Showing papers on "Transverse isotropy published in 1991"

Passive material properties of intact ventricular myocardium determined from a cylindrical model.

Abstract: The equatorial region of the canine left ventricle was modeled as a thick-walled cylinder consisting of an incompressible hyperelastic material with homogeneous exponential properties. The anisotropic properties of the passive myocardium were assumed to be locally transversely isotropic with respect to a fiber axis whose orientation varied linearly across the wall. Simultaneous inflation, extension, and torsion were applied to the cylinder to produce epicardial strains that were measured previously in the potassium-arrested dog heart. Residual stress in the unloaded state was included by considering the stress-free configuration to be a warped cylindrical arc. In the special case of isotropic material properties, torsion and residual stress both significantly reduced the high circumferential stress peaks predicted at the endocardium by previous models. However, a resultant axial force and moment were necessary to cause the observed epicardial deformations. Therefore, the anisotropic material parameters were found that minimized these resultants and allowed the prescribed displacements to occur subject to the known ventricular pressure loads. The global minimum solution of this parameter optimization problem indicated that the stiffness of passive myocardium (defined for a 20 percent equibiaxial extension) would be 2.4 to 6.6 times greater in the fiber direction than in the transverse plane for a broad range of assumed fiber angle distributions and residual stresses. This agrees with the results of biaxial tissue testing. The predicted transmural distributions of fiber stress were relatively flat with slight peaks in the subepicardium, and the fiber strain profiles agreed closely with experimentally observed sarcomere length distributions. The results indicate that torsion, residual stress and material anisotropy associated with the fiber architecture all can act to reduce endocardial stress gradients in the passive left ventricle.

The general problem of elastic wave propagation in multilayered anisotropic media

Abstract: Exact analytical treatment of the interaction of harmonic elastic waves with n-layered anisotropic plates is presented. Each layer of the plate can possess up to as low as monoclinic symmetry and thus allowing results for higher symmetry materials such as orthotropic, transversely isotropic, cubic, and isotropic to be obtained as special cases. The wave is allowed to propagate along an arbitrary angle from the normal to the plate as well as along any azimuthal angle. Solutions are obtained by using the transfer matrix method. According to this method formal solutions for each layer are derived and expressed in terms of wave amplitudes. By eliminating these amplitudes the stresses and displacements on one side of the layer are related to those of the other side. By satisfying appropriate continuity conditions at interlayer interfaces a global transfer matrix can be constructed which relates the displacements and stresses on one side of the plate to those on the other. Invoking appropriate boundary conditions on the plates outer boundaries a large variety of important problems can be solved. Of these mention is made of the propagation of free waves on the plate and the propagation of waves in a periodic media consisting of a periodic repetition of the plate. Confidence is the approach and results are confirmed by comparisons with whatever is available from specialized solutions. A variety of numerical illustrations are included.

Effects of point singularities on shear-wave propagation in sedimentary basins

Abstract: SUMMARY In most directions of propagation in anisotropic solids, seismic shear waves split in regular and predictable ways that, in principle, can be directly related to the degree of anisotropy and the anisotropic symmetry of the rockmass In all anisotropic solids, however, there are directions of propagation, known as shear-wave singularities, where the split shear-waves have the same phase-velocities For directions of propagation near the commonest type of singularity, the point singularity, the relationship between the phase and group-velocities may undergo rapid variations for small changes in direction This results in shear-waves along rays (propagating at the grozq-velocity) behaving anomalously, with irregular polarizations and amplitude changes as if they were propagating near cusps, although the degree of anisotropy may be too small to cause conventional cusps on the group-velocity wave surfaces The effects of propagation near such point singularities have been identified in sedimentary basins where they are features of the well-established phenomenon of azimuthal isotropy (transverse isotropy with a vertical axis of symmetry) caused by horizontal lithology, or by fine layering (PTL anisotropy), combining with the more recently recognized azimuthal anisotropy, caused by distributions of near-parallel near-vertical fluid-filled inclusions (EDA anisotropy) This paper demonstrates these irregular effects by calculating synthetic shear waves in directions near a point singularity in a material simulating a possible sedimentary basin Such anomalies may be important in exploration seismology as point singularities can occur along nearly vertical ray paths in sedimentary basins If not identified correctly, the effects of such point singularities could be mistakenly attributed to structural irregularities, and if correctly identified, the directions of such singularities can place tight constraints on possible combinations of PTL and EDA anisotropy in sedimentary basins

Onset of convection in an anisotropic porous medium with oblique principal axes

Abstract: Summary and conclusions The present work is the first study of Rayleigh-Be'nard convection in an anisotropic porous medium with oblique principal axes. The analysis is restricted to transversely isotropic media with isotropic thermal conductivity. Qualitatively new flow patterns occur at the onset of convection. If the transverse permeability is larger than the longitudinal permeability, the planes of motion are tilted, but the cell walls are vertical as usual. On the other hand, if the longitudinal permeability is the larger one, the flow occurs in vertical planes, but the cell walls are tilted. 'The preference for these different patterns is explained as a preference for flow directions with as small a tangential permeability as possible. This preference also gives rise to a tendency of concentration of the flow along the cell boundaries when the anisotropy increases, in the case when the cell walls are tilted. The tilt is primarily of mechanical origin, because a vertical forcing due

Long-wave anisotropy in stratified media; a numerical test

Abstract: When a seismic signal propagates in a stratified earth, there is anisotropy if the dominant wavelength is long enough compared to the layer thickness. In this situation, the layered medium can be replaced by an equivalent nondispersive transversely isotropic medium. Theoretical and experimental analyses of the required minimum ratio of seismic wavelength to layer spacing based on kinematic considerations yield different results, with a much higher value in the experimental test.The present work investigates the effects of layering by wave simulation and attempts to establish quantitatively the minimum ratio for which the long-wave approximation starts to be valid. We consider two-constituent periodically layered media and analyze the long-wave approximation for different material compositions and different material proportions in 1-D and 2-D media. The evaluation of the minimum ratio compares snapshots and synthetic seismograms visually and through a measure of coherence.Layering induces scattering with wave dispersion or anisotropy depending upon the wavelength-to-layer thickness ratio. The modeling confirms the dispersive characteristics of the wave field, the scattering effects in the form of coda waves at short wavelengths, and the smoothed transversely isotropic behavior at long wavelengths. 1-D numerical tests for different media indicate that the minimum ratio is highest for the midrange of compositions, i.e., equal amount of each material, and for stronger reflection coefficients between the constituents. For epoxy-glass, the value is around R = 8, while for sandstone-limestone, it is between R = 5 and R = 6. Recent wave-propagation experiments done in epoxy-glass also imply a highest minimum ratio for midrange of composition; however, the 1-D numerical tests confirm the long-wave approximation at shorter wavelengths than experimentally. The 2-D case shows that the more anisotropic the equivalent medium, the higher the minimum ratio, and that the approximation depends upon the propagation angle with longer wavelengths required in the direction of the layering.

Development of composite coaxial cylinder stress analysis model and its application to SiC monofilament systems

Abstract: An analytical model is presented allowing prediction of the principal stresses in a system composed of a set of coaxial cylinders, subject to temperature change or applied stress. The materials must exhibit transverse isotropy of stiffness and thermal expansivity. The model represents a development of an analysis published by Mikata and Taya, the modification allowing any number of component cylinders and a finite outer radius. Use of the model is illustrated by means of a series of examples involving SiC monofilaments. Application to the behaviour of composites containing many aligned fibres is demonstrated, using cylinder radii appropriate for the fibre volume fraction in the composite. It is shown by comparison with predictions from an Eshelby model that this is an acceptable approximation, preferable to the surrounding of fibre and matrix by an outer “composite” layer of infinite radius.

A geometrically nonlinear theory of transversely isotropic laminated composite plates and its use in the post-buckling analysis

Abstract: A higher-order, geometrically nonlinear theory of transversely isotropic symmetrically laminated composite plates is formulated and their post-buckling behavior is analysed. The numerical illustrations emphasize the role played by transverse shear deformation, transverse normal stress, higher-order effects and the character of in-plane boundary conditions. The results obtained within the present higher-order theory are compared with those of first-order transverse shear deformation and classical (Kirchhoff) theory, and conclusions on their range of applicability and the influence of various parameters are outlined.

Determination of the elastic constants of anisotropic materials using laser‐generated ultrasonic signals

Abstract: This paper presents the solution of the materials characterization problem in which the elastic constants of an anisotropic material are determined from ultrasonic wavespeed measurements made in nonprincipal directions of a specimen. The ultrasonic waves were generated via the point‐source/point‐receiver technique using a pulsed laser as a source and a miniature, point‐like transducer as a receiver. Data were acquired during a scan of the source along one of the principal acoustic axes of symmetry of the material. In each waveform the arrivals of the quasi‐longitudinal and the two quasi‐shear bulk modes were measured and the elastic constants of the material were then recovered using an optimization algorithm. Experimental results are presented for a transversely isotropic, unidirectional fiberglass/polyester and a single crystal specimen of silicon. It was found that the nonlinear fit between the measured and the recovered longitudinal slowness values is excellent. Some discrepancies are observed in the ...

The range of effects of azimuthal isotropy and EDA anisotropy in sedimentary basins

Abstract: SUMMARY Combinations of bedding- or lithology-induced azimuthal isotropy, with an axis of symmetry perpendicular to the bedding plane, and crack-induced extensivedilatancy anisotropy (EDA), with a horizontal axis of symmetry, are believed to be common in sedimentary basins, and cause the widely observed phenomenon of shear-wave splitting. Combinations of two such transversely isotropic forms of anisotropy with orthogonal axes of cylindrical symmetry lead to orthorhombic symmetry. This has two major effects: (1) the polarizations of the faster split shear waves may no longer be parallel to the strike of the cracks, or fractures, even for near-vertical propagation; and (2) such orthorhombic symmetry systems necessarily have a number of directions, called shear-wave point singularities, where shear waves display disturbed or anomalous behaviour, again possibly in near-vertical directions. Unless these effects are correctly identified, they could be interpreted mistakenly for the effects of structural irregularities or discontinuities. In contrast, recognition of the 3-D geometry of this behaviour places comparatively tight constraints on possible combinations of anisotropy in the rockmass. In order to give some understanding of the geometry of these phenomena, this paper presents 3-D patterns of the behaviour of shear-wave splitting that have been computed for a range of combinations of crack- and bedding-induced anisotropy .

Numerical simulation of ultrasonic wave propagation in anisotropic and attenuative solid materials

Abstract: The axisymmetric elastodynamic finite element code developed is capable of predicting quantitatively accurate displacement fields for elastic wave propagation in isotropic and transversely isotropic materials. The numerical algorithm incorporates viscous damping by adding a time-dependent tensor to Hooke's law. Amplitude comparisons are made between the geometric attenuation in the far field and the corresponding finite element predictions to investigate the quality and validity of the code. Through-transmission experimental measurements made with a 1 MHz L-wave transducer attached to an aluminum sample support the code predictions. The algorithm successfully models geometric beam spreading dispersion and energy absorption due to viscous damping. This numerical model is a viable tool for the study of elastic wave propagation in nondestructive testing applications. >

On a nonlinear theory for muscle shells: Part II--Application to the beating left ventricle.

Abstract: This paper specializes the nonlinear laminated-muscle-shell theory developed in Part I to cylindrical geometry and computes stresses in arteries and the beating left ventricle. The theory accounts for large strain, material nonlinearity, thick-shell effects, torsion, muscle activation, and residual strain. First, comparison with elasticity solutions for pressurized arteries shows that the accuracy of the shell theory increases as transmural stress gradients and the shell thickness decrease. Residual strain reduces the stress gradients, lowering the error in the predicted peak stress in thick-walled arteries (R/t = 2.8) from about 30 to 10 percent. Second, the canine left ventricle is modeled as a thick-walled laminated cylinder with an internal pressure. Each layer is composed of transversely isotropic muscle with a fiber orientation based on anatomical data. Using a single pseudostrain-energy density function (with time-varying coefficients) for passive and active myocardium, the model predicts strain distributions that agree fairly well with published experimental measurements. The results also show that the peak fiber stress occurs subendocardially near the beginning of ejection and that residual strains significantly alter stress gradients within each lamina, but the magnitude of the peak fiber stress changes by less than 20 percent.

A constitutive relation for the viscous flow of an oriented fiber assembly

Abstract: A constitutive relation for an equivalent, homogeneous fluid is developed for the anisotropic viscous flow of an oriented assembly of discontinuous fibers suspended in a viscous fluid. The anisotropic viscous compliance matrix can be expressed in terms of three constants by assuming the equivalent fluid to be incompressible and the microstructure to have axial symmetry (transversely isotropic). By means of a micromechanics analysis, the three terms of the constitutive relation are expressed in terms of the viscosity of the matrix fluid, the fiber aspect ratio, and the fiber volume fraction. A comparison of the viscosity terms reveals that the elongational viscosity in the fiber direction varies as the square of the fiber aspect ratio and a complex function of the fiber volume fraction. Furthermore, the ratio of the axial elongational viscosity to the transverse elongational viscosity and both axial and transverse shear viscosities was shown to be 10 exp 4 - 10 exp 6 for fiber aspect ratio of 100-1000, except at extreme values of the fiber volume fraction.

General Anisotropic Elastic Tensor In Rocks: Approximation, Invariants, And Particular Directions

Abstract: If one accepts physical measurements as absolutely accurate most real media would exhibit the most general anisotropy (hiclinic). In fact, all physical measurements arc blemished by errors. As a consequence, ven if the media to be tested were of much higher symmetry (isotropic, transversely isotropic, orthotropic...), they wonld apparently exhibit the most general anisotmpy due to the imperfection of the measurements. In spite of that. in practical situations, these. media can often be reasonably approximated by media of higher symmem. Focusing on the elastic properties of rocks, we propose a method for approximating an elastic tensor of whatever symmetry by tensors of higher symmetry. The degree of the approximations i quantified. The initial elasticity matrix and iu appromations are rewritten in particular eference axes in which most of the “off-diagonal” terms are minimized. The method is illustrated with experimental results. This paper constitutes an extension of the pioneer-work of F. Fedorov and an application of ideas of S. Cowin and co-authors.

Scaled physical modelling of anisotropic wave propagation: multioffset profiles over an orthorhombic medium

Abstract: SUMMARY In this paper we report further results of scaled physical modelling experiments in the laboratory in which ultrasonic elastic waves are propagated through an anisotropic medium of orthorhombic symmetry. Whereas our earlier experiments consisted for the most part in sending and receiving on opposite faces of a small cube of phenolic laminate, these new results are from multioffset profiles run parallel and at 45" to principal directions on a larger slab of phenolic. The variation of NMO velocity with offset (or angle of incidence) has been determined for compressional and transverse shear waves along profiles in the two principal directions on the 3-face (parallel to laminations) of the slab. These observed group velocities differ from the exact theoretical values by a maximum of about 1 per cent or less and also compare favourably with the theoretical velocities calculated from Thomsen's first-order equations, with maximum differences of about 2 per cent. Differences between the observed and theoretical velocities are attributed to some combination of finite transducer size (geometrical or effective path length effects, array attenuation effects, or interference with the otherwise free surface), sample inhomogeneity and/or anelasticity, and experimental error. The transmission shot gathers acquired for propagation in symmetry planes, and for source-receiver pairs with the same polarization, are similar in form to records acquired over a transversely isotropic medium. The effect of the shear-wave window and the variation of the hyperbolic NMO parameter with offset are clearly seen. Transmission records were also acquired in off-symmetry planes, namely along profiles at 45" to principal directions. On these records, which include all nine possible pairs of source-receiver polarizations, we see clear shear-wave splitting at and near zero offset and more complicated wave effects with increasing offset, such as one or another wave phase dying out. This could be due to cusping of wave surfaces or rapid changes of amplitude and/or polarization with ray direction, possibly as consequences of nearby shear-wave singularities.

Stress analysis of multilayered anisotropic elastic media

Abstract: The stress analysis of media subjected to applied surface tractions is performed. The solutions are obtained based on the Fouier transfrom technique together with the aid of he stiffness matrix approach. It can be uniformly applied to media with transversely isotropic, orthotropic, and monoclinic layers

Applications of perturbation theory to acoustic logging

Abstract: To study the normal modes generated during acoustic logging, the authors have developed a method to calculate (1) phase velocities when the formation has slight, general anisotropy, (2) partial derivatives of either the wave number or frequency with respect to either an elastic modulus or density, (3) group velocities, and (4) phase velocities when the cross section of the borehole is slightly irregular. The method, which is based upon perturbation theory, relates first-order perturbations in frequency, wave number, elastic moduli, densities, and locations of interfaces for a general model with many fluid and solid layers which have any cross-sectional shape and any type of anisotropy. To demonstrate the use of this method, formulas for the four applications are developed for a relatively simple model consisting of a fluid-filled borehole through a transversely isotropic solid with its symmetry axis parallel to the borehole, and some sample calculations are performed. Using these examples, formulas for more complicated borehole models that may be appropriate for some field situations could readily be developed. The significance of this work is that three applications can be used to study the normal modes when the borehole environment is complicated and the other can be used in anmore » inversion for formation properties.« less

P-SV reflection moveouts for transversely isotropic media with a vertical symmetry axis

Abstract: In a recently published short note, F. K. Levin (1989) discusses the relation between the “moveout velocities” of P-P, P-SV, and SV-SV reflections from the bottom of a transversely isotropic layer with a vertical symmetry axis. We refer to such a medium as one exhibiting “polar anisotropy.” Levin’s note was prompted by a paper of Tessmer and Behle (1988), and it is relevant to a paper by Iverson and others (1989), both of which discuss the computation of shear velocities from moveout velocities obtained with P-P and P-S reflections. Levin’s note addresses the practically important question of the use of this method in the presence of polar anisotropy, a phenomenon which we believe occurs almost universally in the sedimentary layers of the real earth. Levin suggests that polar anisotropy of “typical” magnitude must be considered in this problem. He uses as an estimate of typical magnitudes data given by Thomsen (1986) and concludes from numerical examples that the method of estimating shear velocities prop...

Symmetry conditions for third order elastic moduli and implications in nonlinear wave theory

Abstract: Several results are presented concerning symmetry properties of the tensor of third order elastic moduli. It is proven that a set of conditions upon the components of the modulus tensor are both necessary and sufficient for a given direction to be normal to a plane of material symmetry. This leads to a systematic procedure by which the underlying symmetry of a material can be calculated from the 56 third order moduli. One implication of the symmetry conditions is that the nonlinearity parameter governing the evolution of acceleration waves and nonlinear wave phenomena is identically zero for all transverse waves associated with a plane of material symmetry.

Characterization of fiber-matrix interface by guided waves: Axisymmetric case

Abstract: A hybrid method combining the finite-element and eigenfunction expansions is proposed to study the guided waves in composite cylinders. This method is shown to be flexible in modeling generally anisotropic cylinders and computationally efficient. However, most of the analysis is confined to the case of a transversely isotropic cylindrical core surrounded by coaxial isotropic cylinders. The object is to model uniaxial fiber-reinforced materials with interface zones between the fiber (which is anisotropic) and the matrix (which is isotropic). Numerical results are presented for the axisymmetric case for a graphite fiber in magnesium and epoxy matrices. In particular, the measurable and analyzable effects of imperfect bonding are investigated. The reduction of bond stiffness is found to have significant effect on the leaky waves.

Longitudinal Motion of a Thick Transversely Isotropic Hollow Cylinder

Abstract: The propagation of a longitudinal harmonic wave in a transversely isotropic shell has been investigated in the development of ultrasonic techniques for thick hollow composite cylinders. The characteristic equation for satisfying the stress-fee inner and outer cylindrical boundaries has been obtained in an exact form in terms of the wavelength, the cylinder radii and the material constants. The phase velocity of the fundamental mode is calculated for a wide range of the wavelength for various cylinder radii for some typical sample materials. The shell wave speeds for the second mode of vibration are also presented. Comparisons are made between shell wave speeds and plate wave speeds. The spread of the wave speeds for the composite shells is shown to be much wider than that for an isotropic shell.

An exact solution for transversely isotropic, simply supported and layered rectangular plates

Abstract: Levinson's solution for the problem of a simply supported rectangular plate of arbitrary thickness by normal surface loads is extended to the transversely isotropic and layered case. The exact closed form solution is obtained by using the propagator matrix method in a system of vector functions. As a special case of the layered medium, the normal displacement or deflection of a homogeneous plate of arbitrary thickness by normal surface loads is also given. It is shown that it approaches the classical solution for the transversely isotropic thin plate as the thickness approaches zero on the one hand, and on the other hand reduces to the thick plate expression as given by Levinson when the medium is isotropic.