Showing papers on "Isotropy published in 1971"

Short Range Order Effects in the Isotropic Phase of Nematics and Cholesterics

Abstract: We assume that (1) the local state of order in the isotropic phase is a symmetric traceless tensor Qαβ, proportional to the anisotropic part of a tensor property such as the magnetic susceptibility; (2) the free energy may be expanded in powers of Qαβ and of its gradients. This allows a unified description covering the anomalous magnetic birefringence, the intensity of light scattering, and the properties of the nematic/isotropic interface. For a cholesteric, although the optical rotation is huge in the ordered phase, we predict that it should not be anomalous just above the transition point Tc. We also investigate the dynamics of fluctuations of Qαβ, and discuss the flow birefringence, the frequency width of the Rayleigh scattering, and the attenuation of ultrasonic shear waves, in terms of 3 viscosity coefficients.

Fracture mechanics of interfacial cracks

Abstract: This paper contains a two-dimensional analysis of the stress field around a crack on the plane interface between two bonded dissimilar anisotropic elastic half-spaces. This analysis is then combined with the usual local form of the Griffith virtual work argument to give an explicit fracture criterion which involves a suitably defined ‘stress concentration vector’ and the specific surface energy of the bonded surfaces. This criterion has a simple structure and reduces to the conventional form of Irwin when the two half-spaces are isotropic and identical. The analysis is then extended to cracks moving uniformly and a local fracture criterion with the same structure as the static criterion is derived by an energy balance argument. The criterion is specialized to isotropic half-spaces for illustration, when it predicts that the speed of a crack on an interface between such media will be limited by a speed Vc which is slightly greater than the smaller of the two Rayleigh wave speeds. A by-product of the analysis is an expression for the displacement field of an arbitrary interfacial dislocation, either stationary or moving uniformly.

Cube‐Resonance Method to Determine the Elastic Constants of Solids

Abstract: The Rayleigh‐Ritz method of eigenvalue approximation is used to obtain a solution for the free vibrations of an elastic solid. The resonant frequencies of an isotropic cube, calculated by this method, agree with the frequencies of the Lame modes, for which an exact solution is possible, up to eight significant digits. Experimental measurements of the 13 lowest‐frequency modes of a cube of fused quartz are all within 0.5% of the results computed by using the elastic constants determined by another method. These constants differ from the elastic constants determined directly from the experimental frequencies by less than 0.2% for the shear modulus and by less than 0.003 for the value of Poisson's ratio. The cube‐resonance method of elastic constant determination is also applicable to substances of more general crystallographic symmetry.

Second-Harmonic Radiation from Metal Surfaces

Abstract: A calculation is performed of the second-harmonic radiation generated by the reflection of incident radiation from a metal surface in a model that considers only plasma effects and neglects periodic lattice structure in the metallic bulk and any form of surface roughness. The radiation is due to three kinds of currents: one localized to within a few angstroms of the surface and parallel to it, another also localized at the surface but normal to it, and a bulk current in the skin-depth region of the incident light, a region several hundred angstroms thick. Previous expressions for the normal surface current are found to be incorrect, and a new expression is derived. The second-harmonic radiation due to the surface currents is sensitive to surface conditions, while the contribution of the bulk current is only weakly sensitive to them. Theoretical estimates suggest that changes in the radiation intensity with variation in surface condition are caused primarily by changes in the parallel surface currents due to variations in surface scattering. In the course of this investigation general relationships and explicit forms for the second-order response of an isotropic electron gas are given.

Uniqueness Theorems in Linear Elasticity

Abstract: 1 Introduction.- 2 Basic Equations.- 2.1 Formulation of Initial-Boundary Value Problems.- 2.2 The Classical and Weak Solutions.- 2.3 The Homogeneous Isotropic Body. Plane Elasticity.- 2.4 Definiteness Properties of the Elasticities.- 3 Early Work.- 4 Modern Uniqueness Theorems in Three-Dimensional Elastostatics.- 4.1 The Displacement Boundary Value Problem for Bounded Regions.- 4.1.1 General Anisotropy.- 4.1.2 A Homogeneous Anisotropic Material.- 4.1.3 A Homogeneous Isotropic Material.- 4.1.4 The Implication of Strong Ellipticity for Uniqueness.- 4.1.5 The Non-Homogeneous Isotropic Material with no Definiteness Assumptions on the Elasticities.- 4.1.6 The Displacement Boundary Value Problem for a Homogeneous Isotropic Sphere.- 4.1.7 Fichera's Maximum Principle.- 4.2 Exterior Domains.- 4.3 The Traction Boundary Value Problem.- 4.3.1 General Anisotropy.- 4.3.2 A Homogeneous Isotropic Material.- 4.3.3 The Traction Boundary Value Problem for a Homogeneous Isotropic Elastic Sphere.- 4.3.4 Necessary Conditions for Uniqueness in the Traction Boundary Value Problem for Three-Dimensional Homogeneous Isotropic Elastic Bodies.- 4.4 Mixed Boundary Value Problems.- 4.4.1 General Anisotropy.- 4.4.2 A Homogeneous Isotropic Material.- 5 Uniqueness Theorems in Homogeneous Isotropic Two-Dimensional Elastostatics.- 5.1 Kirchhoff's Theorem in Two-Dimensions. The Displacement and Traction Boundary Value Problems.- 5.2 Uniqueness in Plane Problems with Special Geometries.- Appendix: Uniqueness of Three-Dimensional Axisymmetric Solutions.- 6 Problems in the Whole- and Half-Space.- 6.1 Specification of the Various Boundary Value Problems. Continuity onto the Boundary and in the Neighbourhood of Infinity.- 6.2 Uniqueness of Problems (a)-(d). Corollaries for the Space EN.- 6.3 Uniqueness for the Mixed-Mixed Problem of Type (e).- 6.3.1 A Complete Representation of the Biharmonic Displacement in a Homogeneous Isotropic Body Occupying the Half-Space.- 6.3.2 Uniqueness in the Mixed-Mixed Problem (e).- 7 Miscellaneous Boundary Value Problems.- 7.1 Problems for a Sphere.- 7.2 The Cauchy Problem for Isotropic Elastostatics.- 7.3 The Signorini Problem. Other Problems with Ambiguous Conditions.- 8 Uniqueness Theorems in Elastodynamics. Relations with Existence, Stability, and Boundedness of Solutions.- 8.1 The Initial Displacement and Mixed-Boundary Value Problems. Energy Arguments.- 8.2 The Initial-Displacement Boundary Value Problem. Analyticity Arguments.- 8.3 The Initial-Mixed Boundary Value Problem for Bounded Regions. Further Arguments.- 8.4 Summary of Existing Results in the Uniqueness of Elastodynamic Solutions.- 8.5 Non-Standard Problems, including those with Ambiguous Conditions.- 8.6 Stability, Boundedness, Existence and Uniqueness.- References.

Isotropic solutions of the Einstein-Boltzmann equations

Abstract: It is shown that in all solutions of the Einstein-Boltzmann equations in which the particle distribution function is isotropic about some 4-velocity field, the distortion of that velocity field vanishes; further, either its expansion or its rotation vanishes. We discuss briefly further kinetic solutions in which the energy-momentum tensor has a perfect fluid form.

Spatial Resolution of a Resistance Wire Temperature Sensor

Abstract: Effects of sensor length on measured one‐dimensional temperature spectrum and temperature variance dissipation rate are studied. Expressions are evaluated numerically for isotropic turbulence with the Corrsin‐Pao three‐dimensional temperature spectrum, providing response curves.

Specular reflectance and ellipsometric spectroscopy of oriented molecular layers

Abstract: Assuming an oriented molecular layer to behave optically as a homogeneous, uniaxial medium with its optic axis normal to the interface, such a system is treated theoretically to yield equations relevant to both specular reflectance and ellipsometric spectroscopy. Explicit expressions are derived for the empirical quantities (relating them to the optical constants of the media and other system parameters) which are both reasonably simple and correct to second-order terms in the film thickness. From ellipsometric measurements alone, it is not possible to distinguish between very thin uniaxial and isotropic films. However, data for a very thin non-absorbing uniaxial film on an absorbing substrate (e.g., a metal), if analyzed on the assumption of film isotropy, lead to an apparent absorption index for the film of the magnitude of the absorption index found for semi-conductors. A similar result is predicted for specular reflectance measurements, except that in that case the apparent optical constants of the film depend on the angle of incidence.

Isotropy of the 3 K Background

Abstract: THE 3 K background radiation provides a frame of reference equivalent to that defined by the matter by which it was last scattered. Conservative estimates place this last scattering at a redshift of z = 7 while others range up to redshifts of 1,000 or so1,2. Motion of the Earth relative to this extremely distant reference frame can be measured by observing the 24 h anisotropy in the 3 K background. If we assume no intrinsic anisotropy in the radiation, then a velocity of the Earth v would result in a temperature distribution where T0 is the average background temperature and θ is the angle between v and the direction of observation3. Thus a temperature anisotropy of 10−3 K (1 mK) corresponds to a speed of ∼100 km s−1. Previous efforts to measure this anisotropy have yielded only the component in the Earth's equatorial plane4,5. The most precise result to date is that obtained by Conklin, who reports an equatorial component of 1.9±0.8 mK directed toward 10 h right ascension. (This result (unpublished) is based in part on data obtained after publication of ref. 5.) The experiment described here is an attempt to determine the right ascension, declination and magnitude of the Earth's velocity vector by measuring the 24 h anisotropy in the cosmic background.

Fast computation of exchange-broadened isotropic E.S.R. spectra

Abstract: A general line shape equation for intramolecular exchange among many sites has been derived by means of the Liouville density matrix theory and adapted to fast computer calculations. Applications of the computer programme ESREXN, which is based on the formalism of the present paper, are shown by examples.

The theory of elastic composite materials

Abstract: A general analysis is given for the determination of the effective elastic constants of a macroscopically isotropic and homogeneous two-phase composite material. Approximate formulae are given and their validity discussed in terms of the general features of the sample geometry.

Laminated Transversely Isotropic Cylindrical Shells

Abstract: A theory for the analysis of stresses in laminated circular cylindrical shells subjected to arbitrary axisymmetric mechanical and thermal loadings has been developed. This theory, specifically for use with pyrolytic-graphite-type materials, differs from the classical thin shell theory in that it includes the effects of transverse shear deformation and transverse isotropy, as well as thermal expansion through the shell thickness. Solutions in several forms are developed for the governing equations. The form taken by the solution function is governed by geometric considerations. A range in which the various solution forms occur was determined numerically. As a sample problem, the slow cooling of pyrolytic graphite deposited onto a commercial graphite mandrel was considered. Investigation of normal and shear stress behavior at the pyrolytic graphite-mandrel interface showed that these stresses decrease in magnitude with increasing E/Gc ratio and increasing deposit to mandrel thickness (ha /hb ) ratio. This implies that a thin mandrel and a material weak in shear are desirable to minimize the possibilities of flaking and delamination of the pyrolytic graphite.

Wave Propagation in Nematic Liquid Crystals

Abstract: Basic laws of motion of micropolar continuum are presented, and the adequacy of applying micropolar theory to liquid crystals is indicated. A set of constitutive equations is derived for nematic liquid crystals. Wave propagation problems are solved, and it is shown that the theoretical analysis is in good agreement with the experimental data, which indicates the isotropy of the phase velocity of the longitudinal wave and the anisotropy of the damping coefficient. The coupling, although small, is shown to exist between longitudinal and rotational waves.

Reflection of Light by a System of Nonabsorbing Isotropic Film–Nonabsorbing Isotropic Substrate with Randomly Rough Boundaries

Abstract: The reflection of light by a system consisting of a nonabsorbing isotropic film and nonabsorbing isotropic substrate with both boundaries (air–film and film–substrate) rough is considered. The scalar theory of light scattering on such a system has been developed. The formulas characterizing both the coherent and the incoherent components of the reflected-light flux have been derived for the case of the identical film (both boundaries are rough, the air–film boundary is a copy of the film–substrate boundary) as well as for the general film (both boundaries are rough and different). The numerical results of the calculation performed for a system of SiO2–Si are presented and the experimental results for the case of art identical film of SiO2 on a Si single crystal are given. The agreement between the theory and the experiment is fairly good. The correctness of the film thickness and its index of refraction depends to a fairly high degree on the roughness of the air–film and film–substrate boundaries.

Nonlinear Buckling Analysis of Sandwich Arches

Abstract: An incremental displacement formulation for the nonlinear finite element analysis of sandwich arches is presented. The formulation permits the analysis of geometrically nonlinear problems with finite rotations but infinitesimal strains. The finite element model employed is a straight beam-column type, and has a three-layered sandwich construction with similar facings. Homogeneous and isotropic material properties are assumed; and flexural, extensional and shear deformations of all three layers are considered. Therefore, it is possible to analyze homogeneous structures by simply assigning the same material properties to all three layers. The method is applicable to investigation of symmetrical and asymmetrical buckling and post-buckling behavior of arches. For the latter, an augmentation scheme has been used to give a positive definite stiffness matrix in the descending branch of the load-deflection curve. Results compare favorably with analytical solutions available in the literature.

The Propagation of Surface Waves in Anisotropic Media

Abstract: Summary Surface wave propagation in examples of unlayered and multilayered anisotropic media is examined numerically with a programme using an extension of the Thompson-Haskell matrix formulation. Surface wave propagation in an isotropic earth model containing an anisotropic layer in the upper mantle has been found, for the most part, to differ very little from propagation in a purely isotropic model. An exception is the propagation of the third generalized mode (corresponding to the second Rayleigh mode in isotropic structures), which has particle motion differing considerably from motion in isotropic media. Observations of such particle motion in the Earth have been made.

Flexural Vibrations of Rectangular and Other Polygonal Plates

Abstract: The finite strip method is used for the flexural vibration analysis of elastic plates. The plates can be isotropic or orthotropic in property, of constant or variable thickness, and with distributed or concentrated masses. It can have any combination of free, simply-supported and clamped boundary conditions and can also be continuous in one direction. The stiffness matrix of a strip with two opposite ends simply-supported, free or clamped is formed by assuming suitable basic function series in the longitudinal direction which satisfies the end conditions and a simple cubic polynomial in the transverse direction. A consistent mass matrix can also be formed for each strip. The stiffness and mass matrices of all the strips making up a plate are then assembled to form an eigenvalue matrix in the same way as for a beam problem. The method is simple but versatile, and all the natural frequencies and corresponding modal shapes can be obtained rapidly from an intermediate or even small size electronic digital computer.

The elastic constants of a two-phase composite material

Abstract: A new formula for the shear modulus of a macroscopically homogeneous and isotropic composite material is derived Geometries for which this formula is applicable are suggested and correspond to those which may be obtained in two-component composites consisting of two continuous phases

Modulus of short carbon and glass fiber reinforced composites

Abstract: A theoretical model for a short fiber reinforced composite is proposed The composite is assumed to consist of an aggregate of sub-units, each sub-unit possessing the elastic properties of a reinforced composite in which the fibers are continuous and fully aligned The elastic constants of a partially oriented composite are then calculated by the Voigt and Reuss averaging procedures, giving upper and lower bounds respectively for the composite modulus Comparison is made with experimental data for such composites The measured modulus of glass and carbon fiber composites is found to be given by the Reuss or lower bound, to a good approximation compared with the difference between the bounds, for fiber orientations ranging from almost isotropic to highly aligned

Elastic shear moduli and crystal stability at high P and T

Abstract: This paper presents new theoretical expressions for the elastic velocities (of a cubic or isotropic homogeneous solid) as functions of density and temperature (equations 7 and 8). These equations have been derived within the fourth-order anharmonic theory, i.e., the extension of the theory of lattice dynamics into the regime of finite strain, and they include implicitly the effects of noncentral forces, distant-neighbor interaction, and thermal vibration. Application of the equations to garnet, spinel, and olivine (considered as Voigt-Reuss-Hill isotropic bodies) confirms and lends rigor to the present consensus on the elasticity of the mantle. The equations indicate further that the observed instabilities of these materials under upper mantle conditions are not directly related to vanishing of an elastic modulus.

Elastic fields of a dislocation loop in a two-phase material

Abstract: The purpose of this study is to set forth the mathematical framework for an efficient treatment of a plane dislocation loop in a two-phase material, and to carry the elastostatic part of the analysis far enough, so that the results are immediately tractable for applications to solid state. The two-phase material is idealized as two isotropic elastic half-spaces with perfect adhesion, while the loop is placed in a plane parallel to the interface. It is shown that the elastic fields can be obtained by differentiating two types of integrals and, thus, are readily evaluated for any shape of the loop for which the integrals are available from potential theory. Explicit results are given for circular prismatic and glide loops.

The effect of velocity sensitivity on temperature derivative statistics in isotropic turbulence

Abstract: The velocity sensitivity of a resistance-wire temperature sensor is expressed in terms of sensor parameters, and the resulting errors in temperature derivative moments in isotropic turbulence are evaluated. It is shown that velocity sensitivity of a degree completely negligible for most purposes causes severe contamination of the measured third moment. The contamination terms are shown to be production rates of the mean square temperature gradient and vorticity, respectively, and therefore create positive values of measured derivative skewness. The dominant contamination term is related to the temperature spectrum through the balance equation for the mean-square temperature gradient, and calculations based on an assumed spectral form show that under typical conditions the measured skewness is large. This mechanism could provide an alternative to anisotropy as an explanation of the positive skewnesses recently measured in the atmosphere.

Yield surfaces in prestrained aluminum and copper

Abstract: The analysis of strain-hardening materials subjected to multiaxial states of stress requires more detailed experimental information about the effects of previous plastic deformation on the yield surfaces of real materials than is presently available. To provide insight into some of these effects thin walled tubular specimens of 1100-0 aluminum and annealed OFHC copper were subjected to biaxial stresses through the application of simultaneous axial tension and internal pressure, and the effects of the magnitude, direction, and sequence of prestraining operations on subsequent yield surfaces were determined. It was found that the yield surface behavior depends greatly upon the definition of yielding employed. Use of small proof strain definitions resulted in very anisotropic yield surface characteristics which reflected the effect of previous deformation. On the other hand, use of large proof strains resulted in isotropic yield surface characteristics which were devoid of previous deformation influence. The small proof strain yield curves were found, in general, to expand and translate in the direction of prestrain and, for biaxial prestrains, to be distorted in the vicinity of the loading point. Multiple prestrain sequences in normal directions induce a large negative cross effect similar to Bauschinger effect observed under reversed loading. Such anisotropic behavior was found to contradict the two most commonly used continuum mechanics predictions, the isotropic and kinematic hardening rules.