Anti-plane time-harmonic Green's functions for a circular inhomogeneity in piezoelectric medium with a spring- or membrane-type interface

The present work focuses on the two-dimensional anti-plane time-harmonic dynamic Green's functions for a circular inhomogeneity immersed in an infinite matrix with an imperfect interface, where both the inhomogeneity and matrix are assumed to be piezoelectric and transversely isotropic. Two types of imperfect interface, the spring-type interface with electromechanical coupling and the membrane-type interface, are considered. The former type is often used to model the electromechanical damage along the interface while the latter is largely employed to simulate surface/interface effect of nano-sized inhomogeneity. By using the Bessel function expansions, explicit solutions for the electromechanical fields induced by a time-harmonic anti-plane line force and line charge located in an unbounded matrix as well as the circular inhomogeneity are respectively derived. The present solutions can recover the anti-plane Green's functions for some special cases, such as the dynamic or quasi-static Green's functions of piezoelectricity with perfect interface as well as the dynamic or quasi-static Green's functions of pure elasticity with imperfect interface. For detailed discussions, selected analytical results are presented. Dependence of the electromechanical fields on circular frequency as well as interface properties is examined. The size effect related to interfacial imperfection is also discussed.     

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.     

On thermoelastic fields of a multi-phase inhomogeneity system with perfectly/imperfectly bonded interfaces

The stress fields of cylindrical and spherical multi-phase inhomogeneity systems with perfect or imperfect interfaces under uniform thermal and far-field mechanical loading conditions are investigated by use of the Boussinesq displacement potentials. The radius of the core inhomogeneity and the thickness of its surrounding coatings are arbitrary. The discontinuities in the tangential and normal components of the displacement at the imperfect interfaces are assumed to be proportional to the associated tractions. In this work, for the problems where the phases of the inhomogeneity system are homogeneous, the exact closed-form thermo-elastic solutions are presented. These solutions along with a systematic numerical methodology are utilized to solve various problems of physical importance, where the constituent phases of the inhomogeneity system may be made of a number of different functionally graded (FG) and homogeneous materials, and each interface may have a perfect or imperfect boundary condition, as desired. Also, the effect of the interfacial sliding and debonding on the stress field and elastic energy of an FG-coated inhomogeneity is examined.     

Antiplane deformation of a partially bonded elliptical inclusion

A new method that introduces two holomorphic potential functions (the two-phase potentials) is applied to analyze the antiplane deformation of an elliptical inhomogeneity partially-bonded to an infinite matrix. Elastic fields are obtained when either the matrix is subject to a uniform longitudinal shear or the inhomogeneity undergoes a uniform shear transformation. The stress field possesses the square-root singularity of a Mode III interface crack, which, in the special case of a rigid line inhomogeneity, changes in order, as the crack tip approaches the inhomogeneity end. In the latter situation the crack-tip elastic fields are linear in two real stress intensity factors related to a strong and a weak singularity of the stress field.     

Elastic Energies in Circular Inhomogeneities: Imperfect Versus Perfect Interfaces

The change in the total potential energy in a stressed elastic plane system, consisting of an unbounded matrix containing a cylindrical inhomogeneity of circular cross-section, is studied, when an imperfect bonding is formed across the interface. The imperfect bonding is simulated by linearly elastic springs distributed over the interface. Two loading cases are examined: an equilibrium system of fixed uniform tractions acting in the remote boundary of the matrix, and a phase transformation in the inhomogeneity prescribed by stress free uniform eigenstrains distributed in the inhomogeneity region. For both loadings, the fully elastic fields in explicit forms are derived involving the spring compliances and three new two-phase parameters depending on the elastic properties of the two materials. The elastic energies stored in the whole system and in its constituents are determined in simple and compact forms. It is shown that, in both loading cases, the total potential energy of the system is reduced. It is found that, in nanoscale, the ratio of the elastic energy stored in interface to the elastic energy stored in inhomogeneity increases rapidly for small values of the circular radius and tends to zero for large values. Also, this ratio increases as the matrix becomes softer compared to the inhomogeneity.     

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.     

Antiplane electro-mechanical field of a piezoelectric finite wedge under shear loading and at fixed-grounded boundary conditions

This paper presents the general solutions of antiplane electro-mechanical field solutions for a piezoelectric finite wedge subjected to a pair of concentrated forces and free charges. The boundary conditions on the circular segment are considered as fixed and grounded. Employing the finite Mellin transform method, the stress and electrical displacement at all fields of the piezoelectric finite wedge are derived analytically. In addition, the singularity orders and intensity factors of stress and electrical displacement can also be obtained. These parameters can be applied to examine the fracture behavior of the wedge structure. After being reduced to the problem of an antiplane edge crack or an infinite wedge in a piezoelectric medium, the results compare well with those of previous studies.     

压电材料反平面界面裂纹的杂交元分析

基于奇异性电弹场数值特征解开发了一种新型反平面界面裂纹尖端单元。将新型单元与四节点压电P-S单元组装,求解从绝缘到导通的任意电边界条件下,压电结构反平面界面裂纹尖端电弹场的数值解。考察了层厚、载荷类型和裂纹面间电边界条件等对反平面界面裂纹尖端断裂参数的影响。算例证明新型单元能使P-S单元数显著降低,计算结果更为精确。     

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.     

Analysis of a partially debonded elliptic inhomogeneity in piezoelectric materials

A generalized solution was obtained for the partially debonded elliptic inhomogeneity problem in piezoelectric materials under antiplane shear and inplane electric loading using the complex variable method. It was assumed that the interfacial debonding induced an electrically impermeable crack at the interface. The principle of conformal transformation and analytical continuation were employed to reduce the formulation into two Riemann-Hilbert problems. This enabled the determination of the complex potentials in the inhomogeneity and the matrix by means of series of expressions. The resulting solution was then used to obtain the electroeiastic fields and the energy release rate involving the debonding at the inhomogeneity-matrix interface. The validity and versatility of the current general solution have been demonstrated through some specific examples such as the problems of perfectly bonded elliptic inhomogeneity , totally debonded elliptic inhomogeneity, partially debonded rigid and conducting elliptic inhomogeneity, and partially debonded circular inhomogeneity.     

Moving antiplane shear crack in hexagonal piezoelectric crystals

Closed-form solutions are obtained and discussed for the stress and electric displacement fields around a loaded Griffith-type antiplane shear strip crack moving in hexagonal piezoelectric crystals. Representative numerical results are presented for ZnO and PZT-4.     

Dynamic interaction between a matrix crack and a circular inhomogeneity with a distinct interphase

This article provides a theoretical treatment of the dynamic interaction between a matrix crack and an arbitrarily located circular inhomogeneity with a distinct interphase under antiplane loading. The matrix⧹inhomogeneity interphase is characterized by a linear spring model. The theoretical formulations governing the steady state problem are based upon the use of integral transform techniques, Bessel function expansions and a Pseudo-incident wave technique. The closed form expression for the resulting stress intensity factor at the matrix crack is obtained by solving the appropriate singular integral equations using Chebyshev polynomials. Typical examples are provided to show the effect of the location of the inhomogeneity, the material combination and the interface property upon the dynamic stress intensity factor of the matrix crack.     

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.     

反平面V形切口塑性应力奇异性分析

论文提出了用插值矩阵法计算幂硬化塑性材料反平面V形切口和裂纹尖端区域的应力奇异性.首先在切口和裂纹尖端区域采用自尖端径向度量的渐近位移场假设,将其代入塑性全量理论的基本微分方程后,推导出包含应力奇异性特征指数和特征角函数的非线性常微分方程特征值问题.然后采用插值矩阵法迭代求解导出的控制方程,得到一般的塑性材料反平面V形切口和裂纹的前若干阶应力奇异阶和相应的特征角函数,该法的重要优点是以上求解的特征角函数和它们各阶导函数具有同阶精度,并且一次性地求出前若干阶特征对.同时,插值矩阵法计算量小,易于和其他方法联合使用,这些优点在后续求解尖端区域完全应力场非常优越.论文方法的计算结果与现有结果对照,发现吻合良好,表明了论文方法的有效性.     

Thermo-elastic properties of heterogeneous materials with imperfect interfaces: Generalized Levin's formula and Hill's connections

We study the thermo-elastic properties of heterogeneous materials containing spherical particles or cylindrical fibres. The interface between the matrix and second-phase inhomogeneity is imperfect with either the displacement or the stress experiencing a jump across it. We relate the effective coefficient of thermal expansion (CTE) to the effective elastic moduli and thereby generalize Levin's formula, and reveal two connections among the effective elastic moduli, thereby generalizing Hill's connections. In contrast to the classical results, the effective CTE in the presence of an imperfect interface is strongly dependent on the size of the inhomogeneity, besides the interface elastic and thermo-elastic properties. This size dependence has been accurately captured by simple scaling laws.     

Some problems in the antiplane shear deformation of bi-material wedges

The antiplane shear deformation of a bi-material wedge with finite radius is studied in this paper. Depending upon the boundary condition prescribed on the circular segment of the wedge, traction or displacement, two problems are analyzed. In each problem two different cases of boundary conditions on the radial edges of the composite wedge are considered. The radial boundary data are: traction–displacement and traction–traction. The solution of governing differential equations is accomplished by means of finite Mellin transforms. The closed form solutions are obtained for displacement and stress fields in the entire domain. The geometric singularities of stress fields are observed to be dependent on material property, in general. However, in the special case of equal apex angles in the traction–traction problem, this dependency ceases to exist and the geometric singularity shows dependency only upon the apex angle. A result which is in agreement with that cited in the literature for bi-material wedges with infinite radii. In part II of the paper, Antiplane shear deformation of bi-material circular media containing an interfacial edge crack is considered. As a special case of bi-material wedges studied in part I of the paper, explicit expressions are derived for the stress intensity factor at the tip of an edge crack lying at the interface of the bi-material media. It is seen that in general, the stress intensity factor is a function of material property. However, in special cases of traction–traction problem, i.e., similar materials and also equal apex angles, the stress intensity factor becomes independent of material property and the result coincides with the results in the literature.     

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.     

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.     

High frequency scattering due to a pair of time-harmonic antiplane forces on the faces of a finite interface crack between dissimilar anisotropic materials

The high-frequency elastodynamic problem involving the excitation of an interface crack of finite width lying between two dissimilar anisotropic elastic half-planes has been analyzed. The crack surface is excited by a pair of time-harmonic antiplane line sources situated at the middle of the cracked surface. The problem has first been reduced to one with the interface crack lying between two dissimilar isotropic elastic half-planes by a transformation of relevant co-ordinates and parameters. The problem has then been formulated as an extended Wiener–Hopf equation (cf. Noble, 1958) and the asymptotic solution for high-frequency has been derived. The expression for the stress intensity factor at the crack tips has been derived and the numerical results for different pairs of materials have been presented graphically.