THE ELASTIC FIELD INDUCED BY A HEMISPHERICAL INCLUSION IN THE HALF—SPACE

The elastic field induced by a hemispherical inclusion with uniform eigeustralns in asemi-infinite elastic medium is solved by using the Green‘s function method and series expansion tech-nique. The exact solutions axe presented for the displacement and stress fields which can be expressedby complete elliptic integrals of the first, second, and third kinds and hypergeometric functions. Thepresent method can be used to determine the corresponding elastic fields when the shape of the inclusionis a spherical crown or a spherical segment. Finally, numerical results axe given for the displacementand stress fields along the axis of symmetry (x3-axis).     

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.     

The elastic field caused by a general ellipsoidal inclusion and the application to martensite formation 2

The elastic field throughout an ellipsoidal inclusion in an indefinitely-extended anisotropic material is investigated when an eigenstrain (a stress-free transformation strain) is periodically distributed throughout the inclusion. This is an extention of the results obtained by J.D. Eshelby (1961) for uniform eigenstrains and by R.J. Asaro and D.M. Barnett (1975) for polynomial eigenstrains. The solution is applied to the evaluation of elastic strain energies of a disc-shaped martensite with alternating twins and of a spherical precipitate with a banded structure. The significant amount of the elastic strain energies explains the necessity of the supercooling of austenite steel for the martensitic transformation to occur.     

Special properties of Eshelby tensor for a regular polygonal inclusion

When studying the regular polygonal inclusion in 1997, Nozaki and Taya discovered numerically some remarkable properties of Eshelby tensor: Eshelby tensor at the center and the averaged Eshelby tensor over the inclusion domain are equal to that of a circular inclusion and independent of the orientation of the inclusion. Then Kawashita and Nozaki justified the properties mathematically. In the present paper, some other properties of a regular polygonal inclusion are discovered. We find that for an N-fold regular polygonal inclusion except for a square, the arithmetic mean of Eshelby tensors at N rotational symmetrical points in the inclusion is also equal to the Eshelby tensor for a circular inclusion and independent of the orientation of the inclusion. Furthermore, in two corollaries, we point out that Eshelby tensor at the center, the averaged Eshelby tensor over the inclusion domain, and the line integral average of Eshelby tensors along any concentric circle of the inclusion are all identical with the arithmetic mean.The project supported by the National Natural Science Foundation of China (10172003 and 10372003) The English text was polished by Keren Wang.     

The determination of elastic parameters of a half-space using a monochromatic vibration field at the surface

The paper presents a method to determine Lamé parameters λ, μ and density ϱ in a layered half-space, using monochromatic vibrations of its surface, excited by a harmonic source which is assumed to be known. The equations governing the vibrations are reduced to the Sturm-Liouville problem in scalar form for Love-type displacements, and in matrix form for the Rayleigh type. The scalar and matrix potentials of the Sturm-Liouville equations can be recovered from the corresponding impedance. Explicit formulas are given to construct the potentials by amplitudes and wavenumbers of normal (progressive) modes and attenuated standing waves. The potentials then are used to determine the elastic parameters and the density. The method can also be used for the acoustic equation.     

The stress field of a sliding inclusion

This paper discusses the stress fields when a spheroidal inclusion, free to slip along its interface, is subjected to a constant nonshear eigenstrain, and when an elastic body containing the inhomogeneity is under all-around tension or uniaxial tension at infinity. In each case the stress field in the inclusion or the inhomogeneity is not constant, contrary to Eshelby's solution. When sliding takes place, the stress increases locally compared with the perfect bonding case, but the elastic energy decreases due to the relaxation. The relative displacement (slip) along the interface is also evaluated.     

Three-dimensional elastic displacements induced by a dislocation of polygonal shape in anisotropic elastic crystals

Dislocations and the elastic fields they induce in anisotropic elastic crystals are basic for understanding and modeling the mechanical properties of crystalline solids. Unlike previous solutions that provide the strain and/or stress fields induced by dislocation loops, in this paper, we develop, for the first time, an approach to solve the more fundamental problem—the anisotropic elastic dislocation displacement field. By applying the point-force Green’s function for a three-dimensional anisotropic elastic material, the elastic displacement induced by a dislocation of polygonal shape is derived in terms of a simple line integral. It is shown that the singularities in the integrand of this integral are all removable. The proposed expression is applied to calculate the elastic displacements of dislocations of two different fundamental shapes, i.e. triangular and hexagonal. The results show that the displacement jump across the dislocation loop surface exactly equals the assigned Burgers vector, demonstrating that the proposed approach is accurate. The dislocation-induced displacement contours are also presented, which could be used as benchmarks for future numerical studies.     

Size dependent,non-uniform elastic field inside a nano-scale spherical inclusion due to interface stress

The primary objective of the present paper is to analyze the influence of interface stress on the elastic field within a nano-scale inclusion. Special attention is focused on the case of non-hydrostatic eigenstrain. From the viewpoint of practicality, it is assumed that the inclusion is spherically shaped and embedded into an infinite solid, within which an axisymmetric eigenstrain is prescribed. Following Goodier’s work, the elastic fields inside and outside the inclusion are obtained analytically. It is found that the presence of interface stress leads to conclusion that the elastic field in the inclusion is not only dependent on inclusion size but also on non-uniformity. The result is in strong contrast to Eshelby’s solution based on classical elasticity, and it is helpful in the understanding of relevant physical phenomena in nano-structured solids.     

A novel hybrid finite element analysis of inplane singular elastic field around inclusion corners in elastic media

This paper deals with the inplane singular elastic field problems of inclusion corners in elastic media by an ad hoc hybrid-stress finite element method. A one-dimensional finite element method-based eigenanalysis is first applied to determine the order of singularity and the angular dependence of the stress and displacement field, which reflects elastic behavior around an inclusion corner. These numerical eigensolutions are subsequently used to develop a super element that simulates the elastic behavior around the inclusion corner. The super element is finally incorporated with standard four-node hybrid-stress elements to constitute an ad hoc hybrid-stress finite element method for the analysis of local singular stress fields arising from inclusion corners. The singular stress field is expressed by generalized stress intensity factors defined at the inclusion corner. The ad hoc finite element method is used to investigate the problem of a single rectangular or diamond inclusion in isotropic materials under longitudinal tension. Comparison with available numerical results shows the present method is an efficient mesh reducer and yields accurate stress distribution in the near-field region. As applications, the present ad hoc finite element method is extended to discuss the inplane singular elastic field problems of a single rectangular or diamond inclusion in anisotropic materials and of two interacting rectangular inclusions in isotropic materials. In the numerical analysis, the generalized stress intensity factors at the inclusion corner are systematically calculated for various material type, stiffness ratio, shape and spacing position of one or two inclusions in a plate subjected to tension and shear loadings.     

Deformation field of a misfitting inclusion

J.D. Eshelby (1957, 1959) has calculated the deformation field associated with an ellipsoidal inclusion in a state of homogeneous strain within an infinite matrix. Since most real precipitates occur with facets, the strain within such an inclusion is not uniform. Thus, plate precipitates of θ′ in Al-Cu and η in Al-Au have coherent broad faces with mismatches of 1.34 and 4.95 % respect- ively and semicoherent or disordered interfaces at the edges with residual mismatches of about ?4.3 and ?1.00% normal to the broad faces. The deformation field in the matrix around such precipitates has been calculated using Kelvin's (1848) result for the stress field due to a point force. The calculations show the existence of high stresses near the edges of the precipitates where they have an appreciable misfit. Unlike the case of an ellipsoidal inclusion, the stress fields of these precipitates have dilatational components which can affect the diffusion of solute atoms to them and, thus, the kinetics of interface migration. The behavior of alloys containing these precipitates indicates that the moduli of the precipitates are somewhat greater than those of the matrices. The present calculations, based on the assumption that the two moduli are the same, underestimate the actual deformation field in the matrix. In real systems, therefore, the effects of the deformation field on misfit dislocation nucleation and kinetics of interface migration are likely to be somewhat greater in general.     

Interaction of plane elastic harmonic waves with a perfectly bonded elastic inclusion

The interaction of plane harmonic waves with a thin elastic inclusion in the form of a strip in an infinite body (matrix) under plane strain conditions is studied. It is assumed that the bending and shear displacements of the inclusion coincide with the displacements of its midplane. The displacements in the midplane are found from the theory of plates. The priblem-solving method represents the displacements as discontinuous solutions of the Lamé equations and finds the unknown discontinuities solving singular integral equations by the numerical collocation method. Approximate formulas for the stress intensity factors at the ends of the inclusion are derived     

Bending of elastic plates with a physically nonlinear inclusion

An infinite elastic isotropic plate with an elliptical, physically nonlinear inclusion loaded at infinity by uniformly distributed moments is considered. Surface loads are absent. The problem of the stress-strain state of the plate is solved in a closed form. It is shown that, for reasonably general stress-strain relations for the inclusion, the bending-moment field (and the corresponding curvatures) in the inclusion is homogeneous. __________ Translated from Prikladnaya Mekhanika i Tekhnicheskaya Fizika, Vol. 47, No. 6, pp. 152–157, November–December, 2006.     

The torsional oscillations of a rigid spherical inclusion embedded in an elastic stratum

In this paper we investigate the axisymmetric harmonic torsional vibrations of a rigid spherical inclusion, embedded centrally in an elastic stratum of finite thickness. For an infinite solid the problem has a simple closed form solution. Here the stratum problem is formulated as a Fredholm integral equation of the first kind, and a perturbation method employed to solve this equation approximately for large values of the ratio of sphere radius to stratum depth. Approximations are given for quantities of physical interest such as the stress distribution at the inclusion-elastic medium interface, and the stress couple on the sphere.

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Résumé Dans cette étude on recherche les vibrations axisymmétriques harmoniques et torsionales d'une inclusion sphérique et raide, située centralement dans une couche d'une épaisseur bornée. Pour un solide infini le problème a une solution élémentaire.On formule le problème de couche comme une équation intégrale de Fredholm de la première sorte. On se sert d'une méthode de perturbation pour résoudre cette équation d'une façon approximative pour les grandes valeurs de la proportion du rayon de la sphère contre la profondeur de la couche. Les approximations sont calculées pour la distribution d'effort employée sur l'inclusion et pour la couple de torsion employée sur la sphère.

     

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.     

Effect of interface stresses on the elastic deformation of an elastic half-plane containing an elastic inclusion

The effect of the interface stresses is studied upon the size-dependent elastic deformation of an elastic half-plane having a cylindrical inclusion with distinct elastic properties. The elastic half-plane is subjected to either a uniaxial loading at infinity or a uniform non-shear eigenstrain in the inclusion. The straight edge of the half-plane is either traction-free, or rigid-slip, or motionless, which represents three practical situations of mechanical structures. Using two-dimensional Papkovich–Neuber potentials and the theory of surface/interface elasticity, the elastic field in the elastic half-plane is obtained. Comparable with classical result, the new formulation renders the significant effect of the interface stresses on the stress distribution in the half-plane when the radius of the inclusion is reduced to the nanometer scale. Numerical results show that the intensity of the influence depends on the surface/interface moduli, the stiffness ratio of the inclusion to the surrounding material, the boundary condition on the edge of the half-plane and the proximity of the inclusion to the edge.     

The plane frictionless contact of two elastic bodies — the inclusion problem

Summary The mixed boundary value problem of the contact of two plane elastic bodies of arbitrary shape is solved for zero friction in their contact zone. It is reduced to a system of four singular integral equations referred to the contact zone and the remaining parts of the boundaries of the two bodies. The system is complemented by two more equations derived from the single-valuedness of the displacements along the contact boundaries. The solution of these equations yields the distribution of the contact stresses and the contact length. The method is applied to the symmetric case of an infinite elastic plate containing an oversized elastic inclusion, with and without axial forces applied to the plate at infinity. The evolution of contact relaxation and the progress of the gap between the inclusion and the plate is also given.

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Der ebene reibungslose Kontakt von zwei elastischen Körpern — Das Problem der Einlagerung

Übersicht Das gemischte Randwertproblem der Berührung zweier ebener elastischer Körper von beliebiger Form wird im Falle verschwindender Reibung in der Kontaktzone gelöst. Das Problem wird reduziert auf ein System von vier singulären Integralgleichungen, welche sich auf die Kontaktzone und die übrigen Ränder der zwei Körper beziehen. Das System wird mit zwei weiteren Gleichungen vervollständigt, welche von der Eindeutigkeit der Verschiebungen längs der berührenden Ränder hergeleitet werden. Die Lösung dieser Gleichungen gibt die Verteilung der Kontaktspannungen und die Kontaktlänge. Die Methode wird auf den symmetrischen Fall einer unendlichen elastischen Platte (mit und ohne axiale Kräfte auf den unendlich fernen Rändern), welche eine elastische Einlagerung mit Übermaß enthält, angewandt. Die Entwicklung der Kontaktauflösung und der Vorschrift der Lücke zwischen Einlagerung und Platte werden mit Hilfe der Lösung obiger Gleichungen angegeben.