Static equilibrium of an elastic orthotropic medium with an arbitrarily oriented triaxial ellipsoidal inclusion

The stress state of an elastic orthotropic medium with an arbitrarily oriented triaxial ellipsoidal inclusion is analyzed. A solution is obtained using the triple Fourier transform and the Fourier-transformed Green’s function for an infinite anisotropic medium. The high efficiency of the approach is demonstrated by solving the problem for a transversely isotropic material with a spheroidal cavity for which the exact solution is known. A numerical analysis is conducted to study the stress distribution over the surface of the inclusion with different orientations in the orthotropic space. It is revealed that in some cases the orientation of the inclusion has a strong effect on the stress concentration __________ Translated from Prikladnaya Mekhanika, Vol. 43, No. 4, pp. 55–61, April 2007.     

Interaction of Anisotropic Ellipsoidal Inclusions in an Isotropic Medium under Mechanical Loads and Uniform Heating

Eshelby's equivalent-inclusion method is extended to a finite number of arbitrarily oriented anisotropic ellipsoidal inclusions in an elastic isotropic matrix under polynomial mechanical loading and heating. The interaction of two identical and two different triaxial ellipsoidal inclusions in an elastic medium is studied as numerical examples. In special cases, the results are compared with those obtained by other authors     

Stress State of an Orthotropic Electroelastic Medium with an Arbitrarily Oriented Elliptic Inclusion*

International Applied Mechanics - The problem of electric and stress state in an orthotropic electroelastic space containing an arbitrarily oriented ellipsoidal inclusion under homogeneous force...     

Interaction of an Ellipsoidal Inclusion with an Elliptic Crack in an Elastic Material under Triaxial Tension

The interaction of an elastic ellipsoidal inclusion with an elliptic crack in an infinite elastic medium under triaxial loading is analyzed. The stress state in the elastic space is represented as a superposition of the principal state and perturbed states, which are due to the presence and interaction of the inclusion and the crack. The analytical solution of the problem is found using the method of equivalent inclusion, the potential of an inhomogeneous ellipsoid, and a system of harmonic functions for an elliptic crack. The effect of triaxial loading on the stress intensity factors is analyzed     

The effective medium and the average field approximations vis-à-vis the Hashin-Shtrikman bounds. II. The generalized self-consistent scheme in matrix-based composites

The present Part II of this two-part study is concerned with the average field approximation (AFA), and the effective medium approximation (EMA) in two-phase matrix-based dielectric composites through the use of an auxiliary configuration in which a particle of the inclusion phase is first surrounded by some matrix material, and then embedded in the effective medium. Those models will be referred as the generalized self-consistent scheme-average field approximation (GSCS-AFA), and the generalized self-consistent scheme-effective medium approximation (GSCS-EMA). We show that there are four types of the GSCS-AFA and a single type of the GSCS-EMA. In this paper the application of those models to dielectric composites with isotropic constituents and an inclusion phase that consists of randomly oriented ellipsoidal particles will be studied. The analytical solution of the auxiliary problem, which consists of an ellipsoidal particle confocally surrounded by a matrix shell and embedded in the effective medium, is achieved by means of ellipsoidal harmonics. Our results show that the effective property predictions of the GSCS-EMA and GSCS-AFA for the considered systems differ from each other, and more importantly, out of the four GSCS-AFA models, three of them violate the Hashin-Shtrikman bounds. The predictions of the GSCS-EMA obey the bounds. It is then shown that the version of the GSCS-AFA which obeys the Hashin-Shtrikman bounds for an inclusion phase with randomly oriented ellipsoids will violate them in the case of a particle shape which is not simply connected. Moreover, it turns out that the SCS-AFA studied in Part I also violates the Hashin-Shtrikman bounds in that case; the EMA, as expected, owing to its realizability property, continues to obey the bounds. Among the AFA and EMA in matrix-based composites, the GSCS-EMA therefore stands out as the method to be recommended.     

Stress State of a Transversely Isotropic Medium with an Arbitrarily Oriented Spheroidal Inclusion

The stress-concentration problem for an elastic transversely isotropic medium containing an arbitrarily oriented spheroidal inclusion (inhomogeneity) is solved. The stress state in the elastic space is represented as the superposition of the principal state and the perturbed state due to the inhomogeneity. The problem is solved using the equivalent-inclusion method, the triple Fourier transform in space variables, and the Fourier-transformed Green function for an infinite anisotropic medium. Double integrals over a finite domain are evaluated using the Gaussian quadrature formulas. In special cases, the results are compared with those obtained by other authors. The influence of the geometry and orientation of the inclusion and the elastic properties of the medium and inclusion on the stress concentration is studied__________Translated from Prikladnaya Mekhanika, Vol. 41, No. 2, pp. 33–40, February 2005.     

The Stress State of an Elastic Medium with an Elliptic Crack and Two Ellipsoidal Cavities

This paper is a study into the interaction of two triaxial ellipsoidal cavities whose surfaces are under different pressures with an elliptic crack in an infinite elastic medium. The stress state in the elastic space is represented by a superposition of perturbed states due to the presence and interaction of the cavities and the crack. The exact solution of the problem is constructed by using a modified method of equivalent inclusion, the potential of an inhomogeneous ellipsoid, and a system of harmonic functions for the elliptic crack. A numerical analysis is carried out to find how the geometry of the cavities and the crack, the distance between them, and the pressure on their surfaces affect the stress intensity factors     

Eigenvibrations in trapped-energy contoured piezoelectric resonators with a one-sided arbitrarily oriented elliptical convexity

A mathematical model of a piezoelectric plate is discussed with a one-sided convex surface of an arbitrary shape. Generic relations are given to calculate the frequency spectrum and distributions of the vibration amplitudes with normal drive levels. A particular case of the ellipsoidal convexity is studied, that is arbitrarily oriented with respect to the plate axes. A numerical example for such a curvature is also given. Based upon this, we show that the frequency spectrum is critically dependent on the angle between the main ellipsoid axis and the rotated coordinates of a piezoelectric plate and to the ratio of the main radii of an ellipsoid. We notice that such a surface allows for a proper placement of the anharmonics in the frequency spectrum, avoiding their interaction caused by environment. It may also be useful to model the manufacturing imperfection and its influence upon the vibration spectra.     

The Stress–Strain State of an Elastic Ferromagnetic Medium with an Ellipsoidal Inclusion in a Homogeneous Magnetic Field

A stress–strain problem is solved for an infinite elastic magnetically soft medium with an ellipsoidal inclusion in an external magnetic field. The main characteristics of the stress–strain state and induced magnetic fields in the medium and the inclusion are determined and their distribution over the surface of the inclusion is analyzed     

Porous polycrystals built up by uniformly and axisymmetrically oriented needles: homogenization of elastic propertiesPolycristaux poreux constitués d'aiguilles orientées de façon uniforme ou axisymétrique : homogénéisation des propriétés élastiques

Porous polycrystal-type microstructures built up of needle-like platelets or sheets are characteristic for a number of biological and man-made materials. Herein, we consider (i) uniform, (ii) axisymmetrical orientation distribution of linear elastic, isotropic as well as anisotropic needles. Axisymmetrical needle orientation requires derivation of the Hill tensor for arbitrarily oriented ellipsoidal inclusions with one axis tending towards infinity, embedded in a transversely isotropic matrix; therefore, Laws' integral expression of the Hill tensor is evaluated employing the theory of rational functions. For a porosity lower 0.4, the elastic properties of the polycrystal with uniformly oriented needles are quasi-identical to those of a polycrystal with solid spheres. However, as opposed to the sphere-based model, the needle-based model does not predict a percolation threshold. As regards axisymmetrical orientation distribution of needles, two effects are remarkable: Firstly, the sharper the cone of orientations the higher the anisotropy of the polycrystal. Secondly, for a given cone, the anisotropy increases with the porosity. Estimates for the polycrystal stiffness are hardly influenced by the anisotropy of the bone mineral needles. Our results also confirm the very high degree of orientation randomness of crystals building up mineral foams in bone tissues. To cite this article: A. Fritsch et al., C. R. Mecanique 334 (2006).     

Spheroidal inhomogeneity in a transversely isotropic piezoelectric medium

Summary Piezoelectric material containing an inhomogeneity with different electroelastic properties is considered. The coupled electroelastic fields within the inclusion satisfy a system of integral equations solved in a closed form in the case of an ellipsoidal inclusion. The solution is utilized to find the concentration of the electroelastic fields around an inhomogeneity, and to derive the expression for the electric enthalpy of the electroelastic medium with an ellipsoidal inclusion that is relevant for various applications. Explicit closed-form expressions are found for the electroelastic fields within a spheroidal inclusion embedded in the transversely isotropic matrix. Results are specialized for a cylinder, a flat rigid disk and a crack. For a penny-shaped crack, the quantities entering the crack propagation criterion are found explicitly. Received 17 February 2000; accepted for publication 9 May 2000     

Eshelby's problem of non-elliptical inclusions

The Eshelby problem consists in determining the strain field of an infinite linearly elastic homogeneous medium due to a uniform eigenstrain prescribed over a subdomain, called inclusion, of the medium. The salient feature of Eshelby's solution for an ellipsoidal inclusion is that the strain tensor field inside the latter is uniform. This uniformity has the important consequence that the solution to the fundamental problem of determination of the strain field in an infinite linearly elastic homogeneous medium containing an embedded ellipsoidal inhomogeneity and subjected to remote uniform loading can be readily deduced from Eshelby's solution for an ellipsoidal inclusion upon imposing appropriate uniform eigenstrains. Based on this result, most of the existing micromechanics schemes dedicated to estimating the effective properties of inhomogeneous materials have been nevertheless applied to a number of materials of practical interest where inhomogeneities are in reality non-ellipsoidal. Aiming to examine the validity of the ellipsoidal approximation of inhomogeneities underlying various micromechanics schemes, we first derive a new boundary integral expression for calculating Eshelby's tensor field (ETF) in the context of two-dimensional isotropic elasticity. The simple and compact structure of the new boundary integral expression leads us to obtain the explicit expressions of ETF and its average for a wide variety of non-elliptical inclusions including arbitrary polygonal ones and those characterized by the finite Laurent series. In light of these new analytical results, we show that: (i) the elliptical approximation to the average of ETF is valid for a convex non-elliptical inclusion but becomes inacceptable for a non-convex non-elliptical inclusion; (ii) in general, the Eshelby tensor field inside a non-elliptical inclusion is quite non-uniform and cannot be replaced by its average; (iii) the substitution of the generalized Eshelby tensor involved in various micromechanics schemes by the average Eshelby tensor for non-elliptical inhomogeneities is in general inadmissible.     

Interaction of an Ellipsoidal Inclusion with an Elliptic Crack in an Elastic Medium

An approach is considered to how to allow for the interaction between an ellipsoidal heterogeneity (inclusion) and an elliptic crack in an elastic medium. Using the superposition of perturbed stress states, the boundary conditions are satisfied on the ellipsoidal surface by the method of equivalent inclusion and on the crack surface by the least-squares method. A numerical analysis is carried out. Typical mechanical effects are revealed. In the calculations, the stress state near the ellipsoidal heterogeneity is approximated by a polynomial of the second degree in Cartesian coordinates, whereas the load on the crack surface is simulated by a polynomial of the fourth degree in Cartesian coordinates. In particular cases, the results are in good agreement with the data obtained by other authors     

Stress–Strain State of a Ferromagnetic with a Paraboloidal Inclusion in a Homogeneous Magnetic Field

The stress–strain state of an infinite elastic soft ferromagnetic medium with an elliptic paraboloidal inclusion is analyzed. The material of the inclusion is a soft ferromagnetic too. The medium is in a magnetic field directed along the minor axis of the elliptic section of the paraboloid by a plane perpendicular to its axis. The main characteristics of the stress–strain state and induced magnetic fields in the medium and inclusion are determined. The features of the stress distribution over the inclusion boundary are studied     

Propagation and scattering of ultrasonic waves in polycrystals with arbitrary crystallite and macroscopic texture symmetries

A general ultrasonic attenuation model for a polycrystal with arbitrary macroscopic texture and triclinic ellipsoidal grains is described with proper accounting for the anisotropic Green’s function for the reference medium. The texture and the ellipsoidal grain frames in the model are independent and the wave propagation direction is arbitrary. The attenuation coefficients are obtained in the Born approximation accompanied by the Rayleigh and stochastic asymptotes. The scattering model displays statistical anisotropy due to two independent factors: (1) shape of the oriented grains and (2) preferred crystallographic orientation of the grains leading to macroscopic anisotropy of the homogenized reference medium. The model is applicable to most single phase polycrystalline materials that may occur as a result of thermomechanical manufacturing processes leading to different macrotextures and elongated-shaped grains. It predicts the strength of ultrasonic scattering and its dependence on frequency and propagation direction as a function of grain shape, grain crystallographic symmetry and macroscopic texture parameters and provides the texture-induced dependence of macroscopic ultrasonic velocity on propagation angle. It considers proper wave polarizations due to macroscopic anisotropy and scattering-induced transformations of waves with different polarizations. Competing effects of grain shape and texture on the attenuation are observed. In contrast to the macroscopically isotropic case, where in the stochastic regime the attenuation is highest in the direction of the longest ellipsoidal axis of the grain, the wave attenuation in the elongation direction may be suppressed or amplified by the texture with different effects on the quasilongitudinal and quasitransverse waves. The frequency behavior is also interestingly affected by texture: a hump in the total attenuation coefficient is found for the fast quasitransverse wave which is purely the result of macroscopic anisotropy and the existence of two quasitransverse waves; this hump is not observed in the macroscopically isotropic case. Striking differences of the texture effect on the directional dependences of the attenuation coefficients are found at low versus high frequencies.     

Application of results on Eshelby tensor to the determination of effective poroelastic properties of anisotropic rocks-like composites

The present work is devoted to the determination of the macroscopic poroelastic properties of anisotropic elastic porous materials saturated by a fluid under pressure. It makes use of the theoretical results provided by Withers [Withers, P.J., 1989. The determination of the elastic field of an ellipsoidal inclusion in a transversely isotropic medium, and its relevance to composite materials. Philosophical Magazine A 59 (4), 759–781.] for the problem of an ellipsoidal inclusion embedded in a transversely isotropic elastic medium. The particular case of a spherical inclusion is very important for rock-like composites such as argillite and shales. The implementation of these results in a micromechanical theory of poroelasticity allows to quantify the effects of the solid matrix anisotropy and of pore space on the effective poromechanical properties. Closed form expressions of Biot tensor and of Biot modulus are presented as well as numerical applications for anisotropic shales.     

The scattering of electroelastic waves by an ellipsoidal inclusion in piezoelectric medium

The scattering problem for a single ellipsoidal piezoelectric inclusion embedded in piezoelectric medium is investigated. Based on the polarization method, the extended displacements are expressed in terms of integral equations, whose kernels are obtained from the Green’s functions of homogenous matrix. In this paper, the 3D dynamic Green’s functions are derived by means of the Radon transform technique. To illustrate the use of the equations, scattering by a piezoelectric, ellipsoidal inhomogeneity in a piezoelectric medium is considered in the low frequency and an asymptotic formula for this scattering cross-section is obtained. Numerical results of the scattering cross-sections are carried out for a spheroidal BaTiO₃-inclusion in a PZT-5H-matrix.     

The effective medium and the average field approximations vis-à-vis the Hashin-Shtrikman bounds. I. The self-consistent scheme in matrix-based composites

The effective medium approximation (EMA) and the average field approximation (AFA) are two classical micromechanics models for the determination of effective properties of heterogeneous media. They are also known in the literature as ‘self-consistent’ approximations. In the AFA, the basic idea is to estimate the actual average field existing in a phase through a configuration in which a typical particle of that phase is embedded in the homogenized medium. In the EMA, on the other hand, one or more representative microstructural elements of the composite is embedded in the homogenized effective medium subjected to a uniform field, and the demand is made that the dominant part of the far-field disturbance vanishes. Both parts of this study are concerned with two-phase, matrix-based, effectively isotropic composites with an inclusion phase consisting of randomly oriented particles of arbitrary shape in general, and ellipsoidal shape in particular. The constituent phases are assumed to be isotropic. It is shown that in those systems the AFA and EMA give different predictions, with the distinction between them becoming especially striking regarding their standing vis-à-vis the Hashin-Shtrikman (HS-bounds). While due to its realizability property the EMA will always obey the bounds, we show that there are circumstances in which the AFA may violate the bounds. In the AFA for two-phase matrix-based composites, the embedded inclusion is a particle of the inclusion phase. If the particle is directly embedded in the effective medium, the method is called here the self-consistent scheme-average field approximation (SCS-AFA), and will obey the HS-bounds for an inclusion shape that is simply connected. If the embedded entity is a matrix-coated particle, then the method is called the generalized self-consistent scheme-average field approximation (GSCS-AFA), and may violate the HS-bounds. On the other hand, in the EMA for matrix-based composites with well-separated inclusions, we indicate that in view of its premises the embedding with a matrix-coated particle generally becomes the appropriate one, and the method is thus called the generalized self-consistent scheme-effective medium approximation (GSCS-EMA). Part I of this study is concerned with SCS-AFA in dielectrics and elasticity, and Part II with the GSCS-AFA and GSCS-EMA in dielectrics.     

INFLUENCE OF SAMPLE PREPARATION TECHNOLOGY ON SHEAR BEHAVIOR OF INHERENT ANISOTROPIC SAND IN TRIAXIAL TESTS 1)

基于砂土颗粒的毛细效应特性和冻融试验原理,提出一种砂土试样制备方法,能够在0°~90°范围内任意选取沉积方向,且适用于不同颗粒级配的试样。在中围压条件下进行三轴固结排水剪切试验,研究结果表明,毛细效应、冻融相结合的制样方法能够减缓压缩剪切试验的应力-应变关系增长速度,降低试样的抗剪峰值强度,对残余强度的影响相对较弱。在其他应力路径条件下,例如拉伸、减压压缩、等应力压缩等试验中,可以不考虑装样方法的影响。建议的制样方法能够合理应用到砂土初始各向异性的研究工作中。