Showing papers on "Transverse isotropy published in 1981"

Ultrasonic Measurement of Orthotropic Elastic Constants of Bovine Femoral Bone

Abstract: Using a pulse transmission ultrasound method, we have determined the elastic properties of bone samples taken from along the length and around the periphery of a bovine femur. Twenty specimens, in the form of 5-mm cubes, were tested. All nine of the orthotropic elastic constants were determined for each specimen. Analysis of our data indicate that there are statistically significant variations from the usual assumption of transverse isotropy.

Cylindrical waves in transversely isotropic media

Abstract: Because of fine layering, many sedimentary rocks can be characterized as transversely isotropic, possessing an axis of symmetry perpendicular to the layering. The many kinds of measurements made in fluid‐filled holes drilled parallel to this axis prompted this study of axisymmetric cylindrical waves. The literature shows how such waves in an isotropic solid can be expressed in terms of a compressional potential or a shear potential. This paper shows that in a transversely isotropic medium, particular combinations of these potentials are needed simultaneously, yielding either a quasi‐compressional wave or a quasi‐shear wave. The expressions are used to compute the transient response of an acoustic logging tool. The speeds of refracted compressional and shear waves agree with the speeds of plane waves traveling along the axis of symmetry. Dispersive waves in the fluid annulus yield further information about the elastic constants of the solid. The potentials also portray the radiation of elastic waves from a point force or from a transducer on the surface.

Pn velocity anisotropy in southern California

Abstract: We analyze P_n propagation as a function of azimuth across a 28-station, 150-km aperture subarray of the SCARLET network centered near the central Transverse Ranges, California. We selected signals from 81 earthquakes and explosions with epicentral distances ranging from 150 to 400 km, covering all azimuths except a 40° gap from the southwest and a lesser gap from the northeast direction. For each source the apparent velocity of P_n was determined using a one-norm measure of misfit. The apparent P_n velocity does not show any systematic variation with epicentral distance but exhibits a strong azimuthal dependence. Our preferred interpretation calls for a slightly dipping (2° to N40W) planar moho, with 3 to 4 per cent anisotropy of subcrustal material. Transverse isotropy with a nearly horizontal symmetry axis is sufficient to explain the data; the direction of sagittal symmetry is N50W. The isotropic velocity of P_n is 7.8 km/sec. In contrast, a higher (8.1 km/sec) P_n velocity is found in the Mojave block, with no indication of anisotropy. These observations are consistent with a subcrustal model of the Pacific-North America plate boundary where ductile flow is characterized by simple shear in a vertical plane with strike parallel to the direction of relative plate motion.

Systematic classification of layer‐induced transverse isotropy*

Abstract: Waves propagating through a sequence of layers that are thin compared with the wavelength show effects of anisotropy: velocity and displacement direction depend on the angle between the plane of layering and the wave normal, and shear waves split up into two distinct types of different velocity. The layered medium can thus be replaced by a transversely isotrophic medium the parameters of which depend on the parameters of the individual constituent layers. A survey of the anisotropy effects possible in such a medium is generally done by varying the layer parameters in order to obtain different replacement media. This approach guarantees that the replacement medium is realistic, but it does not guarantee adequate sampling of the set of replacement media. To this end one has to begin by selecting the replacement media and then check whether the chosen media possess stable (and eventually realistic) representations by layer sequences. In general, there is an infinite number of layer representations for any transversely isotropic medium that can at all be represented. However, if one restricts the solutions to those requiring the minimal number of layers and the minimum number of different layer parameters, the set of solutions has only one free parameter (i.e., it is a one-dimensional manifold), and an important subset even has a unique solution. A simple algorithm exists for the determination of these “simplest representations”. Aside from sampling the set of representable transversely isotropic media for survey purposes, the method can be applied to the problem of determining the cause of observed anisotropy effects (or lateral changes in such effects). If this method can be applied to real data, it would for instance allow to determine changes in relative thickness or lithology on a scale smaller than the limit of resolution of the seismic method.

Surface motion over a sedimentary valley for incident plane P and SV waves

Abstract: Finite difference techniques are applied to study the response of a sedimentary basin in an isotropic elastic half-space to vertically incident compressional and shear sources. The sedimentary rock of the basin is modeled as a transversely isotropic material. Improved finite difference approximations to the boundary conditions are developed. Of primary interest is the formation of Rayleigh waves over the flank of the basin. Displacement time series at points on the free surface and Rayleigh orbits are plotted. Three-dimensional spectral amplitude graphs of the entire time series up to a period of temporary quiescence are presented. The Rayleigh wave is the dominant scattered phase for the basin geometry considered, having 20 to 30 per cent of the source displacement amplitude at the free surface in a basin with 26° flanks, 0.7-km depth, 6-km width, over a half-space whose acoustic impedance contrast with the basin was two. The effect of varying the degree of compressional anisotropy in the basin on the amplitudes of refracted and scattered phases of engineering interest was minimal.

An endochronic theory for transversely isotropic fibrous composites

Abstract: A rational methodology of modelling both nonlinear and elastic dissipative response of transversely isotropic fibrous composites is developed and illustrated with the aid of the observed response of graphite-polyimide off-axis coupons. The methodology is based on the internal variable formalism employed within the text of classical irreversible thermodynamics and entails extension of Valanis' endochronic theory to transversely isotropic media. Applicability of the theory to prediction of various response characteristics of fibrous composites is illustrated by accurately modelling such often observed phenomena as: stiffening reversible behavior along fiber direction; dissipative response in shear and transverse tension characterized by power-laws with different hardening exponents; permanent strain accumulation; nonlinear unloading and reloading; and stress-interaction effects.

Epicenter and Epicentral-Axis Motion of a Transversely Isotropic Elastic Half-Space

Abstract: A linear, transversely isotropic, elastic half-space is excited by a buried point source. The stress-free surface of the half-space is normal to the axis of material symmetry. By means of integral transforms, an explicit solution is obtained at the epicenter for hexagonal–crystals whose Cagniard inversion path requires contour integration. Solution singularities are discussed and graphical results are presented for the normal component of displacement for the four crystals apatite, beryllium, cadmium and zinc. From this solution the reciprocal theorem is then used to obtain the epicentral-axis displacement caused by a normal surface point load.

Indentation of Anisotropic Halfspace by Yielding Circular Foundation

Abstract: This paper presents an analytical study of the interaction between a transversely isotropic elastic half space and a structurally failing circular foundation plate under axisymmetric loading.

A Mixed Boundary Value Problem of a Transversely-Isotropic Half-Space Under Torsion by a Flat Annular Rigid Stamp

Abstract: The s tudy of the torsional oscillation of an elastic solid is important in several fields like soil mechanics, Reissner and Sagoei [1], and mechanical transmission line theory, Macoy [2]. The oscillation of a half-space, excited by a rigidly attached circular disc, were first considered by Sagoci [3] and an approximate treatment of both the oscillating half-space and stratum was subsequently given by Bycroft [4]. Recently, Collins [5] has formulated exactly both the problem as Fredhelm integral equations of the second kind, utilizing methods developed by him [6] and that the of torsion of elastic half space with penny-shape crack and penny-shape inclusion by Dhawan [7], [8]. On the other hand, the mixed boundary value problem for a flat annular rigid stamp was, at first, treated by Gubenko and Mossakovskii [9]. After that, many papers for triple integral equations have been published with relation to the theory of elasticity, fluid dynamics, magnetics, etc. [10]--[16]. In the present paper, we analyzes the three-par tmixed boundary value problem of a transversely-isotropic half-space under torsion by a flat annular rigid stamp on the basis of three-dimensional theory of elasticity. The tangential stress on the stam p is continuous at all points except the inner and the outer edges of the stamp.

Long-wave elastic anisotropy in transversely isotropic media by J. G. Berryman, and Seismic velocities in transversely isotropic media by F. K. Levin; discussion and replies

Abstract: Berryman and Levin made an assumption about constancy or limited variations of Poisson’s ratio in the thin layers, in their analyses of elastic anisotropy in thin‐layered media. Berryman states (p. 913): “Rare cases can occur with large variations in Poisson’s ratio.” However, on p. 911 Berryman does point out (with reference to Benzing) that range of variations of the parameter γ = VS/VP from 0.45 to 0.65 is typical of rocks. That corresponds to a range of variations of Poisson’s ratio of 0.373 to 0.134 (i.e., almost three times as much).

An asymmetric mixed boundary value problem of a transversely isotropic half-space subjected to a moment by an annular rigid punch

Abstract: The solution, within the classical theory of elasticity, has been presented for the problems of an annular punch on a transversely — isotropic half-space under bending. Numerical results are obtained for Cadmium Crystal. The results indicate that (a) The contact stress becomes infinite asr→a+0 orr→b−0 and the results agree closely with those for the circular punch ina

Stress concentration around cracks in long bones

Abstract: By modeling a long bone (e.g. femur) as a cylindrical shell of transversely isotropic material, the deformation and the concentration of stresses around penny-shaped cracks are studied in the paper. Expressions for the stress intensity factor, the shape of the crack, and the crack energy are obtained. A quantitative analysis is made in respect of the normal stress and displacement in the neighbourhood of the crack. The magnitudes of these quantities have also been calculated for another non-isotropic material (magnesium) and an isotropic material (aluminium) and compared with those for bone material.

Porous materials: The effect of a fluid on the elastic moduli and on the velocity of elastic waves

Abstract: The dependence of the elastic moduli of a material in an actual state on the moduli of a dry and fluid-saturated material is derived with the aid of the principle of superposition of reference states. Wave equations are derived and the dependence of the phase velocity on the values in the reference states and on the contents of the liquid and gas in the pores is expressed.

Flow of a liquid with an anisotropic viscosity tensor: a skew initial orientation

Abstract: In this paper some idealized anisotropic fluids— the liquids D and F presented in [1]—are analysed in simple shearing flows and in channel flows The liquids, transversely isotropic initially, are here taken to have a uniform initial orientation in some arbitrary fixed direction in space The discussion therefore generalizes somewhat the analysis in [1] of the same flow situations when special initial orientations were taken (such as parallel to the streamlines) It is generally found that the flow is unsteady, that stresses are time-dependent, and that transverse isotropy does not persist This can be true even if the initial orientation differs only very slightly from a special orientation that would lead to steady, Newtonian behaviour It is found also that, in simple shearing flow, time-dependent lateral shear stresses can be induced; these stresses must be applied through the plates bounding the flow (in addition to the primary driving shear stress and appropriate normal stresses) if rectilinear flow is to be possible Such lateral shear stresses can also arise in pressure-driven channel flow This is showm to imply that, if longitudinal flow is assumed, the associated stress pattern is incompatible with the equations of motion: except with special initial orientations, rectilinear flow of these liquids in a straight-walled channel is not possible In the special cases where retilinear flow is possible, the velocity profile is not necessarily symmetric about the channel axis

Flow of a liquid with an anisotropic viscosity tensor: elongational flow

Abstract: Some idealized anisotropic viscous liquids, previously analysed in simple-shearing-type flows, are here studied in elongational flows. The liquids, referred to as liquids D and F , are transversely isotropic initially; and in this paper we take the initial orientation to be spatially uniform, with the privileged direction in the material lying in the (1,2)-plane at some angle ψ to the (Cartesian) x 1 -axis along which the material is later extended. Special cases with ψ = 0 and ψ = π/2 are treated first, and then some numerical solutions for other values of ψ are given. In general, the velocity υ i is found to be time-dependent (even under constant stress) and, except when ψ = 0, the transverse isotropy is lost after t = 0. Also a shear stress p 12 is generally induced, i.e. an unexpected shear stress must be applied to the bulk material if elongational flow is to be maintained. Under a constant applied stretching stress p 11 = P , liquid D (with ψ ≠ 0) will extend not only longitudinally, but also in a lateral direction, so that the material tends to spread out into a sheet. In the exceptional case ψ = 0 (obtainable only as a singular limit when ψ → 0), the liquid will contract symmetrically in the transverse plane, with the velocity at each point tending to a non-zero constant, at large times. Liquid F , on the other hand, will in general extend longitudinally and contract laterally, with a velocity that eventually decreases as t −1 everywhere. The case ψ = π/2 is special in that υ 1 and υ 3 both tend to non-zero constants, while υ 2 tends to zero in such a way that the corresponding extension ratio tends to a anon-zero constant (so that a cylinder of this material, parallel to the x 1 -axis, is eventually stretched into a thin strip lying approximately in the (1,2)-plane).