Dynamic propagation of a finite eccentric crack in a functionally graded piezoelectric ceramic strip

The dynamic propagation of an eccentric Griffith crack in a functionally graded piezoelectric ceramic strip under anti-plane shear is analyzed using the integral transform method. A constant velocity Yoffe-type moving crack is considered. Fourier transform is used to reduce the problem to a pair of dual integral equations, which is then expressed in a Fredholm integral equation of the second kind. We assume that the properties of the functionally graded piezoelectric material vary continuously along the thickness. The impermeable crack boundary condition is adopted. Numerical values on the dynamic stress intensity factors are presented for the functionally graded piezoelectric material to show the dependence of the gradient of material properties, crack moving velocity, and eccentricity. The dynamic stress intensity factors of a moving crack in functionally graded piezoelectric material increases when the crack moving velocity, eccentricity of crack location, material property gradient, and crack length increase. This paper was recommended for publication in revised form by Associate Editor Hyeon Gyu Beom Jeong Woo Shin received a B.S. and M.S. degree in Mechanical Engineering from Yonsei University in Seoul, Korea in 1998 and 2000, respectively. A major field of Mr. Shin is fracture mechanics. He is currently working on the KARI (Korea Aerospace Research Institute) as a senior researcher. He conducted load analysis of fixed wing aircraft and full scale airframe static test at the KARI. He is now developing landing gear in the KHP (Korea Helicopter Program) as a performance engineer.     

Anti-plane moving crack in a functionally graded piezoelectric layer between two dissimilar piezoelectric strips

The dynamic propagation of a crack in a functionally graded piezoelectric material (FGPM) interface layer between two dissimilar piezoelectric layers under anti-plane shear is analyzed using integral transform approaches. The properties of the FGPM layers vary continuously along the thickness. The FGPM layer and two homogeneous piezoelectric layers are connected weak-discontinuously. A constant velocity Yoffe-type moving crack is considered. The Fourier transform is used to reduce the problem to two sets of dual integral equations, which are then expressed to the Fredholm integral equations of the second kind. Numerical values on the dynamic energy release rate (DERR) are presented for the FGPM to show the effects on electric loading, gradient of the material properties, crack moving velocity, and thickness of the layers. The following are helpful to increase resistance to crack propagation in the FGPM interface layer: (a) certain direction and magnitude of the electric loading, (b) increasing the thickness of the FGPM interface layer, and (c) increasing the thickness of the homogeneous piezoelectric layer to have larger material properties than those of the crack plane in the FGPM interface layer. The DERR always increases with the increase of crack moving velocity and the gradient of the material properties.     

Dynamic response of an anti-plane shear crack in a functionally graded piezoelectric strip

The dynamic response of a cracked functionally graded piezoelectric material (FGPM) under transient anti-plane shear mechanical and in-plane electrical loads is investigated in the present paper. It is assumed that the electroelastic material properties of the FGPM vary smoothly in the form of an exponential function along the thickness of the strip. The analysis is conducted on the basis of the unified (or natural) crack boundary condition which is related to the ellipsoidal crack parameters. By using the Laplace and Fourier transforms, the problem is reduced to the solutions of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and crack sliding displacement are presented to show the influences of the elliptic crack parameters, the electric field, FGPM gradation, crack length, and electromechanical coupling coefficient.     

Functionally graded piezoelectric strip with eccentric crack under anti-plane shear

In this paper, we examine the singular stresses and electric fields in a functionally graded piezoelectric ceramic strip containing an eccentric crack off the center line under anti-plane shear loading with the theory of linear piezoelectricity. It is assumed that the properties of the functionally graded piezoelectric ceramic strip vary continuously along the thickness. Fourier transforms are used to reduce the problem to the solution of two pairs of dual integral equations, which are then expressed to a Fredholm integral equation of the second kind. Numerical values on the stress intensity factor and the energy release rate are obtained.     

Dynamic propagation of a weak-discontinuous interface crack between two dissimilar functionally graded layers under anti-plane shear

The dynamic propagation of an interface crack between two functionally graded material (FGM) layers under anti-plane shear is analyzed using the integral transform method. The properties of the FGM layers vary continuously along their thicknesses. The properties of the two FGM layers vary and the two layers are connected weak-discontinuously. A constant velocity Yoffe-type moving crack is considered. The Fourier transform is used to reduce the problem to a dual integral equation, which is then expressed to a Fredholm integral equation of the second kind. Numerical values on the dynamic energy release rate (DERR) are presented for the FGM to show the effect of the gradient of material properties, crack moving velocity, and thickness of FGM layers. The following are helpful to increase resistance to interface crack propagation in FGMs: a) increasing the gradient of material properties, b) an increase of shear modulus and density from the interface to the upper and lower free surface, and c) increasing the thickness of the FGM layer. The DERR increases or decreases with increase of the crack moving velocity.     

Anti-plane shear crack in a functionally gradient piezoelectric layer bonded to dissimilar half spaces

In this paper, the problem of a crack located in a functionally gradient piezoelectric interlayer between two dissimilar homogeneous piezoelectric half-planes being subjected to an anti-plane mechanical loading and an in-plane electric loading is considered. The material properties of the interlayer, such as the elastic stiffness, piezoelectric constant and dielectric constant, are assumed to vary continuously along the thickness of the interlayer, and the crack surface condition is assumed to be impermeable or permeable. By using the Fourier transform, the problem is first reduced to two pairs of dual integral equations and then into a Fredholm integral equation of the second kind. Numerical calculations are carried out, and the effects of crack geometric parameters on the stress intensity factor and the energy release rate are shown graphically.     

Analysis of a moving crack in a functionally graded strip between two homogeneous layers

In this paper a finite crack with constant length (Yoffe-type crack) propagating in a functionally graded strip with spatially varying elastic properties between two dissimilar homogeneous layers under in-plane loading was studied. By utilizing the Fourier transformation technique, the mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. The influences of the geometric parameters, the graded parameter, the crack length and speed on the stress intensity factors are investigated. The numerical results show that the graded parameters, the thicknesses of the functionally graded strip and the two homogeneous layers, the crack size and speed have significant effects on the dynamic fracture behavior.     

Transient response of a permeable crack normal to a piezoelectric-elastic interface: Anti-plane problem

In this paper, the anti-plane transient response of a central crack normal to the interface between a piezoelectric ceramics and two same elastic materials is considered. The assumed crack surfaces are permeable. By virtue of integral transform methods, the electroelastic mixed boundary problems are formulated as two set of dual integral equations, which, in turn, are reduced to a Fredholm integral equation of the second kind in the Laplace transform domain. Time domain solutions are obtained by inverting Laplace domain solutions using a numerical scheme. Numerical values on the quasi-static stress intensity factor and the dynamic energy release rate are presented to show the dependences upon the geometry, material combination, electromechanical coupling coefficient and electric field.     

Interface crack of two dissimilar bonded functionally graded strips with arbitrary distributed properties under plane deformations

In this paper the plane elasticity problem of two bonded dissimilar functionally graded strips containing an interface crack with material properties varying arbitrarily is studied. The governing equation in terms of Airy stress function is formulated and exact solutions are obtained for several special variations of material properties in Fourier transformation domain. A multi-layered model is employed to model arbitrary variations of material properties based on two linear-distributed material compliance parameters. The mixed boundary problem is reduced to a system of singular integral equations that are solved numerically. Some numerical examples are given to demonstrate the accuracy, efficiency and versatility of the model. Numerical results show that fracture behavior of materials can be greatly affected by graded variation of elastic modulus and the influence of the specific form of elastic modulus on the fracture behavior of FGM is limited.     

Elastodynamic response of a crack perpendicular to the graded interfacial zone in bonded dissimilar materials under antiplane shear impact

A solution is given for the elastodynamic problem of a crack perpendicular to the graded interfacial zone in bonded materials under the action of antiplane shear impact. The interfacial zone is modeled as a nonhomogeneous interlayer with the power-law variations of its shear modulus and mass density between the two dissimilar, homogeneous half-planes. Laplace and Fourier integral transforms are employed to reduce the transient problem to the solution of a Cauchy-type singular integral equation in the Laplace transform domain. Via the numerical inversion of the Laplace transforms, the values of the dynamic stress intensity factors are obtained as a function of time. As a result, the influences of material and geometric parameters of the bonded media on the overshoot characteristics of the dynamic stress intensities are discussed. A comparison is also made with the corresponding elastostatic solutions, addressing the inertia effect on the dynamic load transfer to the crack tips for various combinations of the physical properties.     

球壳与柱壳之功能梯度压电涂层的热效应分析

针对以球壳和圆柱壳为基体的功能梯度压电涂层,由多场耦合控制方程和层间连续条件导出递推关系,建立了显式的力-电-热多场耦合解。对于多层功能梯度压电涂层,此解为精确解;对于连续功能梯度压电涂层,可将涂层分为若干层,只要层数取得足够大,所得的近似解将收敛于精确解。研究表明：压电效应对基体的应力影响可以忽略,但对涂层的应力影响非常显著;增加层数并不能有效降低最大应力,但增加层数可显著减小相邻层层间周向应力的突变幅度,减小层间应力集中;然而涂层与基体之间的界面径向应力却随层数的增大而使界面强度弱化。     

Closed-form vibration analysis of thick annular functionally graded plates with integrated piezoelectric layers

This paper employs an analytical method to analyze vibration of piezoelectric coupled thick annular functionally graded plates (FGPs) subjected to different combinations of soft simply supported, hard simply supported and clamped boundary conditions at the inner and outer edges of the annular plate on the basis of the Reddy's third-order shear deformation theory (TSDT). The properties of host plate are graded in the thickness direction according to a volume fraction power-law distribution. The distribution of electric potential along the thickness direction in the piezoelectric layer is assumed as a sinusoidal function so that the Maxwell static electricity equation is approximately satisfied. The differential equations of motion are solved analytically for various boundary conditions of the plate. In this study closed-form expressions for characteristic equations, displacement components of the plate and electric potential are derived for the first time in the literature. The present analysis is validated by comparing results with those in the literature and then natural frequencies of the piezoelectric coupled annular FG plate are presented in tabular and graphical forms for different thickness-radius ratios, inner-outer radius ratios, thickness of piezoelectric, material of piezoelectric, power index and boundary conditions.     

Mode I and mode II analyses of a crack normal to the graded interlayer in bonded materials

In this paper, the plane elasticity equations are used to investigate the in-plane normal (mode I) and shear (mode II) behavior of a crack perpendicular to and terminating at the interface in bonded media with a graded interfacial zone. The interfacial zone is treated as a nonhomogeneous interlayer with the continuously varying elastic modulus between the two dissimilar, homogeneous semi-infinite constituents. For each of the individual loading modes, based on the Fourier integral transform technique, a singular integral equation with a Cauchy kernel is derived in a separate but parallel manner. In the numerical results, the values of corresponding modes of stress intensity factors are illustrated for various combinations of material and geometric parameters of the bonded media in conjunction with the effect of the material nonhomogeneity within the graded interfacial zone.     

Vibration amplitude and thermal effects on the nonlinear behavior of thin circular functionally graded plates

In this paper, the nonlinear free axisymmetric vibration of a thin circular functionally graded plate in thermal environment is formulated in terms of von-Karman's dynamic equations, and a semi-analytical approach is developed. The plate thickness is constant and the material properties of the functionally graded plate are assumed to vary continuously through the thickness, according to a power-law distribution of the volume fraction of the constituents. For harmonic vibrations, by using assumed-time-mode method and Kantorovich time averaging technique, governing equations are solved. The nonlinear frequencies and associated stresses are determined at large amplitudes of vibration. Effects of material compositions and thermal loads on the vibration characteristics and stresses are examined. The numerical results obtained here are compared with available published results, based on various approaches.     

Elastic analysis of exponentially graded piezoelectric cylindrical structures as sensors and actuators

An analytical solution is obtained for functionally graded piezoelectric cylindrical structures, and the electroelastic behavior is investigated. The material properties are assumed to vary as an exponential function along the radial direction of the cylinder. The cylindrical structure acts as sensor or actuator and is subjected to mechanical and electrical loads at its inner and outer cylindrical surfaces. Based on the linear piezoelectricity theory, the governing equation in terms of the radial displacement is obtained as a second-order nonhomogeneous ordinary differential equation with variable coefficients, and its solution is obtained by the Frobenius series method. The illustrative examples show that the material nonhomogeneity has significant effect on the electroelastic fields of functionally graded piezoelectric cylindrical sensors and actuators.     

Moving interfacial crack between two dissimilar soft ferromagnetic materials in uniform magnetic field

This paper presents the dynamic magnetoelastic stress intensity factors of a Yoffe-type moving crack at the interface between two dissimilar soft ferromagnetic elastic half-planes. The solids are subjected to a uniform in-plane magnetic field and the crack is opened by internal normal and shear tractions. The problem is considered within the framework of linear magnetoelasticity. By application of the Fourier integral transform, the mixed boundary problem is reduced to a pair of integral equations of the second kind with Cauchy-type singularities. The singular integral equations are solved by means of a Jacobi polynomial expansion method. For a particular case, closed-form solutions are obtained. It is shown that the magnetoelastic stress intensity factors depend on the moving velocity of the crack, the magnetic field and the magnetoelastic properties of the materials.     

Analytical investigation on axisymmetric free vibrations of moderately thick circular functionally graded plate integrated with piezoelectric layers

In this paper, a free vibration analysis of moderately thick circular functionally graded (FG) plate integrated with two thin piezoelectric (PZT4) layers is presented based on Mindlin plate theory. The material properties of the FG core plate are assumed to be graded in the thickness direction, while the distribution of electric potential field along the thickness of piezoelectric layers is simulated by sinusoidal function. The differential equations of motion are solved analytically for two boundary conditions of the plate: clamped edge and simply supported edge. The analytical solution is validated by comparing the obtained resonant frequencies with those of an isotropic host plate. The emphasis is placed on investigating the effect of varying the gradient index of FG plate on the free vibration characteristics of the structure. Good agreement between the results of this paper and those of the finite element analyses validated the presented approach.     

On the plane contact problem of a functionally graded elastic layer loaded by a frictional sliding flat punch

This article is concerned with the contact mechanics of a functionally graded layer loaded by a frictional sliding flat punch. The coefficient of friction is assumed to be constant and the lower side of the graded layer is firmly attached to a rigid foundation. The graded, nonhomogeneous property of the medium is represented in terms of an exponential variation of the shear modulus, while Poisson’s ratio is taken to be constant. Based on the use of plane elasticity equations and the Fourier integral transform technique, the formulation of the current contact mechanics problem lends itself to a Cauchy-type singular integral equation of the second kind for the unknown contact pressure, which is solved numerically. As a result, the effects of several parameters, i.e., the material nonhomogeneity, the friction coefficient, the punch width, and Poisson’s ratio, on the distributions of the contact pressure and the in-plane surface stress component are presented.     

Anti-plane shear behavior of an arbitrarily oriented crack in bonded materials with a nonhomogeneous interfacial zone

The anti-plane shear problem of bonded elastic materials containing a crack at an arbitrary angle to the graded interfacial zone is investigated in this paper. The interfacial zone is modeled as a nonhomogeneous interlayer of finite thickness with the continuously varying shear modulus between the two dissimilar, homogeneous half-planes. Formulation of the crack problem is based upon the use of the Fourier integral transform method and the coordinate transformations of basic field variables. The resulting Cauchy-type singular integral equation is solved numerically to provide the values of modeIII stress intensity factors. A comprehensive parametric study is then presented of the influence of crack obliquity on the stress intensity factors for different crack size and locations and for different material combinations, in conjunction with the material nonhomogeneity within the graded interfacial zone.     

Two-dimensional contact problem for a coating-graded layer-substrate structure under a rigid cylindrical punch

The two-dimensional frictionless contact problem of a coating structure consisting of a surface coating, a functionally graded layer and a substrate under a rigid cylindrical punch is investigated in this paper. The coating and substrate are homogeneous materials with distinct physical properties, while the intermediate layer is inhomogeneous with its shear modulus changing arbitrarily along the thickness direction. To approximate the through-thickness variation, a piecewise linear multi-layer model is used and the graded layer is divided into a number of sub-layers whose shear modulus is assumed to vary linearly. Poisson's ratio, however, is taken as a constant within the structure for simplicity. By using the transfer matrix method and Fourier integral transform technique, the governing equations are reduced to a Cauchy singular integral equation which is numerically solved to determine the normal contact pressure, contact region, the through-thickness stress fields and longitudinal stress distributions at interfaces. A parametric study is conducted, showing that both normal contact pressure and stress fields in the structure are significantly influenced by the shear modulus ratio and the thickness ratio of the exponentially graded layer but are less sensitive to the gradient index of the graded layer whose shear modulus follows a power law variation.