Showing papers on "Transverse isotropy published in 1984"
TL;DR: In this article, the influence of aspect ratio α, from zero to infinity, on the effective elastic moduli of a transversely isotropic composite was examined, where the reinforcing inclusions, which could be flakes or short fibers, are assumed to be spheroidal and unidirectionally aligned.
Abstract: This paper examines the influence of aspect ratio α, from zero to infinity, on the effective elastic moduli of a transversely isotropic composite. The reinforcing inclusions, which could be flakes or short fibers, are assumed to be spheroidal and unidirectionally aligned. Of the five independent elastic constants, the longitudinal Young's modulus E11 and in-plane shear modulus μ12 appear to increase with increasing aspect ratio, while the transverse Young's modulus E22, out-plane shear modulus μ23, and plane-strain bulk modulus K23, generally decrease. It is further noted that E11 is more sensitive to α when α > 1 but the others are more so when α < 1. The present analysis was carried out by the combination of Eshelby's and Mori-Tanaka's theories of inclusions.
636 citations
TL;DR: In this article, the authors proposed a scheme to construct a structure composed of anisotropic strata from reflection and transmission properties of individual interfaces using a slightly modified version of the recursion scheme of Kennett.
Abstract: Summary. The response of a structure composed of anisotropic strata can be built up from the reflection and transmission properties of individual interfaces using a slightly modified version of the recursion scheme of Kennett. This scheme is conveniently described in terms of scatterer operators and scatterer products. The effects of a free surface and the introduction of a simple point source at any depth can be accommodated in a manner directly analogous to the treatment for isotropic structures. As in the isotropic case the results so obtained are stable to arbitrary wavenumbers. For isotropic media, synthetic seismograms can be constructed by computing the structure response as a function of frequency and radial wavenumber, then performing the appropriate Fourier and Hankel transforms to obtain the wavefield in time-distance space. Such a scheme is convenient for any system with cylindrical symmetry (including transverse isotropy). Azimuthally anisotropic structures, however, do not display cylindrical symmetry; for these the transverse component of the wavenumber vector will, in general, be non-zero, with the result that phase, group, and energy velocities may all diverge. The problem is then much more conveniently addressed in Cartesian coordinates, with the frequency-wavenumber to time-distance transformation accomplished by 3-D Fourier transform.
186 citations
TL;DR: In this paper, the authors study the relationship between velocities calculated from t 2 - x 2 -curves and the true velocity of a transversely isotropic layer, and calculate expressions for these terms as a function of the elastic parameters.
Abstract: The arrival-time curve of a reflection from a horizontal interface, beneath a homogeneous isotropic layer, is a hyperbola in the x - t -domain
If the subsurface is one-dimensionally inhomogeneous (horizontally layered), or if some or all of the layers are transversely isotropic with vertical axis of symmetry, the statement is no longer strictly true, though the arrival-time curves are still hyperbola-like In the case of transverse isotropy, however, classical interpretation of these curves fails Interval velocities calculated from t 2 - x 2 -curves do not always approximate vertical velocities and therefore cannot be used to calculate depths of reflectors
To study the relationship between velocities calculated from t 2 - x 2 -curves and the true velocities of a transversely isotropic layer, we approximate t 2 - x 2 -curves over a vertically inhomogeneous transversely isotropic medium by a three-term Taylor series and calculate expressions for these terms as a function of the elastic parameters It is shown that both inhomogeneity and transverse isotropy affect slope and curvature of t 2 - x 2 -curves For P-waves the effect of transverse isotropy is that the t 2 - x 2 -curves are convex upwards; for SV-waves the curves are convex downwards For SH-waves transverse isotropy has no effect on curvature
113 citations
TL;DR: In this article, the elastic constants of the long-wave equivalent transversely isotropic medium can be calculated in several ways, all of which are based on the low-frequency limit: one either determines from the outset the macroscopic strain response of a representative block of the periodically layered medium to (in average homogeneous) stresses, or one determines the dispersion equation for such media and evaluates this equation for low frequencies.
Abstract: Elastic waves propagating in a periodically layered medium exhibit transverse isotropy, provided the wavelength is long compared to the spatial period of the layer sequence. The elastic constants of the long‐wave equivalent transversely isotropic medium can be calculated in several ways, all of which are based on the low‐frequency limit: one either determines from the outset the macroscopic (in average homogeneous) strain response of a representative block of the periodically layered medium to (in average homogeneous) stresses, or one determines the dispersion equation for such media and evaluates this equation for low frequencies. Both approaches yield the same replacement medium. The replacement of a layered medium by a homogeneous transversely isotropic medium is justified if all wavelengths are sufficiently long. High‐resolution techniques, the increasing use of shear waves, and attention to stratigraphic detail require a quantitative evaluation of what is sufficiently long, as well as a study of what...
102 citations
TL;DR: In this paper, the authors presented a method of analysis of seismic records which is very sensitive to azimuthal anisotropy and is applicable at almost any depth range.
Abstract: Summary. Seismic anisotropy has been previously studied at depths usually not exceeding 100 or 150 km. In this paper we present a method of analysis of seismic records which is very sensitive to azimuthal anisotropy and is applicable at almost any depth range. The idea of the method is to detect and analyse the SH-component of the waves, converted from P to S in the mantle. The procedure of record processing includes frequency filtering, axis rotation, transformation of the record to a standard form, stacking the standardized SH-component records of many seismic events, and the harmonic analysis of amplitude as a function of the direction of wave propagation. When applied to the long-period records of NORSAR the procedure detected a converted wave with the properties implying the possibility of its propagation in a transversely isotropic medium with a horizontal axis of symmetry. Our preferred model postulates anisotropy of ∼ 1 per cent in a layer 50 km thick at the base of the upper mantle.
53 citations
TL;DR: In this article, a wave-scattering method was used to derive dispersion relationships for waves propagating perpendicular to continuous fibers that are oriented unidirectionally, and these relationships were compared with others derived by energy methods to obtain upper and lower bounds of the effective static moduli.
50 citations
TL;DR: In this article, a microscopic region of a unidirectional composite is modelled by a finite element micromechanical analysis using a generalized plane strain formulation, but including longitudinal shear loading.
49 citations
48 citations
TL;DR: In this article, the authors made a partial test by making laboratory measurements of compressional-wave velocities parallel and perpendicular to layering in fabricated samples consisting of glass and epoxy.
Abstract: In theory, stratified media in which the layers are elastically homogeneous and isotropic approximate transversely isotropic media with an axis of symmetry perpendicular to layering when the seismic wavelength is sufficiently longer than the layer spacing. The phenomenon has apparently been observed in field measurements, and acoustic anisotropy in deep‐sea sediments, measured in the laboratory, has been attributed to fine‐scale bedding laminations. However, to the best of our knowledge, no rigorous test of the theory has been made. We have made a partial test by making laboratory measurements of compressional‐wave velocities parallel and perpendicular to layering in fabricated samples consisting of glass and epoxy. We found no statistically significant difference between observation and theory in this limited test. Further, having used several frequencies, we found that the velocities progressively change from the long‐wave values toward those predicted by the time‐average relation, as expected. Finally,...
45 citations
TL;DR: In this paper, it was shown that the data can be satisfied equally well by an isotropic model consisting of several thin layers, and that the incompatibility of Love and Rayleigh-wave data might be produced by intrinsic anisotropy.
Abstract: Summary. Various factors can make it difficult to explain observations of Love- and Rayleigh-wave dispersion with the same relatively simple isotropic model. These factors include systematic errors which might occur in determinations of observed group and phase velocities, lateral variations in structure along the path of travel, and the attempt to explain observations with a model comprised of only a small number of thick layers. The last of these factors is illustrated by an inversion of dispersion data in the central United States where shear-wave anisotropy had previously been invoked as one way to explain incompatible Love- and Rayleigh-wave velocities. It is shown that the data can be satisfied equally well by an isotropic model consisting of several thin layers.
In cases where the incompatibility of Love- and Rayleigh-wave data might be produced by intrinsic anisotropy, it is necessary to invert those data using an anisotropic theory rather than by separate isotropic inversions of Love and Rayleigh waves. Inversions of fundamental-mode data for a region of the Pacific, assuming anisotropic media in which the layers are transversely isotropic with a vertical axis of symmetry, lead to models which are highly non-unique. Even if the inversions solve only for shear velocities in the litho-sphere and asthenosphere it is not possible, without supplementary information, to ascertain the depth interval over which anisotropy occurs or to determine the thickness of the lithosphere or asthenosphere with much precision.
34 citations
TL;DR: In this article, the authors discuss how to account for rock anisotropy when measuring in situ stresses by overcoring with instrumented devices such as the CSIRO hollow inclusion cell or U.S. Bureau of Mines (USBM) gauge.
TL;DR: The absence of significant effects of anisotropy in reflection seismics can be explained by three facts: most observations employ rays that make but moderate angles with the axis of symmetry; until recently only compressional waves were used, and, provided the constituent lamellae have S/P-velocity ratios constrained to a narrow range, the wavefront of compressional wave deviates significantly from a sphere only for larger angles as mentioned in this paper.
Abstract: Summary. Lamellations – layering on a scale small compared to the wavelengths – result in anisotropy (transverse isotropy) of elastic wave propagation. Since many sedimentary sequences can be regarded as lamellated, one might expect that exploration seismics would be beset with problems resulting from straightforward processing under the assumption of isotropy. Such problems have rarely been encountered.
The absence of significant effects of anisotropy in reflection seismics can be explained by three facts: in exploration seismics, most observations employ rays that make but moderate angles with the axis of symmetry; until recently only compressional waves were used, and, provided the constituent lamellae have S/P-velocity ratios constrained to a narrow range, the wave-front of compressional waves deviates significantly from a sphere only for larger angles.
Anisotropy effects can, in principle, be inverted to give information concerning the details of lamellation. In order to observe such effects, one has to use shear-wave observations. Any combination of observations that leads, under the assumption of isotropy, to inconsistencies can be used for the determination of anisotropy parameters. The simplest approach is to compare the depths determined with arrival times and stacking velocities from observations with polarization. For data of very high quality, the deviation of the reflection even from a hyperbola (or, more simply, the deviation of the corresponding t2/x2-plot from a straight line) can be exploited. To this end one has to use a technique that is akin to the determination of interval velocities from stacking velocities.
TL;DR: In this paper, an expression for the displacement field due to a point force in an unbounded, anisotropic elastic medium is derived using a radon transform method, which is thus a Green's function for the whole space.
Abstract: An expression for the displacement field due to a point force in an unbounded, anisotropic elastic medium is derived using a radon-transform method. The field is thus a Green's function for the anisotropic elastic whole space. A relatively simple algebraic expression for the asymptotic field is then found by means of the principle of stationary phase together with certain aspects of the differential geometry of the slowness surface. The general results are applied to the transversely isotropic medium; numerical results are presented for the hexagonal crystals of cobalt and apatite. The results of this work should be of value in the disciplines of crystal physics and seismology.
TL;DR: In this paper, the elastic constants of a particle-reinforced-composite wrought plate produced by powder-metallurgy methods were investigated by measuring ultrasonic-wave velocities using a pulse-echo method.
TL;DR: In this paper, the authors inverted multimode Love and Rayleigh wave data related to the Pacific Ocean and to North America using two sets of parameters: 1) the isotropic P and S velocities plus the density and 2), five elastic parameters describing a transversely isotropical medium, plus density.
Abstract: Multimode Love and Rayleigh wave data related to the Pacific Ocean and to North America are inverted using two sets of parameters: 1) the isotropic P and S velocities plus the density and 2), five elastic parameters describing a transversely isotropic medium, plus the density. It is found that the assumption of isotropy leads to very unrealistic upper-mantle models while much more realistic models are obtained under the assumption of transverse isotropy. The lateral variations in density between the Pacific Ocean and North America are then less than 1.5% and, in both regions, a large density gradient is well resolved at a depth near 360 km.
TL;DR: In this article, the axial translation of a rigid elliptical disc inclusion which is embedded in bonded contact at the interface of a transversely isotropic bi-material elastic medium is examined.
Abstract: The present paper examines the problem related to the axial translation of a rigid elliptical disc inclusion which is embedded in bonded contact at the interface of a transversely isotropic bi-material elastic medium. The result for the stiffness of the elliptical disc inclusion is presented in the form of a set of bounds. These bounds are developed, in exact closed form, by imposing displacement and traction constraints at the bi-material interface.
TL;DR: In this article, the effects of various parameters on the natural frequencies of axisymmetric clamped plates and simply supported plates subjected to initial stresses are investigated, where the initial stress is a combination of a pure bending stress plus an extensional stress in the plane of the plate.
TL;DR: In this article, a cylindrical crack in a transversely isotropic elastic body on the macroscopic level is considered, which models a crack-like defect on an interface between fiber and matrix in the fiber reinforced composite materials.
Abstract: This paper considers a cylindrical crack in a transversely isotropic elastic body on the macroscopic level. which models a crack-like defect on an interface between fiber and matrix in the fiber reinforced composite materials. Numerical calculations are carried out for E.glass/epoxy and graphite/epoxy composites when they are subjected to a constant internal pressure or shear traction linearly distributing in the axial direction on the crack surfaces. Stress intensity factors are evaluated for different values of the crack-length and the elastic constants.
TL;DR: A strain energy function is proposed to describe the nonlinear mechanical behavior of arteries and results in an accurate stress-strain relation that can be used readily for the study of wave propagations in the blood vessels.
Abstract: Based on the theory of Green and Adkins [9], a strain energy function is proposed to describe the nonlinear mechanical behavior of arteries. The arterial tissue is assumed to be a nonlinear elastic, incompressible material with local triclinicity and transverse isotropy. Although the arterial tissue shows viscous phenomena, experimental results have indicated that viscosity is only a second-order effect as compared to the nonlinear elasticity of the tissue. The advantage of the formulation presented herein is that it is relatively simple and results in an accurate stress-strain relation that can be used readily for the study of wave propagations in the blood vessels. For nonlinear finite strain elasticity of the order two, ten elastic constants are needed to describe the material nonlinearity of the arterial tissue. Based on the orthogonal, transverse isotropies and the incompressibility conditions, ten constraint equations may be established and the elastic constants can be uniquely determined by correlating with the experimental results. An example of calculating these elastic constants is made by using the experimental data of Patel, et al. [14-17] for the intercoastal arteries in living dogs. The predicted mechanical behavior of canine arteries is quite satisfactory as compared to the experimental data except when the longitudinal and the circumferential stretches exceed 1.60. However, such a strain magnitude may not be expected in in-vivo arteries because of the constraints of peripheral connecting tissues. Otherwise, the strain energy function including higher order strain terms should be used.(ABSTRACT TRUNCATED AT 250 WORDS)
TL;DR: In this paper, the acoustoelastic response of a transversely isotropic body loaded in plane stress is computed for arbitrarily severe anisotropy, and relations for the propagation speeds of longitudinal and shear waves propagating normal to the plane of stress are given as functions of the stress.
Abstract: The acoustoelastic response of a transversely isotropic body loaded in plane stress is computed for arbitrarily severe anisotropy. Relations for the propagation speeds of longitudinal and shear waves propagating normal to the plane of stress are given as functions of the stress. The shear wave results are combined to obtain expressions for the polarization direction, which in general differs from the principal directions of stress or strain, and the magnitude of the ultrasonic birefringence as given by the difference in transit time per unit length of the two shear waves. Similar expressions derived previously for slightly anisotropic bodies are recalled and are compared to the present results. It is shown that the earlier results are special cases of the more general model discussed here and a numerical investigation of the acoustoelastic response of several materials indicates that the assumptions leading to these more restricted models may not be valid, even in materials which at first glance appear to...
TL;DR: In this article, the authors investigated the stress distribution produced by the identation of a penny-shaped crack by an oblate smooth spheroidal rigid inclusion in a transversely isotropic medium.
Abstract: The stress distribution produced by the identation of a penny-shaped crack by an oblate smooth spheroidal rigid inclusion in a transversely isotropic medium is investigated using the method of Hankel transforms. This three-part mixed boundary value problem is solved using the techniques of triple integral equations. The normal contact stress between the crack surface and the indenter is written as the product of the associated half-space contact stress and a nondimensional crack-effect correction function. An exact expression for the stress-intensity is obtained as the product of a dimensional quantity and a nondimensional function. The curves for these nondimensional functions are presented and used to determine the values of the normalized stress-intensity factor and the normalized maximum contact stress. The stress-intensity factor is shown to be dependent on the material constants and increasing with increasing indentation. The stress-intensity factor also increases if the radius of curvature of the indenter surface increases.
TL;DR: In this article, the authors examined the problem of asymmetric rotation of a rigid elliptical disc inclusion embedded in bonded contact with a transversely isotropic elastic solid of infinite extent.
TL;DR: In this paper, an analytical approach for predicting the effective elastic moduli and breaking strength of FRP-FW pipes was presented, where the elastic modulus of each lamina composed of aligned, continuous transversely isotropic fibres was derived based on the equivalent inclusion method of Eshelby.
TL;DR: In this article, a general solution for the theory of elasticity in terms of p -analytic functions for a transversely isotropic body in a non-axisymmetric stress state was given.
TL;DR: In this article, the authors modeled a craze region in glassy polymers as an elastic transversely isotropic homogeneous inclusion of thin elliptical shape with different elastic properties from the bulk polymer.
TL;DR: In this article, the authors adapted the Dugdale hypothesis to the problem of a penny-shaped crack contained in a thick transversely isotropic plate and solved the problem using the techniques of Hankel transform.
TL;DR: In this paper, a dependence is established for the character of the ray velocity of longitudinal low-frequency waves on the ratio of P- and S-wave velocities in thin layers.
Abstract: Summary. Three problems of seismic anisotropy in thinly layered media (TPM) are discussed: (1) A dependence is established for the character of the ray velocity of longitudinal low-frequency waves on the ratio of P- and S-wave velocities in thin layers. (2) Conditions are specified for cusps on SV-wave surfaces. Nomograms are suggested for quick estimation of these conditions. (3) A comparison is made between TPM anisotropy and other types of transversely isotropic media.
01 Jan 1984
TL;DR: The constitutive theory for strongly anisotropic materials is treated elsewhere (Chapter I) in this book as discussed by the authors, and the response of these composites is usually highly anisoust - it is usually very strong and complex, so that isotropic theory would not provide even a rough approximation to their behaviour under most types of loading conditions.
Abstract: The constitutive theory for strongly anisotropic materials is treated elsewhere (Chapter I) in this book. These materials are characterised by their physical property of having ‘strong’ directions, for each of which the extensional modulus is much greater than the shear moduli associated with that direction. This property is particularly true of many of the man-made fibre-reinforced composites which are now coming into widespread use; in these a relatively weak, isotropic matrix with certain desirable properties (such as lightness or ductility) is strengthened throughout in one or more particular directions by introducing strong reinforcing fibres in those directions. Macroscopically, these composite materials will exhibit mechanical properties which are transversely isotropic if reinforced by one family of fibres, for example, or orthotropic if reinforced by two families (refer Chapter I). This will be so even if all the constituents are isotropic themselves. Moreover, the response of these composites is not just anisotropic - it is usually highly anisotropic, so that isotropic theory would not provide even a rough approximation to their behaviour under most types of loading conditions.
TL;DR: In this article, the problem of determining three-dimensional Poisson's ratio effect corrections for two-dimensional plane stress theory is solved approximately by assuming equilibrium stress systems with parabolic variations in thickness direction of the primary face-parallel stresses.
Abstract: The problem of determining three-dimensional Poisson's ratio effect corrections for two-dimensional plane stress theory is solved approximately by assuming equilibrium stress systems with parabolic variations in thickness direction of the primary face-parallel stresses. An application of a variational theorem for stresses and displacements is shown to lead to a tenth-order system of two-dimensional differential equations for stress measures and certain weighted averages of displacements components. It is further shown that the solution of the tenth-order system can be expressed in terms of a bi-harmonic function, in conjunction with the solutions of one second-order and one fourth-order differential equation, involving Laplace operators only.
TL;DR: In this paper, the phase shift of a transverse wave from the surface of a piezoelectric crystal of class 6 is considered for the case when the reflection plane coincides with or is close to the plane of transverse isotropy.