Green's functions for transversely isotropic piezoelectric multilayered half-spaces

This paper presents Green's functions for transversely isotropic piezoelectric and layered half-spaces. The surface of the half-space can be under general boundary conditions and a point source (point-force/point-charge) can be applied to the layered structure at any location. The Green's functions are obtained in terms of two systems of vector functions, combined with the propagator-matrix method. The most noticeable feature is that the homogeneous solution and propagator matrix are independent of the choice of the system of vector functions, and can therefore be treated in a unified manner. Since the physical-domain Green's functions involve improper integrals of Bessel functions, an adaptive Gauss-quadrature approach is applied to accelerate the convergence of the numerical integral. Typical numerical examples are presented for four different half-space models, and for both the spring-like and general traction-free boundary conditions. While the four half-space models are used to illustrate the effect of material stacking sequence and anisotropy, the spring-like boundary condition is chosen to show the effect of the spring constant on the Green's function solutions. In particular, it is observed that, when the spring constant is relatively large, the response curve can be completely different to that when it is small or when it is equal to zero, with the latter corresponding to the traction-free boundary condition.     

Green's functions for the isotropic or transversely isotropic space containing a circular crack

Summary Green's functions for an infinite three-dimensional elastic solid containing a circular crack are derived in terms of integrals of elementary functions. The solid is assumed to be either isotropic or transversely isotropic (with the crack being parallel to the plane isotropy).     

Green’s functions for transversely isotropic magnetoelectroelastic media

Based on the governing equations of transversely isotropic magnetoelectroelastic media, four general solutions on the cases of distinct eigenvalues and multiple eigenvalues are given and expressed in five mono-harmonic displacement functions. Then, based on these general solutions, employing the trial-and-error method, the three-dimensional Green’s functions of infinite, two-phase and semi-infinite magnetoelectroelastic media under point forces, point charge and magnetic monopole are all presented in terms of elementary functions for all cases of distinct eigenvalues and multiple eigenvalues. Numerical results are also presented.     

Deformational theory of plasticity of transversely isotropic media

We propose a version of the theory of plasticity of transversely isotropic media for the case of simple loading. Our version is based on the concept of yield surface. We use a quadratic condition of yield that takes into account the partial effects of equivalent stresses computed according to von Mises and Hill and according to Tresca on the plastic deformation of the material. In the general case, this condition can be interpreted as a singular surface in the space of stresses. On the basis of the assumptions concerning the linearity of trajectories of plastic deformation and their normality to the initial yield surface under simple loading as well as concerning the existence of a relationship between the introduced equivalent stresses and equivalent plastic strains independent of the type of the stressed state, we deduce reversible master equations of plasticity. The adequacy of the proposed model is confirmed by the good agreement between the results of numerical analysis and experimental data. Translated from Problemy Prochnosti, No. 1, pp. 5 – 14, January – February, 1998.     

Moving antiplane shear crack in transversely isotropic magnetoelectroelastic media

An antiplane shear strip crack moving uniformly in transversely isotopic magnetoelectroelastic media when subjected to representative non-constant crack-face loading conditions is studied. Readily calculable explicit closed-form representations are determined and discussed for the components of the stress, electric and magnetic fields created throughout the material. Representative numerical data are presented. Alternative boundary conditions for which corresponding analyses can be derived analogously are listed.     

Influence functions of the displacement discontinuity method for anisotropic bodies

In this study, the boundary element equations are obtained from the influence functions of a displacement discontinuity in an anisotropic elastic medium. For this purpose, Kelvin fundamental solutions for anisotropic media on infinite and semi-infinite planes are used to form dipoles from singular loads. Various combinations of these dipoles are used to obtain the influence functions of the displacement discontinuity. Boundary element equations are then derived analytically by the integration of these influence functions on a constant element which results in a linear system for unknown displacement discontinuities. The boundary integrals are calculated in closed form over constant elements. The obtained formulation is applied to a number of classical engineering problems.Tel.: +90-212-285-65-85, 90-212-285-37-07     

Asymmetric wave propagation in a transversely isotropic half-space in displacement potentials

With the aid of a complete representation using two displacement potentials, an efficient and accurate analytical derivation of the fundamental Green’s functions for a transversely isotropic elastic half-space subjected to an arbitrary, time-harmonic, finite, buried source is presented. The formulation includes a complete set of transformed stress-potential and displacement-potential relations that can be useful in a variety of elastodynamic as well as elastostatic problems. The present solutions are analytically in exact agreement with the existing solutions for a half-space with isotropic material properties. For the numerical evaluation of the inversion integrals, a quadrature scheme which gives the necessary account of the presence of singularities including branch points and pole on the path of integration is implemented. The reliability of the proposed numerical scheme is confirmed by comparisons with existing solutions.     

Some dynamic properties of incompressible,transversely isotropic elastic media

Summary This paper investigates various dynamic properties of incompressible, transversely isotropic elastic media. The propagation condition for such materials allows the wave speeds to be obtained in explicit form. An examination of the slowness surface and direction of energy flux as the extensional modulus along the fibre tends to infinity is then easily carried out. The paper also includes an investigation of the dynamic response of such materials to a particular line impulsive force. This is done using integral transforms. These transforms are invertible in closed form.     

Transducer-modeling in general transversely isotropic media via point-source-synthesis: Theory

Based on a theory of elastic wave propagation in arbitrarily oriented transversely isotropic media, which has been presented recently, the radiation characteristics of ultrasonic transducers in these media are determined. Using the directivity patterns for normal and transverse point sources on the free surface of such (semi-infinite) materials—the derivation is based on the reciprocity theorem—the radiated wave fields are obtained by the method of point-source-synthesis, i.e., by superposing the wave fields of numerous point sources located within the transducer aperture. Since ultrasonic inspection of anisotropic materials, especially weld material in nuclear power plants, suffers from the well-known effects of beam splitting, beam distortion, and beam skewing, valuable information in view of an optimized inspection is provided. Focusing on transversely isotropic weld material specimens, numerical evaluation is performed for several grain orientations with respect to the transducer-normal. The approach presented is particularly useful in view of an appropriate extension to inhomogeneous welds and the consideration of time-dependent RF-impulse functions.     

The method of fundamental solutions for three-dimensional elastostatic problems of transversely isotropic solids

This paper describes the method of fundamental solutions (MFS) to solve three-dimensional elastostatic problems of transversely isotropic solids. The desired solution is represented by a series of closed-form fundamental solutions, which are the displacement fields due to concentrated point forces acting on the transversely isotropic material. To obtain the unknown intensities of the fundamental solutions, the source points are properly located outside the computational domain and the boundary conditions are then collocated. Furthermore, the closed-form traction fields corresponding to the previously published point force solutions are reviewed and addressed explicitly in suitable forms for numerical implementations. Three numerical experiments including Dirichlet, Robin, and peanut-shaped-domain problems are carried out to validate the proposed method. It is found that the method performs well for all the three cases. Furthermore, a rescaling method is introduced to improve the accuracy of Robin problem with noisy boundary data. In the spirits of MFS, the present meshless method is free from numerical integrations as well as singularities.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

The asymmetric displacement of a rigid spheroidal inclusion embedded in a transversely isotropic medium

Summary An explicit analytical solution is presented for the problem of a rigid spheroidal inclusion embedded in bonded contact with an infinite transversely isotropic elastic medium, where the inclusion is given a constant displacement in a direction perpendicular to the axis of symmetry of the material. The displacement potential representation for the equilibrium of three-dimensional transversely isotropic bodies is used to solve the problem. The loadfeflection relationship for the spheroidal inclusion and its limiting configurations are obtained in closed form. Numerical results are presented to show the effect of both the aspect ratio of the spheroid and the anisotropy on the translational stiffness.With 5 Figures     

Boundary element analysis of three-dimensional crack problems in two joined transversely isotropic solids

The present paper presents a boundary element analysis of penny-shaped crack problems in two joined transversely isotropic solids. The boundary element analysis is carried out by incorporating the fundamental singular solution for a concentrated point load in a transversely isotropic bi-material solid of infinite space into the conventional displacement boundary integral equations. The conventional multi-region method is used to analyze the crack problems. The traction-singular elements are employed to capture the singularity around the crack front. The values of the stress intensity factors are obtained by using crack opening displacements. The numerical scheme results are verified with the closed-form solutions available in the literature for a penny-shaped crack parallel to the plane of the isotropy of a homogeneous and transversely isotropic solid of infinite extent. The new problem of a penny-shaped crack perpendicular to the interface of a transversely isotropic bi-material solid is then examined in detail. The crack surfaces are subject to the three normal tractions and the uniform shear traction. The associated stress intensity factor values are obtained and analyzed. The present results can be used for the prediction of the stability of composite structures and the hydraulic fracturing in deep rock strata and reservoir engineering.     

Green's functions and three-dimensional steady-state heat-conduction problems in a two-layered composite

A boundary-value problem for steady-state heat conduction in a three-dimensional, two-layered composite is studied. The method of Green's function is used in the study. Green's functions are constructed as double sums in terms of eigenfunctions in two of the three directions. The eigenfunctions in the direction orthogonal to the layers are unconventional and must be defined appropriately. The use of different forms of the Green's functions leads to different representations of the solutions as double sums with different convergence characteristics and it is shown that the method of Green's functions is superior to the classical method of separation of variables.     

Slicing procedure for approximate three-dimensional Green's functions for cracks in plates of finite thickness

Exact results for the stress intensity factor are presented for an external circular crack with oppositely directed concentrated loads applied to the crack surfaces. This result is specialized to the case of a semi-infinite crack in an infinite body with concentrated loads on the crack. A procedure is then suggested by which one can obtain from the corresponding plane result the approximate three-dimensional Green's function (concentrated load result) for any straight crack in an infinite elastic body. This procedure is used to determine the Green's functions for a finite-length crack in an infinite body, and is then used in conjunction with a suggested slicing procedure to obtain approximate three-dimensional Green's function for plates of finite thickness and infinite extent, containing finite length cracks. Previously existing solutions for crack problems are compared with results obtained by application to plate tension and bending problems of the three-dimensional Green's functions. The results indicate that the procedure yields satisfactory results when stress gradients through the plate thickness are not excessive. However, an accurate assessment of the validity of the slicing procedure awaits further progress in three-dimensional crack analysis.

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Zusammenfasung Es werden exakte Werte für den Spannungsintensitätsfaktor im Falle eines kreisförmigen Oberflächenrisses, wobei die Rißoberflächen konzentrierten und entgegengesetzten Belastungen direkt unterworfen sind. Dieses Ergebnis ist anwendbar auf den Fall eines halbunendlichen Risses in einem Körper unendlicher Abmessungen, wobei der Riß konzentrierten Beanspruchungen unterworfen ist.Anschließend wird ein Verfahren vorgeschlagen, welches es ermöglicht die dreidimensionale angenäherte Funktion von Green aus dem entsprechenden planen Ergebnis zu ermitteln und dies für den Fall eines beliebigen geraden Risses in einem elastischen Körper unendlicher Größe. Nach diesem Verfahren werden die Green'schen Funktionen für einen Riß endlicher Größe in einem unendlichen Körper bestimmt. Dieser wird anschließend dazu benutzt um mit Hilfe eines Unterteilungsverfahrens die angenäherten dreidimensionalen Green'schen Funktionen für Feinbleche unendlicher Oberfläche mit Rissen endlicher Abmessungen zu ermitteln.Die unter Anwendung der dreidimensionalen Green'schen Funktionen auf die Probleme von Zug- und Biegebe-anspruchung von Platten erzielten Ergebnisse, werden mit den schon früher vorgeschlagenen Lösungen verglichen. Dieser Vergleich zeigt, daß der vorgeschlagene Weg befriedigende Lösungen ergibt, sofern die Spannungsgradienten über die Dicke des Bleches nicht übermässig groß sind.Um jedoch die Gültigkeit des angewandten Unterteilungsverfahren exakt zu prüfen, sind bedeutende Fortschritte auf dem Gebiet der dreidimensionalen Analyse von Rissen noch erfordert.

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Résumé On présente les valeurs exactes du facteur d'intensité de contrainte dans le cas d'une fissure circulaire périphérique où des charges concentrées opposées sont directement appliquées sur ses surfaces. Ce résultat s'applique au cas de la fissure semi-infinie, dans un corps infini, des charges concentrées étant appliquées à la fissure.On suggère ensuite une procédure permettant d'obtenir la fonction tridimensionnelle approchée de Green à partir du résultat plan correspondant, et ce pour tout cas de fissure droite dans un corps infini et élastique. On détermine selon cette procédure les fonctions de Green pour une fissure de longueur finie dans un corps infini, et on l'utilise ensuite, à l'aide d'un processus de découpage, à l'obtention des fonctions tridimensionnelles approchées de Green pour des tôles d'épaisseur fine et de surface infinie, comportant des fissures de dimensions finies.Les résultats obtenus par l'application des fonctions tridimensionnelles de Green aux problèmes de traction et de flexion des plaques sont comparés aux solutions proposées antérieurement. Il résulte de cette comparaison que la procédure suivie fournit des résultats satisfaisants pour autant que les gradients de contrainte suivant l'épaisseur ne soient pas excessifs.Toutefois, pour vérifier d'une manière exacte la validité du processus de découpage qui a été adopté, il est nécessaire d'attendre que des progrès plus substantiels aient été accomplis en matière d'analyse tridimensionnelle des fissures.

     

Singularity-reduced integral equations for displacement discontinuities in three-dimensional linear elastic media

A systematic procedure is followed to develop singularity-reduced integral equations for displacement discontinuities in homogeneous linear elastic media. The procedure readily reproduces and generalizes, in a unified manner, various integral equations previously developed by other means, and it leads to a new stress relation from which a general weakly-singular, weak-form traction integral equation is established. An isolated discontinuity is treated first (including, as special cases, cracks and dislocations) after which singularity-reduced integral equations are obtained for cracks in a finite domain. The first step in the development is to regularize Somigliana's identity by utilizing a stress function for the stress fundamental solution to effect an integration by parts. The resulting integral equation is valid irrespective of the choice of stress function (as guaranteed by a certain ‘closure condition’ established for the integral operator), but certain particular forms of the stress function are introduced and discussed, including one which admits an interpretation as a ‘line discontinuity’. A singularity-reduced integral equation for the displacement gradients is then obtained by utilizing a relation between the stress function and the stress fundamental solution along with the closure condition. This construction does not rely upon a particular choice of stress function, and the final integral equation (which is a generalization of Mura's (1963) formula) has a kernel which is a simple function of the stress fundamental solution. From this relation, singularity-reduced integral equations for the stress and traction are easily obtained. The key step in the further development is the construction of an alternative stress integral equation for which a differential operator has been ‘factored out’ of the integral. This is accomplished by, in essence, establishing a stress function for the stress field induced by the discontinuity. A weak-form traction integral equation is then readily obtained and involves a kernel which is only weakly-singular. The nonuniqueness of this kernel is discussed in detail and it is shown that, at least in a certain sense, the kernel which is given is the simplest possible. The results for an isolated discontinuity are then adapted to treat cracks in a finite domain. In doing so, emphasis is given to the development of weakly-singular, weak-form displacement and traction integral equations since these form the basis of an effective numerical procedure for fracture analysis (Li et al., 1998), and such equations are presented for both elastostatics and elastodynamics. A noteworthy aspect of the development is that there is no need to introduce Cauchy principal value integrals much less Hadamard finite part integrals. Finally, the utility of the systematic procedure presented here for use in obtaining singularity-reduced integral equations for other unbounded media (viz. the half-space and bi-material) is indicated. This revised version was published online in July 2006 with corrections to the Cover Date.     

Evaluation of the three-dimensional elastic Green's function in anisotropic cubic media

By using the Fourier transforms method, the three-dimensional Green's function solution for a unit force applied in an infinite cubic material is evaluated in this paper. Although the elastic behavior of a cubic material can be characterized by only three elastic constants, the explicit solutions of Green's function for a cubic material are not available in the literatures. The central problem for explicitly solving the elastic Green's function of anisotropic materials depends upon the roots of a sextic algebraic equation, which results from the inverse Fourier transforms and is composed of the material constants and position vector parameters. The close form expression of Green's function is presented here in terms of roots of the sextic equation. The sextic equation for an anisotropic cubic material is discussed thoroughly and specific results are given for possible explicit solutions.     

Elastostatic Green’s functions for an arbitrary internal load in a transversely isotropic bi-material full-space

The problem of a full-space which is composed of two half-spaces with different transversely isotropic materials with an internal load at an arbitrary distance from the interface is considered. By virtue of Hu-Nowacki-Lekhnitskii potentials, the equations of equilibrium are uncoupled and solved with the aid of Hankel transform and Fourier decompositions. With the use of the transformed displacement- and stress-potential relations, all responses of the bi-material medium are derived in the form of line integrals. By appropriate limit processes, the solution can be shown to encompass the cases of (i) a homogeneous transversely isotropic full-space, and (ii) a homogeneous transversely isotropic half-space under arbitrary surface load. As the integrals for the displacement- and stress-Green’s functions, for an arbitrary point load can be evaluated explicitly, illustrative results are presented for the fundamental solution under different material anisotropy and relative moduli of the half-spaces and compared with existing solutions.