Singular boundary elements for the analysis of cracks in plane strain

Abstract

SUMMARYStraight and curved cracks are modelled by direct formulation boundary elements, of geometry defined by Hermitian cubic shape functions. Displacement and traction are interpolated by the Hermitian functions, supplemented by singular functions which multiply stress intensity factors corresponding to the dominant modes of crack opening in which displacement is proportional to the square root of distance I from the crack tip, and subdominant modes in which it is proportional to rl". The singular functions extend over many boundary elements on each crack face.A nodal collocation scheme is used, in which additional boundary integral equations are obtained by differentiation of the equation obtained from Betti's theorem. The hypersingular kernels of the equations so derived are integrated by consideration of trial displacement fields of subdomains lying to either side of the crack. Examples are shown of the analysis of buried and edge cracks, to demonstrate the effects of modelling subdominant modes and extending singular shape functions over many elements.KEY WORDS stress fracture boundary element

Coupling the BEM/TBEM and the MFS for the numerical simulation of acoustic wave propagation and transient conduction heat transfer

“…The pressure source is placed 4.0 m from the barrier in the horizontal direction, and 0.5 m above the ground, as Figure 3 shows. The barrier, 3.0 m tall, is placed 5.0 m from a semi-circular dome. The pressure response is obtained over a two-dimensional grid of 26347 receivers arranged along the x and y directions at equal intervals and placed in the vicinity of the acoustic barrier and dome from 0.0m x  to 25.0m x  and from y = 0.0m to y = 10.0m.…”

“…The Traction Boundary Element Method (TBEM) is a numerical method that solves the thin-body difficulties that arise when modeling wave propagation in the presence of very thin heterogeneities such as small imperfections, dimensionless cracks or almost imperceptible defects. Different attempts have been made to overcome this difficulty [2,3]. Most of the work published refers to the cases of 2D and, in some cases, 3D geometries.…”

A partition of unity enriched dual boundary element method for accurate computations in fracture mechanics

“…The problem was overcome by partitioning the domain into near and far-field regions at the cost of ease of implementation. More recently, Watson [12] developed a method in which special singular shape functions are created using eigenfunctions from the Williams expansion [13] that describe a crack tip singularity. In the formation of these shape functions, additional unknowns are introduced requiring the use of auxiliary Boundary Integral Equations (BIEs).…”

The simulation of 3D elastic scattering produced by thin rigid inclusions using the traction boundary element method

“…wave propagation, is capable of dealing with heterogeneities with small thickness [32][33][34][35]. Several authors have addressed the problem of evaluating the hypersingular integrals that appear in these formulations (see [36][37][38][39]). Prosper [40] and Prosper and Kausel [41] simulated the wave propagation in the vicinity of flat horizontal empty cracks with no thickness in unbounded elastic media, making use of the traction boundary element method (TBEM).…”