Showing papers on "Isotropy published in 1979"

Solutions for effective shear properties in three phase sphere and cylinder models

Abstract: S olutions are presented for the effective shear modulus of two types of composite material models. The first type is that of a macroscopically isotropic composite medium containing spherical inclusions. The corresponding model employed is that involving three phases: the spherical inclusion, a spherical annulus of matrix material and an outer region of equivalent homogeneous material of unlimited extent. The corresponding two-dimensional, polar model is used to represent a transversely isotropic, fiber reinforced medium. In the latter case only the transverse effective shear modulus is obtained. The relative volumes of the inclusion phase to the matrix annulus phase in the three phase models are taken to be the given volume fractions of the inclusion phases in the composite materials at large. The results are found to differ from those of the well-known Kerner and Hermans formulae for the same models. The latter works are now understood to violate a continuity condition at the matrix to equivalent homogeneous medium interface. The present results are compared extensively with results from other related models. Conditions of linear elasticity are assumed.

Analysis of Properties of Fiber Composites With Anisotropic Constituents

Abstract: Expressions and bounds for the five effective elastic moduli of a unidirectional fiber composite, consisting of transversely isotropic fibers and matrix, are derived on the basis of analogies between isotropic and transversely isotropic elasticity equations. Application of results for determination of the five elastic moduli of graphite fibers is discussed. Effective thermal expansion coefficients are derived on the basis of a general theorem. Effective conductivities, dielectric constants, and magnetic permeabilities are derived by use of certain mathematical analogies.

Theory of Systems of Rodlike Particles: I. Athermal systems

Abstract: An improved treatment of the lattice model proposed in 1956 is presented for a system of hard rods, with axial ratio x, dispersed in a diluent. Interparticle forces, apart from repulsions on contact, are deliberately ignored within the scope of the present paper. The equilibrium distribution of orientations of the rodlike particles in a phase with nematic order is derived and explicitly taken into account. Thermodynamic properties in the asymptotic limit of high degree of order reduce approximately. to those obtained previously. Calculations carried out for equilibrium between anistropic (nematic) and isotropic phases yield ν′x/νx = 1.4653 for the ratio of volume fractions in the respective phases in the limit x → ∞. The product xνx, expressing the combined covolume of solute species in the isotropic phase at coexistence is 7.89 in the same limit. The calculated critical value of the axial ratio for coexistence of the two phases in the neat liquid is xcrit, = 6.417. Results are compared with thos...

Long-wave elastic anisotropy in transversely isotropic media

Abstract: Compressional waves in horizontally layered media exhibit very weak long‐wave anisotropy for short offset seismic data within the physically relevant range of parameters. Shear waves have much stronger anisotropic behavior. Our results generalize the analogous results of Krey and Helbig (1956) in several respects: (1) The inequality (c11-c44)(c33-c44)⩾(c13+c44)2 derived by Postma (1955) for periodic isotropic, two‐layered media is shown to be valid for any homogeneous, transversely isotropic medium; (2) a general perturbation scheme for analyzing the angular dependence of the phase velocity is formulated and readily yields Krey and Helbig’s results in limiting cases; and (3) the effects of relaxing the assumption of constant Poisson’s ratio σ are considered. The phase and group velocities for all three modes of elastic wave propagation are illustrated for typical layered media with (1) one‐quarter limestone and three‐quarters sandstone, (2) half‐limestone and half‐sandstone, and (3) three‐quarters limesto...

Quantum effects in the early universe. I. Influence of trace anomalies on homogeneous, isotropic, classical geometries

Abstract: The use of the effective-action method to calculate quantum effects in the early universe is described. An application is made to the calculation of the effect of one-loop contributions of conformally invariant matter fields on the evolution of homogeneous, isotropic, spatially flat classical geometries containing classical radiation.

Light scattering properties of spheroidal particles

Abstract: In the present paper, the light scattering characteristics of spheroidal particles are evaluated within the framework of a scattering theory developed for a homogeneous isotropic spheroid. This approach is shown to be well suited for computing the scattering quantities of spheroidal particles of fairly large sizes (up to a size parameter of 30). The effects of particle size, shape, index of refraction, and orientation on the scattering efficiency factors and the scattering intensity functions are studied and interpreted physically. It is shown that, in the case of oblique incidence, the scattering properties of a long slender prolate spheroid resemble those of an infinitely long circular cylinder.

Remarks on positive frequency and Hamiltonians in expanding universes.

Abstract: We examine critically the persistent belief that the physical interpretation of a quantum field theory in curved space-time is to be found by introducing a notion of particles at each time in terms of the exact eigenstates of some energylike operator at that time. It seems that all straightforward implementations of this principle suffer from either ambiguity or failure to produce finite particle densities and renormalizable energy-momentum tensors. Calculations are done for a cosmological model of Bianchi type I, containing a scalar fieldφ whose equation includes the term ξRφ. Diagonalization of thetotal energy suggests a certain particle definition, which, however, yields a finite particle creation rate only when ξ=1/6 and the universe is isotropic. Diagonalization of acanonical Hamiltonian is a prescription loose enough to be made consistent with almost any ansatz (including one that does yield physically acceptable expectation values), hence of no theoretical value. An apparently successful naive approach to the isotropic case (for any ξ) is explained as an unconscious first approximation to an adiabatic analysis of the notion of positive frequency.

An effective stress law for anisotropic elastic deformation

Abstract: An effective stress law is derived analytically to describe the effect of pore fluid pressure on the linearly elastic response of saturated porous rocks which exhibit anisotropy. For general anisotropy the difference between the effective stress and the applied stress is not hydrostatic. The effective stress law involves two constants for transversely isotropic response and three constants for orthotropic response; these constants can be expressed in terms of the moduli of the porous material and of the solid material. These expressions simplify considerably when the anisotropy is structural rather than intrinsic, i.e., in the case of an isotropic solid material with an anisotropic pore structure. In this case the effective stress law involves the solid or grain bulk modulus and two or three moduli of the porous material, for transverse isotropy and orthotropy, respectively. The law reduces, in the case of isotropic response, to that suggested by Geertsma (1957) and by Skempton (1961) and derived analytically by Nur and Byerlee (1971).

Anisotropic shear modulus due to stress anisotropy

Abstract: By special measurement technique it is shown that the dynamic shear modulus in homogenous isotropic sand samples depends upon the direction of stress. Furthermore it can be demonstrated that the shear modulus is not related to the first invariant of stress tensor as assumed hitherto, but to the respective stress component itself. Measurements were performed making use of the cross correlation method and stochastic shear wave excitation by a special shear wave exciter that avoids the generation of p waves. Polarized shear waves were generated in a cubical sand sample and the shear wave velocity was measured independent of the value and the direction of the stress in the sample. Due to the sensivity of the correlation method small vibration amplitudes could be applied, such that a linear stress-strain relation could be assumed. It was shown that the placing technique of the sand had no influence upon the test results.

Adiabatic contraction of anisotropic spheres in general relativity

Abstract: For specific equations of state we study the adiabatic contraction of a sphere with anisotropic stresses for different degrees of anisotropy. The equation of motion of particles in the matter is integrated numerically for different initial conditions. It is found that the local pressure-density law for equilibrium is different for different regions in the material. In some examples the models are more stable than the isotropic model; in others they are less stable.

Reflected irradiance indicatrices of natural surfaces and their effect on albedo

Abstract: The indicatrices of solar radiation reflected from characteristic natural surfaces were measured with a Nimbus Medium Resolution Radiometer (MRIR) 3 m above the ground. Results indicated that areas such as salt and alkali flats had only small deviations from isotropic reflections, while others such as sparsely vegetated areas had substantial deviations. The indicatrices were strongly dependent on the sun angle; thus a daily variation was found for most features. Typical indicatrices, normalized to nadir angle of zero degrees, are presented along with their impacts on measured albedo, which varies with solar angle. Our results can (1) improve surface albedo considerations using space-generated data, and (2) serve as a more realistic lower boundary condition for atmospheric transfer determinations based on space data.

Local isotropy and large structures in a heated turbulent jet

Abstract: Recent measurements of the skewness of the derivative of the temperature fluctuation θ, implying the breakdown of local isotropy even in high Reynolds number shear flows, are examined. Using the temperature signal in a slightly heated axisymmetric jet, a detailed quantitative analysis is made of the suggestion that the observed presence of a well-defined large-scale pattern of the temperature signal in these flows is responsible for this breakdown. A selective ensemble averaging technique is used for separating this pattern from fluctuations superposed on it. The technique is extended to extract the large-scale patterns in simultaneously measured axial (u), radial (v) velocity fluctuations, and the products uv, uθ and vθ, so that it is possible to separate contributions of these patterns from those of the superposed fluctuations to several important turbulent quantities. The mean shape of the patterns, their degree of anisotropy and correlation, and their contribution to turbulence intensities and Reynolds shear stress are obtained. Probability densities and spectra of these quasi-homogeneous superposed fluctuations are also obtained. Results show that the fluctuations are consistent with local isotropy and make the dominant contribution to the turbulence intensities, that the large-scale patterns are responsible for the observed skewness values of the derivative of v, and that the fluctuations may be responsible for a significant part of the turbulent momentum and heat transport, especially in the region of the jet where the turbulent energy production is substantial.

Temporal characteristics of single-scatter radiation

Abstract: An efficient and versatile model that describes the temporal characteristics of scattered radiation when single-scatter conditions prevail is proposed and developed. The model is based on the focal radii property of a prolate spheroid, and for an impulsive source located at one focal point and an observer at the other, it associates all scattering events occurring on a given prolate spheriodal surface with the same transit time. All interactions between the radiation and matter are classified as either elastic scattering or absorption, and expressions are obtained for the intensity and the collected energy as a function of time at the observation point. The model is applied to isotropic, Rayleigh, and Mie-type scattering; the single-scatter phase function for Mie-type scattering is approximated by the Henyey-Greenstein function. For simple source and observer geometries, closed-form expressions are obtained for the intensity in the isotropic and Rayleigh cases. Finally, the model is applied to examples which typify middle ultraviolet radiation propagating in the earth’s atmosphere.

Surface‐induced ordering of a liquid crystal in the isotropic phase

Abstract: A detailed account of a measurement of order parameter of a liquid crystal at the boundary by means of the wall‐induced pretransitional birefringence is given. Several surface treatments were studied including surfactants and evaporated films. Although all treatments produced good alignment in the nematic phase, the boundary order parameter (hence the strength of the aligning force) in the isotropic phase differed very much depending on the treatment, indicating the diverse nature of the alignment process.

Discontinuous Deformation Gradients in Plane Finite Elastostatics of Incompressible Materials. (I) General Considerations. (II) An Example

Abstract: This investigation is concerned with the possibility of the change of type of the differential equations governing finite plane elastostatics for incompressible elastic materials, and the related is sue of the existence of equilibrium fields with discontinuous deformation gradients. Explicit necessary and sufficient conditions on the deformation invariants and the material for the ellipticity of the plane displacement equations of equilibrium are established. The issue of the existence, locally, of "elastostatic shocks" -- elastostatic fields with continuous displacements and discontinuous deformation gradients -- is then investigated. It is shown that an elastostatic shock exists only if the governing field equations suffer a loss of ellipticity at some deformation. Conversely, if the governing field equations have lost ellipticity at a given deformation at some point, an elastostatic shock can exist, locally, at that point. The results obtained are valid for an arbitrary homogeneous, isotropic, incompressible, elastic material. In order to illustrate the occurrence of elastostatic shocks in a physical problem, a specific displacement boundary value problem is studied. Here, a particular class of isotropic, incompressible, elastic materials which allow for a loss of ellipticity is considered. It is shown that no solution which is smooth in the classical sense exists to this problem for certain ranges of the applied loading. Next, we admit solutions involving elastostatic shocks into the discussion and find that the problem may then be solved completely. When this is done, however, there results a lack of uniqueness of solutions to the boundary value problem. In order to resolve this non-uniqueness, dissipativity and stability are investigated.

Reflection and transmission coefficients for seismic waves in ellipsoidally anisotropic media

Abstract: The deficiency of an isotropic model of the earth in the explanation of observed traveltime phenomena has led to the mathematical investigation of elastic wave propagation in anisotropic media. A type of anisotropy dealt with in the literature (Potsma, 1955; Cerveny and Psencik, 1972; and Vlaar, 1968) is uniaxial anisotropy or transverse isotropy. A special case of transverse isotropy which assumes the wavefronts to be ellipsoids of revolution has been used by Cholet and Richard (1954) and Richards (1960) in accounting for the observed traveltimes at Berraine in the Sahara and in the foothills of Western Canada. The kinematics of this problem have been treated in a number of papers, the most notable being Gassmann (1964). However, to appreciate fully the effect of anisotropy, the dynamics of the problem must be explored. Assuming a model of the earth consisting of plane transversely isotropic layers with the axes of anisotropy perpendicular to the interfaces, a prime requisite for obtaining amplitude dist...

Spatial correlation of phase-expansion coefficients for propagation through atmospheric turbulence

Abstract: Spatial correlation functions of phase-expansion coefficients are derived for phase fluctuations of a wave that has propagated through a layer of atmospheric turbulence. First, a general expression is derived giving the correlation of the coefficients of phase-expansion functions orthogonal over an arbitrary circularly symmetric weighting function for an isotropic turbulence spectrum. Second, the correlations are evaluated analytically for a Gaussian weight and numerically for a uniform circular weight. Finally, some of the correlations for a layer are integrated over the Hufnagel 1974 structure constant model to obtain results applicable to ground-to-space propagation.

On electromagneto-thermoelastic plane waves

Abstract: The propagation of harmonically time-dependent electromagneto-thermoelastic plane waves of assigned frequency in an unbounded, homogeneous, isotropic, elastic, thermally and electrically conducting medium is considered. The theory of thermoelasticity recently proposed by Green and Lindsay is used to account for the interactions between the elastic and thermal fields. The results pertaining to phase velocity and attenuation coefficient of various types of waves are compared with those of Nayfeh and Nemat-Nasser who have dealt with a theory of thermoelasticity having a thermal relaxation time.

Self‐consistent imbedding and the ellipsoidal model for porous rocks

Abstract: Equations are obtained for the effective elastic moduli for a model of an isotropic, heterogeneous, porous medium. The mathematical model used for computation is abstract in that it is not simply a rigorous computation for a composite medium of some idealized geometry, although the computation contains individual steps which are just that. Both the solid part and the pore space are represented by ellipsoidal or spherical ‘grains’ or ‘pores’ of various sizes and shapes. The strain of each grain, caused by external forces applied to the medium, is calculated in a self-consistent imbedding (SCI) approximation, which replaces the true surrounding of any given grain or pore by an isotropic medium defined by the effective moduli to be computed. The ellipsoidal nature of the shapes allows us to use Eshelby's theoretical treatment of a single ellipsoidal inclusion in an infinite homogeneous medium. Results are compared with the literature, and discrepancies are found with all published accounts of this problem. Deviations from the work of Wu, of Walsh, and of O'Connell and Budiansky are attributed to a substitution made by these authors which, though an identity for the exact quantities involved, is only approximate in the SCI calculation. This reduces the validity of the equations to first-order effects only. Differences with the results of Kuster and Toksoz are attributed to the fact that the computation of these authors is not self consistent in the sense used here. A result seems to be the stiffening of the medium as if the pores are held apart. For spherical grains and pores, their calculated moduli are those given by the Hashin-Shtrikman upper bounds. Our calculation reproduces, in the case of spheres, an early result of Budiansky. An additional feature of our work is that the algebra is simpler than in earlier work. We also incorporate into the theory the possibility that fluid-filled pores are interconnected. This is achieved with the use of Gassmann's equation.

Two-dimensional plane wave theory of strong acousto-optic interaction in isotropic media

Abstract: Strong interaction is the preferred (high-efficiency) mode of operation in acousto-optic devices. Yet most theories of strong interaction use simplifying assumptions, such as sharply bounded sound columns, which are often unrealistic. The theory presented in this paper makes no such assumptions; it describes strong acousto-optic interaction as a multiple scattering of plane waves.