A new approach for modelling discrete cracks in meshfree particle methods in three dimensions is described. The cracks can be arbitrarily oriented, but their growth is represented discretely by activation of crack surfaces at individual particles, so no representation of the crack’s topology is needed. The crack is modelled by a local enrichment of the test and trial functions with a sign function (a variant of the Heaviside step function), so that the discontinuities are along the direction of the crack. The discontinuity consists of cylindrical planes centered at the particles. The method is formulated for large deformations and arbitrary nonlinear and rate-dependent materials; cohesive laws govern the traction-crack opening relations. To reduce computational cost and since more accuracy around the crack tip is needed to obtain adequate results, h-adaptivity is incorporated in the method. The model is applied to several three-dimensional problems, some of which are compared to experimental data.