Periodically distributed parallel cracks in a functionally graded piezoelectric (FGP) strip bonded to a FGP substrate under static electromechanical load

In this paper, the Fourier integral transform–singular integral equation method is presented for the problem of a periodic array of cracks in a functionally graded piezoelectric strip bonded to a different functionally graded piezoelectric material. The properties of two materials, such as elastic modulus, piezoelectric constant and dielectric constant, are assumed in exponential forms and vary along the crack direction. The crack surface condition is assumed to be electrically impermeable or permeable. The mixed boundary value problem is reduced to a singular integral equation over crack by applying the Fourier transform and the singular integral equation is solved numerically by using the Lobatto–Chebyshev integration technique. The analytic expressions of the stress intensity factors and the electric displacement intensity factors are derived. The effects of the loading parameter λ, material constants and the geometry parameters on the stress intensity factor, the energy release ratio and the energy density factor are studied.     

Thermal singular stresses of glassfiber reinforced plastics with surface cracks at cryogenic temperatures

Thermal singular stress problem for glassfiber reinforced plastics with surface cracks at cryogenic temperatures is considered. For the case of the crack which is normal to and ends at the interface between orthotropic elastic materials, the order of stress singularity around the tip of the crack is obtained. Fourier transforms are used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gauss–Jacobi integration formula. Numerical calculations are carried out for the cases of embedded and edge cracks, and the thermal stress intensity factors at different temperatures are shown graphically.     

Axisymmetric crack problem for a hollow cylinder imbedded in a dissimilar medium

The analytical solution for the linear elastic problem of flat annular crack in a transversely isotropic hollow cylinder imbedded in a transversely isotropic medium is considered. The hollow cylinder is assumed to be perfectly bonded to the surrounding medium. This structure, which can represent a cylindrical coating-substrate system, is subjected to uniform crack surface pressure. Because of the geometry and the loading, the problem is axisymmetric. The z = 0 plane on which the crack lies, is also a plane of symmetry. The composite media consisting of the hollow cylinder and the surrounding medium extends to infinity in z and r directions. The mixed boundary value problem is formulated in terms of the unknown derivative of the crack surface displacement by using Fourier and Hankel transforms. By extending the crack to the inner surface and to the interface, the cases of surface crack and crack terminating at the interface are obtained. Asymptotic analyses are performed to derive the generalized Cauchy kernel and associated stress singularities. The resulting singular integral equation is solved numerically. Stress intensity factors for various crack configurations, crack opening displacements and stresses along the interface and on z = 0 plane are presented for sample material combinations and geometric parameters.     

Mode-I crack problem for functionally graded layered structures

This paper deals with two bonded functionally graded finite strips with two collinear cracks. Different layers may have different nonhomogeneity properties in the structure. A bi-parameter exponential function was introduced to simulate the continuous variation of material properties. The problem is solved by using the integral transform, singular integral equation methods and the theory of residues. Various internal cracks and edge crack and crack crossing the interface configurations are investigated, respectively. The asymptotic stress field near the tip of a crack crossing the interface is examined and it is shown that, unlike the corresponding stress field in piecewise homogeneous materials, in this case the “kink” in material property at the interface does not introduce any singularity. Numerical calculations are carried out, and the influences of nonhomogeneity constants, geometric parameters and crack interactions on the stress intensity factors are investigated.     

Fracture of a surface layer bonded to a half space

The plane problem of a cracked elastic surface layer bonded to an elastic half space is considered. The surface layer is assumed to contain a transverse crack whose surface is subjected to uniform compression. The problem is formulated in terms of a singular integral equation, the derivative of the crack surface displacement being the density function. By using appropriate quadrature formulas, the integral equation reduces to a system of linear algebraic equations. This system is solved; the stress intensity factors and the crack surface displacement for various crack geometries, namely for internal crack, edge crack, crack touching the interface, and completely broken layer cases, are obtained.     

Anti-plane problem of periodic interface cracks in a functionally graded coating-substrate structure

In this paper, the interface cracking between a functionally graded material (FGM) and an elastic substrate is analyzed under antiplane shear loads. Two crack configurations are considered, namely a FGM bonded to an elastic substrate containing a single crack and a periodic array of interface cracks, respectively. Standard integral-transform techniques are employed to reduce the single crack problem to the solution of an integral equation with a Cauchy-type singular kernel. However, for the periodic cracks problem, application of finite Fourier transform techniques reduces the solution of the mixed-boundary value problem for a typical strip to triple series equations, then to a singular integral equation with a Hilbert-type singular kernel. The resulting singular integral equation is solved numerically. The results for the cases of single crack and periodic cracks are presented and compared. Effects of crack spacing, material properties and FGM nonhomogeneity on stress intensity factors are investigated in detail.     

Transverse cracks in a strip with reinforced surfaces

The symmetrical problem of two transverse cracks in an elastic strip with reinforced surfaces is formulated in terms of a singular integral equation. The special cases of one central crack or two edge cracks are discussed. Numerical methods for solving the problems with internal cracks are outlined and stress intensity factors are presented for various geometries and degrees of surface reinforcement.     

New Fredholm integral equation for multiple crack problem in plane elasticity and antiplane elasticity

A singular integral equation for the multiple crack problem of plane elasticity is formulated in this paper. In the formulation we choose the crack opening displacement (COD) as unknown function and the resultant force as the right hand term of the equation. After using Vekua's regularization procedure or making inversion of the Cauchy singular integral in the equation, a new Fredholm integral equation is obtainable. The obtained Fredholm integral equation is compact in form and easy for computation. After solving the equation, the CODs of the cracks and the stress intensity factors (SIFs) at the crack tips can be derived immediately. Similar formulation for the multiple crack problem of antiplane elasticity is also presented. Finally, numerical examples are given to demonstrate the use of the proposed integral equation approach.     

The mode I crack problem for layered piezoelectric plates

The plane strain singular stress problem for piezoelectric composite plates having a central crack is considered. For the case of the crack which is normal to and ends at the interface between the piezoelectric plate and the elastic layer, the order of stress singularity around the tip of the crack is obtained. The Fourier transform technique is used to formulate the problem in terms of a singular integral equation. The singular integral equation is solved by using the Gaus–Jacobi integration formula. Numerical calculations are carried out, and the main results presented are the variation of the stress intensity factor as functions of the geometric parameters, the piezoelectric material properties and the electrical boundary conditions of the layered composites.     

The surface crack problem for a layered plate with a functionally graded nonhomogeneous interface

The plane elasticity solution is presented in this paper for the crack problem of a layered plate. A functionally graded interfacial region is assumed to exist as a distinct nonhomogeneous transitional layer with the exponentially varying elastic property between the dissimilar homogeneous surface layer and the substrate. The surface layer contains a crack perpendicular to the boundaries. The Fourier transform technique is used to formulate the problem in terms of a singular integral equation. The main results presented are the variations of stress intensity factors as functions of geometric and material parameters of the layered plate.     

Boundary integral equation method for conductive cracks in two and three-dimensional transversely isotropic piezoelectric media

Using the fundamental solutions and the Somigliana identity of piezoelectric medium, the boundary integral equations are obtained for a conductive planar crack of arbitrary shape in three-dimensional transversely isotropic piezoelectric medium. The singular behaviors near the crack edge are studied by boundary integral equation approach, and the intensity factors are derived in terms of the displacement discontinuity and the electric displacement boundary value sum near the crack edge on crack faces. The boundary integral equations for two dimensional crack problems are deduced as a special case of infinite strip planar crack. Based on the analogy of the obtained boundary integral equations and those for cracks in conventional isotropic elastic material and for contact problem of half-space under the action of a rigid punch, an analysis method is proposed. As an example, the solution to conductive Griffith crack is derived.     

Axisymmetric crack terminating at the interface of transversely isotropic dissimilar media

In this paper, the axisymmetric elasticity problem of an infinitely long transversely isotropic solid cylinder imbedded in a transversely isotropic medium is considered. The cylinder contains an annular or a penny shaped crack subjected to uniform pressure on its surfaces. It is assumed that the cylinder is perfectly bonded to the medium. A singular integral equation of the first kind (whose unknown is the derivative of crack surface displacement) is derived by using Fourier and Hankel transforms. By performing an asymptotic analysis of the Fredholm kernel, the generalized Cauchy kernel associated with the case of `crack terminating at the interface' is derived. The stress singularity associated with this case is obtained. The singular integral equation is solved numerically for sample cases. Stress intensity factors are given for various crack geometries (internal annular and penny-shaped cracks, annular cracks and penny-shaped cracks terminating at the interface) for sample material pairs.     

The gauss-hermite numerical integration method for the solution of the plane elastic problem of semi-infinite periodic cracks

The problem of parallel semi-infinite periodic cracks subjected to a transversely directed load in an infinite isotropic and elastic medium under conditions of plane stress or plane strain can be reduced to the solution of a Cauchy-type singular integral equation along one of the cracks. This equation can be transformed into a system of linear equations by means of an approximation of the integrals through the Gauss-Hermite procedure and application of the equation to distinct points along the faces of the crack. Stress intensity factors thus determined for the crack tips under constant load along the cracks are in satisfactory agreement with corresponding values derived previously.     

Perturbation method for the solution of a Zener–Stroh crack with a slightly curved configuration

This paper investigates the Zener–Stroh crack with curved configuration in plane elasticity. A singular integral equation is suggested to solve the problem. Formulae for evaluating the SIFs and T-stress at the crack tip are suggested. If the curve configuration is a product of a small parameter and a quadratic function, a perturbation method based on the singular integral equation is suggested. In the method, the singular integral equation can be expanded into a series with respect to the small parameter. Therefore, many singular integral equations can be separated from the same power order for the small parameter. These singular integral equations can be solved successively. The solution of the successive singular integral equations will provide results for stress intensity factors and T-stress at the crack tip. It is found that the behaviors for the solution of SIFs and T-stress in the Zener–Stroh crack and the Griffith crack are quite different. This can be seen from the presented comparison results.     

Periodic cracking of functionally graded coatings

The antiplane elasticity problem for a functionally graded coating bonded to a homogeneous half space is considered. The coating is assumed to contain periodic cracks perpendicular to the surface. The problem is formulated in terms of an integral equation with strongly singular kernels. Three dimensionless parameters representing the crack depth, the material nonhomogeneity and the crack periodicity are identified. In addition to the mode III stress intensity factor calculated by varying these three parameters, the results presented include a qualitative discussion of the question of fracture instability, the effect of periodic cracking on the relaxation of stresses on the coating surface, and the comparison of the total strain energy released as a result of surface cracking with that assumed in a simple stress relaxation model.     

Comments on the evaluation of the stress intensity factor for a general re-entrant corner in anisotropic bi-materials

This paper illustrates an efficient contour integral procedure to obtain stress intensity factors in combination of the asymptotic analysis with finite element analysis. Note that this set-up is very general: the material can be anisotropic elastic, and the specimen can be built as a bi-material system, notches of arbitrary opening angle can be analyzed (γ = 0 → crack, γ = 180° → free edge).The purpose of this technical note is to comment on three issues in the notch mechanics: the interpretation of the eigenvalue equation, the definition of stress intensity factors, and the effect of the outer contour location on H-integral evaluations.     

Asymmetrical Cracks Parallel to an Interface between Dissimilar Materials

This paper solves a plane strain problem for two bonded dissimilar planes containing a crack parallel to the interface in each layer. The bimaterial system is loaded by tractions distributed along the crack surfaces. Based on the Fourier transform, the problem is reduced to a system of Cauchy type singular integral equations which contain exact and explicit kernel functions. The solution of these equations is obtained easily by utilizing Gauss–Chebyshev integral formulae for various material combinations and geometrical parameters. Several numerical results of stress intensity factors, energy release rate and stress distribution along the interface are presented to exhibit the interaction among cracks and interface. This revised version was published online in July 2006 with corrections to the Cover Date.     

The practical evaluation of stress intensity factors at semi-infinite crack tips

The solution of crack problems in plane or antiplane elasticity can be reduced to the solution of a singular integral equation along the cracks. In this paper the Radau-Chebyshev method of numerical integration and solution of singular integral equations is modified, through a variable transformation, so as to become applicable to the numerical solution of singular integral equations along semi-infinite intervals, as happens in the case of semi-infinite cracks, and the direct determination of stress intensity factors at the crack tips. This technique presents considerable advantages over the analogous technique based on the Gauss-Hermite numerical integration rule. Finally, the method is applied to the problems of (i) a periodic array of parallel semi-infinite straight cracks in plane elasticity, (ii) a similar array of curvilinear cracks, (iii) a straight semi-infinite crack normal to a bimaterial interface in antiplane elasticity and (iv) a similar crack in plane elasticity; in all four applications appropriate geometry and loading conditions have been assumed. The convergence of the numerical results obtained for the stress intensity factors is seen to be very good.     

Determination of higher order coefficients and zones of dominance using a singular integral equation approach

A method to determine higher order coefficients from the solution of a singular integral equation is presented. The coefficients are defined by , which gives the radial stress at a distance, r, in front of the crack tip. In this asymptotic series the stress intensity factor, k₀, is the first coefficient, and the T-stress, T₀, is the second coefficient. For the example of an edge crack in a half space, converged values of the first 12 mode I coefficients (k_n and T_n, n = 0, … , 5) have been determined, and for an edge crack in a finite width strip, the first six coefficients are presented. Coefficients for an internal crack in a half space are also presented. Results for an edge crack in a finite width strip are used to quantify the size of the k-dominant zone, the kT-dominant zone and the zones associated with three and four terms, taking into account the entire region around the crack tip.     

An integral equation approach to the inclusion-crack interactions in three-dimensional infinite elastic domain

 In this paper, an integral equation method to the inclusion-crack interaction problem in three-dimensional elastic medium is presented. The method is implemented following the idea that displacement integral equation is used at the source points situated in the inclusions, whereas stress integral equation is applied to source points along crack surfaces. The displacement and stress integral equations only contain unknowns in displacement (in inclusions) and displacement discontinuity (along cracks). The hypersingular integrals appearing in stress integral equation are analytically transferred to line integrals (for plane cracks) which are at most weakly singular. Finite elements are adopted to discretize the inclusions into isoparametric quadratic 10-node tetrahedral or 20-node hexahedral elements and the crack surfaces are decomposed into discontinuous quadratic quadrilateral elements. Special crack tip elements are used to simulate the variation of displacements near the crack front. The stress intensity factors along the crack front are calculated. Numerical results are compared with other available methods. Received: 28 January 2002 / Accepted: 4 June 2002 The work described in this paper was partially supported by a grant from the Research Grant Council of the Hong Kong Special Administration Region, China (Project No.: HKU 7101/99 E).