Machine-precision fracture mechanics evaluations with the consistent boundary element method

Resumo

As hitherto proposed in the technical literature, the boundary element modeling of cracks is best carried
out resorting to a hypersingular fundamental solution – in the frame of either the so-called dual formulation or the
displacement discontinuity approach –, since with Kelvin’s fundamental solution it would not be possible to deal
with the ensuing numerical and topological issues. A more natural approach would be the direct representation
of the crack tip singularity in terms of generalized Westergaard stress functions, as already proposed in the frame
of the hybrid boundary element method. Quite recently, we have been able to demonstrate that the conventional
boundary element formulation is in fact able to precisely represent high-stress gradients and deal with extremely
convoluted topologies provided only that the problem’s integrals be properly evaluated, which has turned out
to be the case. The reference literature on the present subject is briefly outlined along with the paper, which
includes the simplest scheme of evaluating stress intensity factors in terms of crack tip opening displacements
as a viable alternative to the J-integral. We propose that independently of configuration a cracked structure be
geometrically represented as it would appear in real-world laboratory experiments, with crack openings in the range
of micrometers or even nanometers, which would still be mathematically feasible – albeit not mechanically. In
fact, machine precision evaluation of all quantities may be always achieved and stress results consistently obtained
at interior points arbitrarily close to crack tips. 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. We
also investigate how different simulations of the crack tip shape affect results, which turn out to be more accurate
with the use of higher-order, such as quartic, boundary elements.