Elastic properties of inhomogeneous transversely isotropic rocks

The problem to determine the effective elastic moduli and velocities of elastic wave propagation in transversely isotropic solid containing aligned spheroidal inhomogeneities (solid grains, vugs and micro-cracks) has been solved using the self-consistent scheme known as effective medium approximation (EMA). Since a solution of so-called one-particle problem is a base for each self-consistent method, we solved this problem as a first step for spheroidal inhomogeneity in a transversely isotropic medium. In contrast to the known solution of this problem by Lin and Mura we obtained the expressions for the strain field inside inclusion in the explicit form (without quadratures). The obtained solution was used then in the symmetric variant of the EMA where each component of the system was considered as spheroid with its own aspect ratio. This approach was applied to simulate the properties of the rocks containing isolated pores and micro-cracks. For connected fluid-filled pores we used the anisotropic variant of the Gassmann theory. The results of the calculations, obtained for the effective elastic moduli, have been compared with the experimental data and theoretical simulations of the other authors. Unlike many other rock mechanics theories, EMA approximation gives correct elastic moduli values even in the nondilute concentration of inhomogeneities. The comparison of the experimental data for oriented crack system with the EMA predictions indicates their good correspondence.     

各向同性和横观各向同性材料的统一点力解

本文根据横观各向同性弹性力学的通解获得了无限体的点力解，由它可以直接退化到各向同性情形的Ｋｅｌｖｉｎ解，利用这个点力解编制的边界元法程序，适用于横观各向同性材料也适用于各向同性材料，因此是真正的统一点力解。还用边界元法计算了两个数值例题。     

Complete solutions of three-dimensional problems in transversely isotropic media

Continuum Mechanics and Thermodynamics - We present a systematic approach to the derivation of complete solutions for three-dimensional transversely isotropic elastic problems. The class of...     

Elastic moduli of transversely isotropic graphite fibers and their composites

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.     

Wave propagation in an incompressible transversely isotropic fibre-reinforced elastic media

The boundary conditions at free surface of an incompressible, transversely isotropic elastic half-space are satisfied to obtain the reflection coefficients for the case when outer slowness section is re-entrant. Two quasi-shear waves will be reflected for an angular range of direction of incident wave. The numerical illustrations of reflection coefficients are presented graphically for three arbitrary materials.     

Absorbing boundary conditions for elastic waves in transversely isotropic media

In the numerical simulation of elastic wave propagation in the solid, it is essential to introduce absorbing boundary conditions to limit the large or unbounded domain of computation. In this paper, the absorbing boundaries for transversely isotropic media are composed of simple first-order partial differential operators, and each of the operators can perfectly absorb a plane wave outgoing at a certain angle. To test the absorbing ability, the reflection coefficient formulas for the quasi-P and quasi-S wave on the absorbing boundary are derived based on the potential functions theory of the elastic wave. Numerical examples show that the absorbing effect is good. The boundary conditions given here have a practical meaning.Supported by National Natural Science Foundation of China.     

Localized acoustic waves at twist boundaries in transversely isotropic media

Existence criteria and basic characteristics are analytically found for elastic waves localized at a twist boundary in transverse isotropic media. The boundary is formed by two identical semi-infinite bodies with non-collinear principal axes parallel to the interface. The analysis is based on the Stroh formalism specified to the case of transverse isotropy. The dispersion equation is presented in a general form and explicitly solved for small misorientations. The waves in the sector situated between the directions of transverse isotropy in the sub-media of the bicrystal are explicitly described.     

Dynamic behavior of piles embedded in transversely isotropic layered media

Dynamic behavior of single pile embedded in transversely isotropic layered media is investigated using the finite element method combined with dynamic stiffness matrices of the soil derived from Green's function for ring loads. The influence of soil anisotropy on the dynamic behavior of piles is examined through a series of parametric studies     

Elastic contact between a sphere and a semi infinite transversely isotropic body

The problem is solved by using a Hankel transformation. The stress and displacement expressions are explicitly given for any point of the medium. Curves of numerical results are presented.     

Characteristic analysis for stress wave propagation in transversely isotropic fluid-saturated porous media

According to generalized characteristic theory, a characteristic analysis for stress wave propagation in transversely isotropic fluid-saturated porous media was performed. The characteristic differential equations and compatibility relations along bicharacteristics were deduced and the analytical expressions for wave surfaces were obtained. The characteristic and shapes of the velocity surfaces and wave surfaces in the transversely isotropic fluid-saturated porous media were discussed in detail. The results also show that the characteristic equations for stress waves in pure solids are particular cases of the characteristic equations for fluid-saturated porous media.     

Interaction of shock-wave fronts with interface of transversely isotropic elastic media

A problem of diffraction of an incident-wave front interacting with an interface is considered within the framework of shock-wave theory for anisotropic elastic media. An approach to geometrical construction of reflected- and refracted-wave fronts using methods of continuation by a parameter is proposed for solution of Snellius generalized nonlinear equations. A numerical example is presented. Translated from Prikladnaya Mekhanika, Vol. 35, No. 4, pp. 30–36, April, 1999.     

Diffraction of discontinuous waves by ellipsoidal interfaces of transversely isotropic elastic media

Discontinuous waves penetrating the interfaces of transversely isotropic elastic media are studied using the ray-path method. The cases where such waves interact with curvilinear reflectors and with biconvex and biconcave lenses are considered. It is shown that discontinuous waves can focus or scatter depending on the physical properties and sequence of the media under consideration.Translated from Prikladnaya Mekhanika, Vol. 40, No. 10, pp. 98–106, October 2004.     

On boundary integral equation formulation for axisymmetric problems of transversely isotropic media

Summary In this paper we deal with the formulation of the axisymmetric boundary integral equation for the transversely isotropic elastic body. We assume that the axis of elastic symmetry is coincident with the axis of rotation and that the elastic body is subjected to arbitrary axisymmetric loading without torsion. Numerical calculations are carried out and compared with results by the finite element method.

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Zur Beschreibung von axialsymmetrischen, elastisch transversal-isotropen Problemen durch Randintegralgleichungen

Übersicht Behandelt wird die Formulierung der Randintegralgleichung für den axialsymmetrischen, elastisch transversal-isotropen Körper. Es wird angenommen, daß die Achse der elastischen Symmetrie mit der Rotationsachse zusammenfällt und der Körper einer beliebigen axialsymmetrischen Belastung ohne Torsion unterworfen ist. Ausgeführte numerische Beispiele werden mit Ergebnissen der Finite-Element-Methode verglichen.

     

Elastic state of a transversely isotropic piezoelectric body with an arbitrarily oriented elliptic crack

The elastic stress state in a piezoelectric body with an arbitrarily oriented elliptic crack under mechanical and electric loads is analyzed. The solution is obtained using triple Fourier transform and the Fourier-transformed Green’s function for an unbounded piezoelastic body. Solving the problem for the case of a crack lying in the isotropy plane, for which there is an exact solution, demonstrates that the approach is highly efficient. The distribution of the stress intensity factors along the front of a crack in a piezoelectric body under uniform mechanical loading is analyzed numerically for different orientations of the crack __________ Translated from Prikladnaya Mekhanika, Vol. 44, No. 2, pp. 39–48, February 2008.     

Transient displacement fields in hexagonal crystals and transversally isotropic media

The structure of the disturbed region and the geometry of the wave front is investigated under the condition that a concentrated source of the instantaneous-pulse type is acting in an unbounded transversally isotropic medium. The regions of permissible values of the anisotropy coefficient introduced in [1] for transversally isotropic media on the basis of conditions of the elastic energy's positive-definiteness and hyperbolicity conditions are determined. It is suggested that motion of the medium occurs under conditions of plane deformation.Translated from Zhurnal Prikladnoi Mekhaniki i Tekhnicheskoi Fiziki, No. 2, pp. 153–160, March–April, 1976.The author expresses his thanks to S. A. Khristianovich for his attention to this work.     

Wave propagation in transversely isotropic cylinders

Various approaches have been used for model1ing problems dealing with interaction of acoustic/elastic waves with transversely isotropic cylinders. The authors developed the first mathematical model for the scattering of acoustic waves from transversely isotropic cylinders [Honarvar, F., Sinclair, A.N., 1996. Acoustic wave scattering from transversely isotropic cylinders. Journal of the Acoustical Society of America 100, 57–63.]. In the current paper, this model is used for derivation of the frequency equations of longitudinal and flexural wave propagation in free transversely isotropic cylinders. Consistency of this model with the physics of the problem is demonstrated and a systematic solution to the corresponding equations is developed. Numerical results obtained for a number of transversely isotopic cylinders are used for verification of the mathematical model.     

General steady-state solutions for transversely isotropic thermoporoelastic media in three dimensions and its application

This paper presents a set of 3D general solutions for thermoporoelastic media for the steady-state problem. By introducing two displacement functions, the equations governing the elastic, pressure and temperature fields are simplified. The operator theory and superposition principle are then employed to express all the physical quantities in terms of two functions, one of which satisfies a quasi–Laplace equation and the other satisfies a differential equation of the eighth order. The generalized Almansi's theorem is used to derive the displacements, pressure and temperature in terms of five quasi-harmonic functions for various cases of material characteristic roots. To show its practical significance, an infinite medium containing a penny-shaped crack subjected to mechanical, pressure and temperature loads on the crack surface is given as an example. A potential theory method is employed to solve the problem. One integro-differential equation and two integral equations are derived, which bear the same structures to those reported in literature. For a penny-shaped crack subjected to uniformly distributed loads, exact and complete solutions in terms of elementary functions are obtained, which can serve as a benchmark for various kinds of numerical codes and approximate solutions.