Stress intensity factors for an oblique edge crack in a coating/substrate system with a graded interfacial zone under antiplane shear

This paper investigates the edge crack problem for a coating/substrate system with a functionally graded interfacial zone under the condition of antiplane deformation. With the interfacial zone being modeled by a nonhomogeneous interlayer having the continuously varying shear modulus between the dissimilar, homogeneous phases of the coated medium, the coating is assumed to contain an edge crack at an arbitrary angle to the interfacial zone. The Fourier integral transform method is used in conjunction with the coordinate transformations of basic field variables. Formulation of the proposed crack problem is then reduced to solving a singular integral equation with a generalized Cauchy kernel. The mode III stress intensity factors are defined and evaluated in terms of the solution to the integral equation. In the numerical results, the values of the stress intensity factors are plotted, illustrating the effects of the crack orientation angle for various material and geometric combinations of the coating/substrate system with the graded interfacial zone.     

Interaction of two offset interfacial cracks in bonded dissimilar media with a functionally graded interlayer: Antiplane deformation

This paper is concerned with the problem of bonded dissimilar, homogeneous media with a functionally graded interlayer, weakened by two offset interfacial cracks under antiplane deformation. Based on the Fourier integral transform method, formulation of the crack problem is reduced to a system of Cauchy-type singular integral equations. The mode III stress intensity factors are defined and evaluated in terms of the solution to the integral equations. Numerical results include the variations of stress intensity factors versus offset distance between the two cracks for various combinations of material and other geometric parameters of the bonded system, addressing the interaction of the two neighboring interfacial cracks spaced apart by the graded interlayer.     

Thermoelastic problem of steady-state heat flows disturbed by a crack at an arbitrary angle to the graded interfacial zone in bonded materials

Plane thermoelasticity solutions are presented for the problem of a crack in bonded materials with a graded interfacial zone. The interfacial zone is treated as a nonhomogeneous interlayer having spatially varying thermoelastic moduli between dissimilar, homogeneous half-planes. The crack is assumed to exist in one of the half-planes at an arbitrary angle to the graded interfacial zone, disturbing uniform steady-state heat flows. The Fourier integral transform method is employed in conjunction with the coordinate transformations of field variables in the basic thermoelasticity equations. Formulation of the current nonisothermal crack problem lends itself to the derivation of two sets of Cauchy-type singular integral equations for heat conduction and thermal stress analyses. The heat-flux intensity factors and the thermal-stress intensity factors are defined and evaluated in order to quantify the singular characters of temperature gradients and thermal stresses, respectively, in the near-tip region. Numerical results include the variations of such crack-tip field intensity factors versus the crack orientation angle for various combinations of material and geometric parameters of the dissimilar media bonded through the thermoelastically graded interfacial zone. The dependence of the near-tip thermoelastic singular field on the degree of crack-surface partial insulation is also addressed.     

Elastodynamic analysis of a crack at an arbitrary angle to the graded interfacial zone in bonded half-planes under antiplane shear impact

A solution is provided for the elastodynamic problem of a crack at an arbitrary angle to the graded interfacial zone in bonded media under the action of antiplane shear impact. The interfacial zone is modeled by a nonhomogeneous interlayer with the spatially varying shear modulus and mass density in terms of power functions between the two dissimilar, homogeneous half-planes. Based on the use of Laplace and Fourier integral transforms and the coordinate transformations of basic field variables, formulation of the transient crack problem is reduced to solving a Cauchy-type singular integral equation in the Laplace transform domain. The crack-tip response in the physical domain is recovered via the inverse Laplace transform and the values of dynamic mode III stress intensity factors are obtained as a function of time. A comprehensive parametric study is then presented of the effects of crack obliquity on the overshoot behavior of the transient crack-tip response, by plotting the peak values of the dynamic stress intensity factors versus the crack orientation angle for various material and geometric combinations of the bonded system.     

Fracture analysis of mode-II crack perpendicular to imperfect bimaterial interface

The problem of a mode-II crack close to and perpendicular to an imperfect interface of two bonded dissimilar materials is investigated.The imperfect interface is modelled by a linear spring with the vanishing thickness.The Fourier transform is used to solve the boundary-value problem and to derive a singular integral equation with the Cauchy kernel.The stress intensity factors near the left and right crack tips are evaluated by numerically solving the resulting equation.Several special cases of the mode-II crack problem with an imperfect interface are studied in detail.The effects of the interfacial imperfection on the stress intensity factors for a bimaterial system of aluminum and steel are shown graphically.The obtained observation reveals that the stress intensity factors are dependent on the interface parameters and vary between those with a fully debonded interface and those with a perfect interface.     

The Anti-plane Dynamic Fracture Analysis of Functionally Graded Materials with Arbitrary Spatial Variations of Material Properties

Anti-plane dynamic fracture analysis is presented for functionally graded materials (FGM) with arbitrary spatial variations of material properties. The FGM with the material properties varying continuously in an arbitrary manner is modeled as a multi-layered medium with the elastic modulus and mass density varying linearly in each sub-layer and continuous at the interfaces between two adjacent sub-layers. With this linearly inhomogeneous multi-layered model, the problem of a crack in a graded interfacial zone bonded to two homogeneous half-spaces or in a coating bonded to a homogeneous half-space subjected to the anti-plane shear impact load is investigated. Laplace and Fourier transforms and transfer matrix are applied to reduce the associated mixed boundary value problem to a Cauchy singular integral equation which is solved numerically in the Laplace transformed domain. The dynamic stress intensity factors (DSIF) are obtained by using the numerical technique of Laplace inversion.     

Scattering of harmonic antiplane shear waves by an arc-shaped interfacial crack in functionally graded annular bi-material guide rail

Abstract

The dynamic behavior of an arc-shaped interfacial crack in an orthotropic functionally graded annular bi-material structure is investigated. In order for the analysis to be executable, the material properties are assumed to vary with the power function of the radial coordinates. By applying the separation variable method, the boundary value problem of the partial differential equation describing the fracture problem of this article can be transformed into a Cauchy kernel singular integral equation with the unknown jump of displacements across the crack surfaces. The obtained integral equation is solved numerically by Lobatto–Chebyshev collocation method to show the effects of the geometric and physical parameters upon the dynamic stress field near the crack tips.

Communicated by Kuang-Hua Chang.     

Dynamic response of a crack in a functionally graded interface of two dissimilar piezoelectric half-planes

Summary  In this paper, the dynamic anti-plane crack problem of two dissimilar homogeneous piezoelectric materials bonded through a functionally graded interfacial region is considered. Integral transforms are employed to reduce the problem to Cauchy singular integral equations. Numerical results illustrate the effect of the loading combination parameter λ, material property distribution and crack configuration on the dynamic stress and electric displacement intensity factors. It is found that the presence of the dynamic electric field could impede of enhance the crack propagation depending on the time elapsed and the direction of applied electric impact. Received 4 December 2001; accepted for publication 9 July 2002 This work is supported by the National Natural Science Foundation of China through Grant No. 10132010.     

Mode III crack problem in a functionally graded magneto-electro-elastic strip

Considering the material properties to be one-dimensionally dependent, this paper studied an anti-plane problem for an embedded crack and edge crack perpendicular to the boundary of a functionally graded magneto-electro-elastic strip. The crack is assumed to be either magneto-electrically impermeable or permeable. Integral transform and dislocation density functions are employed to reduce the problem to the solution of a system of singular integral equations. Numerical results show the effects of the loading combination parameter, material gradient parameter and crack configuration on the field intensity factors and the energy release rates of the functionally graded magneto-electro-elastic strip.     

Stress intensity factor for an edge crack in two bonded dissimilar materials under convective cooling

The transient thermal stress crack problem for two bonded dissimilar materials subjected to a convective cooling on the surface containing an edge crack perpendicular to the interface is considered. The problem is solved using the principle of superposition and the uncoupled quasi-static thermoelasticity. The crack problem is formulated by applying the transient thermal stresses obtained from the uncracked medium with opposite sign on the crack surfaces to be the only external loads. Fourier integral transform is used to solve the perturbation problem resulting in a singular integral equation of Cauchy type in which the derivative of the crack surface displacement is the unknown function. The numerical results of the stress intensity factors are calculated for both the edge crack and the crack terminating at the interface using two different composite materials and illustrated as a function of time, crack length, coefficient of heat transfer, and the thickness ratio.     

Dynamic fracture behaviors of cracks in a functionally graded magneto-electro-elastic plate

In this paper the dynamic anti-plane problem for a functionally graded magneto-electro-elastic plate containing an internal or an edge crack parallel to the graded direction is investigated. The crack is assumed to be magneto-electrically impermeable. Integral transforms and dislocation density functions are employed to reduce the problem to Cauchy singular integral equations. Field intensity factors and energy release rate are derived, analyzed and partially calculated numerically. The effects of material graded index, loading combination parameter (including size and direction) and geometry criterion of the plate on the dynamic energy release rate are shown graphically. Numerical results indicate that increasing the graded index can all retard the crack extension, and that both the applied magnetic field loadings and electric field loadings play a dominant role in the dynamic fracture behaviors of crack tips.     

Cylindrical interface crack in a hollow layered functionally graded cylinder under static torsion

A hollow functionally graded composite cylinder under static torsion, which consists of an inner and outer elastic circular tube with a cylindrical interface crack, is studied in this work. By utilizing Fourier integral transform method, the mixed boundary value problem is reduced to a Cauchy singular integral equation, from which the numerical results of the stress intensity factor are obtained by the Lobatto–Chebyshev quadrature technique. Numerical results demonstrate the coupled effects of geometrical, physical, and functionally graded parameters on the interfacial fracture behavior.     

正交各向异性功能梯度涂层-基底结构共线裂纹问题的断裂分析

论文研究了一正交各向异性功能梯涂层粘结到一均匀基底含共线裂纹的平面I型断裂问题.引入新的双参数指数函数模拟连续改变的材料性质,正交各向异性的主轴方向分别为平行和垂直于带的边界,采用积分变换技术,所求的问题转化为第一类的Cauchy奇异积分方程,获得了共线裂纹尖端应力场,结果显示了材料常数和几何参数对应力强度因子的影响.     

Mechanical modeling and transient anti-plane fracture analysis for the graded inter-diffusion regions in a bonded structure

An inter-diffusion interface model (IDIM) is put forward for a bonded structure. Laplace and Fourier integral transforms are applied to reduce the transient anti-plane fracture problem of the structure as a Cauchy singular integral equation. Lobatto-Chebyshev collocation method and Laplace numerical inversion transform are employed to evaluate transient stress intensity factors (TSIFs). The effects of geometrical and physical parameters on TSIFs are studied. Specially discussed are the influences of the weak/micro-discontinuity of the interfaces. Comparison between IDIM and the graded interlayer model indicates that if the inter-diffusion between the two original materials is prominent, the former should be applied instead of the latter in fracture analyses of bonded structures.     

Dynamic stress intensity factors of two collinear mode-III cracks perpendicular to and on the two sides of a bi-FGM weak-discontinuous interface

The mechanical model was established for the anti-plane dynamic fracture problem for two collinear cracks on the two sides of and perpendicular to a weak-discontinuous interface between two materials with smoothly graded elastic properties, as opposed to a sharp interface with discontinuously changing elastic properties. The problem was reduced as a system of Cauchy singular integral equations of the first kind by Laplace and Fourier integral transforms. The integral equations were solved by Erdogan's collocation method and the dynamic stress intensity factors in the time domain were obtained through Laplace numerical inversion proposed by Miller and Guy. The influences of geometrical and physical parameters on the dynamic stress intensity factors were illustrated and discussed, based on which some conclusions were drawn: (a) to increase the thickness of the FGM strip on either side of the interface will be beneficial to reducing the DSIF of a crack perpendicular to a bi-FGM interface and embedded at the center of one of the FGM strips; (b) To increase the rigidity of the FGM strip where the crack is located will increase the DSIF. However, when the material in one side of the interface is more rigid, the DSIF of the interface-perpendicular embedded crack in the other side will be reduced; (c) To decrease the weak-discontinuity of a bi-FGM interface will not necessarily reduce the stress intensity factor of a crack perpendicular to it, which is different from the case of interfacial crack; (d) For two collinear cracks with equal half-length, when the distance between the two inner tips is less than about three times of the half-length, the interaction of them is intensified, however, when the distance is greater than this the interaction becomes weak.     

Mode III fracture problem of an arbitrarily oriented crack in a FGPM strip bonded to a FGPM half plane

This paper studies the mode III electro-elastic field of a cracked functionally graded piezoelectric strip bonded to a functionally graded piezoelectric half plane. The crack is oriented in arbitrary direction. The material properties along x-axis vary in exponential form. By using the Fourier transform, the problem can be formulated into a system of singular integral equations and solved by applying the Gauss–Chebyshev integration formula. The effects come from the edge, crack orientation and the nonhomogeneous material parameters on intensity factors are discussed graphically.     

Fracture analysis of functionally graded coatings: plane deformation

A new multi-layered model for fracture analysis of functionally graded materials (FGMs) with arbitrarily varying elastic moduli under plane deformation has been developed. In this model, the FGM is divided into several sub-layers and in each sub-layer the shear modulus is assumed to be a linear function of the depth while the Poisson's ratio is assumed to be a constant. With this new model, an interface crack problem of a functionally graded coating bonded to a homogeneous half-plane under normal and shear loading is investigated. Employment of transfer matrix method and Fourier integral transform technique reduces the problem to a system of Cauchy singular integral equations which are solved numerically. Stress intensity factors of an interface crack are obtained for the cases of the shear modulus varying in an exponential manner and in a sinusoidal manner. Comparison of the present new model to other existing models shows that the new one is more efficient.     

Mode III crack problems of two bonded functionally graded strips with internal cracks

This paper deals with the anti-plane problem of two bonded functionally graded finite strips. Each strip contains an internal crack normal to the interface. The material properties of two strips are assumed to vary along the direction of the crack lines. A system of singular integral equations is derived and then solved numerically by using Gauss–Chebyshev integration formula. The influences of nonhomogeneous parameters, crack interactions and two edge conditions on the mode III stress intensity factors are investigated.     

CRACK PROPAGATING IN FUNCTIONALLY GRADED COATING WITH ARBITRARILY DISTRIBUTED MATERIAL PROPERTIES BONDED TO HOMOGENEOUS SUBSTRATE

In this paper, a finite crack with constant length （Yoffe type crack） propagating in a functionally graded coating with spatially varying elastic properties bonded to a homogeneous substrate of finite thickness under anti-plane loading was studied. A multi-layered model is employed to model arbitrary variations of material properties based on two linearly-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. The numerical results show that the graded parameters, the thicknesses of the interfacial layer and the two homogeneous layers, the crack size and speed have significant effects on the dynamic fracture behavior.     

Mode III fracture problem of an arbitrarily oriented crack in an FGPM strip bonded to a homogeneous piezoelectric half plane

This paper studies the Mode III electric-elastic field of a cracked functionally graded piezoelectric strip bonded to a homogeneous piezoelectric half plane. The crack is oriented in arbitrary direction. The material properties of the strip vary along the strip thickness in exponential forms. By using the Fourier transform, the problem can be formulated to a system of singular integral equations and solved by applying the Gauss-Chebyshev integration formula. The effects come from the edge, crack orientations and the nonhomogeneous material parameter on intensity factors are discussed graphically.