Circular inclusions in anti-plane strain couple stress elasticity

The solution for a circular inclusion with a prescribed anti-plane eigenstrain is derived. It is shown that the components of the Eshelby tensor within the inclusion, corresponding to a uniform eigenstrain, can be either uniform or non-uniform, depending on the imposed interface conditions. The stress amplification factors due to circular void or rigid inclusion in an infinite medium under remote anti-plane shear stress are calculated. The failure of the couple stress elasticity to reproduce the classical elasticity solution in the limit of vanishingly small characteristic length is indicated for a particular type of boundary conditions. The solution for a circular inhomogeneity in an infinitely extended matrix subjected to remote shear stress is then derived. The effects of the imposed interface conditions, the shear stress and couple stress discontinuities, and the relationship between the inhomogeneity and its equivalent eigenstrain inclusion problem are discussed.     

ANALYSIS OF A SCREW DISLOCATION INSIDE A CIRCULAR INCLUSION WITH INTERFACIAL CRACKS IN PIEZOELECTRIC COMPOSITES

IntroductionPiezoelectric materials have potentials for use in many modern devices and compositestructures. The presence of various defects, such as inclusions, holes, dislocations andcracks, can greatly influence their characteristics and coupled behavio…     

A circular inclusion with a nonuniform interphase layer in anti-plane shear

An analytical solution is derived for the problem of a nonuniformly coated circular inclusion in an unbounded matrix under anti-plane deformations. The inclusion/interphase/matrix system is subject to (1) remote uniform shear and uniform eigenstrain imposed on the circular inclusion, and (2) a screw dislocation or a point force in the matrix. It is found that the varying interphase thickness will exert a significant influence on the nonuniform stress field within the circular inclusion, and on the direction and magnitude of the image force acting on a screw dislocation. In the course of development, it is found that the presence of certain coated inclusions, which are termed stealth, will not cause change of elastic energy in the body. The derived analytical solution for a screw dislocation is then employed as Green’s function to investigate a radial matrix crack interacting with the nonuniformly coated inclusion. The numerical results show that the varying interphase thickness will also affect the stress intensity factors.     

The material force acting on a screw dislocation in the presence of a multi-layered circular inclusion

In this paper, we study the interaction of a screw dislocation with a multi-layered interphase between a circularly cylindrical inclusion and a matrix. The layers are coaxial cylinders of annular cross-sections with arbitrary radii and different shear moduli. The number of layers may also be arbitrary. Continuity of traction and displacement across all interfaces is assumed. We extend Honein et al.’s solution of circularly cylindrical layered media in anti-plane elastostatics to the case where all the singularities reside inside the inclusion core. The solution to this heterogeneous problem is given explicitly, for arbitrary singularities, as a rapidly convergent Laurent series, whose coefficients are expressed in terms of those of the complex potential of a corresponding homogeneous problem with the same singularities. We then consider the two particular cases of a screw dislocation, where, in the first instance, the dislocation resides inside the matrix, while, in the second instance, it is located in the inclusion core. In both instances, the Peach–Koehler force acting on the dislocation is calculated explicitly as a rapidly convergent series. We present several examples, where the effect of the layers on the material force is examined.     

Interaction of an edge dislocation with a coated elliptic inclusion

This paper presents an analytical solution for plane elasticity problems of an elliptically cylindrical layered media subject to an arbitrary edge dislocation. Based on the technique of conformal mapping and the method of analytical continuation in conjunction with the alternating technique, the general expressions of the displacements and stresses, where an edge dislocation is located in matrix, coating layer and inclusion are obtained. The numerical results of image forces exerted on a generalized edge dislocation are carried out by using the generalized Peach–Koehler equation. As a numerical illustration, both the image forces and equilibrium positions are presented for different material combinations and relative thickness of a coating layer. The result shows that the thickness and the shear modulus of the coating layer have a strong influence on the stability of dislocation.     

On the interaction between a dislocation and a circular inhomogeneity with imperfect interface in antiplane shear

The solution of appropriate elasticity problems involving the interaction between inclusions and dislocations plays a fundamental role in many practical and theoretical applications, namely, it increases the understanding of material defects thereby providing valuable insight into the mechanical behavior of composite materials.Although the problem of a three-phase circular inclusion interacting with a dislocation in antiplane shear has been presented [Xiao and Chen, Mech. Mater. 32 (2000) 485], the analysis is limited to the classical perfect bonding condition. The current paper considers the solution for a homogeneous circular inclusion interacting with a dislocation under thermal loadings in antiplane shear. The bonding along the inhomogeneity–matrix interface is considered to be imperfect with the assumption that the interface imperfections are constant. It is found that when the inhomogeneity is soft, regardless of the level of interface imperfection, the inhomogeneity will always attract the dislocation. As a result, no equilibrium positions are available. Alternatively, when the inhomogeneity is hard, an unstable equilibrium position is found which depends on the imperfect interface condition and the shear moduli ratio μ₂/μ₁.     

A generalized screw dislocation interacting with interfacial cracks along a circular inhomogeneity in magnetoelectroelastic solids

The interaction of a generalized screw dislocation with circular arc interfacial cracks under remote antiplane shear stresses, in-plane electric and magnetic loads in transversely isotropic magnetoelectroelastic solids is dealt with. By using the complex variable method, the general solutions to the problem are presented. The closed-form expressions of complex potentials in both the inhomogeneity and the matrix are derived for a single circular-arc interfacial crack. The intensity factors of stress, electric displacement and magnetic induction are provided explicitly. The image forces acting on the dislocation are also calculated by using the generalized Peach–Koehler formula. For the case of piezoelectric matrix and piezomagnetic inclusion, the shielding and anti-shielding effect of the dislocation upon the stress intensity factors is evaluated in detail. The results indicate that if the distance between the dislocation and the crack tip remains constant, the dislocation in the interface will have a largest shielding effect which retards the crack propagation. In addition, the influence of the interfacial crack geometry and materials magnetoelectroelastic mismatch upon the image force is discussed. Numerical computations show that the perturbation effect of the above parameters upon the image force is significant. The main result shows that a stable or unstable equilibrium point may be found when a screw dislocation approaches the surface of the crack from infinity which differs from the perfect bonded case under the same conditions. The present solutions contain a number of previously known results which can be shown to be special cases.     

On the screw dislocation in a functionally graded piezoelectric plane and half-plane

In this paper closed-form expressions of the electroelastic field induced by a piezoelectric screw dislocation in a functionally graded piezoelectric plane and half-plane are derived. The material properties are assumed to vary exponentially along the x and y-directions. The solution for a screw dislocation in a functionally graded piezoelectric plane is obtained through introduction of two generalized stress functions. The solution for a screw dislocation in a functionally graded piezoelectric half-plane is derived by using the method of image. It is also found that the interaction between a piezoelectric screw dislocation and a circular insulating hole in the functionally graded piezoelectric material can be solved by using series expansion method.     

压电螺型位错和共线界面刚性线夹杂的干涉效应

研究了压电材料中压电螺型位错和共线界面导电刚性线夹杂的电弹干涉效应．运用复变函数解析延拓技术与奇性主部分析方法,获得了该问题的一般解答．作为算例,求出了界面含一条刚性线夹杂时两种压电介质区域广义应力函数的封闭形式解．导出了作用在位错上的像力和刚性线夹杂表面剪应力和电位移的解析表达式．讨论了界面刚性线长度,两种材料的剪切模量比和压电系数比对位错力和刚性线表面剪应力的影响规律．为进一步研究该类问题提供了一个基本解。     

A rigid inclusion in an elastic space under the action of a uniform heat flow in the inclusion plane

A solution is presented for the three dimensional static thermoelastic problem of an absolutely rigid inclusion (anticrack) in the case when a uniform heat flow is directed along the inclusion plane. By using the potential method and the Fourier transform technique, the problem is reduced to a system of coupled two-dimensional singular integral equations for the shear stress jumps across the inclusion. As an illustration, a typical application to the circular anticrack is presented. Explicit expressions for the thermal stresses in the inclusion plane are obtained and discussed from the point of view of material failure.     

Interaction between an edge dislocation and a circular inclusion with an inhomogeneously imperfect interface

This research presents an analytical study of the interaction problem of an edge dislocation with a circular inclusion with a circumferentially inhomogeneously imperfect interface. The interface, which is modeled as a spring (interphase) layer with vanishing thickness, is characterized by that in which there is a displacement jump across the interface in the same direction as the corresponding tractions, and the same degree of imperfection is realized in both the normal and tangential directions. Furthermore, the interface parameter is nonuniform along the interface. In order to arrive at an elementary form solution, we introduce a conformal mapping function. Then the stress field as well as the Peach–Koehler force acting on the edge dislocation can be obtained from the derived complex potentials. Calculations demonstrate that the nonuniform interface parameter has a significant influence on the stress field.     

Electro-elastic interaction between a piezoelectric screw dislocation and circular interfacial rigid lines

The electro-elastic interaction between a piezoelectric screw dislocation located either outside or inside inhomogeneity and circular interfacial rigid lines under anti-plane mechanical and in-plane electrical loads in linear piezoelectric materials is dealt with in the framework of linear elastic theory. Using Riemann–Schwarz’s symmetry principle integrated with the analysis of singularity of complex functions, the general solution of this problem is presented in this paper. For a special example, the closed form solutions for electro-elastic fields in matrix and inhomogeneity regions are derived explicitly when interface containing single rigid line. Applying perturbation technique, perturbation stress and electric displacement fields are obtained. The image force acting on piezoelectric screw dislocation is calculated by using the generalized Peach–Koehler formula. As a result, numerical analysis and discussion show that soft inhomogeneity can repel screw dislocation in piezoelectric material due to their intrinsic electro-mechanical coupling behavior and the influence of interfacial rigid line upon the image force is profound. When the radian of circular rigid line reaches extensive magnitude, the presence of interfacial rigid line can change the interaction mechanism.     

刃型位错和圆弧形刚性线夹杂的干涉效应

位错和夹杂的干涉效应对于理解材料的强化和韧化机理具有十分重要的意义。文中研究了晶体材料中刃型位错和多条共圆弧刚性线夹杂的干涉作用。利用Riemann—Schwarz反照原理和复势函数的奇性主部分析技术，得到了问题的一般解答；对于只含一条刚性线夹杂的情况，给出了复势函数的封闭形式解。由Peach-Koehler公式求出了作用在刃型位错上的位错力，并讨论了圆弧形刚性线夹杂对位错力的影响规律，发现弧形刚性线对刃型位错有很强的排斥作用。本文解答不但可作为格林函数获得任意分布位错的相应解答，而且可以用于研究刚性线夹杂和任意形状裂纹的干涉效应问题。     

A Zener–Stroh crack interacting with a coated inclusion with generalized Irwin plastic zone correction

The elastic–plastic fracture behavior of a Zener–Stroh crack interacting with a coated inclusion in composite materials has been investigated with crack tip plastic zone corrections. With the distributed dislocation method, the crack problem is formulated into a set of singular integral equations which are solved numerically. The plastic zone sizes at the both crack tips are determined by a generalized Irwin model where Von Mises stress yielding criterion is used. The stress intensity factor (SIF), the plastic zone size (PZS), the crack tip opening displacement (CTOD) and the effective stress intensity factor have been evaluated. In the numerical examples, the influence of the inclusion shear modulus, the coating-layer thickness and shear modulus, as well as the distance between the crack and inclusion, on the SIF, the PZS and the CTOD are discussed in detail. Numerical examples show that increasing the shear modulus or the thickness of the coating phase, the influence of the inclusion on the normalized SIF and the normalized PZS will be shielded.     

Stress intensity factors of a crack in a composedcircular cylinder

A composed circular cylinder, formed by a core circular cylinder, containing a crackand enclosed by a layered hollow circular cylinder, is investigated in regard to the evaluation ofstress intensity factors. Analytic solutions to the problem are provided, with which the upper andlower bounds of stress intensity factors in a cracked circular cylinder, the stress distribution in alayered hollow circular cylinder, and the stress intensity factors for a crack in the composedcircular cylinder can precisely be determined. Numerical materials, demonstrating the discretevalues of the stress intensity factors, as well as the general pattern according to which the stressintensity factors vary with the material and geometric constants, are presented. The solutions aredeveloped based on a simplified and modified solution to the Hilbert problem, and the matrixpresentation and manipulation of functions and variables, used in the circuit theory.     

Antiplane problem of circular ARC interfacial rigid line inclusions

The antiplane problem of circular arc rigid line inclusions under antiplane concentrated force and longitudinal shear loading was dealt with. By using RiemannSchwaz‘s symmetry principle integrated with the singularity analysis of complex functions, the general solution of the problem and the closed form solutions for some important practical problems were presented. The stress distribution in the immediate vicinity of circular arc rigid line end was examined in detail. The results show that the singular stress fields near the rigid inclusion tip possess a square-root singularity similar to that for the corresponding crack problem under antiplane shear loading, but no oscillatory character. Furthermore, the stresses are found to depend on geometrical dimension, loading conditions and materials parameters. Some practical results concluded are in agreement with the previous solutions.     

Order of singularity and singular stress field about an axisymmetric interface corner in three-dimensional isotropic elasticity

The spatial axisymmetric problem of an isotropic, elastic, homogeneous body containing an inclusion of a different material with an interface corner point (along a circular contour) and arbitrary joining angles is considered in this paper. It is found that the order of the stress singularity at the interface corner coincides with that of the corresponding plane strain problem, but the dependence of the singular stress field on the material properties depends on both the Poisson ratios (of the two isotropic materials) as well as on the ratio of their shear moduli. The theoretical results have been confirmed by numerical, finite-element-based results in a special bimaterial case.     

压电螺位错与含界面裂纹圆形涂层夹杂的干涉

研究位于基体或夹杂中任意点的压电螺型位错与含界面裂纹圆形涂层夹杂的电弹耦合干 涉问题. 运用复变函数方法,获得了基体,涂层和夹杂中复势函数的一般解答. 典型例 子给出了界面含有一条裂纹时,复势函数的精确级数形式解. 基于已获得的复势函数和广 义Peach-Koehler公式,计算了作用在位错上的像力. 讨论了裂纹几何条件,涂层厚度和材 料特性对位错平衡位置的影响规律. 结果表明,界面裂纹对涂层夹杂附近的位错运动有很大 的影响效应,含界面裂纹涂层夹杂对位错的捕获能力强于完整粘结情况;并发现界面裂纹长 度和涂层材料常数达到某一个临界值时可以改变像力的方向. 解答的特殊情形包含了以 往文献的几个结果.     

Analysis of a cracked concrete containing an inclusion with inhomogeneously imperfect interface

The distributed dislocation technique is applied to determine the behavior of a cracked concrete matrix containing an inclusion. The analysis of cracked concrete in the presence of inclusions such as steel expansions is a practical problem that needs special attention. The solution to the problem of interaction of an edge dislocation with a circular inclusion having circumferentially inhomogeneously imperfect interface is available in the literature. This analytical solution is used in the distributed dislocation technique to obtain the stress intensity factor for the cracked concrete in the presence of inclusion. The interface of the matrix and the inclusion is assumed inhomogeneously imperfect and the stress intensity factor is determined for the cracked concrete for a case of two identical cracks on diametrically opposite sides of the inclusion. Consideration of this general inhomogeneously imperfect interface is the contribution of this paper. The variation of the inhomogeneity parameters is studied and presented. Additionally, the general assumption for the interface is simplified to the special case of perfectly bonded interface. The observations for the perfect interface are coincident with the previously reported results.     

Antiplane study on confocally elliptical inhomogeneity problem using an alternating technique

This paper presents a novel efficient procedure to analyze the two-phase confocally elliptical inclusion embedded in an unbounded matrix under antiplane loadings. The antiplane loadings considered in this paper include a point force and a screw dislocation or a far-field antiplane shear. The analytical continuation method together with an alternating technique is used to derive the general forms of the elastic fields in terms of the corresponding problem subjected to the same loadings in a homogeneous body. This approach could lead to some interesting simplifications in solution procedures, and the derived analytical solution for singularity problems could be employed as a Green's function to investigate matrix cracking in the corresponding crack problems. Several specific solutions are provided in closed form, which are verified by comparison with existing ones. Numerical results are provided to show the effect of the material mismatch, the aspect ratio, and the loading condition on the elastic field due to the presence of inhomogeneities.