Showing papers on "Isotropy published in 1997"

A new integral equation formalism for the polarizable continuum model: Theoretical background and applications to isotropic and anisotropic dielectrics

Abstract: We present a new integral equation formulation of the polarizable continuum model (PCM) which allows one to treat in a single approach dielectrics of different nature: standard isotropic liquids, intrinsically anisotropic medialike liquid crystals and solid matrices, or ionic solutions. The present work shows that integral equation methods may be used with success also for the latter cases, which are usually studied with three-dimensional methods, by far less competitive in terms of computational effort. We present the theoretical bases which underlie the method and some numerical tests which show both a complete equivalence with standard PCM versions for isotropic solvents, and a good efficiency for calculations with anisotropic dielectrics.

Evaluation of Solvent Effects in Isotropic and Anisotropic Dielectrics and in Ionic Solutions with a Unified Integral Equation Method: Theoretical Bases, Computational Implementation, and Numerical Applications

Abstract: We present the full implementation of the integral equation formalism (IEF) we have recently formulated to treat solvent effects. The method exploits a single common approach for dielectrics of very different nature: standard isotropic liquids, intrinsically anisotropic media like liquid crystals, and ionic solutions. We report here an analysis of its both formal and technical details as well as some numerical applications addressed to state the achieved generalization to all kinds of molecular solutes and to show the equally reliable performances in treating such different environmental systems. In particular, we report, for isotropic liquids, data of solvation free energies and static (hyper)polarizabilities of various molecular solutes in water, for anisotropic dielectrics, a study of an SN2 reaction, and finally, for ionic solution, a study of some structural aspects of ion pairing.

Large Elastic Deformations of Isotropic Materials

Abstract: The equilibrium of a cube of incompressible, neo-Hookean material, under the action of three pairs of equal and oppositely directed forces f 1, f 2, f 3, applied normally to, and uniformly distributed over, pairs of parallel faces of the cube, is studied. It is assumed that the only possible equilibrium states are states of pure, homogeneous deformation.

Design of materials with extreme thermal expansion using a three-phase topology optimization method

Abstract: Composites with extremal or unusual thermal expansion coefficients are designed using a three-phase topology optimization method. The composites are made of two different material phases and a void phase. The topology optimization method consists in finding the distribution of material phases that optimizes an objective function (e.g. thermoelastic properties) subject to certain constraints, such as elastic symmetry or volume fractions of the constituent phases, within a periodic base cell. The effective properties of the material structures are found using the numerical homogenization method based on a finite-element discretization of the base cell. The optimization problem is solved using sequential linear programming. To benchmark the design method we first consider two-phase designs. Our optimal two-phase microstructures are in fine agreement with rigorous bounds and the so-called Vigdergauz microstructures that realize the bounds. For three phases, the optimal microstructures are also compared with new rigorous bounds and again it is shown that the method yields designed materials with thermoelastic properties that are close to the bounds. The three-phase design method is illustrated by designing materials having maximum directional thermal expansion (thermal actuators), zero isotropic thermal expansion, and negative isotropic thermal expansion. It is shown that materials with effective negative thermal expansion coefficients can be obtained by mixing two phases with positive thermal expansion coefficients and void.

Collision statistics in an isotropic particle-laden turbulent suspension. Part 1. Direct numerical simulations

Abstract: Direct numerical simulations of heavy particles suspended in a turbulent fluid are performed to study the rate of inter-particle collisions as a function of the turbulence parameters and particle properties. The particle volume fractions are kept small (∼10−4) so that the system is well within the dilute limit. The fluid velocities are updated using a pseudo-spectral algorithm while the particle forces are approximated by Stokes drag. One unique aspect of the present simulations is that the particles have finite volumes (as opposed to point masses) and therefore particle collisions must be accounted for. The collision frequency is monitored over several eddy turnover times. It is found that particles with small Stokes numbers behave similarly to the prediction of Saffman & Turner (1956). On the other hand, particles with very large Stokes numbers have collision frequencies similar to kinetic theory (Abrahamson 1975). For intermediate Stokes numbers, the behaviour is complicated by two effects: (i) particles tend to collect in regions of low vorticity (high strain) due to a centrifugal effect (preferential concentration); (ii) particle pairs are less strongly correlated with each other, resulting in an increase in their relative velocity. Both effects tend to increase collision rates, however the scalings of the two effects are different, leading to the observed complex behaviour. An explanation for the entire range of Stokes numbers can be found by considering the relationship between the collision frequency and two statistical properties of the particle phase: the radial distribution function and the relative velocity probability density function. Statistical analysis of the data, in the context of this relationship, confirms the relationship and provides a quantitative description of how preferential concentration and particle decorrelation ultimately affect the collision frequency.

Analysis of the elastic properties of open-cell foams with tetrakaidecahedral cells

Abstract: The elastic constants of an open-cell foam model, having tetrakaidecahedral cells on a BCC lattice, were found The Young's modulus, shear modulus and Poisson's ratio were derived, as functions of the edge cross section and the foam density, by considering the bending, twisting and extension of the cell edges If edge bending were the only mechanism the moduli would vary with the square of the foam density The other deformation mechanisms are predicted to reduce the power law exponent by 3–5%, and the effect of edge torsion on the modulus level is small The foam bulk modulus is predicted to vary linearly with its relative density, so Poisson's ratio approaches 05 at low densities The lattice model is elastically isotropic, whereas other lattice models of foams are highly anisotropic

Linear Elastic Behavior of a Low-Density Kelvin Foam With Open Cells

Abstract: A micromechanical analysis for the linear elastic behavior of a low-density foam with open cells is presented. The foam structure is based on the geometry of Kelvin soap froth with flat faces: 14-sided polyhedral cells contain six squares and eight hexagons. Four struts meet at every joint in the perfectly ordered, spatially periodic, open-cell structure. All of the struts and joints have identical shape. Strut-level force-displacement relations are expressed by compliances for stretching, bending, and twisting. We consider arbitrary homogeneous deformations of the foam and present analytic results for the force, moment, and displacement at each strut midpoint and the rotation at each joint. The effective stress-strain relations for the foam, which has cubic symmetry, are represented by three elastic constants, a bulk modulus, and two shear moduli, that depend on the strut compliances. When these compliances are evaluated for specific strut geometries, the shear moduli are nearly equal and therefore the elastic response is nearly isotropic. The variational results of Hashin and Shtrikman are used to calculate the effective isotropic shear modulus of a polycrystal that contain grains of Kelvin foam.

Formulation and implementation of three-dimensional viscoelasticity at small and finite strains

Abstract: Purely elastic material models have a limited validity. Generally, a certain amount of energy absorbing behaviour can be observed experimentally for nearly any material. A large class of dissipative materials is described by a time- and frequency-dependent viscoelastic constitutive model. Typical representatives of this type are polymeric rubber materials. A linear viscoelastic approach at small and large strains is described in detail and this makes a very efficient numerical formulation possible. The underlying constitutive structure is the generalized Maxwell-element. The derivation of the numerical model is given. It will be shown that the developed isotropic algorithmic material tensor is even valid for the current configuration in the case of large strains. Aspects of evaluating experimental investigations as well as parameter identification are considered. Finally, finite element simulations of time-dependent deformations of rubber structures using mixed elements are presented.

Four-manifolds with positive isotropic curvature

Abstract: 1. Positive Isotropic Curvature 2 (1) The Result 2 (2) The Algebra of Isotropic Curvature 4 2. Curvature Pinching 6 (1) Pinching Estimates which are Preserved 6 (2) Pinching Estimates which Improve 13 (3) Necklike Curvature Pinching 21 3. The Geometry of Necks 27 (1) Harmonic Parametrizations by Spheres 27 (2) Geometric Necks 30 (3) Curvature Necks 35 (4) The Fundamental Group 41 (5) Finding Necks 44 4. Surgery 47 (1) How to do Surgery 47 (2) Curvature Changes under Surgery .49 (3) Pinching Estimates under Surgery 57 5. Recovering the Manifold from Surgery , 60 Research supported by NSF grant # DMS92-04336

On the large deformation behaviour of reinforced rubber at different temperatures

Abstract: This essay investigates the temperature dependence of the mechanical properties of a filler-loaded tread compound experimentally and proposes a physically based method to represent this behaviour in the framework of non-linear continuum thermomechanics. To this end, we realise a series of monotonic and cyclic strain controlled tests on cylindrical specimens in tension at different temperature levels. The experimental data show the isothermal mechanical behaviour to be mainly influenced by non-linear elasticity in combination with non-linear rate dependence and weak equilibrium hysteresis. We observe that the rate sensitivity of the material depends strongly on the temperature : at low temperature levels, the rate sensitivity is essentially higher than at high temperatures. The elastic properties of the material depend comparatively less on the temperature. Nevertheless, higher temperature levels lead to higher equilibrium stresses. In order to represent the material behaviour, we start with a multiplicative split of the deformation gradient into a mechanical and a thermal part as proposed by Lu and Pister (1975). Physically, this idea corresponds to a stress-free thermal expansion followed by an isothermal stress-producing deformation. We suppose the thermal part of the deformation gradient to be isotropic. As a consequence of this, the velocity gradient decomposes additively into a pure thermal and a pure mechanical part. By using these elements, we exploit the Clausius Duhem inequality and assume the so-called ‘mechanical second Piola Kirchhoff stress tensor’ to be a functional of the ‘mechanical Green's strain tensor’. In a further step, we define this functional by a system of constitutive equations which are based on a rheological model. The evolution equations for the internal variables are formulated by using the concept of dual variables proposed by Haupt and Tsakmakis (1989, 1996). The rate sensitivity is modelled by a stress and temperature dependent viscosity function. The elastic part of the equilibrium stress is described by entropy elasticity in combination with a modified Mooney Rivlin strain energy function. The equilibrium hysteresis effects are represented by rate independent plasticity in arclength representation as proposed by Valanis (1971). The constitutive model is compatible with the dissipation principle of thermodynamics and describes the general trend of the experimental data fairly well.

On the Universality of the Kolmogorov Constant in Numerical Simulations of Turbulence

Abstract: Motivated by a recent survey of experimental data [K.R. Sreenivasan, Phys. Fluids, 2778 (1995)], we examine data on the Kolmogorov spectrum constant in numerical simulations of isotropic turbulence, using results both from previous studies and from new direct numerical simulations over a range of Reynolds numbers (up to 240 on the Taylor scale) at grid resolutions up to 5123. It is noted that in addition to k-5/3 scaling, identification of a true inertial range requires spectral isotropy in the same wavenumber range. We found that a plateau in the compensated three-dimensional energy spectrum at k eta ~ 0.1--0.2 , commonly used to infer the Kolmogorov constant from the compensated three-dimensional energy spectrum, actually does not represent proper inertial range behavior. Rather, a proper, if still approximate, inertial range emerges at k eta ~ 0.02-0.05 when R>sub /sub sub /sub sub /sub sub /sub< ~ 0.53 for C =1.62, in excellent agreement with experiments. However the one- and three-dimensional estimates are not fully consistent, because of departures (due to numerical and statistical limitations) from isotropy of the computed spectra at low wavenumbers. The inertial scaling of structure functions in physical space is briefly addressed. Since DNS is still restricted to moderate Reynolds numbers, an accurate evaluation of the Kolmogorov constant is very difficult. We focus on providing new insights on the interpretation of Kolmogorov 1941 similarity in the DNS literature and do not consider issues pertaining to the refined similarity hypotheses of Kolmogorov.

Fast Hole Transport in a New Calamitic Liquid Crystal of 2-(4′-Heptyloxyphenyl)-6-Dodecylthiobenzothiazole

Abstract: The carrier transport in different phases of a new photoconductive calamitic liquid crystal, 2-(4\ensuremath{'}-heptyloxyphenyl)-6-dodecylthiobenzothiazole was studied by the time-of-flight technique: In the smectic A phase, a fast hole transient photocurrent was obtained in a nondispersive manner, in which the mobility was as high as $5\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}3}\phantom{\rule{0ex}{0ex}}{\mathrm{cm}}^{2}/\mathrm{V}\mathrm{s}$ and independent of applied electric field; in the isotropic phase, however, slower carrier transport was observed, probably due to positive or negative ions, and their mobilities were as low as ${10}^{\ensuremath{-}5}\phantom{\rule{0ex}{0ex}}{\mathrm{cm}}^{2}/\mathrm{V}\mathrm{s}$. These experimental results demonstrate the importance of local molecular alignment in creating the fast electronic conduction in calamitic liquid crystals.

Reflection moveout and parameter estimation for horizontal transverse isotropy

Abstract: Transverse isotropy with a horizontal axis of symmetry (HTI) is the simplest azimuthally anisotropic model used to describe fractured reservoirs that contain parallel vertical cracks. Here, I present an exact equation for normal-moveout (NMO) velocities from horizontal reflectors valid for pure modes in HTI media with any strength of anisotropy. The azimuthally dependent P -wave NMO velocity, which can be obtained from 3-D surveys, is controlled by the principal direction of the anisotropy (crack orientation), the P -wave vertical velocity, and an effective anisotropic parameter equivalent to Thomsen9s coefficient δ. An important parameter of fracture systems that can be constrained by seismic data is the crack density, which is usually estimated through the shear-wave splitting coefficient γ. The formalism developed here makes it possible to obtain the shear-wave splitting parameter using the NMO velocities of P and shear waves from horizontal reflectors. Furthermore, γ can be estimated just from the P -wave NMO velocity in the special case of the vanishing parameter e, corresponding to thin cracks and negligible equant porosity. Also, P -wave moveout alone is sufficient to constrain γ if either dipping events are available or the velocity in the symmetry direction is known. Determination of the splitting parameter from P -wave data requires, however, an estimate of the ratio of the P -to- S vertical velocities (either of the split shear waves can be used). Velocities and polarizations in the vertical symmetry plane of HTI media, that contains the symmetry axis, are described by the known equations for vertical transverse isotropy (VTI). Time-related 2-D P -wave processing (NMO, DMO, time migration) in this plane is governed by the same two parameters (the NMO velocity from a horizontal reflector and coefficient η) as in media with a vertical symmetry axis. The analogy between vertical and horizontal transverse isotropy makes it possible to introduce Thomsen parameters of the “equivalent” VTI model, which not only control the azimuthally dependent NMO velocity, but also can be used to reconstruct phase velocity and carry out seismic processing in off-symmetry planes.

P-SH conversions in a flat-layered medium with anisotropy of arbitrary orientation

Abstract: SUMMARY P-SH conversion is commonly observed in teleseismic P waves, and is often attributed to dipping interfaces beneath the receiver. Our modelling suggests an alternative explanation in terms of flat-layered anisotropy. We use reflectivity techniques to compute three-component synthetic seismograms in a 1-D anisotropic layered medium. For each layer of the medium, we prescribe values of seismic velocities and hexagonally symmetric anisotropy about a common symmetry axis of arbitrary orientation. A compressional wave in an anisotropic velocity structure suffers conversion to both SV-and SH-polarized shear waves, unless the axis of symmetry is everywhere vertical or the wave travels parallel to all symmetry axes. The P-SV conversion forms the basis of the widely used ‘receiver function’ technique. The P-SH conversion occurs at interfaces where one or both layers are anisotropic. A tilted axis of symmetry and a dipping interface in isotropic media produce similar amplitudes of both direct (P) and converted (Ps) phases, leaving the backazimuth variation of the P-Ps delay as the main discriminant. Seismic anisotropy with a tilted symmetry axis leads to complex synthetic seismograms in velocity models composed of just a few flat homogeneous layers. It is possible therefore to model observations of P coda with prominent transverse components with relatively simple 1-D velocity structures. Successful retrieval of salient model characteristics appears possible using multiple realizations of a genetic-algorithm (GA) inversion of P coda from several backazimuths. Using GA inversion, we determine that six P coda recorded at station ARU in central Russia are consistent with models that possess strong (> 10 per cent) anisotropy in the top 5 km and between 30 and 43 km depth. The symmetry axes are tilted, and appear aligned with the seismic anisotropy orientation in the mantle under ARU suggested by SKS splitting.

General theory of small elastic deformations superposed on finite elastic deformations

Abstract: Using tensor notations a general theory is developed for small elastic deformations, of either a compressible or incompressible isotropic elastic body, superposed on a known finite deformation, without assuming special forms for the strain-energy function. The theory is specialized to the case when the finite deformation is pure homogeneous. When two of the principal extension ratios are equal the changes in displacement and stress due to the small superposed deformation are expressed in terms of two potential functions in a manner which is analogous to that used in the infinitesimal deformation of hexagonally aeolotropic materials. The potential functions are used to solve the problem of the infinitesimally small indentation, by a spherical punch, of the plane surface of a semi-infinite body of incompressible isotropic elastic material which is first subjected to a finite pure homogeneous deformation symmetrical about the normal to the force-free plane surface.

A hemispherical model of anisotropic interstellar pickup ions

Abstract: We present an analytical model for the distribution function of interstellar pickup ions in a radially directed heliospheric magnetic field which naturally results in anisotropic particle distributions. The model includes the effects of convection and spatial transport in the solar wind, adiabatic deceleration in the radial flow, adiabatic focusing in the radial field, and pitch angle scattering toward isotropy in the frame of the solar wind. The pitch angle scattering is approximated by the hemispherical assumption: we take the scattering to be very efficient within each pitch angle range μ ≷ 0 (where μ is the cosine of the pitch angle) but inhibited between the hemispheres separated by μ = 0. The analytical solution is obtained for the case where the scattering rate across μ = 0 scales as the particle speed divided by the radial position of the fluid parcel. The model distribution functions can be used to interpret recent observations of anisotropic pickup ions.

Averaging inhomogeneous newtonian cosmologies

Abstract: Idealizing matter as a pressureless fluid and repre- senting its motion by a peculiar-velocity eld superimposed on a homogeneous and isotropic Hubble expansion, we apply (La- grangian) spatial averaging on an arbitrary domain D to the (nonlinear) equations of Newtonian cosmology and derive an exact, general equation for the evolution of the (domain de- pendent) scale factor aD (t). We consider the effect of inhomo- geneities on the average expansion and discuss under which circumstances the standard description of the average motion in terms of Friedmann's equation holds. We nd that this ef- fect vanishes for spatially compact models if one averages over the whole space. For spatially innite inhomogeneous models obeying the cosmological principle of large-scale isotropy and homogeneity, Friedmann models may provide an approxima- tion to the average motion on the largest scales, whereas for hierarchical(Charlier-type)modelsthegeneralexpansionequa- tion shows how inhomogeneities might appreciably affect the expansion at all scales. An averaged vorticity evolution law is also given. Since we employ spatial averaging, the problem of justifying ensemble averaging does not arise. A generalization of the expansion law to general relativity is straightforward for the case of irrotational flows and will be discussed. The effect may have important consequences for a variety of problems in large-scale structure modeling as well as for the interpretation of observations.

Angular dependence of nucleation by curling in a prolate spheroid

Abstract: The nucleation field is calculated for a small ferromagnetic prolate spheroid as a function of its size, its elongation, and the angle between the applied field and the spheroid axis. If there is no other switching mode except for magnetization curling and coherent rotation, these results probably represent the coercivity. The calculation is rigorous only for an isotropic particle, but a first order approximation is also given for a uniaxial or cubic anisotropy whose easy axis is parallel to the long axis of the spheroid.

An analysis of new and existing FDTD methods for isotropic cold plasma and a method for improving their accuracy

Abstract: Over the past few years, a number of different finite-difference time-domain (FDTD) methods for modeling electromagnetic propagation in an isotropic cold plasma have been published. We have analyzed the accuracy and stability of these methods to determine which method provides the greatest accuracy for a given computation time. For completeness, two new FDTD methods for cold plasma, one of which is based on the concept of exponential fitting, are introduced and evaluated along with the existing methods. We also introduce the concept of cutoff modification which can be easily applied to most of the FDTD methods, and which we show can improve the accuracy of these methods with no additional computational cost. Von Neumann's stability analysis is used to evaluate the stability of the various methods, and their accuracy is determined from a straightforward time-and-space harmonic analysis of the dispersion and dissipation errors. Results of numerical experiments to verify the accuracy analysis are presented. It is found that for low-loss plasma, the piecewise linear recursive convolution method (PLRC) method is the most accurate, but the method of Young (see Radio Sci., vol.29, p.1513-22, 1994) can use less memory and is nearly as accurate. In this low-loss plasma regime, cutoff modification can significantly reduce the error near cutoff at the expense of slightly greater error at lower frequencies. For strongly collisional plasmas, the PLRC method also provides the most accurate solution.

Light diffraction at mixed phase and absorption gratings in anisotropic media for arbitrary geometries

Abstract: The coupled wave theory of Kogelnik [H. Kogelnik, Bell Syst. Tech. J. 48, 2909 (1969)] is extended to the case of moderately absorbing thick anisotropic materials with grating vector and medium boundaries arbitrarily oriented with respect to the main axes of the optical indicatrix. Dielectric and absorption modulation with common grating vector and of arbitrary relative phase shift is considered. Solutions for the wave amplitudes, diffraction efficiencies, and angular mismatch sensitivities are given in transmission and reflection geometries. The main difference of the new results with respect to the expressions valid for isotropic media arise due to the walk-off between the wave-front and energy propagation directions. The difference is particularly important in materials with large birefringence, such as organic crystals, ordered polymers, and liquid crystalline cells. The special case of Bragg diffraction and two-beam coupling at holograms recorded in optically inactive photorefractive crystals is analyzed in detail. It is found that the two-beam coupling gain is influenced substantially by an absorption anisotropy.

On the motion of gas bubbles in homogeneous isotropic turbulence

Abstract: This paper is concerned with the motion of small gas bubbles, equivalent diameter about 1.0 mm, in isotropic turbulent flows. Data on the mean velocity of rise and the dispersion of the bubbles have been obtained numerically by simulating the turbulence as a sum of Fourier modes with random phases and amplitudes determined by the Kraichnan and the von Karman–Pao energy-spectrum functions, and by calculating the bubble trajectories from a reasonably well-established equation of motion. The data cover the range β[les ]1, where β is the ratio between the turbulence intensity and the velocity of rise of the bubbles in still fluid. An approximate analysis based on the assumption that β is small yields results that compare favourably with the numerical data, and clarifies the important role played by the lift forces exerted by the fluid.

Poroelastic Solution for an Inclined Borehole

Abstract: The analytical solution for an infinitely long borehole in an isotropic, poroelastic medium, inclined to the far-field principal stresses, is presented. The solution utilizes a loading decomposition scheme which leads to three fundamental problems: a poroelastic plane-strain, an elastic uni-axial, and an elastic antiplane shear problem.

A general boundary element analysis of 2-D linear elastic fracture mechanics

Abstract: This paper presents a boundary element method (BEM) analysis of linear elastic fracture mechanics in two-dimensional solids. The most outstanding feature of this new analysis is that it is a single-domain method, and yet it is very accurate, efficient and versatile: Material properties in the medium can be anisotropic as well as isotropic. Problem domain can be finite, infinite or semi-infinite. Cracks can be of multiple, branched, internal or edged type with a straight or curved shape. Loading can be of in-plane or anti-plane, and can be applied along the no-crack boundary or crack surface. Furthermore, the body-force case can also be analyzed. The present BEM analysis is an extension of the work by Pan and Amadei (1996a) and is such that the displacement and traction integral equations are collocated, respectively, on the no-crack boundary and on one side of the crack surface. Since in this formulation the displacement and/or traction are used as unknowns on the no-crack boundary and the relative crack displacement (i.e. displacement discontinuity) as unknown on the crack surface, it possesses the advantages of both the traditional displacement BEM and the displacement discontinuity method (DDM) and yet gets rid of the disadvantages associated with these methods when modeling fracture mechanics problems. Numerical examples of calculation of stress intensity factors (SIFs) for various benchmark problems were conducted and excellent agreement with previously published results was obtained.