Showing papers on "Isotropy published in 1996"

A micromechanics-based nonlocal constitutive equation and estimates of representative volume element size for elastic composites

Abstract: A variational formulation is employed to derive a micromechanics-based, explicit nonlocal constitutive equation relating the ensemble averages of stress and strain for a class of random linear elastic composite materials. For two-phase composites with any isotropic and statistically uniform distribution of phases (which themselves may have arbitrary shape and anisotropy), we show that the leading-order correction to a macroscopically homogeneous constitutive equation involves a term proportional to the second gradient of the ensemble average of strain. This nonlocal constitutive equation is derived in explicit closed form for isotropic material in the one case in which there exists a well-founded physical and mathematical basis for describing the material's statistics: a matrix reinforced (or weakened) by a random dispersion of nonoverlapping identical spheres. By assessing, when the applied loading is spatially-varying, the magnitude of the nonlocal term in this constitutive equation compared to the portion of the equation that relates ensemble average stresses and strains through a constant “overall” modulus tensor, we derive quantitative estimates for the minimum representative volume element (RVE) size, defined here as that over which the usual macroscopically homogeneous “effective modulus” constitutive models for composites can be expected to apply. Remarkably, for a maximum error of 5% of the constant “overall” modulus term, we show that the minimum RVE size is at most twice the reinforcement diameter for any reinforcement concentration level, for several sets of matrix and reinforcement moduli characterizing large classes of important structural materials. Such estimates seem essential for determining the minimum structural component size that can be treated by macroscopically homogeneous composite material constitutive representations, and also for the development of a fundamentally-based macroscopic fracture mechanics theory for composites. Finally, we relate our nonlocal constitutive equation explicitly to the ensemble average strain energy, and show how it is consistent with the stationary energy principle.

Method for generating long-range correlations for large systems.

Abstract: We propose a new method to generate a sequence of random numbers with long-range power-law correlations that overcomes known difficulties associated with large systems. The new method presents an improvement on the commonly-used methods. We apply the algorithm to generate enhanced diffusion, isotropic and anisotropic self-affine surfaces, and isotropic and anisotropic correlated percolation.

Electroseismic wave properties

Abstract: In a porous material saturated by a fluid electrolyte, mechanical and electromagnetic disturbances are coupled. The coupling is due to an excess of electrolyte ions that exist in a fluid layer near the grain surfaces within the material; i.e., the coupling is electrokinetic in nature. The governing equations controlling the coupled electromagnetic‐seismic (or ‘‘electroseismic’’) wave propagation are presented for a general anisotropic and heterogeneous porous material. Uniqueness is derived as well as the statements of energy conservation and reciprocity. Representation integrals for the various wave fields are derived that require, in general, nine different Green’s tensors. For the special case of an isotropic and homogeneous wholespace, both the plane‐wave and the point‐source responses are obtained. Finally, the boundary conditions that hold at interfaces in the porous material are derived.

Crucial role of the lattice distortion in the magnetism of LaMnO3.

Abstract: The stability of the $A$-type antiferromagnetic order and canted magnetic structure of LaMn${\mathrm{O}}_{3}$ perovskite is explained in the itinerant-electron picture based on the local-spin-density approximation. We demonstrate the crucial role of the observed lattice distortion which strongly affects the magnetocrystalline anisotropy, as well as the anisotropic and isotropic exchange interactions in this compound.

P-wave signatures and notation for transversely isotropic media: An overview

Abstract: Progress in seismic inversion and processing in anisotropic media depends on our ability to relate different seismic signatures to the anisotropic parameters. While the conventional notation (stiffness coefficients) is suitable for forward modeling, it is inconvenient in developing analytic insight into the influence of anisotropy on wave propagation. Here, a consistent description of P -wave signatures in transversely isotropic (TI) media with arbitrary strength of the anisotropy is given in terms of Thomsen notation. The influence of transverse isotropy on P -wave propagation is shown to be practically independent of the vertical S -wave velocity VS0 , even in models with strong velocity variations. Therefore, the contribution of transverse isotropy to P-wave kinematic and dynamic signatures is controlled by just two anisotropic parameters, e and δ, with the vertical velocity VP0 being a scaling coefficient in homogeneous models. The distortions of reflection moveouts and amplitudes are not necessarily correlated with the magnitude of velocity anisotropy. The influence of transverse isotropy on P -wave normal-moveout (NMO) velocity in a horizontally layered medium, on small-angle reflection coefficient, and on point-force radiation in the symmetry direction is entirely determined by the parameter δ. Another group of signatures of interest in reflection seisimology–the dip-dependence of NMO velocity, magnitude of nonhyperbolic moveout, time-migration impulse response, and the radiation pattern near vertical–is dependent on both anisotropic parameters (e and δ) and is primarily governed by the difference between e and δ. Since P -wave signatures are so sensitive to the value of e − δ, application of the elliptical-anisotropy approximation (e = δ) in P -wave processing may lead to significant errors. Many analytic expressions given in the paper remain valid in transversely isotropic media with a tilted symmetry axis. Moreover, the equation for NMO velocity from dipping reflectors, as well as the nonhyperbolic moveout equation, can be used in symmetry planes of any anisotropic media (e.g., orthorhombic).

Anisotropic three-dimensional MHD turbulence

Abstract: Direct spectral method simulation of the three-dimensional magnetohydrodynamics (MHD) equations is used to explore anisotropy that develops from initially isotropic fluctuations as a consequence of a uniform applied magnetic field. Spectral and variance anisotropies are investigated in both compressible and incompressible MHD. The nature of the spectral anisotropy is consistent with the model of Shebalin et al. [1983] in which the spectrum broadens in the perpendicular wavenumber direction, the anisotropy being greater for smaller wavenumbers. Here this effect is seen for both incompressible and polytropic compressible MHD. In contrast, the longitudinal (compressive) velocity fluctuations remain isotropic. Variance anisotropy is observed for low plasma beta compressible MHD but not for incompressible MHD. Solar wind observations are qualitatively consistent with both variance and spectral anisotropies of the type discussed here.

Orientational correlation functions and polarization selectivity for nonlinear spectroscopy of isotropic media. I. Third order

Abstract: The contribution of orientational relaxation to the tensor components of the third‐order nonlinear polarization is evaluated for off‐resonance Raman and dipole resonant experiments in the perturbative limit. Orientational correlation functions are calculated within the model of orientational diffusion for all third‐order tensor components relevant to isotropic media. General expressions for polarization geometries that are selective to particular components of the signal, i.e., magic angles, are derived for collinear and crossed‐beam excitation geometries. It is shown that although limited selectivity exists for Raman spectroscopies, no combination of polarizations will give complete selectivity to the isotropic, anisotropic, or nonresonant contributions to the Raman polarizability tensor. For resonant spectroscopies, the four‐time correlation function that describes the orientational polarization decay can be written as the product of three two‐time correlation functions. While magic angles for orientati...

Beyond the isotropic-model approximation in the theory of thermal conductivity

Abstract: By the use of an iterative method the linearized phonon-Boltzmann equation for a dielectric solid subjected to a thermal gradient is solved in the frame of three-phonon interactions. In this way it is possible to calculate the thermal conductivity of rare-gas solids starting from the pair potential and accounting for the real Brillouin zone of the lattice. The numerical results are in full agreement with experiment and represent a considerable improvement with respect to those previously deduced for an isotropic solid.

Micromechanical Definition of the Strain Tensor for Granular Materials

Abstract: In order to develop constitutive relations for granular materials from the micromechanical viewpoint, general expressions relating macroscopic stress and strain to contact forces and particle displacements are required. Such an expression for the stress tensor under quasi-static conditions is well established in the literature, but a corresponding expression for the strain tensor has been lacking so far. This paper presents such an expression for two-dimensional assemblies. This expression is verified by computer simulations of biaxial and shear tests. As a demonstration of the use of the developed expression, a study is made of the elastic moduli of two-dimensional, isotropic assemblies of bonded, nonrotating disks. Theoretical expressions are given for the elastic moduli in terms of micromechanical parameters, such as coordination number and contact stiffnesses. Comparison with the results from computer simulations show that the agreement is fairly good over a wide range of coordination numbers and contact stiffness ratios.

Nonlinear elasticity and stress-induced anisotropy in rock

Abstract: The elastic nonlinear behavior of rocks as evidenced by deviations from Hooke's law in stress-strain measurements, and attributable to the presence of mechanical defects (compliant features such as cracks, microfractures, grain joints), is a well-established observation. The purpose of this paper is to make the connection between the elastic nonlinearity and stress-induced effects on waves, in this case uniaxial-stress-induced transverse isotropy. The linear and nonlinear elastic coefficients constitute the most condensed manner in which to characterize the elastic behavior of the rock. We present both the second- and the third-order nonlinear elastic constants obtained from experimental data on rock samples assumed homogeneous and isotropic when unstressed. As is normally the case, the third-order (nonlinear) constants are found to be much larger than the second-order (linear) elastic constants. Contrary to results from intact homogeneous solids (materials without mechanical defects), rocks exhibit weak to strong nonlinearity and always in the same manner (i.e., an increase of the moduli with pressure). As a consequence the stress-induced P wave anisotropy and S wave birefringence can be large. The stress-induced P wave anisotropy appears to be much larger than the S wave birefringence. The fast direction is parallel to the stress direction, and the anisotropy goes as sin2 θ, θ being the angle between the propagation direction and the stress direction. Experiments on rocks indicate that at low applied stresses, the proportionality of the stress and the induced S birefringence and P anisotropy predicted by theory is well corroborated.

New developments in direct shape determination from small-angle scattering. 2. Uniqueness

Abstract: The problem of uniqueness of the low-resolution shape determination from small-angle scattering by isotropic monodisperse systems is considered. The particle shape is represented by the envelope function parameterized using spherical harmonics as described in a previous paper [Svergun & Stuhrmann (1991). Acta Cryst. A47, 736–744]. Computer simulations are made on the model bodies with sharp boundaries exactly represented by spherical harmonics. If the number of independent parameters describing the shape is 1 to 1.5 times the number of Shannon channels covered by the data set, the shape restoration is found to be unique and stable with respect to the random and systematic errors. The resolution limits of the straightforward shape determination are connected to the computational accuracy of the model intensities; with current algorithms, shapes described by 15 to 20 independent parameters can be uniquely determined. The results form a basis for an ab initio low-resolution shape determination in terms of spherical harmonics.

A rate-dependent isotropic damage model for the seismic analysis of concrete dams

Abstract: In this paper a rate-dependent isotropic damage model developed for the numerical analysis of concrete dams subjected to seismic excitation is presented. The model is shown to incorporate two features essential for seismic analysis: stiffness degradation and stiffness recovery upon load reversals and strain-rate sensitivity. The issue of mesh objectivity is addressed using the concept of the ‘characteristic length’ of the fracture zone, to show that both the softening modulus and the fluidity parameter must depend on it to provide consistent results as the computational mesh is refined. Some aspects of the numerical implementation of the model are also treated, to show that the model can be easily incorporated in any standard non-linear finite element code. The application of the proposed model to the seismic analysis of a large gravity concrete dam shows that the structural response may vary significantly in terms of the development of damage. The inclusion of rate sensitivity is able to reproduce the experimental observation that the tensile peak strength of concrete can be increased up to 50 percent for the range of strain rates that appear in a structural safety analysis of a dam subjected to severe seismic actions.

Theory of high-NA imaging in homogeneous thin films

Abstract: A description is given of a modeling technique that is used to explore three-dimensional image distributions formed by high numerical aperture (NA > 0.6) lenses in homogeneous, isotropic, linear, and source-free thin films. The approach is based on a plane-wave decomposition in the exit pupil. Factors that are due to polarization, aberration, object transmittance, propagation, and phase terms are associated with each plane-wave component. These are combined with a modified thin-film matrix technique in a derivation of the total field amplitude at each point in the film by a coherent vector sum over all plane waves. One then calculates the image distribution by squaring the electric-field amplitude. The model is used to show how asymmetries present in the polarized image change with the influence of a thin film. Extensions of the model to magneto-optic thin films are discussed.

Composites with Imperfect Interface

Abstract: New variational principles and bounds are introduced, describing the effective conductivity tensor for anisotropic two-phase heat conducting composites with interfacial surface resistance between phases. The new upper bound is given in terms of the two-point correlation function, component volume fractions and moment of inertia tensor for the surface of each heterogeneity. The new lower bound is given in terms of the interfacial surface area, component volume fractions and a scale-free matrix of parameters. This matrix corresponds to the effective conductivity associated with the same geometry but with non-conducting inclusions. The bounds are applied to theoretically predict the occurrence of size effect phenomena. We identify a parameter R$\_{\text{cr}}$ that measures the relative importance of interfacial resistance and contrast between phase resistivities. The scale at which size effects occur is determined by this parameter. For isotropic conducting spheres in a less conducting isotropic matrix we show that for monodisperse suspensions of spheres of radius R$\_{\text{cr}}$ the effective conductivity equals that of the matrix. For polydisperse suspensions of spheres it is shown that, when the mean radius lies below R$_{\text{cr}}$, the effective conductivity lies below that of the matrix.

On the Elliptic Inclusion in Anti-Plane Shear

Abstract: In this paper, we consider the anti-plane shear problem of an elliptic inclusion embedded in an infinite, isotropic, elastic medium, subjected at infinity to a uniform stress field. Using complex v...

Morphologically representative pattern-based bounding in elasticity

Abstract: A general theory for the homogenization of heterogeneous linear elastic materials that relies on the concept of “morphologically representative pattern” is given. It allows the derivation of rigorous bounds for the effective behaviour of the Voigt-Reuss-type, which apply to any distribution of patterns, or of the Hashin-Shtrikman-type, which are restricted to materials whose pattern distributions are isotropic. Particular anisotropic distributions of patterns can also be considered: Hashin-Shtrikman-type bounds for anisotropic media are then generated. The resolution of the homogenization problem leads to a complex composite inclusion problem with no analytical solution in the general case. Here it is solved by a numerical procedure based on the finite element method. As an example of possible application, this procedure is used to derive new bounds for matrix-inclusion composites with cubic symmetry as well as for transversely isotropic materials.

Stress anisotropy and wave propagation: a micromechanical view

Abstract: Wave propagation is a constant-fabric macrophenomenon, suitable to microinterpretation. Both velocity and attenuation characterize state, including inherent and stress-induced anisotropy. The purpose of this research is to study the effect of isotropic and deviatoric stresses on wave propagation in particulate materials at low strains and to interpret results at the microlevel. A resonant-column device was midified to allow for the application of axial extension and axial compression deviatoric loading. The fixed-free boundary condition of the sample was maintained. Data for round, hard-grained sand show that shear wave velocity and attenuation are primarily dependent on the mean stress on the polarization plane, with minimal effect of the deviatoric component, in agreement with prior observations at stress ratios less than 2–3. Attenuation is strongly correlated with the mean stress in the polarization plane and the level of shear strain. Damping does not vanish at low strains, contrary to predictions ba...

High- and low-frequency elastic moduli for a saturated porous/cracked rock-Differential self-consistent and poroelastic theories

Abstract: Although P- and S-wave dispersion is known to be important in porous/cracked rocks, theoretical predictions of such dispersions have never been given. The authors report such calculations and show that the predicted dispersions are high in the case of low aspect ratio cracks ({le}10{sup {minus}3}) or high crack density ({ge}10{sup {minus}1}). Their calculations are derived from first-principle computations of the high- and low-frequency elastic moduli of a rock permeated by an isotropic distribution of pores or cracks, dry or saturated, with idealized geometry (spheres or ellipsoids). Henyey and Pomphrey developed a differential self-consistent model that is shown to be a good approximation. This model is used here, but as it considers cracks with zero thickness, it can not account for fluid content effects. To remove this difficulty, one combines the differential self-consistent approach with a purely elastic calculation of moduli in two cases: that of spherical pores and that of oblate spheroidal cracks with a nonzero volume. This leads to what the authors call the extended differential, self-consistent model (EM). When combining these EM results with the Gassmann equation, it is possible to derive and compare the theoretical predictions for high- and low-frequency effective moduli in the case of amore » saturated rock. Since most laboratory data are ultrasonic measurements and in situ data are obtained at much lower frequencies, this comparison is useful for interpreting seismic data in terms of rock and fluid properties. The predicted dispersions are high, in agreement with previous experimental results. A second comparison is made with the semi-empirical model of Marion and Nur, which considers the effects of a mixed porosity (round pores and cracks together).« less

Interparticle contact behavior and wave propagation

Abstract: The low-strain stiffness and energy dissipation in particulate materials is strongly determined by the behavior of contacts. This paper presents results of a test program designed to study the effect of contact response on the propagation of waves. Wave velocity and attenuation were measured during isotropic loading using a resonant column device at shear strains varying from γ= 10 −5 to γ= 10 −6 . Elastic, viscoplastic and brittle contact behaviors were studied with steel spheres, lead shot, and silica-kaolinite pellets. All measured velocity-stress exponents were b /2 >≈ 1/6, which is the theoretical value for spherical contacts. High-tolerance steel spheres approximated this value. Contact crushing showed the highest exponent. Theoretical analyses confirmed that several phenomena conduce to a velocity-stress exponent b /2 = 0.25: buckling of particle chains and increase in coordination number, elastoplastic behavior, and cone-plane contacts. Load and unload data for viscoplastic lead shot showed that c...

Free-mode surface-wave computations

Abstract: SUMMARY The theory of Love- and Rayleigh-wave dispersion for plane-layered earth models has undergone a number of developments since the initial work of Thomson and Haskell. Most of these were concerned with computational difficulties associated with numerical overflow and loss of precision at high frequencies in the original Thomson-Haskell formalism. Several seemingly distinct approaches have been followed, including the delta matrix, reduced delta matrix, Schwab-Knopoff, fast Schwab-Knopoff, Kennett's Reflection-Transmission Matrix and Abo-Zena methods. This paper analyses all these methods in detail and finds explicit transformations connecting them. It is shown that they are essentially equivalent and, contrary to some claims made, each solves the loss of precision problem equally well. This is demonstrated both theoretically and computationally. By extracting the best computational features of the various methods, we develop a new algorithm (sec Appendix A5), called the fast delta matrix algorithm. To date, this is the simplest and most efficient algorithm for surface-wave dispersion computations (see Fig. 4). The theory given in this paper provides a complete review of the principal methods developed for Love- and Rayleigh-wave dispersion of free modes in plane-layered perfectly elastic, isotropic earth models and puts to rest controversies that have arisen with regard to computational stability.

Optical properties of isotropic chiral media

Abstract: A review is given of the optical properties of isotropic chiral media, based on the symmetrized constitutive relations of Condon. The review includes discussion of wave propagation in chiral media, and derivation of the reflection and transmission amplitudes of an isotropic optically active medium, and of a layer resting on a substrate. Boundary conditions and energy conservation relations are derived. For the chiral layer, simple formulae are given for the reflection and transmission coefficients at normal incidence, in the weak chirality limit, near the critical angles, and for a thin layer. Analytic expressions are given for all the reflection and transmission amplitudes in the general case. An ellipsometric method of measuring the chirality of very small sample volumes is suggested.

One-dimensional wave propagation in the linear theory of thermoelasticity without energy dissipation

Abstract: The linear theory of thermoelasticity without energy dissipation for homogeneous and isotropic materials is employed to study one-dimensional waves in a half-space The waves are supposed to be due to sudden inputs of temperature and stress/strain on the boundary The Laplace transform method is employed to solve the problem Exact solutions, in closed form, for the displacement, temperature, strain, and stress fields are obtained The characteristic features of the underlying theory are analyzed in light of these solutions and their counterparts in earlier works

Elastoplastic Contact Conductance Model for Isotropic Conforming Rough Surfaces and Comparison With Experiments

Abstract: A new thermal elastoplastic contact conductance model for isotropic conforming rough surfaces is proposed. This model is based on surface and thermal models used in the Cooper, Mikic, and Yovanovich plastic model, but it differs in the deformation aspects of the thermal contact conductance model. The model incorporates the recently developed simple elastoplastic model for sphere-flat contacts, and it covers the entire range of material behavior, i.e., elastic, elastoplastic, and fully plastic deformation. Previously data were either compared with the elastic model or the plastic model assuming a type of deformation a priori. The model is used to reduce previously obtained isotropic contact conductance data, which cover a wide range of surface characteristics and material properties. For the first time data can be compared with both the elastic and plastic models on the same plot. This model explains the observed discrepancies noted by previous workers between data and the predictions of the elastic or plastic models.

Wave-induced soil response in a nearly saturated sea-bed of finite thickness

Abstract: Consolidation and storage equations are used to develop a closed-form solution for the wave-induced pore pressure, soil displacements and effective stresses in an elastic sea-bed subject to a system of two intersecting waves. A homogeneous soil matrix of finite thickness in isotropic and saturated conditions only is considered. The three-dimensional general solutions so developed are readily reducible to the conditions for soil of infinite thickness, and also for the limiting cases of two-dimensional progressive and standing waves for soil of finite thickness, for which no explicit solutions have previously been available. Verification of this solution with results of two-dimensional solutions available from a semi-analytical method, a boundary-layer approximation and a numerical model is carried out. Validation is performed by comparison with experimental results. The effects on wave-induced pore pressure of sea-bed thick- ness, shear modulus of soil and grain size are discussed. Les equations de stockag...