Showing papers on "Transverse isotropy published in 1979"

Solutions for effective shear properties in three phase sphere and cylinder models

Abstract: S olutions are presented for the effective shear modulus of two types of composite material models. The first type is that of a macroscopically isotropic composite medium containing spherical inclusions. The corresponding model employed is that involving three phases: the spherical inclusion, a spherical annulus of matrix material and an outer region of equivalent homogeneous material of unlimited extent. The corresponding two-dimensional, polar model is used to represent a transversely isotropic, fiber reinforced medium. In the latter case only the transverse effective shear modulus is obtained. The relative volumes of the inclusion phase to the matrix annulus phase in the three phase models are taken to be the given volume fractions of the inclusion phases in the composite materials at large. The results are found to differ from those of the well-known Kerner and Hermans formulae for the same models. The latter works are now understood to violate a continuity condition at the matrix to equivalent homogeneous medium interface. The present results are compared extensively with results from other related models. Conditions of linear elasticity are assumed.

Analysis of Properties of Fiber Composites With Anisotropic Constituents

Abstract: Expressions and bounds for the five effective elastic moduli of a unidirectional fiber composite, consisting of transversely isotropic fibers and matrix, are derived on the basis of analogies between isotropic and transversely isotropic elasticity equations. Application of results for determination of the five elastic moduli of graphite fibers is discussed. Effective thermal expansion coefficients are derived on the basis of a general theorem. Effective conductivities, dielectric constants, and magnetic permeabilities are derived by use of certain mathematical analogies.

Long-wave elastic anisotropy in transversely isotropic media

Abstract: Compressional waves in horizontally layered media exhibit very weak long‐wave anisotropy for short offset seismic data within the physically relevant range of parameters. Shear waves have much stronger anisotropic behavior. Our results generalize the analogous results of Krey and Helbig (1956) in several respects: (1) The inequality (c11-c44)(c33-c44)⩾(c13+c44)2 derived by Postma (1955) for periodic isotropic, two‐layered media is shown to be valid for any homogeneous, transversely isotropic medium; (2) a general perturbation scheme for analyzing the angular dependence of the phase velocity is formulated and readily yields Krey and Helbig’s results in limiting cases; and (3) the effects of relaxing the assumption of constant Poisson’s ratio σ are considered. The phase and group velocities for all three modes of elastic wave propagation are illustrated for typical layered media with (1) one‐quarter limestone and three‐quarters sandstone, (2) half‐limestone and half‐sandstone, and (3) three‐quarters limesto...

Elastic moduli of transversely isotropic graphite fibers and their composites

Abstract: This paper demonstrates that it is possible to calculate the complete set of elastic mechanical properties for graphite-epoxy fiber-reinforced materials at any fiber-volume fraction by modifying equations previously developed to include transversely isotropic graphite-fiber properties. Experimental verification of the modified equations is demonstrated by using these equations to curve fit elastic-property data obtained ultrasonically over a range of fiber-volume fractions. Material systems under consideration are T300/5208, AS-3501 and Modomor II/LY558 graphite epoxy. Using the modified equations it is possible to extrapolate for fiber properties. From Modomor II/LY558 ultrasonic data, it is shown that five out of seven extrapolated graphite-fiber properties are consistent with the assumption that graphite fibers are transversely isotropic. Elastic properties for T300/5208 and AS-3501 are ultrasonically evaluated by propagating stress waves through six individual specimens but at various angles from a block of unidirectional material. Particular attention is devoted to specimen dimensions. To demonstrate the need for accurately calculating or experimentally measuring all lamina elastic properties, a brief discussion is included on the effect that variations in lamina elastic properties have on calculating interlaminar stresses.

Seismic velocities in transversely isotropic media

Abstract: When a sedimentary earth section is layered on a scale much finer than the wavelength of seismic waves, the waves average the physical properties of the layers; a seismic wave acts as if it were traveling in a single, transversely isotropic solid. We compute the velocities with which P‐waves, SV‐waves, and SH‐waves travel in transversely isotropic solids formed from two‐component solids and find the corresponding moveout velocities from t2-x2 plots. The combinations studied are sandstone and shale, shale and limestone, water sand and gas sand, and gypsum and unconsolidated material, one set of typical physical properties being selected for each component of a combination. A reflector at 1524 m and a geophone spread of 0–3048 m are assumed. The moveout velocity for an SH‐wave is always the velocity for a wave traveling in the horizontal direction. The P‐wave moveout velocity found from surface seismic data can be anywhere from the vertical P‐wave velocity to values between those for vertical and horizontal...

An effective stress law for anisotropic elastic deformation

Abstract: An effective stress law is derived analytically to describe the effect of pore fluid pressure on the linearly elastic response of saturated porous rocks which exhibit anisotropy. For general anisotropy the difference between the effective stress and the applied stress is not hydrostatic. The effective stress law involves two constants for transversely isotropic response and three constants for orthotropic response; these constants can be expressed in terms of the moduli of the porous material and of the solid material. These expressions simplify considerably when the anisotropy is structural rather than intrinsic, i.e., in the case of an isotropic solid material with an anisotropic pore structure. In this case the effective stress law involves the solid or grain bulk modulus and two or three moduli of the porous material, for transverse isotropy and orthotropy, respectively. The law reduces, in the case of isotropic response, to that suggested by Geertsma (1957) and by Skempton (1961) and derived analytically by Nur and Byerlee (1971).

Reflection and transmission coefficients for seismic waves in ellipsoidally anisotropic media

Abstract: The deficiency of an isotropic model of the earth in the explanation of observed traveltime phenomena has led to the mathematical investigation of elastic wave propagation in anisotropic media. A type of anisotropy dealt with in the literature (Potsma, 1955; Cerveny and Psencik, 1972; and Vlaar, 1968) is uniaxial anisotropy or transverse isotropy. A special case of transverse isotropy which assumes the wavefronts to be ellipsoids of revolution has been used by Cholet and Richard (1954) and Richards (1960) in accounting for the observed traveltimes at Berraine in the Sahara and in the foothills of Western Canada. The kinematics of this problem have been treated in a number of papers, the most notable being Gassmann (1964). However, to appreciate fully the effect of anisotropy, the dynamics of the problem must be explored. Assuming a model of the earth consisting of plane transversely isotropic layers with the axes of anisotropy perpendicular to the interfaces, a prime requisite for obtaining amplitude dist...

Analysis of properties of fiber composites with anisotropic constituents

Abstract: Expressions and bounds for the five effective elastic moduli of a unidirectional fiber composite, consisting of transversely isotropic fibers and matrix, are derived on the basis of analogies between isotropic and transversely isotropic elasticity equations. Application of results for determination of the five elastic moduli of graphite fibers is discussed. Effective thermal expansion coefficients are derived on the basis of a general theorem. Effective conductivities, dielectric constants, and magnetic permeabilities are derived by use of certain mathematical analogies.

SH waves in layered transversely isotropic media—An asymptotic expansion approach

Abstract: An asymptotic expansion of the displacement vector in terms of inverse powers of frequency is employed to investigate the dynamic (amplitude) properties of SH waves propagating in plane-layered transversely isotropic media. Both reflected and head waves are considered in terms of the asymptotic expansion, and their ranges of validity and accuracy are discussed. Although the solution of the most general case of wave propagation in an anisotropic medium has been presented in the literature (C˘ervený, 1972), it is instructive to consider this simple case in which many quantities inherent to wave propagation in anisotropic media can be more readily understood and can be solved analytically rather than reverting to numerical methods.

Nearly extensional flows

Abstract: After defining a nearly extensional flow, the stresses which arise in an incompressible simple fluid undergoing such a motion are determined. Since the linear functionals which arise in this perturbation are similar to those in the infinitesimal viscoelasticity theory of a transversely isotropic solid with rotational and reflectional symmetries, the non-zero linear functionals and interralations between them are determined quite easily. It is then shown that self-consistency demands that certain relations exist between the extensional modulus and these linear functionals. As an application, the speed of propagation of an acceleration wave in a fluid undergoing an extensional flow is considered. Finally, the nearly extensional flow theory is cast in terms of small displacements superposed on the extensional flow. In this form, it may be useful in the study of melt spinning.

Second pressure derivatives of the elastic moduli of copper

Abstract: The second pressure derivatives of the elastic moduli for polycrystalline copper have been determined. While the value of the second pressure derivative of the longitudinal modulus is zero or very small, that for the shear modulus is ∼6.3×10−10 Pa−1, hence the second pressure derivative of the bulk modulus turns out to be positive, with a value of 8.4×10−10 Pa−1.

Viscoplasticity of Transversely Isotropic Clays

Abstract: A viscoplastic constitutive relation of the endochronic type (i.e., the inelastic strain increments are characterized by an intrinsic time) is formulated to describe the behavior of transversely isotropic clays produced by one-dimensional consolidation. The formulation contains eight material parameters in addition to those needed for isotropic clays. The hardening and softening functions and the densification-dilatancy function are assumed to be given by the same expressions previously found for isotropic clays, but the invariants involved in these expressions are replaced by the proper transversely isotropic invariants. The pore pressure is determined from the volume change and the compressibility of the water, and the constitutive relation is written in terms of the effective stresses. The elastic moduli are assumed to be functions of hydrostatic stress and inelastic dilatancy, and they are correlated with the consolidation stress. Experimental curves of axial strain for various anisotropically consolidated clays have been fit by a time-independent version of the theory, and a satisfatory agreement has been achieved.

Theory of the Instability and Failure of Viscoelastic Materials in Tension

Abstract: In a tensile test of a bar or fiber formed from a transversely isotropic viscoelastic material, the initial motion in regions away from the clamps is a homogeneous uniaxial extension. If the applied tensile load is “tame” in the sense that it is given by a piecewise smooth function of time, then during the early stages of loading, the homogeneous extension of the specimen is also tame. It is, however, often the case that at a time tc, previous to the instant of fracture, the motion departs from a tame homogeneous extension. The critical time tc is the “failure time” of the specimen, i.e., the time at which the motion first changes character; tc precedes, often by a constant factor, the time at which neck-down is easily visible. The problem of calculating tc for a given loading program is here treated within the framework of the general theory of nonlinear simple materials with fading memory. For such materials the instantaneous tensile modulus, i.e., the derivative of the immediate change in tensile stress with respect to a sudden change in tensile strain, depends upon the previous history of the strain. Reasons are presented here for identifying tc with the earliest time t0 at which the instantaneous tensile modulus becomes zero. It is shown that at a time at which the instantaneous modulus vanishes one cannot arbitrarily assign the rate of change of tensile stress and have the motion remain in the class of tame homogeneous extensions.

Application of the method of caustics to the determination of the ratio of Poisson's ratio to the modulus of elasticity

Abstract: An optical experimental method for the determination of the ratio of Poisson's ratio nu to the modulus of elasticity E in isotropic elastic materials is proposed. This method is based on the method of reflected caustics and is valid both for transparent and opaque materials and in the first case both for optically isotropic and optically anisotropic materials.

A transversely isotropic composite with a penny-shaped crack

Abstract: We consider the problem of determining the stress intensity factor and the crack energy in a transversely isotropic composite medium, containing a penny-shaped crack. We assume that the crack surface is perpendicular to the bond face and the crack is opened by constant internal pressure. By use of integral transform, we reduce the problem to solving a Fredholm integral equation of the second kind. Numerical results are given for the combination of some practical materials such as magnesium and cadmium. The effect of transverse isotropy upon the stress intensity factor, the crack energy and the deformation on the crack surface is discussed.

SH waves in layered transversely isotropic media—a wave approach

Abstract: There are many reports in the literature of anomalies in traveltime data when an isotropic homogeneous Earth model is used to interpret field data. In several cases, the introduction of a layered t...

Traction problems on an elastic jointed rock half-space

Abstract: Traction Problems on an Elastic Jointed Rock Half-Space A rock mass with a major set of parallel, planar joints is considered for the case when the joint spacing is small compared with the length scales of interest, and large compared with the relative displacements across the joints. A continuum theory is adopted and under the additional assumption that the joints have finite elastic stiffnesses associated with their tangential and normal response prior to slip, the rock may be shown to be transversely isotropic about the direction of the joint normal. The class of problems for which analytic solutions are available for a transversely isotropic medium is very limited. However, for the case of low joint stiffnesses an approximate potential theory, analogous to that of inextensible fibre reinforced elastic materials, is available and may be applied to a much more general set of domains. Both the approximate theory and the exact transversely isotropic theory for low joint stiffnesses are applied to plane strain traction problems on a half-space in which the joints are inclined at varying angles to the boundary. Comparisons of the two theories are made for various values of the rock parameters and the joint angle.

Specific directions of plane elastic waves in thin LiTaO3 and LiNbO3 crystal plates

Abstract: When plane clastic waves propagate in specific directions in an anisotropic medium, they can satisfy the conditions of pure longitudinal and transverse plane elastic waves. The purpose of this paper is to find all the specific directions in which pure elastic waves propagate in thin LiTaO3 or LiTaO3 crystal plates. The result, is that all the specific directions for pure plane elastic waves in thin LiTaO3 and LiNbO3 crystal plates have been thoroughly analysed.