Electroelastic properties of cracked piezoelectric materials under longitudinal shear

Analytical solutions are obtained to quantify the influence of cracks on electroelastic properties of piezoelectric materials containing doubly-periodic arrays of cracks. Both the rectangular and diamond-shaped arrays of cracks are considered. Solutions are obtained for the case of an antiplane shear load coupled with an in-plane electrical load. This study makes it possible to understand the multicrack interactions in piezoelectric solids and their effects on the fracture and electroelastic properties. The crack tip field intensity factors and the change in stored electroelastic energy due to the presence of many microcracks are calculated. These calculations enable the prediction of the effective elastic, piezoelectric and dielectric constants of a damaged piezoelectric material. The results of this work can be useful in developing a technique to determine the state of mechanical and electrical damage in piezoelectric materials.     

Impact response of a finite crack in an orthotropic piezoelectric ceramic

Summary The transient dynamic stress intensity factor and dynamic energy release rate were determined for a cracked piezoelectric ceramic under normal impact in this study. A plane step pulse strikes the crack and stress wave diffraction takes place. Laplace and Fourier transforms are employed to reduce the transient problem to the solution of a pair of dual integral equations in the Laplace transform plane. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. A numerical Laplace inversion technique is used to compute the values of the dynamic stress intensity factor and the dynamic energy release rate for some piezoelectric ceramics, and the results are graphed to display the electroelastic interactions.     

Fatigue crack propagation in a piezoelectric ceramic strip subjected to mode III loading

Summary Following the theory of linear piezoelectricity, a forth-power stress intensity factor crack growth equation in an orthotropic piezoelectric ceramic strip is developed under mode III loading. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Dugdale's assumption regarding the plastic zone in metals is applied to estimate the effects of yield around the crack tips. Fourier transforms are used to reduce the electroelastic problem to one involving the numerical solution of a Fredholm integral equation of the second kind. A direct approach based on the accumulated plastic displacement criterion for crack propagation is used to develop the equation to predict the fatigue crack growth. Graphical results showing the effect of electroelastic interactions on the fatigue crack growth rate are presented.     

Electroelastic Analysis of a Piezoelectric Layer with Electrodes

The electroelastic analysis of a piezoelectric layer with two semi-infinite electrodes is made. The layer surfaces are clamped and a voltage is applied between the layer surfaces and the electrodes. By using the integral equation method, the electroelastic field is derived for the cases of crack-type electrodes and of rigid electrodes, respectively. It indicates that the concentration of electric field an electric displacement occurs near the electrode tips for both cases. However, the stresses are bounded for the former, and possess singularity for the latter. The field intensity factors and energy release rate are obtained in explicit form. The counterpart for a single semi-infinite electrode is given.     

Fracture analysis of a piezoelectric layer with a penny-shaped and energetically consistent crack

The problem of an eccentric penny-shaped crack embedded in a piezoelectric layer is addressed by using the energetically consistent boundary conditions. The Hankel transform technique is applied to solve the boundary-value problem. Then two coupling Fredholm integral equations are derived and solved by using the composite Simpson’s rule. The intensity factors of stress, electric displacement, crack opening displacement and electric potential together with the energy release rate are further given. The effects of the thickness of a piezoelectric layer and the discharge field inside the penny-shaped crack on the fracture parameters of concern are discussed through numerical computations. The observations reveal that an increase of the discharge field decreases the stress intensity factor and the energy release rate. An eccentric penny-shaped crack is easier to propagate than a mid-plane one in a piezoelectric layer, and the geometry of the crack along with the layer thickness have significant influences on the electrostatic traction acting on the crack faces. The solutions for a penny-shaped dielectric crack in an infinite or a semi-infinite piezoelectric material can be obtained easily.     

Fracture of a finite piezoelectric layer with a penny-shaped crack

This paper studies a penny-shaped crack in a finite thickness piezoelectric material layer. The piezoelectric medium is subjected to a thermal flux on its top and bottom surfaces. Both thermally insulated crack and heated crack are considered. Numerical solution for the finite layer thickness is obtained through the solution of a pair of dual integral equations. The result reduces to the closed form solution when the thickness of the piezoelectric layer becomes infinite. Exact expressions for the stress and electric displacement at the crack border are given as a function of the stress intensity factor, which is determined by the applied thermal flux. This paper is useful for the reliability design of piezoelectric materials in thermal environments.     

Interactions between N circular cylindrical inclusions in a piezoelectric matrix

Summary This paper studies the interactions between N randomly-distributed cylindrical inclusions in a piezoelectric matrix. The inclusions are assumed to be perfectly bounded to the matrix, which is subjected to an anti-plane shear stress and an in-plane electric field at infinity. Based on the complex variable method, the complex potentials in the matrix and inside the inclusions are first obtained in form of power series, and then approximate solutions for electroelastic fields are derived. Numerical examples are presented to discuss the influences of the inclusion array, inclusion size and inclusion properties on couple fields in the matrix and inclusions. Solutions for the case of an infinite piezoelectric matrix with N circular holes or an infinite elastic matrix containing N circular piezoelectric fibers can also be obtained as special cases of the present work. It is shown that the electroelastic field distribution in a piezoelectric material with multiple inclusions is significantly different from that in the case of a single inclusion.     

Electroelastic response of a flat annular crack in a piezoelectric fiber surrounded by an elastic medium

Following the theory of linear piezoelectricity, the electroelastic problem of a flat annular crack in a piezoelectric fiber embedded in an elastic medium is considered. Fourier and Hankel transform techniques are employed to formulate the mixed-boundary-value problem as a singular integral equation. The stress-intensity factor, energy-release rate and energy-density factor are computed for some piezoelectric composites, and the influence of applied electric fields on the normalized values is displayed graphically.     

The mode I crack problem for layered piezoelectric plates

The plane strain singular stress problem for piezoelectric composite plates having a central crack is considered. For the case of the crack which is normal to and ends at the interface between the piezoelectric plate and the elastic layer, the order of stress singularity around the tip of the crack is obtained. The Fourier transform technique is used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gaus–Jacobi integration formula. Numerical calculations are carried out, and the main results presented are the variation of the stress intensity factor as functions of the geometric parameters, the piezoelectric material properties and the electrical boundary conditions of the layered composites.     

Numerical Green's functions for some electroelastic crack problems

A plane electroelastic problem involving planar cracks in a piezoelectric body is considered. The deformation of the body is assumed to be independent of time and one of the Cartesian coordinates. The cracks are traction free and are electrically either permeable or impermeable. Numerical Green's functions which satisfy the boundary conditions on the cracks are derived using the hypersingular integral approach and applied to obtain a boundary integral solution for the electroelastic crack problem considered here. As the conditions on the cracks are built into the Green's functions, the boundary integral solution does not contain integrals over the cracks. It is used to derive a boundary element procedure for computing the crack tip stress and electrical displacement intensity factors.     

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.     

Antiplane Permeable Edge Cracks in a Piezoelectric Strip of Finite Width

A piezoelectric strip with permeable edge cracks normal to the strip boundaries is analyzed. Under uniform antiplane mechanical shear and inplane electric loading, the distribution of the entire electroelastic field in a cracked piezoelectric strip is determined in explicit analytic form via the conformal mapping technique. It is found that the strain and the electric displacement exhibit the same singularity as the stress near the crack tips, while the electric field is always uniform. The field intensity factors and the energy release rate are independent of the applied electric load for prescribed stress, and related to the applied electric load for prescribed strain.     

Dynamic fracture mechanics study of an electrically impermeable mode III crack in a transversely isotropic piezoelectric material under pure electric load

The dynamic response of an electrically impermeable Mode III crack in a transversely isotropic piezoelectric material under pure electric load is investigated by treating the electric loading process as a transient impact load, which may be more appropriate to mimic the real service environment of piezoelectric materials. The stress intensity factor, the mechanical energy release rate, and the total energy release rate are derived and expressed as a function of time for a given applied electric load. The theoretical results indicate that a purely electric load can fracture the piezoelectric material if the stress intensity factor or the mechanical energy release rate is used as a failure criterion.     

Cylindrical interface cracks in 1-3 piezocomposites

1-3 Piezocomposites are made by embedding piezoelectric fibers/rods in polymer matrix materials. Fiber–matrix interface fracture can affect the performance of piezocomposites. In this paper, axisymmetric interfacial cracks in piezocomposites are studied by considering an idealized model of a single piezoelectric fiber in a matrix material. The displacement discontinuity method is used to formulate the Mode I and II crack problems. The fundamental solutions required for DDM are derived explicitly by using the electroelastic field equations and Fourier integral transforms. The dependence of Mode I and II stress intensity factors of single and multiple interface cracks on fiber and matrix material properties, crack length and distance between cracks are investigated.     

A Moving Mode-III Crack in a Functionally Graded Piezoelectric Strip

In this paper, the problem of a functionally graded piezoelectric strip with a constant-velocity Yoffe-type moving crack is considered. By using the Fourier transforms, the problem is first reduced to dual integral equations and then into Fredholm integral equations of the second kind. The electroelastic field near the crack tip is obtained for electrical impermeable boundary conditions and electrical permeable boundary conditions, respectively. The results obtained show that the gradient of the material properties can increase or decrease the magnitudes of the stress intensity factors, and the velocity can disturb the stress distribution near the crack tip.     

Transient response of a piezoelectric material with a semi-infinite mode-III crack under impact loads

The problem of a semi-infinite impermeable mode-III crack in a piezoelectric material is considered under the action of impact loads. For the case when a pair of concentrated anti-plane impact loads and electric displacements are exerted symmetrically on the upper and lower surfaces of the crack, the asymptotic electroelastic field ahead of the crack tip is determined in explicit form. The dynamic intensity factors of electroelastic field and dynamic mechanical strain energy release rate are obtained. The obtained results can be taken as fundamental solutions, from which general results may directly be evaluated by integration. The method adopted is to reduce the mixed initial-boundary value problem, by using the Laplace and Fourier transforms, into two simultaneous dual integral equations. One may be converted into an Abel's integral equation and the other into a singular integral equation with Cauchy kernel, and the solutions of both equations can be determined in closed-form, respectively. For some particular cases, the present results reduce to the previous results.     

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。     

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。     

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.     

Electroelastic effect of thickness mode langasite resonators

Langasite is a very promising material for resonators due to its good temperature behavior and high piezoelectric coupling, low acoustic loss, and high Q factor. The biasing effect for langasite resonators is crucial for resonator design. In this article, the resonant frequency shift of a thickness-mode langasite resonator is analyzed with respect to a direct current (DC) electric field applied in the thickness direction. The vibration modes of a thin langasite plate fully coated with an electrode are analyzed. The analysis is based on the theory for small fields superposed on a bias in electroelastic bodies and the first-order perturbation integral theory. The electroelastic effect of the resonator is analyzed by both analytical and finite-element methods. The complete set of nonlinear elastic, piezoelectric, dielectric permeability, and electrostrictive constants of langasite is used in the theoretical and numerical analysis. The sensitivity of electroelastic effect to nonlinear material constants is analyzed.