Mixed-type Integral Equations Research Articles

Integral equation method for the torsion of a composite cylinder with crack and inclusion

In this paper, the torsion problem of a composite cylinder with cracks and inclusions is investigated. Firstly, a linear inclusion model is proposed and the fundamental solution of this model is obtained. With the help of this solution together with Muskhelishvili's single-layer potential function and the fundamental solution of a single crack, the torsion problem can be reduced to solving a set of mixed-type integral equations, then the numerical method of these equations is proposed. At the end, several numerical examples are calculated and the results are satisfactory.

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.

Singular integral equation-boundary element method to solve the Saint-Venant bending problem of cracked cylinders

Using the single crack solutions and the regular solution of harmonic function, the bending problem of a cracked cylinder is reduced to solving two sets of mixed-type integral equations which can be solved by combining the numerical method of the singular integral equation with the boundary element method. Several numerical examples are calculated and the stress intensity factors are obtained.

Torsion of composite cylinder containing crack terminating at bimaterial interface

Considered in this paper is the Saint-Venant's torsion of a cylinder containing a dissimilar circular bar and a slanted crack. The authors gave a warping function that satisfies automatically the interface bonded condition, and reduced the torsional problem to a pair of mixed-type integral equations that could be evaluated numerically. The nature of stress singularity for a slanted crack terminating at the interface is studied, and it is shown that the power of singularity is not −0.5 at the interface tip of a crack. The expressions of singular stress around this tip are derived. For two typical composite cylinders, some numerical results giving mode III stress intensity factors, singular stresses, and torsional rigidities are presented.

Saint-Venant's torsion problem for a composite circular cylinder with aninternal edge crack

In this paper the writer uses Muskhelishvili single-layer potential function solution and single crack solution for the torsion problem of a circular cylinder to discuss the torsion problem of a composite cylinder with an internal crack, and the problem is reduced to a set of mixed-type integral equation with generalized Cauchy-kernel. Then, by using the integration formula of Gauss-Jacobi, the numerical method is established and several numerical examples are calculated. The torsional rigidity and the stress intensity factors are obtained. The results of these examples fit the results obtained by the previous papers better.

Stress intensity factors of the eccentric and edge cracked cylinders

Using the single crack solution and the regular solution of harmonic function, the torsion problem of a cracked cylinder is reduced to solving a set of mixed-type integral equations which can be solved by combining the numerical method of singular integral equation with the boundary element method. Several numerical examples are calculated and the stress intensity factors are obtained.

On the problem of crack system with an elliptic hole

The problem of cracks with an elliptic hole in an infinite plane is investigated. By introducing the fictitious loads on the hole edge and using the Muskhelishvili method, the problem is reduced to solving a system of mixed-type integral equations in which some are Fredholm equations but others Cauchy-type singular ones. A numerical method is suggested and can be used for the treatment of other similar equations. The numerical results for some typical examples are given, showing that the method is really effective.