Kinds Of Fundamental Density Functions Research Articles

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

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.

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.

２次元と３次元の介在物の応力集中の干渉効果の比較

This paper deals with stress analysis of elliptical and ellipsoidal inclusions using singular integral equations of the body force method. The stress and displacement fields due to a point force in an infinite plate and a ring force in an infinite body are used as fundamental solutions. On the idea of the body force method, the problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are densities of body forces distributed in the x- and y-directions of infinite plates or in the r- and z-directions of infinite bodies having the same elastic constants of the matrix and inclusions. In order to satisfy the boundary conditions along the inclusions, eight kinds of fundamental density functions proposed in our previous paper are used. Then the body force densities are approximated by a linear combination of the fundamental density functions and polynomials. The present method is found to give rapidly converging numerical results for both of the axial symmetry tension and uniaxial tension. The calculations are carried out systematically for various shape, distance and elastic constant of inclusions and the stress distributions along the boundaries are shown in figures. Then the interaction effects are discussed through the comparison between the elliptical inclusions and ellipsoidal inclusions.

Numerical solution of singular integral equations in stress concentration problems under longitudinal shear loading

The paper deals with numerical solutions of singular integral equations in stress concentration problems for longitudinal shear loading. The body force method is used to formulate the problem as a system of singular integral equations with Cauchy-type singularities, where unknown functions are densities of body forces distributed in the longitudinal direction of an infinite body. First, four kinds of fundamental density functions are introduced to satisfy completely the boundary conditions for an elliptical boundary in the range 0≤φk≤2π. To explain the idea of the fundamental densities, four kinds of equivalent auxiliary body force densities are defined in the range 0≤φk≤π/2, and necessary conditions that the densities must satisfy are described. Then, four kinds of fundamental density functions are explained as sample functions to satisfy the necessary conditions. Next, the unknown functions of the body force densities are approximated by a linear combination of the fundamental density functions and weight functions, which are unknown. Calculations are carried out for several arrangements of elliptical holes. It is found that the present method yields rapidly converging numerical results. The body force densities and stress distributions along the boundaries are shown in figures to demonstrate the accuracy of the present solutions.

Numerical Solution of Singular Integral Equations of the Body Force Method in Stress Concentration Problems for Longitudinal Shear Loading.

This paper deals with numerical solutions of singular integral equations in stress concentration problems for longitudinal shear loading. The body force method is used to formulate the problems as a system of singular integral equations with Cauchy-type singularities, where unknowns are the densities of body forces distributed in the longitudinal direction of an infinite body. In order to satisfy the boundary conditions along elliptical boundaries, four kinds of fundamental density functions are newly introduced. Then the body force densities are approximated by a linear combination of those fundamental density functions and polynomials. The calculations are carried out for several arrangements of elliptical holes, and it is found that the present method yields rapidly converging numerical results. The body force densities and stress distributions along the boundaries are shown in figures to demonstrate the accuracy of the present solutions.

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.

任意分布だ円形介在物の干渉効果の解析法

This paper deals with the numerical solutions of singular integral equations of the body force method in an interaction problem of arbitrarily distributed elliptical inclusions under general loading conditions. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where the densities of body forces distributed in the x- and y-directions of infinite plates having the same elastic constants as those of the matrix and the inclutions 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. 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 conversing numerical results for arbitrarily distributed elliptical inclusions. The numerical results of weight functions and stress distributions along the boundaries are shown in figures to demonstrate the present solution.

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.

Interaction Effect between Ellipsoidal Cavities in an Infinite Body under Uniaxial Tension.

This paper deals with the numerical solution of singular integral equations of the body force method in the interaction problem of ellipsoidal cavities under uniaxial tension. The problem is solved by 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 densities of body forces distributed in the γ, θand z directions are unknown functions. In order to satisfy the boundary conditions along the ellipsoidal boundaries, eight kinds of fundamental density functions proposed in our previous paper are applied. The present method is found to yield rapidly converging numerical results for stress distribution along the boundaries.

Interaction Effect among a Row of Ellipsoidal Cavities in an Infinite Body under Uniaxial Tension.

This paper deals with the numerical solution of singular integral equations of the body force method in the interaction problem for a row of ellipsoidal cavities under uniaxial tension. The problem is solved by 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 densities of body forces distributed in the r, θ and z directions are unknown functions. In order to satisfy the boundary conditions along the ellipsoidal boundaries, eight kinds of fundamental density functions proposed in our previous paper are applied. In the analysis, the number, shape, and distance of cavities are varied systematically; then the magunitude 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 the squared number of cavities.

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 Green's 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.

Singular Integral Equation method in the Analysis of Interaction between Elliptical Inclusions.

This paper deals with numerical solutions of singular integral equations of the body force method in interaction problems of elliptical inclusions under general loading conditions. 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 x- and y-directions of infinite plates having the same elastic constants of the matrix and the 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. Then the body force densities are approximated by a linear combination of the fundamental density functions and polynomials. The accuracy of the present analysis is verified by comparing with the results from previous research. The present method is found to give rapidly converging numerical results for stress distribution along the boundaries of both the matrix and the inclusions.

Optimization of the Position and Size of Auxiliary Hole to Minimize Stress Concentration.

In this paper the optimum position and size of an auxiliary hole are considered in order to minimize the stress concentration due to notches and holes in plates under tension. To solve these problems, the body force method is applied. Then, the problems are formulated as a system of singular integral equations with Cauchy-type singularities, where the densities of body forces distributed in the x- and y-directions are unknown functions. In order to satisfy the boundary conditions along the auxiliary hole and the notch, eight kinds of fundamental density functions proposed in our previous paper are used ; then the body force densities are approximated by a linear combination of the fundamental density functions and polynomials. To obtain the optimum conditions, the direct search method proposed by Hooke-Jeeves is employed. The results show that the position and size of the auxiliary hole are determined efficiently with high accuracy.

Stress Analysis Of Elliptical And EllipsoidalInclusions Using Singular Integral Equations

This paper deals with stress analysis of elliptical and ellipsoidal inclusions using singular integral equations of the body force method. The stress and displacement fields due to a point force in an infinite plate and a ring force in an infinite body are used as fundamental solutions. On the idea of the body force method, the problems are formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknown functions are densities of body forces distributed in the x- and y-directions of infinite plates or in the r- and z-directions of infinite bodies having the same elastic constants of the matrix and inclusions. In order to satisfy the boundary conditions along the inclusions, eight kinds of fundamental density functions proposed in our previous paper are used. Then the body force densities are approximated by a linear combination of the fundamental density functions and polynomials. The present method is found to give rapidly converging numerical results for the problems. The calculations are carried out systematically for various shape, distance and elastic constant of inclusions and the stress distributions along the boundaries of both the matrix and inclusions are shown in figures. Then the interaction effects are discussed through the comparison between the elliptical inclusions and ellipsoidal inclusions.