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.     

General solution for arc crack problem in thermoelastic medium

A problem of a circular arc-shaped crack in an infinite plate under the thermal loading is solved by using the complex variable function and the integral equation method. General solution for arbitrary heat flux along the crack face is obtained. For some particular cases, for example, the constant heat flux case and remote heat flux case, a closed form solution is obtained. The solution technique is effective in derivation and compact in form.     

Scattering of inhomogeneous wave by viscoelastic interface crack

Summary The reflection, refraction and scattering of inhomogeneous plane waves of SH type by an interface crack between two dissimilar viscoelastic bodies are investigated. The singular integral equation method is used to reduce the scattering problem into the Cauchy singular integral equation of first kind by introduction of the crack dislocation density function. Then, the singular integral equation is solved numerically by Kurtz's piecewise continous function method. The crack opening displacement and dynamic stress intensity factor characterizing the scattered near-field are estimated for various incident angles, frequencies and relaxation times. The differences on crack opening displacement and stress intensity factor between elastic and viscoelastic interface crack are contrasted. And the effects of incident angle, incident frequency and relaxation time of the viscoelastic material are analyzed and explained by the features of phase lag and energy dissipation of the viscoelastic wave.     

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.     

A Griffith crack problem for a nonhomogeneous medium

A boundary value problem of two bonded nonhomogeneous similar planes containing a crack is considered. Poisson's ratio is supposed to be uniform whereas the shear modulus varies and is a function of the distance from the plane of the crack. Using Fourier transforms the original problem is reduced to a singular integral equation with a weakly singular kernel. The integral equation is then solved numerically and crack energy and stress intensity factors are calculated.     

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.     

Singular stresses in an infinite solid with a flat annular crack around a spherical cavity under tension

The problem of singular stresses in an infinite elastic solid containing a spherical cavity and a flat annular crack subjected to axial tension is considered. By application of an integral transform method and the theory of triple integral equations the problem is reduced to that of solving a singular integral equation of the first kind. The singular integral equation is solved numerically, and the influence of the spherical cavity upon the stress intensity factor and the influence of the annular crack upon the maximum stress at the surface of the spherical cavity are shown graphically in detail.     

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.     

Dynamic SIF of interface crack between two dissimilar viscoelastic bodies under impact loading*

In this paper, the transient dynamic stress intensity factor (SIF) is determined for an interface crack between two dissimilar half-infinite isotropic viscoelastic bodies under impact loading. An anti-plane step loading is assumed to act suddenly on the surface of interface crack of finite length. The stress field incurred near the crack tip is analyzed. The integral transformation method and singular integral equation approach are used to get the solution. By virtue of the integral transformation method, the viscoelastic mixed boundary problem is reduced to a set of dual integral equations of crack open displacement function in the transformation domain. The dual integral equations can be further transformed into the first kind of Cauchy-type singular integral equation (SIE) by introduction of crack dislocation density function. A piecewise continuous function approach is adopted to get the numerical solution of SIE. Finally, numerical inverse integral transformation is performed and the dynamic SIF in transformation domain is recovered to that in time domain. The dynamic SIF during a small time-interval is evaluated, and the effects of the viscoelastic material parameters on dynamic SIF are analyzed.     

Variation of stress intensity factor along the front of a 3D rectangular crack by using a singular integral equation method

In this paper a singular integral equation method is applied to calculate the distribution of stress intensity factor along the crack front of a 3D rectangular crack. The stress field induced by a body force doublet in an infinite body is used as the fundamental solution. Then, the problem is formulated as an integral equation with a singularity of the form of r ^–3. In solving the integral equation, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function, which expresses stress singularity along the crack front in an infinite body. The calculation shows that the present method gives smooth variations of stress intensity factors along the crack front for various aspect ratios. The present method gives rapidly converging numerical results and highly satisfied boundary conditions throughout the crack boundary.     

Various integral equations for a single crack problem of elastic half-plane

Two kinds of the complex potentials used for the crack problem of the elastic half-plane are suggested. First one is based on the distribution of dislocation along a curve, and second one is based on the distribution of crack opening displacement along a curve. Depending on the use of the complex potentials and the right hand term in the integral equation, two types of the singular integral equation for a single crack problem of elastic half-plane are derived. Regularization of the suggested singular integral equations gives three types of the Fredholm integral equation for the relevant problem. The weaker singular integral equation and the hypersingular integral equation are also introduced. Seven types of the integral equation are finally obtainable. The relation between the kernels of the various integral equations is also discussed.     

Stress intensity factors for a crack in planar disking

Disking is a relatively new manufacturing process for cutting/slicing brittle plates and rods. In the planar disking configuration, a pre-cracked plate is placed against an elastic plate and the two are squeezed together by fluid pressure. At a critical pressure the crack runs across the thickness of the brittle plate producing a clean cut. In this paper a fracture criterion is developed for the process using linear elastic fracture mechanics. The geometry of the process is modeled here as two perfectly bonded, infinite elastic layers with a crack perpendicular to the interface. The problem is formulated in terms of a singular integral equation with the derivative of the crack surface displacement (dislocation density) as the unknown function. Numerical quadrature is used to determine the stress intensity factors as a function of the parameters of the problem.     

Variation of stress intensity factor and crack opening displacement of semi-elliptical surface crack

In this paper a singular integral equation method is applied to calculate the stress intensity factor along crack front of a 3D surface crack. Stress field induced by body force doublet in a semi infinite body is used as a fundamental solution. Then the problem is formulated as an integral equation with a singularity of the form of r ^-3. In solving the integral equations, the unknown functions of body force densities are approximated by the product of a polynomial and a fundamental density function; that is, the exact density distribution to make an elliptical crack in an infinite body. The calculation shows that the present method gives the smooth variation of stress intensity factors along the crack front and crack opening displacement along the crack surface for various aspect ratios and Poisson's ratio. The present method gives rapidly converging numerical results and highly satisfactory boundary conditions throughout the crack boundary.     

A method for the numerical solution of singular integral equations with a principal value integral

A method for the numerical solution of singular integral equations with kernels having a singularity of the Cauchy type is presented. The singular behavior of the unknown function is explicitly built into the solution using the index theorem. The integral equation is replaced by integral relations at a discrete set of points. The integrand is then approximated by piecewise linear functions involving the value of the unknown function at a finite set of points. This permits integration in a closed form analytically. Thus the problem is reduced to a system of linear algebraic equations. The results obtained in this way are compared with the more sophisticated procedures based on Gauss-Chebyshev and Lobatto-Chebyshev quadrature formulae. An integral equation arising in a crack problem of the classical theory of elasticity is used for this purpose.     

Scattering of SH wave by a crack terminating at the interface of a bimaterial

A general method for solving the scattering of plane SH wave by a crack terminating at the interface of a bimaterial is presented. The crack can terminate at the interface in an arbitrary angle. In order to solve the proposed problem, the Greens function for a point harmonic force applied at an arbitrary point of the bimaterial is established by the Fourier transformation method. Using the obtained Greens function and the Betti-Rayleigh reciprocal theorem, the total scattered field of the crack is constructed. The total scattered field of the crack is divided into a regular part and a singular part. The hypersingular integral equation of the crack is obtained in terms of the regular and singular scattered field as well as the free wave field. The stress singularity order and singular stress at the terminating point are analyzed by the hypersingular integral equation and the singular scattered field of the crack. The dynamic stress intensity factor (DSIF) at the terminating point is defined in terms of the singular stresses at the terminating point. Numerical solution of the hypersingular integral equation gives the DSIFs at the crack tips. Comparison of our results with known results confirms the proposed method. Some numerical results and corresponding analysis are given in the paper.Constructive advice from the anonymous reviewers is acknowledged.     

Fracture problems of rubbers: J-integral estimation based upon η factors and an investigation on the strain energy density distribution as a local criterion

The plane elasticity problem studied is of a circular inclusion having a circular arc-crack along the interface and a crack of arbitrary shape in an infinite matrix of different material subjected to uniform stresses at infinity. The solution of the problem is given using Muskhelishvili's complex variable method with sectionally holomorphic functions. First, the solution to the (auxiliary) problem of a dislocation (or force) applied at a point in the matrix with the circular inclusion partially bonded is derived fully in its general form by solving the appropriate Rieman-Hilbert problem. It is subsequently used as the Green's function for the initial problem by introducing an unknown density function associated with a distribution of dislocations along the crack in the matrix. The initial problem is then reduced to a singular integral equation (SIE) over the crack in the matrix only. The SIE is solved numerically by appropriate quadratures and the stress intensity factors reported for the arc-cut and a straight crack in the matrix for a range of values of the geometrical parameters.     

The role of nucleation and growth of microcracks in brittle solids under compression: A numerical study

Summary Microcrack growth and nucleation in brittle microheterogeneous materials is studied by means of the two-dimensional sliding crack model. The problem is treated numerically using a boundary element method (BEM) to solve a singular integral equation for the dislocation density along the crack contour. The evolution of crack patterns is compared for uniaxial and biaxial compression.     

Interaction between a surface crack and a subsurface inclusion

A numerical method for the integration of the singular integral equation resulting from the interaction of a surface crack with a subsurface inclusion is presented. The crack is modelled as a pile-up of dislocations, and the dislocation density function is partitioned into three parts: A singular term due to the load discontinuity imposed by the inclusion, a square root singular term from the crack tip, and a bounded and continuous residual term. By integrating the singular terms explicitly the well behaved residual dislocation density function only has to be determined numerically, together with the intensity of the square root singular term. The method is applied to the determination of the stress intensity factor for a surface crack growing towards and through a circular inclusion whose diameter is equal to the distance from the free surface, and to the determination of the characteristic stress intensity factors when the crack enters the inclusion and leaves it for arbitrary ratios between the inclusion diameter and the distance from the surface.     

Thermoelastic Interaction between a Crack and a Circular Inhomogeneity

A problem of a circular elastic inclusion interacting with a crack under the thermal loading (heat flux at infinity) is revisited, and the system of singular integral equations with logarithmic kernels is obtained. This result corrects some errors found in the previous work by Chao and Lee (1996). Present solution is verified by the finite element method.     

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.