Showing papers on "Isotropy published in 1980"

Elastic wave behavior across linear slip interfaces

Abstract: A model for an imperfectly bonded interface between two elastic media is proposed. Displacement across this surface is not required to be continuous. The displacement discontinuity, or slip, is taken to be linearly related to the stress traction which is continuous across the interface. For isotropic interface behavior, there are two complex frequency dependent interface compliances, ηN and ηT, where the component of the slip normal to the interface is given by ηN times the normal stress and the component tangential to the interface is given by ηT times the shear stress and is in the same direction. Reflection and transmission coefficients for harmonic plane waves incident at arbitrary angles upon a plane linear slip interface are computed in terms of the interface compliances. These coefficients are frequency dependent even when the compliances are real and frequency independent. Examples of the effects of buried slip interfaces on reflection coefficient spectra and on Love‐wave dispersion relations are ...

Isotropic yield surfaces in three dimensions for use in soil mechanics

Abstract: When using numerical methods in soil mechanics, one often needs to define a yield surface in three-dimensional principal-stress space. A special class of yield surfaces, given by J = (p+a)α(1−β sin 3ν)n, where ν is the Lode angle, is considered from the point of view of convexity and agreement with experimental data. Some recently proposed yield functions which belong to this class are compared. It is shown that the model with n = −0.229 is optimal as regards convexity, and can give reasonable agreement with the data.

A penalty plate‐bending element for the analysis of laminated anisotropic composite plates

Abstract: : A C degrees (penalty) finite element is developed for the equations governing the heterogeneous laminated plate theory of Yang, Norris and Stavsky. The YNS theory is a generalization of Mindlin's theory for homogeneous, isotropic plates to arbitrarily laminated anisotropic plates and includes shear deformation and rotary inertia effects. The present element can also be used in the analysis of thin plates by appropriately specifying the penalty parameter. A variety of problems are solved, including those for which solutions are not available in the literature, to show the material effects and the parametric effects of plate aspect ratio, length-to-thickness ratio, lamination scheme, number of layers and lamination angle on the deflections, stresses, and vibration frequencies. Despite its simplicity, the present element gives very accurate results. (Author)

Approximate solution of ellipsometric equations for optically biaxial crystals.

Abstract: Components of the dielectric function of a biaxial crystal are related in a simple first-order approximation to pseudodielectric functions calculated in the isotropic two-phase model from ellipsometric data. If |e| ≫ 1 and one of the principal axes is normal to the plane of incidence, the dominant contribution is shown to arise from the projection of the dielectric tensor onto the line of intersection between surface and plane of incidence.

Temperature fluctuations and scales in grid-generated turbulence

Abstract: To study the mixing of a passive scalar in nearly isotropic turbulence, experiments have been made in isotropically mixed thermal fields with thermal mesh size Mθ (a) equal to the momentum mesh size M, (b) larger than M (obtained by heating only alternate rods of the turbulence generating grid), and (c) smaller than M. This last condition was achieved by inserting a fine heating screen with Mθ M was an increase in the relative intensity of temperature fluctuations compared with the Mθ = M case, and a marginal increase in their decay rate; contrary to expectation, the ratio R of temperature to velocity integral scales in the region of approximate homogeneity did not differ from that corresponding to Mθ = M. In heated screen experiments, the relative decay rate was independent of Mθ/M and ΔT. For the three locations of the heating screen used in these experiments, the decay rate was also independent of the relative distance xs of the heating screen from the turbulence generating grid; however, larger xs was associated with larger relative intensity of fluctuations. To a first approximation, the ratio R approached unity according to the empirical relation R = 1 − A exp [− αxθ/(UT0)], where xθ is downstream distance measured from the heating screen, and T0 is a characteristic turbulence decay time scale at x0 = 0. It was also verified that the skewness of the streamwise temperature derivative is approximately zero sufficiently downstream of the heating screen. Where the present study overlaps with previous measurements, an extensive comparison reveals several points of agreement as well as departure.

Polarization approach to the scattering of elastic waves—I. Scattering by a single inclusion

Abstract: Scattering problems in elastodynamics are formulated in terms of integral equations, whose kernels are obtained from the Green's function for a comparison body. The comparison body will usually be taken as homogeneous and elastic in applications but, at least formally, there is no bar to its being inhomogeneous, viscoelastic and non-local. The novel feature of the formulation is the introduction of a “momentum polarization” to cope with density variations in a way that exactly parallels the stress polarization's correspondence with variations in moduli. To illustrate the use of the equations, scattering by an ellipsoidal inhomogeneity in a generally anisotropic matrix is studied in the Rayleigh limit and an asymptotic formula for its scattering cross-section is given. Detailed results are presented for a spheroidal inhomogeneity in an isotropic matrix, with explicit limiting forms for the scattering cross-sections of penny-shaped cracks, rigid circular discs and rigid needles.

Diffusion of Brownian particles in shear flows

Abstract: The coupling of Brownian displacements and shear-induced convection of spherical colloidal particles in dilute suspensions is examined using solutions of appropriate convective diffusion equations for the time-dependent probability density and also by calculation of relevant statistical quantities for an ensemble of diffusing particles from Langevin equations. Based on a fundamental solution for convective diffusion from a point in a general linear field, analytical expressions for the probability density fα(r; t) are given for the case of an arbitrary, two-dimensional linear flow field. The parameter α, which characterizes the flow, may range from − 1 (pure rotation), through zero (simple shear), to + 1 (pure elongation). The Langevin approach offers interesting insights into the physical mechanism of diffusive–convective coupling, and may also be used to obtain rigorous expressions for moments of the probability density appropriate to a particle diffusing in an unbounded quadratic (Poiseuille) flow. Preliminary experiments are described which qualitatively verify the theoretical predictions for Poiseuille flow, and which suggest a simple, direct method for measuring particle diffusivities. Finally the effect of bounding walls on convective diffusion is considered by means of Monte Carlo calculations. Results show that particle-wall interactions significantly affect the average behaviour of particles located initially within distances of a few particle radii of the wall, since the frictional force is no longer isotropic.

An experimental model for studying the effect of anisotropy on percolative conduction

Abstract: We present preliminary measurements of the electrical conductivity of a composite made of small conducting rods in an insulating matrix. It constitutes a model material for studying the effect of anisotropy on percolation as a function of particle length to diameter ratio. We can interpret our results, especially the lowering of threshold respective to isotropic systems, by means of a geometrical argument when the ratio is high as in our experiments.

Propagation of sound above a porous half‐space

Abstract: An exact solution for the field due to a point source above a homogeneous and isotropic porous ground has been derived in terms of a single branch line integral and a pole residue. The expected properties of the porous ground lead to a less complex solution than that usual to geophysics or underwater acoustics. The method of steepest descents modified by subtraction of the pole enables the derivation of an approximate solution for an externally reacting boundary. The precise nature of the further approximations that follow the assumption of a locally reacting (or impedance) boundary is made explicit and the role of the surface wave pole is clarified. Finally, the existing solutions for a locally reacting boundary are compared, and with one exception they are shown to be identical.

Performance and Analysis of a Reverberating Enclosure with Variable Geometry

Abstract: Mode stirred chambers are discussed. It is shown that, under the basic assumption of isotropy (largely experimentally proven), the coupling equation does not depend on the antenna gain. Chamber losses are examined, both from a general point of view, and in order to determine sensitivity of the measuring technique. An alternative thermodynamic approach is also given.

Crystallographic refinement of the structure of actinidin at 1.7 Å resolution by fast Fourier least‐squares methods

Abstract: The structure of the proteolytic enzyme, actinidin, has been refined by fast Fourier least-squares methods [Agarwal (1978), Acta Cryst. A34, 791-809]. Atomic positions were refined independently by the least-squares program, with the whole protein structure being regularized at intervals. After an initial refinement phase with an overall temperature factor, B, only, individual isotropic B values for all atoms were also refined. Overall, the crystallographic R factor was reduced from 0.429 (for 14 800 reflections to 2.0 A resolution) to 0.171 (for all 23 990 reflections between 10 and 1.7 A resolution), with a final estimated accuracy in atomic positions of <0.1 A. The final model comprises 1657 protein atoms, constrained close to standard geometry, and 163 solvent molecules, the latter identified using somewhat selective criteria. Most of the structure refined automatically with an average shift of 0.45 A for main-chain atoms and 0.56 A for side-chain atoms (maximum shift about 1.5 A). Some larger shifts resulted from manual intervention. Groups of atoms with high B values, or which were not refining well, were removed at intervals for scrutiny in difference maps, and major corrections were made to the conformations of 16 side chains and two peptide units. One correction to the amino-acid sequence was made (Asp 86 → Glx 86) and disordered conformations were introduced for five side chains. The whole refinement was completed in three months.

The structure of the high-temperature solid electrolyte lithium sulphate at 908K

Abstract: The structure of the high-temperature, plastic form of lithium sulphate has been determined from neutron powder diffraction data collected at 908K The structure is face-centred cubic, a=707 AA with the SO42- ion situated at the origin and the oxygen atoms rotationally disordered about the sulphur atom The Li+ ions occupy the +or-(1/4,1/4,1/4) positions The large isotropic temperature factors (B(SO4)=175 AA2 and B(Li)=33 AA2) suggest that the lithium ions occupy a statistical distribution of sites instantaneously displaced from +or-(1/4,1/4,1/4), in short-range correlation with the instantaneous orientations of the surrounding SO42- ions This is the first example of a fast ionic conductor where ionic motion is shown to be enhanced by rotational motion of the translationally static counter ions, although closer study of simple alkali salts may reveal several other examples

Recent Developments in Solution X-Ray Scattering

Abstract: We are concerned here with the structural information that it is possible to retrieve from the X-ray scattering study of macromolecules in solu­ tion. The general law governing X-ray scattering phenomena expresses the angular dependence of the scattered intensity as a function of the space distribution of the interatomic distances in the scatterer and of the nature of the atoms involved. When the scatterer is isotropic, the intensity is a function only of the moduli of the interatomic vectors, not of their orientation. Let r]J(r) be this isotropic distribution of the interatomic distances (a precise definition is given below). For a dilute solution of macromolecules the functionp(r) can be expected to display sharp short-range fluctuations (1 to 5 A) corresponding to pairs of neighboring atoms, followed by more damped fluctuations if middle­ range regularities are present-for example in the 5 to 10 A region for a-helical proteins. Beyond approximately 10 A. the number and variety of the interatomic vectors increases very rapidly with increasing r, and the function per) gradually smoothes out and slowly decreases. When r exceeds the maximal dimensions of the macromolecule, per) becomes uniform, since all the vectors involve only the solvent. Because the scattered intensity i(s) and the distribution p(r) are related by a Fourier transformation, such a separation of the fluctua­ tions of per) into two classes entails the presence in ;(s) of two distinct regions. One, at s small, contains the information relevant to the long-range (macromolecular) organization; the other, at s large, mirrors

Static and dynamic displacements of foundations on heterogeneous multilayered soils

Abstract: An analytical-numerical formulation is presented for dynamic and static analysis of strip foundations on an elastic isotropic medium consisting of heterogeneous layers. Each layer is characterized by an S-wave velocity that increases or decreases linearly with depth, a constant material density, a constant Poisson’s ratio equal to l/4 and a constant linearlyhysteretic critical damping ratio. The solution, based on a transformation that uncouples the wave equations in closed-form, is ‘exact’ in that it properly accounts for the true boundary conditions at the layer interfaces and the surface. Results are presented for two characteristic soil profiles (halfspace and stratum on rigid rock) in the form of normalized load-displacement ratios as functions of key dimensionless factors that influence the foundation behaviour during static and dynamic vertical, horizontal or moment loading. An interesting equivalence is established between a heterogeneous and a homogeneous halfspace, both having the same moduli at a depth equal to the foundation halfwidth (for translational motions) or to l/2 the foundation halfwidth (for rotation), i.e. for low frequency factors, the two media yield displacements of about the same average level, although the occurrence of resonance phenomena due to total wave reflection in the heterogeneous medium leads to fluctuations of the corresponding curves around the mean values. Cet article pr&.ente une formulation analytiquenumerique pour l’analyse dynamique et statique de fondations continues sur un milieu isotrop tlastique form6 de couches hCt&og&es. Chaque couche est

Higher‐order estimates of conditional eddies in isotropic turbulence

Abstract: Conditional eddies in isotropic turbulence are examined by estimating the velocity field around a point where the velocity is specified. The results indicate the low‐order estimates qualitatively resolve the large scale structure for all probable levels of the fluctuation intensity.

The structure of correlation tensors in homogeneous anisotropic turbulence

Abstract: The study of turbulence with spatially homogeneous but anisotropic statistical properties has applications in space physics and laboratory plasma physics. The first step in the systematic study of such fluctuations is the elucidation of the kinematic properties of the relevant statistical objects, which are the correlation tensors. The theory of isotropic tensors, developed by Robertson, Chandrasekhar and others, is reviewed and extended to cover the general case of turbulence with a pseudo-vector preferred direction, without assuming mirror reflection invariance. Attention is focused on two point correlation functions and it is shown that the form of the decomposition into proper and pseudo-tensor contributions is restricted by the homogeneity requirement. It is also shown that the vector and pseudo-vector preferred direction cases yield different results. An explicit form of the two point correlation tensor is presented which is appropriate for analyzing interplanetary magnetic fluctuations. A procedure for determining the magnetic helicity from experimental data is presented.

Stress-acoustic relations for stress measurement by ultrasonic technique

Abstract: New equations are proposed for stress measurements in orthotropic materials by ultrasonic birefringence technique. Intensity of stress-induced anisotropy and its principal direction are related to the applied stress through three coefficients, in contrast to only one coefficient required for isotropic materials. Experiments of ultrasonic birefringence were carried out to determine the three coefficients in a plate of aluminum alloy with a slight orthotropy originated from roll working. Anisotropy induced by uniaxial stress and rotation of the principal axis were observed in several specimens with different directions of rolling. The results are in good agreement with the curves theoretically predicted by the equations proposed here, showing the validity and usefulness of these equations for stress measurements in materials conventionally used.

Discontinuous deformation gradients near the tip of a crack in finite anti-plane shear: an example

Abstract: This investigation aims at the elastostatic field near the edges (tips) of a plane crack of finite width in an all-round infinite body, which — at infinity — is subjected to a state of simple shear parallel to the crack edges. The analysis is carried out within the fully nonlinear equilibrium theory of homogeneous and isotropic, incompressible elastic solids. Further, the particular constitutive law employed here gives rise to a loss of ellipticity of the governing displacement equation of equilibrium in the presence of sufficiently severe anti-plane shear deformations.

Thermohaline Instability in Anisotropie Porous Media

Abstract: The onset of thermohaline convection in a horizontal porous layer is investigated theoretically. The layer is homogeneous, anisotropic, and of infinite horizontal extent. Horizontal isotropy with respect to permeability, thermal diffusivity, and solute diffusivity is assumed. For porous media with thermally insulating solid matrices the stability diagram has the same shape as in the case of isotropy. The critical wave number is constant and equal to that of the one-component case. For thermally conducting matrices, new features may occur. The locus of the direct mode in the stability diagram may not be a straight line, and the corresponding wave number may be nonconstant. The initiation of salt fingers is studied by linear theory. It seems that the width of salt fingers is influenced by anisotropy in the diffusivities. Anisotrophy may or may not favor salt fingers, depending on a dimensionless diffusion parameter being greater than or less than one. 12 references.

Thermal stresses in composite cylinders

Abstract: This paper is concerned with the static problem of the linear theory of thermo-elasticity for a composite cylinder. The cylinder is assumed to be occupied by different inhomogeneous and anisotropic elastic materials. The method described is also used to study the deformation of a circular cylinder made of two different homogeneous and isotropic materials.

Self-similar flows with uniform velocity gradient and their use in modelling the free expansion of polytropic gases

Abstract: The self-similar motion of a polytropic gas with a linear velocity distribution is considered in an arbitrary ν-dimensional space. It is shown that if the initial state of the gas is isotropic the flow has a characteristic ellipsoidal form. Both expanding and compressing flows are shown to exist. The application of such flows as models for the expansion of an initially uniform mass of gas into vacuum is considered by comparison with computationally modelled expansions in one-dimensional cylindrical and spherical geometries. It is found that the accuracy of the representation increases when the heating time is long compared with the characteristic time of expansion.

Computer Simulation Studies of Anisotropic Systems IV. The Effect of Translational Freedom

Abstract: We have simulated the properties of 256 cylindrically symmetric particles interacting via a simple anisotropic potential of the form u$\_2$ (r$\_{12}$) P$\_2$(cos $\beta\_{12}$) and with a scalar Lennard-Jones 12:6 potential, using the Monte Carlo technique. The simulations were performed for two forms of u$\_2$(r$\_{12}$) in the isothermal-isobaric ensemble and yielded values for volume, enthalpy, second-rank orientational order parameter, radial distribution function and second-rank angular correlation function. The specific heat at constant pressure, isothermal compressibility and isobaric expansivity were also obtained but they are subject to considerable error because they were evaluated from fluctuations. The system is found to exhibit a weak, firstorder transition from a nematic to an isotropic phase on increasing the temperature. The isotropic phase possesses short-range spatial and orientational order; it differs from the nematic phase, which has long-range orientational order but only short-range spatial order. The results of these simulations are used to discuss the influence of the range of the anisotropic potential on the behaviour of the nematogen. Previous Monte Carlo simulations of nematic liquid crystals had employed a lattice model with the anisotropic interactions restricted to nearest neighbours. Our results are used to study the effect of these convenient but unrealistic restrictions on the properties of the nematic. The results of our simulations are in reasonable accord with the properties of the nematogen, 4,4$'$-dimethoxyazoxybenzene, although no attempt was made to select a pair potential to mimic the behaviour of any substance. Finally, we use the results of our simulations to test the validity of the molecular field approximation, as applied to nematics. This approximation is one of the foundations of the Maier-Saupe theory and its predictions are compared with the behaviour of the simulated nematics. It would appear that this theory provides a better description of our system than the lattice model, with its enforced spatial order and truncated anisotropic pair potential.

Inverse eigenvalue problems for the mantle

Abstract: We represent the earth as a sphere with radius R and assume that the material is perfectly elastic and isotropic. Thus we ignore ellipticity, rotation, damping, lateral inhomogeneities and anisotropy.

Rheological properties of rodlike polymers in isotropic and liquid crystalline phases

Abstract: A molecular theory is presented for the dynamics of rodlike polymers in concentrated solutions. The theory describes the rotational motion of rods in both the isotropic, and liquid crystalline phases. Combined with the molecular expression of the stress tensor it also gives a unified rheological constitutive equation, which predicts the nonlinear viscoelasticity in both phases.