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.     

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.     

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.     

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.     

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.     

Effect of fiber volume fraction on the stress intensity factors for multi layered composites under arbitrary anti-plane shear loading

A multi-layered orthotropic material with a center crack is subjected to an anti-plane shear loading. The problem is formulated as a mixed boundary value problem by using the Fourier integral transform method. This gives a Fredholm integral equation of the second kind. The integral equation is solved numerically and anti-plane shear stress intensity factors are analyzed in terms of the material orthotropy for each layer, number of layers, crack length to layer thickness and the order of the loading polynomial. Also, the case of monolithic and hybrid composites are investigated in terms of the local fiber volume fraction and the global fiber volume fraction.     

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.     

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.     

Characteristics of a transiently propagating crack in functionally graded materials

When a crack propagates with acceleration, deceleration and time rates of change of stress intensity factors, it is very important for us to understand the effects of acceleration, deceleration and time rates of change of stress intensity factors on the individual stresses and displacements at the crack tip. Therefore, the crack tip stress and displacement fields for a transiently propagating crack along gradient in functionally graded materials (FGMs) with an exponential variation of shear modulus and density are developed and the characteristics of a transiently propagating crack from the fields are analyzed. The effects of the rate of change of the stress intensity factor and the crack tip acceleration on the individual stresses at the crack tip are opposite each other. Specially, the isochromatics (constant maximum shear stress) of Mode I tilt backward around the crack tip with an increase of crack tip acceleration, and tilt forward around the crack tip with an increase of the rate of change of the dynamic mode I stress intensity factor. This paper was recommended for publication in revised form by Associate Editor Chongdu Cho Kwang-Ho Lee received a Ph.D. degree in Yeungnam University in 1993. Dr. Lee is currently a professor at the School of Mechanical and Automotive Engineering at Kyungpook National University in Korea. He also had worked in KOMSCO as an engineer and researcher (1982.3–1996.2). He is interested in the fields of fracture and stress analysis on the composite, interface, nano and functionally graded materials by theoretical and experimental mechanics. Specially, his major interest is analysis of dynamic crack tip fields. Young-Jae Lee received his B.S degree in Agricultural Civil Engineering from Gyeongsang National University (GNU) in 1982. He then received his M.S. and Ph.D. degrees from GNU in 1984 and 1995, respectively. Dr. Lee is currently a professor at the department of Civil Engineering at Kyungpook National University in Korea. From 2005 to 2006, he had served as an editor of Korea Institute for Structure Maintenance and Inspection. His research interests are in the area of evaluation, diagnosis and optimum design of structure. Sang-Bong Cho received a Ph. D. degree from Tokyo University in 1989. Dr. Cho is currently a professor at the division of Mechanical and Automation Engineering at Kyungnam University in Korea. His research interests are in the area of fracture mechanics, FEM stress analysis and fretting fatigue.     

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.     

功能梯度涂层中的Ⅲ型周期裂纹问题

利用分层模型研究功能梯度涂层中的Ⅲ型周期裂纹问题,借助Fourier级数及传递矩阵技术,可将该边值问题化为求解Hilbert奇异积分方程,数值求解该方程即可得到应力强度因子.数值结果表明,当裂纹较密且裂纹相对较短时,裂纹之间的相互作用比较明显;材料剪切模量在厚度方向上的变化对应力强度因子具有较大的影响.     

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.     

弹性半空间中平片裂纹问题的超奇异积分方程方法

使用边界积分方程方法,在有限部积分的意义下,将弹性半空间中垂于自边界面的平片裂纹归结为一组以裂纹面位移间数为示知函数的超奇异积仞氖限部积分蜞 建立了数值人出了用裂纹面位移间尖力强度因子的公式。通过对圆形、菜和矩形等贡型的平片裂纹问题的计算,分析了自由边界面对裂纹前沿应力强度因子的影响。     

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.     

Stress singularities in dissimilar orthotropic composites containing an interlaminar crack

The problem of an interlaminar crack in dissimilar orthotropic composite materials under in-plane and anti-plane loading conditions is investigated. In the analytical model, orthotropic half-spaces are assumed to be bound together by a matrix interlayer which represents the matrix-rich interlaminar region in the fiber-reinforced composite laminate. The crack is embedded within the interlayer. With the utilization of the stiffness matrix approach, a system of singular integral equations of the first kind is derived for the current mixed boundary value problem. Numerical results are obtained for the interlaminar crack in a [0°/90°] fibrous composite laminate subjected to three basic loadings in fracture mechanics. Under each applied loading, variations of major and coupling stress intensity factors with respect to relative crack size, crack location, and fiber volume fraction are illustrated.     

Subsurface crack mechanisms under indentation loading

A linear elastic fracture mechanics analysis of plane-strain indentation of a homogeneous half-space with a subsurface horizontal crack was performed using the finite element method. Stress intensity factor results obtained for an infinite plate with a central crack subjected to far-field tension and a half-space with a frictionless subsurface horizontal crack under a moving surface point load are shown to be in good agreement with corresponding analytical results. Crack mechanism maps illustrating the occurrence of separation, forward and backward slip, stick, and separation at the crack interface are presented for different indentation load positions and crack face friction coefficients. Results for the stresses in the vicinity of the crack tips and the mode I and mode II stress intensity factors are given for different indentation positions, crack face friction coefficients, and both concentrated and distributed surface normal tractions. Although indentation produces a predominantly shear and compressive stress field, mode I loading conditions are shown to occur for certain indentation positions. However, the magnitude of the mode I stress intensity factor is significantly smaller than that of mode II, suggesting that in-plane shear mode crack growth is most likely to occur in the absence of microstructural defects. The significance of crack face friction and sharpness of the indenter on the subsurface shear mode crack propagation rate is interpreted in terms of the mode II stress intensity factor range and material behavior.     

Solution of analytical-variational method on bi-material interface crack and its comparison with experiment

An analytical method is developed to describe the fields of stress and displacement in a bi material strip with an edge interfacial crack. All of the basic governing equations, boundary conditions on crack surfaces, and conditions of continuity along the interface are satisfied by the eigen-function expansion method. The other boundary conditions will be satisfied by the generalized variational principle. Good convergence of generalized stress intensity factors is obtained, and the values of crack opening displacement and energy release rate obtained by this method are close to the experimental results. Some problems regarding oscillatory singularity, contact zone and energy release rate are discussed. Finally, the effects of modulus, thickness and crack length to mode mixity are presented.     

Analysis of a flat annular crack under shear loading

An annular crack in an infinite isotropic elastic solid under shear loading is analyzed. General solution to the Navier's equilibrium equation is expressed in terms of three harmonic functions. Employing the Hankel transform the harmonic functions are represented by the solution of a pair of triple integral equations. The triple integral equations are reduced to a pair of mixed Volterra-Fredholm integral equations, which are numerically solved. The stress intensity factors of the annular crack under various shear loadings such as uniform radial shear, linearly varying radial shear, uniform shear and linearly varying shear are calculated as the Poisson's ratio ν anda/b (a; inner radius,b; outer radius) vary.     

Analysis of a crack approaching a circular hole in cross-ply laminates under biaxial loading

The problem of a crack approaching a circular hole in cross-ply laminates under uniaxial and biaxial loading is investigated in this paper. The effects of material orthotropy, geometry [R/d and a/d], and loading conditions on crack tip singularity are investigated. The stress intensity factors are obtained by the modified mapping collocation method. The present results for an isotropic infinite plate show good agreement with existing solutions. The results for cross-ply laminates show that the stress intensity factors strongly depend on material orthotropy, geometry, and loading condition. The stress intensity factors for cross-ply laminates exist between those for θ=0° and those for θ=90° in the whole range of crack length and decrease as the percentage of 0° plies increases. In the range of small crack length the stress intensity factors for biaxial tension are higher than those for uniaxial tension. In the range of large crack length the stress intensity factors for uniaxial tension are higher than those for biaxial tension.