Special Boundary Integral Equations for Potential Problems in Regions With Circular Holes

1984 ◽  
Vol 51 (4) ◽  
pp. 713-716 ◽  
Author(s):  
D. A. Caulk
Keyword(s):  
Integral Equations ◽  
Infinite System ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Special Boundary ◽  
Laplace's Equation ◽  
Special Cases ◽  
Potential Problems ◽  
Circular Holes

An infinite system of special boundary integral equations is derived for the solution of Laplace’s equation in a general two-dimensional region with circular holes. The solution is shown to converge when the number of holes is finite and no two holes are touching. In special cases, these equations are shown to yield the same results as two more restricted methods, which are based on different approaches.

Analysis of Elastic Torsion in a Bar With Circular Holes by a Special Boundary Integral Method

1983 ◽  
Vol 50 (1) ◽  
pp. 101-108 ◽  
Author(s):  
D. A. Caulk
Keyword(s):  
Integral Equations ◽  
Cross Section ◽  
Boundary Integral Equations ◽  
Torsional Rigidity ◽  
Circular Cross Section ◽  
Boundary Integral ◽  
Boundary Integral Method ◽  
Warping Function ◽  
Special Boundary ◽  
Circular Holes

Special boundary integral equations developed in an earlier paper are generalized here for torsion of an elastic bar with circular holes. In this approach, the solution on the boundary of each hole is represented by a series of circular harmonics, and the coefficients in these series are determined by a special system of boundary integral equations. For a cross section with only one hole, the entire system of equations is reduced without approximation to a single integral equation involving only the warping function on the outer boundary. For multiple holes, approximate equations are derived that retain only the first harmonic in the solution representation on each hole. The latter equations are solved analytically for a circular cross section weakened by a concentric ring of circular holes. Simple expressions are derived for torsional rigidity, warping, and maximum stress. The results for torsional rigidity are an improvement over previous ones obtained by another approximate method.

Closed-form solutions of an elliptical crack subjected to coupled phonon–phason loadings in two-dimensional hexagonal quasicrystal media

2018 ◽  
Vol 24 (6) ◽  
pp. 1821-1848 ◽  
Author(s):  
Yuan Li ◽  
CuiYing Fan ◽  
Qing-Hua Qin ◽  
MingHao Zhao
Keyword(s):  
Closed Form ◽  
Integral Equations ◽  
Analytical Solutions ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Elliptical Crack ◽  
Two Dimensional ◽  
Crack Face ◽  
Penny Shaped Crack ◽  
Crack Problems

An elliptical crack subjected to coupled phonon–phason loadings in a three-dimensional body of two-dimensional hexagonal quasicrystals is analytically investigated. Owing to the existence of the crack, the phonon and phason displacements are discontinuous along the crack face. The phonon and phason displacement discontinuities serve as the unknown variables in the generalized potential function method which are used to derive the boundary integral equations. These boundary integral equations governing Mode I, II, and III crack problems in two-dimensional hexagonal quasicrystals are expressed in integral differential form and hypersingular integral form, respectively. Closed-form exact solutions to the elliptical crack problems are first derived for two-dimensional hexagonal quasicrystals. The corresponding fracture parameters, including displacement discontinuities along the crack face and stress intensity factors, are presented considering all three crack cases of Modes I, II, and III. Analytical solutions for a penny-shaped crack, as a special case of the elliptical problem, are given. The obtained analytical solutions are graphically presented and numerically verified by the extended displacement discontinuities boundary element method.

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