Dynamic characteristics of an eccentric crack in a functionally graded piezoelectric ceramic strip

The dynamic response of an eccentric Griffith crack in functionally graded piezoelectric ceramic strip under anti-plane shear impact loading is analysed using integral transform method. Laplace transform and Fourier transform are used to reduce the problem to two pairs of dual integral equations, which are then expressed to Fredholm integral equations 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 and electric loadings.     

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.     

Central crack in a piezoelectric disc

This study is concerned with the general solution of the field intensity factors and energy release rate for a Griffith crack in a piezoelectric ceramic of finite radius under combined anti-plane mechanical and in-plane electrical loading. Both electrically continuous and impermeable crack surface conditions are considered. Employing Mellin transforms and Fourier series, the problem is reduced to dual integral forms. The solution to the resulting expressions is expressed in terms of Fredholm integral equation of the second kind. The solutions are provided to study the influence of the crack length, the crack surface boundary conditions on the intensity factors and the energy release rate.     

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.     

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.     

Free vibration analysis of functionally graded coupled circular plate with piezoelectric layers

Based on classical plate theory (CLPT), free vibration analysis of a circular plate composed of functionally graded material (FGM) with its upper and lower surfaces bounded by two piezoelectric layers was performed. Assuming that the material properties vary in a power law manner within the thickness of the plate the governing differential equations are derived. The distribution of electric potential along the thickness direction in piezoelectric layers is considered to vary quadratically such that the Maxwell static electricity equation is satisfied. Then these equations are solved analytically for two different boundary conditions, namely clamped and simply supported edges. The validity of our analytical solution was checked by comparing the obtained resonant frequencies with those of an isotropic host plate. Furthermore, for both FGM plate and FGM plate with piezoelectric layers, natural frequencies were obtained by finite element method. Very good agreement was observed between the results of finite element method and the method presented in this paper. Then for the two aforementioned types of boundary conditions, the values of power index were changed and its effect on the resonant frequencies was studied. Also, the effect of piezoelectric thickness layers on the natural frequencies of FGM piezoelectric plate was investigated. This paper was recommended for publication in revised form by Associate Editor Seockhyun Kim Saeed Jafari Mehrabadi received his B.S. in mechanical Engineering from Azad University, Arak, Iran, in 1992. He then received his M.S. from Azad University, Tehran, Iran in 1995. Now he is a faculty member of the department of mechanical engineering in Azad university of Arak, Iran and PhD student of Azad University, Science and Research Campus, Pounak, Tehran, Iran. His interests include computational methods and solid mechanics such as vibration, buckling.     

Numerical calculation of energy release rates by virtual crack closure technique

A seamless analysis of material behavior incorporating complex geometry and crack- tip modeling is one of greatly interesting topics in engineering and computational fracture mechanics fields. However, there are still large gaps between the industrial applications and fundamental academic studies due to a time consuming detailed modeling. In order to resolve this problem, a numerical method to calculate an energy release rate by virtual crack closure technique was proposed in this paper. Both free mesh method and finite element method have been utilized and, thereafter, robust local and global elements for various geometries and boundary conditions were generated. A validity of the proposed method has been demonstrated through a series of fracture mechanics analyses without tedious crack-tip meshing.     

An electrically saturated anti-plane shear crack in a piezoelectric layer constrained between two elastic layers

Within the framework of linear elasticity and nonlinear electroelasticity, the electrical nonlinear behavior of an anti-plane shear crack in a piezoelectric ceramic layer constrained between two orthotropic layers is examined by using an electrical strip saturation model. The analysis is performed based on the electrical unified crack surface condition, which can describe all of the permeable, the impermeable, and the limited permeable ones. Fourier transforms are used to reduce the problem to the solutions of two pairs of dual integral equations, which are then expressed to two Fredholm integral equations of the second kind. Some numerical results for the energy release rates are presented to show the effects of the electrical crack condition parameter relating with elliptic flaw shape parameters and permittivity inside the crack, the electric field, the crack length, the electromechanical coupling coefficient, the crack position, and the bonded elastic materials.     

Creep-fatigue crack growth behavior of a structure with crack like defects at the welds

A study on a creep-fatigue crack growth behavior has been carried out for a cylindrical structure with weldments by using a structural test and an evaluation according to the assessment procedures. The creep-fatigue crack growth behavior following the creep-fatigue crack initiation has been assessed by using the French A16 procedure and the conservatism for the present structural test has been examined. The structural specimen is a welded cylindrical shell made of 316 L stainless steel (SS) for one half of the cylinder and 304 SS for the other half. In the creep-fatigue test, the hold time under a tensile load which produces the primary nominal stress of 45 MPa was one hour at 600°C and creep-fatigue loads of 600 cycles were applied. The evaluation results for the creep-fatigue crack propagation were compared with those of the observed images from the structural test. The assessment results for the creep-fatigue crack behavior according to the French A16 procedure showed that the A16 is overly conservative for the creep-fatigue crack propagation in the present case with a short hold time of one hour.     

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.     

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.     

Stress intensity factors and kink angle of a crack interacting with a circular inclusion under remote mechanical and thermal loadings

A problem of a circular elastic inhomogeneity interacting with a crack under uniform loadings (mechanical tension and heat flux at infinity) is solved. The singular integral equations for edge and temperature dislocation distribution functions are constructed and solved numerically, to obtain the stress intensity factors. The effects of the material property ratio on the stress intensity factor (SIF) are investigated. The computed SIFs are used to predict the kink angle of the crack when the crack grows.     

Analysis of a radial crack in cross-ply laminates under uniaxial tension

The problem of a radial crack in cross-ply laminates under uniaxial tension is investigated in this paper. The normalized stress intensity factors are obtained by the modified mapping collocation method which is based on analytic complex function theory of complex variables. The present results for an isotropic infinite plate show good agreement with existing solutions. In the range of small crack length, the stress intensity factor for a radial crack in cross-ply laminates under uniaxial tension becomes larger as the percentage of 0° plies increases. However in the range of large crack length, it is insensitive to the percentage of 0° plies.     

Exact solution of a compositionally graded piezoelectric layer under uniform stretch, bending and twisting

Electromechanical responses of compositionally graded piezoelectric layers are analysed. The layers consist of polycrystalline piezoelectric ceramics poled along the thickness direction, and thus exhibit material symmetry of a hexagonal crystal in class 6mm. Two cases for layers (i) covered by surface electrodes and (ii) without surface electrodes are considered. Analytical solutions are exact in Saint Venant's sense for both cases. However, solutions are obtained for layers subjected to uniform mechanical loads, including stretch, bending and twisting. Numerical results to show the effects of different compositional gradients are presented.     

Analysis of a conducting crack in an electrostrictive ceramic under combined electric and mechanical loading

A conducting crack in an electrostrictive ceramic under combined electric and mechanical loading is investigated. Analysis based on linear dielectric model predicts that the surfaces of the crack are not open completely but they are contact near the crack tip. The complete solution for the crack with a contact zone in a linear electrostrictive ceramic under combined electric and mechanical loading is obtained by using the complex variable formula. The asymptotic problems for a semi-infinite crack with a partial opening zone as well as for a fully open semi-infinite crack in a nonlinear electrostrictive ceramic are analyzed in order to investigate the effect of the electrical nonlinearity on the stress intensity factor under small scale nonlinear conditions. Particular attention is devoted to a finite crack in the nonlinear electrostrictive ceramic subjected to combined electric and mechanical loading. The stress intensity factor for the finite crack under small scale nonlinear conditions is obtained from the asymptotic analysis.     

内压作用下弯管环向穿透裂纹应力强度因子分析

应力强度因子是缺陷结构安全评定的必需参数。利用有限元分析软件ANSYS建立了内压作用下弯管环向穿透裂纹模型,以管道外径、管道厚度、纵向裂纹角、裂纹半角、环向裂纹角为参数,对不同参数下的应力强度因子进行了计算。结果表明:应力强度因子与纵向裂纹角θ1无关;应力强度因子随裂纹半角θ的增大而增大;应力强度因子随环向裂纹角θ2的增大先减后增。依据计算结果对参数进行了无量纲化,继而拟合了内压作用下弯管环向穿透裂纹应力强度因子计算关系式。     

Consideration of the frictional force on the crack surface and its implications for durability of tires

In order to find out a physical quantity which controls the fatigue life of a structure and to predict the fatigue life of tires, a finite element simulation methodology to use the cracking energy density (CED) and the virtual crack closure technique (VCCT) was proposed and applied to three different tires of a similar size. CED was calculated to predict the location of a crack initiation, and VCCT was used to obtain the strain energy release rate (SERR) at the tip of an initiated crack. Finite element simulations showed that SERR oscillated in the circumferential direction with its minimum occurring just before the contact zone and its maximum occurring just after the center of the contact zone, and SERR was affected significantly by the frictional force acting on the crack surface. In addition, a durability test was conducted to measure the fatigue life of the three tires. The comparison of SERR values with the test data revealed that the fatigue life increased as the amplitude of SERR decreased or as the R-ratio of SERR increased.     

Mode I interaction of a periodic array of parallel cracks in a functionally graded nonhomogeneous plane

The mode I interaction of a periodic array of parallel cracks which are uniformly spaced apart in a functionally graded material is investigated. The two-dimensional theory of nonhomogeneous elasticity is employed as the basic framework for this study. The material nonhomogeneity is represented in terms of the spatial variation of the shear modulus in the exponential form along the direction of cracks, while Poisson’s ratio is assumed to be constant. Formulation of the proposed mixed boundary value problem is reduced to solving a hypersingular integral equation with the crack surface displacement as a new unknown function. As a result, the variations of stress intensity factors are illustrated as a function of possible range of periodic crack spacing in conjunction with the different values of the material nonhomogeneity parameter. Furthermore, crack opening displaccements are presented for various geometric and material combinations.     

Evaluation of stress intensity factor for a partially patched crack using an approximate weight function

A cracked plate with a patch bonded on one side was treated with a crack-bridging model using weight function : assuming continuous distribution of springs acting between th crack surfaces, the stress intensity factor of the patched crack was numerically obtained. Especially in the case of a patched crack subjected to residual non-uniform stress, the stress intensity factor was easily with the corresponding approximate weight function. This paper presented the stress intensity factors for a crack partially patched within a finite plate or a patched crack initiated from a notch.     

Effects of crack geometries on reliabilities using probability theories

The methodologies to estimate the reliability of cracked structures using the fracture toughness and the SIF are developed. The probability theories such as the FORM and the SORM are utilized to calculate the failure probability. It is found that the failure probability increases with the increase of the crack size and the applied stress, and the decrease of the fracture toughness. It is also found that the failure probabilities obtained by the FORM and the SORM are similar each other for the through crack,the edge crack, and the single and the double crack emanated from a hole, and turn out different for the elliptical and semi-elliptical surface cracks. It is noted that the tensile stress affects significantly on the failure probability among the other random variables on the various crack geometries.