Ellipsoidal defect identification in an elastic body from the results of a uniaxial tension (compression) test

We consider the problem of identification of an ellipsoidal cavity or ellipsoidal inclusion (rigid or elastic) in an isotropic linearly elastic body. We solve the problem by a method based on the use of a reciprocity functional. We propose a constructive procedure which allows us to express the defect geometric parameters in terms of the values of the reciprocity functional. These values can be calculated by measuring the displacements on the external surface of the body in a static uniaxial tension (compression) test. The proposed procedure permits exact identification of the parameters of the ellipsoidal defect if it is located in an infinite space. In the case of a bounded elastic body, it can be considered as an approximate procedure.     

Identification of parameters of a plane elliptic crack in an isotropic linearly elastic body from the results of a single uniaxial tension test

The method earlier developed by one of the authors for identifying ellipsoidal defects is numerically tested for the applicability to the problem of identification of a degenerate ellipsoidal defect, i.e., an elliptic crack. The method is based on the reciprocity functional and the assumption that the displacements are measured in a uniaxial tension test of an isotropic linearly elastic body. Calculations show that the earlier developed method is also efficient for identification of an elliptic crack and its parameters (the center coordinates, the normal to the crack plane, and the directions and lengths of the semiaxes) can be determined with high accuracy. Some examples where the crack has a non-elliptic shape are also considered. It is discovered that, in many cases, the ellipsoids that were constructed by formulas reconstructing the ellipsoidal crack from the data on the external boundary of the body that correspond to a nonelliptic crack, approximate the actual defect with sufficient accuracy. The method stability was investigated with respect to noise in the initial data.     

Une transformation du problème d'élasticité linéaire en vue d'application au problème de l'inclusion et aux fonctions de Green

A simple transformation of the problem of the linear elastic structure is presented. The transformed problem corresponds to a new problem of linear elastic structure with different behaviour, geometry and prescribed forces and displacements. The transformed problem can be easier to study, or can correspond to cases with well-known solutions. By means of this transformation, the problem of ellipsoidal inclusion is transformed into a problem of spherical inclusion, the analytical results known for the Eshelby tensor for an isotropic or transversely isotropic matrix are extended to more general cases of matrix behaviour, and finally, close form expressions of the Green function for an infinite medium are derived for some cases of elastic behaviour without transversal isotropy or orthotropy.     

Stress concentrations in the particulate composite with transversely isotropic phases

The accurate series solution have been obtained of the elasticity theory problem for a transversely isotropic solid containing a finite or infinite periodic array of anisotropic spherical inclusions. The method of solution has been developed based on the multipole expansion technique. The basic idea of method consists in expansion the displacement vector into a series over the set of vectorial functions satisfying the governing equations of elastic equilibrium. The re-expansion formulae derived for these functions provide exact satisfaction of the interfacial boundary conditions. As a result, the primary spatial boundary-value problem is reduced to an infinite set of linear algebraic equations. The method has been applied systematically to solve for three models of composite, namely a single inclusion, a finite array of inclusions and an infinite periodic array of inclusions, respectively, embedded in a transversely isotropic solid. The numerical results are presented demonstrating that elastic properties mismatch, anisotropy degree, orientation of the anisotropy axes and interactions between the inclusions can produce significant local stress concentration and, thus, affect greatly the overall elastic behavior of composite.     

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 non-uniform transformation strain problem for an anisotropic ellipsoidal inclusion

The problem of an anisotropic ellipsoidal inclusion which undergoes a stress-free transformation strain (in the sense of J.D. Eshelby) is considered, and the following theorem is proved: If an ellipsoidal region in an infinite anisotropic linear elastic medium undergoes, in the absence of its surroundings, a stress-free transformation strain which is a polynomial of degree M in the position coordinates x_t, then the final stress and strain state in the transformed inclusion, when constrained by its surroundings, is also a polynomial of degree M in x_t.     

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     

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.     

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.     

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.     

Identification of small well-separated defects in an isotropic elastic body using boundary measurements

An inverse problem of identification of a finite number of small, well-separated defects in an isotropic linear elastic body is considered. It is supposed that the defects are cavities or inclusions (rigid or linear elastic). If the defects are cavities then their boundaries are supposed unloaded. If the defects are inclusions it is supposed complete bonding between the matrix and inclusions. It is assumed also that as a result of static test the loads and displacements are measured on the external boundary of the body. A method for determination of centers of the defects projections on an arbitrary plane is developed. If the defects are ellipsoids their geometrical parameters (directions and magnitudes of the ellipsoids axes) are determined also. Numerical examples illustrating efficiency of the developed method are considered.     

The axisymmetrical elastic field of an ellipsoidal inclusion with a slipping interface

This paper is concerned with the axisymmetrical elastic fields caused by an ellipsoidal inclusion with a slipping interface which undergoes a uniform eigenstrain. The problem is solved under a revolving ellipsoidal coordinate with the aid of Papkovich-Neuber general dipacement formula. In contrast to the perfectly bonded interface, when the interface between the inclusion and the matrix cannot sustain shear stress, and is free to slip, the solution cannot be expressed in closed form and involves infinite series. Therefore, the results are illustrated by numerical examples.     

Stress concentration of an ellipsoidal inclusion of revolution in a semi-infinite body under biaxial tension

Summary This paper deals with the stress concentration problem of an ellipsoidal inclusion of revolution in a semi-infinite body under biaxial tension. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknowns are densities of body forces distributed in the r- and z-directions in semi-infinite bodies having the same elastic constants as the ones of the matrix and inclusion. In order to satisfy the boundary conditions along the ellipsoidal boundary, four fundamental density functions proposed in [24, 25] are used. The body-force densities are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield rapidly converging numerical results for stress distribution along the boundaries even when the inclusion is very close to the free boundary. The effect of the free surface on the stress concentration factor is discussed with varying the distance from the surface, the shape ratio and the elastic modulus ratio. The present results are compared with the ones of an ellipsoidal cavity in a semi-infinite body.accepted for publication 11 November 2003     

The rotation of a rigid elliptical disc inclusion embedded in a transversely isotropic elastic solid

This paper examines the problem of asymmetric rotation of a rigid elliptical disc inclusion embedded in bonded contact with a transversely isotropic elastic solid of infinite extent. The moment-rotation relationship for the embedded inclusion is evaluated in explicit closed form.     

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     

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.     

Stress singularities for three-dimensional corners using the boundary integral equation method

The boundary integral equation method is developed to study three-dimensional asymptotic singular stress fields at vertices of a pyramidal notch or inclusion in an isotropic elastic space. Two-dimensional boundary integral equations are used for the infinite body with pyramidal notches and inclusions when either stresses or displacements are specified on its surface. Applying the Mellin integral transformation reduces the problem to one-dimensional singular integral equations over a closed, piece-wise smooth line. Using quadrature formulas for regular and singular integrals with Hilbert and logarithmic kernels, these integral equations are reduced to a homogeneous system of linear algebraic equations. Setting its determinant to zero provides a characteristic equation for the determination of the stress singularity power. Numerical results are obtained and compared against known eigenvalues from the literature for an infinite region with a conical notch or inclusion, for a Fichera vertex, and for a half-space with a wedge-shaped notch or inclusion.     

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     

ANALYSIS OF A TRANSIENT ELASTODYNAMIC CRACK IN PIEZOELECTRIC MATERIALS USING A SINGULAR INTEGRAL EQUATION METHOD

利用广义Betti-Rayleigh 互易公式给出了二维压电材料非渗透裂纹问题的一般解和奇异积分方程,其中未知函数为裂纹上的位移间断和电势间断的导数. 在理论分析的基础上,使用高斯-切比雪夫求积公式及Lubich 卷积积分方法建立了问题的数值求解方法,并给出典型算例的广义动应力强度因子随时间变化的规律.