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.     

Revisit of degenerate scales in the BIEM/BEM for 2D elasticity problems

The boundary integral equation method in conjunction with the degenerate kernel, the direct searching technique (singular value decomposition), and the only two-trials technique (2 × 2 matrix eigenvalue problem) are analytically and numerically used to find the degenerate scales, respectively. In the continuous system of boundary integral equation, the degenerate kernel for the 2D Kelvin solution in the polar coordinates is reviewed and the degenerate kernel in the elliptical coordinates is derived. Using the degenerate kernel, an analytical solution of the degenerate scales for the elasticity problem of circular and elliptical cases is obtained and compared with the numerical result. Further, the triangular case and square case were also numerically demonstrated.     

Edge crack problem in a semi-infinite FGM plate with a bi-directional coefficient of thermal expansion under two-dimensional thermal loading

Summary The temperature distribution in structural elements in practical cases usually changes in two or three directions. Based on such facts, aiming at more effectiveness, a functionally graded material (FGM), whose properties change in two or three directions, is introduced, that investigated here called bi-directional FGM. The current study aims at the formulation, solution and investigation of a semiinfinite edge cracked FGM plate problem with a bi-directional coefficient of thermal expansion under two-dimensional thermal loading. The solution of the boundary value problem that one obtains from the mathematical formulation of the current crack problem under thermal loading reduces to an integral equation with a generalized Cauchy kernel. This integral equation contains many two-dimensional double strongly singular integrals, which can be solved numerically. In order to separate the singular terms and overcome the divergence of the integrals an asymptotic analysis for the singular parts in the obtained integral equation was carried out. Also, the exact solution for many singular integrals is obtained. The obtained numerical results are used in the representation of the thermal stress intensity factor versus the thermal/mechanical nonhomogeneous parameters. The numerical results show that it is possible to reduce and control the thermal stress intensity factor.     

Analysis of the fiber-matrix cylindrical model with a circumferential crack

An axisymmetrical fiber-matrix cylindrical model with a circumferential crack in the matrix of finite diameter is formulated within elastostatic scope. The problem is considered by means of integral transforms and a singular integral equation with a dominant generalized Cauchy kernel is obtained. Following the numerical solution technique developed by Erdogan, Gupta and Cook, the singular integral equation is reduced to a system of linear equations. By solving the linear equations, stress intensity factors associated with the crack length and the material properties are calculated and discussed. The solutions presented in this study are found to be general, including the solutions as special cases of the present formulation for a homogeneous solid cylindrical bar and a thick-walled shell with an outer circumferential crack. This revised version was published online in August 2006 with corrections to the Cover Date.     

Edge cracks in a transversely isotropic hollow cylinder

The analytical solution for the linear elastic, axisymmetric problem of inner and outer edge cracks in a transversely isotropic infinitely long hollow cylinder is considered. The z = 0 plane on which the crack lies is a plane of symmetry. The loading is uniform crack surface pressure. The mixed boundary value problem is reduced to a singular integral equation where the unknown is the derivative of the crack surface displacement. An asymptotic analysis is done to derive the generalized Cauchy kernel associated with edge cracks. It is shown that the stress intensity factor is a function of three material parameters. The singular integral equation is solved numerically. Stress intensity factors are presented for various values of material and geometric parameters.     

A new singular integral equation for the classical crack problem in plane and antiplane elasticity

A new singular integral equation (with a kernel with a logarithmic singularity) is proposed for the crack problem inside an elastic medium under plane or antiplane conditions. In this equation the integral is considered in the sense of a finite-part integral of Hadamard because the unknown function presents singularities of order ?3/2 at the crack tips. The Galerkin and the collocation methods are proposed for the numerical solution of this equation and the determination of the values of the stress intensity factors at the crack tips and numerical results are presented. Finally, the advantages of this equation are also considered.     

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.     

A new boundary integral equation for notch problem of antiplane elasticity

In this paper the notch problem of antiplane elasticity is discussed and a new boundary integral equation is formulated. In the problem, the distributed dislocation density is taken to be the unknown function. Unlike the usual choice, the resultant force function is taken as the right hand term of the integral equation; therefore, a new boundary integral equation for the notch problem of antiplane elasticity with a weaker singular kernel (logarithmic) is obtained. After introducing a particular fundamental solution of antiplane elasticity, the notch problem for the half-plane is discussed and the relevant boundary integral equation is formulated. The integral equations derived are compact in form and convenient for computation. Numerical examples demonstrated that high accuracy can be achieved by using the new boundary equation.     

Singularity extraction technique for integral equation methods with higher order basis functions on plane triangles and tetrahedra

A numerical solution of integral equations typically requires calculation of integrals with singular kernels. The integration of singular terms can be considered either by purely numerical techniques, e.g. Duffy's method, polar co‐ordinate transformation, or by singularity extraction. In the latter method the extracted singular integral is calculated in closed form and the remaining integral is calculated numerically. This method has been well established for linear and constant shape functions. In this paper we extend the method for polynomial shape functions of arbitrary order. We present recursive formulas by which we can extract any number of terms from the singular kernel defined by the fundamental solution of the Helmholtz equation, or its gradient, and integrate the extracted terms times a polynomial shape function in closed form over plane triangles or tetrahedra. The presented formulas generalize the singularity extraction technique for surface and volume integral equation methods with high‐order basis functions. Numerical experiments show that the developed method leads to a more accurate and robust integration scheme, and in many cases also a faster method than, for example, Duffy's transformation. Copyright © 2003 John Wiley & Sons, Ltd.     

The evolution of radial fingers at the interface between two viscous liquids

The problem of determining the evolution of a radial finger in a Hele-Shaw cell filled with two different viscous liquids is formulated in terms of a Fredholm integral equation of the second kind. This integral equation possesses a unique continuous solution for any viscosity ratio β greater than zero and less than or equal to infinity, as long as the curvature is well defined and continuous at each point of the interface curve. The case of β equal to infinity corresponds to the flow configuration of only one liquid (liquid-gas). The requirement of continuous curvature implies that the kernel of the integral equation is nowhere singular.     

Axisymmetric elastodynamic response of a flat annular crack to normal impact waves

The axisymmetric response of a flat annular crack in an infinite medium subjected to normal impact load is investigated in this study. A step stress is applied to the crack surface. The singular solution is equivalent to solutions of the problem of diffraction of normally incident tension wave by a flat annular crack, and the problem of the sudden appearance of a flat annular crack in a uniform tensile stress field. Laplace and Hankel transforms are used to reduce the problem to the solution of a set of triple integral equations in the Laplace transform domain. These equations are solved by using a integral transform technique and the result is expressed in terms of a singular integral equation of the first kind with the kernel which is improved by means of a contour integration on the Riemann surface. A numerical Laplace inversion routine is used to recover the time dependence of the solution. Numerical results of the dynamic stress intensity factor are obtained to show the influence of inertia, the ratio of the inner radius to the outer one and Poisson's ratio on the load transmission to the crack tip.     

A plane contact problem for an elastic orthotropic strip

The contact of a punch with an elastic orthotropic strip is considered. A singular integral equation is derived for the contact pressure. The analytic expression of the associated kernel is unique for all types of orthotropy. An iterative solution method is developed to investigate a thick strip. A direct asymptotic procedure proposed for a thin strip leads to simple explicit formulae. Numerical examples are presented for various values of the relative strip thickness.     

Mode I stress intensity factors at corner points in plane elastic media

A well-known method for the solution of plane elasticity problems consists in reducing them to singular integral equations with Cauchy-type kernels. In this paper a numerical technique is presented for the efficient solution of such equations near corner points and the determination of the corresponding stress intensity factors under symmetry conditions so that only Mode I stress intensity factors exist. The peculiarity of the method consists in taking into account all poles of the kernel of a singular integral equation and at the same time using collocation points outside the integration interval. Two applications of the method to V-notched and cracked media are also made.     

A thin cracked layer bonded to an elastic half-space under an antiplane concentrated load

The antiplane elasticity problem for a thin cracked layer bonded to an elastic half-space under an antiplane concentrated load is considered. The fundamental solution is obtained as a rapidly convergent series in terms of the complex potentials via iterations of Möbius transformation. The singular integral equation with a logarithmic singular kernel is derived to model a crack problem that can be solved numerically in a straightforward manner. The dimensionless mode-III stress intensity factors obtained for various crack inclinations and crack lengths are discussed in detail and provided in graphic form. A strip problem with an arbitrarily oriented crack is also considered.     

Fracture of solids containing arrays of cracks

This survey is largely a collection of the author's recent results on the fracture characteristics of elastic solids containing inhomogeneities in the form of slitlike cracks. The crack tip opening displacement is used as a measure of the fracture behaviour of the solid under three different deformation situations: (a) Opening Mode I; (b) Sliding Mode II; and (c) Tearing Mode III. Various crack geometries have been considered for which closed-form analytical solutions do not seem feasible. To this category belong (a) A stack of cracks (cracks with constant distance of vertical separation) and (b) A doubly-periodic array of cracks forming a rectangular or diamond-shaped pattern. Besides summarizing various results which have been published elsewhere in Scientific Literature, some new results, and necessary amplification of the previous results, are included herein to make the survey as self-contained as possible. It is hoped that it will be found useful by many research workers engaged in crack interaction and propagation problems.The survey is divided into five sections. Section 1 gives a general introduction to the problem at hand and brings out the importance of studying the subject. Sufficient reference is made to the available literature on the subject, without being unduly overemphatic. It is likely that many otherwise good papers have been omitted either through oversight or because they were thought to be peripheral to the subject matter of the survey. Interested readers should, however, be able to find sufficient cross references in the literature cited herein. This section also touches upon the necessity of using the dislocation formalism in solving the problems at hand. For obvious reasons, no attempt has been made to dwell upon this equivalence of slitlike cracks and straight dislocations, the interested reader being again referred to relevant literature.Section 2 formulates the problem in mathematical terms as one consisting in the solution of a singular integral equation. The latter results from the traction-free conditions on the crack faces. The equation is suitably non-dimensionalized and its kernel decomposed into a singular and a non-singular part as dictated by the method of solution.Section 3 presents a perturbation solution for widely spaced cracks. The solution is restricted to a stack of cracks under plane strain conditions.Section 4 deals with an approximate method used to solve the singular integral equation. It is based on an expansion of the non-singular part of the kernel in a series of orthogonal polynomials. The solution of the singular integral equation allows us to calculate the crack tip opening displacement as a function of the externally applied stress and the crack geometry.The results for various loading modes and crack configurations are presented graphically in Section 5 and discussed from the point of fracture initiation from multiple cracks. Where possible, the results are compared with that for an isolated relaxed crack for a better understanding of the change brought about by an array of interacting cracks.In order not to interrupt the text all complicated mathematical expressions have been grouped together and listed at the end of the relevant section.     

Convolution quadrature method‐based symmetric Galerkin boundary element method for 3‐d elastodynamics

The boundary integral equations in 3‐d elastodynamics contain convolution integrals with respect to the time. They can be performed analytically or with the convolution quadrature method. The latter time‐stepping procedure's benefit is the usage of the Laplace‐transformed fundamental solution. Therefore, it is possible to apply this method also to problems where analytical time‐dependent fundamental solutions might not be known. To obtain a symmetric formulation, the second boundary integral equation has to be used which, unfortunately, requires special care in the numerical implementation since it involves hypersingular kernel functions. Therefore, a regularization for closed surfaces of the Laplace‐transformed elastodynamic kernel functions is presented which transforms the bilinear form of the hypersingular integral operator to a weakly singular one. Supplementarily, a weakly singular formulation of the Laplace‐transformed elastodynamic double layer potential is presented. This results in a time domain boundary element formulation involving at least only weakly singular integral kernels. Finally, numerical studies validate this approach with respect to different spatial and time discretizations. Further, a comparison with the wider used collocation method is presented. It is shown numerically that the presented formulation exhibits a good convergence rate and a more stable behavior compared with collocation methods. Copyright © 2008 John Wiley & Sons, Ltd.     

多铁性非均匀空心层合柱圆弧界面的反平面断裂分析

讨论反平面载荷作用下多铁性非均匀空心层合柱的圆弧界面裂纹问题,层合柱由梯度铁电层和梯度铁磁层粘接而成,界面处存在圆弧型裂纹。采用分离变量和Cauchy核奇异积分方程方法求解该断裂问题。通过讨论断裂参数的数值解,分析了梯度非均匀参数、几何与材料参数变动等对应力强度因子的影响。     

A boundary integral approach to analyze the viscous scattering of a pressure wave by a rigid body

The paper provides boundary integral equations for solving the problem of viscous scattering of a pressure wave by a rigid body. By using this mathematical tool uniqueness and existence theorems are proved. Since the boundary conditions are written in terms of velocities, vector boundary integral equations are obtained for solving the problem. The paper introduces single-layer viscous potentials and also a stress tensor. Correspondingly, a viscous double-layer potential is defined. The properties of all these potentials are investigated.By representing the scattered field as a combination of a single-layer viscous potential and a double-layer viscous potential the problem is reduced to the solution of a singular vectorial integral equation of Fredholm type of the second kind.In the case where the stress vector on the boundary is the main quantity of interest the corresponding boundary singular integral equation is proved to have a unique solution.     

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.     

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.