Showing papers on "Isotropy published in 1977"

Renormalization, vortices, and symmetry-breaking perturbations in the two-dimensional planar model

Abstract: The classical planar Heisenberg model is studied at low temperatures by means of renormalization theory and a series of exact transformations. A numerical study of the Migdal recursion relation suggests that models with short-range isotropic interactions rapidly become equivalent to a simplified model system proposed by Villain. A series of exact transformations then allows us to treat the Villain model analytically at low temperatures. To lowest order in a parameter which becomes exponentially small with decreasing temperature, we reproduce results obtained previously by Kosterlitz. We also examine the effect of symmetry-breaking crystalline fields on the isotropic planar model. A numerical study of the Migdal recursion scheme suggests that these fields (which must occur in real quasi-two-dimensional crystals) are strongly relevant variables, leading to critical behavior distinct from that found for the planar model. However, a more exact low-temperature treatment of the Villain model shows that hexagonal crystalline fields eventually become irrelevant at temperatures below the ${T}_{c}$ of the isotropic model. Isotropic planar critical behavior should be experimentally accessible in this case. Nonuniversal behavior may result if cubic crystalline fields dominate the symmetry breaking. Interesting duality transformations, which aid in the analysis of symmetry-breaking fields are also discussed.

The return to isotropy of homogeneous turbulence

Abstract: Three types of homogeneous anisotropic turbulence were produced by the plane distortion, axisymmetric expansion and axisymmetric contraction of grid-generated turbulence, and their behaviour in returning to isotropy was experimentally studied using hot-wire anemometry. It was found that the turbulence trajectory after the plane distortion was highly nonlinear, and did not follow Rotta's linear model in returning to isotropy. The turbulence wanted to become axisymmetric even more than it wanted to return to isotropy. In order to show the rate of return to isotropy of homogeneous turbulence, a map of the ratio of the characteristic time scale for the decay of turbulent kinetic energy to that of the return to isotropy was constructed. This demonstrated that the rate of return to isotropy was much lower for turbulence with a greater third invariant of the anisotropy tensor. The invariant technique was then applied to the experimental results to develop a new turbulence model for the return-to-isotropy term in the Reynolds stress equation which satisfied the realizability conditions. The effect of the Reynolds number on the rate of return to isotropy was also investigated and the results incorporated in the proposed model.

Bounds for effective elastic moduli of disordered materials

Abstract: Recently P.H. Dederichs and R. Zeller (1973) have developed a formal theory of the bounds of odd order n for the effective elastic moduli of linearly elastic, disordered materials. The bounds are established by use of statistical information given in terms of correlation functions up to order n (= 1, 3, 5,…). This theory is extended to include the bounds of even order n. It is indicated how these bounds can be made optimum under the given statistical information. The results for bounds of even and odd order are obtained in forms which resemble Neumann series, containing multiple integrals up to order (n−1). These integrals can be calculated for certain materials which are classified in terms of a gradual statistical homogeneity, isotropy and disorder. Materials which possess these properties up to the correlation functions of nth order are called overall grade n materials. The optimum bounds for overall grade 2 and grade 3 materials are given explicitly. Optimum bounds for materials which are of grade ∞ in homogeneity and isotropy (i.e. (statistically) perfectly homogeneous and isotropic) and, at the same time, disordered of grade 2 or 3 are also derived. Those for grade 2 in disorder are the Z. Hashin and S. Shtrikman's (1963) bounds. Those for grade 3 are the narrowest, explicit bounds so far derived for random elastic materials. They contain within themselves the so-called self-consistent elastic moduli.

An advanced boundary integral equation method for three‐dimensional thermoelasticity

Abstract: The features of an advanced numerical solution capability for boundary value problems of linear, homogeneous, isotropic, steady-state thermoelasticity theory are outlined. The influence on the stress field of thermal gradient, or comparable mechanical body force, is shown to depend on surface integrals only. Hence discretization for numerical purposes is confined to body surfaces. Several problems are solved, and verification of numerical procedures is obtained by comparison with accepted results from the literature.

Path integrals for waves in random media

Abstract: The problem of wave propagation in a random medium is formulated in terms of Feynman’s path integral. It turns out to be a powerful calculational tool. The emphasis is on propagation conditions where the rms (multiple) scattering angle is small but the log‐intensity fluctuations are of order unity—the so‐called saturated regime. It is shown that the intensity distribution is then approximately Rayleigh with calculable corrections. In an isotropic medium, the local or Markov approximation which is commonly used to compute first and second (at arbitrary space–time separation) moments of the wave field is explicitly shown to be valid whenever the rms multiple scattering angle is small. It is then shown that in the saturated regime the third and higher moments can be obtained from the first two by the rules of Gaussian statistics. There are small calculable corrections to the Gaussian law leading to ’’coherence tails.’’ Correlations between waves of different frequencies and the physics of pulse propagation are studied in detail. Finally it is shown that the phenomenon of saturation is physically due to the appearance of many Fermat paths satisfying a perturbed ray equation. For clarity of presentation much of the paper deals with an idealized medium which is statistically homogeneous and isotropic and is characterized by fluctuations of a single typical scale size. However, the extension to inhomogneous, anisotropic, and multiple scale media is given. The main results are summarized at the beginning of the paper.

Formal aspects of the theory of the scattering of ultrasound by flaws in elastic materials

Abstract: An integral equation is used to derive formal expressions for the scattering of a plane wave from a single homogeneous flaw embedded in an isotropic elastic medium. Expressions are found for the scattered amplitudes and differential cross sections. An optical theorem is also derived.

Calculated stresses in multilayered heteroepitaxial structures

Abstract: A useful closed‐form expression for stresses within individual layers of a multilayer composite has been obtained as a function of position within the layer. Equal and isotropic elastic constants were assumed in the calculation, although the error introduced by this assumption is found to vary by no more than the difference in elastic constants. Unequal elastic constants may be handled via computer solutions. The stresses within heterojunction AlxGa1−xAs/GaAs lasers are calculated as an example of the technique. The addition of Al to the active GaAs region is shown to have a drastic effect upon the active‐region stress, changing it from tension to compression. This change of sign in stress is correlated with improvements in operating lifetimes of lasers.

Stresses Around Pin-Loaded Holes in Elastically Orthotropic or Isotropic Plates

Abstract: The stress distribution round a pin-loaded hole in an elastically ortho tropic or isotropic plate is investigated. The hole is loaded frictionless on only a part of its edge by an infinitely rigid pin of the same diameter. The loading force is carried over on the edge by normal stresses, represented by a sine series. It is shown that these stresses depend strongly on the material properties. Infinite plate results are used to estimate the stresses in plates nite width. Numerical results are shown graphically for three laminates n fibre reinforced plastic and for three ratios of width of plate to mole diameter.

Stress-strain relations for materials with different moduli in tension and compression

Abstract: Models in the form of stress-strain, or constitutive, relations are discussed for materials with moduli under tensile loading which are different from those under compressive loading. Criteria for consistent material models are given which are based on the principles of anisotropic elasticity and on the known behavior of such materials. The Ambartsumyan material model is compared with the criteria and found to violate the requirement of symmetric compliances. An improved model, called the weighted compliance matrix (WCM) material model, is shown to satisfy the criteria for isotropic and orthotropic bodies under plane stress. The new model can be extended by deduction to more complicated situations such as anisotropic bodies under general stress states.

Unified Boundary for Finite Dynamic Models

Abstract: The finite element analysis of dynamic problems in an infinite, isotropic medium is examined. To simulate the physically infinite system by a finite model, an energy absorbing boundary is proposed. This boundary is frequency independent and proves to be very efficient in absorbing stress waves. The boundary constants are calculated for the particular cases of plane strain and axisymmetry for isotropic materials.

On yielding of oriented solids

Abstract: Yielding and failure of oriented solids is studied within the framework of tensor representations. Transversely isotropic materials are considered. Using the information available regarding the tensor generators and the set of independent stress and mixed stress-material orientation invariants the general form of constitutive relation for incipient plastic motion of transversely isotropic solid is given and the yield condition is discussed. Specific forms of yield criteria for cohesive materials as well as for materials with internal friction are developed and compared with experimental information. A novel approach to flow and failure of oriented solid is thus explained on example of stratified material, starting with experimental motivation, passing through theoretical development and terminating on comparisons with experiments.

Stochastic closure for nonlinear Rossby waves

Abstract: An extension of the turbulence ‘test-field model’ (Kraichnan 1971 a) is given for two-dimensional flow with Rossby-wave propagation. Such a unified treatment of waves and turbulence is necessary for flows in which the relative strength of nonlinear terms depends upon the length scale considered. We treat the geophysically interesting case in which long, fast Rossby waves propagate substantially without interaction while short Rossby waves are thoroughly dominated by advection. We recover the observations of Rhines (1975) that the tendency of two-dimensional flow to organize energy into larger scales of motion is inhibited by Rossby waves and that an initially isotropic flow develops anisotropy preferring zonal motion. The anisotropy evolves to an equilibrium functional dependence on the isotropic part of the flow spectrum. Theoretical results are found to be in quantitative agreement with numerical flow simulations.

Saint-Venant’s Principle and End Effects in Anisotropic Elasticity

Abstract: The purpose of this paper is to draw attention to the fact that the routine application of Saint-Venant’s principle in the solution of elasticity problems involving highly anisotropic or composite materials is not justified in general. This is illustrated in the context of the plane problem of elasticity for an anisotropic rectangular strip loaded only on the short ends. For highly anisotropic transversely isotropic materials, the slow decay of end effects is demonstrated using a method involving self-equilibrating eigenfunctions. For a graphite/epoxy composite, for example, the characteristic decay length is shown to be approximately four times that for an isotropic material. The results have implications in the accurate measurement of mechanical properties of anisotropic materials.

Effective conductivities of composite materials composed of cubic arrangements of spherical particles embedded in an isotropic matrix

Abstract: Results are presented for the effective conductivities of simple, face-centered, and body-centered cubic arrays of isotropic spherical particles embedded in an isotropic medium possessing a different conductivity. With the use of a multipole expansion technique, numerical and conceptual errors in the prior work of others are corrected. Agreement with existing experimental data is found to be excellent.

Bounds on the conductivity of statistically isotropic polycrystals

Abstract: The problem of the prediction of the effective electrical conductivity of a polycrystal from the conductivity of a single crystal is considered. If the only information known about phase geometry is that the aggregate is statistically homogeneous and isotropic, it is shown that the average of the principal conductivities of the single crystal is the best upper bound on effective conductivity that can possibly be found. A new rigorous lower bound is found for the case of axially symmetric crystals. An exact solution is found for the case of a two-dimensional polycrystal.

Coherent scalar field in pair‐correlated random distributions of aligned scatterers

Abstract: Dispersion equations for coherent propagation of scalar waves in random distributions of pair‐correlated obstacles (aligned or averaged over alignment), are obtained by averaging the functional equations relating the multiple and single scattered amplitudes of the obstacles. The resulting bulk indices of refraction and bulk parameters, for aligned nonradially symmetric scatterers, specify anisotropic media; the anisotropy arises either from the scatterers’ properties (physical parameters or shape, or both) or from their distribution, or from both. The illustrations include both isotropic and anisotropic cases (in one to three dimensions), and the explicit results generalize earlier ones.

The problem of internal and edge cracks in an orthotropic strip

Abstract: The plane elastostatic problem of internal and edge cracks in an infinite orthotropic strip is considered. The problems for the material types I and II are formulated in terms of singular integral equations. For the symmetric case the stress intensity factors are calculated and are compared with the isotropic results. The results show that because of the dependence of the Fredholm kernels on the elastic constants, unlike the crack problem for an infinite plane, in the strip the stress intensity factors are dependent on the elastic constants and are generally different from the corresponding isotropic results.

A theoretical investigation of isotropic raman polarizability correlation functions in dense molecular fluids

Abstract: This paper is concerned with the derivation of a kinetic equation for the description of the relaxation of the isotropic (rotational invariant) part of the Raman polarizability correlation function, with a goal to aid interpretations of experimental results. The isotropic polarizability correlation function is closely associated with the Raman spectral line-shape of a totally symmetric mode of an ensemble of molecules in a liquid. The kinetic equation was derived for an interaction potential consisting of two parts: one associated with a hard-core potential which is assumed to be unaffected by the intramolecular vibrations, and another associated with a potential which is modulated by molecular vibrations. The interaction potential chosen for this work is applicable to the intermolecular dipole-dipole interaction, hydrogen bonding interaction and other types of interactions. While we have included in the derivation the mechanism of the vibrational energy dissipation to nonvibrational degrees of freedom an...

A Theory of Composites Modeled as Interpenetrating Solid Continua

Abstract: The differential equations and boundary conditions describing the behavior of a finitely deformable, heat-conducting composite material are derived by means of a systematic application of the laws of continuum mechanics to a well-defined macroscopic model consisting of interpenetrating solid continua. Each continuum represents one identifiable constituent of the N-constituent composite. The influence of viscous dissipation is included in the general treatment. Although the motion of the combined composite continuum may be arbitrarily large, the relative displacement of the individual constituents is required to be infinitesimal in order that the composite not rupture. The linear version of the equations in the absence of heat conduction and viscosity is exhibited in detail for the case of the two-constituent composite. The linear equations are written for both the isotropic and transversely isotropic material symmetries. Plane wave solutions in the isotropic case reveal the existence of high-frequency (optical type) branches as well as the ordinary low-frequency (acoustic type) branches, and all waves are dispersive. For the linear isotropic equations both static and dynamic potential representations are obtained, each of which is shown to be complete. The solutions for both the concentrated ordinary body force and relative body force are obtained from the static potential representation.

Evidence for anisotropy in the upper mantle beneath Eurasia from the polarization of higher mode seismic surface waves

Abstract: Summary. Analysis of NORSAR records and a number of Soviet microfilms reveals second-mode surface Caves propagating along paths covering a large part of Eurasia. These second modes in the 6-1 5-s period band are frequently disturbed by other surface-wave modes and by body-wave arrivals. However, in all cases, where the modes appear to be undisturbed and show normal dispersion, the Second Rayleigh modes have a slowly varying phase difference with the Second Love modes. This coupling has the particle motion of Inclined Rayleigh waves characteristic of surface-wave propagation in anisotropic media, where the anisotropy possesses a horizontal plane of symmetry. Numerical examination of surface wave propagating in Earth models, with an anisotropic layer in the upper mantle, demonstrate that comparatively small thicknesses of material with weak velocity anisotropy can produce large deviations in the polarizations of Inclined Rayleigh Second modes. In many structures, these inclinations are very sensitive to small changes in anisotropic orientation and to small changes in the surrounding isotropic structure. It is suggested that examination of second mode inclination anomalies of second mode surface waves may be a powerful technique for examining the detailed anisotropic structure of the upper mantle.

Spectral energy transfer in high Reynolds number turbulence

Abstract: The spectral energy transfer of turbulent velocity fields has been examined over a wide range of Reynolds numbers by experimental and empirical methods. Measurements in a high Reynolds number grid flow were used to calculate the energy transfer by the direct Fourier-transform method of Yeh & Van Atta. Measurements in a free jet were used to calculate energy transfer for a still higher Reynolds number. An empirical energy spectrum was used in conjunction with a local self-preservation approximation to estimate the energy transfer at Reynolds numbers beyond presently achievable experimental conditions.Second-order spectra of the grid measurements are in excellent agreement with local isotropy down to low wavenumbers. For the first time, one-dimensional third-order spectra were used to test for local isotropy, and modest agreement with the theoretical conditions was observed over the range of wavenumbers which appear isotropic according to second-order criteria. Three-dimensional forms of the measured spectra were calculated, and the directly measured energy transfer was compared with the indirectly measured transfer using a local self-preservation model for energy decay. The good agreement between the direct and indirect measurements of energy transfer provides additional support for both the assumption of local isotropy and the assumption of self-preservation in high Reynolds number grid turbulence.An empirical spectrum was constructed from analytical spectral forms of von Karman and Pao and used to extrapolate energy transfer measurements at lower Reynolds number to Rλ = 105 with the assumption of local self preservation. The transfer spectrum at this Reynolds number has no wavenumber region of zero net spectral transfer despite three decades of range of the empirical energy spectrum.

An Analytical Method for Evaluation of Impact Damage Energy of Laminated Composites

Abstract: : A higher order beam finite element is developed for dynamic response of beams subjected to impact of elastic spheres. Hertzian law is used to evaluate the contact force. A step by step finite difference method is employed to integrate the time variable. The finite elements are first evaluated for homogeneous isotropic beams and excellent results are found. Impact of glass- epoxy laminates are then considered. The total energy imparted from the projectile to the laminate is computed and compared with experimental data. Good agreement is found. The present finite element procedure also allows one to separate the vibrational energy from the damage energy which is to be related to the residual strength of the composite after impact.