纵向剪切下含共焦刚性核弹性椭圆夹杂中的螺型位错干涉

摘 要 研究了无穷远纵向剪切下无限大基体中含共焦刚性核弹性椭圆夹杂内任意位置螺型位错的干涉问题。运用复变函数保角映射、解析延拓等方法,获得了基体与夹杂区域的应力场的级数形式精确解,并得出了位错像力的解析表达式,导出了纵向剪切下两椭圆界面最大应力及其比值公式。分析结果表明：夹杂内部的刚性核对位错与夹杂的干涉产生明显的扰动效应,排斥硬夹杂内位错,并使之不断趋近弹性夹杂界面。对于软夹杂,夹杂中的位错存在稳定的平衡位置,当位错位于刚性核和平衡位置之间时,位错会趋于弹性夹杂界面;当位错位于平衡位置和弹性夹杂界面之间时,位错会离开界面。结果还显示,夹杂的长轴和短轴之比对位错与夹杂的干涉也有着不可忽视的影响,尤其当位错在刚性核附近时,随着夹杂的长、短轴比值的减小,核对位错的排斥力也明显减弱。本文解答包含了多个以往文献成果。     

The interaction between a screw dislocation and a circular inhomogeneity in gradient elasticity

The interaction between a screw dislocation and a circular inhomogeneity in gradient elasticity is investigated. The screw dislocation is located inside either the inhomogeneity or the matrix. By using the Fourier transform method, closed analytical solutions are obtained when the inhomogeneity and the matrix have the same gradient coefficient. The explicit expressions of image forces exerted on screw dislocations are derived. The motion of the appointed screw dislocation and its equilibrium positions are discussed. The results show that the classical singularity is eliminated. Especially, for the case of a tiny inhomogeneity, the relation of dislocations and inhomogeneities become quite different. The screw dislocation may be attracted by the stiff inhomogeneity and repelled by the soft inhomogeneity when it tends to the interface. So there is an unstable equilibrium position when a dislocation tends to a tiny stiff inhomogeneity and there is a stable equilibrium position when a dislocation tends to a tiny soft inhomogeneity.     

Shielding effect and emission condition of a screw dislocation near a blunt crack in elliptical inhomogeneity

The shielding effect and emission condition of a screw dislocation near a blunt crack in elastic elliptical inhomogeneity is dealt with. Utilizing the Muskhelishvili complex variable method, the explicit series form solutions of the complex potentials in the matrix and the inclusion regions are derived. The stress intensity factor and critical stress intensity factor for dislocation emission are also calculated. The influences of the orientation of the dislocation and morphology of the blunt crack as well as the material elastic dissimilarity upon the shielding effect and emission criterion are discussed in detail. As a result, numerical analysis and discussion show that the positive screw dislocation can reduce the stress intensity factor of the crack tip (shielding effect) only when it is located in the certain region. The shielding effect increases with the increase of the shear modulus of the matrix and the curvature radius of the blunt crack tip, but decreases with the increase of dislocation azimuth angle. The critical loads at infinity for dislocation emission increases with the increment of the emission angle and the curvature radius of the blunt crack tip, and the most probable angle for screw dislocation emission is zero. The present solutions contain previous results as the special cases.     

穿透夹杂界面的半无限楔形裂纹尖端螺型位错的发射研究

摘要：研究了穿透圆形夹杂界面的半无限楔形裂纹与裂纹尖端螺型位错的干涉问题。应用复变函数解析延拓技术与奇性主部分析方法,得到了位错位于半圆形夹杂内部时,半无限基体和半圆形夹杂内复势函数的解析解。然后利用保角映射技术得到了穿透圆形夹杂界面的半无限楔形裂纹尖端螺型位错产生的应力场以及作用在位错上的位错力的解析表达式。主要讨论了螺型位错对裂纹的屏蔽效应以及从楔形裂纹尖端发射位错的临界载荷条件。研究结果表明正的螺型位错可以削弱楔形裂纹尖端的应力强度因子,屏蔽裂纹的扩展,屏蔽效应随位错方位角的增大而减小。位错发射所需的无穷远临界应力随发射角的增加而增大,最可能的位错发射角度为零度,直线裂纹尖端位错的发射比楔形裂纹尖端位错的发射更容易,硬基体抑制位错的发射。     

Edge dislocation interacting with an interfacial crack along a circular inhomogeneity

The elastic interaction of an edge dislocation, which is located either outside or inside a circular inhomogeneity, with an interfacial crack is dealt with. Using Riemann–Schwarz’s symmetry principle integrated with the analysis of singularity of the complex potentials, the closed form solutions for the elastic fields in the matrix and inhomogeneity regions are derived explicitly. The image force on the dislocation is then determined by using the Peach–Keohler formula. The influence of the crack geometry and material mismatch on the dislocation force is evaluated and discussed when the dislocation is located in the matrix. It is shown that the interfacial crack has significant effect on the equilibrium position of the edge dislocation near a circular interface. The results also reveal a strong dependency of the dislocation force on the mismatch of the shear moduli and Poisson’s ratios between the matrix and inhomogeneity.     

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.     

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.     

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.     

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.     

The interaction of a screw dislocation with a circular inhomogeneity near the free surface

The interaction of a screw dislocation with a circular inhomogeneity near the free surface is discussed in this paper. By using the complex potential and conformal mapping technique, an explicit series solution is obtained. Then, the solution is cast into a new expression to separate the interaction effects between the dislocation, inhomogeneity, and free surface. The new expression is not only convenient to reveal the coupling interaction effects, but also helpful to improve the convergence of the solution. As an application of the new expression, a simple approximate formula is presented with high accuracy. Finally, the full-field interaction energy and image force are evaluated and studied graphically. It is found that when the screw dislocation, inhomogeneity, and free surface are close to each other, their interaction effects strongly and intricately couple in the near field. In the case of a soft inhomogeneity or a hole, there is an unstable equilibrium point of the screw dislocation between the inhomogeneity and free surface, whereas in the case of a hard or rigid inhomogeneity, there is an unstable equilibrium point on the opposite side of the inhomogeneity.     

螺型位错偶极子与圆形界面刚性线的干涉效应研究

本文重点研究螺型住错偶极子和圆形夹杂界面刚性线的弹性干涉效应。利用复变函数方法,得到了该问题的一般解;此外还求出了只含一条界面刚性线时的封闭解答,得到了刚性线尖端的应力强度因子以及作用在螺型位错偶板子中心的像力和像力偶矩。研究结果表明：位错偶板子对应力强度因子具有很强的屏蔽或反屏蔽效应;软夹杂吸引位错偶极子,而刚性线排斥位错偶板子,在一定条件下,位错偶极子在刚性线附近出现一个平衡位置;当刚性线的长度争材料剪切模量比达到临界值时,可以改变偶极子和界面之间的干涉机理;刚性线长度对位错偶极子中心像力偶矩也有很大的影响。     

Electroelastic interaction between a piezoelectric screw dislocation and a circular inhomogeneity with interfacial cracks

A piezoelectric screw dislocation in the matrix interacting with a circular inhomogeneity with interfacial cracks under antiplane shear and in-plane electric loading at infinity was dealt with. Using complex variable method, a general solution to the problem was presented. For a typical case, the closed form expressions of complex potentials in the inhomogeneity and the matrix regions and derived explicitly when the interface containsthe electroelastic field intensity factors weresingle crack. The image force acting on the piezoelectric screw dislocation was calculated by using the perturbation technique and the generalized Peach-Koehler formula. As a result, numerical analysis and discussion show that the perturbation influence of the interfacial crack on the interaction effects of the dislocation and the inhomogeneity is significant which indicates the presence of the interfacial crack will change the interaction mechanism when the length of the crack goes up to a critical value. It is also shown that soft inhomogeneity can repel the dislocation due to their intrinsic electromechanical coupling behavior.     

On a semi-infinite crack penetrating a piezoelectric circular inhomogeneity with a viscous interface

We investigate a semi-infinite crack penetrating a piezoelectric circular inhomogeneity bonded to an infinite piezoelectric matrix through a linear viscous interface. The tip of the crack is at the center of the circular inhomogeneity. By means of the complex variable and conformal mapping methods, exact closed-form solutions in terms of elementary functions are derived for the following three loading cases: (i) nominal Mode-III stress and electric displacement intensity factors at infinity; (ii) a piezoelectric screw dislocation located in the unbounded matrix; and (iii) a piezoelectric screw dislocation located in the inhomogeneity. The time-dependent electroelastic field in the cracked composite system is obtained. Particularly the time-dependent stress and electric displacement intensity factors at the crack tip, jumps in the displacement and electric potential across the crack surfaces, displacement jump across the viscous interface, and image force acting on the piezoelectric screw dislocation are all derived. It is found that the value of the relaxation (or characteristic) time for this cracked composite system is just twice as that for the same fibrous composite system without crack. Finally, we extend the methods to the more general scenario where a semi-infinite wedge crack is within the inhomogeneity/matrix composite system with a viscous interface.     

含非完整界面涂层夹杂中螺型位错的弹性分析

研究了含非完整界面圆形涂层夹杂内部一个螺型位错在夹杂、涂层与无限大基体材料中产生的弹性场.运用复变函数函数方法,获得了三个区域复势函数的解析解答.利用求得的应力场和Peach-Koehler公式,得到了作用在螺型位错上位错力的精确表达式.主要讨论了两个非完整界面对位错力的影响规律.结果表明,涂层界面对夹杂内部螺型位错的吸引力随着界面粘结强度的弱化而变大.界面非完整程度增加削弱材料弹性失配对位错力的影响.在一定条件下,非完整界面可以改变夹杂内位错与涂层/基体系统之间的引斥干涉规律,并使位错在夹杂内部产生一个稳定或非稳定的平衡点.     

Interaction between a screw dislocation and an elastic elliptical inhomogeneity with interfacial cracks

The elastic interaction between a screw dislocation and an elliptical inhomogeneity with interfacial cracks is studied. The screw dislocation may be located outside or inside the inhomogeneity. An efficient complex variable method for the complex multiply connected region is developed, and the general solutions to the problem are derived. As illustrative examples, solutions in explicit series form for complex potentials are presented in the case of one or two interfacial cracks. Image forces on the dislocation are calculated by using the Peach-Koehler formula. The influence of crack geometries and material properties on the image forces is evaluated and discussed. It is shown that the interfacial crack has a significant effect on the equilibrium position of the dislocation near an elliptical-arc interface. The main results indicate, when the length of the crack goes up to a critical value, the presence of the interfacial crack can change the interaction mechanism between a screw dislocation and an elliptical inclusion. The present solutions can include a number of previously known results as special cases.The project supported by the National Natural Science Foundation of China(10272009 and 10472030) and the Natural Science Foundation of Hunan Province(02JJY2014)     

增强相/夹杂内螺位错偶极子与含共焦钝裂纹椭圆夹杂的干涉效应

运用弹性力学的复势方法,研究了纵向剪切下增强相/夹杂内螺型位错偶极子与含共焦钝裂纹椭圆夹杂的干涉效应,得到了该问题复势函数的封闭形式解答,由此推导出了夹杂区域的应力场、作用在螺型位错偶极子中心的像力和像力偶矩以及裂纹尖端应力强度因子级数形式解。并分析了位错偶极子倾角 、钝裂纹尺寸和材料常数对位错像力、像力偶矩以及应力强度因子的影响。数值计算结果表明：位错像力、像力偶矩以及应力强度因子均随位错偶极子倾角做周期变化;夹杂内部的椭圆钝裂纹明显增强了硬基体对位错的排斥,减弱了软基体对位错的吸引,且对于硬夹杂,位错出现了一个不稳定平衡位置,该平衡位置随钝裂纹曲率的增大不断向界面靠近;变化 值将出现改变位错偶极子对应力强度因子作用方向的临界值。     

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 screw dislocation near a coated non-elliptical inhomogeneity with internal uniform stresses

We consider the anti-plane shear deformation of a three-phase inhomogeneity-coating-matrix composite containing a coated non-elliptical inhomogeneity whose surrounding matrix is subjected to the action of a screw dislocation and uniform remote anti-plane shear stresses. Our objective is to establish conditions under which the inhomogeneity maintains an internal uniform stress field. Our analysis, which is based on a carefully chosen conformal mapping function, clearly indicates that such an internal uniform stress distribution can be achieved independently of the action of the screw dislocation, which influences the shape of the inhomogeneity depending on its proximity to the dislocation. In fact, we find that when the screw dislocation is located far from the coated inhomogeneity, the corresponding material interfaces become two confocal ellipses as reported previously in the literature. A simple criterion for the convergence of the series in the conformal mapping function is established.     

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…     

位错偶极子和Ⅲ型界面裂纹的干涉效应

研究了多晶体材料中螺型位错偶极子和界面裂纹的弹性干涉作用.利用复变函数方法,得到了该问题复势函数的封闭形式解答.求出了由位错偶极子诱导的应力场和裂纹尖端应力强度应子,分析了偶极子的方向,偶臂和位置以及材料失配对应力强度因子的影响.推导了作用在螺型位错偶极子中心的像力和力偶矩,并讨论了界面裂纹几何条件和不同材料特征组合对位错偶极子平衡位置的影响规律.结果表明,裂纹尖端的螺型位错偶极子对应力强度因子会产生强烈的屏蔽或反屏蔽效应.同时,界面裂纹对螺型位错偶极子在材料中运动有很强的扰动作用.