Tadatoshi Matsuo

Singular Integral Equation Method for Interaction Between Elliptical Inclusions

This paper deals with numerical solutions of singular integral equations in interaction problems of elliptical inclusions under general loading conditions. The stress and displacement fields due to a point force in infinite plates are used as fundamental solutions. Then, the problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the unknowns are the body force densities distributed in infinite plates having the same elastic constants as those of the matrix and inclusions. To determine the unknown body force densities to satisfy the boundary conditions, four auxiliary unknown functions are derived from each body force density. It is found that determining these four auxiliary functions in the range 0 ≤ Φ k ≤ π/2 is equivalent to determining an original unknown density in the range 0 ≤ Φ k ≤ 2π. Then, these auxiliary unknowns are approximated by using fundamental densities and polynomials. Initially, the convergence of the results such as unknown densities and interface stresses are confirmed with increasing collocation points. Also, the accuracy is verified by examining the boundary conditions and relations between interface stresses and displacements. Randomly or regularly distributed elliptical inclusions can be treated by combining both solutions for remote tension and shear shown in this study.

Singular integral equation method in optimization of stress-relieving hole: a new approach based on the body force method

This paper is concerned with a method of decreasing stress concentration due to a notch and a hole by providing additional holes in the region of the principal notch or hole. A singular integral equation method that is useful for this optimization problem is discussed. To formulate the problem the idea of the body force method is applied using the Greens function for a point force as a fundamental solution. Then, the interaction problem between the principal notch and the additional holes is expressed as a system of singular integral equations with a Cauchy-type singular kernel, where densities of the body force distribution in the x- and y-directions are to be unknown functions. In solving the integral equations, eight kinds of fundamental density functions are applied; then, the continuously varying unknown functions of body force densities are approximated by a linear combination of products of the fundamental density functions and polynomials. In the searching process of the optimum conditions, the direction search of Hooke and Jeeves is employed. The calculation shows that the present method gives the stress distribution along the boundary of a hole very accurately with a short CPU time. The optimum position and the optimum size of the auxiliary hole are also determined efficiently with high accuracy.

Numerical solution of singular integral equations in stress concentration problems

Abstract In this paper, the numerical solution of singular integral equations in stress concentration problems is considered. The idea of the body force method stress field induced by a point force in an infinite body is used as a fundamental solution. Then, the problem is formulated as an integral equation with a singularity of the form of r −1 . In solving the integral equations, the boundary conditions satisfied by two types of numerical procedure are examined. Then, it is found that the unknown functions of body force densities should be approximated by the product of a polynomial and several types of fundamental density functions. The calculation shows that this latter method gives a smooth variation of stresses along the elliptical boundary for various geometrical and loading conditions. In addition, this method gives rapidly converging numerical results and highly satisfied boundary conditions along the entire boundary.

Stress analysis of arbitrarily distributed elliptical inclusions under longitudinal shear loading

This paper deals with an interaction problem of arbitrarily distributed elliptical inclusions under longitudinal shear loading. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are the densities of body forces distributed in the longitudinal directions of infinite bodies having the same elastic constants as those of the matrix and inclusions. In order to satisfy the boundary conditions along the elliptical inclusions, four kinds of fundamental density functions are introduced in a similar way of previous papers treating plane stress problems. Then the body force densities are approximated by a linear combination of those fundamental density functions and polynomials. In the analysis, elastic constants of matrix and inclusion are varied systematically; then the magnitude and position of the maximum stress are shown in tables and the stress distributions along the boundary are shown in figures. For any fixed shape, size and elastic constant of inclusions, the relationships between number of inclusions and maximum stress are investigated for several arrangements.

Asymmetric problem of a row of revolutional ellipsoidal cavities using singular integral equations

Abstract This paper deals with numerical solution of singular integral equations of the body force method in an interaction problem of revolutional ellipsoidal cavities under asymmetric uniaxial tension. The problem is solved on the superposition of two auxiliary loads; (i) biaxial tension and (ii) plane state of pure shear. These problems are formulated as a system of singular integral equations with Cauchy-type singularities, where the unknowns are densities of body forces distributed in the r, θ, z directions. In order to satisfy the boundary conditions along the ellipsoidal boundaries, eight kinds of fundamental density functions proposed in our previous papers are applied. In the analysis, the number, shape, and spacing of cavities are varied systematically; then the magnitude and position of the maximum stress are examined. For any fixed shape and size of cavities, the maximum stress is shown to be linear with the reciprocal of squared number of cavities. The present method is found to yield rapidly converging numerical results for various geometrical conditions of cavities.

Analysis of a row of elliptical inclusions in a plate using singular integral equations

This paper deals with the interaction problem of a row of elliptical inclusions under uniaxial tension. The body force method is used to formulate the problem as a system of singular integral equations with Cauchy--type and logarithmic singularities, where the unknowns are densities of body forces distributed in infinite plates that have the same elastic constants as those of the matrix and inclusion. In order to satisfy the boundary conditions along the elliptical boundaries, eight kinds of fundamental density functions, proposed in a previous paper, are applied. In the analysis, the number, shape, and position of inclusions are varied systematically; after which the magnitude and position of the maximum stress are examined. For any fixed shape and size of inclusions, the maximum stress is shown to be linear with the reciprocal of the number of inclusions. The present method is found to yield rapidly converging numerical results for various geometrical conditions of inclusions.

Numerical solutions of singular integral equations having Cauchy-type singular kernel by means of expansion method

This paper is concerned with numerical solutions of singular integral equations with Cauchy-type singular kernel. It is well-known that this type of singular integral equations appears in the analysis of crack problems using the continuously distributed dislocation method. In addition, it also appears in the analysis of notch problems using the body force method. In the present analysis, the unknown function of densities of dislocations and body forces are approximated by the product of the fundamental density functions and polynomials. The accuracy of stress intensity factors and stress concentration factors obtained by the present method is verified through the comparison with the exact solution and the reliable numerical solution obtained by other researchers. The present method is found to give good convergency of the numerical results for notch problem as well as internal and edge crack problems.

Analysis of Interaction between Arbitrarily Distributed Elliptical Inclusions under Longitudinal Shear Loading.

This paper deals with an interaction problem of arbitrarily distributed elliptical inclusions under longitudinal shear loading. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are the densities of body forces distributed in the longitudinal directions of infinite bodies having the same elastic constants as those of the matrix and the inclusions. In order to satisfy the boundary conditions along the inclusions, four kinds of new fundamental density functions are applied. Then the body force densities are approximated by a linear combination of the fundamental density functions and polynomials. The calculations are carried out for several arrangement of the inclusions, and it is found that the present method yields rapidly converging numerical results for arbitrarily distributed elliptical inclusions.

Tension of an Infinite Body Containing a Row of Ellipsoidal Inclusions

This paper deals with a row of equally spaced equal ellipsoidal inclusions in an infinite body subjected to tension. Based on the concepts of the body force method, the problems are formulated as a system of singular integral equations with cauchy-type or logarithmic-type singularities, where the densities of body forces distributed in the γ and z-directions of infinite bodies having the same elastic constants of the matrix and inclusions are unknown functions. In order to satisfy the boundary conditions along the inclusions, eight kinds of fundamental density functions proposed in our previous paper are used. In the analysis, the number, shape and distance of inclusions are varied systematically; then, the magnitude and position of the maximum stress are examined. For any fixed shape and size of inclusions, the maximum stress is shown to be linear with the reciprocal of the squared number of inclusions.