Showing papers on "Isotropy published in 1975"

Eulerian and Lagrangian time microscales in isotropic turbulence

Abstract: In isotropic ‘box’ turbulence without a mean flow, the Lagrangian frequency spectrum extends to frequencies of order behaviour in the inertial subrange, and that it is not governed by Kolmogorov similarity.

Effects of focusing on third-order nonlinear processes in isotropic media. [laser beam interactions

Abstract: A theoretical analysis of third-order nonlinear interactions of focused laser beams is performed for the processes \omega_{1} + \omega_{2} + \omega_{3} \rightarrow \omega_{4}, \omega_{1} + \omega_{2} - \omega_{3} \rightarrow \omega_{4} , and \omega_{1} - \omega_{2} - \omega_{3} \rightarrow \omega_{4} . The total power and far-field beam profile of the generated radiation is related to the total powers of the fundamental beams, to the tightness and location of the focus, and to the value of the difference between the wave vectors of the generated radiation and driving polarization. The optimum degree of wave-vector mismatch as a function of tightness and location of focus is determined for each of the three processes. The process \omega_{1} + \omega_{2} - \omega_{3} \rightarrow \omega_{4} is found to be unique in that it is always optimized by focusing as tightly as possible. Experimental results, which verify the theory for the processes \omega_{1} + \omega_{2} + \omega_{3} \rightarrow \omega_{4} and \omega_{1} + \omega_{2} - \omega_{3} \rightarrow \omega_{4} , are presented.

Boundary-integral equation analysis of cracked anisotropic plates

Abstract: A numerical procedure based on the boundary-integral equation method, is formulated using the fundamental solution (Green's function) for an infinite anisotropic plate containing an exact crack. The boundary-integral equation developed can be solved numerically for the mode 1 and mode 2 stress intensity factors by approximating boundary data on the surface of an arbitrary body, excluding the crack surface. Thus the efficiency and generality of the boundary-integral equation method and the precision of exact crack model analyses are combined in a direct manner. The numerical results reported herein are as accurate as previously published isotropic results. The effects of material anisotropy are reported for center and double-edge cracked geometries. A path independent integral for obtaining mode 1 and mode 2 stress intensity factors directly for arbitrary loading is reported.

Computational Modeling of Turbulent Transport

Abstract: A rational closure technique is presented for the first and second moment-equations in a stratified, contaminated turbulent flow, Following the application of high Reynolds/Peclet number approximations, remaining third moments are expanded about the isotropic, homogeneous state. The stratified, uncontaminated case reduces to seventeen equations in seventeen unknowns. Other authors have suggested some of the terms generated, but Some have been using the wrong terms, or the right terms for the wrong reasons. The approximation is kinetic-theoretic (turbulence/mean motion scales assumed small, and turbulence nearly in equilibrium) and results in a relaxation time, and in generalized gradient transport forms; however, gradients of one quantity can produce fluxes of another. The model relates the time scale for return to isotropy to the Lagrangian integral time scale (reducing to K-theory in a homogeneous parallel flow with orthogonal temperature gradient). Some coefficients are estimated, and preliminary computations arc presented of the unstratified 2-D turbulent wake; only component energies near the centerline are not well reproduced, probably due to the omission of a term with which temporary computational difficulties were being experienced. Stratified, contaminated 3-D calculations appear to be practical.

Coupled-mode theory for anisotropic optical waveguides

Abstract: The well-known coupled-mode theory of waveguides is extended to include dielectric guides made of anisotropic materials. Exact coupled-wave equations for anisotropic dielectric waveguides are derived, and explicit expressions for the coupling coefficients are given. The coupling coefficients for isotropic waveguides are obtained as a special case. A simple approximation for the coupling coefficients in the case of slight anisotropy and slight departure from an ideal waveguide is presented.

Elastic-plastic analysis of an infinite sheet having a circular hole under pressure

Abstract: An exact elastic-plastic solution for the stresses in an infinite sheet having a circular hole subject to pressure is obtained on the basis of J2 deformation theory together with a modified Ramberg-Osgood law. The sheet is orthotropic but isotropic in its plane. The results are assessed on the basis of Budiansky's criterion for the acceptability of J2 deformation theory. By using exact elastic-plastic stresses, the function connecting the pressure at the hole with the radial enlargement is obtained. Upon release of the pressure, residual stresses around the hole are produced.

A state space approach to elasticity

Abstract: The two-dimensional, plane stress problem of linear elasticity is analyzed within a state space framework The medium considered is homogeneous and isotropic Vlasov's mixed formulation of elasticity is used throughout The field equations are derived in closed form, thus avoiding Vlasov's intermediate infinite series solution Finally, all the properties of the transfer matrix are shown to follow directly from embedding the problem into a state space setting

The local electric field. I. The effect on isotropic and anisotropic Rayleigh scattering

Abstract: Isotropic and anisotropic Rayleigh light scattering intensities are reported for seven liquids. Anisotropic intensities as a function of refractive index for four molecules are also reported. We have used spectral analysis to separate the single molecule scattering from the complicating effects of collisional scattering and static pair correlations. Several macroscopic continuum models for the local electric field at a molecule in a liquid phase are summarized and these theoretical models are compared with our experimental results for scattering from single molecules. We have established the equivalence of the local field correction calculated from a fourth‐power local field with that calculated from the dielectric fluctuation method for all models. We further show that of these local field corrections, only that calculated by the ellipsoidal Onsager–Scholte model is consistent with all our experimental data. The size and shape parameters of these models are related to molecular parameters.

Effective conductivity of two−phase material with cylindrical phase boundaries

Abstract: A two−phase material in which the phase boundaries are cylindrical surfaces is considered. The generators of the phase boundaries are parallel to the z axis and the x and y axes are the principal axes of the effective conductivity tensor. If the two phases have conductivities σ1 and σ2, then the effective conductivity σx* (σ1,σ2) (in the x direction) of the material and the effective conductivity σy* (σ2,σ1) (in the y direction) of the material obtained by interchanging the two phases are related by σx* (σ1,σ2) σy* (σ2,σ1) =σ1,σ2. For a statistically isotropic material σx* (σ1,σ2) =σy* (σ1,σ2) =σ* (σ1,σ2). If, in addition, the material is statistically symmetric so that interchange of the two phases yields again the same material, then σ* (σ1,σ2) =σ* (σ2,σ1) = (σ1σ2)1/2.

Elastodynamic Near-Tip Stress and Displacement Fields for Rapidly Propagating Cracks in Orthotropic Materials

Abstract: The near-tip angular variations of elastodynamic stress and displacement fields are investigated for rapid transient crack propagation in isotropic and orthotropic materials. The 2-dimensional near-tip displacement fields are assumed in the general form r/sup P/ T(t, c) K(theta, c), where c is a time-varying velocity of crack propagation, and it is shown that p = 0.5. For isotropic materials, K(theta, c) is determined explicitly by analytical considerations. A numerical procedure is employed to determine K(theta, c) for orthotropic materials. The tendency of the maximum stresses to move out of the plane of crack propagation as the speed of crack propagation increases is more pronounced for orthotropic materials, for the case that the crack propagates in the direction of the larger elastic modulus. The angular variations of the near-tip fields are the same for steady-state and transient crack propagation, and for propagation along straight and curved paths, provided that the direction of crack propagation and the speed of the crack tip vary continuously.

Variable viscosity effects on the onset of convection in porous media

Abstract: The onset of convection in a horizontal, isotropic, water‐saturated porous medium is considered. The temperature difference between the top and bottom is as large as 250 °C. The effects of an eightfold variation in kinematic viscosity are included. The critical Rayleigh number is found to be substantially reduced from the classical value although the associated wavenumber is nearly the same. Neutral mode streamline and isotherm patterns are considerably distorted in the vertical direction in distinction to the symmetric patterns found in the constant viscosity classical calculation.

Nonlinear electroelastic equations cubic in the small field variables

Abstract: The nonlinear differential equations and boundary conditions containing terms up to cubic in the small field variables are obtained from general rotationally invariant nonlinear electroelastic equations derived previously. The electroelastic equations cubic in the small field variables are considerably more tractable than the general electroelastic equations and are applicable in the description of such phenomena as the dependence of wave velocities on wave amplitudes and resonant frequencies on vibration amplitudes in addition to a host of other nonlinear phenomena. The nonlinear constitutive equations for an isotropic purely elastic solid containing terms up to cubic in the small mechanical displacement gradients are presented. The nonlinear equations for the extensional motion of thin isotropic plates containing terms up to cubic in the small mechanical displacement gradients in the plane of the plate are obtained and the influence of the vertical inertia is included, in addition to the extensional sti...

The specific heat of magnetic linear chains

Abstract: Energy spectra and thermodynamical quantities of finite linear chains and polygons of axially symmetrically interacting spins have been numerically calculated. Extrapolation of the molar heat capacities versus the number of spins yielded estimates for infinite systems. Ground-state energies were also obtained. Data are presented for both ferromagnetic and antiferromagnetic chains. The influence of axially symmetric anisotropies in the exchange interaction was studied for S = 1 2 and 1. This influence proved to be qualitatively the same in both cases. For S = 1 to 5 2 , data are included for isotropic exchange in combination with several crystal field anisotropy parameter (D) values, both positive and negative. Further, the effect of the simultaneous presence of a D term and exchange anisotropy is studied for several cases. Agreement of the extrapolated data is found with some existing experimental and theoretical results.

Optical parametric interactions in isotropic materials using a phase‐corrected stack of nonlinear dielectric plates

Abstract: Optical parametric interactions are analyzed for collinear three‐frequency processes in a series of plates of isotropic nonlinear dielectric. Phase‐velocity mismatch caused by refractive‐index dispersion is compensated by making each plate one coherence length thick and orienting consecutive plates so the nonlinear polarization is shifted in phase by π. Parametric transfer of energy then proceeds constructively along the axis of propagation. For second harmonic generation, frequency upconversion and downconversion, and parametric amplification the output intensity is proportional to the square of the number of phase‐corrected plates. The conversion efficiency is the same as for a hypothetical phase‐velocity‐matched interaction in the same nonlinear material, except that the nonlinear susceptibility is effectively reduced by a factor of 2/π.This method can be used to obtain efficient nonlinear interactions over frequency ranges previously precluded by lack of suitable transparent birefringent crystals and for large‐aperture configurations for generating high powers.

Magnetotelluric response of laterally inhomogeneous and anisotropic media

Abstract: The lack of agreement between magnetotelluric field measurements and the calculations based on essentially two‐dimensional models with either anisotropy or lateral inhomogeneity necessitates a more complex model of the earth than has been previously considered. The Galerkin finite‐element method is applied to a two‐dimensional structure with a tensor conductivity. The importance of considering conductivity as a tensor is illustrated by a model consisting of an anisotropic, conducting dike embedded in an anisotropic half‐space. This model can be distinguished from an isotropic model by the nonvanishing diagonal elements of the impedance tensor, the ellipticity indices, and the skew.

Influence of surface energy anisotropy on morphological changes occurring by surface diffusion

Abstract: The influence of surface energy anisotropy on morphological changes occurring by surface diffusion, on simply shaped bodies, is investigated. A preliminary analysis of the equilibrium shape of a two‐dimensional body for arbitrary anisotropy is given, the primary aim being to determine the range of validity of the perturbation scheme used in the subsequent time‐dependent analysis. It is shown that such a scheme is valid for the entire range of shapes later considered if γ (ϑ)+d2γ (ϑ)/dϑ2≳0, where γ (ϑ) is the specific surface free energy of a surface whose normal is oriented at an angle ϑ to the reference crystallographic axes. Under this condition the complete relaxation, from an assumed initial circle to the final equilibrium shape, is derived, providing the surface diffusivity is isotropic. When γ (ϑ)+d2γ/dϑ2 is negative, the perturbation scheme furnishes proof of an initial unstable growth away from the circle, although it cannot be used to derive the complete relaxation behavior. Nevertheless, thereby a proof is provided of the existence of such an instability, which was derived earlier by Mullins for only a semi‐infinite body, in a finite body (in two dimensions). For completeness, a brief discussion is given of the influence of small surface energy anisotropy in three dimensions for surface‐diffusion‐controlled shape changes from an initial sphere. Application of these results to the kinetics of particle shaping during the early stages of thin‐film growth is also presented.

Relativistic distribution functions and applications to electron beams

Abstract: Equilibrium and monoenergetic distribution functions in the laboratory frame are derived from first principles. Thermodynamically consistent macroscopic properties of relativistic beam systems are defined and discussed as a foundation for specific applications. Characteristics of superpinched relativistic electron beams such as isotropy in the lab frame, rms cone angle of the electrons with respect to the beam axis, and the energy flux across a unit surface at an arbitrary angle to the beam axis are obtained. These properties provide important information on the expected symmetry of irradiation of fusion targets by focused electron beams. A linearized kinetic theory is outlined which leads to an integral expression for the general form of the relativistic plasma dispersion function.

Effect of third invariant of stress deviator on plastic deformation of mild steel

Abstract: The procedure discussed by Y. Ohashi and M. Tokuda (1973) for situations involving twodimensional vector space, and which eliminated the effect of the third invariant of the stress deviator from the secondary effect due to the plastic deformation history of initially isotropic mild steel, has been extended to situations involving five-dimensional vector space in general. Thin-walled tubular specimens made of initially isotropic mild steel were tested under combined loadings of internal pressure, axial force and torque by use of an automatic combined-loading mechanical testing machine. Experimental results showed that the effect of the third invariant of the stress deviator was of the same order as the effect of plastic anisotropy. However, the effect of anisotropy due to the plastic deformation history had previously been estimated without separation of the third invariant of the stress deviator which varies with rotation or mirror transformation of the strain trajectory in isotropic vector space. After elimination of the above effect, the plastic anisotropy due to the deformation history of initially isotropic mild steel for strain trajectories of fixed shape is found to remain the same, irrespective of rotation or mirror transformations in the isotropic vector space.

Elastic interaction of point defects in isotropic and anisotropic cubic media

Abstract: Using the Green's tensor function as the solution of the elastic equations for isotropic and anisotropic cubic media the elastic interaction energy is calculated for pairs of defects which have isotropic, tetragonal, or orthorhombic symmetry. Special attention is given to the interaction of a one-dimensionally migrating interstitial, in crowdion configuration, with other defects. The differences of the interaction in elastically isotropic and anisotropic materials, the influence of defect anisotropy on the interaction potential, and the relation of these calculations to experimental results are discussed.

Elastic stresses and self-energies of dislocations of arbitrary orientation in anisotropic media: Olivine, orthopyroxene, calcite, and quartz

Abstract: Elastic stress fields, self-energies, and energy factors of straight edge and screw dislocations in olivine, orthopyroxene, calcite, and α quartz are calculated numerically from the equations of anisotropic elasticity. Energy factor profiles are used to determine which slip systems are preferred on the basis of elastic self-energies, and the effects of elastic anisotropy on the stresses and self-energies of the dislocations are investigated. The moderate anisotropies of olivine and orthopyroxene lead to results that differ by about 10% from the isotropic elastic solutions. Comparison with observations indicates that the slip direction and glide plane preference of dislocations in these orthorhombic minerals cannot be predicted solely on the basis of anisotropic elasticity theory. In quartz and especially in calcite the large elastic anisotropy does have a major influence on slip system preference, which is consistent with the observed systems. Anisotropic calculations for olivine at 1000°C and for α quartz near the α-β transition temperature failed to reveal any significant changes in slip system preference associated with the effects of higher temperatures.

Ultra‐high‐modulus fibers from solution spinning

Abstract: Ultra-high-modulus fibers such as Du Pont PRD-49 (initial modulus up to ∼1000 g/den) and Monsanto X-500 (initial modulus up to ∼600 g/den) are spun from solutions. Both polymers are characterized by a high intrinsic rigidity of individual molecular chains and considerable orientation along the fiber axis. The thermodynamics of solution for rigid and semirigid macromolecules is critically reviewed in order to illustrate conditions under which spontaneous formation of highly oriented fibers is expected. In the case of semirigid polymers, the free energy of (random) mixing pure solvent and parallellized polymer may, according to Flory, become positive for some critical value of a “flexibility parameter.” Formation of an ordered phase for semirigid polymers is not, however, observed by lowering temperature or increasing polymer concentration. In the case of rod-like polymers, still according to Flory, at some critical value of polymer concentration (which decreases with the axial ratio of the macromolecule) the isotropic solution of rods undergoes phase separation with formation of a partly ordered solution. This theoretical prediction is satisfactorily verified by data. While Du Pont fibers are spun from this anisotropic solution, Monsanto's X-500 only yields an isotropic solution at room temperature up to the limit of polymer concentration at which crystallization occurs. This inability of X-500 to form anisotropic solutions at the expected critical concentration is attributed to a partial degree of flexibility. Mechanical properties and orientation of fibers spun from the anisotropic solution appear to be superior to those obtained by spinning from isotropic solution, according to Du Pont's results. When a polymer has a partial degree of flexibility, alteration of physico-chemical variables such as solvent type, solvent composition, temperature, and polymer concentration may still be used in order to increase its rigidity. Theoretical arguments and data supporting this contention are discussed. Moreover, alteration of these variables may also be used to alter the crystallization temperature, allowing formation of the anisotropic solution to occur at a high enough polymer concentration. This expectation was verified in the case of X-500. Finally, the all important role of mechanical orientation of solutions is emphasized. According to Hermans, under high enough shear stress, the difference between the isotropic and the anisotropic solution vanishes. In line with these consideration, drawing techniques are particularly useful in order to achieve almost-perfect orientation and theoretical moduli.

Craze formation and growth in anisotropic polymers

Abstract: Craze formation and craze growth in anisotropic polymers has been studied as a function of the degree of anisotropy and the relation between the testing direction and the primary orientation direction. Tests on PMMA and PC indicate that both the morphology and orientation of the crazes are senstivie functions of testing direction. Crazes form in directions which arenot orthogonal to the principal tensile stress, and the data clearly show that craze growth occurs in directions governed by the major principal strain. The fracture process is identical in nature to that in isotropic polymers, i.e. craze formation, crack nucleation within the craze and subsequent crack propagation through the craze. Thus, the angle of fracture coincides with the craze angle rather than occurring perpendicular to the principal tensile stress.

Density of methoxybenzylidene butylaniline about the isotropic–nematic phase transition

Abstract: We have measured the density and determined the coefficient of expansion of MBBA about its isotropic−nematic transition temperature using a Mettler/Paar density meter. Our results show essential agreement with those of Press and Arrott using a buoyancy method. The small difference (<0.3°C) between the actual transition temperature and the virtual critical temperature differs from that determined by light scattering studies which yield values of about 1°C. Further examination on the nature of the virtual critical temperature is suggested.

Measurements of three velocity components in a model furnace with and without combustion

Abstract: Measured values of three components of mean velocity and the corresponding normal stresses are reported in the flow within an enclosure which is representative of a small-scale furnace with an axi-symmetric, swirling flow configuration. The measurements were obtained in isothermal air flow and in a combusting mixture of air and natural gas, exit swirl numbers of zero and 0.52 were investigated for both the isothermal and combusting cases. A laser anemometer was used to obtain the measurements and comprised an argon-ion laser, a rotating diffraction grating as beam spiltter and frequency shifter, transmission and collection optical components, a photomultiplier and a filter-bank signal processor. The grating provided a frequency shift of ±0.54 MHz and allowed measurements in regions of high turbulence intensity and negative mean velocity. The filter bank consisted of 50 filters spanning the range from 0.6 to 6 MHz which, with the frequency shift indicated above and the optical arrangement employed, corresponded to a measurable velocity range of −2 m/sec to 23 m/sec. Measured values of mean axial velocity and the corresponding normal stress are presented for the four flow conditions of zero and finite swirl, with and without combustion. The turbulent-kinetic energy is presented for the isothermal cases and the regions of near-isotropic turbulence identified, only the axial and tangential normal stresses were measured for the combusting cases. The measurements were designed to be used for the evaluation of turbulent-flow prediction procedures. They demonstrate, for example, that the regions of recirculation are substantially different for the combusting measurements and that the turbulence is far from isotropic over most of the flow fields; similarly, the velocity-probability-density distributions indicate that Gaussian turbulence exists in only negligible regions of the flows.