Closed form solution and numerical analysis for Eshelby’s elliptic inclusion in plane elasticity

This paper presents a closed form solution and numerical analysis for Es- helby＇s elliptic inclusion in an infinite plate. The complex variable method and the confor- real mapping technique are used. The continuity conditions for the traction and displace- ment along the interface in the physical plane are reduced to the similar conditions along the unit circle of the mapping plane. The properties of the complex potentials defined in the finite elliptic region are analyzed. From the continuity conditions, one can separate and obtain the relevant complex potentials defined in the inclusion and the matrix. From the obtained complex potentials, the dependence of the real strains and stresses in the inclusion from the assumed eigenstrains is evaluated. In addition, the stress distribution on the interface along the matrix side is evaluated. The results are obtained in the paper for the first time.     

Two-dimensional Eshelby’s problem for piezoelectric materials with a parabolic boundary

We use complex variable techniques to obtain analytic solutions of Eshelby’s problem consisting of an inclusion of arbitrary shape in an anisotropic piezoelectric plane with a parabolic boundary. The region of the physical plane below the parabola is mapped onto the lower half of the image plane. The problem is then more conveniently studied in the image plane rather than in the physical plane. The critical step in our approach lies in the construction of certain auxiliary functions in the image plane which allow for the technique of analytic continuation to be applied to an inclusion of arbitrary shape.     

Strain gradient solution for a finite-domain Eshelby-type plane strain inclusion problem and Eshelby’s tensor for a cylindrical inclusion in a finite elastic matrix

A solution for the finite-domain Eshelby-type inclusion problem of a finite elastic body containing a plane strain inclusion prescribed with a uniform eigenstrain and a uniform eigenstrain gradient is derived in a general form using a simplified strain gradient elasticity theory (SSGET). The formulation is facilitated by an extended Betti’s reciprocal theorem and an extended Somigliana’s identity based on the SSGET and suitable for plane strain problems. The disturbed displacement field is obtained in terms of the SSGET-based Green’s function for an infinite plane strain elastic body, which differs from that in earlier studies using the three-dimensional Green’s function. The solution reduces to that of the infinite-domain inclusion problem when the boundary effect is suppressed. The problem of a cylindrical inclusion embedded concentrically in a finite plane strain cylindrical elastic matrix of an enhanced continuum is analytically solved for the first time by applying the general solution, with the Eshelby tensor and its average over the circular cross section of the inclusion obtained in closed forms. This Eshelby tensor, being dependent on the position, inclusion size, matrix size, and a material length scale parameter, captures the inclusion size and boundary effects, unlike existing ones. It reduces to the classical elasticity-based Eshelby tensor for the cylindrical inclusion in an infinite matrix if both the strain gradient and boundary effects are not considered. Numerical results quantitatively show that the inclusion size effect can be quite large when the inclusion is very small and that the boundary effect can dominate when the inclusion volume fraction is very high. However, the inclusion size effect is diminishing with the increase of the inclusion size, and the boundary effect is vanishing as the inclusion volume fraction becomes sufficiently low.     

General solution for Eshelby’s problem of 2D arbitrarily shaped piezoelectric inclusions

Eshelby’s problem of piezoelectric inclusions arises sometimes in exploiting the electromechanical coupling effect in piezoelectric media. For example, it intervenes in the nanostructure design of strained semiconductor devices involving strain-induced quantum dot (QD) and quantum wire (QWR) growth. Using the extended Stroh formalism, the present work gives a general analytical solution for Eshelby’s problem of two-dimensional arbitrarily shaped piezoelectric inclusions. The key step toward obtaining this general solution is the derivation of a simple and compact boundary integral expression for the eigenfunctions in the extended Stroh formalism applied to Eshelby’s problem. The simplicity and compactness of the boundary integral expression derived make it much less difficult to analytically tackle Eshelby’s piezoelectric problem for a large variety of non-elliptical inclusions. In the present work, explicit analytical solutions are obtained and detailed for all polygonal inclusions and for the inclusions characterized by Jordan’s curves and Laurent’s polynomials. By considering the piezoelectric material GaAs (110), the analytical solutions provided are illustrated numerically to verify the coincidence between different expressions, and to clarify the jump across the boundary of the inclusion and the singularity around the corner of the inclusion.     

Green’s functions for plane problem under various boundary conditions and applications

Green’s functions of a point dislocation as well as a concentrated force for the plane problem of an infinite plane containing an arbitrarily shaped hole under stress, displacement, and mixed boundary conditions are stated. The Green’s functions are obtained in closed forms by using the complex stress function method along with the rational mapping function technique, which makes it possible to deal with relatively arbitrary configurations. The stress functions for these problems consist of two parts: a principal part containing singular and multi-valued terms, and a complementary part containing only holomorphic terms. These Green’s functions can be derived without carrying out any integration. The applications of the Green’s functions are demonstrated in studying the interaction of debonding and cracking from an inclusion with a line crack in an infinite plane subjected to remote uniform tension. The Green’s functions should have many other potential applications such as in boundary element method analysis. The boundary integral equations can be simplified by using the Green’s functions as the kernels.     

Superposing scheme for the three-dimensional Green’s functions of an anisotropic half-space

This paper presents a method of superposition for the half-space Green’s functions of a generally anisotropic material subjected to an interior point loading. The mathematical concept is based on the addition of a complementary term to the Green’s function in an anisotropic infinite domain. With the two-dimensional Fourier transformation, the complementary term is derived by solving the generalized Stroh eigenrelation and satisfying the boundary conditions on the free surface with the use of Green’s functions in the full-space case. The inverse Fourier transform leads to the contour integrals, which can be evaluated with the application of Cauchy residue theorem. Application of the present results is made to obtain analytical expression for the orthotropic materials which were not reported previously. The closed-form solutions for the transversely isotropic and isotropic materials derived directly from the solutions as being a special case are also given in this paper.     

On Green’s function for a bimaterial elastic half-plane

The problem of a point force acting in a composite, two-dimensional, isotropic elastic half-plane is considered. An exact solution is obtained, using Mellin transforms and the Melan solution for a point force in a homogeneous half-plane.     

Time-harmonic Green’s functions for anisotropic magnetoelectroelasticity

Two-dimensional (2-D) and three-dimensional (3-D) time-harmonic Green’s functions for linear magnetoelectroelastic solids are derived in this paper by means of Radon-transform. Displacement field and electric and magnetic potentials in a fully anisotropic magnetoelectroelastic infinite solid due to a time-harmonic point force, point charge and magnetic monopole are obtained in form of line integrals over a unit circle in 2-D case and surface integrals over a unit sphere in 3-D case. This dynamic fundamental solution is then split into the sum of regular dynamic plus singular terms. The singular terms coincide with the Green’s functions for the static problem and may be further reduced to closed form expressions. The proposed Green’s functions can be used in the corresponding boundary element method (BEM) formulation.     

Some general properties of Eshelby’s tensor fields in transport phenomena and anti-plane elasticity

Consider an infinite thermally conductive medium characterized by Fourier’s law, in which a subdomain, called an inclusion, is subjected to a prescribed uniform heat flux-free temperature gradient. The second-order tensor field relating the gradient of the resulting temperature field over the medium to the uniform heat flux-free temperature gradient is referred to as Eshelby’s tensor field for conduction. The present work aims at deriving the general properties of Eshelby’s tensor field for conduction. It is found that: (i) the trace of Eshelby’s tensor field is equal to the characteristic function of the inclusion, independently of the latter’s shape; (ii) the isotropic part of Eshelby’s tensor field over the inclusion of arbitrary shape is identical to Eshelby’s tensor field over a 2D circular or 3D spherical inclusion; (iii) when the medium is made of an isotropic material and when the inclusion has some specific rotational symmetries, the value of the Eshelby’s tensor field evaluated at the inclusion gravity center and the symmetric average of Eshelby’s tensor fields are both equal to Eshelby’s tensor field for a 2D circular or 3D spherical inclusion. These results are then extended, with the help of a linear transformation, to the general case where the medium consists of an anisotropic conductive material. The method elaborated and results obtained by the present work are directly transposable to the physically analogous transport phenomena of electric conduction, dielectrics, magnetism, diffusion and flow in porous media and to the mathematically identical phenomenon of anti-plane elasticity.     

Stroh formalism in evaluation of 3D Green’s function in thermomagnetoelectroelastic anisotropic medium

The paper presents studies on the Green’s function for thermomagnetoelectroelastic medium and its reduction to the contour integral. Based on the previous studies the thermomagnetoelectroelastic Green’s function is presented as a surface integral over a half-sphere. The latter is then reduced to the double integral, which inner integral is evaluated explicitly using the complex variable calculus and the Stroh formalism. Thus, the Green’s function is reduced to the contour integral. Since the latter is evaluated over the period of the integrand, the paper proposes to use trapezoid rule for its numerical evaluation with exponential convergence. Several numerical examples are presented, which shows efficiency of the proposed approach for evaluation of Green’s function in thermomagnetoelectroelastic anisotropic solids.     

Numerical simulation of shock–vortex interaction in Schardin’s problem

Shock diffraction over a two-dimensional wedge and subsequent shock–vortex interaction have been numerically simulated using the AUSM $+$ + scheme. After the passage of the incident shock over the wedge, the generated tip vortex interacts with a reflected shock. The resulting shock pattern has been captured well. It matches the existing experimental and numerical results reported in the literature. We solve the Navier–Stokes equations using high accuracy schemes and extend the existing results by focussing on the Kelvin–Helmholtz instability generated vortices which follow a spiral path to the vortex core and on their way interact with shock waves embedded within the vortex. Vortex detection algorithms have been used to visualize the spiral structure of the initial vortex and its final breakdown into a turbulent state. Plotting the dilatation field we notice a new source of diverging acoustic waves and a lambda shock at the wedge tip.     

The plane elasticity problem of an isotropic wedge under normal and shear distributed loading––application in the case of a multi-material problem

The problem of a multi-material composite wedge under a normal and shear loading at its external faces is considered with a variable separable solution. The stress and displacement fields are determined using the equilibrium conditions for forces and moments and the appropriate Airy stress function. The infinite isotropic wedge under shear and normal distributed loading along its external faces is examined for different values of the order n of the radial coordinate r. The proposed solution is applied to the elastostatic problem of a composite isotropic k-materials infinite wedge under distributed loading along its external faces. Applications are made in the case of the two-materials composite wedge under linearly distributed loading along its external faces and in the case of a three-materials composite wedge under a parabolically distributed loading along its external faces.     

Green’s function for T-stress of semi-infinite plane crack

Green’s function for the T-stress near a crack tip is addressed with an analytic function method for a semi-infinite crack lying in an elastical, isotropic, and infinite plate. The cracked plate is loaded by a single inclined concentrated force at an interior point. The complex potentials are obtained based on a superposition principle, which provide the solutions to the plane problems of elasticity. The regular parts of the potentials are extracted in an asymptotic analysis. Based on the regular parts, Gre...     

Derivatives of the three-dimensional Green’s functions for anisotropic materials

A concise formulation is presented for the derivatives of Green’s functions of three-dimensional generally anisotropic elastic materials. Direct calculation for derivatives of the Green’s function on the Cartesian coordinate system is a common practice, which, however, usually leads to a complicated course. In this paper the Green’s function derived by Ting and Lee [Ting, T.C.T., Lee, V.G., 1997. The three-dimensional elastostatic Green’s function for general anisotropic linear elastic solids. The Quarterly Journal of Mechanics and Applied Mathematics 50 (3) 407–426] is extended to obtain the derivatives. Using a spherical coordinate system, the Green’s function can be shown as the composition of two independent functions, one depends only on the radial distance of the field point to the origin and the other is in spherical angles. The method of derivation is based on the total differential scheme and then takes its partial differentiation accordingly. With the application of Cauchy residue theorem, the contour integral can be evaluated in terms of the Stroh eigenvalues of a sextic equation. For the degenerate case, evaluation of residues at multiple poles is also given. Applications of the present result are made to examine the Green’s functions and stress components for isotropic and transversely isotropic materials. The results are in exact agreement with existing solutions.     

Galin’s problem for a spatial elastic wedge

We study a three-dimensional contact problem on the indentation of an elliptic punch into a face of a linearly elastic wedge. The wedge is characterized by two parameters of elasticity and its edge is subjected to the action of an additional concentrated force. The other face wedge is free from stresses. The problem is reduced to an integral equation for the contact pressure. An asymptotic solution of this equation is obtained which is effective for a given contact region fairly remote from the edge. Calculations are performed that allow one to evaluate the effect of a force applied outside the contact region on the contact pressure distribution. The problem under study is a generalization of L. A. Galin’s problem on a force applied outside a circular punch on an elastic half-space [1, 2]. In a special case of a wedge with an opening angle of 180° and zero contact ellipse eccentricity, the obtained asymptotic relation coincides with the expansion of Galin’s exact solution in a series. Problems of indentation of an elliptic punch into a spatial wedge with the face not loaded outside the contact region have been studied previously. For example, the paper [3] dealt with the case of a known contact region (asymptotic method) and the paper [4] considered the case of an unknown contact region (numerical method). The solution of Galin’s problem allowed the authors of [2] to reduce the contact problem on the interaction of several punches applied to a half-space to a system of Fredholm integral equations of the second kind (Andreikin-Panasyuk method). A topical direction in contact mechanics is the model of discrete contact as well as related problems on the interaction of several punches [2, 5–8]. The interaction of several punches applied to a face of a wedge can be treated in a similar manner and an asymptotic solution can be obtained for the case where a concentrated force is applied at an arbitrary point of this face beyond the contact region rather than on the edge.     

A resolution of Stewartson’s quarter-infinite plate problem

We revisit a problem originally considered by Stewartson in 1961: the incompressible, high-Reynolds-number flow past a quarter-infinite plate, with a leading edge that is perpendicular to, and a side edge that is parallel to, an undisturbed oncoming freestream. Particular emphasis is placed on the key region close to the side edge, where the flow is (superficially) three-dimensional, although the use of similarity variables reduces the dimensionality of the problem down to two. As noted by Stewartson, this problem has several intriguing features; it includes singularities and is also of a mixed parabolic type, with edge conditions influencing the solution in both directions across the flow domain. These features serve to greatly complicate the (numerical) solution process (the problem is of course also highly non-linear), and computation was clearly infeasible in 1961. In the present paper, a detailed computational study is presented, answering many of the questions that arose from the 1961 study. We present detailed numerical results together with asymptotic analyses of the key locations in the flow.