Multiple elliptical inclusions of arbitrary orientation in composites

A volume integral equation method (VIEM) is introduced for the solution of elastostatic problems in an unbounded isotropic elastic solid containing multiple elliptical inclusions of arbitrary orientation subjected to uniform tensile stress at infinity. The inclusions are assumed to be long parallel elliptical cylinders composed of isotropic and anisotropic elastic material perfectly bonded to the isotropic matrix. The solid is assumed to be under plane strain on the plane normal to the cylinders. A detailed analysis of the stress field at the matrix–inclusion interface for square and hexagonal packing arrays is carried out, taking into account different values for the number, orientation angles and concentration of the elliptical inclusions. The accuracy and efficiency of the method are examined in comparison with results available in the literature.     

Boundary collocation method for multiple defect interactions in an anisotropic finite region

The modified mapping collocation method is extended for the solution of plane problems of anisotropic elasticity in the presence of multiple defects in the form of holes, cracks, and inclusions under general loading conditions. The approach is applied to examine the stress and strain fields in an anisotropic finite region including an elliptical and a circular hole, an elliptical flexible inclusion, and a line crack. It can be readily incorporated into micro-mechanics models, capturing the relative importance of the matrix, the fiber/matrix interface, and reinforcement geometry and arrangement while estimating the effective elastic properties of composite materials. The accuracy and robustness of this method is established through comparison with results obtained from finite element analysis.     

Elastic analysis of a half-plane with multiple inclusions using volume integral equation method

A volume integral equation method (VIEM) is used to calculate the elastostatic field in an isotropic elastic half-plane containing circular inclusions subject to remote loading parallel to the traction-free boundary. The material of the inclusions may be either isotropic or anisotropic and they are assumed to be distributed in square or hexagonal array. A detailed analysis of the stress field at the interface between the matrix and one of the inclusions is carried out for different distances between the inclusion and the surface of the half-plane. The results of the calculations are compared with available results. The VIEM is shown to be very accurate and effective for investigating the local stresses in the presence of multiple inclusions. The method can be applied to multiple inclusions of arbitrary geometry and elastic properties embedded in extended isotropic elastic media.     

Interaction of screw dislocation and anisotropic (or isotropic) circular inclusion in isotropic (or anisotropic) matrix

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.     

任意分布多个椭圆形刚性夹杂的反平面问题

对于硬夹杂与软基体的复合材料,考虑夹杂间的相互影响,采用坐标变换和复变函数的依次保角映射方法,构造任意分布且相互影响的多个椭圆形刚性夹杂模型的复应力函数,同时满足各个夹杂的边界条件,利用围线积分将求解方程化为线性代数方程,推导出了在无穷远双向均匀剪切,椭圆形刚性夹杂任意分布的界面应力解析表达式,算例分析给出了单夹杂模型与多夹杂模型的夹杂形状对界面应力最大值的影响规律,并进行了对比,描绘出了曲线。     

Elastic analysis of a half-plane containing an inclusion and a void using a mixed volume and boundary integral equation method

A mixed volume and boundary integral equation method is used to calculate the plane elastostatic field in an isotropic elastic half-plane containing an isotropic or anisotropic inclusion and a void subject to remote loading parallel to a traction-free boundary. A detailed analysis of the stress field is carried out for three different geometries of the problem. It is demonstrated that the method is very accurate and effective for investigating local stresses in an isotropic elastic half-plane containing multiple isotropic or anisotropic inclusions and multiple voids.     

Plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix

Based on the complex variable method, this paper addresses the plane problems of multiple piezoelectric inclusions in a non-piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is loaded by in-plane mechanical loads while the inclusions are applied by anti-plane electric loads at infinity. The general solutions are first derived for the complex potentials both in the matrix and inside the inclusions, and then numerical results are presented to show the effects of applied electric field, inclusion arrays and material properties on the electroelastic fields around the inclusions. It is shown that the inclusion arrays have a significant influence on the stress distribution at the interface between the matrix and piezoelectric inclusions.     

Uniform antiplane shear stress inside an anisotropic elastic inclusion of arbitrary shape with perfect or imperfect interface bonding

We consider an anisotropic elastic inclusion of arbitrary shape embedded inside an infinite dissimilar anisotropic elastic medium (matrix) subjected to a uniform antiplane shear loading at infinity. In contrast to the corresponding results from linear isotropic elasticity, we show that for certain anisotropic materials, despite the limitation of perfect bonding between the inclusion and its surrounding matrix, it is possible to design an arbitrarily shaped (not necessarily elliptic) inclusion so that the interior stress distribution is uniform provided the shear stress in the matrix (of dissimilar anisotropic material) is also uniform. Further, in the case when the bonding between the inclusion and the matrix is assumed to be imperfect, we show that for the stress distribution inside the inclusion to be uniform, the inclusion must be elliptical.     

Interactions between N circular cylindrical inclusions in a piezoelectric matrix

Summary 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.     

The morphological evolution and migration of inclusions in thin-film interconnects under electric loading

This paper reports the result of an investigation into electromigration-driven morphological evolution of inclusions in finite scale thin-film interconnects using phase field method. The results show that morphological evolution and migration of an inclusion are proportional to the electric field strength applied on thin-film interconnects, and that inclusions with anisotropic diffusion interface move slower than those with isotropic interface under the identical electric field, and show irregular patterns when the inclusion is subjected to perturbation.     

Some elastic properties of reinforced solids,with special reference to isotropic ones containing spherical inclusions

Based on Mori and Tanaka's concept of “average stress” in the matrix and Eshelby's solutions of an ellipsoidal inclusion, an approximate theory is established to derive the stress and strain state of constituent phases, stress concentrations at the interface, and the elastic energy and overall moduli of the composite. Both “stress-free” strain (polarization strain) and “strain-free” stress (polarization stress) are employed in these derivations under the traction- and displacement-prescribed conditions. The theory was developed first for a general multiphase, anisotropic composite with arbitrarily oriented anisotropic inclusions; explicit results are then given for a suspension of uniformly distributed, multiphase isotropic spheres in an isotropic matrix. Numerical results for stress concentrations in the spherical inclusions and at the interface are given for a 2-phase composite. Further, it is shown that the derived moduli are related to the Hashin-Shtrikman bounds and that, when the shear moduli are equal, the overall bulk modulus of a 2-phase composite reduces to Hill's exact solution. As compared with experimental data, the theory also provides reasonably accurate estimates for the Young's modulus of some 2- and 3-phase composites.     

Application of the boundary-domain integral equation in elastic inclusion problems

A boundary-domain integral equation is used to calculate the elastic stress and strain field in a finite or infinite body of isotropic, orthotropic or anisotropic materials characterized with inclusions of arbitrary shapes. Based on the Betti–Rayleigh reciprocal work theorem between the unknown state and a known fundamental solution, the equilibrium of the body with inclusions is formulated in terms of boundary-domain integral equations. The resulting equation involves only the fundamental solution of isotropic medium, and hence the use of complicated fundamental solution for anisotropic materials could be avoided. Numerical examples are given to ascertain the correctness and effectiveness of the boundary-domain integral equation technique for the inclusion problems.     

Interactions between inclusions and various types of cracks

The problems of a crack inside, outside, penetrating or lying along the interface of an anisotropic elliptical inclusion are considered in this paper. Because the crack may be represented by a distribution of dislocation, integrating the analytical solutions of dislocation problems along the crack and applying the technique of numerical solution on the singular integral equation, we can obtain the general solutions to the problems of interactions between cracks and anisotropic elliptical inclusions. Since there are no analytical solutions existing for the general cases of interactions between cracks and inclusions, the comparison is made with the numerical results obtained by other methods or with the analytical results for the special cases which can be reduced from the present problems. These results show that our solutions are correct and universal     

An irregular-shaped inclusion with imperfect interface in antiplane elasticity

Despite extensive studies of inclusions with simple shape, little effort has been devoted to inclusions of irregular shape. In this study, we consider an inclusion of irregular shape embedded within an infinite isotropic elastic matrix subject to antiplane shear deformations. The inclusion–matrix interface is assumed to be imperfect characterized by a single, non-negative, and constant interface parameter. Using complex variable techniques, the analytic function that is defined within the irregular-shaped inclusion is expanded into a Faber series, and in conjunction with the Fourier series, a set of linear algebraic equations for a finite number of unknown coefficients is determined. With this approach and without imposing any constraints on the stress distribution, a semi-analytical solution is derived for the elastic fields within the irregular-shaped inclusion and the surrounding matrix. The method is illustrated using three examples and verified, when possible, with existing solutions. The results from the calculations reveal that the stress distribution within the inclusion is highly non-uniform and depends on the inclusion shape and the weak mechanical contact at the inclusion/matrix boundary. In fact, the results illustrate that the imperfect interface parameter significantly influences the stress distribution.     

The problem of two plastic and heterogeneous inclusions in an anisotropic medium

We demonstrate an integral equation for the total local strain ε^T in an anisotropic heterogeneous medium with incompatible strain ε^p and which is at the same time submitted to an exterior field. The integral equation is solved in the case of an heterogeneous and plastic pair of inclusions, for which we calculate the average fields in each inclusion as well as the different parts of the elastic energy stocked in the medium.The solution is applied to the case of two isotropic and spherical inclusions in an isotropic matrix loaded in shear. The results are compared with those deduced from a more approximate method based on Horn's approximation of the integral equation. In appendix we give a numerical method for calculating the interaction tensors between anisotropic inclusions in an anisotropic medium as well as the analytic solution in the case of two spherical inclusions located in an isotropic medium.     

Interaction between screw dislocations and inclusions with imperfect interfaces in fiber-reinforced composites

A three-phase composite cylinder model is utilized to study the elastic interaction between screw dislocations and embedded multiple circular cross-section inclusions (fibers) with imperfect interfaces in composites. By means of complex variable techniques, the explicit solutions of stress and displacement fields are obtained. With the aid of the Peach–Koehler formula, the explicit expressions of image forces exerted on screw dislocations are derived. The equilibrium positions of the appointed screw dislocation near one of the inclusions are discussed for variable parameters (interface imperfection, material mismatch and dislocation position) and the influence of the nearby inclusions and dislocations is also considered. The results show that, if the inclusion is stiffer than the matrix and the magnitude of the degree of interface imperfection reaches the certain value, a new equilibrium position for the screw dislocation in the matrix can always be produced in comparison with the previous solution (the perfect interface). The effect of elastic constants of the inclusion on the image force and the equilibrium position of the appointed screw dislocation is weak when the interface imperfection is strong. It is also seen that the magnitude of the image force exerted on the appointed dislocation caused by multiple inclusions is always smaller than that produced by a single inclusion. The impact of the closer dislocations on the mobility of the appointed dislocation is very significant.     

Time-domain BEM for 3-D transient elastodynamic problems with interacting rigid movable disc-shaped inclusions

Formulation of time-domain boundary element method for elastodynamic analysis of interaction between rigid massive disc-shaped inclusions subjected to impinging elastic waves is presented. Boundary integral equations (BIEs) with time-retarded kernels are obtained by using the integral representations of displacements in a matrix in terms of interfacial stress jumps across the inhomogeneities and satisfaction of linearity conditions at the inclusion domains. The equations of motion for each inclusion complete the problem formulation. The time-stepping/collocation scheme is implemented for the discretization of the BIEs by taking into account the traveling nature of the generated wave field and local structure of the solution at the inclusion edges. Numerical results concern normal incidence of longitudinal wave onto two coplanar circular inclusions. The inertial effects are revealed by the time dependencies of inclusions’ kinematic parameters and dynamic stress intensity factors in the inclusion vicinities for different mass ratios and distances between the interacting obstacles.     

Equivalent inclusion approach and effective medium approximations for the effective conductivity of isotropic multicomponent materials

Most effective medium approximations for isotropic inhomogeneous materials are based on dilute solutions of some typical inclusions in an infinite matrix medium, while the simplest approximations are those for the composites with spherical and circular inclusions. Practical particulate composites often involve inhomogeneities of more complicated geometry than that of the spherical (or circular) one. In our approach, those inhomogeneities are supposed to be substituted by simple equivalent spherical (circular) inclusions from a comparison of their respective dilute solution results. Then the available simple approximations for the equivalent spherical (circular) inclusion material can be used to estimate the effective conductivity of the original composite. Numerical illustrations of the approach are performed on some 2D and 3D geometries involving elliptical and ellipsoidal inclusions.     

在轴向应力状态下含有扁长椭球形增强物复合材料的平均弹塑性过程

研究了在轴向应力状态下,含有扁长椭球形增强物复合材料的弹塑性过程,进一步提出了判断在屈服点处是基体首先屈服还是增强材料首先屈服的判断基准,寻找到了材料整体的屈服应力与增强材料或基体屈服应力的关系。在本研究中所涉及的材料其基体和增强材料均为理想的各向同性材料,增强材料为扁长椭球体,其体积相对于基体是非常地小并与基体紧密相联。明确地得到了基体或增强材料首先屈服时所对应     

A time domain direct boundary integral method for a viscoelastic plane with circular holes and elastic inclusions

This paper considers the problem of an infinite, isotropic viscoelastic plane containing an arbitrary number of randomly distributed, non-overlapping circular holes and isotropic elastic inclusions. The holes and inclusions are of arbitrary size. All inclusions are assumed to be perfectly bonded to the material matrix but the elastic properties of the inclusions can be different from one another. The Kelvin model is employed to simulate the viscoelastic plane. The numerical approach combines a direct boundary integral method for a similar problem of an infinite elastic plane containing multiple circular holes and elastic inclusions described in [Crouch SL, Mogilevskaya SG. On the use of Somigliana's formula and Fourier series for elasticity problems with circular boundaries. Int J Numer Methods Eng 2003;58:537–578], and a time-marching strategy for viscoelastic material analysis described in [Mesquita AD, Coda HB, Boundary integral equation method for general viscoelastic analysis. Int J Solids Struct 2002;39:2643–2664]. Several numerical examples are given to verify the approach. For benchmark problems with one inclusion, results are compared with the analytical solution obtained using the correspondence principle and analytical Laplace transform inversion. For an example with two holes and two inclusions, results are compared with numerical solutions obtained by commercial finite element software—ANSYS. Benchmark results for a more complicated example with 25 inclusions are also given.