A surface crack in a graded coating bonded to a homogeneous substrate under dynamic loading conditions

The elastodynamic problem of a surface crack in a graded coating bonded to a homogeneous substrate under dynamic loading is considered. The coating is graded along the thickness direction and modeled as a nonhomogeneous medium with an isotropic stress-strain law. The problem is solved under the assumption of plane strain or generalized plane stress conditions. The crack surfaces are subjected to arbitrary dynamic loadings which give rise to mixed fracture modes which turn out to be uncoupled due to the fact that the crack axis is parallel to the material gradient. Using integral transforms, the resulting mixed-boundary value problem is reduced to a set of two uncoupled singular integral equations which are solved numerically to obtain the crack-tip stress intensity factors. The main objective of the paper is to study the effect of the coating thickness and nonhomogeneity parameter on the crack tip dynamic stress intensity factors for the purpose of gaining better understanding on the behavior of graded coatings.     

An anisotropic layered material with a crack

Summary The problem of an anisotropic layered material which contains a plane crack in its interior is considered here. The problem is reduced to a set of Fredholm integral equations of the second kind which may be solved iteratively. Once these integral equations are solved, the crack tip stress intensity factors may be readily computed. Numerical results for some particular examples involving transversely isotropic materials are given here.     

Two bonded half planes with a crack going through the interface

The plane problem of two bonded elastic half planes containing a finite crack perpendicular to and going through the interface is considered. The problem is formulated as a system of singular integral equations with generalized Cauchy kernels. Even though the system has three irregular points, it is shown that the unknown functions are algebraically related at the irregular point on the interface and the integral equations can be solved by a method developed previously. The system of integral equations is shown to yield the same characteristic equation as that for two bonded quarter planes in the general case of the through crack, and the characteristic equation for a crack tip terminating at the interface in the special case. The numerical results given in the paper include the stress intensity factors at the crack tips, the normal and shear components of the stress intensity factors at the singular point on the interface, and the crack surface displacements.     

Thermoelastic wave propagation in an elastic solid containing a finite crack

Investigated in this paper is the scattering of plane harmonic thermoelastic waves around the tip of a finite crack. Integral transform techniques are used to formulate the problem and reduce it to Fredholm integral equations of the second kind. The equations are solved numerically and the singular stress field near the crack tip is determined. In particular, the variation of the stress intensity factor with the frequency of the incoming wave is exhibited graphically. The peak in the magnitude of the stress intensity factor is of paramount interest in the application of fracture mechanics to thermal stress problems.     

A boundary integral approach for plane analysis of electrically semi-permeable planar cracks in a piezoelectric solid

A plane electro-elastostatic problem involving arbitrarily located planar stress free cracks which are electrically semi-permeable is considered. Through the use of the numerical Green's function for impermeable cracks, the problem is formulated in terms of boundary integral equations which are solved numerically by a boundary element procedure together with a predictor–corrector method. The crack tip stress and electric displacement intensity factors can be easily computed once the boundary integral equations are properly solved.     

A Moving Mode-III Crack in a Functionally Graded Piezoelectric Strip

In this paper, the problem of a functionally graded piezoelectric strip with a constant-velocity Yoffe-type moving crack is considered. By using the Fourier transforms, the problem is first reduced to dual integral equations and then into Fredholm integral equations of the second kind. The electroelastic field near the crack tip is obtained for electrical impermeable boundary conditions and electrical permeable boundary conditions, respectively. The results obtained show that the gradient of the material properties can increase or decrease the magnitudes of the stress intensity factors, and the velocity can disturb the stress distribution near the crack tip.     

Trifurcation of a crack due to plane SH-waves in an infinite elastic medium

The dynamic anti-plane problem of trifurcation of a semi-infinite crack due to incidence of two linearly varying plane SH-waves with non-parallel wave fronts in an infinite elastic medium has been considered. The semi-infinite crack is assumed to trifurcate when the plane waves intersect the crack tip. The problem has been solved using the self-similar technique, which is based on the observation that certain field variables show dynamic similarities. The results include the expressions for shear stress in the planes of the cracks and the stress intensity factors at the crack tips. Numerical calculations have been carried out to show the variations of stress intensity factors at the crack tips with the angle of skew for different values of the crack tip velocity and angle of incidence.     

Solution of plate bending problems in fracture mechanics using a specialized finite element technique

This paper deals with the development and application of a special crack-tip finite element to obtain the bending and shear intensity factors for thin elastic plates containing cracks. The bending and shear intensity factors are then used to compute the Strain Energy Density Factor and the direction of crack initiation. The solution procedure is illustrated through several numerical examples. The problem of an axial flow compressor blade containing a crack is solved using a combination of special crack tip plate bending and plane stress elements.     

The scattering of harmonic elastic anti-plane shear waves by a Griffith crack in a piezoelectric material plane by using the non-local theory

In this paper, the dynamic behavior of a Griffith crack in a piezoelectric material plane under anti-plane shear waves is investigated by using the non-local theory for impermeable crack face conditions. For overcoming the mathematical difficulties, a one-dimensional non-local kernel is used instead of a two-dimensional one for the anti-plane dynamic problem to obtain the stress and the electric displacement near the crack tips. By using the Fourier transform, the problem can be solved with the help of two pairs of dual integral equations. These equations are solved using the Schmidt method. Contrary to the classical elasticity solution, it is found that no stress and electric displacement singularity is present near the crack tip. The non-local dynamic elastic solutions yield a finite hoop stress near the crack tip, thus allowing for a fracture criterion based on the maximum dynamic stress hypothesis. The finite hoop stress at the crack tip depends on the crack length, the circular frequency of incident wave and the lattice parameter. For comparison results between the non-local theory and the local theory for this problem, the same problem in the piezoelectric materials is also solved by using local theory.     

Stress analysis of a penny-shaped crack locted between two oblate spheroidal cavities in an infinite solid

The problem of a penny-shaped crack located between two oblate spheroidal cavities in an infinite solid subjected to uniaxial loads is considered. Using transformations between harmonic functions in cylindrical coordinates and those in oblate spheroidal ones, the problem is reduced to non-homogeneous linear equations. The obtained equations are solved numerically and the stress intensity factors at the penny-shaped crack tip under the influence of the two oblate spheroidal cavities are shown graphically.     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semiinfinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

Multiscale continuum modeling of a crack in elastic media with microstructures

Cosserat type continuum theories have been employed by many authors to study cracks in elastic solids with microstructures. Depending on which theory was used, different crack tip stress singularities have been obtained. In this paper, a microstructure continuum theory is used to model a layered elastic medium containing a crack parallel to the layers. The crack problem is solved by means of the Fourier transform. The resulting integrodifferential equations are discretized using the Chebyshev polynomial expansion method for numerical solutions. By using the present theory, the explicit internal length effects upon the crack opening displacement and stress field can be observed. It is found that the stress field near the crack tip is not singular according to the microstructure continuum solution although the level of the opening stress shows an increasing trend until it gets very close to the crack tip. The rising portion of the near tip opening stress is used to project the stress intensity factor which agrees fairly well with that obtained using the FEM to perform stress analyses of the cracked layered medium with the exact geometry. The numerical solutions also indicate that treating the layered medium as an equivalent homogeneous classical elastic solid is not adequate if cracks are present and accurate stress intensity factors in the original layered medium is desired.     

The interface crack problem under a concentrated load for a functionally graded coating–substrate composite system

The plane crack problem for a functionally graded coating–substrate system under a concentrated load is studied in this paper. The medium consists of a functionally graded coating bonded to a homogeneous substrate of finite thickness, containing an interface crack of finite length. With use of the integration transform and differential factor methods, the displacement form can be obtained. By introducing auxiliary functions, the present problem can be turned into solving a group of singular integral equations. The mixed-mode stress intensity factors (SIFs) and strain energy release rates (SERRs) are obtained. The influences of the parameters such as the load location, nonhomogeneity constants and the geometry parameters on the SIFs and SERRs are studied.     

Pull-off test for a cracked layer bonded to a rigid foundation

This paper is concerned with the plane strain problem of an elastic incompressible layer bonded to a rigid foundation. An upward tensile force is applied to the top surface of the layer through a rigid strip of finite thickness. The layer contains either a finite central crack or two semi-infinite external cracks. The analysis leads to a system of singular integral equations. These integral equations are solved numerically and the interface stress distributions, stress intensity factors at the crack tips and at the corners of the rigid strip, probable cleavage angle for the finite crack and strain energy release rate are calculated for various geometries.     

Thermal stresses in a slab containing an annular crack

A linear thermoelastic problem of a slab containing an annular crack is solved. Using integral transform techniques, the problem is reduced to that of solving two singular integral equations of the first kind. The solution of the singular integral equation is obtained in the form of the product of the series of Chebyshev polynomials of the first kind and their weight functions. Thus the essential feature of the singular stress field near the crack is preserved and the crack tip stress intensity factor is easily evaluated. Numerical calculations are also carried out and the variations of the stress intensity factors are plotted against the geometry for various values of physical properties.     

Interface crack problems in graded orthotropic media: Analytical and computational approaches

Interface crack problems in graded orthotropic media are considered using analytical and computational techniques. In the analytical formulation an interface crack between a graded orthotropic coating and a homogeneous orthotropic substrate is considered. The principal axes of orthotropy are assumed to be parallel and perpendicular to the crack plane. Mechanical properties of the medium are assumed to be continuous with discontinuous derivatives at the interface. The problem is formulated in terms of the averaged constants of plane orthotropic elasticity and reduced to a pair of singular integral equations which are solved numerically to compute the mixed mode stress intensity factors and the energy release rate. In the second part of the study, enriched finite elements are formulated and implemented for graded orthotropic materials. Comparisons of the finite element and analytical results show that enriched finite element technique is capable of producing highly accurate results for crack problems in graded orthotropic media. Finally, periodic interface cracking and the four point bending test for graded orthotropic solids are modeled using enriched finite elements and the results are briefly discussed.     

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

In an attempt to simulate buckling of nonuniform coatings, we consider the problem of an embedded crack in a functionally graded 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 loading consists of a uniform compressive strain applied away from the crack region. The graded coating is modeled as a nonhomogeneous medium with an isotropic 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 give the critical buckling strain and the corresponding crack opening displacement shapes. 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 and coating thicknesses.     

Edge crack in an elastic layer resting on Winkler foundation

The paper deals with the stress analysis near a crack tip in an elastic layer resting on Winkler foundation. The edge crack is assumed to be normal to the lower boundary plane. The upper surface of the layer is loaded by given forces normal to the boundary. The considered problem is solved by using the method of Fourier transforms and dual integral equations, which are reduced to a Fredholm integral equation of the second kind. The stress intensity factor is given in the term of solution of the Fredholm integral equation and some numerical results are presented.     

The effect of the lattice parameter of functionally graded materials on the dynamic stress field near crack tips

In this paper, the effect of the lattice parameter of functionally graded materials on the dynamic stress fields near crack tips subjected to the harmonic anti-plane shear waves is investigated by means of non-local theory. By use of the Fourier transform, the problem can be solved with the help of a pair of dual integral equations, in which the unknown variable is the displacement on the crack surfaces. To solve the dual integral equations, the displacement on the crack surfaces is expanded in a series of Jacobi polynomials. Unlike the classical elasticity solutions, it is found that no stress singularities are present near crack tips. The non-local elastic solution yields a finite hoop stress at the crack tip, thus allowing us to use the maximum stress as a fracture criterion in functionally graded materials.     

Thermal resistance inside an interface crack in the functionally graded sandwich structures

This article proposes a thermal facture model of sandwich structures occupying a functionally graded interlayer with thermal resistance inside the crack region introduced. The crack surfaces are partially thermally insulated with thermally insulated crack and thermally conductive crack as limiting cases. A system of singular integral equations in thermo‐elastic field is reduced and solved numerically by using the collocation methods with higher asymptotic terms in order to improve the convergence and accuracy. For a special case, exact solution is derived to validate the theoretical analysis and numerical computation. Numerical analyses are conducted to reveal the influences of the graded interlayer size, graded parameter and dimensionless thermal resistance on the temperatures along the crack plane and the stress intensity factors.