Yoffe-type moving edge crack in a piezoelectric block

Summary We consider an anti-plane edge moving crack problem with the constant velocity in a piezoelectric ceramic block. The far-field anti-plane shear mechanical and in-plane electrical loads are applied to the piezoelectric block. It is expressed to a Fredholm integral equation of the second kind. Expressions for the dynamic field intensity factors and the dynamic energy release rate are obtained. The dynamic stress intensity factor and the dynamic energy release rate depend on the crack propagation speed. Numerical results for several piezoelectric materials are also presented.     

Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

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.     

Singular stress and electric fields of a piezoelectric ceramic strip with a finite crack under longitudinal shear

Summary Following the theory of linear piezoelectricity, we consider the problem of determining the singular stress and electric fields in an orthotropic piezoelectric ceramic strip containing a Griffith crack under longitudinal shear. The crack is situated symmetrically and oriented in a direction parallel to the edges of the strip. Fourier transforms are used to reduce the problem to the solution of a pair of dual integral equations. The solution of the dual integral equations is then expressed in terms of a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate for piezoelectric ceramics are obtained, and the results are graphed to display the influence of the electric field.     

Interfacial fracture analysis of a piezoelectric–polythene composite cylindrical shell patch under axial shear

The present work studies the interfacial fracture in a piezoelectric cylindrical shell patch. The problem is solved by the methods of infinite trigonometric series and Cauchy singular integral equation, and the numerical results of the stress intensity factor (SIF) are obtained. The effects of the interfacial radius and crack’s location on the SIF are explained through the effects of the free surface, interfacial curvature, crack length, and interface end, respectively. An optimal stiffness matching relationship between the piezoelectric layer and dielectric substrate is suggested. The effects of the piezoelectric and dielectric coefficients are explained through the mechanism of piezoelectric stiffening.     

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.     

On a moving dielectric crack in a piezoelectric interface with spatially varying properties

This paper provides a comprehensive theoretical analysis of a finite crack propagating with constant speed along an interface between two dissimilar piezoelectric media under inplane electromechanical loading. The interface is modeled as a graded piezoelectric layer with spatially varying properties (functionally graded piezoelectric materials, i.e., FGPMs). The analytical formulations are developed using Fourier transforms and the resulting singular integral equations are solved with Chebyshev polynomials. Using a dielectric crack model with deformation-dependent electric boundary condition, the dynamic stress intensity factors, electric displacement intensity factor, crack opening displacement (COD) intensity factor, and energy release rate are derived to fully understand this inherent mixed mode dynamic fracture problem. Numerical simulations are made to show the effects of the material mismatch, the thickness of the interfacial layer, the crack position, and the crack speed upon the dynamic fracture behavior. A critical state for the electromechanical loading applied to the medium is identified, which determines whether the traditional impermeable (or permeable) crack model serves as the upper or lower bound for the dielectric model considering the effect of dielectric medium crack filling.     

On the dynamic behavior of a piezoelectric-elastic laminate with a crack in the piezoelectric material under electro-mechanical loads

The dynamic behavior of a piezoelectric-elastic laminate with a crack in the piezoelectric material under in-plane steady-state electro-mechanical loads is considered. Based on the use of integral transform techniques, the problem is reduced to a set of singular integral equations, which are solved using Chebyshev polynomial expansions. Numerical results are provided to show the variation of both the dynamic stress intensity factors and electric displacement intensity factor with frequencies of the applied electro-mechanical loads. A phenomenon similar to “resonance” is observed when the applied loads act in some specific ranges of frequencies, and both the dynamic stress intensity factors and electric displacement intensity factor may increase significantly, which will lead to the failure of piezoelectric material. The effects of applied electric fields, crack geometry and elastic layer thickness on the phenomenon are also discussed.     

加层压电条界面裂纹的稳态扩展

研究当压电条同时与两个不同材料的弹性条粘接在一起,在反平面机械载荷及面内电载荷联合作用下,长度不变的有限Griffith 界面裂纹沿加层压电条界面以常速稳态扩展时裂尖的动态断裂问题。应用Fourier积分变换将问题化为以第二类Fredholm积分方程表示的对偶积分方程,导出了相应的动应力强度因子表达式。给出了动应力强度因子与裂纹传播速度、裂纹长度、压电条及弹性条厚度、电荷载大小及方向的关系曲线。研究结果对结构设计及结构失效的预防具有理论和应用价值。     

SH wave propagation in a cylindrically layered piezoelectric structure with initial stress

Summary SH wave propagation in a cylindrically layered piezoelectric structure with initial stress is investigated analytically. By means of transformation, the governing equations of the coupled waves are reduced to Bessel and Laplace equations. The boundary conditions imply that the displacements, shear stresses, electric potential, and electric displacements are continuous across the interface between the layer and the substrate. The electrically open and short conditions at the cylindrical surface are applied to solve the problem. The phase velocity is numerically calculated for the electrically open and short cases, respectively, for different wavenumber and thickness of the layer. The effect of the initial stress on the phase velocity and the electromechanical coupling factor are discussed in detail for piezoelectric ceramics PZT-5H. We find that the initial stress has an important effect on the SH wave propagation in the cylindrically layered piezoelectric structures. The results also show that the ratio of the layer thickness to the wavelength has a remarkable effect on the SH wave phase velocity and electromechanical coupling factor.     

Dielectric breakdown model for a conductive crack and electrode in piezoelectric materials

The strip dielectric breakdown (DB) model introduced by Zhang and Gao [T.Y. Zhang, C.F. Gao, Fracture behavior of piezoelectric materials, Thero. Appl. Fract. Mech. 41 (2004) 339–379] is used to study the generalized 2D problem of a conductive crack and an electrode in an infinite piezoelectric material. The energy release rate and stress intensity factors are derived based on the Stroh formalism, and then they are applied as failure criteria to predict the critical fracture loads. It is found that the DB strip may take the shielding effect on a conductive crack or electrode. For the case of an electrode, the local energy release rate and stress intensity factor become zero when DB happens ahead of the electrode tip. For the case of a mode-I conductive crack in a transversely isotropic piezoelectric solid, the results based on the DB model show that the critical stress intensity factor linearly increases as the applied electric field parallel to the poling direction increases, while it linearly decreases as the applied electric field anti-parallel to the poling direction increases. Finally, the upper and lower bounds of the actual critical fracture loads are proposed for a conductive crack in a piezoelectric material under combined mechanical–electrical loads.     

Transient analysis of a piezoelectric strip with a permeable crack under anti-plane impact loads

The problem of a through permeable crack situated in the mid-plane of a piezoelectric strip is considered under anti-plane impact loads for two cases. The first is that the strip boundaries are free of stresses and of electric displacements, and the second is that the strip boundaries are clamped rigid electrodes. The method adopted is to reduce the mixed initial-boundary value problem, by using integral transform techniques, to dual integral equations, which are further transformed into a Fredholm integral equation of the second kind by introducing an auxiliary function. The dynamic stress intensity factor and energy release rate in the Laplace transform domain are obtained in explicit form in terms of the auxiliary function. Some numerical results for the dynamic stress intensity factor are presented graphically in the physical space by using numerical techniques for solving the resulting Fredholm integral equation and inverting Laplace transform.     

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.     

Eccentric crack in a rectangular piezoelectric medium under electromechanical loadings

Summary The solutions of an eccentric crack problem in a rectangular piezoelectric ceramic medium under combined anti-plane shear and in-plane electrical loadings are obtained by the continuous electric crack face condition. Fourier transforms and Fourier series are used to reduce the problem to two pairs of dual integral equations, which are then expressed by a Fredholm integral equation of the second kind. Numerical values of the stress intensity factor and the energy release rate are obtained to show the influence of the electric field.     

Dynamic antiplane behaviour of interacting cracks in a piezoelectric medium

In this article, we examine the dynamic interaction between two cracks in a piezoelectric medium under incident antiplane shear wave loading. The theoretical formulations governing the steady-state problem are based upon the use of integral transform techniques and a self-consistent iterative method. The resulting dynamic stress intensity factors at the interacting cracks are obtained by solving the appropriate singular integral equations using Chebyshev Polynomials at different loading frequencies. Numerical examples are provided to show the effect of the geometry of the cracks, the piezoelectric constants of the material and the frequency of the incident wave upon the dynamic stress intensity factor of the cracks. This revised version was published online in August 2006 with corrections to the Cover Date.     

Fracture parameters of a penny-shaped crack at the interface of a piezoelectric bi-material system

A penny-shaped crack at the interface of a piezoelectric bi-material system is considered. Analytical general solutions based on Hankel integral transforms are used to formulate the mixed-boundary value problem corresponding to an interfacial crack and the problem is reduced to a system of singular integral equations. The integral equations are further reduced to two systems of algebraic equations with the aid of Jacobi polynomials and Chebyshev polynomials. Thereafter, the exact expressions for the stress intensity factors and the electric displacement intensity factor at the tip of a crack are obtained. Selected numerical results are presented for various bi-material systems to portray the significant features of crack tip fracture parameters and their dependence on material properties, poling orientation and electric loading.     

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.     

Multiple cracks on the interface between a piezoelectric layer and an orthotropic substrate

In the present work, we investigate the problem of multiple cracks on the interface between a piezoelectric layer and an orthotropic substrate. The method of dislocation simulation and singular integral equation are used to solve the crack problem. The theoretical derivation is verified by the classical result in a special case. Numerical results of the stress intensity factor are obtained, and thereby the effects of geometrical parameters and material orthotropy are surveyed. The optimal stiffness ratio of the orthotropic substrate is suggested for the purpose of interfacial fracture prevention, which is significant for the design and assessment of such a kind of smart structures.     

Love waves in a smart functionally graded piezoelectric composite structure

The propagation of Love waves in a smart functionally graded piezoelectric structure is analyzed by applying elastic wave theory. There is an additional functionally graded layer between the piezoelectric layer and the substrate in this smart structure. When the piezoelectric and dielectric constants vary individually in a functionally graded layer, the asymptotic solutions of Love waves are obtained by applying the WKB method and solving the fourth order differential equation with variable coefficients. The effects of gradient variation on the phase velocity and the coupled electromechanical factor are discussed in detail. The analysis shows that the number of vibration modes is greater than that in the non-graded layer structure, and the coupled electromechanical factor increases with the increase of piezoelectric constant graded variation. Presented results are useful for the improvement of properties of surface acoustic wave (SAW) devices.     

Love wave propagation in layered magneto-electro-elastic structures with initial stress

Summary An analytical approach is taken to investigate Love wave propagation in layered magneto-electro-elastic structures with initial stress, where a piezomagnetic (piezoelectric) material thin layer is bonded to a semi-infinite piezoelectric (piezomagnetic) substrate. The magneto-electrically open and short conditions are applied to solve the problem. The phase velocity of the Love wave is numerically calculated for the magneto-electrically open and short cases, respectively. The effect of the initial stress on the phase velocity and the magneto-electromechanical coupling factor are studied in detail for piezomagnetic ceramics CoFe₂O₄ and piezoelectric ceramics BaTiO₃. We find that the initial stress has an important effect on the Love wave propagation in layered piezomagnetic/piezoelectric structures.