Analytical integrations and SIFs computation in 2D fracture mechanics

Abstract

SUMMARYAnalytical integrations, in the framework of linear elastic problems modelled by means of boundary integral equations, have been considered in a previous publication (Int. J. Numer. Methods Eng. 2002; 53(7):1695-1719): the present note aims at extending the subject to linear elastic fracture mechanics. In such a context, special shape functions have been recently proposed (SIAM J. Appl. Math. 1998; 58: 428-455) in order to increase accuracy in stress intensity factors approximation: the closed form solution for 'integrals' that arise from the boundary element method is a goal of the present work. Exploiting the analytical integrations, asymptotical analysis around the crack tip are made possible, with the purpose of formulating a coherent and accurate correlation between approximated stress intensity factors and crack opening displacements over the crack tip straight special elements.

Analytical integrations in 3D BEM for elliptic problems: Evaluation and implementation

“…[42]. Indeed, closed forms (44), (51), (52) and (61) allow the extension to 3D fracture mechanics of the important result of Gray and Paulino [43]: in a nutshell, it has been shown that-as it happens in two dimensions-the linear term of the expansion of crack opening and sliding about the crack tip vanishes.…”

Hypersingular boundary integral equations and the approximation of the stress tensor

“…[1]. Let * = p + u + + w + − w denote the 'real' boundary of and = p + u + w its idealization: assume to be a piecewise smooth curve in R 2 .…”

“…In the small displacement and strains hypothesis, consider the response of B to the following quasi-static actions: tractionsp(x) on p , displacementsū(x) on u along the boundary, zero bulk forces in . The problem stated above will be termed linear elastic fracture mechanics (LEFM) problem and its numerical approximation is addressed in [1]. In such a framework, the present note aims at studying the approximation of stress vector p(x, n(x)) acting at point x ∈ on a surface of normal n(x).…”

“…The closed form for p h (x, n(x)), x ∈ has been proposed in [1] by analytical integration of (1) in its discrete form. It has been expanded for x about x Tip and compared with Westergaard's asymptotic solution.…”

“…Consider an approximation y h via the Galerkin BEM making use of two quadratic panels and two special elements (enhanced quarter point [1]) at the crack tip. Take a set of points P i / ∈ w but along the crack line (i.e.…”