SOLUTION OF BOUNDARY PROBLEMS FOR A TWO-DIMENSIONAL ELLIPTIC OPERATOR-DIFFERENTIAL EQUATION IN AN ABSTRACT HILBERT SPACE USING THE METHOD OF BOUNDARY INTEGRAL EQUATIONS

2019 ◽  
pp. 11-31
Author(s):  
Ivanov D.Y. ◽  
Keyword(s):  
Differential Equation ◽  
Hilbert Space ◽  
Elliptic Operator ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Boundary Integral ◽  
Operator Differential Equation ◽  
Two Dimensional ◽  
Boundary Problems

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.

Novel Boundary Integral Equations for Two-Dimensional Isotropic Elasticity: An Application to Evaluation of the In-Boundary Stress

2003 ◽  
Vol 70 (6) ◽  
pp. 817-824 ◽  
Author(s):  
V. Manticˇ ◽  
F. J. Calzado ◽  
F. Pari´s
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Anisotropic Elasticity ◽  
Boundary Integral ◽  
Two Dimensional ◽  
Numerical Examples ◽  
Tangential Derivative ◽  
Boundary Stress ◽  
Isotropic Elasticity ◽  
Boundary Integral Representation

A new nonsingular system of boundary integral equations (BIEs) of the second kind for two-dimensional isotropic elasticity is deduced following a recently introduced procedure by Wu (J. Appl. Mech., 67, pp. 618–621, 2000) originally applied for anisotropic elasticity. The physical interpretation of the new integral kernels appearing in these BIEs is studied. An advantageous application of one of these BIEs as a boundary integral representation (BIR) of tangential derivative of boundary displacements on smooth parts of the boundary, and subsequently as a BIR of the in-boundary stress, is presented and analyzed in numerical examples. An equivalent BIR obtained by an integration by parts of the integral including tangential derivative of displacements in the former BIR is presented and analyzed as well. The resulting integral is only apparently hypersingular, being in fact a regular integral on smooth parts of the boundary.

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.

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