Dislocation and point-force-based approach to the special Green's Function BEM for elliptic hole and crack problems in two dimensions

Abstract

SUMMARYIn this paper we give the theoretical foundation for a dislocation and point-force-based approach to the special Green's function boundary element method and formulate, as an example, the special Green's function boundary element method for elliptic hole and crack problems. The crack is treated as a particular case of the elliptic hole. We adopt a physical interpretation of Somigliana's identity and formulate the boundary element method in terms of distributions of point forces and dislocation dipoles in the infinite domain with an elliptic hole. There is no need to model the hole by the boundary elements since the traction free boundary condition there for the point force and the dislocation dipole is automatically satisfied. The Green's functions are derived following the Muskhelishvili complex variable formalism and the boundary element method is formulated using complex variables. All the boundary integrals, including the formula for the stress intensity factor for the crack, are evaluated analytically to give a simple yet accurate special Green's function boundary element method. The numerical results obtained for the stress concentration and intensity factors are extremely accurate.1997 by John Wiley & Sons, Ltd.

A new special element for stress concentration analysis of a plate with elliptical holes

“…Using the complex variable formalism of Muskhelishvili, Green's function solutions for the hole problem can be written in the form [2] …”

“…Among them, the conventional finite element method (FEM) has been widely used for such analysis, in which mesh refinement near the elliptical hole is unavoidably required to capture the stress concentration around it and to achieve the necessary numerical accuracy. To address this point, the boundary element method (BEM) using pointforce-based fundamental solutions or Green's functions was developed to evaluate the stress concentration factor related to an elliptical hole [1,2], and the BEM using special Green's functions that satisfy the particular boundary conditions is referred to as the special Green's function BEM by Denda [2]. In the special Green's function BEM, the special Green's functions related to the elliptical hole are required to satisfy the proper singularity at the source point and the free boundary conditions along the rim of the elliptical hole.…”

Numerical Green's Functions for Line Force and Dislocation around Multiple Cracks

“…Given these fundamental solutions, we establish a technique to determine their Green's functions numerically when multiple cracks are present in two-dimensional isotropic solids. Such Green's functions are called the numerical Green's functions by Telles et al [1,2,3] The majority of the Green's functions are analytical and are concerned about the simplest defect/inhomogeneity geometries possible such as the single center crack(Snider and Cruse [4], Cruse [5], Clements and Haselgrove [6]) or interface crack(Berger and Tewary [7], Yuuki and Cho [8]), the single elliptical hole (Morjaria and Mukherjee [9], Ang and Clements [10], Kamel and Liaw [11], Hwu and Yen [12], Denda and Kosaka [13]), and the half-plane/bimaterial domain (Telles and Brebbia [14], Meek and Dai [15], Dumir and Mehta [16], Pan et al [17], Berger [18], Denda [19]). The only simple geometric configurations have allowed the analytical derivation of these Green's functions.…”

“…This sets the limitation of the numerical Green's function. We perform an extensive error analysis of the numerical Green's functions for the single crack, for which the analytical Green's functions are available (Denda and Kosaka [13]). We will identify the zone of applicability of the proposed numerical Green's functions where they can be confidently applied.…”

“…1, pp. 22-68 (2004) 22 Yen [12], Denda and Kosaka [13]), and the half-plane/bimaterial domain (Telles and Brebbia [14], Meek and Dai [15], Dumir and Mehta [16], Pan et al [17], Berger [18], Denda [19]). The only simple geometric configurations have allowed the analytical derivation of these Green's functions.…”