A variational boundary integral method for the analysis of 3-D cracks of arbitrary geometry modelled as continuous distributions of dislocation loops

A finite element methodology for analysing propagating cracks of arbitrary three-dimensional geometry is developed. By representing the opening displacements of the crack as a distribution of dislocation loops and minimizing the corresponding potential energy of the solid, the kernels of the governing integral equations have mild singularities of the type 1/R. A simple quadrature scheme then suffices to compute all the element arrays accurately. Because of the variational basis of the method, the resulting system of equations is symmetric. By employing six-noded triangular elements and displacing midside nodes to quarter-point positions, the opening profile near the front is endowed with the correct asymptotic behaviour. This enables the direct computation of stress intensity factors from the opening displacements. The special but important cases of periodic and semi-infinite cracks are addressed in some detail. Finally, the geometry of propagating cracks is updated incrementally by recourse to a pseudodynamic crack-tip equation of motion. The crack is continuously remeshed to accommodate the ensuing changes in geometry. The performance of the method is assessed by means of selected numerical examples.