Showing papers on "Isotropy published in 1991"

The spatial structure and statistical properties of homogeneous turbulence

Abstract: A direct numerical simulation at resolution 2403 is used to obtain a statistically stationary three-dimensional homogeneous and isotropic turbulent field at a Reynolds number around 1000 (Rλ ≈ 150). The energy spectrum displays an inertial subrange. The velocity derivative distribution, known to be strongly non-Gaussian, is found to be close to, but not, exponential. The nth-order moments of this distribution, as well as the velocity structure functions, do not scale with n as predicted by intermittency models. Visualization of the flow confirms the previous finding that the strongest vorticity is organized in very elongated thin tubes. The width of these tubes is of the order of a few dissipation scales, while their length can reach the integral scale of the flow.

The effective mechanical properties of nonlinear isotropic composites

Abstract: A new variational structure is proposed that yields a prescription for the effective energy potentials of nonlinear composites in terms of the corresponding energy potentials for linear composites with the same microstructural distributions. The prescription can be used to obtain bounds and estimates for the effective mechanical properties of nonlinear composites from any bounds and estimates that may be available for the effective properties of linear composites. The main advantages of the procedure are the simplicity of its implementation, the generality of its applications and the strength of its results. The general prescription is applied to three special nonlinear composites : a porous material, a two-phase incompressible composite and a rigidly reinforced material. The results are compared with previously available results for the special case of power-law constitutive behavior.

Dissipation of energy in the locally isotropic turbulence

Abstract: In my note (Kolmogorov 1941 a ) I defined the notion of local isotropy and introduced the quantities B d d ( r ) = [ u d ( M ′ ) − u d ( M ) ] 2 , ¯ [ u n ( M ′ ) − u n ( M ) ¯ ] 2 , where r denotes the distance between the points M and M' , ud(M) and ud(M') are the velocity components in the direction MM' ¯¯ at the points M and M' , and un(M) and un(M') are the velocity components at the points M and M' in some direction, perpendicular to MM' .

Passive material properties of intact ventricular myocardium determined from a cylindrical model.

Abstract: The equatorial region of the canine left ventricle was modeled as a thick-walled cylinder consisting of an incompressible hyperelastic material with homogeneous exponential properties. The anisotropic properties of the passive myocardium were assumed to be locally transversely isotropic with respect to a fiber axis whose orientation varied linearly across the wall. Simultaneous inflation, extension, and torsion were applied to the cylinder to produce epicardial strains that were measured previously in the potassium-arrested dog heart. Residual stress in the unloaded state was included by considering the stress-free configuration to be a warped cylindrical arc. In the special case of isotropic material properties, torsion and residual stress both significantly reduced the high circumferential stress peaks predicted at the endocardium by previous models. However, a resultant axial force and moment were necessary to cause the observed epicardial deformations. Therefore, the anisotropic material parameters were found that minimized these resultants and allowed the prescribed displacements to occur subject to the known ventricular pressure loads. The global minimum solution of this parameter optimization problem indicated that the stiffness of passive myocardium (defined for a 20 percent equibiaxial extension) would be 2.4 to 6.6 times greater in the fiber direction than in the transverse plane for a broad range of assumed fiber angle distributions and residual stresses. This agrees with the results of biaxial tissue testing. The predicted transmural distributions of fiber stress were relatively flat with slight peaks in the subepicardium, and the fiber strain profiles agreed closely with experimentally observed sarcomere length distributions. The results indicate that torsion, residual stress and material anisotropy associated with the fiber architecture all can act to reduce endocardial stress gradients in the passive left ventricle.

Induced strain actuation of isotropic and anisotropic plates

Abstract: The development and experimental verification of the induced strain actuation of plate components of an intelligent structure is presented. Equations relating the actuation strains, created by induced strain actuators, to the strains induced in the actuator/substrate system are derived for isotropic and anisotropic plates. Plate strain energy relations are also developed. Several exact solutions are found for simple actuator/substrate systems, and a general procedure for solving the strain energy equations with a Rayleigh-Ritz procedure is formulated. Approximate Ritz solutions lead to both an understanding of the system design parameters, and to detailed models of cantilever plate systems. Simple test articles were used to verify the accuracy of the basic induced strain actuator/substrate system models, and cantilever plate test articles were built and tested to verify the ability of the models to predict the strains induced in systems with extensive stiffness coupling and complicated boundary conditions.

Light scattering by randomly oriented axially symmetric particles

Abstract: Light scattering by ensembles of independently scattering, randomly oriented, axially symmetric particles is considered. The elements of the scattering matrices are expanded in (combinations of) generalized spherical functions; this is advantageous in computations of both single and multiple light scattering. Waterman’s T-matrix approach is used to develop a rigorous analytical method to compute the corresponding expansion coefficients. The main advantage of this method is that the expansion coefficients are expressed directly in some basic quantities that depend on only the shape, morphology, and composition of the scattering axially symmetric particle; these quantities are the elements of the T matrix calculated with respect to the coordinate system with the z axis along the axis of particle symmetry. Thus the expansion coefficients are calculated without computing beforehand the elements of the scattering matrix for a large set of particle orientations and scattering angles, which minimizes the numerical calculations. Like the T-matrix approach itself, the method can be used in computations for homogeneous and composite isotropic particles of sizes not too large compared with a wavelength. Computational aspects of the method are discussed in detail, and some illustrative numerical results are reported for randomly oriented homogeneous dielectric spheroids and Chebyshev particles. Results of timing tests are presented; it is found that the method described is much faster than the commonly used method of numerical angle integrations.

Deformation mechanisms in negative Poisson's ratio materials : structural aspects

Abstract: Poisson's ratio in materials is governed by the following aspects of the microstructure: the presence of rotational degrees of freedom, non-affine deformation kinematics, or anisotropic structure. Several structural models are examined. The non-affine kinematics are seen to be essential for the production of negative Poisson's ratios for isotropic materials containing central force linkages of positive stiffness. Non-central forces combined with pre-load can also give rise to a negative Poisson's ratio in isotropic materials. A chiral microstructure with noncentral force interaction or non-affine deformation can also exhibit a negative Poisson's ratio. Toughness and damage resistance in these materials may be affected by the Poisson's ratio itself, as well as by generalized continuum aspects associated with the microstructure.

A frequency-dependent finite-difference time-domain formulation for transient propagation in plasma

Abstract: Previous FDTD (finite-difference time-domain) formulations were not capable of analyzing plasmas for two reasons. First, FDTD requires that at each time step the permittivity and conductivity be specified as constants that do not depend on frequency, while even for the simplest plasmas these parameters vary with frequency. Second, the permittivity of a plasma can be negative, which can cause terms in FDTD expressions to become singular. A novel FDTD formulation for frequency-dependent materials (FD)/sup 2/TD has been developed. It is shown that (FD)/sup 2/TD can be applied to compute transient propagation in plasma when the plasma can be characterized by a complex frequency-dependent permittivity. While the computational example presented is for a pulse normally incident on an isotropic plasma slab, the (FD)/sup 2/TD formulation is fully three-dimensional. It can accommodate arbitrary transient excitation, with the limitation that the excitation pulse must have no zero frequency energy component. Time-varying electron densities and/or collision frequencies could also be included. The formulation presented is for an isotropic plasma, but extension to anisotropic plasma should be fairly straightforward. >

On Local Isotropy of Passive Scalars in Turbulent Shear Flows

Abstract: An assessment of local isotropy and universality in high-Reynolds-number turbulent flows is presented. The emphasis is on the behaviour of passive scalar fields advected by turbulence, but a brief review of the relevant facts is given for the turbulent motion itself. Experiments suggest that local isotropy is not a natural concept for scalars in shear flows, except, perhaps, at such extreme Reynolds numbers that are of no practical relevance on Earth. Yet some type of scaling exists even at moderate Reynolds numbers. The relation between these two observations is a theme of this paper.

The general problem of elastic wave propagation in multilayered anisotropic media

Abstract: Exact analytical treatment of the interaction of harmonic elastic waves with n-layered anisotropic plates is presented. Each layer of the plate can possess up to as low as monoclinic symmetry and thus allowing results for higher symmetry materials such as orthotropic, transversely isotropic, cubic, and isotropic to be obtained as special cases. The wave is allowed to propagate along an arbitrary angle from the normal to the plate as well as along any azimuthal angle. Solutions are obtained by using the transfer matrix method. According to this method formal solutions for each layer are derived and expressed in terms of wave amplitudes. By eliminating these amplitudes the stresses and displacements on one side of the layer are related to those of the other side. By satisfying appropriate continuity conditions at interlayer interfaces a global transfer matrix can be constructed which relates the displacements and stresses on one side of the plate to those on the other. Invoking appropriate boundary conditions on the plates outer boundaries a large variety of important problems can be solved. Of these mention is made of the propagation of free waves on the plate and the propagation of waves in a periodic media consisting of a periodic repetition of the plate. Confidence is the approach and results are confirmed by comparisons with whatever is available from specialized solutions. A variety of numerical illustrations are included.

Measurements of particle dispersion obtained from direct numerical simulations of isotropic turbulence

Abstract: Measurements of heavy particle dispersion have been made using direct numerical simulations of isotropic turbulence. The parameters affecting the dispersion of solid particles, namely particle inertia and drift due to body forces were investigated separately. In agreement with the theoretical studies of Reeks, and Pismen & Nir, the effect of particle inertia is to increase the eddy diffusivity over that of the fluid (in the absence of particle drift). The increase in the eddy diffusivity of particles over that of the fluid was between 2 and 16%, in reasonable agreement with the increases reported in Reeks, and Pismen & Nir. The effect of a deterministic particle drift is shown to decrease unequally the dispersion in directions normal and parallel to the particle drift direction. Eddy diffusivities normal and parallel to particle drift are shown to be in good agreement with the predictions of Csanady and the experimental measurements of Wells & Stock.

Simplified moment tensor analysis and unified decomposition of acoustic emission source: Application to in situ hydrofracturing test

Abstract: An application of a moment tensor analysis to acoustic emission (AE) is studied to elucidate crack types and orientations of AE sources. In the analysis, simplified treatment is desirable, because hundreds of AE records are obtained from just one experiment and thus sophisticated treatment is realistically cumbersome. Consequently, a moment tensor inversion based on P wave amplitude is employed to determine six independent tensor components. Selecting only P wave portion from the full-space Green's function of homogeneous and isotropic material, a computer code named SiGMA (simplified Green's functions for the moment tensor analysis) is developed for the AE inversion analysis. To classify crack type and to determine crack orientation from moment tensor components, a unified decomposition of eigenvalues into a double-couple (DC) part, a compensated linear vector dipole (CLVD) part, and an isotropic part is proposed. The aim of the decomposition is to determine the proportion of shear contribution (DC) and tensile contribution (CLVD + isotropic) on AE sources and to classify cracks into a crack type of the dominant motion. Crack orientations determined from eigenvectors are presented as crack-opening vectors for tensile cracks and fault motion vectors for shear cracks, instead of stereonets. The SiGMA inversion and the unified decomposition are applied to synthetic data and AE waveforms detected during an in situ hydrofracturing test. To check the accuracy of the procedure, numerical experiments are performed on the synthetic waveforms, including cases with 10% random noise added. Results show reasonable agreement with assumed crack configurations. Although the maximum error is approximately 10% with respect to the ratios, the differences on crack orientations are less than 7°. AE waveforms detected by eight accelerometers deployed during the hydrofracturing test are analyzed. Crack types and orientations determined are in reasonable agreement with a predicted failure plane from borehole TV observation. The results suggest that tensile cracks are generated first at weak seams and then shear cracks follow on the opened joints.

Liquid crystal phase diagram of the Gay-Berne fluid

Abstract: In this paper we report computer simulation results for bulk Gay-Berne fluids with anisotropy parameters κ = 3 and κ′ = 5. Using molecular dynamics simulations in the NVT ensemble, we identify isotropic fluid, nematic and smectic B phases. We observe that the nematic phase is only stable for reduced temperatures T* > 0·80. At lower temperatures, the isotropic phase directly evolves to the smectic B phase via a first order transition. We also give evidence of a weakly first order transition which involves a tilt of the molecular orientations with respect to the smectic planes. The effect of the attractive anisotropic forces in stabilizing the orientationally ordered phases is also studied by performing simulations for a WCA-type Gay-Berne fluid. When combined with previous studies of the vapour-liquid transition by Gibbs ensemble Monte Carlo simulations, and of the isotropic-nematic transition by thermodynamic integration, the results presented here provide quite a complete picture of the phase diagram for...

Computation of absorptive form factors for high-energy electron diffraction

Abstract: An efficient procedure for calculating the contribution of the thermal diffuse scattering to the absorptive form factor is outlined. For an isotropic Einstein model all integrations could be performed analytically by using suitable functions to fit the elastic electron scattering amplitudes. The result is cast into a function subroutine which is available upon request. Computed values are compared with previous calculations and with measurements.

Excitons in anisotropic solids: The model of fractional-dimensional space

Abstract: Wannier-Mott excitons in anisotropic or confined systems are studied using the model of fractional-dimensional space. The excitons in an anisotropic solid are treated as ones in an isotropic fractional-dimensional space, where the dimension is determined by the degree of anisotropy. By solving the simple hydrogenic Schr\"odinger equation in the fractional-dimensional space, exciton wave functions, bound energies, and associated optical spectra are obtained as a function of spatial dimensionality. Dimensional behavior in binding energy, radial density, and angular momentum is discussed. The model provides a quantitative measure of anisotropy by a fractional dimension, as viewed from exciton dynamics, which can be determined experimentally from interband optical spectra. The results obtained here are also applicable to hydrogenic impurities in anisotropic solids.

Locally axisymmetric turbulence

Abstract: The failure of local isotropy to describe the experimentally obtained derivative moments in turbulent shear flows has previously been well-documented, but is briefly reviewed. The same data are then used to evaluate the hypothesis that the turbulence is locally axisymmetric. Locally axisymmetric turbulence is defined herein as turbulence which is locally invariant to rotations about a preferred axis.The derivative moment relations are derived from the general form of the two-point velocity correlation tensor near the origin for axisymmetric turbulence. These are used to derive relations for the rate of dissipation of kinetic energy, the mean-square vorticity, and the components of each. Almost all of the experimental derivative moment data are shown to be consistent with these equations, and thus with local axisymmetry.

Nonlinear viscous fingering in miscible displacement with anisotropic dispersion

Abstract: The effect of anisotropic dispersion on nonlinear viscous fingering in miscible displacements is examined. The formulation admits dispersion coefficient‐velocity field couplings (i.e., mechanical dispersivities) appropriate to both porous media and Hele–Shaw cells. A Hartley transform‐based scheme is used to numerically simulate unstable miscible displacement. Several nonlinear finger interactions were observed. Shielding, spreading, tip splitting, and pairing of viscous fingers were observed here, as well as in isotropic simulations. Multiple coalescence and fading were observed in simulations with weak lateral dispersion, but not for isotropic dispersion. Transversely and longitudinally averaged one‐dimensional concentration histories demonstrate the rate at which the mixing zone broadens and the increase in lateral scale as the fingers evolve when no tip splitting occurs. These properties are insensitive to both the dispersion anisotropy and the Peclet number at high Peclet number and long times. This suggests the dominance of finger interaction mechanisms that are essentially independent of details of the concentration fields and governed fundamentally by pressure fields.

Flow through isotropic granular porous media

Abstract: A generalization of the Navier-Stokes equation is developed to include laminar flow through a rigid isotropic granular porous medium of spatially varying permeability. The model is based on a theory of interspersed continua and the mean geometrical properties of an idealized granular porous microstructure. The derived momentum transport equations are applicable to granular porous media over the entire porosity range from zero through unity. No restriction with respect to flow velocity is imposed, except for the assumption of laminar flow within the pores. The results provide useful and versatile equations and substantiate many of the empirical equations currently in use. One of the major advantages of the generalized momentum equation is its adaptability to numerical simulation.

Microstructure of isotropic materials with negative Poisson' s ratio

Abstract: MATERIALS that expand in the transverse direction under uniaxial extension, or that contract laterally when compressed, are said to have a negative Poisson's ratio, v. For an isotropic elastic material, v is the negative of the ratio of lateral to axial strain under uniaxial extension or compression. Despite the apparently counterintuitive nature of this behaviour, v<0 has been observed for some anisotropic crystals1 and materials comprised of fibrous networks2,3, when loaded in a specific direction. Lakes4 has described a class of foams that constitute perhaps the only known isotropic materials with negative v. Materials of this sort are expected to have interesting mechanical properties, such as high energy absorption and fracture resistance, which may be useful in some applications4. Here we describe a general class of microstructures that lead to a negative Poisson's ratio, and show that some existing and hypothetical materials with this property share features common to this class. Our microstructural model provides insight into why natural materials of this kind are rare, and suggests a general methodology for designing such materials.

The response of isotropic turbulence to isotropic and anisotropic forcing at the large scales

Abstract: The nonlinear interscale couplings in a turbulent flow are studied through direct numerical simulations of the response of isotropic turbulence to isotropic and anisotropic forcing applied at the large scales. Specifically, forcing is applied to the energy‐containing wave‐number range for about two eddy turnover times to fully developed isotropic turbulence at Taylor‐scale Reynolds number 32 on an 1283 grid. When forced isotropically, the initially isotropic turbulence remains isotropic at all wave numbers. However, anisotropic forcing applied through an array of counter‐rotating rectilinear vortices induces high levels of anisotropy at the small scales. At low wave numbers the force term feeds energy directly into two velocity components in the plane of the forced vortices. In contrast, at high wave numbers the third (spanwise) component receives the most energy, producing small‐scale anisotropy very different from that at the large scales. Detailed analysis shows that the development of small‐scale anisotropy is caused primarily by nonlocal wave‐vector triads with one leg in the forced low‐wave‐number range. This latter result is particularly significant because asymptotic analysis of the Fourier‐transformed Navier–Stokes equations shows that distant triadic interactions coupling the energy‐containing and dissipative scales persist at asymptotically high Reynolds numbers, suggesting that the structural couplings between large and small scales in these moderate Reynolds number simulations would also exist in high Reynolds number forced turbulence. The results therefore imply a departure from the classical hypothesis of statistical independence between large‐ and small‐scale structure and local isotropy.

Resonant frequencies of higher order modes in cylindrical anisotropic dielectric resonators

Abstract: A method that calculates the frequency of the lowest order mode in a cylindrical isotropic dielectric is extended to higher order modes in a anisotropic crystal. Four different axial match equations are derived depending on whether they are quasi TE or quasi TM, and have an odd or even axial mode number. A general radial match equation is also derived. Combining it with the relevant axial equation forms a set of two coupled transcendental equations that can be solved numerically. The theory is confirmed by room temperature measurements in two sapphire crystals of different aspect ratios, and in cryogenic sapphire resonators used in high stability fixed and tunable oscillators. The sensitivity of mode frequency to dimensional and permittivity perturbations is analyzed. >

Local Anisotropy in Strained Turbulence at High Reynolds Numbers

Abstract: It is shown that the hypothesis of local isotropy is implausible in the presence of significant mean rates of strain. In fact, it appears that in uniform shear flow near equilibrium, local isotropy can never constitute a systematic approximation, even in the limit of infinite Reynolds number. An estimate of the level of mean strain rate for which local isotropy is formally a good approximation is provided.

Collapse of axially compressed cylindrical shells with random imperfections

Abstract: Using the first-order, second-moment analysis, a stochastic method is presented, whereby the stability of isotropic, orthotropic, and anisotropic nominally circular cylindrical shells under axial compression, external pressure, and/or torsion possessing general nonsymmetric random initial imperfections can be evaluated