Elliptical Inhomogeneity Research Articles

Elastic Fields of Interacting Elliptical Inhomogeneities for Two-Dimensional Problems Based on the Equivalent Inclusion Method

In this paper, the equivalent inclusion method is used to calculate the elastic fields of a two-dimensional plate containing any number of ellipitical inhomogeneities. Both the interior and the exterior Eshelbys tensors are used in this method. Numerical examples are given to assess the performance of the presented method. The solutions obtained with this method have been checked and confirmed by the finite element analysis results.

Null-field integral approach for the piezoelectricity problems with multiple elliptical inhomogeneities

Based on the successful experience of solving anti-plane problems containing multiple elliptical inclusions, we extend to deal with the piezoelectricity problems containing arbitrary elliptical inhomogeneities. In order to fully capture the elliptical geometry, the keypoint of the addition theorem in terms of the elliptical coordinates is utilized to expand the fundamental solution to the degenerate kernel and boundary densities are simulated by the eigenfunction expansion. Only boundary nodes are required instead of boundary elements. Therefore, the proposed approach belongs to one kind of meshless and semi-analytical methods. Besides, the error stems from the number of truncation terms of the eigenfunction expansion and the convergence rate of exponential order is better than the linear order of the conventional boundary element method. It is worth noting that there are Jacobian terms in the degenerate kernel, boundary density and contour integral. However, they would cancel each other out in the process of the boundary contour integral. As the result, the orthogonal property of eigenfunction is preserved and the boundary integral can be easily calculated. For verifying the validity of the present method, the problem of an elliptical inhomogeneity in an infinite piezoelectric material subject to anti-plane shear and in-plane electric field is considered to compare with the analytical solution in the literature. Besides, two circular inhomogenieties can be seen as a special case to compare with the available data by approximating the major and minor axes. Finally, the problem of two elliptical inhomogeneities in an infinite piezoelectric material is also provided in this paper.

Solution of the plane problem for anisotropic media containing an elliptic inhomogeneity with dislocation-like interface

A general form solution of the plane problem for anisotropic media containing an elliptic inhomogeneity with a dislocation-like interface is presented. It is based on the complex series expansion of the stress functions. A general procedure for the determination of coefficients in the series using a continuity condition for the stresses at the imperfect interface and a discontinuity condition for a jump in the normal or tangential displacements at the interface is illustrated. The jump in the displacement components is described by the dislocation-like model based on the assumption that discontinuity of displacement across the interface is linearly proportional to the elastic displacement at the interface in the inhomogeneity. The model can reasonably characterize the imperfect interface from perfect bonding to complete debonding due to separate or combined effect of eigenstrains and far-field tension. Convergence of the results is obtained by truncating a finite number of terms in the series. The present solution is verified with available analytical results for the case of a perfect interface. In this paper, the pattern for the stresses in the heterogeneous anisotropic materials is shown. The effect of imperfect parameters on the distributions of stresses is also discussed. The method and procedure proposed in the paper are useful in analyzing the strength and failure of anisotropic materials containing an inhomogeneity with imperfect interface.

Interface Cracks in Kirchhoff Anisotropic Thin Plates of Dissimilar Materials

This paper investigates the problem of stretching and bending deformations of a Kirchhoff anisotropic thin plate composed of two dissimilar materials bonded along a straight interface containing a crack. Our analysis makes use of the Stroh octet formalism developed recently by Cheng and Reddy (Cheng and Reddy, 2002, “Octet Formalism for Kirchhoff Anisotropic Plates,” Proc. R. Soc. Lond., A458, pp. 1499–1517; Cheng and Reddy, 2003, “Green’s Functions for Infinite and Semi-Infinite Anisotropic Thin Plates,” ASME J. Appl. Mech., 70, pp. 260–267; Cheng and Reddy, 2004, “Laminated Anisotropic Thin Plate With an Elliptic Inhomogeneity,” Mech. Mater., 36, pp. 647–657; Cheng and Reddy, 2005, “Structure and Properties of the Fundamental Elastic Plate Matrix,” J. Appl. Math. Mech., 85, pp. 721–739) for Kirchhoff anisotropic plates. It is found that the interfacial crack-tip field consists of a pair of two-dimensional oscillatory singularities, which are explicitly determined. Two complex intensity factors are proposed to evaluate the two oscillatory singularities.

On the Solution of an Elliptical Inhomogeneity in Plane Elasticity by the Equivalent Inclusion Method

Stress analysis of an elliptical inhomogeneity in an infinite isotropic elastic plane is a classical elasticity problem, which is usually solved by means of the complex variable formulation. In this work, we demonstrate that an alternative method of solution for such a problem, via the equivalent inclusion method, may be more convenient and straightforward without recourse to complex potentials or curvilinear coordinates. The explicit analytical solution can be derived through simple algebraic manipulation, although the longitudinal eigenstrain component should be handled with care in the case of plane strain. Since the ex- terior Eshelby tensor for an elliptical inclusion is available in closed-form, the present study provides a full field stress solution expressed in Cartesian coordinates. Furthermore, the in-plane stress components are represented in terms of Dundurs' parameters. The solution methodology and the convenient formulae of the stress concentration may be of practical use to the engineers in developing benchmarks for design evaluation.

A piezoelectric screw dislocation near an elliptical inhomogeneity containing a confocal rigid line

The interaction between a piezoelectric screw dislocation and an elliptical inhomogeneity in piezoelectric composite material which contains an electrically conductive confocal rigid line is studied, especially analyzing the shielding effect of a piezoelectric screw dislocation near an elliptical inhomogeneity. By applying the complex variable method, the analytical solution to the elastic field and the electric field, the field intensity factors at the tip of the rigid line are derived. The image force acting on the piezoelectric screw dislocation is calculated by using the generalized Peach–Koehler formula. Accordingly, the location and the orientation of the dislocation, the material properties upon the shielding or anti-shielding effect on the stress intensity factors, as well as the effects of the rigid line and the electroelastic properties of the piezoelectric materials on the image force are discussed.

Screw dislocation and elliptic inhomogeneity of a confocal crack in magnetoelectroelastic medium: Comparison of energy release rate and strain energy density

This work is devoted to investigate the magnetoelectroelastic interaction between a generalized screw dislocation and an elliptic inhomogeneity containing a confocal crack in piezoelectric/piezomagnetic composite subjected to remote anti-plane shear stress field, in-plane electric and magnetic field. By using the complex variable method of elasticity, the closed-form expressions of complex potentials of matrix and inhomogeneity are obtained for the dislocation locating both in matrix and inhomogeneity. The expressions of the generalized stress/strain field, image force, the generalized stress intensity factor and energy release rate of crack tip, and strain energy density are derived explicitly. Then, the influence laws of material parameters, the shape of elliptic inhomogeneity and remote loading on these quantities are analyzed. The results show that the image force has different variation laws in magnetoelectroelastic materials with that in elastic materials; stress intensity factor has the same distributing law as electric displacement intensity factor, but is different from magnetic induction intensity factor; the energy release rate (ERR) can be positive and negative depending on the combined action of applied fields, the in-plane electric field and magnetic field. This makes the ERR interpretation unphysical. ERR is not permitted to change sign. The strain energy density function is shown to be positive definite under all conditions.

Conduction degradation in anisotropic multi-cracked materials

The electrical and thermal conduction properties of disordered solids and the possible degradation processes induced by the generation of cracks are central issues in the field of the heterogeneous materials. However, most of the existing theories are unable to consider an arbitrary density of cracks. We obtained an exact result for the fields induced within an elliptic anisotropic inhomogeneity embedded in a different anisotropic (two-dimensional) conductor. Then, we applied it to show that the degradation process strongly depends on the statistical orientational distribution of defects: in particular we theoretically prove that parallel cracks lead to the power law decay log σ ∼ − log N while random oriented cracks lead to the exponential law decay log σ ∼ −N (where σ is the effective conductivity of a region with a large number N of defects), as recently predicted by numerical findings.

Size-dependent interaction of an edge dislocation with an elliptical nano-inhomogeneity incorporating interface effects

The elastic behavior of an edge dislocation, which is positioned outside of a nanoscale elliptical inhomogeneity, is studied within the interface elasticity approach incorporating the elastic moduli and surface tension of the interface. The complex potential function method is used. The dislocation stress field and the image force acting on the dislocation are found and analyzed in detail. The difference between the solutions obtained within the classical-elasticity and interface-elasticity approaches is discussed. It is shown that for the stress field, this difference can be significant in those points of the inhomogeneity-matrix interface, where the radius of curvature is smaller and which are closer to the dislocation. For the image force, this difference can be considerable or dispensable in dependence on the dislocation position, its Burgers vector orientation, and relations between the elastic moduli of the matrix, inhomogeneity and their interface. Under some special conditions, the dislocation can occupy a stable equilibrium position in atomically close vicinity of the interface. The size effect is demonstrated that the normalized image force strongly depends on the inhomogeneity size when it is in the range of several tens of nanometers, in contrast with the classical solution where this force is always constant. The general issue is that the interface elasticity effects become more evident when the characteristic sizes of the problem (inhomogeneity size, interface curvature radius and dislocation-interface spacing) reduce to the nanoscale.

A Piezoelectric Screw Dislocation Interacting with an Elliptical Piezoelectric Inhomogeneity Containing a Confocal Elliptical Rigid Core

The electro-elastic interaction between a piezoelectric screw dislocation and an elliptical piezoelectric inhomogeneity, which contains an electrically conductive confocal elliptical rigid core under remote anti-plane shear stresses and in-plane electrical load is dealt with. The analytical solutions to the elastic field and the electric field, the interfacial stress fields of inhomogeneity and matrix under longitudinal shear and the image force acting on the dislocation are derived by means of complex method. The effect of material properties and geometric configurations of the rigid core on interfacial stresses generated by a remote uniform load, rigid core and material electroelastic properties on the image force is discussed.

About the Porous Media Flow Through Circular and Elliptical Anisotropic Inhomogeneities

In this article, we describe horizontal groundwater flow due to a uniform flow at infinity around a cylindrical or elliptical inhomogeneity, where the permeability inside the inhomogeneity is anisotropic and different from the isotropic domain outside the inhomogeneity. The orientation of the uniform flow with respect to the orientation of the ellipse is arbitrary as well as the orientation of the anisotropy inside the ellipse. We derive an expression for the ratio of the flow through the ellipse with respect to the flow in the homogeneous case.

The Interaction between a Screw Dipole in Matrix and an Elliptical Inhomogeneity Containing a Confocal Crack

The interaction between a screw dipole inside the matrix and an elastic elliptical inhomogeneity containing a confocal crack under longitudinal shear is investigated. By using the complex method of elasticity, singularity analysis of stress function and Riemann boundary problem, the complex functions of matrix and inhomogeneity are obtained in the series form. And the singularity stress field of matrix, the image force and image torque acting on the center of screw dipole and the stress intensity factor of crack tip are deduced. Then the influence of the dip angle of screw dipole, the size of inhomogeneity and the misfit of materials on image force, torque of couple and the stress intensity are analyzed. The results show that the image force varies periodically as and there is two critical value of to make image force getting its maximum and minimum; the decreasing of relative shear modulus and the increasing of inhomogeneity may lead to the increasing of image force; the variation of will change the direction of screw dipole on stress intensity factor.

A Magnetoelectric Screw Dislocation Interacting with an Elliptical Inhomogeneity Containing a Confocal Rigid Line

Based on the complex variable method, the magnetoelectroelastic interaction of a generalized screw dislocation with an elliptical inhomogeneity containing a electrically conductive confocal rigid line under remote anti-plane shear stresses, in-plane electric and magnetic loads is dealt with. The generalized screw dislocation is located inside either the inhomogeneity or the matrix. The analytical-functions of complex potentials for stresses, electric displacement fields and magnetic induction fields in both the inhomogeneity and the matrix are derived. The image force acting on the dislocation are also calculated explicitly. The results show that the influence of the rigid line on the interaction effect between a generalized screw dislocation and an elliptical inhomogeneity is significant. In addition, the material behavior also plays an important role on the image force.

Two-dimensional polynomial eigenstrain formulation of boundary integral equation with numerical verification

The low-order polynomial-distributed eigenstrain formulation of the boundary integral equation (BIE) and the corresponding definition of the Eshelby tensors are proposed for the elliptical inhomogeneities in two-dimensional elastic media. Taking the results of the traditional subdomain boundary element method (BEM) as the control, the effectiveness of the present algorithm is verified for the elastic media with a single elliptical inhomogeneity. With the present computational model and algorithm, significant improvements are achieved in terms of the efficiency as compared with the traditional BEM and the accuracy as compared with the constant eigenstrain formulation of the BIE.

Energy density and release rate study of an elliptic-arc interface crack in piezoelectric solid under anti-plane shear

The anti-plane problem of an elliptical inhomogeneity with an interfacial crack in piezoelectric materials is investigated. The system is subjected to arbitrary singularity loads (point charge and anti-plane concentrated force) and remote anti-plane mechanical and in-plane electrical loads. Using the complex variable method, the explicit series form solutions for the complex potentials in the matrix and the inclusion regions are derived. The electroelastic field intensity factors, the corresponding energy release rates and the generalized strain energy density at the cracks tips are then provided. The influence of the aspect ratio of the ellipse, the crack geometry and the electromechanical coupling coefficient on the energy release rate and the strain energy density is discussed and shown in graphs. The results indicate that the energy release rate increases with increment of the aspect ratio of the ellipse and the influence of electromechanical coupling coefficient on the energy release rate is significant. The strain energy density decreases with increment of the aspect radio of the ellipse and it is always positive for the cases discussed. The energy release rate, however, can be negative when both mechanical and fields are applied.

Analysis of a piezoelectric screw dislocation inside an elliptical inhomogeneity with confocal rigid line in piezoelectric material

This paper deals with the electro-elastic coupling interaction between a piezoelectric screw dislocation which is located inside the elliptical inhomogeneity and an electrically conductive confocal rigid line under remote anti-plane shear stresses and in-plane electrical loads in piezoelectric composite material. The analytical-functions of the complex potentials, stress fields and the image force acting on the piezoelectric screw dislocation are obtained based on the principle of conformal mapping, the method of series expansion, the technical of analytic continuation and the analysis of singularity of complex potentials. The rigid line and the piezoelectric material property combinations upon the image force and the equilibrium position of the dislocation are discussed in detail by the numerical computation.

Electroelastic interaction between a piezoelectric screw dislocation and a confocal elliptic blunt crack with elliptical inhomogeneity

The electroelastic interaction between a piezoelectric screw dislocation and an elliptical inhomogeneity containing a confocal blunt crack under infinite longitudinal shear and in-plane electric field is investigated. Using the sectionally holomorphic function theory, Cauchy singular integral, singularity analysis of complex functions and theory of Rieman boundary problem, the explicit series solution of stress field is obtained when the screw dislocation is located in inhomogeneity. The intervention law of the interaction between blunt crack and screw dislocation in inhomogeneity is discussed. The analytical expressions of generalized stress and strain field of inhomogeneity are calculated, while the image force, field intensity factors of blunt crack are also presented. Moreover, a new matrix expression of the energy release rate and generalized strain energy density (SED) are deduced. With the size variation of blunt crack, the results can be reduced to the case of the interaction between a piezoelectric screw dislocation and a line crack in inhomogeneity. Numerical analysis are then conducted to reveal the effects of the dislocation location, the size of inhomogeneity and blunt crack and the applied load on the image force, energy release rate and strain energy density. The influence of dislocation on energy release rate and strain energy density is also revealed.

The interaction between a screw dislocation and a rigid line in a confocal elliptical inhomogeneity

The interaction between a screw dislocation and an elastic elliptical inhomogeneity which contains a confocal rigid line is investigated. The screw dislocation is located inside either the elliptical inhomogeneity or the infinite matrix. By using the complex potential method, explicit series solutions of complex potentials are obtained. The image force acting on the screw dislocation and the stress intensity factor at the tip of the rigid line are derived. As a result, the analysis and discussion show that the influence of the rigid line on the interaction effects between a screw dislocation and an elliptical inhomogeneity is significant. The rigid line enhances the repulsive force exerted on the dislocation produced by the stiff inhomogeneity and abates the attractive force produced by the soft inhomogeneity. For the soft inhomogeneity, there is an unstable equilibrium position when the dislocation is inside the matrix and there is a stable equilibrium position when the dislocation is inside the inhomogeneity. The stress intensity factor contour around the rigid line tip shows that when a dislocation with positive burgers vector is in the upper half-plane, stress intensity factor will be positive; while in the lower half-plane, stress intensity factor will be negative; and in the x-axis, it will be zero. The absolute value of the stress intensity factor will increase when the dislocation approaches the tip of the rigid line. The stress intensity factor at the rigid line tip is enhanced by a harder matrix and abated by a softer matrix.

N -Phase Elliptical Inhomogeneities With Internal Uniform Stresses in Plane Elasticity

We investigate the problem of an N-phase elliptical inhomogeneity in plane elasticity. The elliptical inhomogeneity is bonded to the unbounded matrix through the intermediate (N−2) interphases, and the matrix is subjected to remote uniform stresses. We observe that the stress field inside the elliptical inhomogeneity is still uniform when the following two conditions are satisfied: (i) The formed interfaces are (N−1) confocal ellipses, and (ii) the interphases and the matrix possess the same shear modulus but different Poisson’s ratios. In Appendixes A xB, we also discuss an arbitrary number of interacting arbitrary shaped inhomogeneities embedded in an infinite matrix, and an N-phase inhomogeneity with (N−1) interfaces of arbitrary shape. Here all the phases comprising the composite possess the same shear modulus but different Poisson’s ratios. The results in the main body and in Appendixes A xB are further extended in Appendix C to finite plane strain deformations of compressible hyperelastic harmonic materials.

Time-Decaying Uniform Stresses Inside an Anisotropic Elliptical Inhomogeneity With Nonuniform Interfacial Slip

This study addresses the problem of an elastically anisotropic elliptical inhomogeneity bonded to an infinite elastically anisotropic matrix through a linear viscous interface. Our results show that uniform, as well as time-decaying stresses, still exist inside the elliptical inhomogeneity when the interface drag parameter, which is varied along the interface, is properly designed, and when the matrix is subjected to remote uniform antiplane shearing. Interestingly, the internal stresses decay with not one but two relaxation times. Some special cases are discussed in detail to demonstrate the obtained solutions.