Showing papers on "Isotropy published in 1992"

From isotropic to anisotropic superconductors: A scaling approach.

Abstract: We present a new scaling approach which allows one to map results obtained for isotropic superconductors to anisotropic materials in a simple and direct way. The scaling rules are obtained on the level of Ginzburg-Landau-- or London--type equations and applied directly to the desired phenomenological quantity. We illustrate the method by calculating the elasticity moduli, the depinning and melting temperatures, the critical current densities, and the activation barriers for classical and quantum creep in anisotropic superconductors for an arbitrary angle between the magnetic field and the axes of anisotropy.

Composite materials with poisson's ratios close to — 1

Abstract: A family of two-dimensional, two-phase, composite materials with hexagonal symmetry is found with Poisson's ratios arbitrarily close to — 1. Letting k∗, k1,k2 and μ∗,μ1,μ2 denote the bulk and shear moduli of one such composite, stiff inclusion phase and compliant matrix phase, respectively, it is rigorously established that when k1 = K2r and μ1 = μ2r there exists a constant c depending only on k2, μ2 and the geometry such that k∗/μ∗

Direct simulation of particle dispersion in a decaying isotropic turbulence

Abstract: Results of a numerical investigation of the dispersion of solid particles in decaying isotropic turbulence are presented. The 3D time-dependent velocity field of a homogeneous nonstationary turbulence is computed using the method of direct numerical simulation (DNS). The dispersion characteristics of three different solid particles (corn, copper, and glass) injected in the flow are obtained by integrating the complete equation of particle motion along the instantaneous trajectories of 22-cubed particles for each particle type, and then by performing ensemble averaging. Good agreement was achieved between the present DNS results and the measured time development of the mean-square displacement of the particles. Questions of how and why the dispersion statistics of a solid particle differ from those of its corresponding fluid point and surrounding fluid and what influences inertia and gravity have on these statistics are also discussed.

Kinematic simulation of homogeneous turbulence by unsteady random Fourier modes

Abstract: The velocity field of homogeneous isotropic turbulence is simulated by a large number (38–1200) of random Fourier modes varying in space and time over a large number (> 100) of realizations. They are chosen so that the flow field has certain properties, namely (i) it satisfies continuity, (ii) the two-point Eulerian spatial spectra have a known form (e.g. the Kolmogorov inertial subrange), (iii) the time dependence is modelled by dividing the turbulence into large- and small-scales eddies, and by assuming that the large eddies advect the small eddies which also decorrelate as they are advected, (iv) the amplitudes of the large- and small-scale Fourier modes are each statistically independent and each Gaussian. The structure of the velocity field is found to be similar to that computed by direct numerical simulation with the same spectrum, although this simulation underestimates the lengths of tubes of intense vorticity.Some new results and concepts have been obtained using this kinematic simulation: (a) for the inertial subrange (which cannot yet be simulated by other means) the simulation confirms the form of the Eulerian frequency spectrum , where e,U0,ω are the rate of energy dissipation per unit mass, large-scale r.m.s. velocity, and frequency. For isotropic Gaussian large-scale turbulence at very high Reynolds number, CE ≈ 0.78, which is close to the computed value of 0.82; (b) for an observer moving with the large eddies the ‘Eulerian—Lagrangian’ spectrum is ϕEL11 = CELeω−2, where CEL ≈ 0.73; (c) for an observer moving with a fluid particle the Lagrangian spectrum ϕL11 = CLeω−2, where CL ≈ 0.8, a value consistent with the atmospheric turbulence measurements by Hanna (1981) and approximately equal to CEL; (d) the mean-square relative displacement of a pair of particles 〈Δ2〉 tends to the Richardson (1926) and Obukhov (1941) form 〈Δ2〉 = GΔet3, provided that the subrange extends over four decades in energy, and a suitable origin is chosen for the time t. The constant GΔ is computed and is equal to 0.1 (which is close to Tatarski's 1960 estimate of 0.06); (e) difference statistics (i.e. displacement from the initial trajectory) of single particles are also calculated. The exact result that Y2 = GYet3 with GY = 2πCL is approximately confirmed (although it requires an even larger inertial subrange than that for 〈Δ2〉). It is found that the ratio [Rscr ]G = 2〈Y2〉/〈Δ2〉≈ 100, whereas in previous estimates [Rscr ]G≈ 1, because for much of the time pairs of particles move together around vortical regions and only separate for the proportion of the time (of O(fc)) they spend in straining regions where streamlines diverge. It is estimated that [Rscr ]G ≈ O(fc−3). Thus relative diffusion is both a ‘structural’ (or ‘topological’) process as well as an intermittent inverse cascade process determined by increasing eddy scales as the particles separate; (f) statistics of large-scale turbulence are also computed, including the Lagrangian timescale, the pressure spectra and correlations, and these agree with predictions of Batchelor (1951), Hinzc (1975) and George et al. (1984).

Quantitative determination of molecular structure in multilayered thin films of biaxial and lower symmetry from photon spectroscopies. I. Reflection infrared vibrational spectroscopy

Abstract: A semitheoretical formalism based on classical electromagnetic wave theory has been developed for application to the quantitative treatment of reflection spectra from multilayered anisotropic films on both metallic and nonmetallic substrates. Both internal and external reflection experiments as well as transmission can be handled. The theory is valid for all wavelengths and is appropriate, therefore, for such experiments as x‐ray reflectivity, uv–visible spectroscopic ellipsometry, and infrared reflection spectroscopy. Further, the theory is applicable to multilayered film structures of variable number of layers, each with any degree of anisotropy up to and including full biaxial symmetry. The reflectivities (and transmissivities) are obtained at each frequency by solving the wave propagation equations using a rigorous 4×4 transfer matrix method developed by Yeh in which the optical functions of each medium are described in the form of second rank (3×3) tensors. In order to obtain optical tensors for materials not readily available in single crystal form, a method has been developed to evaluate tensor elements from the complex scalar optical functions (n) obtained from the isotropic material with the limitations that the molecular excitations are well characterized and obey photon–dipole selection rules.This method is intended primarily for infrared vibrational spectroscopy and involves quantitative decomposition of the isotropic imaginary optical function (k) spectrum into a sum of contributions from fundamental modes, the assignment of a direction in molecular coordinates to the transition dipole matrix elements for each mode, the appropriate scaling of each k vector component in surface coordinates according to a selected surface orientation of the molecule to give a diagonal im(n) tensor, and the calculation of the real(n) spectrum tensor elements by the Kramers–Kronig transformation. Tensors for other surface orientations are generated by an appropriate rotation matrix operation. To test the viability of this approach, three sets of experimentally derived infrared spectra of oriented monolayer assemblies on quite distinctively different substrates were chosen for simulation: (1) n‐alkanethiols self‐ assembled onto gold, (2) n‐alkanoic acid salt Langmuir–Blodgett (LB) monolayers on carbon, and (3) n‐alkanoic acid salt LB monolayers on silica glass. The formalism developed was used to simulate the spectral response and to derive structural features of the monolayers. Good agreement was found where comparisons with independent studies could be made and, in general, the method appears quite useful for structural studies of highly organized thin films.

The isector: a simple and direct method for generating isotropic, uniform random sections from small specimens

Abstract: SUMMARY The very simple and strong principle of vertical sections devised by Baddeley et al. has been a major advance in stereology when any kind of anisotropy is present in the specimen under study. On the other hand, some important stereological estimators still require isotropic, uniform random sections. This paper deals with a simple technique for embedding specimens in rubber moulds with spherical cavities. After the embedding, any handling of the resulting sphere independent of the specimen will induce isotropy of the final histological sections.

Nonlinear Cyclic Behavior of Reinforcing Bars Including Buckling

Abstract: The effects of inelastic buckling on monotonic and cyclic behavior of reinforcing steel bars are studied. Experimental tests show that this phenomenon occurs when the ratio between length and diameter of the bar exceeds 5, and that it leads to a postyield softening branch in compression that strongly influences the cyclic behavior of the bar. An analytical model of rebars accounting for inelastic buckling is presented, suitable for inclusion in programs for the analysis of r/c sections using the fiber type of discretization. The model employs four hardening rules (kinematic, isotropic, memory, and saturation) as functions of four independent parameters (yield stress, elastic modulus, hardening ratio, and a weighing coefficient). An explicit stress‐strain relation in finite terms for loading branches is utilized. The model yields accurate results in predicting the cyclic behavior of rebars, both with and without inelastic buckling, for deformation paths of general nature.

The decay of homogeneous isotropic turbulence

Abstract: A new theory for the decay of homogeneous, isotropic turbulence is proposed in which truly self‐preserving solutions to the spectral energy equation are found that are valid at all scales of motion. The approach differs from the classical approach in that the spectrum and the nonlinear spectral transfer terms are not assumed a priori to scale with a single length and velocity scale. Like the earlier efforts, the characteristic velocity scale is defined from the turbulence kinetic energy and the characteristic length scale is shown to be the Taylor microscale, which grows as the square root of time (or distance). Unlike the earlier efforts, however, the decay rate is shown to be of power‐law form, and to depend on the initial conditions so that the decay rate constants cannot be universal except possibly in the limit of infinite Reynolds number. Another consequence of the theory is that the velocity derivative skewness increases during decay, at least until a limiting value is reached. An extensive review of the experimental evidence is presented and used to evaluate the relative merits of the new theory and the more traditional views.

New variational principles in plasticity and their application to composite materials

Abstract: I n this paper , a variational method for bounding the effective properties of nonlinear composites with isotropic phases, proposed recently by ponte castaneda (J. Mech. Phys. Solids 39, 45, 1991), is given full variational principle status. Two dual versions of the new variational principle are presented and their equivalence to each other, and to the classical variational principles, is demonstrated. The variational principles are used to determine bounds and estimates for the effective energy functions of nonlinear composites with prescribed volume fractions in the context of the deformation theory of plasticity. The classical bounds of Voigt and Reuss for completely anisotropic composites are recovered from the new variational principles and are given alternative, simpler forms. Also, use of a novel identity allows the determination of simpler forms for nonlinear Hashin-Shtrikman bounds, and estimates, for isotropic, particle-reinforced composites, as well as for transversely isotropic, fiber-reinforced composites. Additionally, third-order bounds of the Beran type are determined for the first time for nonlinear composites. The question of the optimality of these bounds is discussed briefly.

Isotropic Majority-Vote Model on a Square Lattice

Abstract: The stationary critical properties of the isotropic majority vote model on a square lattice are calculated by Monte Carlo simulations and finite size analysis. The critical exponentsν, γ, andβ are found to be the same as those of the Ising model and the critical noise parameter is found to beqc=0.075±0.001.

Linear finite element methods for planar linear elasticity

Abstract: A linear nonconforming (conforming) displacement finite element method for the pure displacement (pure traction) problem in two-dimensional linear elasticity for a homogeneous isotropic elastic material is considered. In the case of a convex polygonal configuration domain, error estimates in the energy (L[sup 2]) norm are obtained. The convergence rate does not deteriorate for nearly incompressible material. Furthermore, the convergence analysis does not rely on the theory of saddle point problems. 22 refs.

Prediction of radiative transfer in cylindrical enclosures with the finite volume method

Abstract: This article shows how the finite volume method can be implemented to solve three-dimensional radiation problems in cylindrical enclosures. The medium is considered to be gray, and absorption, emission, and either isotropic or nonisotropic scattering are included. For the special case of axisymmetric radiation, a mapping is described that yields a complete solution by solving the intensity in a single azimuthal direction. The method is shown to rapidly converge to the solution of the radiation transfer equation as the spatial and directional grid is refined. Results from the solution of axisymmetric bench mark problems show that the method is stable, accurate, and computationally efficient. 25 refs.

A model for finite strain elasto-plasticity based on logarithmic strains: computational issues

Abstract: In the context of general isothermal processes, issues related to the constitutive modelling and computational treatment of finite deformation elasto-plasticity are examined employing logarithmic stretches as strain measures. A strain-energy function for isotropic elastic materials is proposed, which leads to a linear stress-strain relation and constant and isotropic elastic modulus in material setting. It is assumed that isotropy is maintained in the intermediate configuration which necessitates a representation of the plastic flow based on the scalar internal variables. By exploiting the main features of the present approach, expressed through the simple hyperelastic constitutive model in conjunction with notions of multiplicative decomposition of the deformation gradient and unstressed configuration, a computationally effective framework is formulated. It is pointed out that in this context, an algorithm could be proposed for rate-independent finite strain elasto-plasticity, which is exact for elastic processes and in the limit of non-hardening, deviatoric elasto-plasticity is in accordance with physical observations. Large elasto-plastic deformations at moderate elastic strains are examined within the approximation theory and displacement based finite element formulation of the boundary value problem proposed. Numerical analysis is performed for a realistic example capturing shear band localisation and the results are compared with experimental data.

Deriving the equations of motion for porous isotropic media

Abstract: The equations of motion and stress/strain relations for the linear dynamics of a two‐phase, fluid/solid, isotropic, porous material have been derived by a direct volume averaging of the equations of motion and stress/strain relations known to apply in each phase. The equations thus obtained are shown to be consistent with Biot’s equations of motion and stress/strain relations; however, the effective fluid density in the equation of relative flow has an unambiguous definition in terms of the tractions acting on the pore walls. The stress/strain relations of the theory correspond to ‘‘quasistatic’’ stressing (i.e., inertial effects are ignored). It is demonstrated that using such quasistatic stress/strain relations in the equations of motion is justified whenever the wavelengths are greater than a length characteristic of the averaging volume size.

Traveltime tomography in anisotropic media—II. Application

Abstract: SUMMARY Cross-borehole seismic data have traditionally been analysed by inverting the arrival times for velocity structure (traveltime tomography). The presence of anisotropy requires that tomographic methods be generalized to account for anisotropy. This generalization allows geological structure to be correctly imaged and allows the anisotropy to be evaluated. In a companion paper we developed linear systems for 2-D traveltime tomography in anisotropic media. In this paper we analyse the properties of the linear system for quasi-compressional waves and invert both synthetic and real data. Solutions to the linear systems consist of estimates of the spatial distributions of five parameters, each corresponding to a linear combination of a small subset of the 21 elastic, anisotropic velocity parameters. The parameters describe the arrival times in the presence of weak anisotropy with arbitrary symmetries. However, these parameters do not, in general, describe the full nature of the anisotropy. The parameters must be further interpreted using additional information on the symmetry system. In the examples in this paper we assume transverse isotropy (TI) in order to interpret our inversions, but it should be noted that this final interpretation could be reformulated in more general terms. The singular value decomposition of the linear system for traveltime tomography in anisotropic media reveals the (expected) ill-conditioning of these systems. As in isotropic tomography, ill-conditioning arises due to the limited directional coverage that can be achieved when sources and receivers are located in vertical boreholes. In contrast to isotropic tomography, the scalelength of the parametrization controls the nature of the parameter space eigenvectors: with a coarse grid all five parameters are required to model the data; with a fine grid some of the parameters appear only in the null space. The linear systems must be regularized using external, a priori information. An important regularization is the expectation that the elastic properties vary smoothly (an ad hoc recognition of the insensitivity of the arrival times to the fine-grained properties of the medium). The expectation of smoothness is incorporated by using a regularization matrix that penalizes rough solutions using finite difference penalty terms. The roughness penalty sufficiently constrains the solutions to allow the smooth eigenvectors in the null space of the unconstrained problem to contribute to the solutions. Hence, the spatial distribution of all five parameters is recovered. The level of regularization required is difficult to estimate; we advocate the analysis of a suite of solutions. Plots of the solution roughness against the data residuals can be used to find ‘knee points’, but for the fine tuning of the regularization one has little recourse but to examine a suite of images and use geological plausibility as an additional criterion. The application of the regularized numerical scheme to the synthetic data reveals that the roughness penalty should include terms that penalize high gradients addition to penalizing high second derivatives. Only when this constraint was included were the features of the original model recovered. The inversions of the field data yield good images of the expected stratigraphy and confirm previous estimates of the magnitude of the anisotropy and the orientation of the symmetry axis. The solutions further indicate an increase in anisotropy from the top to the bottom of the survey region that was not previously detected.

Anisotropic spheres in general relativity

Abstract: Static spheres are studied to investigate the link between the surface value of the potential (and therefore the observable redshift) and the highest occurring ratio of the trace of the pressure tensor to the local density. The transverse pressure is permitted to differ from the radial one. Both Newtonian and relativistic models are examined. Considerably larger redshift values are found when anisotropic pressures are allowed than in the isotropic case

The influence of inclusion shape on the overall viscoelastic behavior of composites

Abstract: The Eshelby-Mori-Tanaka method is extended into the Laplace domain to examine the linearly viscoelastic behavior in two types of composite materials: a transversely isotropic one with aligned spheroidal inclusions and an isotropic one with randomly oriented inclusions. Though approximate in nature, the method offers both simplicity and versatility, with explicit results for the sphere, disk, and fiber reinforcements in the transformed domain. The results coincide with some exact solutions for the composite sphere and cylinder assemblage models and, with spherical voids or rigid inclusions, the effective shear property also lies between Christensen’s bounds. Consistent with the known elastic behavior, the inverted creep compliances in the time domain indicate that, along the axial direction, aligned needles or fibers provide the most effective improvement for the creep resistance of the aligned composite, but that in the transverse plane the disk reinforcement is far superior. For the isotropic composite disks are always the most effective shape, whereas spheres are the poorest. Comparison with the experimental data for the axial creep strains of a glass/ED-6 resin composite containing 54 percent of aligned fibers indicates that the theory is remarkably accurate in this case.

Variable-angle correlation spectroscopy in solid-state nuclear magnetic resonance

Abstract: We describe here a new solid‐state nuclear‐magnetic‐resonance (NMR) experiment for correlating anisotropic and isotropic chemical shifts of inequivalent nuclei in powdered samples. Spectra are obtained by processing signals arising from a spinning sample, acquired in independent experiments as a function of the angle between the axis of macroscopic rotation and the external magnetic field. This is in contrast to previously proposed techniques, which were based on sudden mechanical flippings or multiple‐pulse sequences. We show that the time evolution of variable‐angle‐spinning signals is determined by a distribution relating the isotropic frequencies of the spins with their corresponding chemical shift anisotropies. Fourier transformation of these data therefore affords a two‐dimensional NMR spectrum, in which line shapes of isotropic and anisotropic interactions are correlated. Theoretical and experimental considerations involved in the extraction of this spectral information are discussed, and the techn...

Viscous fingering in miscible displacements: Unification of effects of viscosity contrast, anisotropic dispersion, and velocity dependence of dispersion on nonlinear finger propagation

Abstract: The two‐dimensional (2‐D) isotropic simulations of Tan and Homsy [Phys. Fluids 31, 1330 (1988)] are extended to much broader and longer domains, and the 2‐D anisotropic simulations of Zimmerman and Homsy are extended to include a general velocity dependence. The mechanisms of nonlinear interaction of viscous fingers found for the first time in the anisotropic simulations recur in isotropic simulations, but at weaker levels of dispersion. An appropriate scaling to unify the average long time growth of the instability with both anisotropy in geometry and dispersion is provided. The long time growth of the instability from simulations agrees with acoustic measurements in 3‐D porous media, Bacri et al. [Phys. Rev. Lett. 67, 2005 (1991)], elucidating the effects of viscosity contrast, anisotropy, and velocity dependence of longitudinal dispersion. The combination of sufficiently high viscosity contrast, weak transverse dispersion, and strong dependence of longitudinal dispersion on velocity results in an augmentation to the long time growth of the instability. The associated critical parameter found by linear stability theory of Yortsos and Zeybek [Phys. Fluids 31, 3511 (1988)] predicts accurately this same long time growth increase.

The design of isotropic manipulator architectures in the presence of redundancies

Abstract: The design of redundant isotropic architectures for robotic nia nipulators is the subject of this article. A manipulator is said to have a redundant isotropic architecture if (1) its number of controlled axes is greater than the dimension of its task space. and (2) it is possible for the manipulator to attain configura tions at which all the singular values of its Jacobian matrix are identical and nonzero. The concept of isotropy, which has already been applied to the design of nonredundant manip ulators, is applied to the design of redundant ones. General geometric conditions on the manipulator parameters and on its configuration variables under which isotropy is attained are derived.

Time-dependent models of magnetized pair plasmas

Abstract: Attention is given to a numerical code developed to study the time evolution of electron-positron plasmas. The code solves in a self-consistent manner kinetic equations describing the effects of Compton scattering, two-photon pair production, pair annihilation, cooling of pairs via Coulomb scattering, e-e bremsstrahlung, and synchrotron radiation. The kinetic equations are derived under the approximation of homogeneous and isotropic particle distributions on the basis of a study by Coppi and Blandford (1993). Both stationary and time-varying output radiation spectra are computed. Good qualitative agreement with previous calculations is found, except where the differences are attributable to the improved treatment of the microphysics.

The energy decay in self-preserving isotropic turbulence revisited

Abstract: The assumption of self-preservation allows for an analytical determination of the energy decay in isotropic turbulence. Here, the self-preserving isotropic decay problem is analyzed, yielding a more complete picture of self-serving isotropic turbulence. It is proven rigorously that complete self-serving isotropic turbulence admits two general types of asymptotic solutions: one where the turbulent kinetic energy K approximately t (exp -1) and one where K approximately t (sup alpha) with an exponent alpha greater than 1 that is determined explicitly by the initial conditions. By a fixed point analysis and numerical integration of the exact one-point equations, it is demonstrated that the K approximately t (exp -1) and where K approximately t (sup -alpha) with an exponent alpha greater than 1 that is determined explicitly by the initial conditions. By a fixed point analysis and numerical integration of the exact one point equations, it is demonstrated that the K approximately t (exp -1) power law decay is the asymptotically consistent high Reynolds number solution; the K approximately 1 (sup - alpha) decay law is only achieved in the limit as t yields infinity and the turbulence Reynolds number vanishes. Arguments are provided which indicate that a K approximately t (exp -1) power law decay is the asymptotic state towards which a complete self-preseving isotropic turbulence is driven at high Reynolds numbers in order to resolve the imbalance between vortex stretching and viscous diffusion.

Impact Response of Orthotropic Composite Plates Predicted from a One-Parameter Differential Equation

Abstract: This paper presents an approximate analytical solution for the dynamic response of an infinite specially orthotropic plate impacted by an impactor with a semispherical tip. Thus, the solution is valid for low mass impacts. The analysis is an extension and rederivation of a solution for isotropic plates proposed by Zener. The analysis assumes a Hertzian contact law and is based on Kirchhoff's plate equation. The plate response is expressed in terms of contact force, contact pressure, central displacement, central curvature, and size of the impact affected area. The response is computed from a dimensionless differential equation in time, which is only dependent on the inelasticity parameter lambda. Lambda is a function of the impact velocity and variables describing the impactor and the plate. For a given lambda, the response can be interpolated from the solution plots for a number of representativ e values of lambda. Results computed from the model are compared with published numerical analyses and a number of experiments, and a close agreement is noted. Finally, the analysis shows the time-dependent velocity of a flexural wave propagating from the impact center.

New results in the theory of elasticity for two-dimensional composites

Abstract: We bring together and discuss a number of exact relationships in two-dimensional (or plane) elasticity, that are useful in studying the effective elastic constants and stress fields in two-dimensional composite materials. The first of these dates back to Michell (1899) and states that the stresses, induced by applied tractions, are independent of the elastic constants in a two-dimensional material containing holes. The second involves the use of Dundurs constants which, for a composite consisting of two isotropic elastic phases, reduce the dependence of stresses on the elastic constants from three independent dimensionless parameters to two. It is shown that these two results are closely related to a recently proven theorem by Cherkaev, Lurie and Milton, which we use to show that the effective Young's modulus of a sheet containing holes is independent of the Poisson's ratio of the matrix material. We also show that the elastic moduli of a composite can be found exactly if the shear moduli of the components are all equal; a previously known result. We illustrate these results with computer simulations, where appropriate. Finally we conjecture on generalizations to multicomponent composite materials and to situations where the bonding between the phases is not perfect.

Stability and Instability of an Uniaxial Alignment Against Biaxial Distortions in the Isotropic and Nematic Phases of Liquid Crystals

Abstract: The nonlinear relaxation equation for the five components of the (2nd rank) alignment tensor is used to analyze the stability or instability of its uniaxial stationary solutions against biaxial distortions. The equation is valid for both the isotropic and nematic phases of a liquid crystal. The influence of an external field is taken into account. For liquid crystals with positive dielectric anisotropy in the presence of an electric field, the uniaxial state is stable against biaxial distortions. The situation is different for the case of a negative dielectric anisotropy. At temperatures above the field-free transition temperature isotropic-nematic, application of an electric field leads to a uniaxial state which becomes unstable against biaxial distortions if the strength of the field exceeds a critical value. Properties of the biaxial state which can exist in a limited range of temperatures are also presented. Introduction The dynamic behavior of the second rank alignment tensor specifying the molecular orientation in the isotropic and nematic phases of liquid crystals has recently been analyzed [1] for the case where the alignment is uniaxial. Point of departure was a nonlinear relaxation equation for the alignment tensor, which, in general, can also become biaxial. It is the purpose of this article to study the stability or instability of the uniaxial solutions (in the presence of a field with uniaxial symmetry) against biaxial distortions. In particular, it is demonstrated that a transition from a uniaxial to.a biaxial state can occur in liquid crystals with negative dielectric anisotropy in the presence of an electric field and in the vicinity of the phase transition isotropic-nematic. The transition from a uniaxial to a biaxial positional order, observed in molecular dynamics computer simulations of orientated ferro-fluids [2] can be treated by a similar approach [3]. J. Non-Equilib. Thermodyn., Vol. 17, 1992, No. 2 © Copyright 1992 Walter de Gruyter · Berlin · New York 154 P. Kaiser et al. This article proceeds as follows. In section 1, the nonlinear relaxation equation is stated, firstly, in its tensorial form. Reduced variables are msed for the time, the temperature and the strength of an external orienting field. Cartesian basis tensors and the associated components of the alignment tensor are introduced (section 1.2). After a few remarks on uniaxial and biaxial symmetry, the relaxation equations are given for the components of the alignment tensor (section 1.4). Section 2 is devoted to the stability analysis. It is assumed that the system is in a stationary uniaxial state. The time dependence of small biaxial distortions is inferred from the relaxation equations of the relevant components of the alignment tensor. The limits of stability in the presence of an external field are discussed in sections 2.3 and 2.4. For particles with negative dielectric anisotropy in the presence of an electric field an instability can be encountered in a certain temperature intervall which is above the nematic-isotropic transition temperature for the field free case if the field exceeds a critical value. Properties of the biaxial state are presented in section 3. The appendix contains some mathematical details concerning the basis tensors which were needed for the derivation of the relaxation equations of the tensor components. 1. The relaxation equation for the alignment tensor 1.1 Tensorial form The nonlinear relaxation equation for the second rank alignment tensor specifying the molecular orientation assumes a convenient form if this tensor is expressed in units of the magnitude of the alignment at the nematic to isotropic phase transition temperature TNi. This scaled alignment tensor is denoted by $μν where the Greek subscripts refer to cartesian components. With the time t scaled in appropriate units, this relaxation equation can be written as [1] d δσ where ó is a (dimensionless) Landau-de Gennes potential. Here we choose [1] ó = i $8μν5μν 1/6 5μν8νλ8λμ + \ (3μν3μν) Γμν8μν . (2) The summation convention is used for Greek subscripts. J. Non-Equilib. Thermodyn., Vol. 17, 1992, No. 2 Unaxial and biaxial alignment 155 The quantity -(· -30-0 is a dimensionless temperature variable. The characteristic temperature T* is somewhat below the transition temperature TNi. Notice that & = 1 and & = 0 correspond to T = TNi and Τ = Ã*, respectively. The field tensor F v causes the orientation due to an external electric (E) or magnetic (B) field. In the case of an electric field, e.g. one has Ρμν~ΕμΕν where the symbol ... refers to the symmetric traceless part of a tensor. With the choice (2) for ó, equation (1) becomes 3 J/6 3μλ5λν + 2(^^)5^ F„v = 0 . (4) The tensorial nonlinear relaxation equation (4) applies to both the isotropic and nematic phases of liquid crystals. However, it is restricted to spatially homogeneous systems and the possible coupling with a flow process underlying the flow birefringence is disregarded here. For further discussion of the physical meaning of the variables occurring in (4) and for the relation of the present approach to de Gennes' theory for the pretransitional behavior [4—6] as well as to the Ericksen-Leslie theory for nematics [4-6] see [1] and the literature quoted there. It is the advantage of the scaled form (4) of the tensorial relaxation equation that in addition to the field tensor, only the temperature variable S occurs as a control parameter. The symmetric traceless second rank tensor $μν has 5 independent components. One choice of independent components are the spherical components S2m with m = 0, +1, +2 where a reference axis, e.g. parallel to the unit vector e has to be specified. Here we prefer a set of (pseudospherical) components which follow from an expansion with respect to (cartesian) basis tensors to be stated next. 1.2 Basis tensors Let T&\ k = 0, 1, 2, 3, 4 be 5 orthonormalized symmetric traceless basis tensors with the property = #*; k, / = 0, 1,2, 3, 4 (5) where S is the Kronecker symbol. Then the alignment tensor 8μν can be decomposed according to

Ultrasonic wave interaction with a thin anisotropic layer between two anisotropic solids: Exact and asymptotic‐boundary‐condition methods

Abstract: Boundary conditions introduced to model a thin anisotropic layer between two generally anisotropic solids are given. The model can be used to describe an imperfect anisotropic interface. The present results for anisotropic boundary conditions are a generalization of previous work valid for either an isotropic viscoelastic layer [J. Acoust. Soc. Am. 89, 503–515 (1991)] or an orthotropic layer with a plane of symmetry coinciding with the incident plane [J. Acoust. Soc. Am. 91, 1875–1887 (1992)]. The boundary conditions are represented by a 6×6 transfer matrix which relates six‐dimensional vectors formed from stresses and displacements on each side of the interface. The transfer matrix is obtained as an asymptotic representation of the three‐dimensional solution for a thin orthotropic layer of arbitrary orientation between two solids. Such boundary conditions couple the in‐plane and out‐of‐plane stresses and displacements on the interface even for isotropic bodies. Interface imperfections are modeled by an i...