Method of Boundary Integral Equations for Analysis of Three Dimensional Crack Problems

1981 ◽  
pp. 183-205 ◽  
Author(s):  
J. Balaš ◽  
J. Sládek
Keyword(s):  
Integral Equations ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Boundary Integral ◽  
Crack 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.

scholarly journals NUMERICAL ANALYSIS OF THE DYNAMICS OF THREE-DIMENSIONAL ANISOTROPIC BODIES BASED ON NON-CLASSICAL BOUNDARY INTEGRAL EQUATIONS

2021 ◽  
Vol 83 (1) ◽  
pp. 76-86
Author(s):  
A.A. Belov ◽  
A.N. Petrov
Keyword(s):  
Integral Equation ◽  
Boundary Element ◽  
Integral Equations ◽  
Boundary Integral Equations ◽  
Three Dimensional ◽  
Integral Equation Method ◽  
Boundary Integral ◽  
Classical Approach ◽  
Classical Boundary ◽  
Nonclassical Formulation

The application of non-classical approach of the boundary integral equation method in combination with the integral Laplace transform in time to anisotropic elastic wave modeling is considered. In contrast to the classical approach of the boundary integral equation method which is successfully implemented for solving three-dimensional isotropic problems of the dynamic theory of elasticity, viscoelasticity and poroelasticity, the alternative nonclassical formulation of the boundary integral equations method is presented that employs regular Fredholm integral equations of the first kind (integral equations on a plane wave). The construction of such boundary integral equations is based on the structure of the dynamic fundamental solution. The approach employs the explicit boundary integral equations. The inverse Laplace transform is constructed numerically by the Durbin method. A numerical solution of the dynamic problem of anisotropic elasticity theory based on the boundary integral equations method in a nonclassical formulation is presented. The boundary element scheme of the boundary integral equations method is built on the basis of a regular integral equation of the first kind. The problem is solved in anisotropic formulation for the load acting along the normal in the form of the Heaviside function on the cube face weakened by a cubic cavity. The obtained boundary element solutions are compared with finite element solutions. Numerical results prove the efficiency of using boundary integral equations on a single plane wave in solving three-dimensional anisotropic dynamic problems of elasticity theory. The convergence of boundary element solutions is studied on three schemes of surface discretization. The achieved calculation accuracy is not inferior to the accuracy of boundary element schemes for classical boundary integral equations. Boundary element analysis of solutions for a cube with and without a cavity is carried out.

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