Showing papers on "Isotropy published in 1976"

On a Lagrangean for classical field dynamics and renormalization group calculations of dynamical critical properties

Abstract: From the path probability density for nonlinear stochastic processes a Lagrangean for classical field dynamics is derived. This formulation provides a convenient approach to the mode coupling equations and the renormalization group theory of critical dynamics. An application is given for the time-dependent isotropic Heisenberg ferromagnet. The dynamical exponent $$z = \frac{{d + 2 - \eta }}{2}$$ is derived aboveT c for all dimensionsd>2.

Measurement of acoustoelastic and third-order elastic constants for rail steel

Abstract: Measurements of the stress‐induced changes in ultrasonic wave speeds in several samples of steels typically used in railroad rails are presented. All of the five possible relative changes in wave speeds for an initially isotropic material subjected to a uniaxial state of stress have been determined and agree to within the limits of accuracy of the measurement with the second‐order theory of Hughes and Kelly. The third‐order elastic constants are calculated from the acoustoelastic data.

On the failure of ellipticity of the equations for finite elastostatic plane strain

Abstract: In this paper we establish necessary and sufficient conditions, in terms of the local principal stretches, for ordinary and strong ellipticity of the equations governing finite plane equilibrium deformations of a compressible hyperelastic solid. The material under consideration is assumed to be homogeneous and isotropic, but its strain-energy density is otherwise unrestricted. We also determine the directions of the characteristic curves appropriate to plane elastostatic deformations that are accompanied by a failure of ellipticity.

Unified boundary for finite dynamic models

Abstract: The finite element analysis of dynamic problems in an infinite, isotropic medium is discussed. To simulate the physically infinite system by a finite model, an energy absorbing boundary is proposed. This boundary is frequency independent and proves to be very efficient in absorbing stress waves. The boundary constants are calculated for the particular cases of plane strain and axisymetry. /Author/TRRL/

On finite anti-plane shear for imcompressible elastic materials

Abstract: This paper is concerned with deformations corresponding to antiplane shear in finite elastostatics. The principal result is a necessary and sufficient condition for a homogeneous, isotropic, incompressible material to admit nontrivial states of anti-plane shear. The condition is given in terms of the strain energy density characteristic of the material and is illustrated by means of special examples.

On two methods of determination of particle size distribution functions by means of small-angle X-ray scattering

Abstract: The total X-ray intensity as a function of h (h is the radial coordinate in reciprocal space), scattered by an isotropic system of particles of equal shapes but of different sizes R, can, under certain conditions, be expressed as an integral over the particle size distribution function D(R), multiplied by a common single-particle function of hR which can be calculated from the assumed particle shape. In the first method D(R) is calculated from this relation by the method of least squares, in which values of D at a limited number of particle sizes are the unknowns. To avoid oscillations in the D curve, constraints are imposed on the D values. The proper weight to be assigned to these constraints must be determined by trial and error. The method has been adapted to suit various assumptions and requirements as to the shape of the particles, the type of distribution function to be calculated, and experimental conditions (slit or pinhole focusing). The second method is essentially the one described by Schmidt, Weil & Brill [X-ray & Electron Methods of Analysis, pp. 86–100. (1968), New York: Plenum], which, however, is adapted to the use of slit-smeared intensities. Both methods may give rise to artefacts in the calculated distribution functions in the range of the smallest particle sizes, which are sensitive to the setting of the various parameters and to experimental errors. However, the position and shape of the main maxima can usually be determined quite well. The agreement between the results obtained by the two methods is satisfactory.

Anisotropic Correlation Functions for Objective Analysis

Abstract: Covariance models used in the data assimilation step of operational forecasting generally assume isotropy of height field correlations on constant pressure levels. Because of the evidence that this assumption is a significant source of forecast error, especially In regions of low density data, a two-dimensional anisotropic correlation model has been derived. Using a simple autoregressive scheme, cumbersome extension of the modeling problem has been avoided and much of the direction-dependent variability of observed statistics is resolved. Compared to deviations of observed correlation values around the best fitting isotropic model, the residual variance has been reduced by 56&%.

Orthogonality relation for the Rayleigh-Lamb modes of vibration of a plate

Abstract: An orthogonality relation for the Rayleigh‐Lamb modes of vibration of an isotropic, homogeneous, elastic plate is derived.Subject Classification: [43]40.24.

Effect of solute size and shape on orientational order in liquid crystal systems

Abstract: Wide-line n.m.r. has been used to obtain the order parameter, S, as a function of temperature, for nematic p-methoxybenzylidine-p-n-butylaniline (MBBA) containing solutes of differing size and shape. These consisted of : (i) anisotropic molecules, capable of correlating their orientations with MBBA molecules, viz n-hexane, n-hexadecane and trans-decalin, and (ii) isomers of more isotropic shape, apparently incapable of orientational correlation, viz 2,2-dimethylbutane, 2,2,4,4,6,8,8-heptamethylnonane, and cis-decalin. With the same mole fraction, x2, of anisotropic solute, there is the same lowering of S while the lowering with the isotropic solutes increases with solute size. The parameter S is > 0.36 in the one-phase nematic region in the phase diagram and is equal to 0.36 along both the (T, x2)N line where the isotropic phase first appears on heating and the (T, x2)I line where the nematic phase appears on cooling, i.e., S= 0.36 through the two-phase region. For all systems, S is a single function of the ratio T/TN. Phase diagrams have been obtained for 21 MBBA + solute systems. Solutes include the anisotropic, orientation-correlating normal alkanes, n= 6, 8, 12, 16, 20, 24 and trans-decalin and a variety of more isotropic, non-correlating solutes : the highly branched alkane isomers of Cn, n= 6, 8, 12 and 16; cycloalkanes; cis-decalin; SnX4, where X = methyl, butyl, octyl and lauryl; Pb(butyl)4 and dimethylsiloxane oligomers. Identical (T, x2) phase diagrams are obtained for all n-alkane solutes, their effect on the nematic-isotropic transition being independent of their molecular dimensions. This implies that the molecular cross-sectional area is the relevant factor for these anisotropic solutes. For the isotropic solutes, in contrast, there is a strong dependence on solute size of the (T, x2)N and (T, x2)I lines. Values of the slopes of TN and TI against solute mole fraction are corrected to infinite solute dilution. The results are compared with predictions of lattice model, virial expansion and other theories. They are also used to give the difference of solute activity coefficients in the nematic and isotropic MBBA(γN∞2/γI∞2– 1). This quantity is independent of size for the normal alkane solutes but increases rapidly for the branched alkanes and other isotropic solutes.

Elastic and optical properties of stishovite

Abstract: The single-crystal elastic constants, the Raman and IR frequencies, the isotropic average of the elastic constants, and the first pressure derivatives of these quantities have been calculated for stishovite from a modified rigid ion model. In addition to Coulomb interaction, short-range repulsive interaction between first and second nearest neighbors and angle-bending forces for the oxygen-silicon-oxygen angles are included. The parameters of the model are determined from a least squares fit to the available zero-pressure elastic and optical data in the isostructural sequence SiO2, GeO2, and SnO2. The results for the isotropic bulk modulus and for the shear modulus are K = 3.145 Mbar and μ = 2.03 Mbar and for their pressure derivatives, (∂K/∂P) = 7.0 and (∂μ/∂P) = 1.13.

Mechanical Properties of Non-Metallic Thin Films

Abstract: The importance of mechanical properties of thin films has been established over the past several decades. In the first place, many properties are substantially modified in a condensed flm. The literature contains many examples (1) illustrating shifts in band-gap in semiconductors, transition temperature for super-conducting films, or expected magnetic anisotropy. Any property which itself is strain sensitive, may well be modified in a deposited film. Even a property which is originally isotropic may have lower symmetry when the effects of the strain are considered. The control of the stresses by changing deposition conditions is one of the goals now partially reached in the search to develop new properties.

Surface and interfacial waves of arbitrary form in isotropic elastic media

Abstract: A method due to Friedlander of accommodating disturbances of arbitrary form into the theory of surface waves in a semi-infinite isotropic elastic body is extended and shown to yield a simple closed form solution for the displacement field. An analogous treatment of interfacial waves of arbitrary form at a plane contact discontinuity separating different isotropic elastic materials is also given.

Local anisotropy in heated and cooled turbulent boundary layers

Abstract: Turbulent temperature as well as streamwise and vertical velocity components have been measured in relatively high Reynolds number (Rλ∼1100) heated and cooled boundary layers of the Air–Sea Interaction Simulation Tunnel at the Institut de Mecanique Statistique de la Turbulence. Spectra and probability distributions are compared with various consequences of Kolmogoroff’s assumption of local isotropy. The spectral functions present a first approximation agreement with the assumption. Yet, evidence of local anisotropy is shown by the values of the skewness of ∂T/∂x1 which are very different from zero, either positive or negative. Possible errors due to cold wire velocity sensitivity are examined. An apparent connection between small scale and large scale anisotropy is discussed.

Prediction of Buckling Loads Based on Experimentally Measured Initial Imperfections

Abstract: Correlation studies between experimental buckling loads and analytical predictions based on experimentally measured initial imperfections were carried out for axially compressed isotropic and stiffened cylindrical shells. By expanding the response of a cylindrical shell in truncated Fourier series, the nonlinear Donnell type shell equations for imperfect stiffened shells were reduced to a set of linear equations in the correction terms by Newton’s method of quasilinearization. Solutions were obtained for isotropic and for ring and stringer stiffened shells. The amplitudes of the initial imperfections used in the analysis were calculated from the corresponding Imbert-Donnell imperfection models. The free parameters in this imperfection model were obtained by least square fitting the harmonics of the experimentally measured initial imperfections. It was possible in all cases to achieve satisfactory correlation using only a few suitably chosen deflection and imperfection modes.

On a phase interchange relationship for composite materials

Abstract: A theorem exists relating the transverse conductivity of a fiber reinforced material in a determinate manner to the conductivity of the composite with the phase properties interchanged. It is shown that no such theorem can exist in the three‐dimensional case, e.g., for a statistically isotropic composite material. However, an inequality is established relating the two effective conductivities.

Body force equivalents for stress-drop seismic sources

Abstract: The equivalent body forces for a stress-drop seismic source are found. When the isotropic stress drop and one of the three principal stress drops are zero, then the equivalent body forces are the same double couple without moment which would result from a shear dislocation. In general however, all six stress-drop components must be specified as independent functions of time.

First order phase transition in the plane rotator ferromagnetic model in two dimensions

Abstract: We show that the two-dimensional isotropic ferromagnetic rotator model exhibits a first order phase transition if the interaction decays asr −α with 2<α<4

Erratum: 13C–{1H} nuclear Overhauser enhancement and 13C spin–lattice relaxation in molecules undergoing multiple internal rotations

Abstract: The effects of multiple internal rotations on the 13C–{1H} nuclear Overhauser enhancement (NOE) and 13C spin lattice relaxation has been considered. It is shown that the 5×5 matrices which arise from a treatment using the Wigner rotation matrices of order 2 can be replaced by 3×3 matrices by making use of the inherent symmetry of the problem without any loss of generality. The cross correlations which arise in chains of methylene carbons are also taken into account. Calculations are given for the NOE values for carbons in a methylene chain for several values of the internal diffusion coefficients Di, assumed constant along the chain, as a function of D0. It is found that the effect of cross correlations on the NOE values is relatively small except in the region where the isotropic diffusion coefficient of the end of the chain satisfies the condition 6D0∼ωc. In this region the effect of cross correlations depends on the internal diffusion coefficient Di, being insignificant for Di=109 sec−1 and somewhat mo...

Crossover behaviour and effective critical exponents in isotropic and anisotropic Heisenberg systems

Abstract: Renormalisation group methods are used to determine, to second order in epsilon =4-d, the scaling function describing the crossover from Gaussian to Heisenberg behaviour in the susceptibility of an isotropic n-component spin system. The results are used in conjunction with an earlier Feynman graph calculation to obtain an O( epsilon 2) representation of the n-to-m-component susceptibility crossover function, and the corresponding effective exponents, for an anisotropic n-component system.

Comments on the classical theory of energy transfer. II. Extension to higher multipoles and anisotropic media

Abstract: The classical electromagnetic theory of the fluorescence emission and energy transfer in layered systems is extended to describe magnetic dipole and electric quadrupole radiation and anisotropic media. A general formulation is developed for energy transfer from various emitter types and orientations to isotropic acceptors. The description is exact within the classical framework and requires none of the usual assumptions as to the nature of the acceptor layer. The theory is further extended to describe one‐ and two‐dimensional acceptors and electric dipole radiation in an anisotropic medium. The latter case is the actual situation in the fatty‐acid layer experiments. The coupling to the surface plasmon modes of the acceptor is discussed as it relates to the various systems described here.

Brillouin scattering from isotropic metals

Abstract: Brillouin scattering from isotropic metals is investigated experimentally and theoretically. The observation of highly asymmetric Brillouin spectra from liquid mercury and liquid gallium is reported. A comparison is made between theory and experiment; the theory includes the damping of the acoustic phonons. It is found that the present state of the theory only qualitatively describes the Brillouin scattering of light from highly opaque materials. The velocities of sound of mercury and gallium deduced from the measurements are found to be 38% larger than the corresponding values determined by ultrasonic techniques. The observed intensity is at least an order of magnitude larger than the theory predicts, and the line shape, especially in the case of liquid gallium, differs appreciably from the theoretical line shape. The major origin of the discrepancy between theory and experiment is believed to be the simple model employed for the description of the dynamics of the density fluctuations near the surface. The spectra observed may therefore contribute to further understanding of the surface physics of liquid metals.

Fifth-harmonic generation in isotropic media

Abstract: This paper presents a theoretical study of fifth-harmonic generation (FHG) in an isotropic media. Two schemes are discussed-cascade generation in which the fifth harmonic results from successive nonlinear interactions in two separate elements, and direct generation where the fifth harmonic is produced in an isotropic media with third-and fifth-order nonlinear susceptibilities. In the plane-wave approximation almost full conversion of the pumping energy into the fifth harmonic is found to be possible. FHG with a focused Gaussian beam is also investigated. To determine the optimum conditions for FHG in the cascade scheme, the theory of four-wave mixing of light beams with arbitrary confocal parameters and waist locations is developed. In media with third- and fifth-order nonlinearity, the fifth harmonic results from step and direct processes. The interference between these two processes is discussed. Numerical calculations are presented for metal vapor-gas mixtures and Nd:glass laser pumping radiation.