Hypersingular Integral Equation Method Research Articles

Scattering of seismic waves by cracks in multi-layered geological regions: II. Numerical results

The mechanical model for plane strain, time-harmonic seismic wave propagation problems in cracked, multi-layered geological regions with surface topography and non-parallel interfaces was described in the first part of this work. Here, this model is used to investigate the response of such a region to the presence of traveling elastic waves generated by a seismic source. The computational methodology that was developed in the first part is based on a combination of both the regular (displacement-based) and the hypersingular (traction-based) Boundary Integral Equation Method (BIEM). First, the accuracy and convergence characteristics of this hybrid BIEM are studied. Then, a series of problems involving four different configurations of a reference geological deposit with both interface and internal cracks are solved, for a loading that is due to a seismically-induced pressure wave propagating upwards from the underlying rigid half-plane. The purpose of the numerical study is to investigate the influence of various key parameters of the problem, such as frequency and incidence angle of the incoming wave, size of the surface relief, location and size of the buried cracks, interaction effects between cracks and finally the presence of layers, on both the scattered displacement field and the stress concentration field.

Application of Hypersingular Integral Equation Method to a Three-Dimensional Crack Terminating at an Interface.

In this paper a three dimensional crack perpendicular to and terminating at a bimaterial interface is considered under general mixed-mode loading. The solution utilizes the body force method and requires Green's functions for perfectly bonded elastic half planes. The formulation leads to a system of hypersingular integral equations whose unknowns are three modes of crack opening displacements. The main-part analytical method is used to investigate the characteristics of stress singularity around the crack front touching the interface. An explicit closed-form expression is given for the singular stress field, which is controlled by general stress intensity factors. The relations between three modes of crack opening displacements and stress intensity factors are also discussed.

218 Application of Hypersingular Integral Equation Method to a Three-dimensional Crack Meeting an Interface

218 Application of Hypersingular Integral Equation Method to a Three-dimensional Crack Meeting an Interface

Hypersingular formulation for boundary stress evaluation

Instead of using shape function derivatives and Hooke's law, the full stress tensor is evaluated at boundary points by direct application of boundary integral identities for the displacement derivatives. It is first shown that integral equations with singular or hypersingular kernels do not give rise to unbounded terms, even when the source point is on the boundary. A general method for performing the integration is also described. Numerical results are quite interesting, since the stress components evaluated through the hypersingular integral equation method show very good accuracy even on coarse meshes.

Analysis of Stress Intensity Factors for Curved Cracks.

In this paper, the hypersingular integral equation method (HIEM) is applied to calculate the stress intensity factors of curved crack problems. The stress fields induced by two kinds of displacement discontinuity are used as fundamental solutions. Then, the problem is formulated as a system of integral equations with the singularity of the form of r-2. In the numerical calculation, two kinds of unknown functions are approximated by the products of the fundamental density functions and polynomials. The calculation shows that the present method gives rapidly converging numerical results for curved cracks under various geometrical conditions. In addition, a method of evaluation of the stress intensity factors for arbitrary shaped curved cracks is proposed using the approximate replacement to a simple straight crack.

Three‐dimensional fracture simulation with a single‐domain, direct boundary element formulation

AbstractAn efficient hypersingular boundary integral equation method for three‐dimensional fracture mechanics was presented in a previous paper. The details of the numerical implementation of this method are further discussed herein. In particular, an algorithm for achieving the required differentiability of the crack surface displacement function is discussed. To illustrate the utility of the method, computational results for several strongly interacting multiple‐crack geometries are presented. The calculated stress intensity factors are in excellent agreement with those obtained by an approximate analytical method due to Kachanov and Laures.

The Hypersingular Integral Equation Method in the Analysis of Bent and Branched Cracks.

In the previous papers, the hypersingular integral equation method (HIEM) was shown to be useful for exactly analyzing straight crack problems in the two-dimensional plane. The present paper concerns the application of the HIEM for the solution of the problems with intricate crack shapes. As examples, the stress intensity factors of bent and branched cracks under uniform tension are treated. Even in the case of that the bent part of the crack is extremely short, the accurate numerical results are obtained by selecting a convenient set of collocation points. The calculation shows that the HIEM, in which unknown functions are approximated by using Chebyshev polynomials, gives results of better accuracy compared with the previous method using such as stepped functions.

Efficient Inverse Analysis to Search for the Edge Crack in the Semi-Infinite Plate.

In this study, the location of the edge crack in the semi-infinite plate and the loading stress at the infinity are detected by the data of strains measured around the region of the crack. In searching for the location of the edge crack, the method of gradient search is employed to minimize the square sum of the residuals between the measured strain distributions and the computed ones for the assumed crack locations. The strain field induced by the assumed edge crack is exactly analyzed by using the hypersingular integral equation method (HIEM). All integrals used in the present analysis are expressed in closed forms and the results can be given with short CPU time. The starting point for the gradient search can be estimated efficiently on the basis of the concept of the body force method where the crack can be represented by a pair of point forces. The results show the location of the crack are determined efficiently with good accuracies.

Teh hypersingular lntegral equation method in the analysis of mixed mode crack problems. (The extended body force method using chebychev polynomials.)

In this paper, the hypersingular integral equation method (HIEM) is applied to calculate the stress intensity factors of the mixed mode crack problems. The stress fields induced by "two kinds of displacement discontinuity" are used as fundamental solutions. Then, the problem is formulated as a system of integral equations with the singularity of the form of (ζ-χ)-2. In the numerical calculation, two kinds of unknown-functions are approximated by the products of the weight functions and Chebychev polynomials. As examples, the stress intensity factors of the oblique edge crack in the semi-infinite plate are treated. The calculation shows that the HIEM gives rapidly converging numerical results even for extremely oblique edge crack.

The singular integral equation method in the analysis of a crack perpendicular to the interface. The hypersingular integral equation method and the body force method.

In this paper, the plane problem of two bonded elastic half planes containing a crack perpendicular to and going through the interface is considered. The problem is formulated as a system of singular integral equations by using the stress field induced by "the displacement discontinuity". In solving the equations, the hypersingular integral equation method (HIEM) and the body force method (BFM) are compared. The calculation shows that the HIEM, in which the unknown function is approximated by the product of the weight function and chebychev polynominals, gives rapidly converging numerical results in this analysis.