Showing papers on "Isotropy published in 1988"

Valence Bond Ground States in Isotropic Quantum Antiferromagnets

Abstract: Haldane predicted that the isotropic quantum Heisenberg spin chain is in a “massive” phase if the spin is integral. The first rigorous example of an isotropic model in such a phase is presented. The Hamiltonian has an exactSO(3) symmetry and is translationally invariant, but we prove the model has a unique ground state, a gap in the spectrum of the Hamiltonian immediately above the ground state and exponential decay of the correlation functions in the ground state. Models in two and higher dimension which are expected to have the same properties are also presented. For these models we construct an exact ground state, and for some of them we prove that the two-point function decays exponentially in this ground state. In all these models exact ground states are constructed by using valence bonds.

Light scattering from a sphere arbitrarily located in a Gaussian beam, using a Bromwich formulation

Abstract: We present a theoretical description of the scattering of a Gaussian beam by a spherical, homogeneous, and isotropic particle. This theory handles particles with arbitrary size and nature having any location relative to the Gaussian beam. The formulation is based on the Bromwich method and closely follows Kerker’s formulation for plane-wave scattering. It provides expressions for the scattered intensities, the phase angle, the cross sections, and the radiation pressure.

Effective permittivity of dielectric mixtures

Abstract: General mixing formulas are derived for discrete scatterers immersed in a host medium. The inclusion particles are assumed to be ellipsoidal. The electric field inside the scatterers is determined by quasi-static analysis, assuming the diameter of the inclusion particles to be much smaller than one wavelength. The results are applicable to general multiphase mixtures, and the scattering ellipsoids of the different phases can have different sizes and arbitrary ellipticity distribution and axis orientation, i.e. the mixture may be isotropic or anisotropic. The resulting mixing formula is nonlinear and is suitable for iterative solutions. The formula contains a quantity called the apparent permittivity, and with different choices of this quantity, the result leads to the generalized Lorentz-Lorenz formula, the generalized Polder-van Santen formula, and the generalized coherent potential-quasicrystalline approximation formula. The results are applied to calculating the complex effective permittivity of dry and wet snow, and sea ice. >

Elastic wave propagation in media with parallel fractures and aligned cracks

Abstract: A model of parallel slip interfaces simulates the behaviour of a fracture system composed of large, closely spaced, aligned joints. The model admits any fracture system anisotropy: triclinic (the most general), monoclinic, orthorhombic or transversely isotropic, and this is specified by the form of the 3 × 3 fracture system compliance matrix. The fracture system may be embedded in an anisotropic elastic background with no restrictions on the type of anisotropy. To compute the long wavelength equivalent moduli of the fractured medium requires at most the inversion of two 3 × 3 matrices. When the fractures are assumed on average to have rotational symmetry (transversely isotropic fracture system behaviour) and the background is assumed isotropic, the resulting equivalent medium is transversely isotropic and the effect of the additional compliance of the fracture system may be specified by two parameters (in addition to the two isotropic parameters of the isotropic background). Dilute systems of flat aligned microcracks in an isotropic background yield an equivalent medium of the same form as that of the isotropic medium with large joints, i.e. there are two additional parameters due to the presence of the microcracks which play roles in the stress-strain relations of the equivalent medium identical to those played by the parameters due to the presence of large joints. Thus, knowledge of the total of four parameters describing the anisotropy of such a fractured medium tells nothing of the size or concentration of the aligned fractures but does contain information as to the overall excess compliance due to the fracture system and its orientation. As the aligned microcracks, which were assumed to be ellipsoidal, with very small aspect ratio are allowed to become non-fiat, i.e. have a growing aspect ratio, the moduli of the equivalent medium begin to diverge from the standard form of the moduli for flat cracks. The divergence is faster for higher crack densities but only becomes significant for microcracks of aspect ratios approaching 0.3.

Micromechanical Aspects of Isotropic Granular Assemblies With Linear Contact Interactions

Abstract: The paper presents a micromechanical analysis of plane granular assemblies of discs with a range of diameters, and interacting according to linear contact force-interparticle compliance relationships. Contacts are assumed to be fixed and indestructible. Macroscopically, the system is described in terms of a two-dimensional analogue of generalized Hooke’s law. Explicit expressions for elastic constants in terms of microstructure are derived for dense isotropic assemblies. It is shown that Poisson’s ratio for dense systems depends on the ratio of tangential to normal contact stiffnesses. The derived expression for Poisson’s ratio is verified by numerically simulating plane assemblies comprising 1000 particles. The effect of density on Poisson’s ratio is investigated using numerical simulations. The theory of dense plane systems is extended to dense three-dimensional systems comprising spheres. Finally, it is shown that Poisson’s result ν=1/4 is recovered for spherical particles with central interactions.

The differential scheme and its application to cracked materials

Abstract: A Differential Scheme (DS) approximation for elastic properties of cracked materials is established by a limiting process on the basis of the DS for porous materials. The method is applied to obtain stiffness reduction due to randomly oriented elliptical and penny-shaped crack distributions in isotropic matrix, and to the case of aligned plane cracks in orthotropic sheets.

Variational bounds on the effective moduli of anisotropic composites

Abstract: The vritional inequalities of Hashin and Shtrikman are transformed to a simple and concise form. They are used to bound the effective conductivity tensor σ∗ of an anisotropic composite made from an arbitrary number of possibly anisotropic phases, and to bound the effective elasticity tensor C ∗ of an anisotropic mixture of two well-ordered isotropic materials. The bounds depend on the conductivities and elastic moduli of the components and their respective volume fractions. When the components are isotropic the conductivity bounds, which constrain the eigenvalues of σ∗, include those previously obtained by Hashin and Shtrikman, Murat and Tartar, and Lurie and Cherkaev. Our approach can also be used in the context of linear elasticity to derive bounds on C ∗ for composites comprised of an arbitrary number of anisotropic phases. For two-component composites our bounds are tighter than those obtained by Kantor and Bergman and by Francfort and Murat, and are attained by sequentially layered laminate materials.

The equivalent isotropic displacement factor

Abstract: A check of recent articles in Acta Crystallographica Section C shows that some confusion exists about the definition of the equivalent isotropic displacement factor Ueq. A common error is the use of the non-orthogonalized tensor U for the calculation of Ueq in non-orthogonal crystal systems. In addition, a number of cases have been found where at* is confused with a t or B with fl, or where the wrong factors are used to relate Ulj or fllj to Btj or vice versa. Ue~'S for the different crystal systems are derived from the general Y Y U a a a a expression Ueq l • , ~-'~ i j i j l j l\" J\" 0108-2701/88/040775-02503.00 Introduction Since anisotropic displacement factors* are to be deposited the equivalent isotropic displacement factors are published together with the atomic coordinates. Browsing through the structure papers in A eta Crystallographica one can find some fifty different definitions for Ueq or Beq, many of which are definitely wrong. Consequently, the Commission on Journals (1986) recommended use of the definitions given by * We follow here the recommendation by Brock (1984) and use the expression 'displacement factor' instead of'temperature factor'. © 1988 International Union of Crystallography 776 S H O R T C O M M U N I C A T I O N S Hamilton (1959) or by Willis & Pryor (1975). Unfortunately, it is not always stated clearly that the eigenvalues of the displacement tensor are derived from the orthogonalized tensor. Willis & Pryor (1975) define (u 2) = ] trace B, an equation which is often incorrectly applied to nonorthogonalized tensors. Also Prince (1982) does not point out explicitly in his chapter on equivalent isotropic temperature factors that Beq is derived from the orthogonalized tensor. A survey of all articles in Acta Crystallographica Section C which have been accepted for publication after the recommendation of the Commission on Journals (1986) still shows many cases of wrongly defined displacement factors. This short note should help to clarify the confusion, which also trapped the present authors on one occasion (Fischer & TiUmanns, 1983).

Wave propagation in laminated composite plates

Abstract: A stiffness method has been used in this article to study dispersive wave propagation in a laminated anisotropic plate. The advantage of this method is in its usefulness in obtaining numerical results for the dispersion characteristics of waves propagating in a plate with an arbitrary number of arbitrarily anisotropic laminae. This method has been applied here, as a way of illustration, to a plate made up of transversely isotropic laminae with the axis of isotropy of each lamina lying in the plane of the lamina. Results thus obtained are shown to agree well with the exact solutions for isotropic and transversely isotropic single layered plates. Numerical results are presented for cross‐ply (0°/90°/0°) laminated composite plates and show that the frequency spectrum in this case differs considerably from that for a single layered (0°) plate.

Mathematical Modelling of Flow Through Consolidated Isotropic Porous Media

Abstract: A new mathematical model is proposed for time-independent laminar flow through a rigid isotropic and consolidated porous medium of spatially varying porosity. The model is based upon volumetric averaging concepts. Explicit assumptions regarding the mean geometric properties of the porous microstructure lead to a relationship between tortuosity and porosity. Microscopic inertial effects are introduced through consideration of flow development within the pores. A momentum transport equation is derived in terms of the fluid properties, the porous medium porosity and a characteristic length of the microstructure. In the limiting cases of porosity unity and zero, the model yields the required Navier-Stokes equation for free flow and no flow in a solid, respectively.

Light dosimetry in optical phantoms and in tissues: I. Multiple flux and transport theory

Abstract: A simple multiple flux model, which is equivalent to the diffusion approximation, is derived from the equation of transfer in a plane as well as in a spherical geometry. The equations obtained are similar to those of the Kubelka-Munk and related heuristic models. This permits conclusions regarding the limitations of these models and the values of their constants. A simple calculation is also presented of the radiance as a function of direction in the diffusion domain. This, together with the effective attenuation coefficient, permits indirect experimental determination of both the albedo and the anisotropy factor (g) of the scattering function. Similarity relations are discussed, as they result from the so called delta-Eddington approximation, leading to the conclusion that far from boundaries and sources light propagation characteristics do not change very much when g and sigma s are varied, provided sigma s(1-g) is kept constant ( sigma s=scattering coefficient). Therefore, only two optical constants are required to approximately describe light propagation in homogeneous and isotropic media in the diffusion approximation.

Composite technology for total hip arthroplasty.

Abstract: Composite materials, which can be very strong while having a low modulus of elasticity, are being studied because such materials have potential to be made into isoelastic hip prostheses. Composites intended for medical applications incorporate carbon or polyamide as a fiber component, while polysulfone, polyetheretherketone, or polyethylene is used as a matrix component. Mechanical properties (especially the modulus of elasticity) are emphasized because of the desire to match those properties of the proximal femur. Many of the variables that affect the mechanical properties of these materials are explained. The application of stress to different fiber orientations demonstrates the mechanical properties of the composite, and this is proved mathematically. It is shown that in composites with fibers oriented in the same direction, the modulus of elasticity in the direction of the fibers generally approaches that of the fibers as the amount of matrix decreases. Perpendicular to the fibers, the modulus of elasticity of the composite is only slightly greater than that of the matrix material. For isotropic chopped-fiber composites, the modulus of elasticity approaches that of the matrix as the fiber content decreases; at high-fiber content, the modulus is significantly less than that of oriented long-fiber composites. In general, the modulus of elasticity and fiber content have a linear relationship. Composites have fatigue properties that vary with direction and approach ultimate strength in tension but are lower in compression. The fatigue properties of proposed composites are discussed. Abrasion as a cause of stress concentration sites and wear particles is considered.

Three-dimensional elastic modeling by the Fourier method

Abstract: Earlier work on three-dimensional forward modeling is extended to elastic waves using the equations of con­ servation of momentum and the stress-strain relations for an isotropic elastic medium undergoing infinitesimal deformation. In addition to arbitrary compressional (or P-wave) velocity and density variation in lateral and vertical directions, elastic modeling permits shear (or S-wave) velocity variation as well. The elastic wave equation is solved using a generalization of the method for the acoustic case. Computation of each time step begins by computing six strain components by per­ forming nine spatial partial differentiation operations on the three displacement components from the pre­ vious time step. The six strains and two Lame constants are linearly combined to yield six stress components.

X‐ray diffraction line broadening due to dislocations in non‐cubic materials. I. General considerations and the case of elastic isotropy applied to hexagonal crystals

Abstract: Use is made of the theory of dislocation-induced X-ray diffraction line broadening in the form presented by Krivoglaz, Martynenko & Ryaboshapka [Fiz. Metall. Metalloved. (1983), 55, 5–171 to express the so-called orientation factors occurring in the relations of diffraction profile parameters (e.g. Fourier coefficients, line widths) in a form which systematically takes into account both the lattice geometry and the elastic behaviour of the scattering crystals. The formalism can be used, in principle, for any materials and types of dislocations. In the case of elastically isotropic media the orientation factors can be described by analytical expressions. The application of the formalism is demonstrated in some detail for various slip systems in hexagonal polycrystals with random orientation of grains.

Scaled field particle theory of the structure and the thermodynamics of isotropic hard particle fluids

Abstract: The pair structure and the thermodynamics of isotropic fluids composed of hard particles of different shapes and concentrations are considered by a new approach which (i) unifies the diagramatic Percus–Yevick theory and the geometric scaled particle theory, and (ii) leads to analytic accurate approximations for the direct correlation functions and cavity distribution functions. The scaled‐particle interpolation idea of Reiss et al. is achieved diagramatically, approximating the full expansion of the direct correlation functions by appropriately renormalized low order graphs, in which the size of the field particle (‘‘black circle’’) is scaled. It is shown that the lowest order scaled field particle approximation, involving only pair excluded volumes as variables, and pair overlap volumes as functions, contains the exact solution of the Percus–Yevick integral equation for the general hard sphere mixture in D dimensions. An exact geometric relation obeyed by the excluded volume (covolume) of two fused conve...

Depression of nonlinearity in decaying isotropic turbulence

Abstract: Simulations of decaying isotropic Navier–Stokes turbulence exhibit depression of the normalized mean‐square nonlinear term to 57% of the value for a Gaussianly distributed velocity field with the same instantaneous velocity spectrum. Similar depression is found for dynamical models with random coupling coefficients (modified Betchov models). This suggests that the depression is dynamically generic rather than specifically driven by alignment of velocity and vorticity.

Finite strain solutions in compressible isotropic elasticity

Abstract: Three classes of compressible isotropic elastic solids are introduced, for each of which the strain energy, expressed as a function of the three principal invariants of the stretch tensors, is linear in two of its arguments and nonlinear in the third argument. One of these is the class of harmonic materials. Several deformation fields are examined, for which the governing equations, for general compressible isotropic elastic response, reduce to a nonlinear ordinary differential equation. For the three special classes of materials, this differential equation may be solved in closed form, giving a deformation field which is independent of the material response function, or its solution may be reduced to a single quadrature, involving the nonlinear material response function.

Two‐dimensional exchange NMR of powder samples. II. The dynamic evolution of two‐time distribution functions

Abstract: Theoretical as well as experimental examples concerning the evolution of the two‐time distribution S2‖0 (ω1,ω2 ;tm) as a function of the mixing time tm are presented, where S2‖0 is identical with the two‐dimensional (2D) absorption spectrum rendered by 2D exchange NMR spectroscopy of static powder samples. The model calculations comprise standard models like isotropic rotational diffusion or overall isotropic reorientation combined with discrete internal rotational jumps to simulate the chain dynamics of polymers. In any case, the 2D spectrum directly reflects the main aspects of the underlying motional mechanism. An axially symmetric coupling (η=0) between spin and lattice is assumed throughout. Thus, the angular information contained in a 2D spectrum is completely specified by a one‐dimensional jump angle distribution supplied with each spectrum. In connection with the simulations the numerical mapping of a discrete distribution function into a space of new variables is discussed. In the experimental se...

On the effective conductivity of polycrystals and a three‐dimensional phase‐interchange inequality

Abstract: We derive optimal bounds on the effective conductivity tensor of polycrystalline aggregates by introducing an appropriate null‐Lagrangian that is rotationally invariant. For isotropic aggregates of uniaxial crystals an outstanding conjecture of Schulgasser is proven, namely that the lowest possible effective conductivity of isotropic aggregates of uniaxial crystals is attained by a composite sphere assemblage, in which the crystal axis is directed radially outwards in each sphere. By laminating this sphere assemblage with the original crystal we obtain anisotropic composites that are extremal, i.e., attaining our bounds. These, together with other results established here, give a partial characterization of the set of all possible effective tensors of polycrystalline aggregates. The same general method is used to prove a conjectured phase interchange inequality for isotropic composites of two isotropic phases. This inequality correlates the effective conductivity of the composite with the effective tensor when the phases are interchanged. It leads to optimal bounds on the effective conductivity when another effective constant, such as the effective diffusion coefficient, has been measured, or when one has information about ζ1 which is a parameter characteristic of the microgeometry, or when one knows the material is symmetric, i.e., invariant under phase interchange like a three‐dimensional checkerboard.

Failure of Quasi-Isotropic Composite Laminates with Free Edges

Abstract: Failure in quasi-isotropic (π/4 and π/3) laminates with free edges under on-axis and off-axis loads has been studied experimentally and analytically. Although these laminates are isotropic in stiff...

Transportation theory of multiple scattering and its application to seismic coda waves of impulse source

Abstract: The energy distribution of elastic waves in an infinite elastic medium with uniformly and randomly distributed scatterers has been researched. The scattering process is assumed to be isotropic and without conversions between wave types. We get the equation on the distribution of energe density in time and space covering single as well as multiple scattering. Taking physical symmetry of the field into account, it can be simplified. In the case of small earthquakes, the energy source of elastic waves can be assumed as a short pulse emitted isotropically at t=0. The first-order approximate solution in the 3-dimensional space can be obtained, and it is equivalent to Sato's solution for single scattering. In the 2-dimensional space the complete analytical solution has been derived by the mathematical inductance which leads to a conclusion that the codas of surface waves can give the Q-factor related to intrinsic absorption. The equation obtained in this paper is more general.

On the Overall Properties of Nonlinearly Viscous Composites

Abstract: The overall mechanical properties of a power-law viscous material, as weakened by an isotropic distribution of voids, or reinforced by rigid particles, are investigated; the overall stress potential being estimated from a variational principle recently developed by Talbot & Willis ( IMA J. appl. Math. 35, 39 (1985)). ‘Self-consistent’ estimates are generated by two alternative conditions which coincide for linear behaviour; the general superiority of one is demonstrated. A rigorous upper bound for the stress potential of the voided material is established, which reduces to the classical Hashin–Shtrikman bound in the particular case of linear constitutive behaviour. It always lies below the elementary ‘Voigt’ bound, which is obtained from the classical principle of minimum energy. Dual results, based on the strain-rate potential, are also obtained and discussed. Comparisons are made between previous estimates of overall properties and those obtained in this work.

X‐ray diffraction line broadening due to dislocations in non‐cubic materials. II. The case of elastic anisotropy applied to hexagonal crystals

Abstract: The calculation of orientation factors determining the anisotropy of dislocation-induced broadening of X-ray diffraction lines in crystals is treated for elastically anisotropic materials in terms of a general formalism. The application of the procedure is demonstrated by representation of a computer program and numerical calculations of orientation factors for different slip systems in hexagonal polycrystals with randomly oriented grains. A comparison of the results with those obtained in the approximation of elastic isotropy shows that the anisotropy of the diffraction-line broadening is essentially caused by the geometrical part of the orientation factor. In most cases the elastic anisotropy of the crystal leads only to small corrections.

Computation of the amplitude of stress singular terms for cracks and reentrant corners

Abstract: : The theoretical basis and performance characteristics of two new methods for the computation of the coefficients of the terms of asymptotic expansions at reentrant corners from finite element solutions are presented The methods, called the contour integral method and the cutoff function method, are very efficient: The coefficients converge to their true values as fast as the strain energy, or faster In order to make the presentation as simple as possible, we assume that the elastic body is homogeneous and isotropic, is loaded by boundary tractions only and, in the neighborhood of the reentrant corner, its boundaries are stress free The methods described herein can be adapted to cases without such restrictions Keywords: Finite element methods, P-extension, Fracture mechanics, Elasticity, Stress intensity factors, Mixed mode, Extraction methods, Convergence, Error estimate