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.     

An axisymmetric problem of an embedded crack in a graded layer bonded to a homogeneous half-space

In an attempt to simulate non-uniform coating delamination, the elasto-static problem of a penny shaped axisymmetric crack embedded in a functionally graded coating bonded to a homogeneous substrate subjected to crack surface tractions is considered. The coating’s material gradient is parallel to the axisymmetric direction and is orthogonal to the crack plane. The graded coating is modeled as a non-homogeneous medium with an isotropic constitutive law. Using Hankel transform, the governing equations are converted into coupled singular integral equations, which are solved numerically to yield the crack tip stress intensity factors. The Finite Element Method was additionally used to model the crack problem. The main objective of this paper is to study the influence of the material non-homogeneity and the crack position on the stress intensity factors for the purpose of gaining better understanding on the behavior of graded coatings.     

Antiplane crack at the interface of two bonded dissimilar graded piezoelectric materials

The problem of an antiplane crack situated in the interface of two bonded dissimilar graded piezoelectric half-spaces is considered under the permeable crack assumption. The mechanical and electrical properties of the half-spaces are considered for a class of functional forms for which the equilibrium equation has analytic solutions. By using an integral transform technique, the problem is reduced to dual integral equations which are transformed into a Fredholm integral equation by introducing an auxiliary function. The stress intensity factors are obtained in explicit form in terms of auxiliary functions. By solving the Fredholm integral equation numerically, the numerical results for stress intensity factors are obtained which have been displayed graphically to show the influence of the graded piezoelectric materials.     

Anti-plane shear of an arbitrary oriented crack in a functionally graded strip bonded with two dissimilar half-planes

An internal crack located within a functionally graded material (FGM) strip bonded with two dissimilar half-planes and under an anti-plane load is considered. The crack is oriented in an arbitrary direction. The material properties of strip are assumed to vary exponentially in the thickness direction and two half-planes are assumed to be isotropic. Governing differential equations are derived and to reduce the difficulty of the problem dealing with solution of a system of singular integral equations Fourier integral transform is employed. Semi closed form solution for the stress distribution in the medium is obtained and mode III stress intensity factor (SIF), at the crack tip is calculated and its validity was verified. Finally, the effects of nonhomogeneous material parameter and crack orientation on the stress intensity factor are studied.     

Numerical solution of the Cauchy problem for steady-state heat transfer in two-dimensional functionally graded materials

The application of the method of fundamental solutions to the Cauchy problem for steady-state heat conduction in two-dimensional functionally graded materials (FGMs) is investigated. The resulting system of linear algebraic equations is ill-conditioned and, therefore, regularization is required in order to solve this system of equations in a stable manner. This is achieved by employing the zeroth-order Tikhonov functional, while the choice of the regularization parameter is based on the L-curve method. Numerical results are presented for both smooth and piecewise smooth geometries. The convergence and the stability of the method with respect to increasing the number of source points and the distance between the source points and the boundary of the solution domain, and decreasing the amount of noise added into the input data, respectively, are analysed.     

Modeling method for a crack problem of functionally graded materials with arbitrary properties—piecewise-exponential model

A new model, piecewise-exponential model (PE model), is developed to investigate the crack problem of the functionally graded materials (FGMs) with arbitrary properties. In the PE model, the functionally graded material is divided into some nonhomogeneous layers along the gradient direction of the properties, with each layer’s properties varying exponentially. By this way, the real material properties can be approached by a series of exponential functions. Since the real material properties are used on both surfaces of each nonhomogeneous layer, the nature of continuously varying properties of FGMs can be approached accurately. The influences of the local nonhomogeneity on the crack-tip fields can be fully considered. By using the new model, the fracture problem of a functionally graded strip with arbitrary properties and a crack vertical to the free surfaces is studied. The integral transform method, the theory of residues and the theory of singular integral equation are applied. Some representative samples with different kinds of nonhomogeneous properties are analyzed and the corresponding stress intensity factors (SIFs) are presented. It is shown that the PE mode is effective for investigating the crack problems of the FGMs with arbitrary properties.     

Mode III stress intensity factors in an interfacial crack in dissimilar bonded materials

The antiplane shear deformation problem of two edge-bonded dissimilar isotropic wedges is considered. In the case when the sum of the two apex angles is equal to 2π, the problem reduces to that of two edge-bonded dissimilar materials with an interfacial crack subjected to concentrated antiplane shear tractions on the crack faces. An explicit expression is extracted for the stress intensity factor at the crack tip. In the special cases of different combinations of the apex angles, the obtained expression for the stress intensity factor may be simplified and relations of a simpler form are given for the stress intensity factor. It is shown that the stress intensity factor is dependent on the material properties as well as the geometry and loading. However, in special cases of equal apex angles as well as the case of similar materials the dependency of the stress intensity factor on the material properties disappears.     

Stress intensity factors around a cylindrical crack in an interfacial zone in composite materials

Axisymmetric stresses around a cylindrical crack in an interfacial cylindrical layer between an infinite elastic medium with a cylindrical cavity and a circular elastic cylinder made of another material have been determined. The material constants of the layer vary continuously from those of the infinite medium to those of the cylinder. Tension surrounding the cylinder and perpendicular to the axis of the cylinder is applied to the composite materials. To solve this problem, the interfacial layer is divided into several layers with different material properties. The boundary conditions are reduced to dual integral equations. The differences in the crack faces are expanded in a series so as to satisfy the conditions outside the crack. The unknown coefficients in the series are solved using the conditions inside the crack. Numerical calculations are performed for several thicknesses of the interfacial layer. Using these numerical results, the stress intensity factors are evaluated for infinitesimal thickness of the layer.     

Stress concentration around a spheroidal crack caused by a harmonic wave incident at an arbitrary angle

A method is proposed to study the stress concentration around a shallow spheroidal crack in an infinite elastic body. The stress concentration is due to the diffraction of a low-frequency plane longitudinal wave by the crack. The direction of wave propagation is established in which the combined concentration of mode I and mode II stresses is maximum __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 1, pp. 70–77, January 2006.     

Buckling of an orthotropic graded coating with an embedded crack bonded to a homogeneous substrate

To simulate buckling of nonuniform coatings, we consider the problem of an embedded crack in a graded orthotropic coating bonded to a homogeneous substrate subjected to a compressive loading. The coating is graded in the thickness direction and the material gradient is orthogonal to the crack direction which is parallel with the free surface. The elastic properties of the material are assumed to vary continuously along the thickness direction. The principal directions of orthotropy are parallel and perpendicular to the crack orientation. The loading consists of a uniform compressive strain applied away from the crack region. The graded coating is modeled as a nonhomogeneous medium with an orthotropic stress–strain law. Using a nonlinear continuum theory and a suitable perturbation technique, the plane strain problem is reduced to an eigenvalue problem describing the onset of buckling. Using integral transforms, the resulting plane elasticity equations are converted analytically into singular integral equations which are solved numerically to yield the critical buckling strain. The Finite Element Method was additionally used to model the crack problem. The main objective of the paper is to study the influence of material nonhomogeneity on the buckling resistance of the graded layer for various crack positions, coating thicknesses and different orthotropic FGMs.     

弹性功能梯度材料板条中周期裂纹的反平面问题

讨论了弹性功能梯度材料板条中裂纹的反平面问题. 用Fourier 变换方法得到了一个基本解. 这个基本解表示了实轴上一点作用有点位错时引起的影响. 利 用此基本解可得单裂纹和周期裂纹问题的奇异积分方程. 在周期裂纹求解时, 远处裂纹对于中央裂纹的影响作了有效的近似处理. 最后, 给出了数值结果, 它表示了材料性质对于裂纹端应力强度因子的影响.     

Asymptotic fields around an interfacial crack with a cohesive zone ahead of the crack tip

This paper considers an interfacial crack with a cohesive zone ahead of the crack tip in a linearly elastic isotropic bi-material and derives the mixed-mode asymptotic stress and displacement fields around the crack and cohesive zone under plane deformation conditions (plane stress or plane strain). The field solution is obtained using elliptic coordinates and complex functions and can be represented in terms of a complete set of complex eigenfunction terms. The imaginary portion of the eigenvalues is characterized by a bi-material mismatch parameter ε = arctanh(β)/π, where β is a Dundurs parameter, and the resulting fields do not contain stress singularity. The behaviors of “Mode I” type and “Mode II” type fields based on dominant eigenfunction terms are discussed in detail. For completeness, the counterpart for the Mode III solution is included in an appendix.     

The mixed mode crack problem in an FGM layer bonded to a homogeneous half-plane

In this paper, the plane elasticity problem of an arbitrarily oriented crack in a FGM layer bonded to a homogeneous half-plane is considered. The problem is modeled by assuming that the elastic properties of the FGM layer are exponential functions of the thickness coordinate and are continuous at the interface of the FGM layer and the half-plane.The Fourier transform technique is used to reduce the problem to the solution of a system of Cauchy-type singular integral equations, which are solved numerically. The stress intensity factors are computed for various crack orientations, crack locations and material parameters. The results show that crack length, crack orientation and the non-homogeneity parameter of the strip material have significant effect on the fracture of the FGM layer.     

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.     

Moving eccentric crack in a piezoelectric strip bonded to elastic materials

Summary  The steady-state of a propagation eccentric crack in a piezoelectric ceramic strip bonded between two elastic materials under combined anti-plane mechanical shear and in-plane electrical loadings is considered in this paper. The analysis based on the integral transform approach is conducted on the permeable crack condition. Field intensity factors and energy release rate are obtained in terms of a Fredholm integral equation of the second kind. It is shown for this geometry that the crack propagation speed has influence on the dynamic energy release rate. The initial crack branching angle for a PZT-5H piezoceramic structure is predicted by the maximum energy release rate criterion. Received 23 January 2001; accepted for publication 18 October 2001     

Mode III fracture of an arbitrary oriented crack in two dimensional functionally graded material

In this paper, a two dimensional functionally graded material (2D-FGM) under an anti-plane load with an internal crack is considered. The crack is oriented in an arbitrary direction. The material properties are assumed to vary exponentially in two planar directions. The problem is analyzed and solved by two different methods namely Fourier integral transforms with singular integral equation technique, and then by the finite element method. The effects of crack orientation, material non-homogeneity, and other parameters on the value of stress intensity factor (SIF) are studied. Finally, the obtained results for Mode III stress intensity factor of different methods are compared.     

An anti-plane propagating crack in a functionally graded piezoelectric strip bonded to a homogeneous piezoelectric strip

Summary A finite crack propagating at constant speed in a functionally graded piezoelectric strip (FGPS) bonded to a homogeneous piezoelectric strip is considered. It is assumed that the electroelastic material properties of the FGPS vary exponentially across the thickness of the strip, and that the bimaterial strip is under combined anti-plane mechanical shear and in-plane electrical loads. The analysis is conducted for the electrically unified crack boundary condition, which includes both the traditional permeable and the impermeable ones. By using the Fourier transform, the problem is reduced to the solution of Fredholm integral equations of the second kind. Numerical results for the stress intensity factor and the crack sliding displacement are presented to show the influences of the crack propagation speed, electric loads, FGPS gradation, crack length, electromechanical coupling coefficient, properties of the bonded homogeneous piezoelectric strip and crack location.     

On the direction of development of a thin fracture process zone at the tip of an interfacial crack between dissimilar media

The Wiener-Hopf method is used to study, under the conditions of plane strain, the direction of development of a thin fracture process zone at the tip of an interfacial crack in a piecewise homogeneous isotropic elastoplastic body. The zone is modeled by a straight line of tangential displacement discontinuity that emerges from the crack tip at an angle to the interface. The dependences of the zone length and the angle on the load and other parameters of the problem are investigated __________ Translated from Prikladnaya Mekhanika, Vol. 42, No. 2, pp. 14–23, February 2006.     

Driving forces and kinking of an oblique crack in bonded nonhomogeneous materials

Summary  Plane elasticity solutions are presented for the problem of an oblique crack in two bonded media. The material model under consideration consists of a homogeneous half-plane with an arbitrarily oriented crack and a nonhomogeneous half-plane. The Fourier integral transform method is employed in conjunction with the coordinate transformations of field variables in the basic elasticity equations. Formulation of the crack problem results in having to solve a system of singular integral equations for arbitrary crack surface tractions. A crack perpendicular to or along the bonded interface between the homogeneous and nonhomogeneous constituents arises as a limiting case. In the numerical results, the values of mixed-mode stress intensity factors are provided for various combinations of relevant geometric and material parameters of the bonded media. Subsequently, the infinitesimal kinks from the tips of a main crack are presumed, with the corresponding local driving forces being evaluated in terms of the stress intensities of the main crack. The criterion of maximum energy release rate is applied with the aim of making some conjectures concerning the likelihood of kinking and the probable kink direction based on the approximation of local homogeneity and brittleness of the crack-tip behavior. Received 25 September 2001; accepted for publication 13 February 2002     

Stress intensity factors for an edge interface crack in a bonded semi-infinite plate for arbitrary material combination

Although a lot of interface crack problems were previously treated, few solutions are available under arbitrary crack lengths and material combinations. In this paper the stress intensity factors of an edge interface crack in a bonded strip are considered under tension with varying the crack length and material combinations systematically. Then, the limiting solutions are provided for an edge interface crack in a bonded semi-infinite plate under arbitrary material combinations. In order to calculate the stress intensity factors accurately, exact solutions in an infinite bonded plate are also considered to produce proportional singular stress fields in the analysis of FEM by superposing specific tensile and shear stresses at infinity. The details of this new numerical solution are described with clarifying the effect of the element size on the stress intensity factor. It is found that for the edge interface crack the normalized stress intensity factors are not always finite depending upon Dunders’ parameters. This behavior can be explained from the condition of the singular stress at the end of bonded strip. Convenient formulas are also given by fitting the computed results.