Problem Of Circular Inclusion Research Articles

Virtual element method for modeling the deformation of multiphase composites

In this paper, we study applications of the virtual element method (VEM) for simulating the deformation of multiphase composites. The VEM is a Galerkin approach that is applicable to meshes that consist of arbitrarily-shaped polygonal and polyhedral (simple and nonsimple) elements. In the VEM, the basis functions are defined as the solution of a local elliptic partial differential equation, and are never explicitly computed in the implementation of the method. The stiffness matrix of each element is built by using the elliptic projection operator of the internal virtual work (bilinear form) and it consists of two terms: a consistency term that is exactly computed (linear patch test is satisfied) and a correction term (ensures stability) that is orthogonal to affine displacement fields and has the right scaling. The VEM simplifies mesh generation for a multiphase composite: a stiff inclusion can be modeled using a single polygonal or polyhedral element. Attributes of the virtual element approach are highlighted through comparisons with Voronoi-cell lattice models, which provide discrete representations of material structure. The comparisons involve a suite of two-dimensional linear elastic problems: patch test, axisymmetric circular inclusion problem, and the deformation of a three-phase composite. The simulations demonstrate the accuracy and flexibility of the virtual element method.

Circular inclusion problem of two‐dimensional decagonal quasicrystals with interfacial rigid lines under concentrated force

AbstractThe plane problem of two‐dimensional decagonal quasicrystals with a rigid circular arc inclusion was investigated under infinite tension and concentrated force. Based on complex representations of stresses and displacements of two‐dimensional decagonal quasicrystals, the above problem is transformed into Riemann boundary problem by using the analytic continuation principle of complex functions. The general solutions of two‐dimensional decagonal quasicrystals under the action of plane concentrated force and infinite uniform tension are derived. The closed solutions of complex potential functions in several typical cases are obtained, and the formula of singular stress field at the tip of rigid line inclusions is given. The results show that the stress field at the tip of circular arc rigid line inclusions has singularity of oscillation under plane load. Numerical examples are given to analyze the effects of inclusion radius, different inclusions, the coupling coefficient and phason field parameter on stress singularity coefficients.

Analytical solution for circular inhomogeneous inclusion problems with non-uniform axisymmetric eigenstrain distribution

This paper presents the analytical solution of a class of plane elasticity problems for circular inhomogeneous inclusions with non-uniform axisymmetric eigenstrain distribution, which includes both radial and hoop eigenstrains. The complex variable potential solution for a point-wise eigenstrain in an infinite plane solid is presented first, in which two principal strains and their directions are taken into account. Directly employing it as influence function, the complex variable potential for the circular homogeneous inclusion problem is formulated with Green's function method. The novelty of this approach is that it is able to take intrinsic advantage of complex variable approach and effectively tackle the mathematical difficulties encountered during formulation. Next, by using the principle of equivalent eigenstrain, the main challenge in solving inhomogeneous inclusion problems is overcome, allowing the general explicit analytical solution to be derived. Based on these results, three illustrative examples of practical significance are given: (i) dissimilar cylinder interference-fits within an infinite body, (ii) a pure dilatational eigenstrain problem within a circular inclusion, and (iii) a circular inclusion problem with a wedge disclination eigenstrain distribution. The fundamental formulation introduced here will find application in other aspects in the mechanics of fiber composites, thermoelasticity, and nano-mechanics of defects in solids.

A representation theorem for the circular inclusion problem

A representation theorem is obtained for an arbitrarily loaded elastic bimaterial solid consisting of an infinite plane containing a circular inhomogeneity. The elastic image method is used for the analysis. The theorem expresses the Airy stress functions that generate the elastic fields for the composite solid explicitly in terms of the Airy stress function for the corresponding homogeneous infinite solid. It shows that if the solution for the homogeneous infinite solid is available, then the solutions for the corresponding bimaterial solid can be deduced by the process of differentiation and integration. The result could provide the important advantage of economy of effort in the determination of the elastic fields for composite planes with circular interfaces.

Strain gradient solution for the Eshelby-type polygonal inclusion problem

The Eshelby-type problem of an arbitrary-shape polygonal inclusion embedded in an infinite homogeneous isotropic elastic material is analytically solved using a simplified strain gradient elasticity theory (SSGET) that contains one material length scale parameter. The Eshelby tensor for a plane strain inclusion with an arbitrary polygonal cross section is derived in a general form in terms of three potential functions, two of which have the same definitions as the ones involved in the counterpart Eshelby tensor based on classical elasticity. These potential functions, as area integrals over the polygonal cross section, are first converted to three line (contour) integrals using Green’s theorem, which are then evaluated analytically by direct integration. The newly derived Eshelby tensor is separated into a classical part and a gradient part. The former contains Poisson’s ratio only, while the latter includes the material length scale parameter additionally, thereby enabling the interpretation of the inclusion size effect. For homogenization applications, the area average of the new Eshelby tensor over the polygonal cross section is also provided in a general form. To illustrate the newly obtained Eshelby tensor and its area average, three types of polygonal inclusions are quantitatively studied by directly using the general formulas derived. The numerical results show that the components of the SSGET-based Eshelby tensor for all inclusion shapes considered vary with both the position and the inclusion size. It is also observed that the components of the averaged Eshelby tensor based on the SSGET change with the inclusion size. By applying the newly derived solution, a homogenization analysis of a composite reinforced by polygon-shaped fibers is performed. The induced strain in the composite is evaluated, and the effective stiffness of the composite is predicted. It is found that the values of the induced strain components approach from below those in a corresponding composite reinforced by circular fibers when the fiber size or the number of sides of the polygonal fiber increases. Also, the analysis reveals that the components of the effective stiffness tensor vary slightly with the fiber shape, and thus the simpler solution for the circular inclusion problem may be used to estimate the effective stiffness of a composite reinforced by regular polygon-shaped fibers.

A boundary smoothing algorithm for image-based modeling and its application to micromechanical analysis of multi-phase materials

Abstract A micromechanical modeling framework is developed to analyze mechanical behavior of multi-phase materials. The framework includes novel and efficient numerical algorithms for describing realistic geometry of microstructures in finite element models. One of the difficulties of the image-based modeling is that the pixel images have jagged phase boundaries. A phase function is introduced to define the smooth phase boundaries from the jagged pixel boundaries of the micrographs. The phase function is used to construct finite element meshes conforming to the phase boundaries. The effectiveness of smooth boundary meshes is verified by the analytical solution of a circular inclusion problem. For verification, image-based RVE models are constructed from the scanning electron micrographs of a dual-phase steel. The predicted uniaxial stress–strain behavior of the RVE models compared well with the measurements. Highly localized regions or shear bands are observed in the ferritic phase of the dual-phase steel.

Scattering of flexural wave in a thin plate with multiple circular inclusions by using the null-field integral equation approach

The subject of scattering flexural wave in a thin plate with multiple circular inclusions under the incident flexural wave is studied in this paper. A semi-analytical approach is proposed to solve this problem which can be decomposed into several interior circular inclusion problems and an exterior plate problem subject to the incident wave. The scattered field in the associated exterior problem is solved by using the null-field integral formulation in conjunction with degenerate kernels, tensor transformation and Fourier series. All dynamic kernels of plate in the direct formulation are expanded into degenerate forms to avoid the integral singularity and further the rotated degenerate kernels have been derived to consider the general case of multiple circular inclusions. The proposed results for an infinite plate with one circular inclusion are compared with the available analytical solutions to verify the validity of the proposed method. To demonstrate the generality of the proposed method, the cases of multiple inclusions are studied and their quasi-static results are verified by static data of FEM using ABAQUS. Numerical results indicate that the DMCF of two inclusions is apparently larger than that of one when two inclusions are close to each other. Fictitious frequency appearing in the exterior problem can be suppressed by using the more number of Fourier series terms. Numerical results show that the space between scatterers has the opposite effect on the near-field DMCF in comparison with the far-field scattering pattern.

Stress Intensity Factor for an Interphase Griffith Crack Interacting with Two Imperfect Interfaces

The solution for the elastic three-phase circular inclusion problem plays a fundamental role in many practical and theoretical applications. In particular, it offers the fundamental solution for the generalized self-consistent method in the mechanics of composites materials. In this work, a semi-analytical method is presented for the problem of a pre-existing radial Griffith crack embedded within the interphase layer surrounding a circular inclusion. Novel to this work is that the bonding at the inclusion—interphase interface and the interphase-matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. Employing complex variable techniques, we derive series representations for the corresponding stress functions inside the inclusion, in the interphase layer and the surrounding matrix. The governing boundary value problem is then formulated in such a way that these stress distributions simultaneously satisfy the traction-free condition along the crack face, the imperfect interface conditions and the prescribed asymptotic loading conditions. The advantage of the series method over other methods, such as the dislocation density method, is that in the former case the resulting expressions are linear and can be solved readily whereas in the latter case the method leads to cumbersome integral equations which are often numerically difficult to solve. Stress intensity factor (SIF) calculations are performed at the crack tips for different material property combinations, imperfect interface conditions and crack locations under mode I loading. The results not only provide for a quantitative description of the interaction between a radial interphase crack and a three-phase inclusion with imperfect interfaces but the results clearly demonstrate the significance of how two imperfect boundaries can influence crack behavior.

On multiple circular inclusions in plane magnetoelasticity

A general series solution to the magnetoelastic problem of interacting circular inclusions in plane magnetoelasticity is provided in this paper. By the use of complex variable theory and Laurent series expansion method, the general expression of the magnetic and the magnetoelastic complex potentials for the circular inclusion problem is derived. Expanding the definition of the Airy’s stress function of pure elastic field into the magnetoelastic field and applying the superposition method, the general expression then can be reduced to a set of linear algebraic equations and solved in a series form. An approximate closed form solution for the case of two arbitrarily located inclusions is also provided. For illustrating the effect of the pertinent parameters, the numerical results of the interfacial magnetoelastic stresses are displayed in graphic form.

Interaction between a radial matrix crack and a three-phase circular inclusion with imperfect interface in plane elasticity

The solution for the elastic three-phase circular inclusion problem plays a fundamental role in many practical and theoretical applications. In particular, it offers the fundamental solution for the generalized self-consistent method in the mechanics of composites materials. In this paper, a general method is presented for evaluating the interaction between a pre-existing radial matrix crack and a three-phase circular inclusion. The bonding at the inclusion-interphase interface is considered to be imperfect with the assumption that the interface imperfections are constant. On the remaining boundary, that being the interphase-matrix interface, the bonding is considered to be perfect. Using complex variable techniques, we derive series representations for the corresponding stress functions inside the inclusion, in the interphase layer and the surrounding matrix. The governing boundary value problem is then formulated in such a way that these stress distributions simultaneously satisfy the traction free condition along the crack face, the imperfect interface conditions and the prescribed asymptotic loading conditions. Stress intensity factor (SIF) calculations are performed at the crack tips for different material property combinations, imperfect interface conditions and crack positions. The results illustrate convincingly the role of an interphase layer as well as the effects of an imperfect interface on crack behavior. For instance, when the interphase layer is softer than the inclusion and matrix, the results show that the radial matrix crack will propagate from the nearby crack tip regardless of the level of the imperfect (spring-layer) interface parameter. In comparison, when the interphase layer is stiffer than the inclusion and matrix, the interphase layer will shield the crack from effects of the imperfect (spring-layer) interface. Hence, these results provide a quantitative description of the interaction problem between a three-phase inclusion with interface imperfections and a radial matrix crack.

A Three-phase Circular Inclusion With A SlidingInterface And A Radial Matrix Crack

The solution for the elastic three-phase circular inclusion problem plays a fundamental role in many practical and theoretical applications. In particular, it offers the fundamental solution for the generalized self-consistent method in the mechanics of composites materials. In this paper, a general method is presented for evaluating the interaction between a pre-existing matrix crack and a threephase circular inclusion. The bonding at the inclusion-interphase interface is considered to be the sliding imperfect interface. Physically, this is important for modelling certain features of material behaviour (such as grain-boundary sliding and damage occurring in the circumferential direction at constituent interfaces). On the remaining boundary, that being the interphase-matrix interface, the bonding is considered to be perfect. Using complex variable techniques, we derive series representations for the corresponding stress functions inside the inclusion, in the interphase layer and in the surrounding matrix. The governing boundary value problem is then formulated in such a way that these stress distributions simultaneously satisfy the traction free condition along the crack face, the sliding imperfect interface condition and the prescribed asymptotic loading conditions. Stress intensity factor (SIF) calculations are performed at the crack tips for different material property combinations, interface imperfections and crack positions. The results illustrate convincingly the role of an interphase layer as well as the effects of a sliding interface on crack behaviour.

On Perturbation Solutions for Nearly Circular Inclusion Problems in Plane Thermoelasticity

An approximate analytical solution to the nearly circular inclusion problems of arbitrary shape in plane thermoelasticity is provided. The shape of the inclusion boundary considered in the present study is assumed to have the form r=a0[1+Aθ], where a0 is the radius of the unperturbed circle and Aθ is the radius perturbation magnitude that is represented by a Fourier series expansion. The proposed method in this study is based on the complex variable theory, analytical continuation theorem, and the boundary perturbation technique. Originating from the principle of superposition, the solution of the present problem is composed of the reference and the perturbation terms that the reference term is the known exact solution pertaining to the case with circular inclusion. First-order perturbation solutions of both temperature and stress fields are obtained explicitly for elastic inclusions of arbitrary shape. To demonstrate the derived general solutions, two typical examples including elliptical and smooth polygonal inclusions are discussed in detail. Compared to other existing approaches for elastic inclusion problems, our methodology presented here is remarked by its efficiency and applicability to inclusions of arbitrary shape in a plane under thermal load.

On multiple circular inclusions in plane thermoelasticity

A general series solution to the problem of interacting circular inclusions in plane thermoelasticity is provided in this paper. Based upon the complex variable theory and the use of Laurent series expansion, the general expression of the stress functions is derived explicitly for the circular inclusion problem under remote uniform heat flow. By applying the use of the superposition, the problem dealing with any number of arbitrarily located inclusions can be then reduced to a set of linear algebraic equations which are solved with the aid of a perturbation technique. For illustrating the use of the present approach, an approximate closed form solution of the stress functions is derived explicitly for the problem containing two arbitrarily located inclusions. Numerical results of the interfacial stresses around a rigid circular inclusion or hoop stress along a circular hole due to the presence of an elastic inclusion are provided to demonstrate the dependence of the solution upon the pertinent parameters.

Two-circle theorem of flow in porous media

In [1], using the method of conformai mapping, an attempt was made to extend the known circle theorem of flow in porous media [2, 3] to the case of two circular regions of the same radius occupied by the same medium. Below, using bipolar coordinates, the theorem is further extended to cover the problem of circular inclusions of different radii and different permeabilities. A corresponding theorem for two contiguous inclusions is presented as a limiting case. The exposition is illustrated with examples from which the results described in the literature follow as special cases.

An inhomogeneity problem in couple stress theory

Using Hilbert theory and Mindlin’s couple stress theory, the problem of two-dimensional circular inhomogeneity (when the inserted material is of different size than the size of the cavity and having different elastic constants) is studied in this paper. Stresses could be bounded at infinity. The formulation is valid also for regions other than the circular ones when the elastic constants of inclusion (inserted material) and matrix (outer material) are the same. The problem of circular inclusion when the matrix is finite has also been tackled. Numerical results are in conformity with the fact that the effect of couple stresses is negligible when the ratio of the smallest dimension of the body to the characteristic length is large.

Multiple circular inclusion problems in longitudinal shear deformation

An exact analytic solution is given for a case of antiplane deformation of an clastic solid containing an arbitrary number of circular cylindrical inclusions. The solution takes into account nonuniform spacing between the inclusions, which can have different radii. An example of interest to workers in composite materials is discussed.

A two-dimensional circular inclusion problem

AbstractThe elastic fields in an elastic circular inclusion and its surrounding infinite dissimilar elastic matrix, are determined when either the matrix or inclusion is subject to a concentrated force or edge dislocation.

Circular inclusion in an infinite elastic medium with a circular inhomogeneity

AbstractIn a previous paper ((l)) the authors gave the solution for the two-dimensional circular inclusion problem in a medium containing a circular cavity. This paper seeks to solve the more general problem of a similar inclusion when the cavity is replaced by an inhomogeneity which could be of a different elastic material. The solution consists in finding three sets of suitable complex potential functions ø(z) and ψ(z) for three regions: the inhomogeneity, the inclusion and the rest of the material. The solution depends upon the evaluation of the complex potentials for a material containing the inhomogeneity when on the former a finite force is acting at some fixed point. It may be noted that two sets of ø(z) and ψ(z) have to be found in this case: one for the inhomogeneity and the other for the rest of the material. This may be taken as an auxiliary problem.