The 3-D BEM implementation of a numerical Green's function for fracture mechanics applications

Abstract

SUMMARYThe use of Green's functions has been considered a powerful technique in the solution of fracture mechanics problems by the boundary element method (BEM). Closed-form expressions for Green's function components, however, have only been available for few simple 2-D crack geometry applications and require complex variable theory. The present authors have recently introduced an alternative numerical procedure to compute the Green's function components that produced BEM results for 2-D general geometry multiple crack problems, including static and dynamic applications. This technique is not restricted to 2-D problems and the computational aspects of the 3-D implementation of the numerical Green's function approach are now discussed, including examples.

Solution of stress intensity factors of multiple cracks in plane elasticity with eigen COD formulation of boundary integral equation

“…If the criterion (16) does not meet, return to (13) to renew the vector x for the next iteration. Otherwise, proceed to the post process stage such as computing the SIF, the stresses and displacements at the interested locations or computing the overall properties of the cracked body.…”

“…Since analytical Green's functions are not available for the majority of the applications, the numerical Green's function (NGF) procedure suggested by Telles, et al [11][12][13][14] has gained an attention as an efficient option to solve LEFM engineering problems to avoid the crack tip discretization.…”

Potential discontinuity simulations with the numerical Green’s function in steady and transient problems

“…This produces degeneration in the standard boundary integral equation, which either imposes displacement continuity across the crack, if the limit is not properly taken, or even a singularity of the system matrix.A well established alternative for fracture mechanic problems is the use of the respective Green's function, for the crack geometry, eliminating the problem unknowns over its boundaries. An alternative and more general procedure called the numerical Green's function (NGF) [4,14] has been introduced for elastostatic applications to accommodate any shape and number of cracks, employing an efficient hypersingular numerical generation of its components for 2-D and 3-D linear elasticity problems.…”

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

“…The only simple geometric configurations have allowed the analytical derivation of these Green's functions. The numerical Green's function technique of Telles [1,2,3], aimed for crack problems, has broken this limitation and enabled the derivation of the Green's functions for more complex twodimensional and three-dimensional crack configurations. The Green's function is decomposed into the singular and the image terms, the latter being the solution with the negative of the traction loading on the crack surfaces induced by the former.…”

“…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.…”