Finite element computations of the stresses ahead of a tortuous microcrack in a polycrystal (including crack flanks friction) are coupled with simulations of crystallographic crack growth based on discrete dislocations dynamics. An incubation period for crack growth beyond grain boundaries is introduced. The model reproduces the shape of experimental crack growth curves obtained on 316LN stainless steel and the decrease in arrest periods at grain boundaries as the crack grows. It predicts a large scatter in growth rates related to the variety of local textures. It also describes the fact that overloads allowing arrested cracks to cross the grain boundaries can make small cycles damaging.