Elastic Crack Problems Research Articles

New singular integral equation for curve crack problem in plane elasticity

The three possibilities for formulating the integral equation are listed in Table 1. The first type of the integral equation, type A, was suggested in [1,2], and it belongs to a singular integral equation. The second type, type B, was mainly developed in [3], and it belongs to a weakly singular integral equation. The third type, type C, will be the topic of this report. To facilitate the following discussion, we introduce the following integrals

Image method for curved crack problem in antiplane elasticity

Image method for curved crack problem in antiplane elasticity

A Numerical Integration Formula for the Solution of the Singular Integral Equation for Classical Crack Problems in Plane and Antiplane Elasticity

A numerical integration formula for the solution of the singular integral equation for classical crack problems in plane and antiplane elasticity is developed. The method is based on a modification of the Gauss-Chebyshev quadrature and the derivation of a finite part integral having an algebraic singularity of (3/2) at the limits of integration. The procedure is applied to determine the finite part integrals which have analytical solutions and the results are compared. Finally, the integration formula is applied to an actual crack problem and the stress intensity factors are computed and presented.

Numerical estimation of the coefficient of the homogenous Riemann-Hilbert problem on the basis of boundary data

The coefficient of the classical homogeneous Riemann-Hilbert boundary value problem in complex analysis is evaluated by using an appropriate complex integral on a closed contour surrounding the arc where the problem is defined. Numerical interaction rules for complex contour integrals are used. Only the numerical values of the sectionally analytic function which is the solution of the problem are required (at the nodes of the quadrature rule). Numerical results in a few applications are displayed, and the rapid convergence of the approach is clearly observed. The present results are applicable to a large class of physical and engineering problems, e.g., to the interface crack problem in classical plane elasticity.

強磁場下の構造材料の力学的挙動と電磁破壊力学

Design and development of superconducting structures require basic research on electromagnetic fracture mechanics. In the theory of brittle fracture in a strong magnetic field, usually we consider linear electromagnetic elastic solutions of crack problems. Examples including analysis of electromagnetic elastic crack problems are reviewed to show the effect of magnetic field on the mechanical behavior of cracked materials. Many electrically conducting materials are used in magnetic and electric current fields. First, the electromagnetic elastic problems of conducting cracked materials are analized to show the effect of electromagnetic force. The current flow is disturbed by the presence of the crack and the electromagnetic body force is caused by the interaction between the magnetic field and the disturbed current. If the electrically conducting material is used in the strong magnetic field, we must consider the effect of induced current. Secondly, scattering of time harmonic waves by a crack in an electrically conducting material under a magnetic field is analized to show this effect. If a ferromagnetic material is used in a magnetic field, we must consider the effect of induced magnetization. Lastly, the fracture and deformation of a soft ferromagnetic material in a magnetic field is investigated to show this effect.

New integral equation for curve crack problem in plane elasticity with arbitrary loading condition

New integral equation for curve crack problem in plane elasticity with arbitrary loading condition

A mixed hybrid finite element for three-dimensional elastic crack analysis

A new three-dimensional crack tip element is proposed, which is based on a mixed hybrid stress/displacement model. A truncated series expansion of eigenfunctions for the straight semi-infinite crack is deduced and assumed for the internal stress and displacement fields in the element. The basic approach of constructing these hybrid elements is outlined. Their good capability, efficiency and accuracy for analyzing three-dimensional elastic crack problems are demonstrated by first numerical examples.

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

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.

Three‐dimensional crack analysis using singular boundary elements

AbstractA multi‐domain method of solving three‐dimensional elastic crack problems in an infinite elastic body using the boundary element method is proposed. The displacement and traction behaviours near a crack front are incorporated in special crack elements. The elimination of singularities arising from the term combined with Kelvin's kernel for displacement in the integrals is discussed in detail. Stress intensity factors of modes I, II and III are obtained directly from crack‐front nodal values, without any extrapolation as in some other methods. No differentiation of conventional boundary integral equations (with Kelvin's tensor kernels) is necessary in the current approach. This method is applicable to cracks of arbitrary shape. Infinite bodies are modelled precisely as such, not approximated as large finite bodies. Numerical solutions of stress intensity factors are given for several problems involving a penny‐shaped crack.

Crack problem in plane elasticity under antisymmetric loading

Crack problem in plane elasticity under antisymmetric loading

Determination of stress distributions around notches by boundary element method

Abstract Based on the complex potentials formulation and conformal mapping approach, we derive the fundamental solution for a point force acting in an infinite plane containing an arbitrary shaped notch. The solution satisfies the traction‐free boundary condition along the notch. Therefore, in the boundary element computation of the stress distribution for a notch problem the notch boundary can be excluded from the path of boundary integration, and the high stress gradient around the notch is not disturbed. Several examples are given to show the efficiency and accuracy resulting from the implementation of the present fundamental solution in the boundary element computation of the stress concentrations around notches and the stress intensity factors for elastic crack problems.

On the completeness of the Westergaard stress functions

The Westergaard method yields a succinct solution to linear elastic crack problems. Here it is shown that the superposition of the Westergaard functions with a uniform uniaxial stress field provides a complete solution to the equations of two-dimensional linear elasticity for the infinite plane.

Crack problems in plane and antiplane elasticity using a singular integral equation

A numerical integration formula for the investigation of the singular integral of loakimidis for classical crack problems in plane and antiplane elasticity is developed. The method is based on a modification of the Gauss-Chebyshev quadrature and the definition of finite part integral having an algebraic sigularity of (−3/2) at the limits of integration. Once developed the procedure is applied to the determination of finite part integrals which have analytical solutions and the results are compared. Finally the integration for mula is applied to an actual crack problem and the stress intonsity factors are computed and presented.

Weight functions for crack problems using boundary element analysis

The use of weight functions is a well-established technique for the solution of mechanics problems in structural engineering. Increasing use of these techniques is being made to solve crack problems. The special feature of crack problems in elasticity is the presence of a singular stress field at the crack-tip. The incorporation of certain fundamental singular fields into the boundary element formulation enables weight functions for the stress intensity factors to be obtained on all the boundaries of a cracked body from a single boundary element calculation. With these weight functions the stress intensity factor for any loading on any boundary of the body can be obtained from a simple integration.

Analytical Approach to Some Non‐Classical Elastic Contact and Crack Problems

L'auteur etend ces travaux anterieurs (Fabrikant, 1986a, 1986b, 1987), concernant le contact elastique et la fissuration, au cas des charges de cisaillement de domaines tres generaux

Mangler-type principal value integrals in hypersingular integral equations for crack problems in plane elasticity

The Mangler-type principal value integrals play an important role in the formulation and solution of the related hypersingular integral equations for crack problems in plane elasticity. Here we consider the classical, fundamental definition of this class of integrals, the most important of their properties and the existing methods for the solution of Mangier-type hypersingular integral equations. Several crack problems in plane elasticity have been already solved by this approach.

On the location of straight discontinuity intervals of arbitrary sectionally analytic functions by using complex path-independent integrals

The classical method of locating zeros and poles of analytic and meromorphic functions in the complex plane via the evaluation of complex path-independent integrals on closed contours surrounding the sought zeros and poles is generalized to become applicable (for the first time) to the location of straight discontinuity intervals of completely general sectionally analytic functions possessing such an interval. Such is the case e.g. in crack problems in plane elasticity or in airfoils in plane fluid dynamics. This generalization makes use of appropriate or arbitrary quadrature rules and, in this way, it leads to a completely numerical method. The method is illustrated in the case of a simple sectionally analytic function, where the classical Chebyshev and Legendre polynomials are used and numerical results are presented. Possible generalizations of the present results are also reported in brief and are strongly expected in future.

An improved boundary element Galerkin method for three-dimensional crack problems

In this paper we analyze the solution of crack problems in three-dimensional linear elasticity by equivalent integral equations of the first kind on the crack surface. Besides existence and uniqueness we give sharp regularity results for the solution of these pseudodifferential equations. Two versions of Eskin's Wiener-Hopf technique are presented: the first one requires the factorization of matrix-valued symbols which is avoided in the second case. Based on these regularity results we show how to improve the boundary element Galerkin method for our integral equations by using special singular trial functions. We apply the approximation property and inverse assumption of these elements together with duality arguments and derive quasi-optimal asymptotic error estimates in a scale of Sobolev spaces.

New integral equation for plane elasticity crack problems

In the usual formulation of singular equation approach for crack problems in plane elasticity [1,2], if one changes the right-hand term of the integral equation from tractions to resultant forces, a new integral equation can be obtained and is presented in this paper. The newly obtained integral equation has a log singular kernel. Interpolation equation for the dislocation functions (the undetermined functions in the integral equations) is proposed. Numerical examination is used to demonstrate the efficiency of the present technique, and a number of numerical examples are given.

An application of potential theory for solving the plane elastic crack problem

An elastic field of displacements in the vicinity of a plane rectilinear crack inside an infinite plate is determined using the potential method. It is shown that the solution of this problem obtained by Muskhelishvili is not the unique one. The paper gives the general solution of this problem which depends upon a finite number of constants. Physically, the constants will take those values which provide for a minimum of the elastic energy of a plate with a crack of a given length. [Russian Text Ignored]