Showing papers on "Isotropy published in 2000"

Analysis of functionally graded plates

Abstract: Theoretical formulation, Navier's solutions of rectangular plates, and finite element models based on the third-order shear deformation plate theory are presented for the analysis of through-thickness functionally graded plates. The plates are assumed to have isotropic, two-constituent material distribution through the thickness, and the modulus of elasticity of the plate is assumed to vary according to a power-law distribution in terms of the volume fractions of the constituents. The formulation accounts for the thermomechanical coupling, time dependency, and the von Karman-type geometric non-linearity. Numerical results of the linear third-order theory and non-linear first-order theory are presented to show the effect of the material distribution on the deflections and stresses. Copyright © 2000 John Wiley & Sons, Ltd.

High-resolution tensor MR elastography for breast tumour detection.

Abstract: MR elastography is a novel imaging technique for the visualization of elastic properties of tissue. It is expected that this method will have diagnostic value for the clarification of suspicious breast lesions. Low-frequency mechanical waves are coupled into the tissue and visualized via an MR sequence which is phase-locked to the mechanical excitation. Commonly, elasticity is assumed to be isotropic and reconstruction is performed in only two dimensions. The technique is extended to three dimensions such that the entire symmetric elasticity tensor is assessed. This is achieved by measuring different phases of the mechanical wave during one oscillatory cycle. Thereby it is possible to provide information about the anisotropy of the elasticity tensor. Finite-element simulations as well as phantom experiments are performed to demonstrate the feasibility of the method. Initial clinical results of a breast carcinoma are presented. The analysis of the eigenvalues of the elasticity tensor support the hypothesis that breast carcinoma might exhibit an anisotropic elasticity distribution. The surrounding benign tissue appears isotropic. Thereby new and additional diagnostic information is provided which might help in distinguishing between benign and malignant breast diseases.

Mechanical Response of Polymers: An Introduction

Abstract: Preface 1. Discussion of response of a viscoelastic material 2. Constitutive equations for one-dimensional response of viscoelastic materials: mechanical analogs 3. Constitutive equations for one-dimensional linear response of a viscoelastic material 4. Some features of the linear response of viscoelastic materials 5. Histories with constant strain or stress rates 6. Sinusoidal oscillations 7. Constitutive equation for three dimensional linear isotropic viscoelastic materials 8. Axial load, bending and torsion 9. Dynamics of bodies with viscoelastic support 10. Boundary value problems for linear isotropic viscoelastic materials 11. Influence of temperature Appendices References Index.

A transfer-matrix approach for estimating the characteristic impedance and wave numbers of limp and rigid porous materials

Abstract: A method for evaluating the acoustical properties of homogeneous and isotropic porous materials that may be modeled as fluids having complex properties is described here. To implement the procedure, a conventional, two-microphone standing wave tube was modified to include: a new sample holder; a section downstream of the sample holder that accommodated a second pair of microphone holders and an approximately anechoic termination. Sound-pressure measurements at two upstream and two downstream locations were then used to estimate the two-by-two transfer matrix of porous material samples. The experimental transfer matrix method has been most widely used in the past to measure the acoustical properties of silencer system components. That procedure was made more efficient here by taking advantage of the reciprocal nature of sound transmission through homogeneous and isotropic porous layers. The transfer matrix of a homogeneous and isotropic, rigid or limp porous layer can easily be used to identify the material’s characteristic impedance and wave number, from which other acoustical quantities of interest can be calculated. The procedure has been used to estimate the acoustical properties of a glass fiber material: good agreement was found between the estimated acoustical properties and those predicted by using the formulas of Delany and Bazley.

An affine formulation for the prediction of the effective properties of nonlinear composites and polycrystals

Abstract: Variational approaches for nonlinear elasticity show that Hill’s incremental formulation for the prediction of the overall behaviour of heterogeneous materials yields estimates which are too stiff and may even violate rigorous bounds. This paper aims at proposing an alternative ‘affine’ formulation, based on a linear thermoelastic comparison medium, which could yield softer estimates. It is first described for nonlinear elasticity and specified by making use of Hashin–Shtrikman estimates for the linear comparison composite; the associated affine self-consistent predictions are satisfactorily compared with incremental and tangent ones for power-law creeping polycrystals. Comparison is then made with the second-order procedure (Ponte Castaneda, P., 1996. Exact second-order estimates for the effective mechanical properties of nonlinear composite materials. J. Mech. Phys. Solids, 44 (6), 827–862) and some limitations of the affine method are pointed out; explicit comparisons between different procedures are performed for isotropic, two-phase materials. Finally, the affine formulation is extended to history-dependent behaviours; application to the self-consistent modelling of the elastoplastic behaviour of polycrystals shows that it offers an improved alternative to Hill’s incremental formulation.

A new class of extremal composites

Abstract: The paper presents a new class of two-phase isotropic composites with extremal bulk modulus. The new class consists of micro geometrics for which exact solutions can be proven and their bulk moduli are shown to coincide with the Hashin–Shtrikman bounds. The results hold for two and three dimensions and for both well- and non-well-ordered isotropic constituent phases. The new class of composites constitutes an alternative to the three previously known extremal composite classes: finite rank laminates, composite sphere assemblages and Vigdergauz microstructures. An isotropic honeycomb-like hexagonal microstructure belonging to the new class of composites has maximum bulk modulus and lower shear modulus than any previously known composite. Inspiration for the new composite class comes from a numerical topology design procedure which solves the inverse homogenization problem of distributing two isotropic material phases in a periodic isotropic material structure such that the effective properties are extremized.

Phase singularities in isotropic random waves

Abstract: The singularities of complex scalar waves are their zeros; these are dislocation lines in space, or points in the plane. For waves in space, and waves in the plane (propagating in two dimensions, o...

Properties of spherical galaxies and clusters with an NFW density profile

Abstract: Using the standard dynamical theory of spherical systems, we calculate the properties of spherical galaxies and clusters whose density profiles obey the universal form first obtained in high resolution cosmological N-body simulations by Navarro, Frenk & White. We adopt three models for the internal kinematics: isotropic velocities, constant anisotropy and increasingly radial Osipkov-Merritt anisotropy. Analytical solutions are found for the radial dependence of the mass, gravitational potential, velocity dispersion, energy and virial ratio and we test their variability with the concentration parameter describing the density profile and amount of velocity anisotropy. We also compute structural parameters, such as half-mass radius, effective radius and various measures of concentration. Finally, we derive projected quantities, the surface mass density and line-of-sight as well as aperture velocity dispersion, all of which can be directly applied in observational tests of current scenarios of structure formation. On the mass scales of galaxies, if constant mass-to-light is assumed, the NFW surface density profile is found to fit well Hubble-Reynolds laws. It is also well fitted by Sersic R^(1/m) laws, for m ~ 3, but in a much narrower range of m and with much larger effective radii than are observed. Assuming in turn reasonable values of the effective radius, the mass density profiles imply a mass-to-light ratio that increases outwards at all radii.

Multiphase composites with extremal bulk modulus

Abstract: This paper is devoted to the analytical and numerical study of isotropic elastic composites made of three or more isotropic phases. The ranges of their effective bulk and shear moduli are restricted by the Hashin–Shtrikman–Walpole (HSW) bounds. For two-phase composites, these bounds are attainable, that is, there exist composites with extreme bulk and shear moduli. For multiphase composites, they may or may not be attainable depending on phase moduli and volume fractions. Sufficient conditions of attainability of the bounds and various previously known and new types of optimal composites are described. Most of our new results are related to the two-dimensional problem. A numerical topology optimization procedure that solves the inverse homogenization problem is adopted and used to look for two-dimensional three-phase composites with a maximal effective bulk modulus. For the combination of parameters where the HSW bound is known to be attainable, new microstructures are found numerically that possess bulk moduli close to the bound. Moreover, new types of microstructures with bulk moduli close to the bound are found numerically for the situations where the aforementioned attainability conditions are not met. Based on the numerical results, several new types of structures that possess extremal bulk modulus are suggested and studied analytically. The bulk moduli of the new structures are either equal to the HSW bound or higher than the bulk modulus of any other known composite with the same phase moduli and volume fractions. It is proved that the HSW bound is attainable in a much wider range than it was previously believed. Results are readily applied to two-dimensional three-phase isotropic conducting composites with extremal conductivity. They can also be used to study transversely isotropic three-dimensional three-phase composites with cylindrical inclusions of arbitrary cross-sections (plane strain problem) or transversely isotropic thin plates (plane stress or bending of plates problems).

On the decay of homogeneous isotropic turbulence

Abstract: Decaying homogeneous, isotropic turbulence is investigated using a phenomenological model based on the three-dimensional turbulent energy spectra. We generalize the approach first used by Comte-Bellot and Corrsin [J. Fluid Mech. 25, 657 (1966)] and revised by Saffman [J. Fluid Mech. 27, 581 (1967); Phys. Fluids 10, 1349 (1967)]. At small wave numbers we assume the spectral energy is proportional to the wave number to an arbitrary power. The specific case of power 2, which follows from the Saffman invariant, is discussed in detail and is later shown to best describe experimental data. For the spectral energy density in the inertial range we apply both the Kolmogorov −5/3 law, E(k)=Ce2/3k−5/3, and the refined Kolmogorov law by taking into account intermittency. We show that intermittency affects the energy decay mainly by shifting the position of the virtual origin rather than altering the power law of the energy decay. Additionally, the spectrum is naturally truncated due to the size of the wind tunnel tes...

Estimation of fracture parameters from reflection seismic data—Part III: Fractured models with monoclinic symmetry

Abstract: Existing geophysical and geological data indicate that orthorhombic media with a horizontal symmetry plane should be rather common for naturally fractured reservoirs. Here, we consider two orthorhombic models: one that contains parallel vertical fractures embedded in a transversely isotropic background with a vertical symmetry axis (VTI medium) and the other formed by two orthogonal sets of rotationally invariant vertical fractures in a purely isotropic host rock. Using the linear‐slip theory, we obtain simple analytic expressions for the anisotropic coefficients of effective orthorhombic media. Under the assumptions of weak anisotropy of the background medium (for the first model) and small compliances of the fractures, all effective anisotropic parameters reduce to the sum of the background values and the parameters associated with each fracture set. For the model with a single fracture system, this result allows us to eliminate the influence of the VTI background by evaluating the differences between t...

Density functional for hard sphere crystals: A fundamental measure approach

Abstract: A new free energy density functional for hard spheres is presented, along the lines of the fundamental measure theory, which reproduces the Percus-Yevick equation of state and direct correlation function for the fluid, with a tensor weighted density. The functional, based on the zero-dimension limit, is exact for any one-dimensional density distribution of the spheres. The application to the hard sphere crystals gives excellent results, solving all of the qualitative problems of previous density functional approximations, including the unit cell anisotropy in the fcc lattice and the description of the metastable bcc lattice. The hard sphere (HS) model is central to the study of molecular packing effects in fluids [1]. The Percus-Yevick (PY) approximation for the correlation structure in the HS fluid [2] is a keystone in the theory of liquids, and the development of density functional (DF) approximations for the Helmholtz free energy of inhomogeneous HS systems, F r, opened the study of fluids near walls or inside pores [3]. The transition from the dense HS fluid to the fcc crystal has been studied since the 1980s with nonlocal approximations for F r, based on the previous knowledge of the thermodynamics and the direct correlation function, cr, r, in the bulk fluid [4,5]. The solid phase is treated as a self-maintained inhomogeneous fluid, corresponding to a local minimum of the grand potential, V F r 2m N, which is compared with that for the homogeneous fluid in the same approximation. The results for the coexisting fluid and solid densities and for the properties of the solid phase at higher density are quite good with different types of nonlocal DF, such as the weighted density (WDA) [6,7], the effective liquid [8] approximations (ELA) and their modified forms (MWDA, GELA) [9– 11]. However, this general success has limitations: a good description of the solid phase is obtained only within a variational minimization of V restricted to normalized Gaussian peaks centered at the lattice positions; the relaxation of the normalization of each peak produces unphysical values of the unit cell occupancy and spoils the good results for the equation of state. Also there is a small but rather systematic error predicting too low values for the Lindemann ratio (the mean square deviation of the molecules from the lattice positions); attempts to remove this problem by relaxing the isotropic Gaussian unit cell density distributions gave qualitatively bad results, with the wrong sign for the anisotropy [12]. A new problem appears when the DF are used to describe less packed crystal structures, such as the bcc lattice which is never the equilibrium state for hard spheres but still may be useful when the HS are a reference system for softer interactions. Constrained computer simulations have been used to describe the hard sphere bcc crystal as a metastable state [13] and it is also possible to find local minima of the WDA free energy when the density is restricted to normalized peaks at the bcc lattice [13,14], but the results of the later are fully unphysical, with the Lindemann ratio increasing with the density and very high compressibility even at the close-packed density, when the molecules should be fixed at their lattice positions. To understand the reasons for both the remarkable success of the DF theories of crystallization, within the restricted minimization of normalized isotropic peaks in the fcc lattice, and their qualitative failure when the restrictions are removed, we have to look at the way in which the DF approximations are built. The nonlocal forms for F r reproduce, through its second order functional derivative, a given approximation (PY usually) for cr, r in the fluid. In the fcc crystal, with twelve nearest neighbor normalized Gaussian peaks around each particle, the short-range

Phase behavior of two-dimensional hard rod fluids

Abstract: Monte Carlo simulations are used to study two-dimensional hard rod fluids consisting of spherocylinders confined to lie in a plane. The phase behavior is mapped out as a function of the aspect ratio (L/D) of the particles, from the hard disc limit at one extreme (L/D=0) to the thin hard needle limit at the other (L/D=∞). For long rods, a 2D nematic phase is observed at high density in which the orientational correlation functions decay algebraically, indicating that the phase does not possess true long range orientational order. The simulation data indicate that the transition from this phase to the low density isotropic phase is continuous, via a Kosterlitz–Thouless disclination unbinding type mechanism, rather than first order. For short rods the nematic phase disappears so that, on expansion, the solid phase undergoes a first order transition directly to an isotropic phase. Although the latter phase is globally isotropic, we find evidence for strong local positional and orientational correlations between the particles. Where possible, the simulation results are compared and contrasted to experimental, simulation and theoretical data for other two-dimensional liquid crystalline systems.

Simulation of anisotropic wave propagation based upon a spectral element method

Abstract: We introduce a numerical approach for modeling elastic wave propagation in 2-D and 3-D fully anisotropic media based upon a spectral element method. The technique solves a weak formulation of the wave equation, which is discretized using a high-order polynomial representation on a finite element mesh. For isotropic media, the spectral element method is known for its high degree of accuracy, its ability to handle complex model geometries, and its low computational cost. We show that the method can be extended to fully anisotropic media. The mass matrix obtained is diagonal by construction, which leads to a very efficient fully explicit solver. We demonstrate the accuracy of the method by comparing it against a known analytical solution for a 2-D transversely isotropic test case, and by comparing its predictions against those based upon a finite difference method for a 2-D heterogeneous, anisotropic medium. We show its generality and its flexibility by modeling wave propagation in a 3-D transversely isotropic medium with a symmetry axis tilted relative to the axes of the grid.

Particle shape effects on the effective permittivity of anisotropic or isotropic media consisting of aligned or randomly oriented ellipsoidal particles

Abstract: The effective permittivity (dielectric constant) of anisotropic or isotropic porous media is affected by the shape of particles composing the mixture. Directional permittivities are influenced by extreme aspect ratio particles, often found aligned with the bedding plane of rock or soil. Our objectives were to determine the effects of particle shape and preferential orientation on the effective permittivity of porous media. Confocal spheroids (ellipsoids of revolution) were used to mathematically describe a range of particle shapes from disks to spheres to needles. Dielectric mixing models which account for the polarization due to inclusion shape and axial alignment were used to estimate the shape effect. Permittivity measurements in an anisotropic packing of disk-shaped mica particles using time domain reflectometry showed an alteration of the permittivity due to the shape effect. Two- and three-phase predictions based on Maxwell-Garnett (1904) showed trends similar to measurements in anisotropic packings of mica. Particle shape effects can be a significant factor in dielectric permittivity measurements and should be a consideration especially where particle aspect ratio deviates by more than an order of magnitude from that of a sphere (unity). As the particle shape is less spherical, the resulting effective permittivity of the mixture is more similar to the inclusion permittivity and differs more from the permittivity of the background. Ellipsoid size and surface area provide an estimate of the combined effects of bound water and particle shape on the effective mixture permittivity. For high aspect ratio particles, shape effects on the effective permittivity appear to be comparable in magnitude to those of bound water prevalent in clay-sized media.

Development and geometry of isotropic and directional shrinkage-crack patterns.

Abstract: We have studied shrinkage-crack patterns which form when a thin layer of an alumina/water slurry dries. Both isotropic and directional drying were studied. The dynamics of the pattern formation process and the geometric properties of the isotropic crack patterns are similar to what is expected from recent models, assuming weak disorder. There is some evidence of a gradual increase in disorder as the drying layer become thinner, but no sudden transition, in contrast to what has been seen in previous experiments. The morphology of the crack patterns is influenced by drying gradients and front propagation effects, with sharp gradients having a strong orienting and ordering effect.

Diffusion of General Data on Non-Flat Manifolds viaHarmonic Maps Theory: The Direction Diffusion Case

Abstract: In a number of disciplines, directional data provides a fundamental source of information. A novel framework for isotropic and anisotropic diffusion of directions is presented in this paper. The framework can be applied both to denoise directional data and to obtain multiscale representations of it. The basic idea is to apply and extend results from the theory of harmonic maps, and in particular, harmonic maps in liquid crystals. This theory deals with the regularization of vectorial data, while satisfying the intrinsic unit norm constraint of directional data. We show the corresponding variational and partial differential equations formulations for isotropic diffusion, obtained from an L_2 norm, and edge preserving diffusion, obtained from an L norm in general and an L_1 norm in particular. In contrast with previous approaches, the framework is valid for directions in any dimensions, supports non-smooth data, and gives both isotropic and anisotropic formulations. In addition, the framework of harmonic maps here described can be used to diffuse and analyze general image data defined on general non-flat manifolds, that is, functions between two general manifolds. We present a number of theoretical results, open questions, and examples for gradient vectors, optical flow, and color images.

An invariant basis for natural strain which yields orthogonal stress response terms in isotropic hyperelasticity

Abstract: A novel constitutive formulation is developed for finitely deforming hyperelastic materials that exhibit isotropic behavior with respect to a reference configuration The strain energy per unit reference volume, W, is defined in terms of three natural strain invariants, K1–3, which respectively specify the amount-of-dilatation, the magnitude-of-distortion, and the mode-of-distortion Distortion is that part of the deformation that does not dilate Moreover, pure dilatation (K2=0), pure shear (K3=0), uniaxial extension (K3=1), and uniaxial contraction (K3=−1) are tests which hold a strain invariant constant Through an analysis of previously published data, it is shown for rubber that this new approach allows W to be easily determined with improved accuracy Albeit useful for large and small strains, distinct advantage is shown for moderate strains (eg 2–25%) Central to this work is the orthogonal nature of the invariant basis If η represents natural strain, then {K1,K2,K3} are such that the tensorial contraction of (∂Ki/∂η) with (∂Kj/∂η) vanishes when i≠j This result, in turn, allows the Cauchy stress t to be expressed as the sum of three response terms that are mutually orthogonal In particular (summation implied) t=Ai∂W/∂Ki, where the ∂W/∂Ki are scalar response functions and the Ai are kinematic tensors that are mutually orthogonal

Isostatic response of the Australian lithosphere: Estimation of effective elastic thickness and anisotropy using multitaper spectral analysis

Abstract: Gravity and topography provide important insights regarding the degree and mechanisms of isostatic compensation. The azimuthally isotropic coherence function be- tween the Bouguer gravity anomaly and topography evolves from high to low for increasing wavenumber, a diagnostic that can be predicted for a variety of lithospheric loading models and used in inversions for flexural rigidity thereof. In this study we investigate the isostatic response of continental Australia. We consider the effects of directionally anisotropic plate strength on the coherelce. The anisotropic coherence function is calculated for regions of Australia that have distinctive geological and geophysical properties. The coherence estimation is performed by the Thomson multiple-Slepian-taper spectral analysis method extended to two-dimensional fields. Our analysis reveals the existence of flexural anisotropy in central Australia, indicative of a weaker N-S direction of lower Te. This observation is consistent with the suggestion that the parallel faults in that area act to make the lithosphere weaker in the direction perpendicular to them. It can. also be related to the N-S direction of maximum stress and possibly the presence of E-W running zones weakened due to differential sediment burial rates. We also demonstrate that the multitaper method has distinct advantages for computing the isotropic coherence function. The ability to make many independent estimates of the isostatic response that are minimally affected by spectral leakage results in a coherence that is more robust than with modified periodogram methods, particularly at low wavenumbers. Our analysis elucidates the reasons for discrepancies in previous estimates of effective elastic thickness Te of the Australian lithosphere. In isotropic inversions for Te, we obtain values that are as much as a factor of 2 less than those obtained in standard inversions of the periodogram coherence using Bouguer gravity and topography but greater than those obtained by inversions that utilize free-air rather than Bouguer gravity and ignore the presence of subsurface loads. However, owing to the low spectral power of the Australian topography, the uncertainty on any estimate of Te is substantial.

Controlling Anisotropy in Mass-Spring Systems

Abstract: This paper presents a deformable model that offers control of the isotropy or anisotropy of elastic material, independently of the way the object is tiled into volume elements. The new model is as easy to implement and almost as efficient as mass-spring systems, from which it is derived. In addition to controlled anisotropy, it contrasts with those systems in its ability to model constant volume deformations. We illustrate the new model by animating objects tiled with tetrahedral and hexahedral meshes.

A theory of stress-softening in incompressible isotropic materials

Abstract: A general theory of isotropic stress-softening in incompressible isotropic materials is developed. The principal idea is that a stress-softening material is an inelastic material that has selective memory of only the maximum previous deformation to which it is subjected. This memory dependence is incorporated within general material response functions that are monotone decreasing functions of a stress-softening variable, which is a monotone increasing function of the maximum previous strain experienced by the material. A loading criterion is introduced to identify when the material is loaded along its virgin deformation path where the maximum previous strain is its current value, and to identify when it is unloaded to deform subsequently as an ideal isotropic elastic material in both elastic loading and unloading, so long as the maximum previous strain is not exceeded. The effect of loading from a configuration of maximum previous strain is to further stress-soften the material. Results demonstrating the effects of stress-softening are obtained for general isotropic stress-softening materials in simple uniaxial extension and in simple shear. A simplified analytical model together with a special softening function are introduced to illustrate some general results and to provide specific analytical and graphical examples. Both general and model-specific analytical results obtained for simple uniaxial extension are shown to be consistent with the overall ideal phenomenological behavior exhibited in experiments by others on stress-softening in simple tension and compression. Similar but totally new results for simple shear are derived, and their relation to effects in simple tension are discussed. It is demonstrated that the larger effect of softening occurs in the simple uniaxial extension, the effect in even a gross equivalent simple shear being small. All results are obtained from general three-dimensional constitutive equations.

Radiating gravitational collapse with shear viscosity

Abstract: A model is proposed of a collapsing radiating star consisting of an isotropic fluid with shear viscosity undergoing radial heat flow with outgoing radiation. The pressure of the star, at the beginning of the collapse, is isotropic but owing to the presence of the shear viscosity the pressure becomes more and more anisotropic. The behaviour of the density, pressure, mass, luminosity and the effective adiabatic index is analysed. Our work is compared to the case of a collapsing shearing fluid of a previous model, for a star with 6 M⊙.

Anisotropic scaling contributions to high-order structure functions in high-reynolds-number turbulence

Abstract: We make an attempt at obtaining the scaling exponents for the anisotropic components of structure functions of order 2 through 6. We avoid mixing these components with their isotropic counterparts for each order by using tensor components that are entirely anisotropic. We do this by considering terms of the isotropic sector corresponding to j=0 in the SO(3) decomposition of each tensor, and then constructing components that are explicitly zero in the isotropic sector. We use an interpolation formula to compensate for the large-scale encroachment of inertial-range scales. This allows us to examine the lowest order anisotropic scaling behavior. The resulting anisotropic exponents for a given tensorial order are larger than those known for the corresponding isotropic part. One conclusion that emerges is that the anisotropy effects diminish with decreasing scale, although much more slowly than previously thought.

Nonresonance and global existence of prestressed nonlinear elastic waves

Abstract: The nonlinear hyperbolic system of pde's governing the evolution of the deformation of isotropic hyperelastic materials is considered. In the absence of boundaries and with an additional nonresonance or null condition, the system has global smooth solutions starting close to a one-parameter family of homogeneous dilations. The proof combines energy estimates with new decay estimates for the linear problem.