Machine-precise evaluation of stress intensity factors with the consistent boundary element method

Abstract

As classically proposed in the technical literature, the boundary element modeling of cracks is best carried out by resorting to a hypersingular fundamental solution – in the frame of the so-called dual formulation – since with the singular fundamental solution alone, the ensuing topological issues would not be adequately tackled. A more natural approach might rely on the direct representation of the crack tip singularity, as already proposed in the frame of the hybrid boundary element method, with implementation of generalized Westergaard stress functions. On the other hand, recent mathematical assessments indicate that the conventional boundary element formulation – based on Kelvin’s fundamental solution – is, in fact, able to precisely represent high stress gradients and deal with extremely convoluted topologies provided only that the numerical integrations be properly resolved. We propose in this paper that independent of the configuration, a cracked structure is geometrically represented as it would appear in real-world laboratory experiments, with crack openings in the range of micrometers. (The nanometer range is actually mathematically feasible, but not realistic in terms of continuum mechanics.) Owing to the newly developed numerical integration scheme, machine precision evaluation of all quantities may be achieved and stress results consistently evaluated at interior points arbitrarily close to crack tips. Importantly, no artificial topological issues are introduced, linear algebra conditioning is kept well under control, and arbitrarily high convergence of results is always attainable. The present developments apply to two-dimensional problems. Some numerical illustrations show that highly accurate results are obtained for cracks represented with just a few quadratic, generally curved, boundary elements – and a few Gauss–Legendre integration points per element – and that the numerical evaluation of the J-integral turns out to be straightforward and actually the most reliable means of obtaining stress intensity factors. Higher-order boundary elements lead to still better results.