Numerical solution of singular integral equations in stress concentration problems (original) (raw)
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Dynamic stress concentration studies by boundary integrals and Laplace transform
International Journal for Numerical Methods in Engineering, 1981
The dynamic stress field and its concentrations around holes of arbitrary shape in infinitely extended bodies under plane stress or plane strain conditions are numerically determined. The material may be linear elastic or viscoelastic, while the dynamic load consists of plane compressionnl waves of harmonic or general transient nature. The method consists of applying the Laplace transform with respect to time to the governing equations of motion and formulating and solving the problem numerically in the transformed domain by the boundary integral equation method. The stress field can then be obtained by a numerical inversion of the transformed solution. The correspondence principle is invoked for the case of viscoelastic material behaviour. The method is simplified for the case of harmonic waves where no numerical inversion is involved.
International Journal of Fracture, 1995
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.
Stress analysis by local integral equations
Boundary Elements and Other Mesh Reduction Methods XXIX, 2007
This paper is a comparative study for various numerical implementations of local integral equations developed for stress analysis in plane elasticity of solids with functionally graded material coefficients. Besides two meshless implementations by the point interpolation method and the moving least squares approximation, the element based approximation is also utilized. The numerical stability, accuracy, convergence of accuracy and cost efficiency (assessed by CPU-times) are investigated in numerous test examples with exact benchmark solutions.
A general numerical approximation of the stress characteristic field at a singular point
PROCEEDINGS OF THE INTERNATIONAL CONFERENCE ON MATHEMATICAL SCIENCES AND TECHNOLOGY 2018 (MATHTECH2018): Innovative Technologies for Mathematics & Mathematics for Technological Innovation
A general numerical approximation for the construction of the stress characteristic field in the deformation of ideal granular material at a singular point is considered. The self weight of the granular material is neglected and the stresses are assumed to obey the Coulomb yield criterion. A numerical approximation using finite difference method is used to solve a boundary value problem that leads to the construction of a complete stress characteristic field. The method is presented in this paper and tests, within the MATLAB program at each stage of the construction.
Archive of Applied Mechanics, 2004
This paper deals with the stress concentration problem of an ellipsoidal inclusion of revolution in a semi-infinite body under biaxial tension. The problem is formulated as a system of singular integral equations with Cauchy-type or logarithmic-type singularities, where unknowns are densities of body forces distributed in the r-and z-directions in semi-infinite bodies having the same elastic constants as the ones of the matrix and inclusion. In order to satisfy the boundary conditions along the ellipsoidal boundary, four fundamental density functions proposed in are used. The body-force densities are approximated by a linear combination of fundamental density functions and polynomials. The present method is found to yield rapidly converging numerical results for stress distribution along the boundaries even when the inclusion is very close to the free boundary. The effect of the free surface on the stress concentration factor is discussed with varying the distance from the surface, the shape ratio and the elastic modulus ratio. The present results are compared with the ones of an ellipsoidal cavity in a semi-infinite body.
A note on the Gauss-Jacobi quadrature formulae for singular integral equations of the second kind
International Journal of Fracture, 2000
A fast and efficient numerical method based on the Gauss-Jacobi quadrature is described that is suitable for solving Fredholm singular integral equations of the second kind that are frequently encountered in fracture and contact mechanics. Here we concentrate on the case when the unknown function is singular at both ends of the interval. Quadrature formulae involve fixed nodal points and provide exact results for polynomials of degree 2n − 1, where n is the number of nodes. Finally, an application of the method to a plane problem involving complete contact is presented.
A New Numerical Method for the Analysis of Stress Concentration in Tension Specimen
Analyzing the stress concentration in the critical region of a mixed boundary value elastic problem has always been an important area of interest in applied mechanics. In most of the cases, the problems were circumvented using a numerical solution. A numerical approach has been used to solve elastic problems of arbitrary shaped bodies subjected to mixed boundary conditions. The formulation of displacement potential function is used in finite difference method to determine the stress concentration factors in the critical point of fillet region of tension specimen. In this paper, ASTM D 638 type І, II, III, IV and specimens of other proposed dimensions are analyzed. A relationship between stress concentration factor and width-fillet ratio (w/r) has been established. It is also observed that width ratio (W/w) has no effect on stress concentration factor for higher fillet radius. But for lower fillet radius, width ratio has significant effect on stress concentration factor. It is suggested that single arc tension specimen (ASTM D 638 type I, II, and III) can be used instead of double arc specimen (ASTM D 638 type IV) since complicated geometry of double arc tension specimen cannot help to reduce stress concentration in the specimen during tensile test. The results obtained have been presented graphically and are in good agreement with the available results in the literature.
Modeling of Stress Distribution in a Semi-infinite Piecewise-homogeneous Body
In this paper the Fourier vector integral transforms method with discontinuous coefficients developed by authors is used for elasticity theory problems solving. The analytical solving dynamic problems for theory of elasticity in piecewise homogeneous half-space is found. The explicit construction of direct and inverse Fourier vector transforms with discontinuous coefficients is presented. Unknown tension in the boundary conditions and in the internal conjugation conditions don't commit splitting in a considered dynamic problem, so the application of the scalar Fourier integral transforms with piece-wise constant coefficients does not lead to success. Conformable theoretical bases of a method are presented in this paper. The technique of applying Fourier vector transforms for solving problems of the dynamic problems the elasticity theory.
Journal of Engineering Mathematics
We consider the elastic stress near a hole with corners in an infinite plate under biaxial stress. The elasticity problem is formulated using complex Goursat functions, resulting in a set of singular integro-differential equations on the boundary. The resulting boundary integral equations are solved numerically using a Chebyshev collocation method which is augmented by a fractional power term, derived by asymptotic analysis of the corner region, to resolve stress singularities at corners of the hole. We apply our numerical method to the test case of the hole formed by two partially-overlapping circles, which can include either a corner pointing into the solid or a corner pointing out of the solid. Our numerical results recover the exact stress on the boundary to within relative error 10 −3 for modest computational effort. Keywords elasticity • Goursat functions • boundary integral equations • numerical methods • corners • stress singularities This work was supported by a grant from the Simons Foundation (Award #354717, BJS).