压电材料反平面应变状态的任意形状夹杂问题

应用复函数的Ｆａｂｅｒ级数展开方法，分析了含任意形状夹杂的压电材料反平面应变问题，给出了问题的复势函数解。利用这个解，具体讨论了椭圆形夹杂及其极限（几何方面与物理方面）问题。并给出了三角形、正方形夹杂的近似结果。其特例结果与早期工作一致     

Eshelby problem of polygonal inclusions in anisotropic piezoelectric full- and half-planes

This paper presents an exact closed-form solution for the Eshelby problem of polygonal inclusion in anisotropic piezoelectric full- and half-planes. Based on the equivalent body-force concept of eigenstrain, the induced elastic and piezoelectric fields are first expressed in terms of line integral on the boundary of the inclusion with the integrand being the Green's function. Using the recently derived exact closed-form line-source Green's function, the line integral is then carried out analytically, with the final expression involving only elementary functions. The exact closed-form solution is applied to a square-shaped quantum wire within semiconductor GaAs full- and half-planes, with results clearly showing the importance of material orientation and piezoelectric coupling. While the elastic and piezoelectric fields within the square-shaped quantum wire could serve as benchmarks to other numerical methods, the exact closed-form solution should be useful to the analysis of nanoscale quantum-wire structures where large strain and electric fields could be induced by the misfit strain.     

Analysis of a Partially Debonded Conducting Rigid Elliptical Inclusion in a Piezoelectric Matrix

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General solution for Eshelby’s problem of 2D arbitrarily shaped piezoelectric inclusions

Eshelby’s problem of piezoelectric inclusions arises sometimes in exploiting the electromechanical coupling effect in piezoelectric media. For example, it intervenes in the nanostructure design of strained semiconductor devices involving strain-induced quantum dot (QD) and quantum wire (QWR) growth. Using the extended Stroh formalism, the present work gives a general analytical solution for Eshelby’s problem of two-dimensional arbitrarily shaped piezoelectric inclusions. The key step toward obtaining this general solution is the derivation of a simple and compact boundary integral expression for the eigenfunctions in the extended Stroh formalism applied to Eshelby’s problem. The simplicity and compactness of the boundary integral expression derived make it much less difficult to analytically tackle Eshelby’s piezoelectric problem for a large variety of non-elliptical inclusions. In the present work, explicit analytical solutions are obtained and detailed for all polygonal inclusions and for the inclusions characterized by Jordan’s curves and Laurent’s polynomials. By considering the piezoelectric material GaAs (110), the analytical solutions provided are illustrated numerically to verify the coincidence between different expressions, and to clarify the jump across the boundary of the inclusion and the singularity around the corner of the inclusion.     

Interacting circular inclusions in antiplanepiezoelectricity

The problem of multiple piezoelectric circular inclusions, which are perfectly bondedto a piezoelectric matrix, is analyzed in the framework of linear piezoelectricity. Both the matrixand the inclusions are assumed to possess the symmetry of a hexagonal crystal in the 6 mm classand subject to electromechanical loadings (singularities) which produce in-plane electric fieldsand out-of-plane displacement. Based upon the complex variable theory and the method ofsuccessive approximations, the solution of electric field and displacement field either in theinclusions or in the matrix is expressed in terms of explicit series form. Stress and electric fieldconcentrations are studied in detail which are dependent on the mismatch in the materialconstants, the distance between two circular inclusions, and the magnitude of electromechanicalloadings. It is shown that, when the two inclusions approach each other, the oscillatory behaviorof the stress and electric field can be induced in the inclusion as the matrix and the inclusions arepoled in the opposite directions. This important phenomenon can be utilized to build a verysensitive sensor in a piezoelectric composite material system. The present derived solution canalso be applied to the inclusion problem with straight boundaries. The problem associated withthree-material media under electromechanical sources is also considered.     

General coupled solution of anisotropic piezoelectric materials with an elliptic inclusion

In this investigation, the Stroh formalism is used to develop a general solution for an infinite, anisotropic piezoelectric medium with an elliptic inclusion. The coupled elastic and electric fields both inside the inclusion and on the interface of the inclusion and matrix are given. The project supported by the National Natural Science Foundation of China     

Analysis of Beams with Piezoelectric Actuators

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The effective thermoelectroelastic properties of microinhomogeneous materials

The paper is concerned with composite materials which consist of a homogeneous matrix phase with a set of inclusions uniformly distributed in the matrix. The components of these materials are considered to be ideally elastic and exhibit piezoelectric properties. One of the variants of the self-consistent scheme, the Effective Field Method (EFM) is applied to calculate effective dielectric, piezoelectric and thermoelastic properties of such materials, taking into account the coupled electroelastic effects. At first the coupled thermoelectroelastic problem for a homogeneous medium with an isolated inclusion is solved. For an ellipsoidal inclusion and constant external field the solution of this problem is found in a closed analytic form. This solution is then used in the EFM to derive the effective thermoelectroelastic operator for the composite containing a random array of ellipsoidal inclusions. Explicit formulae for the electrothermoelastic constants are given for composites, reinforced by spheroidal inclusions.     

含椭圆形刚性夹杂的压电材料平面问题

应用复变函数的Ｆａｂｅｒ级数展开方法，本文研究了含椭圆形刚性夹杂的压电材料平面问题，给出了问题的封闭解。解签表明，夹杂内的电场强度和电位移为常量。并通过算例分析，讨论了正，逆压电效应在基体孔周处的机电行为。     

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.     

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.     

The scattering of electroelastic waves by an ellipsoidal inclusion in piezoelectric medium

The scattering problem for a single ellipsoidal piezoelectric inclusion embedded in piezoelectric medium is investigated. Based on the polarization method, the extended displacements are expressed in terms of integral equations, whose kernels are obtained from the Green’s functions of homogenous matrix. In this paper, the 3D dynamic Green’s functions are derived by means of the Radon transform technique. To illustrate the use of the equations, scattering by a piezoelectric, ellipsoidal inhomogeneity in a piezoelectric medium is considered in the low frequency and an asymptotic formula for this scattering cross-section is obtained. Numerical results of the scattering cross-sections are carried out for a spheroidal BaTiO₃-inclusion in a PZT-5H-matrix.     

Analysis of a Circular Piezoelectric Semiconductor Embedded in a Piezoelectric Semiconductor Substrate

We analyze anti-plane electromechanical fields associated with a circular piezoelectric semiconductor of 6 mm symmetry embedded in a matrix of a different piezoelectric semiconductor. An exact solution is obtained. The solution shows the presence of field concentration near the interface. It is also found that the strain and electric fields inside the inclusion are not uniform.     

压电复合材料中的Eshelby夹杂问题

通过采用解析延拓和共形映射技术，获得了压电复合材料中有关Eshelby夹杂几个典型问题的精确弹性解答，即横观各向同性压电介质中任意形状的Eshelby夹杂与圆柱异相夹杂间相互作用；一般各向异性压电介质中任意形状的Eshelby夹杂与双压电材料所形成界面的相互作用．成功求解这些问题的关健在于构造一个辅助函数．与Ru所采用的方法不同，所引入的辅助函数在无穷远点不存在极点，从而使得所展开的分析更加自然合理．分析结果清楚地揭示出Eshelby夹杂的存在对压电复合材料机电耦合响应将产生不容被忽视的影响．很典型的一个例于是当一个Eshelby椭圆夹杂与圆柱异相夹杂相互作用时，每个夹杂体内部的应力场和电场都将是不均匀的；另一个例于是位于界面附近的Eshelby夹杂有可能是界面发生损伤的一个重要原因．     

Analysis of wave propagation in piezoelectric coupled cylinder affected by transverse shear and rotary inertia

In this paper, we examine the wave propagation in a piezoelectric coupled cylindrical shell affected by the shear effect and rotary inertia. A complete mathematical analysis of wave propagation solution in this piezoelectric coupled cylindrical shell is provided. The dispersion characteristics are derived through the solving an eigenvalue problem. Results are validated by the classical solution of a metallic cylinder. Besides providing and discussing the dispersion curves for different wave modes, we also examine the piezoelectric effects on the dispersion curves. Further to the above investigation, comparison of dispersion solutions from different shell theories is also conducted. This work may serve as a benchmark for wave propagation in piezoelectric coupled cylindrical shells.     

Piezoelectric Study on Elliptic Inhomogeneity Problem Using Alternating Technique

This paper presents a novel efficient procedure to analyze the elliptical inhomogeneity problem in piezoelectric materials under electromechanical loadings. The electromechanical loadings considered in this paper include a point force and a point charge or a far-field anti-plane shear and in-plane electric field. The analytical continuation method together with alternating technique is used to derive the electroelastic fields in terms of the corresponding homogeneous solution. Compared to existing related papers, this approach could lead to some interesting simplifications in solution procedure and the derived analytical solution for singularity problems can be employed as a Green's function to investigate matrix cracking in the inclusion/matrix system. Numerical results are provided to show the effect of the material mismatch, the aspect ratio and the loading condition on the electroelastic field due to the presence of the inhomogeneity.     

Plane problems in piezoelectric media with elliptic inclusion

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Electroelastic interaction between piezoelectric screw dislocation and circularly layered inclusion with imperfect interfaces

The interaction between a piezoelectric screw dislocation and an interphase layer in piezoelectric solids is theoretically investigated.Here,the dislocation located at arbitrary points inside either the matrix or the inclusion and the interfaces of the interphase layer are imperfect.By the complex variable method,the explicit solutions to the complex potentials are given,and the electroelastic fields can be derived from them.The image force acting on the dislocation can be obtained by the generalized PeachKoehler formula.The motion of the piezoelectric screw dislocation and its equilibrium positions are discussed for variable parameters.The important results show that,if the inner interface of the interphase layer is imperfect and the magnitude of degree of the interface imperfection reaches the certain value,two equilibrium positions of the piezoelectric screw dislocation in the matrix near the interface are found for the certain material combination which has never been observed in the previous studies(without considering the interface imperfection).     

Piezoelectric screw dislocations interacting with a circular inclusion with imperfect interface

The electroelastic coupling interaction between multiple screw dislocations and a circular inclusion with an imperfect interface in a piezoelectric solid is investigated. The appointed screw dislocation may be located either outside or inside the inclusion and is subjected to a line charge and a line force at the core. The analytic solutions of electroelastic fields are obtained by means of the complex-variable method. With the aid of the generalized Peach–Koehler formula, the explicit expressions of image forces exerted on the piezoelectric screw dislocations are derived. The motion and the equilibrium position of the appointed screw dislocation near the circular interface are discussed for variable parameters (interface imperfection, material electroelastic mismatch, and dislocation position), and the influence of the nearby parallel screw dislocations is also considered. It is found that the piezoelectric screw dislocation is always attracted by the electromechanical imperfect interface. When the interface imperfection is strong, the impact of material electroelastic mismatch on the image force and the equilibrium position of the dislocation becomes weak. Additionally, the effect of the nearby dislocations on the mobility of the appointed dislocation is very important.     

The effective properties of piezocomposites, part I: Single inclusion problem

The problem of a piezoelectric ellipsoidal inclusion in an infinite nonpiezoelectric matrix is very important in engineering. In this paper, it is solved via Green's function technique. The closed-form solutions of the electroelastic Eshelby's tensors for this kind of problem are obtained. The electroelastic Eshelby's tensors can be expressed by the Eshelby's tensors of the perfectly elastic inclusion problem and the perfectly dielectric inclusion problem. Since the closed-form solutions of the Eshelby's tensors of the perfectly elastic inclusion problem and the perfectly dielectric inclusion problem can be given by theory of elasticity and electrodynamics, respectively, the electroelastic Eshelby's tensors can be obtained conveniently. Using these results, the closed-form solutions of the constraint elastic fields and the constraint electric fields inside the piezoelectric ellipsoidal inclusion are also obtained. These expressions can be readily utilized in solutions of numerous problems in the micromechanics of piezoelectric solids, such as the deformation and energy analysis, damage evolution and fracture of the piezoelectric materials. The project supported by the National Natural Science Foundation of China