A natural approach to the introduction of finite-part integrals into crack problems of three-dimensional elasticity

The classical singular integral equation for the problem of a plane crack inside an infinite isotropic elastic medium and under an arbitrary normal pressure distribution was recently modified and written without the use of the Laplace operator Δ or the derivatives of the unknown function, but with the use of a finite-part integral. In this paper, a second complete derivation of the same equation is made (not based on previous forms of this equation) by using a limiting procedure, which makes it clear why the finite-part integral results in this equation. It is believed that the present results will be used in future for the introduction of finite-part integrals into a lot of crack problems in the theory of three-dimensional elasticity.     

Complex hypersingular integrals and integral equations in plane elasticity

Summary Complex hypersingular (finite-part) integrals and integral equations are considered in the functional class of N. Muskhelishvili. The appropriate definition is given. Three regularization (equivalence) formulae follow from this definition. They reduce hypersingular integrals to singular ones and allow to derive hypersingular analogues for Sokhotsky-Plemelj's formulae and for conditions that are necessary and sufficient for the function to be piecewise holomorphic. Two approaches to get and investigate complex hypersingular equations follow from these results: one of them is based on the equivalence formulae; as to the other, it is based on above-mentioned conditions. As an example, authors' equation for plane elasticity is studied. The existence of a unique solution is stated and some advantages over singular equations are outlined. To solve hypersingular equations the quadrature rules are presented. The accuracy of different quadrature formulae is compared, the examples being used. They confirm the need to take into account asymptotics and to carry out a thorough analytical investigation to get safe numerical results.     

Complex potentials and integral equations for curved crack and curved rigid line problems in plane elasticity

Summary In this paper, a simple layer potential and a double layer potential are suggested to solve the curved crack problem. The complex potentials in the simple layer case are formulated on the distributed dislocation along the curve. Meantime, the complex potentials in the double layer case are formulated on the crack displacement opening distribution. Behaviors of the complex potentials, for example the behaviors of increments of some physical quantities around a large circle, are analyzed in detail. Continuity and discontinuity of some physical quantities in the normal direction of the curve are analyzed, which are key points for formulating the integral equations of the problems. One weaker singular, two singular and one hypersingular integral equations are suggested to solve the problems. The relations between the kernels in different integral equations are addressed. Similarly, a simple layer potential and a double layer potential are suggested to solve the curved rigid line problem. The complex potentials in the simple layer case are formulated on the distributed forces along the curve. Meantime, the complex potentials in the double layer case are formulated on the resultant force function. One weaker singular, two singular and one hypersingular integral equations are suggested to solve the problems. When the resultant forces and moment are applied on the deformable line, the constraint equations are suggested. For more general cases, for example, in the case that the tractions applied on the two crack faces are not same in magnitude and opposite in direction, a singular integral equation is suggested. The equation is obtained by a superposition of two kinds of single layer potentials.     

The general type of finite-part singular integrals and integral equations with logarithmic singularities used in fracture mechanics

Summary A new method is proposed, by using some special quadrature rules, for the numerical evaluation of the general type of finite-part singular integrals and integral equations with logarithmic singularities. In this way the system of such equations can be numerically solved by reduction to a system of linear equations. For this reduction, the singular integral equation is applied to a number of appropriately selected collocation points on the integration interval, and then a numerical integration rule is used for the approximation of the integrals in this equation. An application is given, to the determination of the intensity of the logarithmic singularity in a simple crack inside an infinite, isotropic solid.With 1 Figure     

Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part I: Hypersingular integral equation and theoretical analysis

This paper deals with some basic linear elastic fracture problems for an arbitrary-shaped planar crack in a three-dimensional infinite transversely isotropic piezoelectric media. The finite-part integral concept is used to derive hypersingular integral equations for the crack from the point force and charge solutions with distinct eigenvalues s _i(i=1,2,3) of an infinite transversely isotropic piezoelectric media. Investigations on the singularities and the singular stress fields and electric displacement fields in the vicinity of the crack are made by the dominant-part analysis of the two-dimensional integrals. Thereafter the stress and electric displacement intensity factor K-fields and the energy release rate G are exactly obtained by using the definitions of stress and electric displacement intensity factors and the principle of virtual work, respectively. The hypersingular integral equations under axially symmetric mechanical and electric loadings are solved analytically for the case of a penny-shaped crack.     

Two-dimensional principal value hypersingular integrals for crack problems in three-dimensional elasticity

Summary A principal value definition of the basic hypersingular integral in the fundamental integral equation for two-dimensional cracks in three-dimensional isotropic elasticity is proposed. As is the case with the corresponding definitions of Cauchy-type one-dimensional and two-dimensional principal value singular integrals, as well as Mangler-type one-dimensional principal value hypersingular integrals, the present definition is based on the special consideration of an appropriate region around the singular point. The cases of circular, square and equilateral triangular regions are considered in some detail.     

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.     

The mixed-type integral equations for transient elastodynamic plane crack problems

In the present paper, by use of the boundary integral equation method and the techniques of Green fundamental solution and singularity analysis, the dynamic infinite plane crack problem is investigated. For the first time, the problem is reduced to solving a system of mixed-typed integral equations in Laplace transform domain. The equations consist of ordinary boundary integral equations along the outer boundary and Cauchy singular integral equations along the crack line. The equations obtained are strictly proved to be equivalent with the dual integral equations obtained by Sih in the special case of dynamic Griffith crack problem. The mixed-type integral equations can be solved by combining the numerical method of singular integral equation with the ordinary boundary element method. Further use the numerical method for Laplace transform, several typical examples are calculated and their dynamic stress intensity factors are obtained. The results show that the method proposed is successful and can be used to solve more complicated problems.     

Application of finite-part integrals to three-dimensional fracture problems for piezoelectric media Part II: Numerical analysis

In this paper, elliptical cracks and rectangular cracks embedded in a three-dimensional infinite transversely isotropic piezoelectric solid are analyzed under combined mechanical tension and electric fields. The hypersingular integral equation method is used to solve the mentioned problems. The unknown function in the hypersingular integral equations is approximated with a product of the fundamental density function and polynomials. The hypersingular integrals can be numerically evaluated by using a method of Taylor series expansion. Therefore, the hypersingular integral equations for the crack problems can be solved immediately. Finally, numerical examples of the stress and electric displacement intensity factors as well as the energy release rates for these crack configurations are presented. The numerical results demonstrate the present approach to be very efficient.     

A hybrid-element approach to crack problems in plane elasticity

The hybrid-element concept and the complex variable technique have been adopted for constructing a special super-element to be used jointly with conventional finite elements for the analysis of elastic stress intensity factors for plane cracks. The use of the complex variable technique permits the proper consideration of the stress intensity at the crack tip, and it also leads to very efficient programming. The use of such a super-element in the finite element solution has been shown to be highly accurate when only a very coarse element mesh is used near the crack.     

Solution of plane anisotropic elastostatical boundary value problems by singular integral equations

Summary Basis for the presented method is the knowledge of certain fundamental solutions for theinfinite anisotropic medium. By superimposing these singular solutions in a suitable fashion, the given boundary value problem can be formulated as a tensorial integral equation (singularity method).We start from two basic singularities in the anisotropicplane, viz. the single force and the edge dislocation, and then consider the corresponding integral equations and a method for approximating their solution.With 2 Figures