Showing papers on "Isotropy published in 1985"

Two-Dimensional Modelling of Compressive Failure in Delaminated Laminates

Abstract: An analytical model is developed to assess the compressive strength criticality of near-surface interlaminar defects in laminated composites. The delaminated region is elliptic in shape, separating a thick isotropic plate from a thin orthotropic layer whose material axes coincide with the ellipse axes. The growth conditions and growth behavior of this defect are studied by breaking the overall problem into an elastic stability problem and a fracture problem. Post-buckling solution for the elliptic section is obtained using the Rayleigh-Ritz method while an energy balance criterion based on a self-similar disbond growth governs the fracture. The parameters controlling the growth or arrest of the delamination damage are identified as the fracture energy, disbond depth and elastic properties of the materials from both sides of the delaminating interface. By varying the degree of material anisotropy relative to the loading axis a range in growth behavior was found including stable or unstable crack growth parallel to or normal to the loading axis.

Shear-wave polarizations on a curved wavefront at an isotropic free surface

Abstract: Summary. We present polarization diagrams of the particle motions at the free surface of an isotropic half-space generated by incident shear waves from a local buried point source. The reflectivity technique is used to calculate synthetic seismograms from which the particle motions are plotted. The particle motions are examined over a range of epicentral distances in a uniform isotropic half-space for different source frequencies and polarization angles, and for different Poisson's ratios. The particle motions due to a curved wavefront possess different characteristics from those generated by plane wavefronts at corresponding incidence angles. A curved wavefront generates a local SP-phase: a P-headwave which propagates along the free surface, and arrives shortly before the direct S-wave. These two arrivals give rise to cruciform particle motions in the sagittal and horizontal planes, which could be misinterpreted as anisotropy-induced shear-wave splitting. An examination of the particle motion in the transverse plane, mutually orthogonal to the sagittal and horizontal planes, can be used to discriminate between isotropic and anisotropic interpretations. The amplitude of the SP-phase is enhanced when it propagates in a low-velocity surface layer overlying the source layer, and may then become the dominant phase on radial-component seismograms. The presence of even a single surface layer may introduce considerable complexity into the seismogram, and we examine the effects of layer thickness, velocity contrast, and source depth on the corresponding polarization diagrams. Reliable information on the source and propagation path characteristics of shear waves from a buried local point source can only be obtained from free-surface records if they are recorded within a very limited epicentral distance range.

Spherical stellar systems with spheroidal velocity distributions

Abstract: A new method is described for deriving families of anisotropic distribution functions consistent with any spherically symmetric density profile. The algorithm is straightforward and sufficiently simple that analytic solutions can often be obtained. Each solution is defined by a single free parameter r/sub a/, the ''anisotropy radius''; the ratio of radial to tangential velocity dispersions is given by sigma/sup 2//sub r/ /sigma/sup 2//sub t/ = 1 +- r/sup 2//r/sup 2//sub a/. The models are isotropic in the center, and become either radially or tangentially anisotropic at large radii. Superposition of two or more solutions with different degrees of anisotropy allows one to construct models with almost any desired velocity structure. The algorithm is applied to a model galaxy with a de Vaucouleurs density law, and families of line-of-sight velocity dispersion profiles are obtained. The algorithm produces physically acceptable solutions that are nearly as radially anisotropic as those found by the more general linear programming method, and with considerably less computational effort.

Hydrodynamics of icosahedral quasicrystals

Abstract: The equations governing long-wavelength, low-frequency excitations in icosahedral quasicrystals are derived. It is found that while the speeds of the propagating modes are isotropic, the attenuations are not, implying that purely macroscopic experiments can in principle distinguish quasicrystals from crystals, glasses, or conventional incommensurate systems. The coefficient of the anisotropy is, regrettably, quite small. The complete spectrum consists of three diffusive phasons, two pairs of transverse and one pair of longitudinal sound modes, a vacancy diffusion mode, a heat diffusion mode, and, in a material with n atomic species, n-1 additional particle diffusion modes. The diffusion times of the vacancy and phason modes are expected to be comparable and very long. It is shown that propagating phasons, even at short wavelength, are an unlikely prospect. The static, equilibrium elastic properties are also anisotropic, but are approached very slowly, and in many situations, the elastic response is isotropic on experimentally accessible time scales. Our results also imply that nonlinear fluctuation corrections to the linearized hydrodynamics presented here are finite as q and \ensuremath{\omega}\ensuremath{\rightarrow}0, i.e., there is no breakdown of conventional hydrodynamics in icosahedral quasicrystals.

Shear-wave polarizations near the North Anatolian Fault – II. Interpretation in terms of crack-induced anisotropy

Abstract: Summary. The polarizations of shear waves recorded by networks of digital thxee-component seismometers immediately above small earthquakes near the North Anatolian Fault in Turkey display shear-wave splitting on almost all shear-wave seismograms recorded within the shear-wave window. This splitting is incompatible with source radiation-patterns propagating through simple isotropic structures but is compatible with effective anisotropy of the internal structure of the rock along the ray paths. This paper interprets the phenomena in terms of widespread crack-induced anisotropy. Distributions of stress-induced cracks model many features of the observations, and synthetic polarization diagrams calculated for propagation through simulated cracked rock are similar to the observed patterns. This evidence for widespread crack-induced anisotropy lends strong support to the hypothesis of extensive-dilatancy anisotropy (EDA) suggested by laboratory experiments in subcritical crack-growth. The crucial evidence confirming some form of EDA would be observations of temporal changes in shear-wave splitting as the stress field alters the crack density and crack geometry. There is some weak evidence for such temporal changes at one site, but further analysis of suitable digital three-component seismometer networks in seismic areas is required to confirm EDA.

Reflection and refraction of elastic waves on a plane interface between two generally anisotropic media

Abstract: A unified approach to the study of reflection and refraction of elastic waves in general anisotropic media is presented. Christoffel equations and boundary conditions for both anisotropic media in coordinate systems formed by incident and interface planes, rather than in crystallographic coordinates, are considered. Consideration of wave propagation in an acoustic‐axis direction is included in the general algorithm, so results can be obtained both generally and for planes of symmetry, including planes of isotropy. General features of the numerical results are discussed. Energy conversion coefficients are shown to satisfy reciprocity relations which are formulated. It is much more natural to consider intensity–conversion ratios, rather than amplitude–conversion ratios, showing the important role of ray (rather than wave‐vector) directions in describing phenomena such as grazing angles. In particular, it is shown that the incident wave vector for grazing incidence may be greater or less than 90°: The domain of incident wave‐vector angles can actually split into disjoint pieces. The reflection coefficient at grazing incidence is shown to be unity, as in the isotropic case. Critical‐angle phenomena are described naturally by this approach.

A Beam Theory for Anisotropic Materials

Abstract: Beam theory plays an important role in structural analysis. The basic assumption is that initially plane sections remain plane after deformation, neglecting out-of-plane warpings. Predictions based on these assumptions are accurate for slender, solid, cross-sectional beams made out of isotropic materials. The beam theory derived in this paper from variational principles is based on the sole kinematic assumption that each section is infinitely rigid in its own plane, but free to warp out of plane. After a short review of the Bernoulli and Saint-Venant approaches to beam theory, a set of orthonormal eigenwarpings is derived. Improved solutions can be obtained by expanding the axial displacements or axial stress distribution in series of eigenwarpings and using energy principles to derive the governing equations. The improved Saint-Venant approach leads to fast converging solutions and accurate results are obtained considering only a few eigenwarping terms.

A uniform GTD formulation for the diffraction by a wedge with impedance faces

Abstract: A uniform high-frequency solution is presented for the diffraction by a wedge with impedance faces illuminated by a plane wave perpendicularly incident on its edge. Arbitrary uniform isotropic impedance boundary conditions may be imposed on the faces of the wedge, and both the transverse electric (TE) and transverse magnetic (TM) cases are considered. This solution is formulated in terms of a diffraction coefficient which has the same structure as that of the uniform geometrical theory of diffraction (UTD) for a perfectly conducting wedge. Its extension to the present case is achieved by introducing suitable multiplying factors, which have been derived from an asymptotic evaluation of the exact solution given by Maliuzhinets. When the field point is located on the surface near the edge, a more accurate asymptotic evaluation is employed to obtain a high-frequency expression for the diffracted field, which is suitable for several specific applications. The formulation described in this paper may provide a useful, rigorous basis to search for a more numerically efficient but yet accurate approximation.

Efficient Monte Carlo method for simulation of fluctuating conformations of native proteins.

Abstract: A powerful Monte Carlo method is described to simulate thermal conformational fluctuations in native proteins by using an empirical conformational energy function in which bond lengths and bond angles are kept fixed and only dihedral angles are independent variables. In this method, collective variables corresponding to eigenvectors of the second-derivative matrix of the energy function at its minimum point are scaled according to corresponding eigenvalues in such a way that the energy function in terms of the scaled collective variables is isotropic at the minimum point. Simulation is carried out with an isotropic step size in the space of these scaled collective variables. This simulation method is applied to a small protein, bovine pancreatic trypsin inhibitor (BPTI), and its model harmonic system defined by a quadratic energy function with the same second-derivative matrix as that of BPTI at its minimum point. Efficiency of the simulation method with an isotropic step size in the space of the scaled collective variables is found to be about 500–50 times greater than the conventional method with with an isotropic step in the space of the usual nonscaled variables. One step of this new method generates conformational changes that occur in the real-time range of 0.05 ps. In a record of 5 × 105 step simulation, the BPTI molecule is observed to migrate beyond a single minimum-energy region.

Solutions for planar optical waveguide equations by selecting zero elements in a characteristic matrix

Abstract: The propagation properties of optical planar waveguides with multilayer index profiles are analyzed by the transfer matrix of transmitted and reflected beam amplitudes in multilayers. The propagation wave number for guided-wave modes is obtained from the condition that certain elements in the transfer matrix must be zero. This numerical technique requires much shorter computer times compared with the usual method of solving the eigenvalue equations, obtained by setting the characteristic determinant to zero. The analysis is also applicable either to waveguides that have losses or to certain cases of uniaxial dielectric anisotropy. All waveguides are assumed to be magnetically isotropic. Some examples of the analysis of graded-index profiles and calculations of the effect of metal claddings and prism perturbations on guided modes are given.

Isotropic homogeneous universe with viscous fluid

Abstract: Exact solutions are obtained for the isotropic homogeneous cosmological model with viscous fluid. The fluid has only bulk viscosity and the viscosity coefficient is taken to be a power function of the mass density. The equation of state assumed obeys a linear relation between mass density and pressure. The models satisfying Hawking’s energy conditions are discussed. Murphy’s model is only a special case of this general set of solutions and it is shown that Murphy’s conclusion that the introduciton of bulk viscosity can avoid the occurrence of space‐time singularity at finite past is not, in general, valid.

On isotropic brownian motions

Abstract: This paper includes a detailed study of isotropic brownian motion on matrices and brownian flows associated with isotropic gaussian fields. Characteristic exponents, stability, asymptotic behaviour, statistical equilibrium are the topics on which the main results are obtained.

Theory of homogenous turbulence

Abstract: The properties of homogeneous isotropic turbulence is investigated, and the authors consider the influence of a magnetic field on those properties in a conducting fluid. The authors discuss the physical mechanism underlying the decay of fluctuation energy in the k-locality interval.

Theory of Fermi-liquid effects in high-field tunneling.

Abstract: Fermi-liquid effects are incorporated into the quasiclassical theory of superconductivity to give a quantitative theory for understanding tunneling experiments on real metals in high magnetic fields. The theory is formulated for arbitrary impurity scattering and anisotropy. Numerical calculations are done for the dirty, isotropic limit. The calculations for low spin-orbit scattering show that the energy difference in a magnetic field between spin-up and spin-down electrons is proportional to the l=0 antisymmetric Fermi-liquid parameter as confirmed by recent experiments. Furthermore, this same Fermi-liquid parameter is shown to renormalize the Pauli-limiting field in the calculation of the upper critical field. Recent experiments on thin films of Al are compared with the theoretical calculations.

Anisotropy in the oceanic lithosphere — theory and observations from the Ngendei seismic refraction experiment in the south‐west Pacific

Abstract: Summary. P-wave travel-time data from a seismic refraction experiment on the 1983 Ngendei expedition to the south Pacific indicate anisotropy at two levels in the oceanic lithosphere. In the upper mantle, P-wave velocities vary between 8.0 and 8.5 km s-1 with the fast direction at N30°E. Crustal anisotropy within layer 2 is characterized by azimuthal P-wave velocity differences of about 0.2–0.4 km s-1, with the fast direction at N120°E, orthogonal to the upper mantle anisotropy. The observed anisotropy is consistent with a model in which aligned olivine crystals cause anisotropy in the upper mantle and aligned cracks within layer 2 cause anisotropy in the crust. We examine a model of P-wave propagation from a point source in an isotropic layer above an anisotropic half-space and show that under these conditions anisotropy can cause seismic ray-paths to deviate from the vertical plane connecting source and receiver. This deviation has a negligible effect on seismic travel times but must be taken into account in modelling P-wave polarization anomalies. P-wave polarization anomalies within the anisotropic half-space are the sum of the particle motion deviation from the ray-path, and ray-path deviation from the source-receiver azimuth, and are typically range-dependent. P-wave polarization anomalies will also occur in the overlying isotropic layer, purely as a result of the ray-path deviation, with greater deviations at long ranges. A simple model of upper mantle anisotropy (6kms-1 crust above 8.0–8.6 km s-1 anisotropic mantle) produces surface polarization anomalies of 1.5–3.5°. In practice, P-wave polarization anomalies are difficult to observe because they are small compared to observed scatter in polarization data and are sensitive to differences in horizontal seismometer gain levels. We examine the implications of anisotropic ray-path deviations on travel times and P-wave polarizations for the Ngendei data as well as some previous marine seismic refraction experiments.

Cosmic Background Anisotropy Induced by Isotropic Flat-Spectrum Gravitational-Wave Perturbations

Abstract: A calculation is made of the temperature anisotropy that would be produced in the cosmic microwave background by an isotropic, stochastic ensemble of primordial gravitational waves having a flat initial spectrum. On angular scales THETA > 2 the anisotropy autocorrelation function has practically the same multipole dependence as previously established for the case of flat-spectrum adiabatic perturbations, while on scales THETA < 1 the anisotropy becomes insignificant. Upper limits are placed on the gravitational-wave amplitude and on the expected quadrupole anisotropy.

On the Theory of a Cosserat Point and Its Application to the Numerical Solution of Continuum Problems

Abstract: The theory of a Cosserat point is developed to describe motion of a body that is essentially a material point surrounded by some small volume. The development of this theory is motivated mainly by its applicability to the numerical solution of continuum problems. Attention is confined to the purely mechanical theory and nonlinear balance laws are proposed for Cosserat points with arbitrary constitutive properties. The linearized theory is developed and constitutive equations for an elastic material are discussed within the context of both the nonlinear and linear theories. Explicit constitutive equations for a linear-elastic isotropic Cosserat point are developed to model a parallelepiped composed of a linear-elastic homogeneous isotropic material.

A normal stress criterion for crack extension direction in orthotropic composite materials

Abstract: A criterion for predicting the direction of crack extension in orthotropic composite materials is presented. The criterion is based upon the normal stress and the anisotropic tensile strength on arbitrary planes about the tip of a crack. Results are obtained, via finite element solutions, for: (1) isotropic mixed mode fracture, (2) cracks in unidirectional off-axis slotted composite tensile coupons and (3) cracks in cross plied laminates. Comparisons are made with other theories and experimental results.

Theory of backscattering enhancement of random discrete isotropic scatterers based on the summation of all ladder and cyclical terms

Abstract: Backscattering enhancement of random discrete scatterers was previously investigated by a second-order theory. In this paper it is shown that the second-order theory is inadequate for problems with appreciable albedo and optical thickness. Multiple-scattering effects are included by summing all the ladder terms and the cyclical terms for isotropic point particles. The summation of the ladder terms leads to the classical Schwarzchild–Milne integral equation. The summation of the cyclical terms leads to a two-variable cyclical-transfer integral equation. The cyclical-transfer integral equation is then solved numerically, and the results include all the multiple-scattering effects associated with cyclical terms. Numerical results are illustrated as a function of scattering angle, albedo, and optical thickness. Results demonstrate that higher-order multiple-scattering effects are important for appreciable albedo and optical thickness. The multiple-scattering solution also gives a sharper backscattering peak than the second-order solution.

Observations of laterally inhomogeneous anisotropy in the continental lithosphere

Abstract: Elastic anisotropy may be an important factor when seismic data are used to determine the structure, composition and dynamics of the Earth's interior1,2. Azimuthal anisotropy has been found in the oceanic and, although less definitely, in the continental lithosphere from field experiments where the Pn-wave velocity was observed to vary with direction3–5. These studies suggest a 180° periodicity in travel-time variations as the only criterion of the presence of azimuthal anisotropy. This criterion, however, may fail to distinguish between anisotropy and lateral inhomogeneity of the isotropic medium. A new method for measuring azimuthal anisotropy has been described recently6, based on the phenomenon of shear-wave splitting in anisotropic media, which suggests three criteria instead of one to identify azimuthal anisotropy. We have now applied this method to the records of permanent seismograph stations in southern Germany. Our results provide strong evidence for the presence of azimuthal anisotropy in the lithosphere of the region and reveal pronounced lateral variations in the parameters of anisotropy on a scale of about 200 km.

The yield surface of textured polycrystals

Abstract: T he plastic anisotropy of a material is characterized in part by its yield surface. It is shown that conventional descriptions, based on extensions of the von Mises hypothesis for isotropic materials, are experimentally and theoretically inadequate in many instances. Symmetry arguments are used to derive the dimensionality and extent of the space necessary for representing the yield surface under various conditions of anisotropy. A useful concept is introduced: “closed” subspaces, in which sections and projections of the yield surface are identical and in which, therefore, normality is complete. Yield surfaces of heavily rolled or sheared sheets are derived from a computer simulation of polycrystal plasticity. It is found that even mild textures give rise to significant departures from “oval” yield surfaces: they develop sharp ridges and extensive flats. The anisotropy coefficients for in-plane tension of rolled sheets have been calculated. For torsion testing under fixed and free end conditions, respectively, the axial force and the length change have been calculated, as well as the change in the ratio of wall thickness to diameter.

p-polarized nonlinear surface polaritons in materials with intensity-dependent dielectric functions

Abstract: We investigate the properties of p-polarized nonlinear surface polaritons (NLSP) propagating along the interfaces of optically nonlinear materials. We show that Maxwell's equations for the NLSP can be solved exactly in quadratures for optically isotropic media with dielectric functions which can be an arbitrary function of the field intensity. The required boundary conditions can be imposed readily, and a form of the dispersion relation for the NLSP is obtained without the need to solve for the field profile first. The general results are then applied to a specific model in which the material has a nonlinear dielectric function proportional to the electric field intensity. Both the self-focusing and self-defocusing cases are studied, as well as different values of the linear dielectric functions inside and outside the material. The physically allowed regions in parameter space and the nonlinear surface-plasmon resonance conditions are examined. The field profile in each region is also investigated.