An approximate solution of the interaction between an edge dislocation and an inclusion of arbitrary shape

An approximate solution of the interaction force between an edge dislocation and an inclusion of arbitrary shape is derived, from which a set of succinct formulas for several special inclusion shapes are obtained. As compared with several classical solutions to special inclusion shapes, the present approximate solution has fairly good accuracy.     

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.     

Edge dislocation interacting with a nonuniformly coated circular inclusion

This paper investigated the interaction between an edge dislocation and a nonuniformly coated circular inclusion. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with alternating technique, the solutions to plane elasticity problems for three dissimilar media are derived explicitly in a series form. For a limiting case when the thickness of the interphase layer is uniform, the derived analytical solutions of this paper are reduced to exactly the same results available in the literature. The image force acting on the dislocation is then determined by using the Peach–Koehler formula. It is found that the combination of material constants and nonuniformity of the interphase thickness will exert a significant influence on the dislocation force.     

Interaction of an edge dislocation with a coated elliptic inclusion

This paper presents an analytical solution for plane elasticity problems of an elliptically cylindrical layered media subject to an arbitrary edge dislocation. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the general expressions of the displacements and stresses, where an edge dislocation is located in matrix, coating layer and inclusion are obtained. The numerical results of image forces exerted on a generalized edge dislocation are carried out by using the generalized Peach–Koehler equation. As a numerical illustration, both the image forces and equilibrium positions are presented for different material combinations and relative thickness of a coating layer. The result shows that the thickness and the shear modulus of the coating layer have a strong influence on the stability of dislocation.     

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.     

Vibrations of a free-edge variable-thickness plate of arbitrary shape in plan

A numerical algorithm without saturation which provides reliable results on a coarse grid was developed to solve the problem of free vibrations of a free-edge variable-thickness plate of arbitrary shape in plan. The results were compared with the results of calculations performed in other studies.     

Hamiltonian structure for a neutrally buoyant rigid body interacting with N vortex rings of arbitrary shape: the case of arbitrary smooth body shape

We present a (noncanonical) Hamiltonian model for the interaction of a neutrally buoyant, arbitrarily shaped smooth rigid body with N thin closed vortex filaments of arbitrary shape in an infinite ideal fluid in Euclidean three-space. The rings are modeled without cores and, as geometrical objects, viewed as N smooth closed curves in space. The velocity field associated with each ring in the absence of the body is given by the Biot–Savart law with the infinite self-induced velocity assumed to be regularized in some appropriate way. In the presence of the moving rigid body, the velocity field of each ring is modified by the addition of potential fields associated with the image vorticity and with the irrotational flow induced by the motion of the body. The equations of motion for this dynamically coupled body-rings model are obtained using conservation of linear and angular momenta. These equations are shown to possess a Hamiltonian structure when written on an appropriately defined Poisson product manifold equipped with a Poisson bracket which is the sum of the Lie–Poisson bracket from rigid body mechanics and the canonical bracket on the phase space of the vortex filaments. The Hamiltonian function is the total kinetic energy of the system with the self-induced kinetic energy regularized. The Hamiltonian structure is independent of the shape of the body, (and hence) the explicit form of the image field, and the method of regularization, provided the self-induced velocity and kinetic energy are regularized in way that satisfies certain reasonable consistency conditions.      

Stress state of a transversely-isotropic body with elliptical inclusion

We present an exact solution for the problem in elasticity theory of a transversely Isotropic body containing an elliptical inclusion. We assume that the tensile stresses act at a distance sufficiently far away from the inclusion, along the axes of the ellipse and perpendicular to the plane of the ellipse. We find that two fracture mechanisms are possible under the action of the type of force under consideration: detachment of the material from the inclusion, and fracture near the stress concentrator. We obtain formulas for the stress intensity factors for each case.     

Stress state of a transversely isotropic medium with an anisotropic inclusion. Arbitrary linear force field at infinity

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27, No. 7, pp. 25–35, July, 1991.     

Elastic field perturbation by a misfitting ring inclusion

The plane elasticity problem of a circular ring inhomogeneity, with either a hollow or a rigid core, is confronted. A solution is obtained under a wide class of loading conditions, the main limitation being that internal stress sources (if any) are located in the matrix. In order to take into account interfacial residual stresses, misfit between matrix and inhomogeneity is allowed. Loading by misfit alone, by uniform remote stresses, and by an edge dislocation, are explicitly treated as special cases.     

Scattering of monochromatic elastic waves on a planar crack of arbitrary shape

Scattering of monochromatic elastic waves on an isolated planar crack of arbitrary shape is considered. The 2D-integral equation for the crack opening vector is discretized by Gaussian approximating functions. For such functions, the elements of the matrix of the discretized problem have forms of standard one-dimensional integrals that can be tabulated. For regular grids of approximating nodes, the matrix of the discretized problem has the Toeplitz structure, and the corresponding matrix–vector products can be calculated by the fast Fourier transform technique. The latter strongly accelerates the process of iterative solution of the discretized problem. Examples of calculations of crack opening vectors, dynamic stress-intensity factors, and differential cross-sections of circular (penny-shaped) and non-circular cracks for various incident wave fields are presented. For a penny-shaped crack and longitudinal incident waves normal to the crack plane, an efficient semi-analytical method of the solution of the scattering problem is developed. The results of both methods are compared in a wide frequency region of the incident field.     

Fluid sloshing in an ice-hole of arbitrary shape

A numerical method is proposed for determining the natural frequencies and modes of the small oscillations of an ideal fluid in a half-space bounded above by a rigid plane with an aperture of arbitrary shape. Considering the monotonic dependence of the eigenvalues on the geometry, it can be stated that the eigenvalues for a half-space are universal upper limits for the corresponding eigenvalues of tanks with an arbitrary boundary but the same free surface.Translated from Izvestiya Rossiiskoi Akademii Nauk, Mekhanika Zhidkosti i Gaza, No.4, pp. 108–112, July–August, 1992.     

Stress state of a transversely isotropic medium with an anisotropic spheroidal inclusion. Arbitrary uniform stress field at infinity

Institute of Mechanics, Academy of Sciences of the Ukrainian SSR, Kiev. Translated from Prikladnaya Mekhanika, Vol. 27. No. 6, pp. 22–30, June, 1991.     

A crack of arbitrary shape inside an infinite transversely isotropic cylinder under arbitrary load

Summary  In the first part of the article an infinite circular cylinder is considered, made of transversely isotropic elastic material and weakened by a plane crack perpendicular to its axis O z. The crack is opened by an arbitrary normal stress. The second part is devoted to the same crack loaded by an arbitrary tangential stress. The complete solution in both cases is presented as a sum of the solution of a similar problem of a crack in an infinite space and an integral transform term, the parameters of which are determined from a set of linear algebraic equations derived from the boundary conditions. Governing integral equations with respect to the yet unknown crack displacement discontinuities are obtained. In the case of a circular crack, these equations can be inverted and solved by the method of consecutive interations. Received 30 November 2000; accepted for publication 3 May 2001     

Diffusion to a reacting particle of arbitrary shape having a liquid flowing around it

The field of concentrations and the rate of diffusion toward a drop or particle of arbitrary shape situated in a stream of liquid are determined for the case in which a chemical action of the first order is taking place on the surface. The solution is derived by matching asymptotic expansions in terms of the Péclet number; the Reynolds number is regarded as being small. In the particular case of an infinitely high chemical reaction rate the results derived in an earlier paper [1] follow directly from this solution. On the other hand, for a particle of spherical form the results agree with the solution presented in [2].     

Theory of scattering from multilayered bodies of arbitrary shape

Green 's functions and boundary integral equation methods are used to derive a matrix set of equations for scattering from a multilayered homogeneous elastic body embedded in an infinite elastic material. The surfaces separating the layers have arbitrary shape. The formalism for the three-layer material is derived in detail and generalized to N-layers. A matrix factorization method (MFM) is shown to considerably simplify the computational problem. The relation to the problems of acoustic waves in fluids and electromagnetic waves in a dielectric material is briefly indicated.     

Bending of a stretched plate containing an eccentrically plate reinforced hole of arbitrary shape

In this paper we consider the problem of a stretched plate containing a hole of arbitrary shape which is reinforced by thickening the plate, on one side only, in a region surrounding the hole. Due to the eccentricity of the reinforcement a bending boundary layer occurs in the neighbourhood of the junction between the plate and the reinforcement. The equations for the moments at the junction are found to be identical to those for the circular hole in Ref. [1]. The boundary layer occurring at a clamped edge of arbitrary shape is also discussed.