A piezoelectric screw dislocation in a three-phase composite cylinder model with an imperfect interface

The electroelastic coupling interaction between a piezoelectric screw dislocation and the embedded circular cross-section inclusions with imperfect interfaces in piezoelectric solids is investigated by using a three-phase composite cylinder model. By means of a complex variable technique, the explicit solutions of electroelastic fields are obtained. With the aid of the Peach-Koehler formula, the explicit expression for the image force exerted on the piezoelectric screw dislocation is derived. The image force on the dislocation and its equilibrium positions near one of the inclusions are discussed for variable parameters (interface imperfection and material electroelastic dissimilarity) and the influence of nearby inclusions is also considered. The results show that when compared with the previous solution (the perfect interface), more equilibrium positions of the screw dislocation in the matrix may be available due to the effect of the interface imperfection when the dislocation is close to the electroelastic stiff inclusion. It is also found that the magnitude of the image force exerted on the piezoelectric screw dislocation produced by multiply inclusions is always smaller than that produced by a single inclusion and the impact of nearby inclusions on the mobility of the screw dislocation is very important.     

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.     

Closed form solutions for partially debonded circular inclusion in piezoelectric materials

Summary This paper deals with the problem of a partially debonded piezoelectric circular inclusion in a piezoelectric matrix. This boundary value problem is reduced to two Riemann-Hilbert problems through the use of the analytical continuation theory.Closed form solutions are obtained by considering the behavior of the complex field potentials at origin and infinity. The formulae for the electro-elastic field intensity factors of the interfacial crack are derivedexplicitly. Several particular cases are provided to show the effect of the crack angle, the mechanical and electrical properties and the loads on the electroelastic field singularities.     

Local Electroelastic Field and Effective Electroelastic Moduli of Piezoelectric Nanocomposites with Interface Effect

Due to the large ratio of surface area to volume in nanoscale objects, the property of surfaces and interfaces likely becomes a prominent factor in controlling the behavior of nano-heterogeneous materials. In this work, based on the Gurtin-Murdoch surface/interface elastic theory, a distinct expression is derived for embedded nano-inclusion in an infinite piezoelectric matrix coupled with interface effect. For the problem of a spherical inclusion in transversely isotropic piezoelectric medium, we reach a conclusion that the elastic and electric field are uniform when eigen-strain and eigen-electric field imposed on the inclusion are uniform even in the presence of the interface influence. The electroelastic fields in the inclusion are related to both interface electroelastic parameters and the radius of the inclusion. Then overall properties of the composites are estimated by using the dilute distribution model. Numerical results reveal that the effective electroelastic moduli are function of the interface parameters and the size of the nano-inhomogeneities.     

压电螺型位错与椭圆夹杂界面导电刚性线的干涉效应

研究了无限大压电基体材料中压电螺型位错与含界面导电刚性线椭圆夹杂的电弹耦合干涉问题。运用求解复杂多连通域问题的复变函数方法,获得了椭圆夹杂和基体区域复势函数以及电弹性场的精确级数形式解答。利用广义Peach-Koehler公式导出作用于压电螺型位错上的位错力公式。主要讨论了刚性线几何尺寸和椭圆曲率对位错力的影响规律。分析结果表明:界面刚性线排斥基体中的位错,对靠近椭圆夹杂界面的螺型位错的运动和平衡位置有重要的影响。当刚性线的长度达到临界值,界面刚性线的存在会改变螺型位错与压电椭圆夹杂的干涉规律。椭圆夹杂的压缩系数变大,刚性线尺寸对位错力的影响也越大。     

Two circular inclusions with inhomogeneous interfaces interacting with a circular Eshelby inclusion in anti-plane shear

Summary An analytical solution in infinite series form for two circular cylindrical elastic inclusions embedded in an infinite matrix with two circumferentially inhomogeneous imperfect interfaces interacting with a circular Eshelby inclusion in anti-plane shear is derived by employing complex variable techniques. All of those coefficients in the series can be uniquely determined in a simple and transparent way. Numerical examples are given to illustrate the effect of imperfection and circumferential inhomogeneity of the two interfaces as well as the size, location and elastic properties of the two circular inclusions on the stress fields induced within the two circular inclusions and the Eshelby inclusion.     

Closed-form solution for a coated circular inclusion under uniaxial tension

A coated circular inclusion embedded in an infinite matrix is analyzed in the framework of two-dimensional isotropic linear elasticity. A closed-form solution is obtained for the case of far-field uniaxial tension using Muskelishvilis complex potential method. The solutions for the stress and strain distributions for all three regions, that is, matrix, coating, and inclusion, were obtained for various coating-to-matrix shear modulus ratios, while keeping the fiber and matrix shear moduli the same. Test cases for an inclusion without the coating and hollow inclusion were also studied. The energy release rate was evaluated using the path-independent M-integral, which is used to calculate the energy release rate for the self-similar expansion of defects surrounded by the closed contour of the integral. The results for the stress and strain concentrations along with the energy release rate due to this material inhomogeneity were analyzed to yield a better understanding of the mechanics of materials with circular inclusions. This can be helpful in designing intelligent composite structures with embedded optical fiber sensors.     

Neutral-coated circular piezoelectric inclusions

This study is concerned with the neutrality of a coated circular piezoelectric inclusion when the matrix is subjected to remote uniform electroelastic loadings. Two design schemes are proposed to make the coated inclusion neutral. In the first scheme, the thickness of the coating can be designed for given electroelastic constants of the composite so as to make the inclusion neutral to certain remote uniform loadings. In the second scheme, the electroelastic constants of the coating can be designed for given electroelastic constants of the inclusion and the matrix, and given coating thickness so as to make the inclusion neutral to any remote uniform loadings. The analysis indicates that even in the absence of the coating, the piezoelectric inclusion can be made neutral if the electroelastic constants of the two-phase composite satisfy a restriction.     

压电材料中螺型位错与含界面刚性线圆形涂层夹杂的干涉

研究了压电材料中位于基体的螺型位错与含界面刚性线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精确级数形式解答。基于广义Peach-Koehler公式,计算了作用在位错上的像力。讨论了刚性线几何条件、界面层厚度和材料电弹特性对位错力和位错平衡位置的影响规律。结果表明:对于软夹杂和软涂层的情况,刚性线长度存在一个临界值改变像力的方向。螺型位错先被吸引后被排斥,在夹杂附近有一个稳定的平衡点。对于硬夹杂和硬涂层的情况,位错一直被排斥,刚性线对位错力的影响较小。     

Null-field approach for piezoelectricity problems with arbitrary circular inclusions

In this paper, we derive the null-field integral equation for piezoelectricity problems with arbitrary piezoelectric circular inclusions under remote anti-plane shears and in-plane electric fields. Separable expressions of fundamental solutions and Fourier series for boundary densities are adopted to solve the piezoelectric problem with circular inclusions. Four gains are obtained: (1) well-posed model, (2) singularity free, (3) boundary-layer effect free and (4) exponential convergence. The solution is formulated in a manner of semi-analytical form since error purely attributes to the truncation of Fourier series. Two piezoelectric problems with two piezoelectric circular inclusions are revisited and compared with the Chao and Chang's solutions to demonstrate the validity of our method. The limiting case that the two inclusions separate far away leads to the Pak's solution of a single inclusion. Stress and electric field concentrations are calculated and are dependent on the distance between the two inclusions, the mismatch in the material constants and the magnitude of mechanical and electromechanical loadings. The results for the shear and electric loadings in two directions are also compared well with the Wang and Shen's results.     

Effect of an interphase layer on the electroelastic stresses within a three-phase elliptic inclusion

Electroelastic stresses induced by electromechanical loadings and lattice mismatch between components and surrounding materials are found to significantly influence the electronic performance of devices and, in some cases, are identified as a major cause of failure and degradation. To reduce electromechanical failure an effective method is to apply an intermediate layer, with appropriate geometry and material properties, between the components of dissimilar piezoelectric materials. In this paper, the effect of an intermediate layer on the electroelastic stresses within an elliptical inhomogeneity is examined within the framework of linear piezoelectricity. Exact closed-form solutions are obtained for the electroelastic stresses in the inclusion, the interphase layer and the matrix, respectively, under remote mechanical antiplane shear and inplane electric field, by means of the complex variable method. It is shown that the electroelastic stresses depend on only two complex coefficients. Simple formulae and numerical examples are used to illustrate the effects of the interphase layer on the electroelastic stresses within the inclusion, and the dependency of this effect on the aspect ratio of the elliptical inclusion.     

Elliptical inclusion problem in antiplane piezoelectricity: Implications for fracture mechanics

An elliptical piezoelectric inclusion embedded in an infinite piezoelectric matrix is analyzed in the framework of linear piezoelectricity. Using the conformal mapping technique, a closed-form solution is obtained for the case of a far-field antiplane mechanical load and an inplane electrical load. The solution to a permeable elliptical hole problem is obtained as a limiting case of vanishing elastic modulus of the inclusion. This enables the study of the nature of crack tip electric field singularity which is shown to depend on the electrical boundary condition imposed on the crack faces. The energy release rate of a self-similarly expanding slender crack in the presence of electric fields is obtained by using the generalized M-integral. The energy release rate expression indicates that the electric field has a crack-arresting influence. This effect is inferred to have a more fundamental physical origin in the interaction between the applied electric field and the induced surface charges on the crack faces. An experimental result contradicting the theoretical prediction on the crack-arresting effect is also discussed.     

Micromechanics determination of electroelastic properties of piezoelectric materials containing voids

This paper examines the electroelastic properties of piezoelectric materials that contain voids. A unified micromechanics approach is adopted for determining the properties. Voids are treated as spheroidal inclusions with zero elastic moduli. The surrounding material is assumed to be linearly piezoelastic and transversely isotropic. The electroelastic Eshelby tensors for spheroidal inclusions have been evaluated numerically for different aspect ratios. Utilizing these tensors and applying the Mori–Tanaka mean field theory that accounts for the interaction between inclusions and matrix, the effective electroelastic properties of the materials are obtained. Numerical examples are given based on PZT-5H and BaTiO₃. Influences of the volume fraction and aspect ratio of voids on the material properties have been studied. Emphasis has been placed on the piezoelastic coupling effect of the material. For both materials, the piezoelastic coupling provides a stiffening effect on the materials, and the influence is more pronounced when void volume increases and when the aspect ratio of voids becomes shorter.     

圆环形截面压电纤维复合材料有效电弹性性能研究

研究含周期分布压电纤维的压电复合材料的有效电弹性性能。通过在材料代表性体积单元边界上施加位移和电势周期边界条件,利用有限元法求得了代表性体积单元内的电弹性场。由平均电弹性场和压电复合材料有效电弹性性能定义,预测了圆环形截面压电纤维复合材料的有效电弹性系数。通过算例,比较了相同压电材料体积分数下圆环形截面压电纤维复合材料与圆截面压电纤维复合材料有效电弹性性能的差异,讨论了圆环形截面压电纤维内部非压电填充物的力学性质对有效压电系数的影响。该文结论可为高灵敏度压电复合材料设计提供 参考。     

Anti-plane electro-elastic fields in an infinite matrix with N coated-piezoelectric inclusions

Electro-elastic fields in an infinite matrix with N coated-piezoelectric inclusions are studied based on the complex variable method. Based on the assumption that the coated inclusions are completely bounded the matrix which is subjected to remote anti-plane shears and in-plane electric fields, the general solutions for complex potentials are first derived for arbitrary arrays of inclusions. Then, the numerical results are obtained for special cases in which the inclusions are located in different ways. Discussions are made about the influence of the coating thickness, the array type of inclusions and the material properties on the interface stresses and electric displacements in the matrix. It is found that both the array types of inclusions and the Young’s modulus of the three-phase material system have remarkable effects on stresses, but not on electric displacements, while the piezoelectric constants of the coating and its thickness have significant effects on electric displacements, but not on stresses.     

Interactions of interfacial arc cracks

The interaction of two interfacial arc cracks around a circular elastic inclusion embedded in an elastic matrix is examined. New results for stress intensity factors for a pair of interacting cracks are derived for a concentrated force acting in the matrix. For verifying the point load solutions, stress intensity factors under uniform loading are obtained by superposing point force results. For achieving this objective, a general method for generating desired stress fields inside a test region using point loads is described. The energetics of two interacting interfacial arc cracks is discussed in order to shed more light on the debonding of hard or soft inclusions from the matrix. The analysis based on complex variables is developed in a general way to handle the interactions of multiple interfacial arc cracks/straight cracks.     

Antiplane electro‐mechanical field of a piezoelectric finite wedge under a pair of concentrated forces and free charges

Abstract

This paper presents general 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 traction free and insulated. Using finite Mellin transform methods, the stress and electrical displacement in all fields of the piezoelectric finite wedge are derived analytically. Singularity orders and intensity factors of stress and electrical displacement can be obtained too. After being reduced to a problem of an antiplane edge crack or an infinite wedge in a piezoelectric medium, the results compare well with those of previous studies.     

Effects of electric boundary conditions on electroelastic field in a cracked piezoelectric strip

Effects of electric boundary conditions on electroelastic field in a cracked piezoelectric strip are examined. Attention is focussed on an antiplane shear central crack normal to the strip surfaces. By decoupling equations and using the conformal mapping technique, expressions for electroelastic field in the piezoelectric strip are determined under the assumptions of an impermeable, permeable, or conducting crack, respectively. Comparison for the singularity near the crack tips among the obtained electroelastic fields is made.     

On diffraction of waves by the infinite grating of circular dielectric cylinders at oblique incidence: Floquet representation

Abstract

We investigate the diffraction of plane E-polarized electromagnetic waves by an infinite array of infinitely long circular dielectric cylinders for the most generalized case of oblique incidence. Employing the Sommerfeld integral representation of the Hankel function, and exploiting Poisson's summation formula in separation-of-variables solution, we have acquired a new representation for the diffraction of waves by the infinite grating of the circular dielectric cylinders at oblique incidence. The exact solutions for the external electric and magnetic field intensities have been derived in terms of the diffraction angles of the grating which are obtained by solving the grating equation. In addition, the transmitted and reflected fields of the infinite grating have been presented in terms of propagating and evanescent Floquet modes.     

Surface/interface Energy Effect on Electromechanical Responses Around a Nanosized Elliptical Inclusion under Far-field Loading at an Arbitrary Angle

Electro-elastic surface/interface around nano-sized piezoelectric inclusions shows great effect on the response of piezoelectric nano-structures. In this paper, a theoretical model is proposed to examine the surface/interface effect on the electromechanical responses around a nano-sized elliptical piezoelectric inclusion embedded in an infinite piezoelectric matrix under far-field loading with an arbitrary angle, and the effect of loading angle is considered Combining the conformal mapping technique and electro-elastic surface/interface theory, a closed form solution of this problem is obtained and the interactive effect between the surface/interface and the aspect ratio of the elliptical inclusion is examined.