Green’s function for an elastic plane with a circular inclusion incorporating surface energy

An analytical solution in infinite series form for two circular cylindrical elastic inclusions embedded in an infinite matrix with two circumferentially inhomogeneous imperfect interfaces interacting with a circular Eshelby inclusion in anti-plane shear is derived by employing complex variable techniques. All of those coefficients in the series can be uniquely determined in a simple and transparent way. Numerical examples are given to illustrate the effect of imperfection and circumferential inhomogeneity of the two interfaces as well as the size, location and elastic properties of the two circular inclusions on the stress fields induced within the two circular inclusions and the Eshelby inclusion.

The plane deformation of an infinite elastic matrix enclosing a single circular inclusion incorporating stretching and bending resistance for the inclusion–matrix interface is revisited using a refined linearized version of the Steigmann–Ogden model. This refined version of the Steigmann–Ogden model differs from other linearized counterparts in the literature mainly in that the tangential force of the interface defined in this version depends not only on the stretch of the interface but also on the bending moment and initial curvature of the interface (the corresponding bending moment relies on the change in the real curvature of the interface during deformation). Closed-form results are derived for the full elastic field in inclusion–matrix structure induced by an arbitrary uniform in-plane far-field loading. It is identified that with this refined version of the Steigmann–Ogden model a uniform stress distribution could be achieved inside the inclusion for any non-hydrostatic far-field loading when R = 3 χ int / λ int (where R is the radius of the inclusion, while λ int and χ int are the stretching and bending stiffness of the interface). Explicit expressions are also obtained for the effective transverse properties of composite materials containing a large number of unidirectional circular cylindrical inclusions using, respectively, the dilute and Mori–Tanaka homogenization methods. Numerical examples are presented to illustrate the differences between the refined version and two typical counterparts of the Steigmann–Ogden model in evaluating the stress field around a circular nanosized inclusion and the effective properties of the corresponding homogenized composites.

The plastic relaxation of a shear crack situated at the interface inside a circular inclusion in an infinite matrix has been analyzed by treating it as a double pile-up of screw dislocations and the plastic zones at either tip of the crack as giant screw dislocations. The ratio of applied stress to yield stress and the magnitude of the Burgers vector of the giant screw dislocations which represent the relative displacement of the crack faces at the tips have been related to the crack parameters. Using a critical relative displacement of the crack faces at the tip of the crack as the criterion for brittle extension of the crack, the tendency of the shear crack to extend into the inclusion or into the matrix has been determined. The effect of shear modulus and size of the inclusion on the behaviour of the plastic zones at either tip of the crack has been discussed. Conclusions are made on the fracture behaviour of a circular inclusion in an infinite matrix.

This paper deals with the problems of flexural wave propagation in an infinite thin elastic plate with circular elastic inclusions. As numerical examples, the problems of plates having two and three circular inclusions with same radius are discussed and influences of incident wave number, radius, separation, and flexural rigidity of the inclusions on the dynamical moment concentration are explored. The results obtained are compared with those of a plate with one inclusion.

A concentrated couple inside or outside a circular inclusion in finite plane elastostatics

The heterogenization technique, recently developed by the authors, is applied to the problem, in antiplane elastostatics, of two circular inclusions of arbitrary radii and of different shear moduli, and perfectly bonded to a matrix, of infinite extent, subjected to arbitrary loading. The solution is formulated in a manner which leads to some exact results. Universal formulae are derived for the stress field at the point of contact between two elastic inclusions. It is also discovered that the difference in the displacement field, at the limit points of the Apollonius family of circles to which the boundaries of the inclusions belong, is the same for the heterogeneous problem as for the corresponding homogeneous one. This discovery leads to a universal formula for the average stress between two circular holes or rigid inclusions. Moreover, the asymptotic behavior of the stress field at the closest points of two circular holes or rigid inclusions approaching each other is also studied and given by universal formulae, i.e., formulae which are independent of the loading being considered.

The antiplane interaction problem for an elastic circular inclusion embedded in an elastic half plane with an arbitrarily located crack is considered in this paper. Based on the technique of analytical continuation and the structure of Moebius transformation, a rapidly convergent series solution pertaining to either the circular inclusion or half-plane matrix is derived in an explicit form. By introducing a distribution of screw dislocations for modeling an arbitrarily located crack, a system of singular integral equations with a logarithmic kernel is established, which can be solved numerically by applying the appropriate interpolation functions. Several numerical examples are given to illustrate the effects of geometrical parameters and material properties on the mode III stress intensity factors as well as the local stress along the interface.

Interaction of a screw dislocation (or an out-of-plane force) and anisotropic circular inclusion in isotropic matrix is studied. Similar problems for an anisotropic circular inclusion in an anisotropic matrix or the isotropic circular inclusion in the isotropic matrix have been solved, however, the anisotropic/isotropic problem (we will here after use this notation, meaning anisotropic circular inclusion in isotropic matrix) has not been solved yet. Recently, Choi et al.(2003) proposed a method based on ‘equivalence theorem’ to deal with a bimaterial interface (straight interface such as x2 = 0) of anisotropic material bonded onto isotropic material. We apply this method to the stated problem.

In the following analysis, we present a rigorous solution for the problem of a circular elastic inclusion surrounded by an infinite elastic matrix in finite plane elastostatics. The inclusion and matrix are separated by a circumferentially inhomogeneous imperfect interface characterized by the linear spring-type imperfect interface model where the interface is such that the same degree of imperfection is realized in both the normal and tangential directions. Through the use of analytic continuation, a set of first-order coupled ordinary differential equations with variable coefficients are developed for two analytic potential functions. The unknown coefficients of the potential functions are determined from their analyticity requirements and some additional problem-specific constraints. An example is then presented for a specific class of interface where the inclusion mean stress is contrasted between the homogeneous interface and inhomogeneous interface models. It is shown that, for circumstances where a homogeneously imperfect interface may not be warranted, the inhomogeneous model has a pronounced effect on the mean stress within the inclusion.

Thermal analysis of plates with circular inclusions

Interaction between a crack and a circular elastic inclusion under remote uniform heat flow

Energetics of two circular inclusions in anti-plane elastostatics

Strain gradient solution for a finite-domain Eshelby-type anti-plane strain inclusion problem

This paper studies the interactions between N randomly-distributed cylindrical inclusions in a piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is subjected to an anti-plane shear stress and an in-plane electric field at infinity. Based on the complex variable method, the complex potentials in the matrix and inside the inclusions are first obtained in form of power series, and then approximate solutions for electroelastic fields are derived. Numerical examples are presented to discuss the influences of the inclusion array, inclusion size and inclusion properties on couple fields in the matrix and inclusions. Solutions for the case of an infinite piezoelectric matrix with N circular holes or an infinite elastic matrix containing N circular piezoelectric fibers can also be obtained as special cases of the present work. It is shown that the electroelastic field distribution in a piezoelectric material with multiple inclusions is significantly different from that in the case of a single inclusion.

A rigorous solution is presented for a problem associated with a circular inclusion embedded within an infinite matrix in plane elastostatics. The bonding at the inclusion-matrix interface is assumed to be imperfect. Specifically, the jump in the normal displacement is assumed to be proportional to the normal traction with the proportionality parameter taken to be circumferentially inhomogeneous. In addition, we assume that displacements in the tangential direction are continuous. This type of interface is generally referred to as an inhomogeneous non-slip interface. Using the principle of analytic continuation, the basic boundary-value problem for four analytic functions is reduced to a first-order differential equation for a single analytic function defined inside the circular inclusion. The resulting closed-form solutions include a finite number of unknown constants determined by analyticity requirements and certain other supplementary conditions. The method is illustrated using several specific examples of a particular class of inhomogeneous non-slip interface. The results from these calculations are compared with the corresponding results when the interface imperfections are homogeneous. These comparisons indicate that the circumferential variation of interface damage has a significant effect on even the average stresses induced within a circular inclusion.