-ray wavelengths are near the absorption edges because in this case the X-ray susceptibility may be anisotropic. The most general form of the spatially dependent tensor of X-ray susceptibility is obtained for all cubic space groups. This tensor is anisotropic at any point of a unit cell except those with cubic point symmetry (being averaged over a unit cell the tensor becomes isotropic providing the macro- scopic isotropy of cubic crystals). From the tensor of susceptibility the structure amplitudes and new extinction rules are derived for the glide-plane and screw-axis forbidden reflections (such reflections are forbidden if the susceptibility is isotropic). For example, the hhh forbidden reflections remain extinguished even if the anisotropy is taken into account. Further restrictions on the structure amplitudes of forbidden reflections are obtained with the natural assumption that the anisotropy of susceptibility is localized at the special atomic positions. The tensor form of the structure amplitudes of nonforbidden reflections is also discussed. The general methods are illustrated by their application to the A15 structure (space group Pm{\bar 3}n).

Formation of Singularities in Elastic Waves

Abstract: This paper deals with the radial solutions of the dynamic equations for an isotropic homogeneous hyperelastic medium. It is shown that nontrivial solutions “blow up” (cease to exist in the proper sense) after a finite time, if: (a) The equations satisfy a certain “genuine nonlinearity condition”. (b) The initial data have compact support and are “sufficiently small”.

Conservation laws in elasticity - II. Linear homogeneous isotropic elastostatics

Abstract: Application a l'elasticite statique lineaire, homogene isotrope. Determination des lois de conservation tridimensionnelles et bidimensionnelles

Inhomogeneous Plane Waves

Abstract: This paper deals with the propagation of elliptically polarised inhomogeneous, time-harmonic plane waves. Such waves arise in many areas. Examples include Rayleigh, Love and Stoneley waves in classical linear isotropic elasticity theory, gravity waves in ideal fluids, TE and TM waves in electromagnetism, and viscoelastic waves. For the most part even though the applications given here are in the theory of isotropic and anisotropic elastic bodies, it should be apparent that the results have application in other areas, such as electromagnetism. The purpose of the paper is to show how the theory of complex vectors, or “bivectors”as Hamilton and Gibbs called them, may be used to give results on the polarisations of inhomogeneous plane waves. Also, the use of bivectors leads to a simple direct formulation of the eigenvalue problem for these waves.

Anisotropic Plasticity of Plane-Isotropic Sheets

Abstract: A general equation is written for the plane-stress yield condition of orthotropic sheet material that is isotropic in its plane. A set of constitutive relations for planar-isotropic strain hardening is then formulated on the basis of the usual normality rule for strain rates, together with the assumption of a scalar relation between an effective stress and a work-equivalent effective strain. It is shown how particular laws proposed earlier for sheet-metal plasticity fit into the general scheme. Solutions to the problem of balanced biaxial stretching of a sheet containing a circular hole illustrate a wide range of effects produced by various realizations of the general constitutive relations.

Characterizations of optical surfaces by measurement of scattering distribution

Abstract: The micropolish of good quality optical surfaces can be characterized by measuring the scattered light distribution. Very often the surface defects are not isotropic but display preferred orientations that are translated into an anisotropy of the scattered light distribution. The total amount of light scattered by very high quality surfaces, coated or uncoated, scarcely exceeds a few hundred parts per million. Precise measurement of the distribution of the scattered light is always a task requiring great care and attention to detail. The apparatus is described. All the necessary degrees of freedom have been included so that the scattering may be completely analyzed. It is possible to make measurements out of the plane of incidence so that the complete spatial distribution of the scattered light can be obtained, whatever the angle of incidence of the primary beam. Thus to characterize the geometry of the system we use four fundamental parameters: the angle of incidence i, the two angles θ and ϕ that define the scattering direction, and the angle α that defines the orientation of the scattering surface in its own plane. Only two free parameters need exist because the surface roughness itself, which is the source of the scattered light, only depends on two variables. We have verified experimentally the validity of the relationships linking i, θ, ϕ, and α. In these relationships the expression for the intensity scattered in a particular direction (θ,ϕ) for an uncoated surface at angle of incidence i can be written in the form of the product of a coefficient, depending only on illumination and observation conditions, and of the 2-D Fourier transform of the autocorrelation functions of the surface roughness. Experimental measurements with uncoated surfaces of black glass have accorded with the theory. When the surfaces are coated with one or several layers the problem is more complicated, but it should be possible to derive information on the autocorrelation functions of each of the interfaces and the degree of correlation between them.

The effect of elastic anisotropy on contrast in the scanning acoustic microscope

Abstract: The contrast obtained with polycrystalline specimens in the scanning acoustic microscope is computed by taking account of the anisotropic stiffness tensor. The resulting variation of signal with defocus is fundamentally different from that which could be obtained with an isotropic material regardless of the values of the elastic properties. In many cases this results largely from the excitation of pseudo-surface waves which effect the response of the microscope in a similar manner to ‘leaky’ Rayleigh waves. The reasons why some materials fail to give good acoustic images of grain structure are discussed.